Notation by work

Physics

All notation · Glossary · Source coverage

2593 symbol spellings in 4175 mathematical passages across 39 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.

The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.

All 2593 entries are available below.

*Notation table · 180 passages

Search aliases: *

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

=Notation table · 1774 passages

Search aliases: =

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

aNotation table · 459 passages

Search aliases: a

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

bNotation table · 218 passages

Search aliases: b

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Notation table · orientation.tex, line 30
Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.
Exact mathematical passage
```
L_{ba}=P_bLP_a
```

\DomNotation table · 23 passages

Search aliases: Dom

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Notation table · orientation.tex, line 36
Unbounded action · An expression such as Hψ includes the condition ψ∈Dom(H).
Exact mathematical passage
```
\psi\in\Dom(H)
```

\dot pNotation table · 20 passages

Search aliases: dot p dot

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
\dot p_b=\sum_aJ_{b\leftarrow a}
```

\dot p_bNotation table · 13 passages

Search aliases: dot p_b dot

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
\dot p_b=\sum_aJ_{b\leftarrow a}
```

\dot\psiNotation table · 16 passages

Search aliases: dotψ dot psi

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
\dot\psi=K\psi
```

HNotation table · 359 passages

Search aliases: H

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
K=-iH
```

hNotation table · 468 passages

Search aliases: h

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Notation table · orientation.tex, line 30
Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.
Exact mathematical passage
```
\sin(hp_j)/h
```

iNotation table · 298 passages

Search aliases: i

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
K=-iH
```

\innerNotation table · 55 passages

Search aliases: inner

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Notation table · orientation.tex, line 28
midrule Inner product · innerxy is conjugate-linear in x and linear in y.
Exact mathematical passage
```
\inner{x}{y}
```

JNotation table · 128 passages

Search aliases: J

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
J_{b\leftarrow a}
```

jNotation table · 293 passages

Search aliases: j

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Notation table · orientation.tex, line 30
Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.
Exact mathematical passage
```
\sin(hp_j)/h
```

J_{b\leftarrow a}Notation table · 11 passages

Search aliases: J_b← a leftarrow

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
J_{b\leftarrow a}
```

KNotation table · 116 passages

Search aliases: K

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
\dot\psi=K\psi
```

K^*Notation table · 3 passages

Search aliases: K^*

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
K^*=-K
```

LNotation table · 170 passages

Search aliases: L

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Notation table · orientation.tex, line 30
Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.
Exact mathematical passage
```
L_{ba}=P_bLP_a
```

L_{ba}Notation table · 6 passages

Search aliases: L_ba

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Notation table · orientation.tex, line 30
Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.
Exact mathematical passage
```
L_{ba}=P_bLP_a
```

\normNotation table · 158 passages

Search aliases: norm

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Notation table · orientation.tex, line 32
Packet weight · p_a=normP_aψ^2. It is a probability for a normalized state and a complete orthogonal resolution. It is not an energy expectation by definition.
Exact mathematical passage
```
p_a=\norm{P_a\psi}^2
```

PNotation table · 246 passages

Search aliases: P

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

pNotation table · 543 passages

Search aliases: p

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Notation table · orientation.tex, line 30
Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.
Exact mathematical passage
```
\sin(hp_j)/h
```

P_aNotation table · 59 passages

Search aliases: P_a

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

p_aNotation table · 26 passages

Search aliases: p_a

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Notation table · orientation.tex, line 32
Packet weight · p_a=normP_aψ^2. It is a probability for a normalized state and a complete orthogonal resolution. It is not an energy expectation by definition.
Exact mathematical passage
```
p_a=\norm{P_a\psi}^2
```

P_a^*Notation table · 2 passages

Search aliases: P_a^*

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

P_a^2Notation table · 2 passages

Search aliases: P_a^2

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Notation table · orientation.tex, line 29
Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.
Exact mathematical passage
```
P_a=P_a^*=P_a^2
```

P_bNotation table · 36 passages

Search aliases: P_b

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Notation table · orientation.tex, line 30
Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.
Exact mathematical passage
```
L_{ba}=P_bLP_a
```

p_jNotation table · 17 passages

Search aliases: p_j

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Notation table · orientation.tex, line 30
Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.
Exact mathematical passage
```
\sin(hp_j)/h
```

\sinNotation table · 47 passages

Search aliases: sin

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Notation table · orientation.tex, line 30
Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.
Exact mathematical passage
```
\sin(hp_j)/h
```

\operatorname{tr}Notation table · 7 passages

Search aliases: tr operatorname

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Notation table · orientation.tex, line 34
Mixed state · ρ≥0, trρ=1 in the finite-dimensional sections.
Exact mathematical passage
```
\operatorname{tr}\rho=1
```

xNotation table · 321 passages

Search aliases: x

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Notation table · orientation.tex, line 28
midrule Inner product · innerxy is conjugate-linear in x and linear in y.
Exact mathematical passage
```
\inner{x}{y}
```

yNotation table · 100 passages

Search aliases: y

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Notation table · orientation.tex, line 28
midrule Inner product · innerxy is conjugate-linear in x and linear in y.
Exact mathematical passage
```
\inner{x}{y}
```

\rhoNotation table · 61 passages

Search aliases: ρ rho

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Notation table · orientation.tex, line 34
Mixed state · ρ≥0, trρ=1 in the finite-dimensional sections.
Exact mathematical passage
```
\rho\geq0
```

\psiNotation table · 180 passages

Search aliases: ψ psi

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
\dot\psi=K\psi
```

\hbarNotation table · 12 passages

Search aliases: ℏ hbar

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Notation table · orientation.tex, line 31
State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.
Exact mathematical passage
```
\hbar=1
```

\leftarrowNotation table · 29 passages

Search aliases: ← leftarrow

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
J_{b\leftarrow a}
```

\inNotation table · 161 passages

Search aliases: ∈ in

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Notation table · orientation.tex, line 36
Unbounded action · An expression such as Hψ includes the condition ψ∈Dom(H).
Exact mathematical passage
```
\psi\in\Dom(H)
```

\sumNotation table · 278 passages

Search aliases: ∑ sum

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
\dot p_b=\sum_aJ_{b\leftarrow a}
```

\sum_aNotation table · 39 passages

Search aliases: ∑_a sum

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Notation table · orientation.tex, line 33
Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.
Exact mathematical passage
```
\dot p_b=\sum_aJ_{b\leftarrow a}
```

\geqNotation table · 72 passages

Search aliases: ≥ geq

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Notation table · orientation.tex, line 34
Mixed state · ρ≥0, trρ=1 in the finite-dimensional sections.
Exact mathematical passage
```
\rho\geq0
```

+Definition passage · 559 passages

Search aliases: +

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Definition passage: The actual refinement tower · refinement.tex, line 11
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
 T_{m+1}f(h)=\frac{b^{m+1}T_mf(h/b)-T_mf(h)}{b^{m+1}-1}.
```

ADefinition passage · 240 passages

Search aliases: A

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Definition passage: Finite effect atlas · extensions.tex, line 9
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
A
```

b^{m+1}Definition passage · 4 passages

Search aliases: b^m+1

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Definition passage: The actual refinement tower · refinement.tex, line 11
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
 T_{m+1}f(h)=\frac{b^{m+1}T_mf(h/b)-T_mf(h)}{b^{m+1}-1}.
```

FDefinition passage · 167 passages

Search aliases: F

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Definition passage: Finite effect atlas · extensions.tex, line 10
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
 F_a=F_a^*\geq0,\qquad \sum_{a\in A}F_a=I.
```

\FDefinition passage · 14 passages

Search aliases: F

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

fDefinition passage · 205 passages

Search aliases: f

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Definition passage: The actual refinement tower · refinement.tex, line 10
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
T_0f(h)=f(h)
```

\F^{-1}Definition passage · 6 passages

Search aliases: F^-1 F

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

F_aDefinition passage · 15 passages

Search aliases: F_a

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Definition passage: Finite effect atlas · extensions.tex, line 10
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
 F_a=F_a^*\geq0,\qquad \sum_{a\in A}F_a=I.
```

F_a^*Definition passage · 1 passage

Search aliases: F_a^*

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Definition passage: Finite effect atlas · extensions.tex, line 10
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
 F_a=F_a^*\geq0,\qquad \sum_{a\in A}F_a=I.
```

\HHDefinition passage · 59 passages

Search aliases: HH

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Definition passage: Finite effect atlas · extensions.tex, line 9
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
\HH
```

IDefinition passage · 235 passages

Search aliases: I

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Definition passage: Finite orthogonal atlas · core.tex, line 8
[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.
Exact mathematical passage
```
(P_a)_{a\in I}
```

\idDefinition passage · 11 passages

Search aliases: id

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Definition passage: Finite orthogonal atlas · core.tex, line 9
[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.
Exact mathematical passage
```
\label{f:atlas-relations}
 P_a^*=P_a,\qquad P_aP_b=\delta_{ab}P_a,
 \qquad \sum_{a\in I}P_a=\id.
```

\mathcal MDefinition passage · 6 passages

Search aliases: M mathcal

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

mDefinition passage · 273 passages

Search aliases: m

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Definition passage: The actual refinement tower · refinement.tex, line 10
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
m\geq0
```

P_\LambdaDefinition passage · 16 passages

Search aliases: P_Λ Lambda

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

QDefinition passage · 140 passages

Search aliases: Q

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

Q^2Definition passage · 25 passages

Search aliases: Q^2

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

RDefinition passage · 170 passages

Search aliases: R

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

R'Definition passage · 12 passages

Search aliases: R'

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

TDefinition passage · 85 passages

Search aliases: T

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Definition passage: The actual refinement tower · refinement.tex, line 10
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
T_0f(h)=f(h)
```

tDefinition passage · 305 passages

Search aliases: t

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

T_0Definition passage · 1 passage

Search aliases: T_0

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Definition passage: The actual refinement tower · refinement.tex, line 10
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
T_0f(h)=f(h)
```

T_mDefinition passage · 9 passages

Search aliases: T_m

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Definition passage: The actual refinement tower · refinement.tex, line 11
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
 T_{m+1}f(h)=\frac{b^{m+1}T_mf(h/b)-T_mf(h)}{b^{m+1}-1}.
```

T_{m+1}Definition passage · 2 passages

Search aliases: T_m+1

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Definition passage: The actual refinement tower · refinement.tex, line 11
[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.
Exact mathematical passage
```
 T_{m+1}f(h)=\frac{b^{m+1}T_mf(h/b)-T_mf(h)}{b^{m+1}-1}.
```

UDefinition passage · 184 passages

Search aliases: U

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

\mathcal UDefinition passage · 14 passages

Search aliases: U mathcal

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

U_hDefinition passage · 12 passages

Search aliases: U_h

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

\mathcal U_hDefinition passage · 2 passages

Search aliases: U_h mathcal

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Definition passage: Physical propagators · continuum.tex, line 109
[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to every f∈HH without an additional constant.
Exact mathematical passage
```
\mathcal U(t)=\F^{-1}U(t)\F,
\qquad
\mathcal U_h(t)=\F^{-1}U_h(t)\F.
```

\deltaDefinition passage · 125 passages

Search aliases: δ delta

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Definition passage: Finite orthogonal atlas · core.tex, line 9
[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.
Exact mathematical passage
```
\label{f:atlas-relations}
 P_a^*=P_a,\qquad P_aP_b=\delta_{ab}P_a,
 \qquad \sum_{a\in I}P_a=\id.
```

\delta_{ab}Definition passage · 5 passages

Search aliases: δ_ab delta

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Definition passage: Finite orthogonal atlas · core.tex, line 9
[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.
Exact mathematical passage
```
\label{f:atlas-relations}
 P_a^*=P_a,\qquad P_aP_b=\delta_{ab}P_a,
 \qquad \sum_{a\in I}P_a=\id.
```

\LambdaDefinition passage · 55 passages

Search aliases: Λ Lambda

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Definition passage: Charged mass coordinate · strong_fields.tex, line 44
[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.
Exact mathematical passage
```
 \mathcal M=\frac R2\left[1-\left(\frac{R'}\Lambda\right)^2
                    +\left(\frac{P_\Lambda}R\right)^2\right]
             +\frac{Q^2}{2R}.
\label{c:strong:mass-coordinate}
```

\psi^2Definition passage · 1 passage

Search aliases: ψ^2 psi

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Definition passage: Finite effect atlas · extensions.tex, line 13
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
\sum_a p_a=\norm\psi^2
```

\psi_aDefinition passage · 11 passages

Search aliases: ψ_a psi

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Definition passage: Finite orthogonal atlas · core.tex, line 13
[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.
Exact mathematical passage
```
\psi_a=P_a\psi
```

\sum_{a\in A}Definition passage · 2 passages

Search aliases: ∑_a∈ A sum in

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Definition passage: Finite effect atlas · extensions.tex, line 10
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
 F_a=F_a^*\geq0,\qquad \sum_{a\in A}F_a=I.
```

\sum_{a\in I}Definition passage · 1 passage

Search aliases: ∑_a∈ I sum in

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Definition passage: Finite orthogonal atlas · core.tex, line 9
[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.
Exact mathematical passage
```
\label{f:atlas-relations}
 P_a^*=P_a,\qquad P_aP_b=\delta_{ab}P_a,
 \qquad \sum_{a\in I}P_a=\id.
```

\leqDefinition passage · 86 passages

Search aliases: ≤ leq

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Definition passage: Finite effect atlas · extensions.tex, line 13
[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.
Exact mathematical passage
```
0\leq F_a\leq I
```

\mathcal AIntroduction passage · 25 passages

Search aliases: A mathcal

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

A^*Introduction passage · 5 passages

Search aliases: A^*

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Introduction passage · refinement.tex, line 9
use the forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). …
Exact mathematical passage
```
A=A^*
```

A^2Introduction passage · 5 passages

Search aliases: A^2

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Introduction passage · refinement.tex, line 9
forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The chan…
Exact mathematical passage
```
A^2=1
```

\mathcal A_{3,1}Introduction passage · 1 passage

Search aliases: A_3,1 mathcal

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Introduction passage · spectral.tex, line 958
LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
\mathcal A_{3,1}=\{2/3,4/3\}
```

\mathcal A_{L,j}Introduction passage · 4 passages

Search aliases: A_L,j mathcal

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

a_nIntroduction passage · 47 passages

Search aliases: a_n

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Introduction passage · continuum.tex, line 30
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
(U(t)a)_n=V_n(t)a_n,\qquad (U_h(t)a)_n=V_{h,n}(t)a_n.
```

A_rIntroduction passage · 2 passages

Search aliases: A_r

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Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

\mathrm{arc}Introduction passage · 1 passage

Search aliases: arc mathrm

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Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

BIntroduction passage · 164 passages

Search aliases: B

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Introduction passage · extensions.tex, line 447
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,
Exact mathematical passage
```
B=B^*
```

B^*Introduction passage · 18 passages

Search aliases: B^*

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Introduction passage · extensions.tex, line 447
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,
Exact mathematical passage
```
B=B^*
```

b_1Introduction passage · 5 passages

Search aliases: b_1

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Introduction passage · spectral.tex, line 953
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
{}_0F_s(;b_1,\ldots,b_s;w)
```

b_rIntroduction passage · 2 passages

Search aliases: b_r

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Introduction passage · spectral.tex, line 954
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
\sum_{q\geq0}w^q/[q!\prod_r(b_r)_q]
```

b_sIntroduction passage · 1 passage

Search aliases: b_s

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Introduction passage · spectral.tex, line 953
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
{}_0F_s(;b_1,\ldots,b_s;w)
```

\bar sIntroduction passage · 3 passages

Search aliases: bar s bar

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Introduction passage · constraints.tex, line 63
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
 Q=\mathrm{id}_F-IP,\qquad \bar s_C=Ps_FI.
```

\bar s_CIntroduction passage · 2 passages

Search aliases: bar s_C bar

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Introduction passage · constraints.tex, line 63
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
 Q=\mathrm{id}_F-IP,\qquad \bar s_C=Ps_FI.
```

\bar s_C^2Introduction passage · 1 passage

Search aliases: bar s_C^2 bar

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Introduction passage · constraints.tex, line 67
raded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no reason for it to vanish merely because the fine dif…
Exact mathematical passage
```
 \bar s_C^2
 =Ps_F(IP)s_FI
 =-Ps_FQs_FI.
 \label{c:constraints:leakage}
```

\mathrm{bright}Introduction passage · 3 passages

Search aliases: bright mathrm

Link to this entry ·

Introduction passage · finite.tex, line 782
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance
Exact mathematical passage
```
(H_{\mathrm{bright}}-\overline E I)^2=\Omega^2I
```

CIntroduction passage · 153 passages

Search aliases: C

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Introduction passage · limit.tex, line 854
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
\phi^n=C\psi^n
```

\CIntroduction passage · 46 passages

Search aliases: C

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Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

\mathcal CIntroduction passage · 7 passages

Search aliases: C mathcal

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
(\mathcal C_F,s_F)
```

cIntroduction passage · 160 passages

Search aliases: c

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Introduction passage · finite.tex, line 792
. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,…
Exact mathematical passage
```
|c\rangle
```

\C^4Introduction passage · 12 passages

Search aliases: C^4 C

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Introduction passage · refinement.tex, line 9
on of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogo…
Exact mathematical passage
```
\C^4
```

\C^{\tau_p}Introduction passage · 1 passage

Search aliases: C^τ_p C tau

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Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

\mathcal C_CIntroduction passage · 3 passages

Search aliases: C_C mathcal

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
P:\mathcal C_F\to\mathcal C_C
```

\mathcal C_FIntroduction passage · 3 passages

Search aliases: C_F mathcal

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
(\mathcal C_F,s_F)
```

C_NIntroduction passage · 7 passages

Search aliases: C_N

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Introduction passage · limit.tex, line 933
[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary …
Exact mathematical passage
```
C_N\square T_{p,h}
```

\cosIntroduction passage · 43 passages

Search aliases: cos

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Introduction passage · finite.tex, line 783
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it redu…
Exact mathematical passage
```
 e^{-\mathrm{i}tH_{\mathrm{bright}}}
 =e^{-\mathrm{i}t\overline E}
 \left[I\cos(\Omega t)
 -\mathrm{i}\frac{H_{\mathrm{bright}}-\overline E I}{\Omega}
              \sin(\Omega t)\right].
 \label{a-finite-bright-propagator}
```

DIntroduction passage · 191 passages

Search aliases: D

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Introduction passage · extensions.tex, line 132
Now let H:D(H)⊂HH→HH be a densely defined symmetric operator. To write HP_aψ for every domain state, require P_aD(H)⊂ D(H). Define norm x_H^2=norm x^2+normHx^2. A domain-compatible atlas is graph compatible when S_Fx⟶ x in the graph norm for every x∈ D(H). Equivalently, both S_Fx→ x and HS_Fx→ Hx hold in HH. The second convergence does not follow from the first for an unbounded H.
Exact mathematical passage
```
H:D(H)\subset\HH\to\HH
```

\mathrm DIntroduction passage · 15 passages

Search aliases: D mathrm

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Introduction passage · constraints.tex, line 657
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
\{\cdot,\cdot\}_{\mathrm D}
```

dIntroduction passage · 141 passages

Search aliases: d

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Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

d_0Introduction passage · 5 passages

Search aliases: d_0

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Introduction passage · spectral.tex, line 555
la-spectral-path-inverse For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse entries by cofactors. For …
Exact mathematical passage
```
d_0=1
```

d_1Introduction passage · 2 passages

Search aliases: d_1

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Introduction passage · spectral.tex, line 555
-path-inverse For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse entries by cofactors. For the endpoin…
Exact mathematical passage
```
d_1=x
```

d_nIntroduction passage · 3 passages

Search aliases: d_n

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Introduction passage · spectral.tex, line 555
t and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse entries by cofactors. For t…
Exact mathematical passage
```
d_n=xd_{n-1}-d_{n-2}
```

d_{n-1}Introduction passage · 1 passage

Search aliases: d_n-1

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Introduction passage · spectral.tex, line 555
t and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse entries by cofactors. For t…
Exact mathematical passage
```
d_n=xd_{n-1}-d_{n-2}
```

d_{n-2}Introduction passage · 1 passage

Search aliases: d_n-2

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Introduction passage · spectral.tex, line 555
t and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse entries by cofactors. For t…
Exact mathematical passage
```
d_n=xd_{n-1}-d_{n-2}
```

d_rIntroduction passage · 2 passages

Search aliases: d_r

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Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

d_vIntroduction passage · 1 passage

Search aliases: d_v

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Introduction passage · limit.tex, line 833
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
d_v
```

d_v^{-1/2}Introduction passage · 1 passage

Search aliases: d_v^-1/2

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Introduction passage · limit.tex, line 834
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
s_v=d_v^{-1/2}(1,\ldots,1)
```

\detIntroduction passage · 11 passages

Search aliases: det

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Introduction passage · spectral.tex, line 545
[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse e…
Exact mathematical passage
```
\det K_n(x)=U_n(x/2)
```

\dot PIntroduction passage · 5 passages

Search aliases: dot P dot

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Introduction passage · core.tex, line 300
The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present or…
Exact mathematical passage
```
 \dot P_a=[V,P_a],\qquad
 \frac{d}{dt}(P_bXP_a)
 =[V,P_b]XP_a+P_b\dot XP_a+P_bX[V,P_a].
```

\dot P_aIntroduction passage · 1 passage

Search aliases: dot P_a dot

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Introduction passage · core.tex, line 300
The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present or…
Exact mathematical passage
```
 \dot P_a=[V,P_a],\qquad
 \frac{d}{dt}(P_bXP_a)
 =[V,P_b]XP_a+P_b\dot XP_a+P_bX[V,P_a].
```

\dot UIntroduction passage · 8 passages

Search aliases: dot U dot

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Introduction passage · core.tex, line 299
The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present or…
Exact mathematical passage
```
V=\dot UU^*
```

\dot XIntroduction passage · 1 passage

Search aliases: dot X dot

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Introduction passage · core.tex, line 300
The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present or…
Exact mathematical passage
```
 \dot P_a=[V,P_a],\qquad
 \frac{d}{dt}(P_bXP_a)
 =[V,P_b]XP_a+P_b\dot XP_a+P_bX[V,P_a].
```

EIntroduction passage · 149 passages

Search aliases: E

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Introduction passage · finite.tex, line 800
tood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radial.
Exact mathematical passage
```
E_c=\epsilon
```

\mathsf EIntroduction passage · 38 passages

Search aliases: E mathsf

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Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

eIntroduction passage · 286 passages

Search aliases: e

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Introduction passage · finite.tex, line 783
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it redu…
Exact mathematical passage
```
 e^{-\mathrm{i}tH_{\mathrm{bright}}}
 =e^{-\mathrm{i}t\overline E}
 \left[I\cos(\Omega t)
 -\mathrm{i}\frac{H_{\mathrm{bright}}-\overline E I}{\Omega}
              \sin(\Omega t)\right].
 \label{a-finite-bright-propagator}
```

e^{-\mathrm{i}tH_{\mathrm{bright}}}Introduction passage · 1 passage

Search aliases: e^-itH_bright mathrm mathrm

Link to this entry ·

Introduction passage · finite.tex, line 783
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it redu…
Exact mathematical passage
```
 e^{-\mathrm{i}tH_{\mathrm{bright}}}
 =e^{-\mathrm{i}t\overline E}
 \left[I\cos(\Omega t)
 -\mathrm{i}\frac{H_{\mathrm{bright}}-\overline E I}{\Omega}
              \sin(\Omega t)\right].
 \label{a-finite-bright-propagator}
```

e^{-itH_{h,n}}Introduction passage · 1 passage

Search aliases: e^-itH_h,n

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

e^{-itH_n}Introduction passage · 1 passage

Search aliases: e^-itH_n

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

e^{-\mathrm{i}t\overline E}Introduction passage · 1 passage

Search aliases: e^-itoverline E mathrm overline

Link to this entry ·

Introduction passage · finite.tex, line 783
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it redu…
Exact mathematical passage
```
 e^{-\mathrm{i}tH_{\mathrm{bright}}}
 =e^{-\mathrm{i}t\overline E}
 \left[I\cos(\Omega t)
 -\mathrm{i}\frac{H_{\mathrm{bright}}-\overline E I}{\Omega}
              \sin(\Omega t)\right].
 \label{a-finite-bright-propagator}
```

e_0Introduction passage · 16 passages

Search aliases: e_0

Link to this entry ·

Introduction passage · finite_core.tex, line 808
Let e_0=(1,0,0,0)^T denote the first spinor basis vector, and choose
Exact mathematical passage
```
e_0=(1,0,0,0)^T
```

E_{3,1}Introduction passage · 1 passage

Search aliases: E_3,1

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Introduction passage · spectral.tex, line 959
hypergeometric-lane Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
E_{3,1}(z)=z\,{}_0F_2(;2/3,4/3;z^3/27)
```

E_cIntroduction passage · 6 passages

Search aliases: E_c

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Introduction passage · finite.tex, line 800
tood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radial.
Exact mathematical passage
```
E_c=\epsilon
```

e_jIntroduction passage · 14 passages

Search aliases: e_j

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Introduction passage · limit.tex, line 937
it-walk-refinement-counterexample Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its inter…
Exact mathematical passage
```
e_j
```

E_{L,j}Introduction passage · 9 passages

Search aliases: E_L,j

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

\mathsf E_mIntroduction passage · 14 passages

Search aliases: E_m mathsf

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Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

e_xIntroduction passage · 8 passages

Search aliases: e_x

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Introduction passage · refinement.tex, line 19
Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to σ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.
Exact mathematical passage
```
e_x
```

e_{\tau x}Introduction passage · 2 passages

Search aliases: e_τ x tau

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Introduction passage · refinement.tex, line 20
Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to σ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.
Exact mathematical passage
```
 \sigma e_x=m_x,\qquad \sigma m_x=e_{\tau x}.
```

\expIntroduction passage · 11 passages

Search aliases: exp

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Introduction passage · finite.tex, line 322
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
\omega_p=\exp(2\pi\mathrm{i}/p)
```

F_2Introduction passage · 3 passages

Search aliases: F_2

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Introduction passage · spectral.tex, line 959
hypergeometric-lane Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
E_{3,1}(z)=z\,{}_0F_2(;2/3,4/3;z^3/27)
```

F_{L-1}Introduction passage · 2 passages

Search aliases: F_L-1

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

F_sIntroduction passage · 1 passage

Search aliases: F_s

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Introduction passage · spectral.tex, line 953
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
{}_0F_s(;b_1,\ldots,b_s;w)
```

F_{u\to v}^nIntroduction passage · 3 passages

Search aliases: F_u→ v^n to

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Introduction passage · limit.tex, line 862
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
\rho_v^{n+1}=\sum_u F_{u\to v}^n
```

F_{v\to u}^nIntroduction passage · 4 passages

Search aliases: F_v→ u^n to

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Introduction passage · limit.tex, line 855
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
 \rho_v^n=\sum_{u\sim v}\norm{\psi^n(v,u)}^2,
 \qquad F_{v\to u}^n=\norm{\phi^n(v,u)}^2.
```

GIntroduction passage · 73 passages

Search aliases: G

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Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

gIntroduction passage · 118 passages

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Introduction passage · refinement.tex, line 6
We use the forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x…
Exact mathematical passage
```
 u^g(x)=g(\tau x)u(x)g(x)^{-1}.
```

G_{p,w}Introduction passage · 13 passages

Search aliases: G_p,w

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Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

G_vIntroduction passage · 2 passages

Search aliases: G_v

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Introduction passage · limit.tex, line 835
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
G_v=2|s_v\rangle\langle s_v|-I
```

H_{\mathrm{bright}}Introduction passage · 3 passages

Search aliases: H_bright mathrm

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Introduction passage · finite.tex, line 782
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance
Exact mathematical passage
```
(H_{\mathrm{bright}}-\overline E I)^2=\Omega^2I
```

H_{h,n}Introduction passage · 20 passages

Search aliases: H_h,n

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

H_{h,S}Introduction passage · 1 passage

Search aliases: H_h,S

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Introduction passage · finite_core.tex, line 1102
Let H_S,H_h,S denote the finite block diagonal Hamiltonians. In coefficient coordinates, squared norm and continuum Hamiltonian expectation are
Exact mathematical passage
```
H_S,H_{h,S}
```

H_nIntroduction passage · 33 passages

Search aliases: H_n

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

H_SIntroduction passage · 1 passage

Search aliases: H_S

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Introduction passage · finite_core.tex, line 1102
Let H_S,H_h,S denote the finite block diagonal Hamiltonians. In coefficient coordinates, squared norm and continuum Hamiltonian expectation are
Exact mathematical passage
```
H_S,H_{h,S}
```

\HH_{\mathrm{arc}}Introduction passage · 1 passage

Search aliases: HH_arc HH mathrm

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Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

\HH_pIntroduction passage · 13 passages

Search aliases: HH_p HH

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Introduction passage · limit.tex, line 74
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
f\in\HH_p
```

\mathrm{i}Introduction passage · 9 passages

Search aliases: i mathrm

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Introduction passage · finite.tex, line 322
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
\omega_p=\exp(2\pi\mathrm{i}/p)
```

\mathrm{id}Introduction passage · 15 passages

Search aliases: id mathrm

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
PI=\mathrm{id}_C
```

\mathrm{id}_CIntroduction passage · 4 passages

Search aliases: id_C mathrm

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
PI=\mathrm{id}_C
```

\mathrm{id}_FIntroduction passage · 8 passages

Search aliases: id_F mathrm

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Introduction passage · constraints.tex, line 63
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
 Q=\mathrm{id}_F-IP,\qquad \bar s_C=Ps_FI.
```

K_nIntroduction passage · 3 passages

Search aliases: K_n

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Introduction passage · spectral.tex, line 545
[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse e…
Exact mathematical passage
```
\det K_n(x)=U_n(x/2)
```

K_{n-1}Introduction passage · 1 passage

Search aliases: K_n-1

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Introduction passage · spectral.tex, line 557
nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse entries by cofactors. For the endpoint entry, the cofactor consists of the product of n-1 entries -1 along the unique path from the first en…
Exact mathematical passage
```
K_{n-1}
```

\KKIntroduction passage · 33 passages

Search aliases: KK

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Introduction passage · continuum.tex, line 33
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
\KK
```

L^1Introduction passage · 1 passage

Search aliases: L^1

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Introduction passage · limit.tex, line 80
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
L^1
```

L^2Introduction passage · 57 passages

Search aliases: L^2

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Introduction passage · limit.tex, line 80
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
L^2
```

L^LIntroduction passage · 2 passages

Search aliases: L^L

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

MIntroduction passage · 187 passages

Search aliases: M

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Introduction passage · extensions.tex, line 447
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,
Exact mathematical passage
```
M_N(\C)
```

M_NIntroduction passage · 1 passage

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Introduction passage · extensions.tex, line 447
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,
Exact mathematical passage
```
M_N(\C)
```

m_xIntroduction passage · 7 passages

Search aliases: m_x

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Introduction passage · refinement.tex, line 19
Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to σ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.
Exact mathematical passage
```
m_x
```

m_{\tau x}Introduction passage · 1 passage

Search aliases: m_τ x tau

Link to this entry ·

Introduction passage · refinement.tex, line 23
Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to σ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.
Exact mathematical passage
```
\sigma^2m_x=m_{\tau x}
```

NIntroduction passage · 142 passages

Search aliases: N

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Introduction passage · limit.tex, line 932
[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary …
Exact mathematical passage
```
N\geq3
```

nIntroduction passage · 374 passages

Search aliases: n

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Introduction passage · limit.tex, line 853
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉
Exact mathematical passage
```
\psi^n(v,u)
```

N^2Introduction passage · 3 passages

Search aliases: N^2

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Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

N^rIntroduction passage · 10 passages

Search aliases: N^r

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Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

\overline EIntroduction passage · 3 passages

Search aliases: overline E overline

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Introduction passage · finite.tex, line 782
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance
Exact mathematical passage
```
(H_{\mathrm{bright}}-\overline E I)^2=\Omega^2I
```

p^{-(h-r+1)/2}Introduction passage · 1 passage

Search aliases: p^-(h-r+1)/2

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Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

p^mIntroduction passage · 8 passages

Search aliases: p^m

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Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

p^{r-1}Introduction passage · 14 passages

Search aliases: p^r-1

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Introduction passage · finite.tex, line 323
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
a\in\{0,\ldots,p^{r-1}-1\}
```

p^wIntroduction passage · 3 passages

Search aliases: p^w

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Introduction passage · finite.tex, line 802
inequation Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radial.
Exact mathematical passage
```
\Pr(R_{p,w};t)/p^w
```

P_+Introduction passage · 15 passages

Search aliases: P_+

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Introduction passage · refinement.tex, line 10
x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogonal. A bijective site translation and unit links therefore make this an …
Exact mathematical passage
```
 (S_{A,\tau,u}\psi)(x)
 =u(\tau^{-1}x)P_+\psi(\tau^{-1}x)
  +u(x)^{-1}P_-\psi(\tau x).
 \label{g:source-shift}
```

P_-Introduction passage · 17 passages

Search aliases: P_-

Link to this entry ·

Introduction passage · refinement.tex, line 10
x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogonal. A bijective site translation and unit links therefore make this an …
Exact mathematical passage
```
 (S_{A,\tau,u}\psi)(x)
 =u(\tau^{-1}x)P_+\psi(\tau^{-1}x)
  +u(x)^{-1}P_-\psi(\tau x).
 \label{g:source-shift}
```

P_\pmIntroduction passage · 9 passages

Search aliases: P_± pm

Link to this entry ·

Introduction passage · refinement.tex, line 9
refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogonal. A bijective …
Exact mathematical passage
```
P_\pm=(1\pm A)/2
```

\pmodIntroduction passage · 9 passages

Search aliases: pmod

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Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

\PrIntroduction passage · 3 passages

Search aliases: Pr

Link to this entry ·

Introduction passage · finite.tex, line 793
e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radia…
Exact mathematical passage
```
 \Pr(R_{p,w};t)=
 \frac{|G_{p,w}|^2}{\delta^2+|G_{p,w}|^2}
       \sin^2\!\left(t\sqrt{\delta^2+|G_{p,w}|^2}\right).
 \label{a-finite-radial-probability}
```

qIntroduction passage · 135 passages

Search aliases: q

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Introduction passage · finite.tex, line 324
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
q\in\{1,\ldots,p-1\}
```

rIntroduction passage · 307 passages

Search aliases: r

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Introduction passage · finite.tex, line 322
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
1\leq r\leq h
```

R^2Introduction passage · 16 passages

Search aliases: R^2

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Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

R_jIntroduction passage · 10 passages

Search aliases: R_j

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Introduction passage · core.tex, line 369
j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.
Exact mathematical passage
```
\label{g:row-holonomies}
 R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},
 \qquad j=0,1.
```

R_pIntroduction passage · 12 passages

Search aliases: R_p

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Introduction passage · spectral.tex, line 634
The polynomial map R_p is defined at every x_j, while the Schur coefficients in [source reference] are not. Cancelling denominators in a determinant can cross this chart boundary; dividing by them in an eigenvector reconstruction cannot. A correct description there keeps the resonant cell modes or solves the full finite system. The fixed poles of the eliminated path are not all automatically eigenvalues of the full cyclic Hamiltonian.
Exact mathematical passage
```
R_p
```

R_{p,w}Introduction passage · 8 passages

Search aliases: R_p,w

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Introduction passage · finite.tex, line 793
e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radia…
Exact mathematical passage
```
 \Pr(R_{p,w};t)=
 \frac{|G_{p,w}|^2}{\delta^2+|G_{p,w}|^2}
       \sin^2\!\left(t\sqrt{\delta^2+|G_{p,w}|^2}\right).
 \label{a-finite-radial-probability}
```

\mathrm{red}Introduction passage · 3 passages

Search aliases: red mathrm

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Introduction passage · constraints.tex, line 658
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
 \Omega_{\mathrm{red}}=\iota^*\Omega_\Sigma,
 \qquad
 \{\Omega_{\mathrm{red}},\Omega_{\mathrm{red}}\}_{\mathrm D}=0.
```

SIntroduction passage · 156 passages

Search aliases: S

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Introduction passage · finite_core.tex, line 1102
Let H_S,H_h,S denote the finite block diagonal Hamiltonians. In coefficient coordinates, squared norm and continuum Hamiltonian expectation are
Exact mathematical passage
```
H_S,H_{h,S}
```

sIntroduction passage · 140 passages

Search aliases: s

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Introduction passage · limit.tex, line 834
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
s_v=d_v^{-1/2}(1,\ldots,1)
```

s^2Introduction passage · 11 passages

Search aliases: s^2

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Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

S_{A,\tau,u}Introduction passage · 7 passages

Search aliases: S_A,τ,u tau

Link to this entry ·

Introduction passage · refinement.tex, line 10
x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogonal. A bijective site translation and unit links therefore make this an …
Exact mathematical passage
```
 (S_{A,\tau,u}\psi)(x)
 =u(\tau^{-1}x)P_+\psi(\tau^{-1}x)
  +u(x)^{-1}P_-\psi(\tau x).
 \label{g:source-shift}
```

S_FIntroduction passage · 13 passages

Search aliases: S_F

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Introduction passage · extensions.tex, line 125
Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no unbounded operator.
Exact mathematical passage
```
S_F=\sum_{a\in F}P_a
```

s_FIntroduction passage · 5 passages

Search aliases: s_F

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
(\mathcal C_F,s_F)
```

s_F^2Introduction passage · 1 passage

Search aliases: s_F^2

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Introduction passage · constraints.tex, line 62
What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no …
Exact mathematical passage
```
s_F^2=0
```

s_vIntroduction passage · 3 passages

Search aliases: s_v

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Introduction passage · limit.tex, line 834
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
s_v=d_v^{-1/2}(1,\ldots,1)
```

\sin^2Introduction passage · 9 passages

Search aliases: sin^2 sin

Link to this entry ·

Introduction passage · finite.tex, line 793
e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radia…
Exact mathematical passage
```
 \Pr(R_{p,w};t)=
 \frac{|G_{p,w}|^2}{\delta^2+|G_{p,w}|^2}
       \sin^2\!\left(t\sqrt{\delta^2+|G_{p,w}|^2}\right).
 \label{a-finite-radial-probability}
```

\squareIntroduction passage · 19 passages

Search aliases: square

Link to this entry ·

Introduction passage · limit.tex, line 933
[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary …
Exact mathematical passage
```
C_N\square T_{p,h}
```

T^{-1}Introduction passage · 8 passages

Search aliases: T^-1

Link to this entry ·

Introduction passage · finite_core.tex, line 782
The source's transported Clifford action is defined by v↦ T(A(T^-1v)). The same is true for the displayed diagonal beta action. Conjugation preserves real coefficients and finite sums, proving the Hamiltonian equality. Apply the exponential identity with z=-it. Substituting each coefficient equality into the finite Fourier sum then proves equality of the packets. square
Exact mathematical passage
```
v\mapsto T(A(T^{-1}v))
```

t^2Introduction passage · 12 passages

Search aliases: t^2

Link to this entry ·

Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

T_{p,h}Introduction passage · 6 passages

Search aliases: T_p,h

Link to this entry ·

Introduction passage · limit.tex, line 933
[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary …
Exact mathematical passage
```
C_N\square T_{p,h}
```

T_{p,h+1}Introduction passage · 1 passage

Search aliases: T_p,h+1

Link to this entry ·

Introduction passage · limit.tex, line 933
[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary …
Exact mathematical passage
```
C_N\square T_{p,h+1}
```

\operatorname{Tr}Introduction passage · 23 passages

Search aliases: Tr operatorname

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Introduction passage · extensions.tex, line 452
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY, (δ_BX)^* · =δ_B(X^*), Tr((δ_BX)Y) · =-Tr(Xδ_BY). For the first identity, the middle terms -iXBY and iXBY cancel. For the second, taking the adjoint reverses the product order and conjugates i, with B^*=B restoring the stated sign. For the third, cyclicity moves the lea…
Exact mathematical passage
```
 \delta_B(XY)&=(\delta_BX)Y+X\delta_BY,\label{eq:foundation-matrix-leibniz}\\
 (\delta_BX)^*&=\delta_B(X^*),\label{eq:foundation-matrix-star}\\
 \operatorname{Tr}((\delta_BX)Y)&=-\operatorname{Tr}(X\delta_BY).
 \label{eq:foundation-matrix-ibp}
```

uIntroduction passage · 144 passages

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Introduction passage · limit.tex, line 853
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉
Exact mathematical passage
```
\psi^n(v,u)
```

U^*Introduction passage · 12 passages

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Introduction passage · core.tex, line 299
The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present or…
Exact mathematical passage
```
V=\dot UU^*
```

u^gIntroduction passage · 3 passages

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Introduction passage · refinement.tex, line 6
We use the forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x…
Exact mathematical passage
```
 u^g(x)=g(\tau x)u(x)g(x)^{-1}.
```

U_nIntroduction passage · 6 passages

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Introduction passage · spectral.tex, line 545
[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse e…
Exact mathematical passage
```
\det K_n(x)=U_n(x/2)
```

U_{n-1}Introduction passage · 8 passages

Search aliases: U_n-1

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Introduction passage · spectral.tex, line 546
[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagonal inverse e…
Exact mathematical passage
```
 (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}
       =\frac{U_{n-1}(x/2)}{U_n(x/2)},\qquad
 (K_n(x)^{-1})_{1n}=\frac1{U_n(x/2)}.
```

VIntroduction passage · 101 passages

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

vIntroduction passage · 171 passages

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Introduction passage · limit.tex, line 833
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
d_v
```

V_{h,n}Introduction passage · 2 passages

Search aliases: V_h,n

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

V_nIntroduction passage · 6 passages

Search aliases: V_n

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Introduction passage · continuum.tex, line 26
Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+t)=U_h(s)U_h(t). This construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient f…
Exact mathematical passage
```
V_n(t)=e^{-itH_n},\qquad V_{h,n}(t)=e^{-itH_{h,n}}.
```

\vec EIntroduction passage · 2 passages

Search aliases: vec E vec

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Introduction passage · limit.tex, line 828
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
\vec E
```

WIntroduction passage · 35 passages

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Introduction passage · core.tex, line 364
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
\label{g:fine-plaquette}
 W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},
 \qquad i,j\in\{0,1\}.
```

wIntroduction passage · 97 passages

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Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

w^qIntroduction passage · 1 passage

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Introduction passage · spectral.tex, line 954
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
\sum_{q\geq0}w^q/[q!\prod_r(b_r)_q]
```

W_{0j}Introduction passage · 1 passage

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Introduction passage · core.tex, line 369
j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.
Exact mathematical passage
```
\label{g:row-holonomies}
 R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},
 \qquad j=0,1.
```

W_{1j}Introduction passage · 1 passage

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Introduction passage · core.tex, line 369
j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.
Exact mathematical passage
```
\label{g:row-holonomies}
 R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},
 \qquad j=0,1.
```

W_{ij}Introduction passage · 2 passages

Search aliases: W_ij

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Introduction passage · core.tex, line 364
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
\label{g:fine-plaquette}
 W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},
 \qquad i,j\in\{0,1\}.
```

XIntroduction passage · 103 passages

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Introduction passage · core.tex, line 300
The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present or…
Exact mathematical passage
```
 \dot P_a=[V,P_a],\qquad
 \frac{d}{dt}(P_bXP_a)
 =[V,P_b]XP_a+P_b\dot XP_a+P_bX[V,P_a].
```

X^*Introduction passage · 7 passages

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Introduction passage · extensions.tex, line 452
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY, (δ_BX)^* · =δ_B(X^*), Tr((δ_BX)Y) · =-Tr(Xδ_BY). For the first identity, the middle terms -iXBY and iXBY cancel. For the second, taking the adjoint reverses the product order and conjugates i, with B^*=B restoring the stated sign. For the third, cyclicity moves the lea…
Exact mathematical passage
```
 \delta_B(XY)&=(\delta_BX)Y+X\delta_BY,\label{eq:foundation-matrix-leibniz}\\
 (\delta_BX)^*&=\delta_B(X^*),\label{eq:foundation-matrix-star}\\
 \operatorname{Tr}((\delta_BX)Y)&=-\operatorname{Tr}(X\delta_BY).
 \label{eq:foundation-matrix-ibp}
```

x^2Introduction passage · 12 passages

Search aliases: x^2

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Introduction passage · extensions.tex, line 126
Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no unbounded operator.
Exact mathematical passage
```
 \norm{S_Fx}^2=\sum_{a\in F}\norm{P_ax}^2,\qquad
 \sum_a\norm{P_ax}^2=\norm x^2.
```

X_{0j}Introduction passage · 3 passages

Search aliases: X_0j

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Introduction passage · core.tex, line 369
j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.
Exact mathematical passage
```
\label{g:row-holonomies}
 R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},
 \qquad j=0,1.
```

X_{0j}^{-1}Introduction passage · 3 passages

Search aliases: X_0j^-1

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Introduction passage · core.tex, line 369
j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.
Exact mathematical passage
```
\label{g:row-holonomies}
 R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},
 \qquad j=0,1.
```

x_H^2Introduction passage · 1 passage

Search aliases: x_H^2

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Introduction passage · extensions.tex, line 137
Now let H:D(H)⊂HH→HH be a densely defined symmetric operator. To write HP_aψ for every domain state, require P_aD(H)⊂ D(H). Define norm x_H^2=norm x^2+normHx^2. A domain-compatible atlas is graph compatible when S_Fx⟶ x in the graph norm for every x∈ D(H). Equivalently, both S_Fx→ x and HS_Fx→ Hx hold in HH. The second convergence does not follow from the first for an unbounded H.
Exact mathematical passage
```
\norm x_H^2=\norm x^2+\norm{Hx}^2
```

X_{i,j+1}^{-1}Introduction passage · 1 passage

Search aliases: X_i,j+1^-1

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Introduction passage · core.tex, line 364
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
\label{g:fine-plaquette}
 W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},
 \qquad i,j\in\{0,1\}.
```

X_{ij}Introduction passage · 2 passages

Search aliases: X_ij

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Introduction passage · core.tex, line 363
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
X_{ij}
```

x_jIntroduction passage · 5 passages

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Introduction passage · spectral.tex, line 634
The polynomial map R_p is defined at every x_j, while the Schur coefficients in [source reference] are not. Cancelling denominators in a determinant can cross this chart boundary; dividing by them in an eigenvector reconstruction cannot. A correct description there keeps the resonant cell modes or solves the full finite system. The fixed poles of the eliminated path are not all automatically eigenvalues of the full cyclic Hamiltonian.
Exact mathematical passage
```
x_j
```

YIntroduction passage · 49 passages

Search aliases: Y

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Introduction passage · extensions.tex, line 452
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY, (δ_BX)^* · =δ_B(X^*), Tr((δ_BX)Y) · =-Tr(Xδ_BY). For the first identity, the middle terms -iXBY and iXBY cancel. For the second, taking the adjoint reverses the product order and conjugates i, with B^*=B restoring the stated sign. For the third, cyclicity moves the lea…
Exact mathematical passage
```
 \delta_B(XY)&=(\delta_BX)Y+X\delta_BY,\label{eq:foundation-matrix-leibniz}\\
 (\delta_BX)^*&=\delta_B(X^*),\label{eq:foundation-matrix-star}\\
 \operatorname{Tr}((\delta_BX)Y)&=-\operatorname{Tr}(X\delta_BY).
 \label{eq:foundation-matrix-ibp}
```

Y_{i+1,j}Introduction passage · 1 passage

Search aliases: Y_i+1,j

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Introduction passage · core.tex, line 364
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
\label{g:fine-plaquette}
 W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},
 \qquad i,j\in\{0,1\}.
```

Y_{ij}Introduction passage · 1 passage

Search aliases: Y_ij

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Introduction passage · core.tex, line 363
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
Y_{ij}
```

Y_{ij}^{-1}Introduction passage · 1 passage

Search aliases: Y_ij^-1

Link to this entry ·

Introduction passage · core.tex, line 364
Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right plaquette into the left base fiber before multiplying it with the le…
Exact mathematical passage
```
\label{g:fine-plaquette}
 W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},
 \qquad i,j\in\{0,1\}.
```

\ZIntroduction passage · 25 passages

Search aliases: Z

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Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

zIntroduction passage · 196 passages

Search aliases: z

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

z^3Introduction passage · 2 passages

Search aliases: z^3

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Introduction passage · spectral.tex, line 959
hypergeometric-lane Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
E_{3,1}(z)=z\,{}_0F_2(;2/3,4/3;z^3/27)
```

z^jIntroduction passage · 1 passage

Search aliases: z^j

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

z^LIntroduction passage · 1 passage

Search aliases: z^L

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Introduction passage · spectral.tex, line 947
· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an…
Exact mathematical passage
```
 E_{L,j}(z)=\frac{z^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                              \frac{z^L}{L^L}\right).
 \label{a-spectral-hypergeometric-lane}
```

\Z_pIntroduction passage · 6 passages

Search aliases: Z_p Z

Link to this entry ·

Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

|Introduction passage · 243 passages

Search aliases: |

Link to this entry ·

Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

|^2Introduction passage · 61 passages

Search aliases: |^2

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Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

\pmIntroduction passage · 20 passages

Search aliases: ± pm

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Introduction passage · refinement.tex, line 9
refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogonal. A bijective …
Exact mathematical passage
```
P_\pm=(1\pm A)/2
```

\cdotIntroduction passage · 12 passages

Search aliases: · cdot

Link to this entry ·

Introduction passage · constraints.tex, line 657
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
\{\cdot,\cdot\}_{\mathrm D}
```

\delta^2Introduction passage · 15 passages

Search aliases: δ^2 delta

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Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

\delta_BIntroduction passage · 8 passages

Search aliases: δ_B delta

Link to this entry ·

Introduction passage · extensions.tex, line 448
Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,
Exact mathematical passage
```
 \delta_BX=i[B,X]=i(BX-XB).
```

\epsilonIntroduction passage · 93 passages

Search aliases: ϵ epsilon

Link to this entry ·

Introduction passage · finite.tex, line 800
tood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radial.
Exact mathematical passage
```
E_c=\epsilon
```

\epsilon_0Introduction passage · 4 passages

Search aliases: ϵ_0 epsilon

Link to this entry ·

Introduction passage · strong_fields.tex, line 6
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
G=c=4\pi\epsilon_0=1
```

\iotaIntroduction passage · 10 passages

Search aliases: ι iota

Link to this entry ·

Introduction passage · constraints.tex, line 657
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
\iota
```

\iota^*Introduction passage · 1 passage

Search aliases: ι^* iota

Link to this entry ·

Introduction passage · constraints.tex, line 658
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
 \Omega_{\mathrm{red}}=\iota^*\Omega_\Sigma,
 \qquad
 \{\Omega_{\mathrm{red}},\Omega_{\mathrm{red}}\}_{\mathrm D}=0.
```

\Lambda^2Introduction passage · 4 passages

Search aliases: Λ^2 Lambda

Link to this entry ·

Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

\muIntroduction passage · 70 passages

Search aliases: μ mu

Link to this entry ·

Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

\mu_pIntroduction passage · 12 passages

Search aliases: μ_p mu

Link to this entry ·

Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

\piIntroduction passage · 100 passages

Search aliases: π pi

Link to this entry ·

Introduction passage · finite.tex, line 322
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
\omega_p=\exp(2\pi\mathrm{i}/p)
```

\rho_v^nIntroduction passage · 4 passages

Search aliases: ρ_v^n rho

Link to this entry ·

Introduction passage · limit.tex, line 855
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
 \rho_v^n=\sum_{u\sim v}\norm{\psi^n(v,u)}^2,
 \qquad F_{v\to u}^n=\norm{\phi^n(v,u)}^2.
```

\rho_v^{n+1}Introduction passage · 3 passages

Search aliases: ρ_v^n+1 rho

Link to this entry ·

Introduction passage · limit.tex, line 862
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
\rho_v^{n+1}=\sum_u F_{u\to v}^n
```

\SigmaIntroduction passage · 40 passages

Search aliases: Σ Sigma

Link to this entry ·

Introduction passage · constraints.tex, line 658
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
 \Omega_{\mathrm{red}}=\iota^*\Omega_\Sigma,
 \qquad
 \{\Omega_{\mathrm{red}},\Omega_{\mathrm{red}}\}_{\mathrm D}=0.
```

\sigmaIntroduction passage · 67 passages

Search aliases: σ sigma

Link to this entry ·

Introduction passage · refinement.tex, line 20
Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to σ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.
Exact mathematical passage
```
 \sigma e_x=m_x,\qquad \sigma m_x=e_{\tau x}.
```

\sigma^2Introduction passage · 6 passages

Search aliases: σ^2 sigma

Link to this entry ·

Introduction passage · refinement.tex, line 23
Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to σ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.
Exact mathematical passage
```
\sigma^2e_x=e_{\tau x}
```

\tauIntroduction passage · 71 passages

Search aliases: τ tau

Link to this entry ·

Introduction passage · limit.tex, line 828
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
\tau_2=2
```

\tau^{-1}Introduction passage · 6 passages

Search aliases: τ^-1 tau

Link to this entry ·

Introduction passage · refinement.tex, line 10
x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is (S_A,τ,uψ)(x) =u(τ^-1x)P_+ψ(τ^-1x) +u(x)^-1P_-ψ(τ x). The channels are orthogonal. A bijective site translation and unit links therefore make this an …
Exact mathematical passage
```
 (S_{A,\tau,u}\psi)(x)
 =u(\tau^{-1}x)P_+\psi(\tau^{-1}x)
  +u(x)^{-1}P_-\psi(\tau x).
 \label{g:source-shift}
```

\tau_2Introduction passage · 1 passage

Search aliases: τ_2 tau

Link to this entry ·

Introduction passage · limit.tex, line 828
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
\tau_2=2
```

\tau_pIntroduction passage · 2 passages

Search aliases: τ_p tau

Link to this entry ·

Introduction passage · limit.tex, line 829
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
\tau_p=p-1
```

\PhiIntroduction passage · 50 passages

Search aliases: Φ Phi

Link to this entry ·

Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

\phiIntroduction passage · 36 passages

Search aliases: ϕ phi

Link to this entry ·

Introduction passage · limit.tex, line 854
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
\phi^n=C\psi^n
```

\phi^nIntroduction passage · 2 passages

Search aliases: ϕ^n phi

Link to this entry ·

Introduction passage · limit.tex, line 854
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
\phi^n=C\psi^n
```

\Phi_eIntroduction passage · 4 passages

Search aliases: Φ_e Phi

Link to this entry ·

Introduction passage · strong_fields.tex, line 7
The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift.…
Exact mathematical passage
```
 ds^2=-N^2dt^2+\Lambda^2(dr+N^r dt)^2+R^2d\Omega^2,
 \qquad \mathcal A=A_rdr+\Phi_e dt.
```

\psi^nIntroduction passage · 3 passages

Search aliases: ψ^n psi

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Introduction passage · limit.tex, line 853
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉
Exact mathematical passage
```
\psi^n(v,u)
```

\psi_{r,a,q}^{(h)}Introduction passage · 2 passages

Search aliases: ψ_r,a,q^(h) psi

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Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

\OmegaIntroduction passage · 45 passages

Search aliases: Ω Omega

Link to this entry ·

Introduction passage · finite.tex, line 781
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalent
Exact mathematical passage
```
\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}
```

\omegaIntroduction passage · 37 passages

Search aliases: ω omega

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Introduction passage · finite.tex, line 322
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
\omega_p=\exp(2\pi\mathrm{i}/p)
```

\Omega^2Introduction passage · 4 passages

Search aliases: Ω^2 Omega

Link to this entry ·

Introduction passage · finite.tex, line 782
G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance
Exact mathematical passage
```
(H_{\mathrm{bright}}-\overline E I)^2=\Omega^2I
```

\omega_pIntroduction passage · 2 passages

Search aliases: ω_p omega

Link to this entry ·

Introduction passage · finite.tex, line 322
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
\omega_p=\exp(2\pi\mathrm{i}/p)
```

\omega_p^{q d_r(x)}Introduction passage · 1 passage

Search aliases: ω_p^q d_r(x) omega

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Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

\Omega_{\mathrm{red}}Introduction passage · 1 passage

Search aliases: Ω_red Omega mathrm

Link to this entry ·

Introduction passage · constraints.tex, line 658
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
 \Omega_{\mathrm{red}}=\iota^*\Omega_\Sigma,
 \qquad
 \{\Omega_{\mathrm{red}},\Omega_{\mathrm{red}}\}_{\mathrm D}=0.
```

\Omega_\SigmaIntroduction passage · 3 passages

Search aliases: Ω_Σ Omega Sigma

Link to this entry ·

Introduction passage · constraints.tex, line 658
Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missi…
Exact mathematical passage
```
 \Omega_{\mathrm{red}}=\iota^*\Omega_\Sigma,
 \qquad
 \{\Omega_{\mathrm{red}},\Omega_{\mathrm{red}}\}_{\mathrm D}=0.
```

\ldotsIntroduction passage · 25 passages

Search aliases: … ldots

Link to this entry ·

Introduction passage · finite.tex, line 323
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,
Exact mathematical passage
```
a\in\{0,\ldots,p^{r-1}-1\}
```

\ellIntroduction passage · 50 passages

Search aliases: ℓ ell

Link to this entry ·

Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

\ell^2Introduction passage · 26 passages

Search aliases: ℓ^2 ell

Link to this entry ·

Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

\toIntroduction passage · 75 passages

Search aliases: → to

Link to this entry ·

Introduction passage · limit.tex, line 855
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
 \rho_v^n=\sum_{u\sim v}\norm{\psi^n(v,u)}^2,
 \qquad F_{v\to u}^n=\norm{\phi^n(v,u)}^2.
```

\mapstoIntroduction passage · 9 passages

Search aliases: ↦ mapsto

Link to this entry ·

Introduction passage · finite_core.tex, line 782
The source's transported Clifford action is defined by v↦ T(A(T^-1v)). The same is true for the displayed diagonal beta action. Conjugation preserves real coefficients and finite sums, proving the Hamiltonian equality. Apply the exponential identity with z=-it. Substituting each coefficient equality into the finite Fourier sum then proves equality of the packets. square
Exact mathematical passage
```
v\mapsto T(A(T^{-1}v))
```

\prodIntroduction passage · 10 passages

Search aliases: ∏ prod

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Introduction passage · spectral.tex, line 954
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
\sum_{q\geq0}w^q/[q!\prod_r(b_r)_q]
```

\prod_rIntroduction passage · 2 passages

Search aliases: ∏_r prod

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Introduction passage · spectral.tex, line 954
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
\sum_{q\geq0}w^q/[q!\prod_r(b_r)_q]
```

\sum_{a\in F}Introduction passage · 3 passages

Search aliases: ∑_a∈ F sum in

Link to this entry ·

Introduction passage · extensions.tex, line 125
Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no unbounded operator.
Exact mathematical passage
```
S_F=\sum_{a\in F}P_a
```

\sum_{q\geq0}Introduction passage · 2 passages

Search aliases: ∑_q≥0 sum geq

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Introduction passage · spectral.tex, line 954
mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).
Exact mathematical passage
```
\sum_{q\geq0}w^q/[q!\prod_r(b_r)_q]
```

\sum_uIntroduction passage · 2 passages

Search aliases: ∑_u sum

Link to this entry ·

Introduction passage · limit.tex, line 860
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
\sum_uF_{v\to u}^n=\rho_v^n
```

\sum_{u\sim v}Introduction passage · 2 passages

Search aliases: ∑_u∼ v sum sim

Link to this entry ·

Introduction passage · limit.tex, line 855
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
 \rho_v^n=\sum_{u\sim v}\norm{\psi^n(v,u)}^2,
 \qquad F_{v\to u}^n=\norm{\phi^n(v,u)}^2.
```

\intIntroduction passage · 56 passages

Search aliases: ∫ int

Link to this entry ·

Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

\int_{x+p^m\Z_p}Introduction passage · 1 passage

Search aliases: ∫_x+p^mZ_p int Z

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Introduction passage · limit.tex, line 75
For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.
Exact mathematical passage
```
 (\mathsf E_m f)(x)
 =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
 \label{a-limit-expectation}
```

\simIntroduction passage · 3 passages

Search aliases: ∼ sim

Link to this entry ·

Introduction passage · limit.tex, line 855
Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A,…
Exact mathematical passage
```
 \rho_v^n=\sum_{u\sim v}\norm{\psi^n(v,u)}^2,
 \qquad F_{v\to u}^n=\norm{\phi^n(v,u)}^2.
```

\equivIntroduction passage · 9 passages

Search aliases: ≡ equiv

Link to this entry ·

Introduction passage · finite.tex, line 325
Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
 \psi_{r,a,q}^{(h)}(x)=
 \begin{cases}
 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
       &x\equiv a\pmod{p^{r-1}},\\
 0,&\text{otherwise},
 \end{cases}
 \label{a-finite-wavelet}
```

\subsetIntroduction passage · 10 passages

Search aliases: ⊂ subset

Link to this entry ·

Introduction passage · extensions.tex, line 125
Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no unbounded operator.
Exact mathematical passage
```
F\subset A
```

\otimesIntroduction passage · 37 passages

Search aliases: ⊗ otimes

Link to this entry ·

Introduction passage · limit.tex, line 830
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
 \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
```

\langleIntroduction passage · 49 passages

Search aliases: ⟨ langle

Link to this entry ·

Introduction passage · limit.tex, line 835
Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^…
Exact mathematical passage
```
G_v=2|s_v\rangle\langle s_v|-I
```

\rangleIntroduction passage · 93 passages

Search aliases: ⟩ rangle

Link to this entry ·

Introduction passage · finite.tex, line 792
. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,…
Exact mathematical passage
```
|c\rangle
```

\longrightarrowIntroduction passage · 15 passages

Search aliases: ⟶ longrightarrow

Link to this entry ·

Introduction passage · extensions.tex, line 138
Now let H:D(H)⊂HH→HH be a densely defined symmetric operator. To write HP_aψ for every domain state, require P_aD(H)⊂ D(H). Define norm x_H^2=norm x^2+normHx^2. A domain-compatible atlas is graph compatible when S_Fx⟶ x in the graph norm for every x∈ D(H). Equivalently, both S_Fx→ x and HS_Fx→ Hx hold in HH. The second convergence does not follow from the first for an unbounded H.
Exact mathematical passage
```
 S_Fx\longrightarrow x\quad\text{in the graph norm for every }x\in D(H).
 \label{eq:foundation-graph-compatible}
```

+_pUse in source · 2 passages

Search aliases: +_p

Link to this entry ·

Use in source · finite.tex, line 379
One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an id…
Exact mathematical passage
```
f\mapsto|+_p\rangle\otimes f
```

+_xUse in source · 2 passages

Search aliases: +_x

Link to this entry ·

Use in source · limit.tex, line 687
Z_sign have eigenvalues +1,-1 on a dyadic sign qubit, and let B=B^* act on the remaining factors. Assume there are orthonormal states |b_0⟩,|b_1⟩ with B|b_0⟩=0 and B|b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sign⊗ B, |Ψ(0)⟩=|+_x⟩⊗ frac|b_0⟩+|b_1⟩sqrt2. Evolution under this explicit interaction alone gives |Ψ(t)⟩=frac1sqrt2 (|+_x⟩⊗|b_0⟩ +e^-igtZ_sign|+_x⟩⊗|b_1…
Exact mathematical passage
```
 H_{\mathrm{int}}=gZ_{\mathrm{sign}}\otimes B,
 \qquad
 |\Psi(0)\rangle=|+_x\rangle\otimes
 \frac{|b_0\rangle+|b_1\rangle}{\sqrt2}.
```

\boldsymbol1Use in source · 1 passage

Search aliases: 1 boldsymbol

Link to this entry ·

Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

\mathbf1Use in source · 5 passages

Search aliases: 1 mathbf

Link to this entry ·

Use in source · finite.tex, line 644
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber
Exact mathematical passage
```
 E_r^{(h)}(x,y)=p^{-(h-r)}
              \mathbf1_{\{x\equiv y\, (\mathrm{mod}\,p^r)\}}.
```

\boldsymbol1_{\mathrm c}Use in source · 1 passage

Search aliases: 1_ c boldsymbol mathrm

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Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

\boldsymbol1_{\mathrm f}Use in source · 1 passage

Search aliases: 1_ f boldsymbol mathrm

Link to this entry ·

Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

\mathbf1_{B_{m,a}}Use in source · 2 passages

Search aliases: 1_B_m,a mathbf

Link to this entry ·

Use in source · limit.tex, line 25
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
 \qquad \mathcal V_m=\operatorname{span}
 \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
 \label{a-limit-cylinder-embedding}
```

\mathbf1_{B_{m,a+d p^{m-1}}}Use in source · 1 passage

Search aliases: 1_B_m,a+d p^m-1 mathbf

Link to this entry ·

Use in source · limit.tex, line 174
For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω_p^(q'-q)d=δ_qq'. The number of these orthonormal functions is exactly d_m, so they span that level. Together with ψ_0=1, they form a complete orthono…
Exact mathematical passage
```
 \psi_{m,a,q}=p^{(m-1)/2}\sum_{d=0}^{p-1}
 \omega_p^{qd}\mathbf1_{B_{m,a+d p^{m-1}}},
 \quad
 \begin{matrix}m\geq1,\quad 0\leq a<p^{m-1},\\1\leq q<p.\end{matrix}
 \label{a-limit-haar-wavelets}
```

\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}Use in source · 1 passage

Search aliases: 1_x≡ y (mod p^r) mathbf equiv mathrm

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}Use in source · 1 passage

Search aliases: 1_x≡ y (mod p^r-1) mathbf equiv mathrm

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

\mathbf1_{\{x\equiv y\, (\mathrm{mod}\,p^r)\}}Use in source · 1 passage

Search aliases: 1_{x≡ y (mod p^r)} mathbf equiv mathrm

Link to this entry ·

Use in source · finite.tex, line 644
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber
Exact mathematical passage
```
 E_r^{(h)}(x,y)=p^{-(h-r)}
              \mathbf1_{\{x\equiv y\, (\mathrm{mod}\,p^r)\}}.
```

A^{\mathcal H}Use in source · 3 passages

Search aliases: A^ H mathcal

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Use in source · finite_core.tex, line 748
Proposition PH09 (coordinate conjugacy of generators and exponentials). Let A be a complex-linear endomorphism of A, and let A^ H=TAT^-1. For every complex z,
Exact mathematical passage
```
A^{\mathcal H}=TAT^{-1}
```

a^{(r)}Use in source · 2 passages

Search aliases: a^(r)

Link to this entry ·

Use in source · continuum.tex, line 176
Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If a^(r)→ a and widehat H_ma^(r)→ b in KK, then convergence holds at each coordinate. Since the finite matrix H_n is continuous, b_n=lim_rH_na_n^(r)=H_na_n. The output sequence equals b∈KK, so a belongs to the maximal domain and widehat H_ma=b. This direct argument is included as an ordinary explanatory deduction.
Exact mathematical passage
```
a^{(r)}\to a
```

\mathcal A^*Use in source · 4 passages

Search aliases: A^* mathcal

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Use in source · spectral.tex, line 29
section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt …
Exact mathematical passage
```
 X=\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\qquad
 (\mathcal A X\mathcal A^*)_{rs}=X_{rs}.
 \label{a-spectral-block-reconstruction}
```

a^{-1}Use in source · 2 passages

Search aliases: a^-1

Link to this entry ·

Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

a^2Use in source · 2 passages

Search aliases: a^2

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Use in source · core.tex, line 188
_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These orders concern the chosen pointwise comparison and …
Exact mathematical passage
```
\theta=O(a^2)
```

a^3Use in source · 1 passage

Search aliases: a^3

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Use in source · core.tex, line 188
Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutio…
Exact mathematical passage
```
O(a^3)
```

a^4Use in source · 2 passages

Search aliases: a^4

Link to this entry ·

Use in source · core.tex, line 180
^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term f…
Exact mathematical passage
```
\label{g:centered-plaquette}
 \theta=-qa_\mu a_\nu F_{\mu\nu}(x_c)+O(a^4).
```

a^5Use in source · 1 passage

Search aliases: a^5

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Use in source · core.tex, line 188
abelg:wilson-expansion 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or it…
Exact mathematical passage
```
O(a^5)
```

a^6Use in source · 2 passages

Search aliases: a^6

Link to this entry ·

Use in source · core.tex, line 184
u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluat…
Exact mathematical passage
```
\label{g:wilson-expansion}
 1-\operatorname{Re}W
 =\tfrac12q^2a_\mu^2a_\nu^2 F_{\mu\nu}(x_c)^2+O(a^6).
```

a^8Use in source · 1 passage

Search aliases: a^8

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Use in source · core.tex, line 188
ered-plaquette θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justif…
Exact mathematical passage
```
O(a^8)
```

\mathcal A^{\mathrm{ad}}_{p,h}Use in source · 3 passages

Search aliases: A^ad_p,h mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 425
For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+…
Exact mathematical passage
```
 \mathcal A^{\mathrm{std}}_{p,h}|pk+r\rangle
 &=|r\rangle\otimes|k\rangle,
 &0\leq r<p,\quad0\leq k<m,\nonumber\\
 \mathcal A^{\mathrm{ad}}_{p,h}|a+d p^{h-1}\rangle
 &=|d\rangle\otimes|a\rangle,
 &0\leq d<p,\quad0\leq a<m.
 \label{a-finite-digit-splits}
```

a^gUse in source · 3 passages

Search aliases: a^g

Link to this entry ·

Use in source · refinement.tex, line 40
Coarse endpoint gauges g(x) and midpoint gauges q(x) may be chosen independently. The two sublinks transform by a^g(x)=q(x)a(x)g(x)^-1, b^g(x)=g(τ x)b(x)q(x)^-1. The midpoint factors cancel in b^ga^g, leaving the required coarse transformation. At the midpoint, a^g(x)g(x)ψ(x)=q(x)a(x)ψ(x), while endpoint covariance is immediate. Hence interpolation commutes exactly with the corresponding gauge actions. This argument allows independent new gauge data at every inserted midpoint.
Exact mathematical passage
```
 a^g(x)=q(x)a(x)g(x)^{-1},\qquad
 b^g(x)=g(\tau x)b(x)q(x)^{-1}.
```

A^kUse in source · 3 passages

Search aliases: A^k

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Use in source · finite_core.tex, line 762
Proof. Induction gives (TAT^-1)^k=TA^kT^-1. For k=0, use TT^-1=I; for the induction step, cancel the adjacent T^-1T. Every exponential partial sum therefore satisfies
Exact mathematical passage
```
(TAT^{-1})^k=TA^kT^{-1}
```

a^kUse in source · 1 passage

Search aliases: a^k

Link to this entry ·

Use in source · spectral.tex, line 1029
is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane by the full tail. Since (Q+1+k)!≥(Q+1)!k!, ∑_n=Q+1^∞fraca^nn! ≤fraca^Q+1(Q+1)!∑_k=0^∞fraca^kk!, which proves [source reference].
Exact mathematical passage
```
 \sum_{n=Q+1}^\infty\frac{a^n}{n!}
 \leq\frac{a^{Q+1}}{(Q+1)!}\sum_{k=0}^\infty\frac{a^k}{k!},
```

a^nUse in source · 1 passage

Search aliases: a^n

Link to this entry ·

Use in source · spectral.tex, line 1029
is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane by the full tail. Since (Q+1+k)!≥(Q+1)!k!, ∑_n=Q+1^∞fraca^nn! ≤fraca^Q+1(Q+1)!∑_k=0^∞fraca^kk!, which proves [source reference].
Exact mathematical passage
```
 \sum_{n=Q+1}^\infty\frac{a^n}{n!}
 \leq\frac{a^{Q+1}}{(Q+1)!}\sum_{k=0}^\infty\frac{a^k}{k!},
```

a^{Q+1}Use in source · 2 passages

Search aliases: a^Q+1

Link to this entry ·

Use in source · spectral.tex, line 1015
∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subseries of this sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of …
Exact mathematical passage
```
 e^a\frac{a^{Q+1}}{(Q+1)!},\qquad a=|t|\norm{A_0}.
 \label{a-spectral-lane-tail-bound}
```

\mathcal A^{\mathrm{std}}_{p,h}Use in source · 2 passages

Search aliases: A^std_p,h mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 425
For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+…
Exact mathematical passage
```
 \mathcal A^{\mathrm{std}}_{p,h}|pk+r\rangle
 &=|r\rangle\otimes|k\rangle,
 &0\leq r<p,\quad0\leq k<m,\nonumber\\
 \mathcal A^{\mathrm{ad}}_{p,h}|a+d p^{h-1}\rangle
 &=|d\rangle\otimes|a\rangle,
 &0\leq d<p,\quad0\leq a<m.
 \label{a-finite-digit-splits}
```

a^\daggerUse in source · 3 passages

Search aliases: a^† dagger

Link to this entry ·

Use in source · spectral.tex, line 1246
Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number norm. Truncation of a vector in either domain converges in the corresponding graph norm, so the identities extend to those domains. For the isolated …
Exact mathematical passage
```
 Q_r^{(L)}aQ_s^{(L)}=0\quad\text{unless }r\equiv s-1\pmod L,
 \qquad
 Q_r^{(L)}a^\dagger Q_s^{(L)}=0\quad\text{unless }r\equiv s+1\pmod L.
 \label{a-spectral-ladder-routing}
```

a_{\mathbf d}Use in source · 2 passages

Search aliases: a_ d mathbf

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Use in source · finite.tex, line 756
). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_ d=0. Hence H_int|A⟩=0. Only the radial vector is bright under this equal-leaf interaction.
Exact mathematical passage
```
|A\rangle=\sum_{\mathbf d}a_{\mathbf d}|\mathbf d\rangle
```

A_{\mathbf r}Use in source · 6 passages

Search aliases: A_ r mathbf

Link to this entry ·

Use in source · limit.tex, line 596
[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber
Exact mathematical passage
```
A_{\mathbf r}
```

A_0Use in source · 8 passages

Search aliases: A_0

Link to this entry ·

Use in source · spectral.tex, line 1001
[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber
Exact mathematical passage
```
A_0\in\mathcal B(\HH)
```

a_0Use in source · 1 passage

Search aliases: a_0

Link to this entry ·

Use in source · refinement.tex, line 430
Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero need not converge; summability controls the accumulated tail.
Exact mathematical passage
```
h_n=a_0 2^{-n}
```

a_0^2Use in source · 1 passage

Search aliases: a_0^2

Link to this entry ·

Use in source · refinement.tex, line 431
Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero need not converge; summability controls the accumulated tail.
Exact mathematical passage
```
 \varepsilon_n=C4^{-n},\qquad C=a_0^2\mathcal A_{\partial}.
```

A_0^jUse in source · 1 passage

Search aliases: A_0^j

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Use in source · spectral.tex, line 1037
If A_0^L=α I, the hypergeometric expression reduces every lane to a scalar function multiplying A_0^j. The propagator lies in the finite span of I,A_0,…,A_0^L-1. A more general polynomial identity gives the corresponding finite algebra of powers. These statements follow from the supplied algebraic relation; they do not assert one for an arbitrary Hamiltonian.
Exact mathematical passage
```
A_0^j
```

A_0^LUse in source · 1 passage

Search aliases: A_0^L

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Use in source · spectral.tex, line 1036
If A_0^L=α I, the hypergeometric expression reduces every lane to a scalar function multiplying A_0^j. The propagator lies in the finite span of I,A_0,…,A_0^L-1. A more general polynomial identity gives the corresponding finite algebra of powers. These statements follow from the supplied algebraic relation; they do not assert one for an arbitrary Hamiltonian.
Exact mathematical passage
```
A_0^L=\alpha I
```

A_0^{L-1}Use in source · 1 passage

Search aliases: A_0^L-1

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Use in source · spectral.tex, line 1038
If A_0^L=α I, the hypergeometric expression reduces every lane to a scalar function multiplying A_0^j. The propagator lies in the finite span of I,A_0,…,A_0^L-1. A more general polynomial identity gives the corresponding finite algebra of powers. These statements follow from the supplied algebraic relation; they do not assert one for an arbitrary Hamiltonian.
Exact mathematical passage
```
I,A_0,\ldots,A_0^{L-1}
```

a_{2n+1,2n}Use in source · 1 passage

Search aliases: a_2n+1,2n

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Use in source · extensions.tex, line 231
For a simple warning, the antisymmetric scalar array with a_2n,2n+1=1/n, a_2n+1,2n=-1/n for n≥1, and all other entries zero has finite support in each row but infinite total absolute sum. This array is only a summation example, not an asserted physical realization of the current. It shows why rowwise convergence alone cannot authorize arbitrary rearrangement.
Exact mathematical passage
```
a_{2n+1,2n}=-1/n
```

a_{2n,2n+1}Use in source · 1 passage

Search aliases: a_2n,2n+1

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Use in source · extensions.tex, line 231
For a simple warning, the antisymmetric scalar array with a_2n,2n+1=1/n, a_2n+1,2n=-1/n for n≥1, and all other entries zero has finite support in each row but infinite total absolute sum. This array is only a summation example, not an asserted physical realization of the current. It shows why rowwise convergence alone cannot authorize arbitrary rearrangement.
Exact mathematical passage
```
a_{2n,2n+1}=1/n
```

A_aUse in source · 6 passages

Search aliases: A_a

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Use in source · constraints.tex, line 243
[The resolved algebraic transfer step] Assume the continuum polynomial constraint identities in this convention and 16K<N. Their full-grid sampled evaluations obey {G[λ],G[μ]}_N · =G[λ×μ], {D[ξ],G[λ]}_N · =G[ξ^aδ_aλ], {D[ξ],D[η]}_N · =D[[ξ,η]], {G[λ],H[n]}_N · =0, {D[ξ],H[n]}_N · =H[ L_ξ^(-1)n], {H[n],H[m]}_N · =V[S(n,m)] =D[S(n,m)]+G[S^a(n,m)A_a]. Including the smearing, the Gauss, improved momentum, and Hamiltonian functionals contain at most three, four, and five band-K factors. Their canonic…
Exact mathematical passage
```
 \{G[\lambda],G[\mu]\}_N&=G[\lambda\times\mu],\\
 \{D[\xi],G[\lambda]\}_N&=G[\xi^a\delta_a\lambda],\\
 \{D[\xi],D[\eta]\}_N&=D[[\xi,\eta]],\\
 \{G[\lambda],H[n]\}_N&=0,\\
 \{D[\xi],H[n]\}_N&=H[\mathcal L_\xi^{(-1)}n],\\
 \{H[n],H[m]\}_N&=V[S(n,m)]
                  =D[S(n,m)]+G[S^a(n,m)A_a].
 \label{c:constraints:resolved-algebra}
```

A_a^iUse in source · 3 passages

Search aliases: A_a^i

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Use in source · constraints.tex, line 203
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,
Exact mathematical passage
```
A_a^i
```

A_a^jUse in source · 1 passage

Search aliases: A_a^j

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

A_b^iUse in source · 1 passage

Search aliases: A_b^i

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

A_b^kUse in source · 1 passage

Search aliases: A_b^k

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

A_hUse in source · 3 passages

Search aliases: A_h

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Use in source · strong_fields.tex, line 251
Direct and split radius evolution. On the positive branch, restriction of the radius equation to the reconstruction manifold gives dot R_j=g(R_j)+b_j(D_4R)_j, g(R)=-fracsqrt2MR-Q^2R, where b=N^r is prescribed in the electrovac benchmark. A split alternative is dot R=g(R)+A_h[b]R, A_h[b]u=frac12(bD_4u+D_4(bu)-(D_4b)u). On smooth data these formulas differ by half of the discrete product-rule defect, which is fourth order away from boundaries. Their exact finite-grid operators differ.
Exact mathematical passage
```
 \dot R=g(R)+A_h[b]R,\qquad
 A_h[b]u=\frac12\bigl(bD_4u+D_4(bu)-(D_4b)u\bigr).
\label{c:strong:split-radius}
```

A_iUse in source · 7 passages

Search aliases: A_i

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Use in source · refinement.tex, line 327
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
 \theta_i(p,h)=\int_0^h A_i(p+t e_i)\,dt,
 \qquad u_i(p,h)=e^{i\theta_i(p,h)}.
 \label{g:integrated-links}
```

a_{ij}Use in source · 2 passages

Search aliases: a_ij

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Use in source · core.tex, line 645
· -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h
Exact mathematical passage
```
B_i(a)=\delta_ja_{ij}
```

A_jUse in source · 12 passages

Search aliases: A_j

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Use in source · core.tex, line 196
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,
Exact mathematical passage
```
A_j,E_j\in\R
```

a_jUse in source · 1 passage

Search aliases: a_j

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Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

a_{j+1/2}Use in source · 1 passage

Search aliases: a_j+1/2

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Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

a_{Lq+j}Use in source · 1 passage

Search aliases: a_Lq+j

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Use in source · spectral.tex, line 909
, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follo…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
 \quad
 \mathfrak R_{L,j}\mathfrak R_{L,k}
   =\delta_{jk}\mathfrak R_{L,j},\quad
 \sum_j\mathfrak R_{L,j}=I.
 \label{a-spectral-analytic-lanes}
```

A_{m,0}Use in source · 2 passages

Search aliases: A_m,0

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Use in source · refinement.tex, line 55
[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower step gives normT_m+1R(h)≤ fracb^m+1normT_mR(h/b)+normT_mR(h)b^m+1-1. Starting with the stated remainder bound and proceeding by induction multiplies i…
Exact mathematical passage
```
 \norm{T_mg(h)-F}\leq A_{m,p}C|h|^p+A_{m,0}\varepsilon.
```

A_{m,p}Use in source · 3 passages

Search aliases: A_m,p

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Use in source · refinement.tex, line 51
[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower step gives normT_m+1R(h)≤ fracb^m+1normT_mR(h/b)+normT_mR(h)b^m+1-1. Starting with the stated remainder bound and proceeding by induction multiplies i…
Exact mathematical passage
```
 A_{m,p}=\prod_{k=1}^{m}\frac{b^{k-p}+1}{b^k-1}.
```

A_NUse in source · 3 passages

Search aliases: A_N

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Use in source · extensions.tex, line 375
Two other bounded multipliers approach one along this positive spectrum: b(n)=fracnsqrt1+n^2, q(n)=fracn-in+i. Their correct anchored cutoff is A_N(f)=S_Nf(H_λ)S_N+(I-S_N). The errors are then normb(H_λ)-A_N(b) =sup_n>N|fracnsqrt1+n^2-1|⟶0, and normq(H_λ)-A_N(q) =sup_n>Nfrac2sqrtn^2+1⟶0. The identity tail is an analytic choice justified by these asymptotic multipliers. It is not a universal prescription for every two-sided or non-diagonal spectrum.
Exact mathematical passage
```
 A_N(f)=S_Nf(H_\lambda)S_N+(I-S_N).
```

a_n^{(r)}Use in source · 1 passage

Search aliases: a_n^(r)

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Use in source · continuum.tex, line 177
Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If a^(r)→ a and widehat H_ma^(r)→ b in KK, then convergence holds at each coordinate. Since the finite matrix H_n is continuous, b_n=lim_rH_na_n^(r)=H_na_n. The output sequence equals b∈KK, so a belongs to the maximal domain and widehat H_ma=b. This direct argument is included as an ordinary explanatory deduction.
Exact mathematical passage
```
b_n=\lim_rH_na_n^{(r)}=H_na_n.
```

a_{n_0}Use in source · 1 passage

Search aliases: a_n_0

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Use in source · finite_core.tex, line 810
a_n_0=frac1sqrt2e_0, a_n_A=frac12e_0, a_n_D=frac12e_0.
Exact mathematical passage
```
a_{n_0}=\frac1{\sqrt2}e_0,\qquad
a_{n_A}=\frac12e_0,\qquad
a_{n_D}=\frac12e_0.
```

a_{n_A}Use in source · 1 passage

Search aliases: a_n_A

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Use in source · finite_core.tex, line 810
a_n_0=frac1sqrt2e_0, a_n_A=frac12e_0, a_n_D=frac12e_0.
Exact mathematical passage
```
a_{n_0}=\frac1{\sqrt2}e_0,\qquad
a_{n_A}=\frac12e_0,\qquad
a_{n_D}=\frac12e_0.
```

a_{n_D}Use in source · 1 passage

Search aliases: a_n_D

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Use in source · finite_core.tex, line 810
a_n_0=frac1sqrt2e_0, a_n_A=frac12e_0, a_n_D=frac12e_0.
Exact mathematical passage
```
a_{n_0}=\frac1{\sqrt2}e_0,\qquad
a_{n_A}=\frac12e_0,\qquad
a_{n_D}=\frac12e_0.
```

a_rUse in source · 1 passage

Search aliases: a_r

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Use in source · core.tex, line 237
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
a_r=-\Phi'
```

a_{\rm tr}Use in source · 1 passage

Search aliases: a_rm tr rm

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Use in source · core.tex, line 647
· -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h · -innerA(a)b_rm tr_h -innera_rm trA(b)_h -∑_kinnerδ_ka_rm trδ_kb_rm tr_h. This expression is symmetric in a,b.
Exact mathematical passage
```
 2\inner{a}{\mathcal G(b)}_h
 ={}&-2\inner{B(a)}{B(b)}_h
 +\sum_k\inner{\delta_ka}{\delta_kb}_h\\
 &-\inner{A(a)}{b_{\rm tr}}_h
 -\inner{a_{\rm tr}}{A(b)}_h
 -\sum_k\inner{\delta_ka_{\rm tr}}{\delta_kb_{\rm tr}}_h.
```

A_xUse in source · 2 passages

Search aliases: A_x

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Use in source · refinement.tex, line 326
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
A_x,A_y,A_z
```

A_yUse in source · 5 passages

Search aliases: A_y

Link to this entry ·

Use in source · refinement.tex, line 326
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
A_x,A_y,A_z
```

A_zUse in source · 2 passages

Search aliases: A_z

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Use in source · refinement.tex, line 326
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
A_x,A_y,A_z
```

a_\muUse in source · 2 passages

Search aliases: a_μ mu

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Use in source · core.tex, line 179
Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Conse…
Exact mathematical passage
```
a_\mu,a_\nu=O(a)
```

a_\mu^2Use in source · 1 passage

Search aliases: a_μ^2 mu

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Use in source · core.tex, line 184
u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluat…
Exact mathematical passage
```
\label{g:wilson-expansion}
 1-\operatorname{Re}W
 =\tfrac12q^2a_\mu^2a_\nu^2 F_{\mu\nu}(x_c)^2+O(a^6).
```

a_\nuUse in source · 2 passages

Search aliases: a_ν nu

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Use in source · core.tex, line 179
Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Conse…
Exact mathematical passage
```
a_\mu,a_\nu=O(a)
```

a_\nu^2Use in source · 1 passage

Search aliases: a_ν^2 nu

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Use in source · core.tex, line 184
u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluat…
Exact mathematical passage
```
\label{g:wilson-expansion}
 1-\operatorname{Re}W
 =\tfrac12q^2a_\mu^2a_\nu^2 F_{\mu\nu}(x_c)^2+O(a^6).
```

\mathcal A_{\partial}Use in source · 5 passages

Search aliases: A_∂ mathcal partial

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Use in source · refinement.tex, line 398
[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same connection at a chart point p. For the two-stage finite Hilbert operator of Section~[source reference], with the same contractive mass factors on bot…
Exact mathematical passage
```
 \mathcal A_{\partial}={}&B_{y\mid x}+B_{x\mid y}
 +2(B_{z\mid x}+B_{x\mid z})\notag\\
 &+2(B_{z\mid y}+B_{y\mid z}).
 \label{g:derivative-amplitude}
```

a_\perpUse in source · 2 passages

Search aliases: a_⊥ perp

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Use in source · constraints.tex, line 422
An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 e^I=e_\perp^I\mathrm dt+E^I,\qquad
 \omega^{IJ}=\omega_\perp^{IJ}\mathrm dt+\Omega^{IJ},\qquad
 a=a_\perp\mathrm dt+A.
```

\mathrm{ad}Use in source · 3 passages

Search aliases: ad mathrm

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Use in source · finite.tex, line 425
For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+…
Exact mathematical passage
```
 \mathcal A^{\mathrm{std}}_{p,h}|pk+r\rangle
 &=|r\rangle\otimes|k\rangle,
 &0\leq r<p,\quad0\leq k<m,\nonumber\\
 \mathcal A^{\mathrm{ad}}_{p,h}|a+d p^{h-1}\rangle
 &=|d\rangle\otimes|a\rangle,
 &0\leq d<p,\quad0\leq a<m.
 \label{a-finite-digit-splits}
```

\operatorname{area}Use in source · 1 passage

Search aliases: area operatorname

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Use in source · core.tex, line 192
[Constant curvature density] For ϕ(t,x)=b(t^2-x^2)/2, one has ϕ_tt-ϕ_xx=2b, R=4b e^-2ϕ, W_square=e^2b area(square)K. The curvature density e^2ϕR/2 is constant although R varies with position. Adding oriented subrectangle integrals reconstructs the coarse rapidity exactly.
Exact mathematical passage
```
 \phi_{tt}-\phi_{xx}=2b,
 \qquad R=4b\,e^{-2\phi},
 \qquad W_\square=e^{2b\,\operatorname{area}(\square)K}.
```

\operatorname{asinh}Use in source · 1 passage

Search aliases: asinh operatorname

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Use in source · core.tex, line 317
N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b) · =frac2c_0^2|∫_-L^L∂_bΦ(x,b) dx| =frac4GM_0Lc_0^2bsqrtL^2+b^2. Here α is the magnitude of the deflection toward the source. The limit L→∞ gives 4GM…
Exact mathematical passage
```
 \Delta t_{\rm Shapiro}
 &=-\frac2{c_0^3}\int_{-L}^L\Phi(x,b)\,dx
 =\frac{4GM_0}{c_0^3}\operatorname{asinh}\left(\frac Lb\right),\\
 \alpha(L,b)
 &=\frac2{c_0^2}\left|\int_{-L}^L\partial_b\Phi(x,b)\,dx\right|
 =\frac{4GM_0L}{c_0^2b\sqrt{L^2+b^2}}.
```

\mathcal BUse in source · 9 passages

Search aliases: B mathcal

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Use in source · spectral.tex, line 1001
[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber
Exact mathematical passage
```
A_0\in\mathcal B(\HH)
```

b^{-1}Use in source · 2 passages

Search aliases: b^-1

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Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

b^{-r}Use in source · 1 passage

Search aliases: b^-r

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Use in source · refinement.tex, line 26
ition[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
 \frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.
```

b^2Use in source · 2 passages

Search aliases: b^2

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Use in source · core.tex, line 315
+(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
\Phi(x,b)=-GM_0/\sqrt{x^2+b^2}
```

\mathcal B^aUse in source · 1 passage

Search aliases: B^a mathcal

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Use in source · constraints.tex, line 572
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
 \mathcal H_{\mathrm M}=\frac1{2e}q_{ab}
       (\mathcal E^a\mathcal E^b+\mathcal B^a\mathcal B^b),
 \qquad \mathcal B^a=\tfrac12\epsilon^{abc}\mathcal F_{bc}.
```

\mathcal B^bUse in source · 1 passage

Search aliases: B^b mathcal

Link to this entry ·

Use in source · constraints.tex, line 572
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
 \mathcal H_{\mathrm M}=\frac1{2e}q_{ab}
       (\mathcal E^a\mathcal E^b+\mathcal B^a\mathcal B^b),
 \qquad \mathcal B^a=\tfrac12\epsilon^{abc}\mathcal F_{bc}.
```

b^gUse in source · 2 passages

Search aliases: b^g

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Use in source · refinement.tex, line 40
Coarse endpoint gauges g(x) and midpoint gauges q(x) may be chosen independently. The two sublinks transform by a^g(x)=q(x)a(x)g(x)^-1, b^g(x)=g(τ x)b(x)q(x)^-1. The midpoint factors cancel in b^ga^g, leaving the required coarse transformation. At the midpoint, a^g(x)g(x)ψ(x)=q(x)a(x)ψ(x), while endpoint covariance is immediate. Hence interpolation commutes exactly with the corresponding gauge actions. This argument allows independent new gauge data at every inserted midpoint.
Exact mathematical passage
```
 a^g(x)=q(x)a(x)g(x)^{-1},\qquad
 b^g(x)=g(\tau x)b(x)q(x)^{-1}.
```

b^kUse in source · 2 passages

Search aliases: b^k

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Use in source · refinement.tex, line 18
[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
 T_m(h^rv)=c_{m,r}h^rv,\qquad
 c_{m,r}=\prod_{k=1}^{m}\frac{b^{k-r}-1}{b^k-1}.
```

b^{k-p}Use in source · 1 passage

Search aliases: b^k-p

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Use in source · refinement.tex, line 51
[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower step gives normT_m+1R(h)≤ fracb^m+1normT_mR(h/b)+normT_mR(h)b^m+1-1. Starting with the stated remainder bound and proceeding by induction multiplies i…
Exact mathematical passage
```
 A_{m,p}=\prod_{k=1}^{m}\frac{b^{k-p}+1}{b^k-1}.
```

b^{k-r}Use in source · 1 passage

Search aliases: b^k-r

Link to this entry ·

Use in source · refinement.tex, line 18
[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
 T_m(h^rv)=c_{m,r}h^rv,\qquad
 c_{m,r}=\prod_{k=1}^{m}\frac{b^{k-r}-1}{b^k-1}.
```

b^{m+1-p}Use in source · 1 passage

Search aliases: b^m+1-p

Link to this entry ·

Use in source · refinement.tex, line 65
nterval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower step gives normT_m+1R(h)≤ fracb^m+1normT_mR(h/b)+normT_mR(h)b^m+1-1. Starting with the stated remainder bound and proceeding by induction multiplies its constant at step m+1 by (b^m+1-p+1)/(b^m+1-1). Hence normT_mR(h)≤ A_m,pC|h|^p. Apply the same argument with p=0 to g-f. Linearity and the triangle inequality separate the two contributions.
Exact mathematical passage
```
(b^{m+1-p}+1)/(b^{m+1}-1)
```

b_0Use in source · 4 passages

Search aliases: b_0

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Use in source · limit.tex, line 682
Let Z_sign have eigenvalues +1,-1 on a dyadic sign qubit, and let B=B^* act on the remaining factors. Assume there are orthonormal states |b_0⟩,|b_1⟩ with B|b_0⟩=0 and B|b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sign⊗ B, |Ψ(0)⟩=|+_x⟩⊗ frac|b_0⟩+|b_1⟩sqrt2. Evolution under this explicit interaction alone gives |Ψ(t)⟩=frac1sqrt2 (|+_x⟩⊗|b_0⟩ +e^-igtZ_sign|+_x⟩⊗…
Exact mathematical passage
```
|b_0\rangle,|b_1\rangle
```

B_{0i}Use in source · 1 passage

Search aliases: B_0i

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Use in source · constraints.tex, line 556
c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ term. Those directions lie in the k…
Exact mathematical passage
```
 B_{ij}=0,\qquad
 B_{0i}=\tfrac1{4\kappa}\epsilon_{ijk}E^j\wedge E^k,
 \qquad
 \int_\Sigma B_{IJ}\wedge\delta\Omega^{IJ}
      =\frac1\kappa\int_\Sigma\widetilde E^a{}_i\delta K_a{}^i\,\mathrm d^3x.
 \label{c:constraints:simplicity}
```

B_{ba}Use in source · 1 passage

Search aliases: B_ba

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Use in source · extensions.tex, line 492
Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.
Exact mathematical passage
```
B_{ba}=P_bBP_a
```

B_{bc}Use in source · 1 passage

Search aliases: B_bc

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Use in source · extensions.tex, line 493
Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.
Exact mathematical passage
```
 P_b\delta_B(X)P_a
 =i\sum_c\left(B_{bc}X_{ca}-X_{bc}B_{ca}\right).
 \label{eq:foundation-corner-path}
```

B_{ca}Use in source · 1 passage

Search aliases: B_ca

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Use in source · extensions.tex, line 493
Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.
Exact mathematical passage
```
 P_b\delta_B(X)P_a
 =i\sum_c\left(B_{bc}X_{ca}-X_{bc}B_{ca}\right).
 \label{eq:foundation-corner-path}
```

B_iUse in source · 4 passages

Search aliases: B_i

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Use in source · extensions.tex, line 479
For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely when that X commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additiona…
Exact mathematical passage
```
B_i,B_j
```

B_{IJ}Use in source · 4 passages

Search aliases: B_IJ

Link to this entry ·

Use in source · constraints.tex, line 434
IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A
Exact mathematical passage
```
 B_{IJ}&=\tfrac1{4\kappa}\epsilon_{KLIJ}E^K\wedge E^L,\\
 \Pi&=\tfrac12\varphi_{IJ}E^I\wedge E^J,\qquad
 \sigma_I=\tfrac1{3!}\epsilon_{IJKL}E^J\wedge E^K\wedge E^L.
```

B_{ij}Use in source · 1 passage

Search aliases: B_ij

Link to this entry ·

Use in source · constraints.tex, line 556
c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ term. Those directions lie in the k…
Exact mathematical passage
```
 B_{ij}=0,\qquad
 B_{0i}=\tfrac1{4\kappa}\epsilon_{ijk}E^j\wedge E^k,
 \qquad
 \int_\Sigma B_{IJ}\wedge\delta\Omega^{IJ}
      =\frac1\kappa\int_\Sigma\widetilde E^a{}_i\delta K_a{}^i\,\mathrm d^3x.
 \label{c:constraints:simplicity}
```

B_jUse in source · 7 passages

Search aliases: B_j

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Use in source · extensions.tex, line 479
For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely when that X commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additiona…
Exact mathematical passage
```
B_i,B_j
```

b_jUse in source · 1 passage

Search aliases: b_j

Link to this entry ·

Use in source · strong_fields.tex, line 245
Direct and split radius evolution. On the positive branch, restriction of the radius equation to the reconstruction manifold gives dot R_j=g(R_j)+b_j(D_4R)_j, g(R)=-fracsqrt2MR-Q^2R, where b=N^r is prescribed in the electrovac benchmark. A split alternative is dot R=g(R)+A_h[b]R, A_h[b]u=frac12(bD_4u+D_4(bu)-(D_4b)u). On smooth data these formulas differ by half of the discrete product-rule defect, which is fourth order away from boundaries. Their exact finite-grid operators differ.
Exact mathematical passage
```
 \dot R_j=g(R_j)+b_j(D_4R)_j,
 \qquad g(R)=-\frac{\sqrt{2MR-Q^2}}R,
\label{c:strong:direct-radius}
```

B_{j-1}Use in source · 1 passage

Search aliases: B_j-1

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Use in source · core.tex, line 78
Put B_j=c_j+1/2Δ_j. Using the equation of motion, the derivative of the kinetic and mass terms at site j is fracdotφ_jh(B_j-B_j-1) -frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2. The right edge contribution differentiates to fracdot c_j+1/2Δ_j^24h +fracB_j2h(dotφ_j+1-dotφ_j), and the left edge has the analogous expression. Adding these terms leaves B_j(dotφ_j+dotφ_j+1)/(2h) on the right and its negative counterpart on the left, plus s_j. These are exactly the flux terms in [source reference]. Peri…
Exact mathematical passage
```
 \frac{\dot\varphi_j}{h}(B_j-B_{j-1})
 -\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2.
```

B_{j\mid i}Use in source · 4 passages

Search aliases: B_jmid i mid

Link to this entry ·

Use in source · refinement.tex, line 363
formal connection structure records the three components, the six cross derivatives ∂_iA_j for i≠ j, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants |∂_iA_j(p)|≤ B_jmid i (i≠ j). For example, B_ymid x bounds the derivative of A_y in the x direction. These explicit …
Exact mathematical passage
```
 |\partial_iA_j(p)|\leq B_{j\mid i}\qquad(i\ne j).
```

B_{L,j}Use in source · 4 passages

Search aliases: B_L,j

Link to this entry ·

Use in source · spectral.tex, line 1061
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -
Exact mathematical passage
```
B_{L,j}(y)=\sum_{q\geq0}(-1)^{Lq+j}y^{Lq+j}/[(Lq+j)!]^2
```

B_{L,j-1\bmod L}Use in source · 1 passage

Search aliases: B_L,j-1bmod L bmod

Link to this entry ·

Use in source · spectral.tex, line 1077
One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^2B_L,j(y)=-yB_L,j-1bmod L(y). For a term of positive degree n, its new coefficient is n^2(-1)^n/(n!)^2=-(-1)^n-1/[(n-1)!]^2. The degree-zero term disa…
Exact mathematical passage
```
 \Theta^2B_{L,j}(y)=-yB_{L,j-1\bmod L}(y).
 \label{a-spectral-bessel-euler}
```

B_{m+1,a+d p^m}Use in source · 1 passage

Search aliases: B_m+1,a+d p^m

Link to this entry ·

Use in source · limit.tex, line 50
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
B_{m,a}=\bigsqcup_d B_{m+1,a+d p^m}
```

B_{m,a}Use in source · 7 passages

Search aliases: B_m,a

Link to this entry ·

Use in source · limit.tex, line 12
Fix a prime p. The compact additive group Z_p has normalized Haar measure μ_p, with B_m,a=a+p^mZ_p, 0≤ a<p^m, μ_p(B_m,a)=p^-m. Write HH_p=L^2(Z_p,μ_p;C). We use the complex inner product innerfg=∫overline f g dμ_p, linear in its second argument. The p-adic coordinate belongs to the measurable space. Amplitudes and the Born probabilities use ordinary complex Hilbert space.
Exact mathematical passage
```
 B_{m,a}=a+p^m\Z_p,\qquad 0\leq a<p^m,\qquad
 \mu_p(B_{m,a})=p^{-m}.
```

B_{m,a+d p^{m-1}}Use in source · 1 passage

Search aliases: B_m,a+d p^m-1

Link to this entry ·

Use in source · limit.tex, line 174
For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω_p^(q'-q)d=δ_qq'. The number of these orthonormal functions is exactly d_m, so they span that level. Together with ψ_0=1, they form a complete orthono…
Exact mathematical passage
```
 \psi_{m,a,q}=p^{(m-1)/2}\sum_{d=0}^{p-1}
 \omega_p^{qd}\mathbf1_{B_{m,a+d p^{m-1}}},
 \quad
 \begin{matrix}m\geq1,\quad 0\leq a<p^{m-1},\\1\leq q<p.\end{matrix}
 \label{a-limit-haar-wavelets}
```

B_{m,b}Use in source · 1 passage

Search aliases: B_m,b

Link to this entry ·

Use in source · limit.tex, line 32
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \inner{\mathcal U_{p,m}e_a}{\mathcal U_{p,m}e_b}
 =p^m\mu_p(B_{m,a}\cap B_{m,b})=\delta_{ab}.
```

b_nUse in source · 9 passages

Search aliases: b_n

Link to this entry ·

Use in source · continuum.tex, line 9
Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of amplitudes embeds by assigning zero to every absent mode.
Exact mathematical passage
```
\inner{a}{b}=\sum_n\inner{a_n}{b_n}
```

\mathcal B_{r,j}Use in source · 3 passages

Search aliases: B_r,j mathcal

Link to this entry ·

Use in source · core.tex, line 447
The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the separated Taylor channels are coupled. Their sum satisfies the Bessel equation. This decomposition resolves series order and does not introduce addition…
Exact mathematical passage
```
 \mathcal B_{r,j}(y)=\sum_{q=0}^{\infty}
 \frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},
 \qquad j=0,\ldots,r-1.
```

\mathcal B_{r,j-1}Use in source · 1 passage

Search aliases: B_r,j-1 mathcal

Link to this entry ·

Use in source · core.tex, line 454
The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the separated Taylor channels are coupled. Their sum satisfies the Bessel equation. This decomposition resolves series order and does not introduce addition…
Exact mathematical passage
```
 \vartheta^2\mathcal B_{r,j}+y\mathcal B_{r,j-1}=0,
 \qquad j\pmod r,
```

b_{\rm tr}Use in source · 1 passage

Search aliases: b_rm tr rm

Link to this entry ·

Use in source · core.tex, line 647
· -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h · -innerA(a)b_rm tr_h -innera_rm trA(b)_h -∑_kinnerδ_ka_rm trδ_kb_rm tr_h. This expression is symmetric in a,b.
Exact mathematical passage
```
 2\inner{a}{\mathcal G(b)}_h
 ={}&-2\inner{B(a)}{B(b)}_h
 +\sum_k\inner{\delta_ka}{\delta_kb}_h\\
 &-\inner{A(a)}{b_{\rm tr}}_h
 -\inner{a_{\rm tr}}{A(b)}_h
 -\sum_k\inner{\delta_ka_{\rm tr}}{\delta_kb_{\rm tr}}_h.
```

B_{x\mid y}Use in source · 2 passages

Search aliases: B_xmid y mid

Link to this entry ·

Use in source · refinement.tex, line 387
· -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancellation inside F_ij.
Exact mathematical passage
```
 |\Theta_{xy}(p,h)|\leq
   (B_{y\mid x}+B_{x\mid y})h^2,
 \qquad
 |\Omega_{xy}(p,h)-1|\leq
   (B_{y\mid x}+B_{x\mid y})h^2.
```

B_{x\mid z}Use in source · 1 passage

Search aliases: B_xmid z mid

Link to this entry ·

Use in source · refinement.tex, line 398
[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same connection at a chart point p. For the two-stage finite Hilbert operator of Section~[source reference], with the same contractive mass factors on bot…
Exact mathematical passage
```
 \mathcal A_{\partial}={}&B_{y\mid x}+B_{x\mid y}
 +2(B_{z\mid x}+B_{x\mid z})\notag\\
 &+2(B_{z\mid y}+B_{y\mid z}).
 \label{g:derivative-amplitude}
```

B_{y\mid x}Use in source · 4 passages

Search aliases: B_ymid x mid

Link to this entry ·

Use in source · refinement.tex, line 366
oss derivatives ∂_iA_j for i≠ j, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants |∂_iA_j(p)|≤ B_jmid i (i≠ j). For example, B_ymid x bounds the derivative of A_y in the x direction. These explicit hypotheses are sufficient for the coordinate arguments below. They a…
Exact mathematical passage
```
B_{y\mid x}
```

B_{y\mid z}Use in source · 1 passage

Search aliases: B_ymid z mid

Link to this entry ·

Use in source · refinement.tex, line 398
[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same connection at a chart point p. For the two-stage finite Hilbert operator of Section~[source reference], with the same contractive mass factors on bot…
Exact mathematical passage
```
 \mathcal A_{\partial}={}&B_{y\mid x}+B_{x\mid y}
 +2(B_{z\mid x}+B_{x\mid z})\notag\\
 &+2(B_{z\mid y}+B_{y\mid z}).
 \label{g:derivative-amplitude}
```

B_{z\mid x}Use in source · 1 passage

Search aliases: B_zmid x mid

Link to this entry ·

Use in source · refinement.tex, line 398
[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same connection at a chart point p. For the two-stage finite Hilbert operator of Section~[source reference], with the same contractive mass factors on bot…
Exact mathematical passage
```
 \mathcal A_{\partial}={}&B_{y\mid x}+B_{x\mid y}
 +2(B_{z\mid x}+B_{x\mid z})\notag\\
 &+2(B_{z\mid y}+B_{y\mid z}).
 \label{g:derivative-amplitude}
```

B_{z\mid y}Use in source · 1 passage

Search aliases: B_zmid y mid

Link to this entry ·

Use in source · refinement.tex, line 398
[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same connection at a chart point p. For the two-stage finite Hilbert operator of Section~[source reference], with the same contractive mass factors on bot…
Exact mathematical passage
```
 \mathcal A_{\partial}={}&B_{y\mid x}+B_{x\mid y}
 +2(B_{z\mid x}+B_{x\mid z})\notag\\
 &+2(B_{z\mid y}+B_{y\mid z}).
 \label{g:derivative-amplitude}
```

B_\partialUse in source · 1 passage

Search aliases: B_∂ partial

Link to this entry ·

Use in source · strong_fields.tex, line 12
da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ_e G)+B_∂. The boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamilton…
Exact mathematical passage
```
 S=\int dt\int dr\,
 \bigl(P_\Lambda\dot\Lambda+P_R\dot R+Q\dot A_r
       -N\mathcal H-N^r\mathcal D-\Phi_e\mathcal G\bigr)+B_\partial.
```

\bar pUse in source · 2 passages

Search aliases: bar p bar

Link to this entry ·

Use in source · constraints.tex, line 731
-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpoints on the unit sphere can have bar q=0 if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an a…
Exact mathematical passage
```
\bar p=(p_0+p_1)/2
```

\bar qUse in source · 4 passages

Search aliases: bar q bar

Link to this entry ·

Use in source · constraints.tex, line 731
-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpoints on the unit sphere can have bar q=0 if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an a…
Exact mathematical passage
```
\bar q=(q_0+q_1)/2
```

\bar\rhoUse in source · 2 passages

Search aliases: barρ bar rho

Link to this entry ·

Use in source · core.tex, line 227
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-def}
 \rho_j=\psi_j^*\psi_j,\qquad
 \bar\rho=\frac1N\sum_j\rho_j,\qquad
 \mathcal G_j=\frac{E_j-E_{j-1}}h+\rho_j-\bar\rho.
```

\bar\chiUse in source · 1 passage

Search aliases: barχ bar chi

Link to this entry ·

Use in source · constraints.tex, line 586
the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
 \{\chi_A(x),\bar\chi_B(y)\}_{\mathrm P}
            =-ie(x)\delta_{AB}\delta^3(x-y).
```

\bar\chi_BUse in source · 1 passage

Search aliases: barχ_B bar chi

Link to this entry ·

Use in source · constraints.tex, line 586
the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
 \{\chi_A(x),\bar\chi_B(y)\}_{\mathrm P}
            =-ie(x)\delta_{AB}\delta^3(x-y).
```

\bar\psiUse in source · 16 passages

Search aliases: barψ bar psi

Link to this entry ·

Use in source · constraints.tex, line 287
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,
Exact mathematical passage
```
\psi,\bar\psi
```

\bigsqcupUse in source · 2 passages

Search aliases: bigsqcup

Link to this entry ·

Use in source · limit.tex, line 50
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
B_{m,a}=\bigsqcup_d B_{m+1,a+d p^m}
```

\bigsqcup_dUse in source · 1 passage

Search aliases: bigsqcup_d bigsqcup

Link to this entry ·

Use in source · limit.tex, line 50
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
B_{m,a}=\bigsqcup_d B_{m+1,a+d p^m}
```

\bigsqcup_{\alpha}Use in source · 1 passage

Search aliases: bigsqcup_α bigsqcup alpha

Link to this entry ·

Use in source · extensions.tex, line 216
For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.
Exact mathematical passage
```
A=\bigsqcup_{\alpha}C_\alpha
```

\bmodUse in source · 7 passages

Search aliases: bmod

Link to this entry ·

Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

\BoxUse in source · 3 passages

Search aliases: Box

Link to this entry ·

Use in source · constraints.tex, line 321
The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor i, the flat-space operators iγ^I∂_I± m have product (iγ^I∂_I+m)(iγ^J∂_J-m) =-Box-m^2, Box=η^IJ∂_I∂_J. Thus its squared field equation is Box+m^2=0, whereas the conventional real massive dispersion in this mostly-plus metric uses Box-m^2=0. A consistent translation of the Clifford, k…
Exact mathematical passage
```
 (i\gamma^I\partial_I+m)(i\gamma^J\partial_J-m)
       =-\Box-m^2,
 \qquad \Box=\eta^{IJ}\partial_I\partial_J.
```

\mathrm{BV}Use in source · 2 passages

Search aliases: BV mathrm

Link to this entry ·

Use in source · constraints.tex, line 387
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
 S_{\mathrm{BV}}=S_0+
     \int\sum_\Xi\langle\Xi^\ddagger,s\Xi\rangle.
 \label{c:constraints:bv-action}
```

\mathbb CUse in source · 8 passages

Search aliases: C mathbb

Link to this entry ·

Use in source · finite_core.tex, line 22
Components of a spinor are numbered 0,1,2,3, in the exact order of the source representation. For v,w∈ C^4,
Exact mathematical passage
```
v,w\in\mathbb C^4
```

\mathrm cUse in source · 11 passages

Search aliases: c mathrm

Link to this entry ·

Use in source · core.tex, line 197
[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
 p_A^{\mathrm c}&=\norm{Q_A\psi}^2=\sum_{a\in C_A}p_a,\\
 J_{B\leftarrow A}^{\mathrm c}
 &=2\operatorname{Re}\inner{Q_B\psi}{KQ_A\psi}
 =\sum_{b\in C_B}\sum_{a\in C_A}J_{b\leftarrow a}.
```

c'Use in source · 1 passage

Search aliases: c'

Link to this entry ·

Use in source · spectral.tex, line 400
in and all the chains [source reference] form an orthogonal decomposition of ℓ^2(T_p,h). A chain starting at depth r has length h-r+1 and appears with multiplicity (p-1)p^r-1, for 1≤ r≤ h. Within one chain, different s have disjoint level supports. Chains based at the same u and different contrast vectors have inner product ⟨ c,c'⟩=0 at a common level. If two vertices are in disjoint subtrees, their chains have disjoint support. If one vertex lies below another, the higher chain is constant, up …
Exact mathematical passage
```
\langle c,c'\rangle=0
```

C^*Use in source · 1 passage

Search aliases: C^*

Link to this entry ·

Use in source · limit.tex, line 848
|s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph edges, with no continuum limit assumed.
Exact mathematical passage
```
U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I
```

C^{-1}Use in source · 2 passages

Search aliases: C^-1

Link to this entry ·

Use in source · spectral.tex, line 188
det(z-H) · =det(z-D)det S(z). For E∈R∖σ(D), an eigenvector of H is equivalent to a nonzero solution of S(E)u=0, with hidden component v=(E-D)^-1B^*u. Put C=z-D. Direct block multiplication gives z-H= I · -BC^-1 0 · I S(z) · 0 0 · C I · 0 -C^-1B^* · I. Both outer factors are invertible and have determinant one. This proves the invertibility equivalence and determinant formula. Inverting the factors gives the stated upper left block. Alternatively, in (z-H)(u,v)^ T=(f,0)^ T, the lower equation giv…
Exact mathematical passage
```
 z-H=
 \begin{pmatrix}I&-BC^{-1}\\0&I\end{pmatrix}
 \begin{pmatrix}\mathcal S(z)&0\\0&C\end{pmatrix}
 \begin{pmatrix}I&0\\-C^{-1}B^*&I\end{pmatrix}.
```

C^2Use in source · 3 passages

Search aliases: C^2

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Use in source · extensions.tex, line 302
For the faster normalized tail c_n=C/n^2, both errors are controlled: normc-S_Nc^2=C^2∑_n>Nn^-4, normH_λ c-S_NH_λ c^2=C^2∑_n>Nn^-2. The state tail is O(N^-3/2) in norm, while the velocity tail is O(N^-1/2). A small state error can therefore coexist with a much larger derivative error even inside the true domain.
Exact mathematical passage
```
 \norm{c-S_Nc}^2=C^2\sum_{n>N}n^{-4},\qquad
 \norm{H_\lambda c-S_NH_\lambda c}^2=C^2\sum_{n>N}n^{-2}.
```

\C^2Use in source · 5 passages

Search aliases: C^2 C

Link to this entry ·

Use in source · core.tex, line 162
[An exactly solved exchange] On C^2, take the coordinate atlas, H=g0 · 1
Exact mathematical passage
```
\C^2
```

c^2Use in source · 2 passages

Search aliases: c^2

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Use in source · limit.tex, line 704
e⊗|b_1⟩). Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors with equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bits is S_sign(t) =h_2(frac1+|cos(gt)|2), h_2(u)=-ulog_2u-(1-u)log_2(1-u), with 0log_2 0=0. At t_* =π/(2|g|) the two sign states are orthogona…
Exact mathematical passage
```
(1-c^2)/4
```

\C^2_AUse in source · 1 passage

Search aliases: C^2_A C

Link to this entry ·

Use in source · core.tex, line 91
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1
Exact mathematical passage
```
\C^4=\C^2_A\oplus\C^2_B
```

\C^2_BUse in source · 1 passage

Search aliases: C^2_B C

Link to this entry ·

Use in source · core.tex, line 91
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1
Exact mathematical passage
```
\C^4=\C^2_A\oplus\C^2_B
```

\C^2_{\mathrm{sign}}Use in source · 2 passages

Search aliases: C^2_sign C mathrm

Link to this entry ·

Use in source · finite.tex, line 94
In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They do not assert that a generic state is a product state, or that a generic Hamiltonian is a sum of independent register Hamiltonians. Entanglement and in…
Exact mathematical passage
```
 \ell^2(U_{p,h})&\cong
 \C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^h\Z),
 &&p\text{ odd},\nonumber\\
 \ell^2(U_{2,h})&\cong
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^h\Z).
 \label{a-finite-unit-hilbert}
```

C^3Use in source · 1 passage

Search aliases: C^3

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Use in source · core.tex, line 292
The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let A be a real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription.
Exact mathematical passage
```
C^3
```

\mathbb C^4Use in source · 7 passages

Search aliases: C^4 mathbb

Link to this entry ·

Use in source · finite_core.tex, line 22
Components of a spinor are numbered 0,1,2,3, in the exact order of the source representation. For v,w∈ C^4,
Exact mathematical passage
```
v,w\in\mathbb C^4
```

c^AUse in source · 2 passages

Search aliases: c^A

Link to this entry ·

Use in source · constraints.tex, line 146
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
c^A
```

c^BUse in source · 1 passage

Search aliases: c^B

Link to this entry ·

Use in source · constraints.tex, line 147
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
 \Omega_F=c^AC_A-\tfrac12c^Ac^BU_{AB}{}^C\varpi_C
             +\Omega_F^{(\ge3)}.
 \label{c:constraints:bfv-charge}
```

\C^{h+1}Use in source · 1 passage

Search aliases: C^h+1 C

Link to this entry ·

Use in source · spectral.tex, line 352
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · ·
Exact mathematical passage
```
R:\C^{h+1}\to\ell^2(T_{p,h})
```

C^{IJ}Use in source · 1 passage

Search aliases: C^IJ

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Use in source · constraints.tex, line 614
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
 T^I=\tfrac\kappa4\epsilon^I{}_{JKL}J_5^Je^K\wedge e^L,
 \qquad
 C^{IJ}=\tfrac\kappa4\epsilon^{IJ}{}_{KL}J_5^Ke^L,
```

\C^MUse in source · 2 passages

Search aliases: C^M C

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Use in source · spectral.tex, line 21
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
J_r:\C^M\to\C^N
```

\C^NUse in source · 7 passages

Search aliases: C^N C

Link to this entry ·

Use in source · spectral.tex, line 21
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
J_r:\C^M\to\C^N
```

\C^nUse in source · 2 passages

Search aliases: C^n C

Link to this entry ·

Use in source · core.tex, line 313
For this section take HH=C^n. A density matrix is a positive semidefinite Hermitian matrix ρ with Trρ=1. Its probability in a sharp packet is p_a=Tr(P_aρ). Hermiticity makes this trace real. Positivity makes it nonnegative, since cyclicity and P_a^2=P_a give Tr(P_aρ)=Tr(P_aρ P_a)≥0. Completeness gives ∑_ap_a=Trρ=1. These are separate uses of the density-matrix assumptions.
Exact mathematical passage
```
\HH=\C^n
```

\C^pUse in source · 1 passage

Search aliases: C^p C

Link to this entry ·

Use in source · spectral.tex, line 374
There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-c…
Exact mathematical passage
```
c=(c_0,\ldots,c_{p-1})\in\C^p
```

\C^{p-1}_{\mathrm{tors}}Use in source · 2 passages

Search aliases: C^p-1_tors C mathrm

Link to this entry ·

Use in source · finite.tex, line 94
In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They do not assert that a generic state is a product state, or that a generic Hamiltonian is a sum of independent register Hamiltonians. Entanglement and in…
Exact mathematical passage
```
 \ell^2(U_{p,h})&\cong
 \C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^h\Z),
 &&p\text{ odd},\nonumber\\
 \ell^2(U_{2,h})&\cong
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^h\Z).
 \label{a-finite-unit-hilbert}
```

\C^rUse in source · 1 passage

Search aliases: C^r C

Link to this entry ·

Use in source · core.tex, line 5
Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from ri…
Exact mathematical passage
```
\C^r
```

\C^sUse in source · 2 passages

Search aliases: C^s C

Link to this entry ·

Use in source · core.tex, line 196
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,
Exact mathematical passage
```
\C^s
```

\C^\timesUse in source · 1 passage

Search aliases: C^× C times

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Use in source · core.tex, line 139
The genuine unit-circle hypothesis is essential. A nonzero complex scalar is invertible, but multiplication by 2 multiplies squared norm by 4. The phase must therefore lie in the unit circle, not merely in C^×. Norm preservation depends on that mathematical hypothesis. The formalization companion identifies its precise type and transport conventions.
Exact mathematical passage
```
\C^\times
```

c^\muUse in source · 1 passage

Search aliases: c^μ mu

Link to this entry ·

Use in source · constraints.tex, line 363
Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,
Exact mathematical passage
```
 sc^\mu=c^\nu\partial_\nu c^\mu,\qquad
 s\chi=\mathcal L_c\chi+\chi^2,\qquad
 s\upsilon=\mathcal L_c\upsilon.
 \label{c:constraints:ghost-differential}
```

c^\nuUse in source · 1 passage

Search aliases: c^ν nu

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Use in source · constraints.tex, line 363
Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,
Exact mathematical passage
```
 sc^\mu=c^\nu\partial_\nu c^\mu,\qquad
 s\chi=\mathcal L_c\chi+\chi^2,\qquad
 s\upsilon=\mathcal L_c\upsilon.
 \label{c:constraints:ghost-differential}
```

c_+Use in source · 1 passage

Search aliases: c_+

Link to this entry ·

Use in source · strong_fields.tex, line 453
dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
 c_+=-\sigma(v+1),\qquad c_-=\sigma(1-v).
\label{c:strong:characteristics}
```

c_-Use in source · 1 passage

Search aliases: c_-

Link to this entry ·

Use in source · strong_fields.tex, line 453
dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
 c_+=-\sigma(v+1),\qquad c_-=\sigma(1-v).
\label{c:strong:characteristics}
```

c_0Use in source · 1 passage

Search aliases: c_0

Link to this entry ·

Use in source · spectral.tex, line 374
There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-c…
Exact mathematical passage
```
c=(c_0,\ldots,c_{p-1})\in\C^p
```

c_0^2Use in source · 5 passages

Search aliases: c_0^2

Link to this entry ·

Use in source · core.tex, line 302
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 ds^2=-\left(1+\frac{2\Phi}{c_0^2}\right)c_0^2dt^2
       +\left(1-\frac{2\Phi}{c_0^2}\right)d\mathbf x^2,
 \qquad \left|\frac{\Phi}{c_0^2}\right|\ll1,
```

c_0^3Use in source · 1 passage

Search aliases: c_0^3

Link to this entry ·

Use in source · core.tex, line 317
N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b) · =frac2c_0^2|∫_-L^L∂_bΦ(x,b) dx| =frac4GM_0Lc_0^2bsqrtL^2+b^2. Here α is the magnitude of the deflection toward the source. The limit L→∞ gives 4GM…
Exact mathematical passage
```
 \Delta t_{\rm Shapiro}
 &=-\frac2{c_0^3}\int_{-L}^L\Phi(x,b)\,dx
 =\frac{4GM_0}{c_0^3}\operatorname{asinh}\left(\frac Lb\right),\\
 \alpha(L,b)
 &=\frac2{c_0^2}\left|\int_{-L}^L\partial_b\Phi(x,b)\,dx\right|
 =\frac{4GM_0L}{c_0^2b\sqrt{L^2+b^2}}.
```

c_0^4Use in source · 1 passage

Search aliases: c_0^4

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

C_1Use in source · 3 passages

Search aliases: C_1

Link to this entry ·

Use in source · refinement.tex, line 249
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.
```

C_{1,j}^nUse in source · 2 passages

Search aliases: C_1,j^n

Link to this entry ·

Use in source · core.tex, line 65
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,
Exact mathematical passage
```
C_{1,j}^n,C_{2,j}^n\in U(2)
```

C_2Use in source · 1 passage

Search aliases: C_2

Link to this entry ·

Use in source · refinement.tex, line 249
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.
```

C_{2,j}^nUse in source · 2 passages

Search aliases: C_2,j^n

Link to this entry ·

Use in source · core.tex, line 65
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,
Exact mathematical passage
```
C_{1,j}^n,C_{2,j}^n\in U(2)
```

C_3Use in source · 2 passages

Search aliases: C_3

Link to this entry ·

Use in source · core.tex, line 430
[Local logarithmic refinement] On the common analytic logarithm branch, in any fixed matrix norm, F_h · =[A,B]+hC_3(A,B)+O(h^2), C_3(A,B) · =tfrac12[A+B,[A,B]], F_h^[1] · :=2F_h/2-F_h=[A,B]+O(h^2). For constant G∈ SU(2), simultaneous conjugation of A,B conjugates both F_h and F_h^[1] by G.
Exact mathematical passage
```
 F_h&=[A,B]+hC_3(A,B)+O(h^2),\label{g:log-series}\\
 C_3(A,B)&=\tfrac12[A+B,[A,B]],\label{g:cubic-coefficient}\\
 F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).
 \label{g:richardson-formula}
```

C_AUse in source · 5 passages

Search aliases: C_A

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Use in source · core.tex, line 189
Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a different physical system.
Exact mathematical passage
```
C_A=f^{-1}(A)
```

C_aUse in source · 3 passages

Search aliases: C_a

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Use in source · refinement.tex, line 472
Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2 =∑_a∫_C_av^-1normψ_a^2 =∑_anormψ_a^2. Equivalently, physical cell values carry the weighted norm v∑_anormq_a^2. The adjoint averaging map yields the …
Exact mathematical passage
```
C_a
```

c_aUse in source · 4 passages

Search aliases: c_a

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Use in source · limit.tex, line 37
_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
c=\sum_a c_ae_a
```

c_{ab}Use in source · 1 passage

Search aliases: c_ab

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Use in source · core.tex, line 494
For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).
Exact mathematical passage
```
c_{ba}=c_{ab}\ge0
```

C_BUse in source · 2 passages

Search aliases: C_B

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Use in source · core.tex, line 197
[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
 p_A^{\mathrm c}&=\norm{Q_A\psi}^2=\sum_{a\in C_A}p_a,\\
 J_{B\leftarrow A}^{\mathrm c}
 &=2\operatorname{Re}\inner{Q_B\psi}{KQ_A\psi}
 =\sum_{b\in C_B}\sum_{a\in C_A}J_{b\leftarrow a}.
```

c_{ba}Use in source · 6 passages

Search aliases: c_ba

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Use in source · core.tex, line 494
For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).
Exact mathematical passage
```
c_{ba}=\gamma_{ba}\pi_a
```

c_dUse in source · 3 passages

Search aliases: c_d

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Use in source · spectral.tex, line 374
There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-c…
Exact mathematical passage
```
\sum_dc_d=0
```

C_hUse in source · 13 passages

Search aliases: C_h

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Use in source · core.tex, line 484
dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.
Exact mathematical passage
```
 C_h(z)=2h\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].
 \label{c:core:gowdy-momentum}
```

C_iUse in source · 6 passages

Search aliases: C_i

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Use in source · core.tex, line 618
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].
Exact mathematical passage
```
 C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
 &C_i&=-2\delta_j\pi_{ij},\nonumber\\
 \mathcal G_{ij}(h)&=\frac12\big[
 \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
 -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
 -\delta_{ij}C\big].
 \label{c:core:adm-operators}
```

\mathcal C_IUse in source · 2 passages

Search aliases: C_I mathcal

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Use in source · constraints.tex, line 451
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,
Exact mathematical passage
```
\mathcal C_I=\mathcal C_I^{\mathrm G}+\mathcal C_I^{\mathrm M}+\mathcal C_I^{\mathrm D}
```

C_{i,x}Use in source · 1 passage

Search aliases: C_i,x

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Use in source · core.tex, line 737
For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.
Exact mathematical passage
```
 E_A^{\rm con}=\sum_{x\in A}h^3
 \left(|C_x|^2+\sum_i|C_{i,x}|^2\right)
```

\mathcal C_I^{\mathrm D}Use in source · 2 passages

Search aliases: C_I^ D mathcal mathrm

Link to this entry ·

Use in source · constraints.tex, line 451
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,
Exact mathematical passage
```
\mathcal C_I=\mathcal C_I^{\mathrm G}+\mathcal C_I^{\mathrm M}+\mathcal C_I^{\mathrm D}
```

\mathcal C_I^{\mathrm G}Use in source · 2 passages

Search aliases: C_I^ G mathcal mathrm

Link to this entry ·

Use in source · constraints.tex, line 451
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,
Exact mathematical passage
```
\mathcal C_I=\mathcal C_I^{\mathrm G}+\mathcal C_I^{\mathrm M}+\mathcal C_I^{\mathrm D}
```

\mathcal C_I^{\mathrm M}Use in source · 2 passages

Search aliases: C_I^ M mathcal mathrm

Link to this entry ·

Use in source · constraints.tex, line 451
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,
Exact mathematical passage
```
\mathcal C_I=\mathcal C_I^{\mathrm G}+\mathcal C_I^{\mathrm M}+\mathcal C_I^{\mathrm D}
```

C_jUse in source · 3 passages

Search aliases: C_j

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Use in source · refinement.tex, line 248
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
C_j:X\to X
```

c_{j+1/2}Use in source · 5 passages

Search aliases: c_j+1/2

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Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

C_{j-1}Use in source · 1 passage

Search aliases: C_j-1

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Use in source · refinement.tex, line 253
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_N\cdots F_1I-IC_N\cdots C_1
 =\sum_{j=1}^N F_N\cdots F_{j+1}D_jC_{j-1}\cdots C_1.
 \label{g:telescoping}
```

c_{j-1/2}Use in source · 2 passages

Search aliases: c_j-1/2

Link to this entry ·

Use in source · core.tex, line 38
equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms in the action produce the two terms in the spatial numerator. The time integration by parts differentiates w_jdotφ_j, including the dot w_jdotφ_j term.
Exact mathematical passage
```
 w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
 =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
 -v_j\varphi_j.
 \label{c:core:scalar-evolution}
```

C_LUse in source · 3 passages

Search aliases: C_L

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Use in source · spectral.tex, line 968
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
(C_Lv)_0=v_{L-1}
```

C_L^LUse in source · 1 passage

Search aliases: C_L^L

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Use in source · spectral.tex, line 970
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
 \mathbf E_L'=C_L\mathbf E_L,\qquad
 \mathbf E_L(0)=e_0,\qquad
 C_L^L=I,\qquad
 \mathbf E_L(z)=e^{zC_L}e_0.
 \label{a-spectral-companion-system}
```

C_L^nUse in source · 1 passage

Search aliases: C_L^n

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Use in source · spectral.tex, line 977
gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=frac1L∑_ℓ=0^L-1 ω_L^-jℓe^ω_L^ℓ z. Thus factorial, hypergeometric, companion-state, and root-filter forms describe the same analytic function. Each f…
Exact mathematical passage
```
C_L^ne_0=e_{n\bmod L}
```

c_{m,0}Use in source · 1 passage

Search aliases: c_m,0

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Use in source · refinement.tex, line 22
[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
c_{m,0}=1
```

C_{M,j}Use in source · 1 passage

Search aliases: C_M,j

Link to this entry ·

Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

c_{m,r}Use in source · 2 passages

Search aliases: c_m,r

Link to this entry ·

Use in source · refinement.tex, line 18
[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
 T_m(h^rv)=c_{m,r}h^rv,\qquad
 c_{m,r}=\prod_{k=1}^{m}\frac{b^{k-r}-1}{b^k-1}.
```

c_nUse in source · 17 passages

Search aliases: c_n

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Use in source · continuum.tex, line 168
equals its Hilbert adjoint. Let b∈Dom(widehat H_m^*) and write c=widehat H_m^*b. By the adjoint relation, innerwidehat H_mab=innerac (a∈Dom(widehat H_m)). Fix a mode n and choose a with the single nonzero coefficient a_n=v, where v∈C^4 is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes innerH_nvb_n=innervc_n. Hermiticity gives innervH_nb_n=innervc_n for every v. Taking v=H_nb_n-c_n shows c_n=H_nb_n. Because c∈KK, the sequence (H_nb_n)_n is square summable. Thu…
Exact mathematical passage
```
\inner{H_nv}{b_n}=\inner{v}{c_n}.
```

C_{P,j}Use in source · 1 passage

Search aliases: C_P,j

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Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

c_{p-1}Use in source · 1 passage

Search aliases: c_p-1

Link to this entry ·

Use in source · spectral.tex, line 374
There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-c…
Exact mathematical passage
```
c=(c_0,\ldots,c_{p-1})\in\C^p
```

C_{\mathrm{phase}}Use in source · 2 passages

Search aliases: C_phase mathrm

Link to this entry ·

Use in source · finite.tex, line 192
The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel that discards a subsystem. For example it sends the normalized state (|0,a⟩+|1,a⟩)/sqrt2 to sqrt2|a⟩. Its adjoint product therefore does not equal th…
Exact mathematical passage
```
 C_{\mathrm{phase}}|s,a\rangle=|a\rangle
 \label{a-finite-forgetting}
```

c_qUse in source · 2 passages

Search aliases: c_q

Link to this entry ·

Use in source · finite_core.tex, line 934
The error also has an exact expression. For fixed mode, let r=p(n), q=q_h(n), with the corresponding c_r,s_r,c_q,s_q. Coordinate subtraction gives
Exact mathematical passage
```
c_r,s_r,c_q,s_q
```

C_{Q,j}Use in source · 1 passage

Search aliases: C_Q,j

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Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

c_rUse in source · 5 passages

Search aliases: c_r

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Use in source · finite_core.tex, line 860
Set E_r=sqrt1+|r|^2, c_r(t)=cos(E_rt), and s_r(t)=sin(E_rt)/E_r. PH03 gives
Exact mathematical passage
```
c_r(t)=\cos(E_rt)
```

C_xUse in source · 1 passage

Search aliases: C_x

Link to this entry ·

Use in source · core.tex, line 737
For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.
Exact mathematical passage
```
 E_A^{\rm con}=\sum_{x\in A}h^3
 \left(|C_x|^2+\sum_i|C_{i,x}|^2\right)
```

C_\alphaUse in source · 7 passages

Search aliases: C_α alpha

Link to this entry ·

Use in source · extensions.tex, line 216
For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.
Exact mathematical passage
```
A=\bigsqcup_{\alpha}C_\alpha
```

C_\betaUse in source · 3 passages

Search aliases: C_β beta

Link to this entry ·

Use in source · extensions.tex, line 223
Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately: J^rm c_βα · =2IminnerQ_βψHQ_αψ · =∑_b∈ C_β(∑_a∈ C_αJ_ba) =∑_a∈ C_α(∑_b∈ C_βJ_ba). For example, first summing a moves H onto P_bψ, then the outer sum pairs ∑_bP_bψ with the fixed vector HQ_αψ. The reverse order has its own symmetric argument. Flattening these expressions to a single unordered sum on C_β× C_α requires joint summability. Equality of two justified iterated sums does not supp…
Exact mathematical passage
```
 J^{\rm c}_{\beta\alpha}
 &=2\operatorname{Im}\inner{Q_\beta\psi}{HQ_\alpha\psi}\\
 &=\sum_{b\in C_\beta}\left(\sum_{a\in C_\alpha}J_{ba}\right)
 =\sum_{a\in C_\alpha}\left(\sum_{b\in C_\beta}J_{ba}\right).
```

C_{\Lambda,j}Use in source · 1 passage

Search aliases: C_Λ,j Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

c_\ellUse in source · 2 passages

Search aliases: c_ℓ ell

Link to this entry ·

Use in source · spectral.tex, line 140
The same construction works for any circulant polynomial H=∑_ℓ c_ℓ S^ℓ, with symbol ∑_ℓ c_ℓ P_p(θ)^ℓ. It preserves all bands, whether or not their energies coincide. For N=p^h, digit reversal relates this standard residue split to the newest-digit adic split developed earlier. It is still necessary to distinguish the retained residue-zero sites from the retained Haar average channel. They are different subspaces.
Exact mathematical passage
```
H=\sum_\ell c_\ell S^\ell
```

\mathrm{coarse}Use in source · 1 passage

Search aliases: coarse mathrm

Link to this entry ·

Use in source · refinement.tex, line 186
lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass substeps also intertwine, the identities compose through the entire execution order M-X-Y-Z-Y-X-M. The refined walk here replaces each coarse spatial fac…
Exact mathematical passage
```
 S_{i,\mathrm{fine}}^2 I_r=I_rS_{i,\mathrm{coarse}}.
```

\cos^2Use in source · 3 passages

Search aliases: cos^2 cos

Link to this entry ·

Use in source · finite_core.tex, line 914
u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.
Exact mathematical passage
```
\|u_r(t)\|_2^2
=c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)
=\cos^2(E_rt)+\sin^2(E_rt)=1.
```

\coshUse in source · 3 passages

Search aliases: cosh

Link to this entry ·

Use in source · spectral.tex, line 1046
The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lane is an analytic contribution to it.
Exact mathematical passage
```
E_{2,0}(tA_0)=\cosh(tA_0)
```

\mathrm{CRT}Use in source · 3 passages

Search aliases: CRT mathrm

Link to this entry ·

Use in source · finite.tex, line 126
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 \mathcal U_{\mathrm{CRT}}:
 \ell^2\bigl((\Z/M\Z)^\times\bigr)
 \longrightarrow
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^{h_2}\Z)
 \otimes\bigotimes_{p\in\mathcal P_{\mathrm o}}
 \left(\C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^{h_p}\Z)\right).
 \label{a-finite-crt}
```

\mathbf dUse in source · 11 passages

Search aliases: d mathbf

Link to this entry ·

Use in source · finite.tex, line 411
D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 V_{p,h+w-1}\cdots V_{p,h}|a\rangle
 =p^{-w/2}\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}|a,\mathbf d\rangle,
 \label{a-finite-nested-refinement}
```

\mathcal DUse in source · 21 passages

Search aliases: D mathcal

Link to this entry ·

Use in source · core.tex, line 407
Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected bas…
Exact mathematical passage
```
\label{f:jump-operators}
 L_{ba}=\sqrt{\gamma_{ba}}\,|e_b\rangle\langle e_a|,
 \qquad
 \mathcal D_L(\rho)=L\rho L^*-	frac12(L^*L\rho+\rho L^*L).
```

\mathrm dUse in source · 21 passages

Search aliases: d mathrm

Link to this entry ·

Use in source · constraints.tex, line 193
If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j), τ_N(∏_jS_Nf_j) · =(2π)^-3∫_ T^3∏_jf_j d^3x. The Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled pointwise products. This is how the resolved construction avoids the finite pointwise derivation obstruction: it retains enough detail instead of demandi…
Exact mathematical passage
```
 \delta_a\prod_jS_Nf_j
 &=S_N\partial_a\left(\prod_jf_j\right),\\
 \tau_N\left(\prod_jS_Nf_j\right)
 &=(2\pi)^{-3}\int_{\mathbb T^3}\prod_jf_j\,\mathrm d^3x.
 \label{c:constraints:alias-free}
```

\mathsf DUse in source · 1 passage

Search aliases: D mathsf

Link to this entry ·

Use in source · limit.tex, line 148
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
\mathsf D_m
```

D^{\mathsf T}Use in source · 1 passage

Search aliases: D^ T mathsf

Link to this entry ·

Use in source · core.tex, line 855
Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], and the two defect terms give the second.
Exact mathematical passage
```
D^{\mathsf T}=-D
```

D^*Use in source · 1 passage

Search aliases: D^*

Link to this entry ·

Use in source · spectral.tex, line 158
Let a finite Hilbert space be the orthogonal sum HH_L⊕HH_D, and write a self-adjoint Hamiltonian as H=A · B B^* · D, A=A^*, D=D^*. Here B:HH_D→HH_L. Let ι_L embed HH_L into the full space. This notation fixes the coordinate type of every compression.
Exact mathematical passage
```
 H=\begin{pmatrix}A&B\\B^*&D\end{pmatrix},\qquad A=A^*,\quad D=D^*.
 \label{a-spectral-block-hamiltonian}
```

d^*Use in source · 3 passages

Search aliases: d^*

Link to this entry ·

Use in source · strong_fields.tex, line 227
A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,
Exact mathematical passage
```
 R_j\ge R_*>0,\quad 2MR_j-Q^2\ge p_*^2>0,
 \quad 0<d_*\le D_4R_j\le d^*.
```

D^+_{\mu,U}Use in source · 3 passages

Search aliases: D^+_μ,U mu

Link to this entry ·

Use in source · core.tex, line 27
[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
 (D^+_{\mu,U}\psi)_x
 &=\frac{U_{x,\mu}\psi_{x+\widehat\mu}-\psi_x}{h},\label{g:forward-difference}\\
 (\Delta_U\psi)_x
 &=\frac1{h^2}\sum_\mu\left(
 U_{x,\mu}\psi_{x+\widehat\mu}
 +U_{x-\widehat\mu,\mu}^{-1}\psi_{x-\widehat\mu}-2\psi_x\right).
 \label{g:laplacian}
```

D^+_{\mu,U^g}Use in source · 1 passage

Search aliases: D^+_μ,U^g mu

Link to this entry ·

Use in source · core.tex, line 36
(Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
D^+_{\mu,U^g}(g\psi)=gD^+_{\mu,U}\psi
```

D^{-1/2}Use in source · 2 passages

Search aliases: D^-1/2

Link to this entry ·

Use in source · constraints.tex, line 798
x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gaug…
Exact mathematical passage
```
\rho_C\propto D^{-1/2}
```

d^{-1/2}Use in source · 1 passage

Search aliases: d^-1/2

Link to this entry ·

Use in source · constraints.tex, line 805
Eliminating z first gives the effective coefficient μ for y^2 and the factor (m+n)^-1/2. Eliminating y next gives d^-1/2. Their product is D^-1/2, exactly as in direct elimination. For the Euclidean weight e^-S/ℏ, using a flat coarse measure instead records this determinant as S_C,ℏ=S_C+tfracℏ2log D, up to an additive constant. A nonlinear symmetry that preserves the classical action can still require a nonconstant induced measure.
Exact mathematical passage
```
d^{-1/2}
```

D^2Use in source · 1 passage

Search aliases: D^2

Link to this entry ·

Use in source · core.tex, line 862
f Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], and the two defect terms give the second.
Exact mathematical passage
```
\xi\eta D^2q
```

d^2Use in source · 1 passage

Search aliases: d^2

Link to this entry ·

Use in source · finite_core.tex, line 1080
(tfrac14d^2+tfrac14 3d^2)^1/2=d,
Exact mathematical passage
```
\left(\tfrac14d^2+\tfrac14\,3d^2\right)^{1/2}=d,
```

\mathrm d^3Use in source · 8 passages

Search aliases: d^3 mathrm

Link to this entry ·

Use in source · constraints.tex, line 193
If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j), τ_N(∏_jS_Nf_j) · =(2π)^-3∫_ T^3∏_jf_j d^3x. The Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled pointwise products. This is how the resolved construction avoids the finite pointwise derivation obstruction: it retains enough detail instead of demandi…
Exact mathematical passage
```
 \delta_a\prod_jS_Nf_j
 &=S_N\partial_a\left(\prod_jf_j\right),\\
 \tau_N\left(\prod_jS_Nf_j\right)
 &=(2\pi)^{-3}\int_{\mathbb T^3}\prod_jf_j\,\mathrm d^3x.
 \label{c:constraints:alias-free}
```

D^TUse in source · 1 passage

Search aliases: D^T

Link to this entry ·

Use in source · strong_fields.tex, line 286
For an interval summation-by-parts derivative with a diagonal positive norm matrix H and HD+D^TH=B, multiplication by b commutes with H. The uncancelled term is instead inneruA_h[b]u_H =frac12u^TB diag(b)u -frac12inneru(Db)u_H. Its boundary flux must be controlled using an actual boundary closure and appropriate boundary data. Saying that a stencil is centered in the interior does not establish this property for the complete interval operator.
Exact mathematical passage
```
HD+D^TH=B
```

d_*Use in source · 1 passage

Search aliases: d_*

Link to this entry ·

Use in source · strong_fields.tex, line 227
A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,
Exact mathematical passage
```
 R_j\ge R_*>0,\quad 2MR_j-Q^2\ge p_*^2>0,
 \quad 0<d_*\le D_4R_j\le d^*.
```

D_+Use in source · 3 passages

Search aliases: D_+

Link to this entry ·

Use in source · strong_fields.tex, line 488
The characteristic evolution uses the second-order one-sided derivatives (D_+u)_j=frac-3u_j+4u_j+1-u_j+22h, (D_-u)_j=frac3u_j-4u_j-1+u_j-22h. The inward channel uses D_+ and the outgoing channel uses D_- outside a horizon, switching to D_+ inside. These interior matter stencils have radius two. The source also uses a radius-two fourth-order interior derivative for v' and separate one-sided boundary formulas. Finite stencil radius for these derivatives must not be promoted to a finite dependency …
Exact mathematical passage
```
 (D_+u)_j=\frac{-3u_j+4u_{j+1}-u_{j+2}}{2h},\qquad
 (D_-u)_j=\frac{3u_j-4u_{j-1}+u_{j-2}}{2h}.
```

D_-Use in source · 2 passages

Search aliases: D_-

Link to this entry ·

Use in source · strong_fields.tex, line 488
The characteristic evolution uses the second-order one-sided derivatives (D_+u)_j=frac-3u_j+4u_j+1-u_j+22h, (D_-u)_j=frac3u_j-4u_j-1+u_j-22h. The inward channel uses D_+ and the outgoing channel uses D_- outside a horizon, switching to D_+ inside. These interior matter stencils have radius two. The source also uses a radius-two fourth-order interior derivative for v' and separate one-sided boundary formulas. Finite stencil radius for these derivatives must not be promoted to a finite dependency …
Exact mathematical passage
```
 (D_+u)_j=\frac{-3u_j+4u_{j+1}-u_{j+2}}{2h},\qquad
 (D_-u)_j=\frac{3u_j-4u_{j-1}+u_{j-2}}{2h}.
```

D_0Use in source · 1 passage

Search aliases: D_0

Link to this entry ·

Use in source · finite.tex, line 299
For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.
Exact mathematical passage
```
D_0
```

\mathbf d_0Use in source · 3 passages

Search aliases: d_0 mathbf

Link to this entry ·

Use in source · limit.tex, line 883
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
 d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
 \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
 \label{a-limit-product-cone}
```

D_0^{(h)}Use in source · 2 passages

Search aliases: D_0^(h)

Link to this entry ·

Use in source · finite.tex, line 274
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 D_0^{(h)}=E_0^{(h)},\qquad
 D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\quad(1\leq r\leq h).
```

D_1Use in source · 1 passage

Search aliases: D_1

Link to this entry ·

Use in source · refinement.tex, line 249
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.
```

D_2Use in source · 1 passage

Search aliases: D_2

Link to this entry ·

Use in source · refinement.tex, line 249
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.
```

D_4Use in source · 14 passages

Search aliases: D_4

Link to this entry ·

Use in source · strong_fields.tex, line 193
On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,Λ,P_Λ,P_R,Q), rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining R, …
Exact mathematical passage
```
 (D_4u)_j=\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.
```

\mathcal D_aUse in source · 3 passages

Search aliases: D_a mathcal

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

d_{C_N}Use in source · 1 passage

Search aliases: d_C_N

Link to this entry ·

Use in source · limit.tex, line 883
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
 d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
 \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
 \label{a-limit-product-cone}
```

D_hUse in source · 13 passages

Search aliases: D_h

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Use in source · extensions.tex, line 437
[The precise defect of a forward difference] On a cyclic grid, let D_hf_j=(f_j+1-f_j)/h, with h≠0. Expanding the product gives D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j =h(D_hf)_j(D_hg)_j. Thus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go theorem, because that theorem concerns an exact identity for every vector at fixed finite resolution.
Exact mathematical passage
```
D_hf_j=(f_{j+1}-f_j)/h
```

\mathcal D_hUse in source · 6 passages

Search aliases: D_h mathcal

Link to this entry ·

Use in source · core.tex, line 831
Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.
Exact mathematical passage
```
 \mathcal D_h[\xi]=h\sum_a\xi_ap_a(Dq)_a,
 \qquad
 \mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).
```

d_hUse in source · 5 passages

Search aliases: d_h

Link to this entry ·

Use in source · core.tex, line 498
g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
d_h=\norm{g_U}_2^2+\norm{g_V}_2^2
```

D_{h+1}^{(h+1)}Use in source · 2 passages

Search aliases: D_h+1^(h+1)

Link to this entry ·

Use in source · finite.tex, line 391
is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an identity on the coarse space. The second is a projector on the fine space, not a fine-space identity.
Exact mathematical passage
```
 V_{p,h}^*V_{p,h}=I,\qquad
 V_{p,h}V_{p,h}^*=E_h^{(h+1)},\qquad
 I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.
 \label{a-finite-refinement-projectors}
```

D_{h,j}Use in source · 3 passages

Search aliases: D_h,j

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Use in source · finite_core.tex, line 240
(D_h,jf)(x)=fracf(x+he_j)-f(x-he_j)2h.
Exact mathematical passage
```
(D_{h,j}f)(x)=\frac{f(x+he_j)-f(x-he_j)}{2h}.
```

d_{h-1}Use in source · 2 passages

Search aliases: d_h-1

Link to this entry ·

Use in source · finite.tex, line 444
ue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_0+⋯+d_h-2p^h-2. This proves the identity on an orthonormal basis.
Exact mathematical passage
```
d_0+d_1p+\cdots+d_{h-1}p^{h-1}
```

d_{h-2}Use in source · 1 passage

Search aliases: d_h-2

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Use in source · finite.tex, line 447
-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_0+⋯+d_h-2p^h-2. This proves the identity on an orthonormal basis.
Exact mathematical passage
```
a=d_0+\cdots+d_{h-2}p^{h-2}
```

D_h^{\mathsf T}Use in source · 1 passage

Search aliases: D_h^ T mathsf

Link to this entry ·

Use in source · core.tex, line 474
Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber
Exact mathematical passage
```
D_h^{\mathsf T}=-D_h
```

D_h^{(h)}Use in source · 1 passage

Search aliases: D_h^(h)

Link to this entry ·

Use in source · finite.tex, line 470
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*, W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] an…
Exact mathematical passage
```
\begin{aligned}
 (\langle0|\otimes I_m)\mathcal W_{p,h}&=V_{p,h-1}^*,\\
 \mathcal W_{p,h}^*
 \bigl[(I_p-|0\rangle\langle0|)\otimes I_m\bigr]
 \mathcal W_{p,h}&=D_h^{(h)}.
\end{aligned}
\label{a-finite-child-detail}
```

D_h^AUse in source · 2 passages

Search aliases: D_h^A

Link to this entry ·

Use in source · core.tex, line 293
The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let A be a real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription.
Exact mathematical passage
```
\label{g:integrated-link}
 U_h(x)=\exp\left(-i\int_x^{x+h}A(s)\,ds\right),\qquad
 (D_h^A\psi)(x)
 =\frac{U_h(x)\psi(x+h)-U_h(x-h)^{-1}\psi(x-h)}{2h}.
```

D_jUse in source · 4 passages

Search aliases: D_j

Link to this entry ·

Use in source · refinement.tex, line 248
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
D_j=F_jI-IC_j
```

\mathcal D_LUse in source · 1 passage

Search aliases: D_L mathcal

Link to this entry ·

Use in source · core.tex, line 407
Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected bas…
Exact mathematical passage
```
\label{f:jump-operators}
 L_{ba}=\sqrt{\gamma_{ba}}\,|e_b\rangle\langle e_a|,
 \qquad
 \mathcal D_L(\rho)=L\rho L^*-	frac12(L^*L\rho+\rho L^*L).
```

\mathcal D_{L_{ba}}Use in source · 1 passage

Search aliases: D_L_ba mathcal

Link to this entry ·

Use in source · core.tex, line 413
of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected basis. A general dissipative operator or a higher-rank atlas does not automatically produce the same closed rate equation.
Exact mathematical passage
```
\label{f:lindblad}
 \dot\rho=\sum_{b\ne a}\mathcal D_{L_{ba}}(\rho).
```

\mathsf D_mUse in source · 1 passage

Search aliases: D_m mathsf

Link to this entry ·

Use in source · limit.tex, line 148
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
\mathsf D_m
```

d_mUse in source · 2 passages

Search aliases: d_m

Link to this entry ·

Use in source · limit.tex, line 170
Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.
Exact mathematical passage
```
d_m=(p-1)p^{m-1}
```

D_m^{(M)}Use in source · 1 passage

Search aliases: D_m^(M)

Link to this entry ·

Use in source · limit.tex, line 149
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
D_m^{(M)}
```

D_rUse in source · 1 passage

Search aliases: D_r

Link to this entry ·

Use in source · finite.tex, line 296
For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.
Exact mathematical passage
```
(E_r-E_{r-1})^2=E_r-2E_{r-1}+E_{r-1}=D_r
```

D_r^{(h)}Use in source · 15 passages

Search aliases: D_r^(h)

Link to this entry ·

Use in source · finite.tex, line 274
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 D_0^{(h)}=E_0^{(h)},\qquad
 D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\quad(1\leq r\leq h).
```

D_r^{(h+1)}Use in source · 1 passage

Search aliases: D_r^(h+1)

Link to this entry ·

Use in source · finite.tex, line 402
Replicated vectors are constant in the new digit. All old cylinder averages commute with replication, giving E_r^(h+1)V_p,h · =V_p,hE_r^(h) (0≤ r≤ h), nonumber D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 E_r^{(h+1)}V_{p,h}&=V_{p,h}E_r^{(h)}\quad(0\leq r\leq h),
 \nonumber\\
 D_r^{(h+1)}V_{p,h}&=V_{p,h}D_r^{(h)}\quad(0\leq r\leq h),
 &D_{h+1}^{(h+1)}V_{p,h}&=0.
 \label{a-finite-detail-intertwining}
```

D_{\rm fine}Use in source · 1 passage

Search aliases: D_rm fine rm

Link to this entry ·

Use in source · refinement.tex, line 195
A flat cube need not reconstruct the prescribed edge. Conversely, take an arbitrary ordered half-link pair (a,b) and call its actual product v=ba. It is always a subdivision of v, even when the separately prescribed coarse link is u≠ v. The exact directional defect is D_rm fineψ · :=S_A,σ,(a,b)^2I_aψ-I_aS_A,τ,uψnotag · =I_a(S_A,τ,v-S_A,τ,u)ψ. The equality follows by applying the exact theorem to the product v and using linearity of I_a. Endpoint restriction makes I_a injective, so the fine defec…
Exact mathematical passage
```
 D_{\rm fine}\psi
  &:=S_{A,\sigma,(a,b)}^2I_a\psi-I_aS_{A,\tau,u}\psi\notag\\
  &=I_a(S_{A,\tau,v}-S_{A,\tau,u})\psi.
 \label{g:exact-mismatch}
```

\mathcal D_{\rm matt}Use in source · 1 passage

Search aliases: D_rm matt mathcal rm

Link to this entry ·

Use in source · strong_fields.tex, line 344
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
 \mathcal H_{\rm matt}=\frac{P_\varphi^2}{2\Lambda R^2}
                         +\frac{R^2\varphi'^2}{2\Lambda},
 \qquad \mathcal D_{\rm matt}=P_\varphi\varphi'.
```

D_{\rm total}Use in source · 5 passages

Search aliases: D_rm total rm

Link to this entry ·

Use in source · refinement.tex, line 300
[Combined finite Hilbert estimate] With the common-space walks and contractive mass factors from [source reference], define D_rm total=Q_fW_v-Q_cW_u. Then normD_rm total ≤ 2δ_x+2δ_y+δ_z+κ(ℓ). If v_i=u_i for all three directions and the cube is flat, the defect is exactly zero. Insert and subtract Q_fW_u: D_rm total=Q_f(W_v-W_u)+(Q_f-Q_c)W_u. All phase factors are unitary, and W_u is a product of contractions. Taking norms bounds the first term by the straight-edge budget and the second by the pa…
Exact mathematical passage
```
 D_{\rm total}=Q_fW_v-Q_cW_u.
```

D_s^{(h)}Use in source · 1 passage

Search aliases: D_s^(h)

Link to this entry ·

Use in source · finite.tex, line 279
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber rankD_0^(h) · =1, · rankD_r^(h) · =(p-1)p^r-1 (r≥1).
Exact mathematical passage
```
 \sum_{r=0}^hD_r^{(h)}&=I,
 &D_r^{(h)}D_s^{(h)}&=\delta_{rs}D_r^{(h)},\nonumber\\
 \operatorname{rank}D_0^{(h)}&=1,
 &\operatorname{rank}D_r^{(h)}&=(p-1)p^{r-1}\quad(r\geq1).
 \label{a-finite-detail-ranks}
```

d_TUse in source · 2 passages

Search aliases: d_T

Link to this entry ·

Use in source · limit.tex, line 883
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
 d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
 \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
 \label{a-limit-product-cone}
```

\mathcal D_\SigmaUse in source · 1 passage

Search aliases: D_Σ mathcal Sigma

Link to this entry ·

Use in source · constraints.tex, line 452
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ,
Exact mathematical passage
```
 \mathcal C_I^{\mathrm G}
 &=\tfrac1{2\kappa}\epsilon_{IJKL}E^J\wedge\mathcal R^{KL}
                  -\tfrac\Lambda\kappa\sigma_I,\\
 \mathcal C_I^{\mathrm M}
 &=\varphi_{IJ}\mathcal F\wedge E^J-\tfrac14\varphi_{KL}\varphi^{KL}\sigma_I,\\
 \mathcal C_I^{\mathrm D}
 &=\tfrac i4\epsilon_{JIKL}E^K\wedge E^L\wedge
       \bigl(\bar\psi\gamma^J\mathcal D_\Sigma\psi
                  -(\mathcal D_\Sigma\bar\psi)\gamma^J\psi\bigr)
                      -m\sigma_I\bar\psi\psi.
```

\mathrm d_\SigmaUse in source · 5 passages

Search aliases: d_Σ mathrm Sigma

Link to this entry ·

Use in source · constraints.tex, line 427
An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
\mathcal R=\mathrm d_\Sigma\Omega+\Omega\wedge\Omega
```

D_\OmegaUse in source · 3 passages

Search aliases: D_Ω Omega

Link to this entry ·

Use in source · constraints.tex, line 428
lit On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 R=\mathrm dt\wedge(\dot\Omega-D_\Omega\omega_\perp)+\mathcal R,
 \qquad
 F=\mathrm dt\wedge(\dot A-\mathrm d_\Sigma a_\perp)+\mathcal F.
```

D_\omegaUse in source · 1 passage

Search aliases: D_ω omega

Link to this entry ·

Use in source · constraints.tex, line 345
Gauge transformations and graded conventions. For even parameters (ξ,λ,u), the coframe transforms by its Lie derivative and Lorentz rotation, the connection by L_ξω-D_ωλ, and the Abelian potential by L_ξ a+ du. The spinor transforms by L_ξψ+tfrac14λ^IJγ_IJψ+iquψ, with the dual action on barψ. All internal contractions in~[source reference] are invariant. A diffeomorphism changes its four-form density by a Lie derivative, which integrates to zero on a periodic domain or under stated boundary cond…
Exact mathematical passage
```
\mathcal L_\xi\omega-D_\omega\lambda
```

\ddaggerUse in source · 2 passages

Search aliases: ddagger

Link to this entry ·

Use in source · constraints.tex, line 386
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
\Xi^\ddagger
```

\ddot hUse in source · 1 passage

Search aliases: ddot h ddot

Link to this entry ·

Use in source · core.tex, line 755
Choose n=n_i=0 and impose the transverse and traceless conditions δ_jh_ij=0, h_rm tr=0, δ_jπ_ij=0, π_rm tr=0. They are preserved by the evolution because G_ij(h)=-tfrac12δ^2h_ij, dot h=2π, dotπ=tfrac12δ^2h, ddot h=δ^2h. On this reduced sector the energy is H_rm TT=normπ_h^2 +frac14∑_knormδ_kh_h^2≥0. Its derivative is zero: substituting the two first-order equations reduces it to ⟨π,δ^2h⟩_h +∑_k⟨δ_kh,δ_kπ⟩_h=0.
Exact mathematical passage
```
 \mathcal G_{ij}(h)=-\tfrac12\delta^2h_{ij},\qquad
 \dot h=2\pi,\qquad\dot\pi=\tfrac12\delta^2h,
 \qquad\ddot h=\delta^2h.
```

\ddotsUse in source · 1 passage

Search aliases: ddots

Link to this entry ·

Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\ddot\varphiUse in source · 1 passage

Search aliases: ddotφ ddot varphi

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Use in source · core.tex, line 38
equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms in the action produce the two terms in the spatial numerator. The time integration by parts differentiates w_jdotφ_j, including the dot w_jdotφ_j term.
Exact mathematical passage
```
 w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
 =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
 -v_j\varphi_j.
 \label{c:core:scalar-evolution}
```

\ddot\varphi_jUse in source · 1 passage

Search aliases: ddotφ_j ddot varphi

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Use in source · core.tex, line 38
equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms in the action produce the two terms in the spatial numerator. The time integration by parts differentiates w_jdotφ_j, including the dot w_jdotφ_j term.
Exact mathematical passage
```
 w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
 =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
 -v_j\varphi_j.
 \label{c:core:scalar-evolution}
```

\operatorname{diag}Use in source · 5 passages

Search aliases: diag operatorname

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Use in source · core.tex, line 41
[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights …
Exact mathematical passage
```
 \psi=\frac1{\sqrt7}(1,i,2,-i)^{\mathsf T},\qquad
 P_A=\operatorname{diag}(1,1,0,0),\quad
 P_B=\operatorname{diag}(0,0,1,1).
```

\dimUse in source · 1 passage

Search aliases: dim

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Use in source · finite.tex, line 199
The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel that discards a subsystem. For example it sends the normalized state (|0,a⟩+|1,a⟩)/sqrt2 to sqrt2|a⟩. Its adjoint product therefore does not equal th…
Exact mathematical passage
```
 \operatorname{rank}C_{\mathrm{phase}}=2^h,
 \qquad\dim\ker C_{\mathrm{phase}}=2^h.
```

\operatorname{dist}Use in source · 1 passage

Search aliases: dist operatorname

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Use in source · limit.tex, line 754
normQe^-itH_λP · ≤min{1,|tλ|normQBP}. For z∉σ(QH_λ Q|_QHH), the coarse self-energy is Σ_P(z)=PH_λ Q (z-QH_λ Q|_QHH)^-1QH_λ P, where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e…
Exact mathematical passage
```
 \norm{\Sigma_P(z)}
 \leq\frac{|\lambda|^2\norm{PBQ}^2}
 {\operatorname{dist}(z,\sigma(QH_\lambda Q|_{Q\HH}))}.
 \label{a-limit-feshbach-bound}
```

\operatorname{div}Use in source · 1 passage

Search aliases: div operatorname

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Use in source · constraints.tex, line 799
stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gauge transformation.
Exact mathematical passage
```
 \operatorname{div}R_C=\frac{x_1}{d},\qquad
 R_C\log\rho_C=-\frac{x_1}{d}.
```

\dot AUse in source · 8 passages

Search aliases: dot A dot

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Use in source · core.tex, line 205
βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out…
Exact mathematical passage
```
\label{g:action}
 \mathcal S_h=\int\left[
 \frac{ih}{2}\sum_j(\psi_j^*\dot\psi_j-\dot\psi_j^*\psi_j)
 +h\sum_jE_j\dot A_j-H_h\right]dt.
```

\dot aUse in source · 1 passage

Search aliases: dot a dot

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Use in source · core.tex, line 111
[Time dependent expansion] Let the lapse equal one, let a(t)>0 be spatially uniform, and let m=0. Then w=a, c=1/a, and dot E_h=-frac h2dot a∑_jdotφ_j^2 -fracdot a2ha^2∑_jΔ_j^2. An increasing a transfers energy out of this coordinate-time field Hamiltonian. The supplied background receives no evolution equation from this calculation. A self-consistent matter--geometry model must also evolve its gravitational variables.
Exact mathematical passage
```
 \dot E_h=-\frac h2\dot a\sum_j\dot\varphi_j^2
 -\frac{\dot a}{2ha^2}\sum_j\Delta_j^2.
```

\dot A_jUse in source · 5 passages

Search aliases: dot A_j dot

Link to this entry ·

Use in source · core.tex, line 205
βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out…
Exact mathematical passage
```
\label{g:action}
 \mathcal S_h=\int\left[
 \frac{ih}{2}\sum_j(\psi_j^*\dot\psi_j-\dot\psi_j^*\psi_j)
 +h\sum_jE_j\dot A_j-H_h\right]dt.
```

\dot A_rUse in source · 2 passages

Search aliases: dot A_r dot

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Use in source · strong_fields.tex, line 12
da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ_e G)+B_∂. The boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamilton…
Exact mathematical passage
```
 S=\int dt\int dr\,
 \bigl(P_\Lambda\dot\Lambda+P_R\dot R+Q\dot A_r
       -N\mathcal H-N^r\mathcal D-\Phi_e\mathcal G\bigr)+B_\partial.
```

\dot CUse in source · 3 passages

Search aliases: dot C dot

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Use in source · core.tex, line 692
[Exact finite linearized closure] For every solution of [source reference], dot C=-δ_iC_i, dot C_i=0. Thus initially vanishing constraints remain zero. With the canonical Frobenius bracket and smearings C[n]=⟨ n,C⟩_h, D[ξ]=⟨ξ_i,C_i⟩_h, all three brackets {C[n],C[m]}, {D[ξ],D[η]}, and {C[n],D[ξ]} vanish identically.
Exact mathematical passage
```
 \dot C=-\delta_iC_i,\qquad\dot C_i=0.
```

\dot cUse in source · 2 passages

Search aliases: dot c dot

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Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

\dot C_iUse in source · 2 passages

Search aliases: dot C_i dot

Link to this entry ·

Use in source · core.tex, line 692
[Exact finite linearized closure] For every solution of [source reference], dot C=-δ_iC_i, dot C_i=0. Thus initially vanishing constraints remain zero. With the canonical Frobenius bracket and smearings C[n]=⟨ n,C⟩_h, D[ξ]=⟨ξ_i,C_i⟩_h, all three brackets {C[n],C[m]}, {D[ξ],D[η]}, and {C[n],D[ξ]} vanish identically.
Exact mathematical passage
```
 \dot C=-\delta_iC_i,\qquad\dot C_i=0.
```

\dot c_{j+1/2}Use in source · 2 passages

Search aliases: dot c_j+1/2 dot

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

\dot c_{j-1/2}Use in source · 1 passage

Search aliases: dot c_j-1/2 dot

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

\dot EUse in source · 7 passages

Search aliases: dot E dot

Link to this entry ·

Use in source · core.tex, line 210
y conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1,
Exact mathematical passage
```
h\dot E_j=-\partial H_h/\partial A_j
```

\dot eUse in source · 1 passage

Search aliases: dot e dot

Link to this entry ·

Use in source · core.tex, line 66
s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.
Exact mathematical passage
```
 \dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.
 \label{c:core:scalar-balance}
```

\dot E_hUse in source · 2 passages

Search aliases: dot E_h dot

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Use in source · core.tex, line 71
s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.
Exact mathematical passage
```
\dot E_h=\sum_js_j=-\partial_tL_h
```

\dot E_jUse in source · 5 passages

Search aliases: dot E_j dot

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Use in source · core.tex, line 210
y conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1,
Exact mathematical passage
```
h\dot E_j=-\partial H_h/\partial A_j
```

\dot e_jUse in source · 1 passage

Search aliases: dot e_j dot

Link to this entry ·

Use in source · core.tex, line 66
s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.
Exact mathematical passage
```
 \dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.
 \label{c:core:scalar-balance}
```

\dot E_{j-1}Use in source · 1 passage

Search aliases: dot E_j-1 dot

Link to this entry ·

Use in source · core.tex, line 243
2Re(ψ_j^*dotψ_j). The mass term contributes zero, since ψ_j^*βψ_j is real. The forward hopping term contributes -J_j/h. The backward scalar is the conjugate of ψ_j-1^*α U_j-1ψ_j, using α=α^* and the scalar nature of U. Its contribution is J_j-1/h. This proves continuity. Periodic summation gives dotbarρ=0. Finally, dot G_j =fracdot E_j-dot E_j-1h+dotρ_j-dotbarρ =fracJ_j-J_j-1h-fracJ_j-J_j-1h=0.
Exact mathematical passage
```
 \dot{\mathcal G}_j
 =\frac{\dot E_j-\dot E_{j-1}}h+\dot\rho_j-\dot{\bar\rho}
 =\frac{J_j-J_{j-1}}h-\frac{J_j-J_{j-1}}h=0.
```

\dot FUse in source · 3 passages

Search aliases: dot F dot

Link to this entry ·

Use in source · extensions.tex, line 47
[Effect continuity] Under these hypotheses, dot p_b=∑_a J^F_ba, ∑_bdot p_b=0. If F_b=F_b(t) is differentiable in operator norm, the first identity becomes dot p_b=s_b+∑_a J^F_ba, s_b=innerψdot F_bψ, ∑_b s_b=0. Self-adjointness of F_b and differentiation give dot p_b=2ReinnerF_bψKψ for a fixed effect. Summing the two terms of [source reference] over a and using ∑_aF_a=I gives ∑_aJ^F_ba=ReinnerF_bψKψ -ReinnerψKF_bψ. The second real part is the negative of the first. The moving case adds the produc…
Exact mathematical passage
```
 \dot p_b=s_b+\sum_a J^F_{ba},\qquad
 s_b=\inner{\psi}{\dot F_b\psi},\qquad \sum_b s_b=0.
```

\dot F_bUse in source · 2 passages

Search aliases: dot F_b dot

Link to this entry ·

Use in source · extensions.tex, line 47
[Effect continuity] Under these hypotheses, dot p_b=∑_a J^F_ba, ∑_bdot p_b=0. If F_b=F_b(t) is differentiable in operator norm, the first identity becomes dot p_b=s_b+∑_a J^F_ba, s_b=innerψdot F_bψ, ∑_b s_b=0. Self-adjointness of F_b and differentiation give dot p_b=2ReinnerF_bψKψ for a fixed effect. Summing the two terms of [source reference] over a and using ∑_aF_a=I gives ∑_aJ^F_ba=ReinnerF_bψKψ -ReinnerψKF_bψ. The second real part is the negative of the first. The moving case adds the produc…
Exact mathematical passage
```
 \dot p_b=s_b+\sum_a J^F_{ba},\qquad
 s_b=\inner{\psi}{\dot F_b\psi},\qquad \sum_b s_b=0.
```

\dot{\mathcal G}Use in source · 2 passages

Search aliases: dot G dot mathcal

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Use in source · core.tex, line 233
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-balance}
 \dot\rho_j+\frac{J_j-J_{j-1}}h=0,
 \qquad\dot{\bar\rho}=0,
 \qquad\dot{\mathcal G}_j=0.
```

\dot{\mathcal G}_jUse in source · 2 passages

Search aliases: dot G_j dot mathcal

Link to this entry ·

Use in source · core.tex, line 233
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-balance}
 \dot\rho_j+\frac{J_j-J_{j-1}}h=0,
 \qquad\dot{\bar\rho}=0,
 \qquad\dot{\mathcal G}_j=0.
```

\dot HUse in source · 2 passages

Search aliases: dot H dot

Link to this entry ·

Use in source · core.tex, line 486
equation dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with zero populations require their own limiting …
Exact mathematical passage
```
\dot H_q=\dot M_q/((1-q)M_q)
```

\dot hUse in source · 4 passages

Search aliases: dot h dot

Link to this entry ·

Use in source · core.tex, line 664
Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber
Exact mathematical passage
```
 S_h^{(2)}=\int\left[
 \inner{\pi}{\dot h}_h-T_h-U_h
 +\inner{n}{C}_h-\inner{n_i}{C_i}_h\right]dt.
 \label{c:core:adm-action}
```

\dot H_{\mathrm m}Use in source · 1 passage

Search aliases: dot H_ m dot mathrm

Link to this entry ·

Use in source · core.tex, line 257
[Electric work and total energy] Along the same solutions, dot H_E=h∑_jE_jJ_j, dot H_ m=-h∑_jE_jJ_j, dot H_h=0.
Exact mathematical passage
```
\label{g:work-balance}
 \dot H_E=h\sum_jE_jJ_j,\qquad
 \dot H_{\mathrm m}=-h\sum_jE_jJ_j,\qquad
 \dot H_h=0.
```

\dot H_EUse in source · 1 passage

Search aliases: dot H_E dot

Link to this entry ·

Use in source · core.tex, line 257
[Electric work and total energy] Along the same solutions, dot H_E=h∑_jE_jJ_j, dot H_ m=-h∑_jE_jJ_j, dot H_h=0.
Exact mathematical passage
```
\label{g:work-balance}
 \dot H_E=h\sum_jE_jJ_j,\qquad
 \dot H_{\mathrm m}=-h\sum_jE_jJ_j,\qquad
 \dot H_h=0.
```

\dot H_hUse in source · 1 passage

Search aliases: dot H_h dot

Link to this entry ·

Use in source · core.tex, line 257
[Electric work and total energy] Along the same solutions, dot H_E=h∑_jE_jJ_j, dot H_ m=-h∑_jE_jJ_j, dot H_h=0.
Exact mathematical passage
```
\label{g:work-balance}
 \dot H_E=h\sum_jE_jJ_j,\qquad
 \dot H_{\mathrm m}=-h\sum_jE_jJ_j,\qquad
 \dot H_h=0.
```

\dot h_{ij}Use in source · 1 passage

Search aliases: dot h_ij dot

Link to this entry ·

Use in source · core.tex, line 671
innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic gradient is 2π_ij-δ_ijπ_rm tr. Lemma~[source reference] makes the potential gradient G(h). The lapse gradient follows by two spatial integrations by pa…
Exact mathematical passage
```
 \dot h_{ij}&=2\pi_{ij}-\delta_{ij}\pi_{\rm tr}
             +\delta_in_j+\delta_jn_i,\nonumber\\
 \dot\pi_{ij}&=-\mathcal G_{ij}(h)
              +\delta_i\delta_jn-\delta_{ij}\delta^2n.
 \label{c:core:adm-evolution}
```

\dot H_qUse in source · 1 passage

Search aliases: dot H_q dot

Link to this entry ·

Use in source · core.tex, line 486
equation dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with zero populations require their own limiting …
Exact mathematical passage
```
\dot H_q=\dot M_q/((1-q)M_q)
```

\dot MUse in source · 2 passages

Search aliases: dot M dot

Link to this entry ·

Use in source · core.tex, line 481
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
 \dot M_q=\frac q2\sum_{b,a}J_{b\leftarrow a}
 (p_b^{q-1}-p_a^{q-1}).
```

\dot mUse in source · 4 passages

Search aliases: dot m dot

Link to this entry ·

Use in source · strong_fields.tex, line 400
Exact mass flux and compatibility. Put A=Π+vΦ and B=Φ+vΠ. The remaining Einstein momentum evolution gives dot m=r^2 J, J=σ AB. One can verify its origin directly from the P_Λ equation. In the flat-slice gauge, the gravitational terms give σ vP_Λ'-σ v^2/2-rσ'. The lapse-gradient term is essential. The matter variation contributes σ r^2(Π^2+Φ^2)/2. As dot m=vdot P_Λ, their sum becomes dot m=σ v[m'+fracr^22(Π^2+Φ^2)]-rvσ'. Using [source reference] and [source reference] yields dot m=r^2σ[v(Π^2+Φ^2)…
Exact mathematical passage
```
 \dot m=r^2\mathcal J,\qquad \mathcal J=\sigma AB.
\label{c:strong:scalar-flux}
```

\dot M_qUse in source · 2 passages

Search aliases: dot M_q dot

Link to this entry ·

Use in source · core.tex, line 481
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
 \dot M_q=\frac q2\sum_{b,a}J_{b\leftarrow a}
 (p_b^{q-1}-p_a^{q-1}).
```

\dot p_1Use in source · 1 passage

Search aliases: dot p_1 dot

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Use in source · core.tex, line 445
[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding -i[H,ρ] to [source reference] adds the coherent current to the jump current. Th…
Exact mathematical passage
```
 \dot p_1=k(1-2p_1),\qquad
 p_1(t)=\tfrac12+\bigl(p_1(0)-\tfrac12\bigr)e^{-2kt},
 \qquad \rho_{12}(t)=e^{-kt}\rho_{12}(0).
```

\dot p_2Use in source · 2 passages

Search aliases: dot p_2 dot

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Use in source · core.tex, line 173
-isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instantaneous flow must be kept separate.
Exact mathematical passage
```
 J_{2\leftarrow1}
 =2\operatorname{Im}\bigl(i g\sin(gt)\cos(gt)\bigr)
 =g\sin(2gt)=\dot p_2.
```

\dot p_A^{\mathrm c}Use in source · 1 passage

Search aliases: dot p_A^ c dot mathrm

Link to this entry ·

Use in source · core.tex, line 232
J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence dot p_A^ c=-3/2 and dot p_B^ c=3/2.
Exact mathematical passage
```
\dot p_A^{\mathrm c}=-3/2
```

\dot p_B^{\mathrm c}Use in source · 3 passages

Search aliases: dot p_B^ c dot mathrm

Link to this entry ·

Use in source · core.tex, line 203
J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
\dot p_B^{\mathrm c}=\sum_AJ_{B\leftarrow A}^{\mathrm c}
```

\dot P_RUse in source · 1 passage

Search aliases: dot P_R dot

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Use in source · strong_fields.tex, line 127
ginequation dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2], dot P_R · =fracΛ(P_Λ^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_…
Exact mathematical passage
```
 \dot P_\Lambda&=N^rP_\Lambda'
 +\frac12\left[1-\frac{P_\Lambda^2+Q^2}{R^2}
                 -\left(\frac{R'}\Lambda\right)^2\right],\\
 \dot P_R&=\frac{\Lambda(P_\Lambda^2+Q^2)}{R^3}
 -\frac{P_\Lambda P_R}{R^2}+(N^rP_R)'
 -\frac{R''}\Lambda+\frac{\Lambda'R'}{\Lambda^2}.
```

\dot P_\LambdaUse in source · 2 passages

Search aliases: dot P_Λ dot Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 127
ginequation dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2], dot P_R · =fracΛ(P_Λ^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_…
Exact mathematical passage
```
 \dot P_\Lambda&=N^rP_\Lambda'
 +\frac12\left[1-\frac{P_\Lambda^2+Q^2}{R^2}
                 -\left(\frac{R'}\Lambda\right)^2\right],\\
 \dot P_R&=\frac{\Lambda(P_\Lambda^2+Q^2)}{R^3}
 -\frac{P_\Lambda P_R}{R^2}+(N^rP_R)'
 -\frac{R''}\Lambda+\frac{\Lambda'R'}{\Lambda^2}.
```

\dot QUse in source · 2 passages

Search aliases: dot Q dot

Link to this entry ·

Use in source · core.tex, line 474
Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber
Exact mathematical passage
```
V=\dot Q
```

\dot RUse in source · 5 passages

Search aliases: dot R dot

Link to this entry ·

Use in source · strong_fields.tex, line 12
da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ_e G)+B_∂. The boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamilton…
Exact mathematical passage
```
 S=\int dt\int dr\,
 \bigl(P_\Lambda\dot\Lambda+P_R\dot R+Q\dot A_r
       -N\mathcal H-N^r\mathcal D-\Phi_e\mathcal G\bigr)+B_\partial.
```

\dot R_jUse in source · 1 passage

Search aliases: dot R_j dot

Link to this entry ·

Use in source · strong_fields.tex, line 245
Direct and split radius evolution. On the positive branch, restriction of the radius equation to the reconstruction manifold gives dot R_j=g(R_j)+b_j(D_4R)_j, g(R)=-fracsqrt2MR-Q^2R, where b=N^r is prescribed in the electrovac benchmark. A split alternative is dot R=g(R)+A_h[b]R, A_h[b]u=frac12(bD_4u+D_4(bu)-(D_4b)u). On smooth data these formulas differ by half of the discrete product-rule defect, which is fourth order away from boundaries. Their exact finite-grid operators differ.
Exact mathematical passage
```
 \dot R_j=g(R_j)+b_j(D_4R)_j,
 \qquad g(R)=-\frac{\sqrt{2MR-Q^2}}R,
\label{c:strong:direct-radius}
```

\dot SUse in source · 1 passage

Search aliases: dot S dot

Link to this entry ·

Use in source · core.tex, line 477
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
\label{f:entropy-current}
 \dot S=-\frac12\sum_{b,a}J_{b\leftarrow a}\log\frac{p_b}{p_a}.
```

\dot uUse in source · 2 passages

Search aliases: dot u dot

Link to this entry ·

Use in source · spectral.tex, line 279
[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber
Exact mathematical passage
```
i\dot u=Au+Bv
```

\dot U^*Use in source · 3 passages

Search aliases: dot U^* dot

Link to this entry ·

Use in source · core.tex, line 270
Differentiate U^*U=id to obtain dot U^*U+U^*dot U=0. This proves Γ^*=-Γ. Multiplication on the right by U^* gives dot U^*=-Γ U^*. Therefore dotϕ=dot U^*ψ+U^*dotψ =-Γϕ+U^*KUϕ. Unitary conjugation preserves skew-adjointness. The difference of the two skew-adjoint operators in [source reference] is skew-adjoint, so the transfer theorem applies.
Exact mathematical passage
```
\dot U^*U+U^*\dot U=0
```

\dot U_{\rm raw}Use in source · 1 passage

Search aliases: dot U_rm raw dot rm

Link to this entry ·

Use in source · core.tex, line 578
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
\mathcal E_{P,h}=\dot U-\dot U_{\rm raw}
```

\dot VUse in source · 2 passages

Search aliases: dot V dot

Link to this entry ·

Use in source · core.tex, line 476
Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber dot U · =γΔ_hP+e^2P[V^2-γ(D_hQ)^2],nonumber dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity va…
Exact mathematical passage
```
 \dot P&=U,&\dot Q&=V,\nonumber\\
 \dot U&=\gamma\Delta_hP+e^{2P}[V^2-\gamma(D_hQ)^2],\nonumber\\
 \dot V&=\gamma\Delta_hQ-2[UV-\gamma(D_hP)(D_hQ)].
 \label{c:core:gowdy-raw}
```

\dot vUse in source · 5 passages

Search aliases: dot v dot

Link to this entry ·

Use in source · spectral.tex, line 279
[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber
Exact mathematical passage
```
i\dot v=Dv+B^*u
```

\dot v_jUse in source · 2 passages

Search aliases: dot v_j dot

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

\dot V_{\rm raw}Use in source · 1 passage

Search aliases: dot V_rm raw dot rm

Link to this entry ·

Use in source · core.tex, line 579
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
\mathcal E_{Q,h}=\dot V-\dot V_{\rm raw}
```

\dot wUse in source · 9 passages

Search aliases: dot w dot

Link to this entry ·

Use in source · spectral.tex, line 66
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
 \dot w_r=\sum_s\mathcal J_{r\leftarrow s},\qquad
 \mathcal J_{r\leftarrow s}
   =2\operatorname{Im}\langle\psi_r,H_{rs}\psi_s\rangle,
 \qquad
 \mathcal J_{r\leftarrow s}=-\mathcal J_{s\leftarrow r}.
```

\dot w_+Use in source · 1 passage

Search aliases: dot w_+ dot

Link to this entry ·

Use in source · strong_fields.tex, line 444
Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
 \dot w_+={}&\sigma(v+1)w_+'
 +[\sigma v'+\sigma'(v+1)]w_+
 +\frac\sigma r[(v+1)w_++(v-1)w_-],\\
 \dot w_-={}&\sigma(v-1)w_-'
 +[\sigma v'+\sigma'(v-1)]w_-
 +\frac\sigma r[(v-1)w_-+(v+1)w_+].
```

\dot w_-Use in source · 1 passage

Search aliases: dot w_- dot

Link to this entry ·

Use in source · strong_fields.tex, line 444
Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
 \dot w_+={}&\sigma(v+1)w_+'
 +[\sigma v'+\sigma'(v+1)]w_+
 +\frac\sigma r[(v+1)w_++(v-1)w_-],\\
 \dot w_-={}&\sigma(v-1)w_-'
 +[\sigma v'+\sigma'(v-1)]w_-
 +\frac\sigma r[(v-1)w_-+(v+1)w_+].
```

\dot w_jUse in source · 4 passages

Search aliases: dot w_j dot

Link to this entry ·

Use in source · core.tex, line 38
equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms in the action produce the two terms in the spatial numerator. The time integration by parts differentiates w_jdotφ_j, including the dot w_jdotφ_j term.
Exact mathematical passage
```
 w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
 =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
 -v_j\varphi_j.
 \label{c:core:scalar-evolution}
```

\dot w_rUse in source · 2 passages

Search aliases: dot w_r dot

Link to this entry ·

Use in source · spectral.tex, line 66
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
 \dot w_r=\sum_s\mathcal J_{r\leftarrow s},\qquad
 \mathcal J_{r\leftarrow s}
   =2\operatorname{Im}\langle\psi_r,H_{rs}\psi_s\rangle,
 \qquad
 \mathcal J_{r\leftarrow s}=-\mathcal J_{s\leftarrow r}.
```

\dot zUse in source · 1 passage

Search aliases: dot z dot

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Use in source · core.tex, line 542
[Fixed-mesh second-order time accuracy] Fix h. Suppose a smooth exact solution of dot z=widehat f_h(τ,z) and its numerical steps remain in a compact regular neighborhood where d_h≥ d_0>0. Assume the mixed time and state derivatives of widehat f_h through order three and the state derivatives of C_h through order four are bounded there. Apply the endpoint correction of Proposition~[source reference] after explicit midpoint. Then its local error is O(Δτ^3) and its global error on a fixed time inte…
Exact mathematical passage
```
\dot z=\widehat f_h(\tau,z)
```

\dot{\bar\rho}Use in source · 3 passages

Search aliases: dotbarρ dot bar rho

Link to this entry ·

Use in source · core.tex, line 233
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-balance}
 \dot\rho_j+\frac{J_j-J_{j-1}}h=0,
 \qquad\dot{\bar\rho}=0,
 \qquad\dot{\mathcal G}_j=0.
```

\dot\thetaUse in source · 2 passages

Search aliases: dotθ dot theta

Link to this entry ·

Use in source · core.tex, line 289
sinθ(t) · cosθ(t). With the coordinate projectors as P_a^(0), ϕ=cosθ -sinθ, Γ=dotθ0 · -1 1 · 0, K_eff=-Γ. The moving weights are p_1=cos^2θ and p_2=sin^2θ, with J_2←1=dotθsin(2θ)=dot p_2. The state has not moved in the original Hilbert space. The measured sectors have moved around it. The connection term accounts for the entire change.
Exact mathematical passage
```
 \phi=\begin{pmatrix}\cos\theta\\-\sin\theta\end{pmatrix},\quad
 \Gamma=\dot\theta\begin{pmatrix}0&-1\\1&0\end{pmatrix},\quad
 K_{\mathrm{eff}}=-\Gamma.
```

\dot\LambdaUse in source · 4 passages

Search aliases: dotΛ dot Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 12
da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ_e G)+B_∂. The boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamilton…
Exact mathematical passage
```
 S=\int dt\int dr\,
 \bigl(P_\Lambda\dot\Lambda+P_R\dot R+Q\dot A_r
       -N\mathcal H-N^r\mathcal D-\Phi_e\mathcal G\bigr)+B_\partial.
```

\dot\PiUse in source · 2 passages

Search aliases: dotΠ dot Pi

Link to this entry ·

Use in source · strong_fields.tex, line 388
E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.
Exact mathematical passage
```
 \dot\varphi&=\sigma(\Pi+v\Phi),\\
 \dot\Phi&=\partial_r[\sigma(\Pi+v\Phi)],\\
 \dot\Pi&=r^{-2}\partial_r[r^2\sigma(\Phi+v\Pi)].
\label{c:strong:scalar-evolution}
```

\dot\piUse in source · 2 passages

Search aliases: dotπ dot pi

Link to this entry ·

Use in source · core.tex, line 671
innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic gradient is 2π_ij-δ_ijπ_rm tr. Lemma~[source reference] makes the potential gradient G(h). The lapse gradient follows by two spatial integrations by pa…
Exact mathematical passage
```
 \dot h_{ij}&=2\pi_{ij}-\delta_{ij}\pi_{\rm tr}
             +\delta_in_j+\delta_jn_i,\nonumber\\
 \dot\pi_{ij}&=-\mathcal G_{ij}(h)
              +\delta_i\delta_jn-\delta_{ij}\delta^2n.
 \label{c:core:adm-evolution}
```

\dot\pi_{ij}Use in source · 1 passage

Search aliases: dotπ_ij dot pi

Link to this entry ·

Use in source · core.tex, line 671
innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic gradient is 2π_ij-δ_ijπ_rm tr. Lemma~[source reference] makes the potential gradient G(h). The lapse gradient follows by two spatial integrations by pa…
Exact mathematical passage
```
 \dot h_{ij}&=2\pi_{ij}-\delta_{ij}\pi_{\rm tr}
             +\delta_in_j+\delta_jn_i,\nonumber\\
 \dot\pi_{ij}&=-\mathcal G_{ij}(h)
              +\delta_i\delta_jn-\delta_{ij}\delta^2n.
 \label{c:core:adm-evolution}
```

\dot\rhoUse in source · 6 passages

Search aliases: dotρ dot rho

Link to this entry ·

Use in source · core.tex, line 323
Suppose a differentiable matrix trajectory obeys the von Neumann equation dotρ=-i[H,ρ], H=H^*. For its coherent transfer define J^(ρ)_b← a =2ImTr(P_bHP_aρ).
Exact mathematical passage
```
\label{f:von-neumann}
 \dot\rho=-i[H,\rho],\qquad H=H^*.
```

\dot\rho_{ij}Use in source · 1 passage

Search aliases: dotρ_ij dot rho

Link to this entry ·

Use in source · core.tex, line 438
In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable …
Exact mathematical passage
```
\label{f:coherence-decay}
 \dot\rho_{ij}=-\tfrac12(r_i+r_j)\rho_{ij}.
```

\dot\rho_jUse in source · 3 passages

Search aliases: dotρ_j dot rho

Link to this entry ·

Use in source · core.tex, line 233
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-balance}
 \dot\rho_j+\frac{J_j-J_{j-1}}h=0,
 \qquad\dot{\bar\rho}=0,
 \qquad\dot{\mathcal G}_j=0.
```

\dot\PhiUse in source · 2 passages

Search aliases: dotΦ dot Phi

Link to this entry ·

Use in source · strong_fields.tex, line 388
E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.
Exact mathematical passage
```
 \dot\varphi&=\sigma(\Pi+v\Phi),\\
 \dot\Phi&=\partial_r[\sigma(\Pi+v\Phi)],\\
 \dot\Pi&=r^{-2}\partial_r[r^2\sigma(\Phi+v\Pi)].
\label{c:strong:scalar-evolution}
```

\dot\phiUse in source · 2 passages

Search aliases: dotϕ dot phi

Link to this entry ·

Use in source · core.tex, line 260
[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.
Exact mathematical passage
```
\label{f:effective-generator}
 \dot\phi=K_{\mathrm{eff}}\phi,\qquad
 K_{\mathrm{eff}}=U^*KU-\Gamma,\qquad
 \Gamma^*=-\Gamma,\quad K_{\mathrm{eff}}^*=-K_{\mathrm{eff}}.
```

\dot\varphiUse in source · 11 passages

Search aliases: dotφ dot varphi

Link to this entry ·

Use in source · core.tex, line 31
w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring…
Exact mathematical passage
```
 L_h=\frac h2\sum_jw_j\dot\varphi_j^2
 -\frac1{2h}\sum_jc_{j+1/2}\Delta_j^2
 -\frac h2\sum_jv_j\varphi_j^2.
 \label{c:core:scalar-action}
```

\dot\varphi_jUse in source · 7 passages

Search aliases: dotφ_j dot varphi

Link to this entry ·

Use in source · core.tex, line 38
equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms in the action produce the two terms in the spatial numerator. The time integration by parts differentiates w_jdotφ_j, including the dot w_jdotφ_j term.
Exact mathematical passage
```
 w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
 =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
 -v_j\varphi_j.
 \label{c:core:scalar-evolution}
```

\dot\varphi_{j+1}Use in source · 3 passages

Search aliases: dotφ_j+1 dot varphi

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

\dot\varphi_j^2Use in source · 4 passages

Search aliases: dotφ_j^2 dot varphi

Link to this entry ·

Use in source · core.tex, line 31
w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring…
Exact mathematical passage
```
 L_h=\frac h2\sum_jw_j\dot\varphi_j^2
 -\frac1{2h}\sum_jc_{j+1/2}\Delta_j^2
 -\frac h2\sum_jv_j\varphi_j^2.
 \label{c:core:scalar-action}
```

\dot\chiUse in source · 2 passages

Search aliases: dotχ dot chi

Link to this entry ·

Use in source · core.tex, line 284
This is the residual symmetry of the stated temporal-gauge model. If χ_j depends on time, the matter time derivative acquires idotχ_jψ_j, while dot A_j acquires (dotχ_j+1-dotχ_j)/h. With E_j fixed and no scalar potential in [source reference], these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete linked walk uses its temporal link T; it must not be transferred to this action after that degree of freedom has been removed.
Exact mathematical passage
```
i\dot\chi_j\psi_j
```

\dot\chi_jUse in source · 2 passages

Search aliases: dotχ_j dot chi

Link to this entry ·

Use in source · core.tex, line 284
This is the residual symmetry of the stated temporal-gauge model. If χ_j depends on time, the matter time derivative acquires idotχ_jψ_j, while dot A_j acquires (dotχ_j+1-dotχ_j)/h. With E_j fixed and no scalar potential in [source reference], these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete linked walk uses its temporal link T; it must not be transferred to this action after that degree of freedom has been removed.
Exact mathematical passage
```
i\dot\chi_j\psi_j
```

\dot\chi_{j+1}Use in source · 1 passage

Search aliases: dotχ_j+1 dot chi

Link to this entry ·

Use in source · core.tex, line 284
This is the residual symmetry of the stated temporal-gauge model. If χ_j depends on time, the matter time derivative acquires idotχ_jψ_j, while dot A_j acquires (dotχ_j+1-dotχ_j)/h. With E_j fixed and no scalar potential in [source reference], these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete linked walk uses its temporal link T; it must not be transferred to this action after that degree of freedom has been removed.
Exact mathematical passage
```
(\dot\chi_{j+1}-\dot\chi_j)/h
```

\dot\psi_jUse in source · 7 passages

Search aliases: dotψ_j dot psi

Link to this entry ·

Use in source · core.tex, line 205
βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out…
Exact mathematical passage
```
\label{g:action}
 \mathcal S_h=\int\left[
 \frac{ih}{2}\sum_j(\psi_j^*\dot\psi_j-\dot\psi_j^*\psi_j)
 +h\sum_jE_j\dot A_j-H_h\right]dt.
```

\dot\psi_j^*Use in source · 1 passage

Search aliases: dotψ_j^* dot psi

Link to this entry ·

Use in source · core.tex, line 205
βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out…
Exact mathematical passage
```
\label{g:action}
 \mathcal S_h=\int\left[
 \frac{ih}{2}\sum_j(\psi_j^*\dot\psi_j-\dot\psi_j^*\psi_j)
 +h\sum_jE_j\dot A_j-H_h\right]dt.
```

\dot\psi_rUse in source · 1 passage

Search aliases: dotψ_r dot psi

Link to this entry ·

Use in source · spectral.tex, line 64
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. endpr
Exact mathematical passage
```
i\dot\psi_r=\sum_sH_{rs}\psi_s
```

\dot\OmegaUse in source · 1 passage

Search aliases: dotΩ dot Omega

Link to this entry ·

Use in source · constraints.tex, line 428
lit On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 R=\mathrm dt\wedge(\dot\Omega-D_\Omega\omega_\perp)+\mathcal R,
 \qquad
 F=\mathrm dt\wedge(\dot A-\mathrm d_\Sigma a_\perp)+\mathcal F.
```

\downarrowUse in source · 3 passages

Search aliases: downarrow

Link to this entry ·

Use in source · core.tex, line 230
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
\lim_{r\downarrow0}r^2\Phi'(r)=0
```

\mathbf EUse in source · 2 passages

Search aliases: E mathbf

Link to this entry ·

Use in source · spectral.tex, line 967
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
\mathbf E_L=(E_{L,0},\ldots,E_{L,L-1})^{\mathsf T}
```

\mathcal EUse in source · 28 passages

Search aliases: E mathcal

Link to this entry ·

Use in source · finite_core.tex, line 1106
N(a)=∑_n∈ S a_n _2^2, E(a)=∑_n∈ S⟨ a_n,H_na_n⟩,
Exact mathematical passage
```
N(a)=\sum_{n\in S}\|a_n\|_2^2,\qquad
\mathcal E(a)=\sum_{n\in S}\langle a_n,H_na_n\rangle,
```

E'_{L,0}Use in source · 1 passage

Search aliases: E'_L,0

Link to this entry ·

Use in source · spectral.tex, line 966
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
E'_{L,0}=E_{L,L-1}
```

E'_{L,j}Use in source · 1 passage

Search aliases: E'_L,j

Link to this entry ·

Use in source · spectral.tex, line 964
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
E'_{L,j}=E_{L,j-1}
```

e^{(L_g+S_1/2)t}Use in source · 1 passage

Search aliases: e^(L_g+S_1/2)t

Link to this entry ·

Use in source · strong_fields.tex, line 268
djoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R(0)_h, normDb_∞≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonli…
Exact mathematical passage
```
 \norm{R(t)-\widetilde R(t)}_h
 \le e^{(L_g+S_1/2)t}\norm{R(0)-\widetilde R(0)}_h,
 \qquad \norm{Db}_\infty\le S_1.
\label{c:strong:semidiscrete-bound}
```

e^{-(r-10)^2}Use in source · 1 passage

Search aliases: e^-(r-10)^2

Link to this entry ·

Use in source · strong_fields.tex, line 502
The initial scalar packet is φ(0,r)=A e^-(r-10)^2, Φ(0,r)=∂_rφ(0,r), Π(0,r)=Φ(0,r)+fracφ(0,r)r. The last equation is the flat-space ingoing condition for χ=rφ, used to initialize the scalar. The nonlinear gravitational constraints are then solved. It is therefore an ingoing packet prescription, rather than exactly freezing the geometry to Minkowski space.
Exact mathematical passage
```
 \varphi(0,r)=A e^{-(r-10)^2},\qquad
 \Phi(0,r)=\partial_r\varphi(0,r),\qquad
 \Pi(0,r)=\Phi(0,r)+\frac{\varphi(0,r)}r.
```

e^{-(r_i+r_j)t/2}Use in source · 1 passage

Search aliases: e^-(r_i+r_j)t/2

Link to this entry ·

Use in source · core.tex, line 441
on closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable ways.
Exact mathematical passage
```
\rho_{ij}(t)=e^{-(r_i+r_j)t/2}\rho_{ij}(0)
```

e^{-1}Use in source · 1 passage

Search aliases: e^-1

Link to this entry ·

Use in source · constraints.tex, line 667
There is also a spectral boundary. Even when e_a^i are trigonometric polynomials, e^-1, e^1/2, and the torsion-free connection are generally not finite Fourier polynomials. The unreduced degree rules 16K<N and 32K<N therefore cannot simply be reapplied to the reduced chart. One may evaluate its finite-dimensional regular functions directly, or retain auxiliary polynomial variables, but either choice needs its own approximation and support analysis.
Exact mathematical passage
```
e^{-1}
```

e^{-2kt}Use in source · 1 passage

Search aliases: e^-2kt

Link to this entry ·

Use in source · core.tex, line 445
[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding -i[H,ρ] to [source reference] adds the coherent current to the jump current. Th…
Exact mathematical passage
```
 \dot p_1=k(1-2p_1),\qquad
 p_1(t)=\tfrac12+\bigl(p_1(0)-\tfrac12\bigr)e^{-2kt},
 \qquad \rho_{12}(t)=e^{-kt}\rho_{12}(0).
```

e^{-2P}Use in source · 1 passage

Search aliases: e^-2P

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

e^{-2\beta\epsilon_m}Use in source · 1 passage

Search aliases: e^-2βϵ_m beta epsilon

Link to this entry ·

Use in source · limit.tex, line 507
For β>0, the multiplier e^-β H_ϵ is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in …
Exact mathematical passage
```
\{f:\sum_m e^{-2\beta\epsilon_m}\norm{\mathsf F_mf}_2^2<\infty\}
```

e^{-2\pi ijk/M}Use in source · 1 passage

Search aliases: e^-2π ijk/M pi

Link to this entry ·

Use in source · spectral.tex, line 95
Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1,
Exact mathematical passage
```
(F_M)_{jk}=M^{-1/2}e^{-2\pi ijk/M}
```

e^{-2\tau}Use in source · 1 passage

Search aliases: e^-2τ tau

Link to this entry ·

Use in source · core.tex, line 342
Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber
Exact mathematical passage
```
\gamma(\tau)=e^{-2\tau}
```

e^{-2\phi}Use in source · 2 passages

Search aliases: e^-2ϕ phi

Link to this entry ·

Use in source · core.tex, line 157
1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.
Exact mathematical passage
```
 \Omega=d\omega=(\phi_{tt}-\phi_{xx})K\,dt\wedge dx,
 \qquad R=2e^{-2\phi}(\phi_{tt}-\phi_{xx}).
 \label{c:core:conformal-curvature}
```

e^{-hA}Use in source · 1 passage

Search aliases: e^-hA

Link to this entry ·

Use in source · core.tex, line 417
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
\label{g:commutator-loop}
 W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.
```

e^{-hB}Use in source · 1 passage

Search aliases: e^-hB

Link to this entry ·

Use in source · core.tex, line 417
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
\label{g:commutator-loop}
 W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.
```

e^{-i(s+t)E}Use in source · 1 passage

Search aliases: e^-i(s+t)E

Link to this entry ·

Use in source · finite_core.tex, line 387
U_H(s)U_H(t) =e^-i(s+t)EP_++e^i(s+t)EP_-=U_H(s+t).
Exact mathematical passage
```
U_H(s)U_H(t)
=e^{-i(s+t)E}P_++e^{i(s+t)E}P_-=U_H(s+t).
```

e^{-i(t-s)A}Use in source · 3 passages

Search aliases: e^-i(t-s)A

Link to this entry ·

Use in source · finite_core.tex, line 993
The same holds for A. Since A commutes with every power of itself, it commutes with e^-iuA. Fix t and define F(s)=e^-i(t-s)Ae^-isB. The product rule gives
Exact mathematical passage
```
F(s)=e^{-i(t-s)A}e^{-isB}
```

e^{-i(t-s)D}Use in source · 1 passage

Search aliases: e^-i(t-s)D

Link to this entry ·

Use in source · spectral.tex, line 288
K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D …
Exact mathematical passage
```
 v(t)=e^{-itD}v_0-i\int_0^te^{-i(t-s)D}B^*u(s)\,ds.
 \label{a-spectral-hidden-reconstruction}
```

e^{-i(t-s)H_\lambda}Use in source · 1 passage

Search aliases: e^-i(t-s)H_λ lambda

Link to this entry ·

Use in source · limit.tex, line 765
and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP e^-isH_0P. The two propagators and the exterior projections are contractions. Integrating the norm bound over an interval of length |t| gives the sec…
Exact mathematical passage
```
 Qe^{-itH_\lambda}P
 =-i\lambda\int_0^t
 Qe^{-i(t-s)H_\lambda}B e^{-isH_0}P\,ds.
```

e^{-igtZ_{\mathrm{sign}}}Use in source · 1 passage

Search aliases: e^-igtZ_sign mathrm

Link to this entry ·

Use in source · limit.tex, line 694
b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sign⊗ B, |Ψ(0)⟩=|+_x⟩⊗ frac|b_0⟩+|b_1⟩sqrt2. Evolution under this explicit interaction alone gives |Ψ(t)⟩=frac1sqrt2 (|+_x⟩⊗|b_0⟩ +e^-igtZ_sign|+_x⟩⊗|b_1⟩). Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors …
Exact mathematical passage
```
 |\Psi(t)\rangle=\frac1{\sqrt2}
 \left(|+_x\rangle\otimes|b_0\rangle
 +e^{-igtZ_{\mathrm{sign}}}|+_x\rangle\otimes|b_1\rangle\right).
 \label{a-limit-entangled-state}
```

e^{-ih^2}Use in source · 1 passage

Search aliases: e^-ih^2

Link to this entry ·

Use in source · core.tex, line 323
[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The formula U_j=e^-ihA_j by itself does not identify a second-order sampling scheme.
Exact mathematical passage
```
 \frac{\widehat U_h(0)-\widehat U_h(-h)^{-1}}{2h}
 =\frac{1-e^{-ih^2}}{2h}=\frac{i}{2}h+O(h^3).
```

e^{-ihA(x)}Use in source · 1 passage

Search aliases: e^-ihA(x)

Link to this entry ·

Use in source · core.tex, line 322
[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The formula U_j=e^-ihA_j by itself does not identify a second-order sampling scheme.
Exact mathematical passage
```
\widehat U_h(x)=e^{-ihA(x)}
```

e^{-ihA(x+h/2)}Use in source · 1 passage

Search aliases: e^-ihA(x+h/2)

Link to this entry ·

Use in source · core.tex, line 319
A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).
Exact mathematical passage
```
\widetilde U_h(x)=e^{-ihA(x+h/2)}
```

e^{-ihA_j}Use in source · 2 passages

Search aliases: e^-ihA_j

Link to this entry ·

Use in source · core.tex, line 196
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,
Exact mathematical passage
```
U_j=e^{-ihA_j}
```

e^{-ihp_j(n)}Use in source · 1 passage

Search aliases: e^-ihp_j(n)

Link to this entry ·

Use in source · finite_core.tex, line 248
D_h,je_n =e_nfrace^ihp_j(n)-e^-ihp_j(n)2h =ifracsin(hp_j(n))he_n.
Exact mathematical passage
```
D_{h,j}e_n
=e_n\frac{e^{ihp_j(n)}-e^{-ihp_j(n)}}{2h}
=i\frac{\sin(hp_j(n))}{h}e_n.
```

e^{-\operatorname{Im}z\,\tau}Use in source · 1 passage

Search aliases: e^-Imz τ operatorname tau

Link to this entry ·

Use in source · spectral.tex, line 305
itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D is self-adjoint, normK(τ)≤normB^2. The factor e^-Imz τ makes the transform absolutely convergent. On a spectral eigenvalue of D, ∫_0^∞ e^i(z-λ)τ dτ=i/(z-λ). The finite spectral sum then proves [source re…
Exact mathematical passage
```
e^{-\operatorname{Im}z\,\tau}
```

e^{-isB}Use in source · 5 passages

Search aliases: e^-isB

Link to this entry ·

Use in source · finite_core.tex, line 977
Proof. First justify differentiation. On |s|≤ R, the series for e^-isB converges uniformly, and its derivative series is norm-dominated by
Exact mathematical passage
```
e^{-isB}
```

e^{-isH_0}Use in source · 2 passages

Search aliases: e^-isH_0

Link to this entry ·

Use in source · limit.tex, line 765
and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP e^-isH_0P. The two propagators and the exterior projections are contractions. Integrating the norm bound over an interval of length |t| gives the sec…
Exact mathematical passage
```
 Qe^{-itH_\lambda}P
 =-i\lambda\int_0^t
 Qe^{-i(t-s)H_\lambda}B e^{-isH_0}P\,ds.
```

e^{-it}Use in source · 2 passages

Search aliases: e^-it

Link to this entry ·

Use in source · finite_core.tex, line 873
In particular, u_0(t)=e^-ite_0. For the centered axis and diagonal modes,
Exact mathematical passage
```
u_0(t)=e^{-it}e_0
```

e^{-it\mathcal A H\mathcal A^*}Use in source · 1 passage

Search aliases: e^-it A H A^* mathcal mathcal

Link to this entry ·

Use in source · spectral.tex, line 45
For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
 e^{-it\mathcal A H\mathcal A^*}
   =\mathcal A e^{-itH}\mathcal A^*.
```

e^{-it_0\epsilon_m}Use in source · 1 passage

Search aliases: e^-it_0ϵ_m epsilon

Link to this entry ·

Use in source · limit.tex, line 326
The modulus-one multipliers give U(t)^*U(t)=I and U(t+s)=U(t)U(s). They also give strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint operator is the generator. A general L^2 state evolves unitarily even i…
Exact mathematical passage
```
 \norm{(U(t)-U(t_0))f}_2^2
 =\sum_m|e^{-it\epsilon_m}-e^{-it_0\epsilon_m}|^2
 \norm{\mathsf F_mf}_2^2\longrightarrow0.
```

e^{-itA}Use in source · 3 passages

Search aliases: e^-itA

Link to this entry ·

Use in source · finite_core.tex, line 970
e^-itA-e^-itB ≤ |t| A-B .
Exact mathematical passage
```
\|e^{-itA}-e^{-itB}\|\le |t|\,\|A-B\|.
```

e^{-itB}Use in source · 3 passages

Search aliases: e^-itB

Link to this entry ·

Use in source · finite_core.tex, line 970
e^-itA-e^-itB ≤ |t| A-B .
Exact mathematical passage
```
\|e^{-itA}-e^{-itB}\|\le |t|\,\|A-B\|.
```

e^{-itD}Use in source · 3 passages

Search aliases: e^-itD

Link to this entry ·

Use in source · spectral.tex, line 281
[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution i…
Exact mathematical passage
```
 i\dot u(t)&=Au(t)+B e^{-itD}v_0
              -i\int_0^tK(t-s)u(s)\,ds,\nonumber\\
 K(\tau)&=B e^{-i\tau D}B^*,\qquad \tau\geq0,
 \label{a-spectral-memory-equation}
```

e^{-itE}Use in source · 2 passages

Search aliases: e^-itE

Link to this entry ·

Use in source · finite_core.tex, line 320
U_H(t)=e^-itEP_++e^itEP_- =cos(tE)I_4-ifracsin(tE)EH.
Exact mathematical passage
```
U_H(t)=e^{-itE}P_++e^{itE}P_-
=\cos(tE)I_4-i\frac{\sin(tE)}{E}H.
```

e^{-itH}Use in source · 2 passages

Search aliases: e^-itH

Link to this entry ·

Use in source · spectral.tex, line 45
For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
 e^{-it\mathcal A H\mathcal A^*}
   =\mathcal A e^{-itH}\mathcal A^*.
```

e^{-itH_0}Use in source · 1 passage

Search aliases: e^-itH_0

Link to this entry ·

Use in source · limit.tex, line 764
quation where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP e^-isH_0P. The two propagators and the exterior projections are contractions. Integrating the norm bound …
Exact mathematical passage
```
Qe^{-itH_0}P=0
```

e^{-itH_A}Use in source · 1 passage

Search aliases: e^-itH_A

Link to this entry ·

Use in source · limit.tex, line 619
ge reduces H_A, and its finite restriction H_A^( h) satisfies exact embedding intertwining with every larger rectangular restriction. For bounded Borel F, F(H_A)f=∑_ rF(A_ r) F_ rf, F(H_A) E_ h→ F(H_A) strongly. In particular, norme^-itH_A(I- E_ h)f =norm(I- E_ h)f. Finite rectangular sums belong to the domain and are dense. Symmetry holds by symmetry on every block. If g belongs to the adjoint domain and k=H_A^*g, testing against an arbitrary vector in KK_ r gives F_ rk=A_ r F_ rg. Square summa…
Exact mathematical passage
```
 \norm{e^{-itH_A}(I-\mathsf E_{\mathbf h})f}
 =\norm{(I-\mathsf E_{\mathbf h})f}.
 \label{a-limit-block-unitary-error}
```

e^{-\mathrm{i}tH_c}Use in source · 1 passage

Search aliases: e^-itH_c mathrm

Link to this entry ·

Use in source · finite.tex, line 561
[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.
Exact mathematical passage
```
 e^{-\mathrm{i}tH_f}V&=Ve^{-\mathrm{i}tH_c},\nonumber\\
 e^{-\beta H_f}V&=Ve^{-\beta H_c},\nonumber\\
 (z-H_f)^{-1}V&=V(z-H_c)^{-1}.
 \label{a-finite-evolution-intertwining}
```

e^{-\mathrm{i}tH_f}Use in source · 2 passages

Search aliases: e^-itH_f mathrm

Link to this entry ·

Use in source · finite.tex, line 561
[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.
Exact mathematical passage
```
 e^{-\mathrm{i}tH_f}V&=Ve^{-\mathrm{i}tH_c},\nonumber\\
 e^{-\beta H_f}V&=Ve^{-\beta H_c},\nonumber\\
 (z-H_f)^{-1}V&=V(z-H_c)^{-1}.
 \label{a-finite-evolution-intertwining}
```

e^{-itH_m^{\mathrm{phys}}}Use in source · 1 passage

Search aliases: e^-itH_m^phys mathrm

Link to this entry ·

Use in source · boundary.tex, line 24
Connecting the group to its generator. The next chapter proves that the constructed group is the spectral evolution of this Hamiltonian: U(t)=e^-itH_m^phys. The proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal generator has exactly that domain. The equality is therefore a conclusion about the complete Hilbert-space group, not an inference from agreement on finitel…
Exact mathematical passage
```
\mathcal U(t)=e^{-itH_m^{\mathrm{phys}}}.
```

e^{-\mathrm{i}tH_{p,h}^{\mathrm{Haar}}}Use in source · 1 passage

Search aliases: e^-itH_p,h^Haar mathrm mathrm

Link to this entry ·

Use in source · finite.tex, line 638
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber
Exact mathematical passage
```
 U_{p,h}(t)=e^{-\mathrm{i}tH_{p,h}^{\mathrm{Haar}}}
 =\sum_{r=0}^he^{-\mathrm{i}t\epsilon_{p,r}}D_r^{(h)}.
 \label{a-finite-shell-propagator}
```

e^{-itH_\epsilon}Use in source · 1 passage

Search aliases: e^-itH_ϵ epsilon

Link to this entry ·

Use in source · limit.tex, line 312
For bounded Borel F:R→C, define F(H_ϵ)f=∑_m≥0F(ϵ_m) F_mf. The series converges by Parseval and normF(H_ϵ)f_2≤sup_m|F(ϵ_m)|norm f_2. The same level decomposition verifies sums, products, and adjoints of these multipliers directly. In particular, U(t) · =e^-itH_ϵ =∑_m≥0e^-itϵ_m F_m, (z-H_ϵ)^-1 · =∑_m≥0(z-ϵ_m)^-1 F_m, z∉σ(H_ϵ). The projections in these sums converge strongly, not absolutely in operator norm in general.
Exact mathematical passage
```
 U(t)&=e^{-itH_\epsilon}
 =\sum_{m\geq0}e^{-it\epsilon_m}\mathsf F_m,
 \label{a-limit-unitary}\\
 (z-H_\epsilon)^{-1}
 &=\sum_{m\geq0}(z-\epsilon_m)^{-1}\mathsf F_m,
 \quad z\notin\sigma(H_\epsilon).
 \label{a-limit-resolvent}
```

e^{-itH_\lambda}Use in source · 3 passages

Search aliases: e^-itH_λ lambda

Link to this entry ·

Use in source · limit.tex, line 740
[Defect, leakage, and a second-order self-energy] Under these hypotheses, normH_λ V-VH_c · =|λ|normBV, normQe^-itH_λP · ≤min{1,|tλ|normQBP}. For z∉σ(QH_λ Q|_QHH), the coarse self-energy is Σ_P(z)=PH_λ Q (z-QH_λ Q|_QHH)^-1QH_λ P, where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-…
Exact mathematical passage
```
 \norm{H_\lambda V-VH_c}&=|\lambda|\norm{BV},
 \label{a-limit-defect-norm}\\
 \norm{Qe^{-itH_\lambda}P}
 &\leq\min\{1,|t\lambda|\norm{QBP}\}.
 \label{a-limit-duhamel-bound}
```

e^{-itH_\infty}Use in source · 3 passages

Search aliases: e^-itH_∞ infty

Link to this entry ·

Use in source · spectral.tex, line 1112
E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say that each lane acts on every vector.
Exact mathematical passage
```
 \sum_jf_j(H_\infty)\psi=e^{-itH_\infty}\psi,
 \qquad \psi\in\bigcap_j\Dom(f_j(H_\infty)).
 \label{a-spectral-unbounded-reconstruction}
```

e^{-it\widetilde H_M}Use in source · 1 passage

Search aliases: e^-itwidetilde H_M widetilde

Link to this entry ·

Use in source · limit.tex, line 394
An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber
Exact mathematical passage
```
 e^{-it\widetilde H_M}
 =U_M^{\mathrm{emb}}(t)+(I-\mathsf E_M),
 \qquad
 \norm{(e^{-it\widetilde H_M}-U(t))f}_2
 \leq2\norm{(I-\mathsf E_M)f}_2.
 \label{a-limit-zero-hamiltonian-extension}
```

e^{-it\epsilon}Use in source · 1 passage

Search aliases: e^-itϵ epsilon

Link to this entry ·

Use in source · limit.tex, line 334
U(t)^*U(t)=I and U(t+s)=U(t)U(s). They also give strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint operator is the generator. A general L^2 state evolves unitarily even if this derivative does not exist.
Exact mathematical passage
```
|(e^{-it\epsilon}-1)/t|\leq|\epsilon|
```

e^{-it\epsilon_m}Use in source · 4 passages

Search aliases: e^-itϵ_m epsilon

Link to this entry ·

Use in source · limit.tex, line 312
For bounded Borel F:R→C, define F(H_ϵ)f=∑_m≥0F(ϵ_m) F_mf. The series converges by Parseval and normF(H_ϵ)f_2≤sup_m|F(ϵ_m)|norm f_2. The same level decomposition verifies sums, products, and adjoints of these multipliers directly. In particular, U(t) · =e^-itH_ϵ =∑_m≥0e^-itϵ_m F_m, (z-H_ϵ)^-1 · =∑_m≥0(z-ϵ_m)^-1 F_m, z∉σ(H_ϵ). The projections in these sums converge strongly, not absolutely in operator norm in general.
Exact mathematical passage
```
 U(t)&=e^{-itH_\epsilon}
 =\sum_{m\geq0}e^{-it\epsilon_m}\mathsf F_m,
 \label{a-limit-unitary}\\
 (z-H_\epsilon)^{-1}
 &=\sum_{m\geq0}(z-\epsilon_m)^{-1}\mathsf F_m,
 \quad z\notin\sigma(H_\epsilon).
 \label{a-limit-resolvent}
```

e^{-\mathrm{i}t\epsilon_{p,0}}Use in source · 1 passage

Search aliases: e^-itϵ_p,0 mathrm epsilon

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

e^{-\mathrm{i}t\epsilon_{p,r}}Use in source · 2 passages

Search aliases: e^-itϵ_p,r mathrm epsilon

Link to this entry ·

Use in source · finite.tex, line 638
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber
Exact mathematical passage
```
 U_{p,h}(t)=e^{-\mathrm{i}tH_{p,h}^{\mathrm{Haar}}}
 =\sum_{r=0}^he^{-\mathrm{i}t\epsilon_{p,r}}D_r^{(h)}.
 \label{a-finite-shell-propagator}
```

e^{-it\lambda}Use in source · 2 passages

Search aliases: e^-itλ lambda

Link to this entry ·

Use in source · spectral.tex, line 1111
Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say that each lane acts on every vector.
Exact mathematical passage
```
\sum_jf_j(\lambda)=e^{-it\lambda}
```

e^{-it\lambda_n}Use in source · 3 passages

Search aliases: e^-itλ_n lambda

Link to this entry ·

Use in source · extensions.tex, line 243
For real (λ_n) define (U(t)c)_n=e^-itλ_nc_n, c∈ℓ^2. The modulus of every multiplier is one. Hence U(t) is a unitary operator, U(t+s)=U(t)U(s), and U(-t)=U(t)^*. For each fixed state it is strongly continuous. To prove the latter claim, split c into a finite head and a small ℓ^2 tail. The finite head is continuous coordinatewise, while the norm of the difference on the tail is at most twice its initial norm.
Exact mathematical passage
```
 (U(t)c)_n=e^{-it\lambda_n}c_n,\qquad c\in\ell^2.
 \label{eq:foundation-spectral-flow}
```

e^{-iu}Use in source · 1 passage

Search aliases: e^-iu

Link to this entry ·

Use in source · extensions.tex, line 257
ability domain] The orbit t↦ U(t)c is differentiable in ℓ^2 at one, equivalently every, time precisely when c∈ D(H_λ). On that domain, fracddtU(t)c=-iH_λ U(t)c, H_λ U(t)c=U(t)H_λ c. If c∈ D(H_λ), each coordinate difference quotient tends to -iλ_ne^-itλ_nc_n. The elementary bound |e^-iu-1|≤|u| bounds the quotient magnitude by |λ_nc_n|. The squared difference from its limit is bounded by 4λ_n^2|c_n|^2, a summable sequence. Dominated convergence therefore gives convergence in ℓ^2.
Exact mathematical passage
```
|e^{-iu}-1|\leq|u|
```

e^{-iuA}Use in source · 3 passages

Search aliases: e^-iuA

Link to this entry ·

Use in source · finite_core.tex, line 992
The same holds for A. Since A commutes with every power of itself, it commutes with e^-iuA. Fix t and define F(s)=e^-i(t-s)Ae^-isB. The product rule gives
Exact mathematical passage
```
e^{-iuA}
```

e^{-iuK}Use in source · 1 passage

Search aliases: e^-iuK

Link to this entry ·

Use in source · limit.tex, line 793
The small-time leading term is -itλ QBP. More precisely, normQe^-itH_λP+itλ QBP ≤frac|λ|t^22 (normH_λ+normH_0)normQBP. To verify this, subtract the constant integrand QBP in Duhamel's formula and use norme^-iuK-I≤|u|norm K for a finite self-adjoint K. The two deviations are bounded by (|t-s|normH_λ+|s|normH_0)normQBP on the oriented integration interval. Their integrals are both t^2/2. For |λ|≤1, this is a uniform O(t^2|λ|) remainder. A generic O(t^2λ^2) remainder would omit the first-order inte…
Exact mathematical passage
```
\norm{e^{-iuK}-I}\leq|u|\norm K
```

e^{-i\alpha_q}Use in source · 1 passage

Search aliases: e^-iα_q alpha

Link to this entry ·

Use in source · spectral.tex, line 130
pa cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine Fourier index is uniquely k=j+qM. Its angle satisfies 2π k/(pM)=…
Exact mathematical passage
```
P_p(\theta)v_q=e^{-i\alpha_q}v_q
```

e^{-i\theta}Use in source · 3 passages

Search aliases: e^-iθ theta

Link to this entry ·

Use in source · spectral.tex, line 102
M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1, e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
 P_p(\theta)e_r=
 \begin{cases}
 e_{r+1},&r<p-1,\\
 e^{-i\theta}e_0,&r=p-1.
 \end{cases}
 \qquad
 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
 \label{a-spectral-bloch-symbol}
```

e^{-i\theta_j}Use in source · 1 passage

Search aliases: e^-iθ_j theta

Link to this entry ·

Use in source · spectral.tex, line 101
rier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1,
Exact mathematical passage
```
e^{-i\theta_j}
```

e^{-i\tau D}Use in source · 1 passage

Search aliases: e^-iτ D tau

Link to this entry ·

Use in source · spectral.tex, line 281
[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution i…
Exact mathematical passage
```
 i\dot u(t)&=Au(t)+B e^{-itD}v_0
              -i\int_0^tK(t-s)u(s)\,ds,\nonumber\\
 K(\tau)&=B e^{-i\tau D}B^*,\qquad \tau\geq0,
 \label{a-spectral-memory-equation}
```

e^{-i\tau J_{1:h}}Use in source · 1 passage

Search aliases: e^-iτ J_1:h tau

Link to this entry ·

Use in source · spectral.tex, line 509
The scalar causal kernel is K(τ)=γ_0^2⟨1,e^-iτ J_1:h1⟩, multiplying I_2. It uses the same radial coefficients as the rational self-energy. Its exact causal relation is [source reference]. adicfigure08_radial_self_energyA branch-normalized finite tree has the same root response as its radial Jacobi chain. The complex-energy resolvent avoids real poles. The oscillatory memory is the return amplitude of a finite hidden chain.
Exact mathematical passage
```
K(\tau)=\gamma_0^2\langle1,e^{-i\tau J_{1:h}}1\rangle
```

e^{-kt}Use in source · 1 passage

Search aliases: e^-kt

Link to this entry ·

Use in source · core.tex, line 445
[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding -i[H,ρ] to [source reference] adds the coherent current to the jump current. Th…
Exact mathematical passage
```
 \dot p_1=k(1-2p_1),\qquad
 p_1(t)=\tfrac12+\bigl(p_1(0)-\tfrac12\bigr)e^{-2kt},
 \qquad \rho_{12}(t)=e^{-kt}\rho_{12}(0).
```

e^{-r^2/(2\sigma^2)}Use in source · 2 passages

Search aliases: e^-r^2/(2σ^2) sigma

Link to this entry ·

Use in source · core.tex, line 283
For the normalized Gaussian density ρ(r)=fracM_0(2π)^3/2σ^3e^-r^2/(2σ^2), σ>0, direct differentiation verifies M_rm enc(r) · =M_0[ erf(frac rsqrt2σ) -sqrtfrac2πfrac rσ e^-r^2/(2σ^2)],
Exact mathematical passage
```
 \rho(r)=\frac{M_0}{(2\pi)^{3/2}\sigma^3}e^{-r^2/(2\sigma^2)},
 \qquad \sigma>0,
```

e^{-S/\hbar}Use in source · 1 passage

Search aliases: e^-S/ℏ hbar

Link to this entry ·

Use in source · constraints.tex, line 805
Eliminating z first gives the effective coefficient μ for y^2 and the factor (m+n)^-1/2. Eliminating y next gives d^-1/2. Their product is D^-1/2, exactly as in direct elimination. For the Euclidean weight e^-S/ℏ, using a flat coarse measure instead records this determinant as S_C,ℏ=S_C+tfracℏ2log D, up to an additive constant. A nonlinear symmetry that preserves the classical action can still require a nonconstant induced measure.
Exact mathematical passage
```
e^{-S/\hbar}
```

e^{-t(N+1)}Use in source · 1 passage

Search aliases: e^-t(N+1)

Link to this entry ·

Use in source · extensions.tex, line 367
For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.
Exact mathematical passage
```
e^{-t(N+1)}
```

e^{-tn}Use in source · 1 passage

Search aliases: e^-tn

Link to this entry ·

Use in source · extensions.tex, line 367
For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.
Exact mathematical passage
```
e^{-tn}
```

e^{-|\alpha|^2/2}Use in source · 1 passage

Search aliases: e^-|α|^2/2 alpha

Link to this entry ·

Use in source · spectral.tex, line 1201
For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber
Exact mathematical passage
```
 |\alpha\rangle=e^{-|\alpha|^2/2}
        \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}|n\rangle,
 \qquad \mu=|\alpha|^2\geq0,
```

e^{-\beta H}Use in source · 1 passage

Search aliases: e^-β H beta

Link to this entry ·

Use in source · finite.tex, line 577
The Euclidean operator e^-β H in this corollary is a spectral multiplier. It is not generally unitary, and a normalized state after Euclidean evolution requires its own normalization. The identity does not compare normalized Gibbs states of the entire fine and coarse systems, since the fine partition function also counts the detail block. For a compatible finite tower, the coarse propagator and its embedded fine propagator agree exactly. This does not make an arbitrary interacting fine Hamiltoni…
Exact mathematical passage
```
e^{-\beta H}
```

e^{-\beta H_c}Use in source · 1 passage

Search aliases: e^-β H_c beta

Link to this entry ·

Use in source · finite.tex, line 561
[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.
Exact mathematical passage
```
 e^{-\mathrm{i}tH_f}V&=Ve^{-\mathrm{i}tH_c},\nonumber\\
 e^{-\beta H_f}V&=Ve^{-\beta H_c},\nonumber\\
 (z-H_f)^{-1}V&=V(z-H_c)^{-1}.
 \label{a-finite-evolution-intertwining}
```

e^{-\beta H_f}Use in source · 1 passage

Search aliases: e^-β H_f beta

Link to this entry ·

Use in source · finite.tex, line 561
[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.
Exact mathematical passage
```
 e^{-\mathrm{i}tH_f}V&=Ve^{-\mathrm{i}tH_c},\nonumber\\
 e^{-\beta H_f}V&=Ve^{-\beta H_c},\nonumber\\
 (z-H_f)^{-1}V&=V(z-H_c)^{-1}.
 \label{a-finite-evolution-intertwining}
```

e^{-\beta H_\epsilon}Use in source · 2 passages

Search aliases: e^-β H_ϵ beta epsilon

Link to this entry ·

Use in source · limit.tex, line 504
For β>0, the multiplier e^-β H_ϵ is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in …
Exact mathematical passage
```
e^{-\beta H_\epsilon}
```

e^{-\beta\epsilon_0}Use in source · 1 passage

Search aliases: e^-βϵ_0 beta epsilon

Link to this entry ·

Use in source · limit.tex, line 511
silon is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It…
Exact mathematical passage
```
 \operatorname{Tr}(e^{-\beta H_\epsilon})
 =e^{-\beta\epsilon_0}
 +\sum_{m\geq1}(p-1)p^{m-1}e^{-\beta\epsilon_m},
 \label{a-limit-thermal-trace}
```

e^{-\beta\epsilon_m}Use in source · 1 passage

Search aliases: e^-βϵ_m beta epsilon

Link to this entry ·

Use in source · limit.tex, line 511
silon is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It…
Exact mathematical passage
```
 \operatorname{Tr}(e^{-\beta H_\epsilon})
 =e^{-\beta\epsilon_0}
 +\sum_{m\geq1}(p-1)p^{m-1}e^{-\beta\epsilon_m},
 \label{a-limit-thermal-trace}
```

e^{-\mu}Use in source · 2 passages

Search aliases: e^-μ mu

Link to this entry ·

Use in source · spectral.tex, line 1207
For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series proves ∑_jp_L,j=1. Their equivalent hypergeometric form is e^-μ times [source reference] at z=μ.
Exact mathematical passage
```
 p_{L,j}(\mu)
 &=\langle\alpha,Q_j^{(L)}\alpha\rangle
   =e^{-\mu}E_{L,j}(\mu)\nonumber\\
 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
                      e^{\mu(\omega_L^\ell-1)}.
 \label{a-spectral-coherent-packets}
```

e^{-\mu(1-\cos(2\pi/L))}Use in source · 1 passage

Search aliases: e^-μ(1-cos(2π/L)) mu cos pi

Link to this entry ·

Use in source · spectral.tex, line 1223
[Approach to uniform residue probability] For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L), giving the second.
Exact mathematical passage
```
 \left|p_{L,j}(\mu)-\frac1L\right|
 \leq\frac1L\sum_{\ell=1}^{L-1}
       e^{-\mu(1-\cos(2\pi\ell/L))}
 \leq\frac{L-1}{L}
       e^{-\mu(1-\cos(2\pi/L))}.
 \label{a-spectral-packet-mixing-bound}
```

e^{-\mu(1-\cos(2\pi\ell/L))}Use in source · 1 passage

Search aliases: e^-μ(1-cos(2πℓ/L)) mu cos pi ell

Link to this entry ·

Use in source · spectral.tex, line 1223
[Approach to uniform residue probability] For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L), giving the second.
Exact mathematical passage
```
 \left|p_{L,j}(\mu)-\frac1L\right|
 \leq\frac1L\sum_{\ell=1}^{L-1}
       e^{-\mu(1-\cos(2\pi\ell/L))}
 \leq\frac{L-1}{L}
       e^{-\mu(1-\cos(2\pi/L))}.
 \label{a-spectral-packet-mixing-bound}
```

e^{-\tau}Use in source · 3 passages

Search aliases: e^-τ tau

Link to this entry ·

Use in source · constraints.tex, line 273
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
t=e^{-\tau}>0
```

e^{-\chi}Use in source · 1 passage

Search aliases: e^-χ chi

Link to this entry ·

Use in source · core.tex, line 185
If the integrated coefficient is χ, then e^χ K=coshχ I+sinhχ K. This preserves the Lorentz form diag(-1,1) and has positive eigenvalues e^χ and e^-χ. Its real principal matrix logarithm is χ K. These are Lorentz boosts. The positive Hilbert norm used for unitary color transport in the gauge monograph is a different structure.
Exact mathematical passage
```
e^{-\chi}
```

E^0Use in source · 1 passage

Search aliases: E^0

Link to this entry ·

Use in source · constraints.tex, line 543
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 E^0=0,\qquad E^i=e_a{}^i\,\mathrm dx^a,
 \qquad e:=\det(e_a{}^i)>0.
 \label{c:constraints:regular-triad}
```

e^0Use in source · 2 passages

Search aliases: e^0

Link to this entry ·

Use in source · constraints.tex, line 335
n of φ gives the unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitut…
Exact mathematical passage
```
F=E e^0\wedge e^1
```

e^1Use in source · 2 passages

Search aliases: e^1

Link to this entry ·

Use in source · constraints.tex, line 335
n of φ gives the unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitut…
Exact mathematical passage
```
F=E e^0\wedge e^1
```

e^{1/2}Use in source · 4 passages

Search aliases: e^1/2

Link to this entry ·

Use in source · constraints.tex, line 591
al becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
\Psi=e^{1/2}\psi
```

e^2Use in source · 1 passage

Search aliases: e^2

Link to this entry ·

Use in source · constraints.tex, line 336
unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitution repairs the H…
Exact mathematical passage
```
\star F=-E e^2\wedge e^3
```

e^{2b\,\operatorname{area}(\square)K}Use in source · 1 passage

Search aliases: e^2b area(square)K operatorname square

Link to this entry ·

Use in source · core.tex, line 192
[Constant curvature density] For ϕ(t,x)=b(t^2-x^2)/2, one has ϕ_tt-ϕ_xx=2b, R=4b e^-2ϕ, W_square=e^2b area(square)K. The curvature density e^2ϕR/2 is constant although R varies with position. Adding oriented subrectangle integrals reconstructs the coarse rapidity exactly.
Exact mathematical passage
```
 \phi_{tt}-\phi_{xx}=2b,
 \qquad R=4b\,e^{-2\phi},
 \qquad W_\square=e^{2b\,\operatorname{area}(\square)K}.
```

e^{2P}Use in source · 10 passages

Search aliases: e^2P

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

e^{2P_j}Use in source · 1 passage

Search aliases: e^2P_j

Link to this entry ·

Use in source · core.tex, line 484
dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.
Exact mathematical passage
```
 C_h(z)=2h\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].
 \label{c:core:gowdy-momentum}
```

e^{2|t|E_n}Use in source · 1 passage

Search aliases: e^2|t|E_n

Link to this entry ·

Use in source · finite_core.tex, line 703
f_h(t)-f(t) _L^2 ≤ h^2( ∑_n∈ S|t|^2e^2|t|E_nC(n)^2 a_n _2^2 )^1/2.
Exact mathematical passage
```
\|f_h(t)-f(t)\|_{L^2}
\le h^2\left(
\sum_{n\in S}|t|^2e^{2|t|E_n}C(n)^2\|a_n\|_2^2
\right)^{1/2}.
```

e^{2|t||\lambda|}Use in source · 1 passage

Search aliases: e^2|t||λ| lambda

Link to this entry ·

Use in source · spectral.tex, line 1136
A sufficient common-domain condition is ∫_R e^2|t||λ| d⟨ψ,E(λ)ψ⟩<∞, since |f_j(λ)|≤ e^|t||λ|. Spectral cutoffs P_R=E([-R,R]) supply a dense set satisfying this condition. On Ran P_R, the restricted Hamiltonian is bounded, the norm-series identities apply, and ∑_jf_j(H_∞)P_Rψ=e^-itH_∞P_Rψ ⟶ e^-itH_∞ψ. On the common domain of the individual lanes, each term also converges to f_j(H_∞)ψ by its spectral-domain integrability. For an arbitrary vector only convergence of the summed cutoff expression is …
Exact mathematical passage
```
 \int_{\R} e^{2|t||\lambda|}
              \,d\langle\psi,E(\lambda)\psi\rangle<\infty,
 \label{a-spectral-exponential-domain}
```

e^{2\pi i n\cdot x}Use in source · 2 passages

Search aliases: e^2π i n· x pi cdot

Link to this entry ·

Use in source · continuum.tex, line 89
Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation, normF f_KK=normf_HH, F^-1F f=f, FF^-1a=a. The inverse is understood as …
Exact mathematical passage
```
e_n(x)=e^{2\pi i n\cdot x}
```

e^{2\pi i(k-n)\cdot x}Use in source · 1 passage

Search aliases: e^2π i(k-n)· x pi cdot

Link to this entry ·

Use in source · finite_core.tex, line 437
⟨ J_nv,J_kw⟩_L^2 =⟨ v,w⟩∫_ T^3e^2π i(k-n)· x dx.
Exact mathematical passage
```
\langle J_nv,J_kw\rangle_{L^2}
=\langle v,w\rangle\int_{\mathbb T^3}e^{2\pi i(k-n)\cdot x}\,dx.
```

e^{2\pi i(x_1+x_2+x_3)}Use in source · 2 passages

Search aliases: e^2π i(x_1+x_2+x_3) pi

Link to this entry ·

Use in source · finite_core.tex, line 898
f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).
Exact mathematical passage
```
f_h(t,x)=\frac{e^{-it}}{\sqrt2}e_0
+\frac12e^{2\pi ix_1}u_A(t)
+\frac12e^{2\pi i(x_1+x_2+x_3)}u_D(t).
```

e^{2\pi i/L}Use in source · 1 passage

Search aliases: e^2π i/L pi

Link to this entry ·

Use in source · spectral.tex, line 899
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in a-spec
Exact mathematical passage
```
\omega_L=e^{2\pi i/L}
```

e^{2\pi ir}Use in source · 1 passage

Search aliases: e^2π ir pi

Link to this entry ·

Use in source · finite_core.tex, line 444
The integral is the product of three one-dimensional integrals. For integer r, the integral of e^2π irx over [0,1) is 1 when r=0; for r≠0, it is (e^2π ir-1)/(2π ir)=0. The product is therefore 1 exactly when n=k. Linearity is immediate, and setting k=n,w=v proves isometry. Expansion of the squared norm of the finite sum gives a double sum of pairings. The off-diagonal pairings vanish and the diagonal pairings are a_n _2^2. square
Exact mathematical passage
```
(e^{2\pi ir}-1)/(2\pi ir)=0
```

e^{2\pi irx}Use in source · 1 passage

Search aliases: e^2π irx pi

Link to this entry ·

Use in source · finite_core.tex, line 443
The integral is the product of three one-dimensional integrals. For integer r, the integral of e^2π irx over [0,1) is 1 when r=0; for r≠0, it is (e^2π ir-1)/(2π ir)=0. The product is therefore 1 exactly when n=k. Linearity is immediate, and setting k=n,w=v proves isometry. Expansion of the squared norm of the finite sum gives a double sum of pairings. The off-diagonal pairings vanish and the diagonal pairings are a_n _2^2. square
Exact mathematical passage
```
e^{2\pi irx}
```

e^{2\pi i\widehat N/L}Use in source · 1 passage

Search aliases: e^2π iwidehat N/L pi widehat

Link to this entry ·

Use in source · spectral.tex, line 1188
On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
V=e^{2\pi i\widehat N/L}
```

e^{2\pi ix_1}Use in source · 2 passages

Search aliases: e^2π ix_1 pi

Link to this entry ·

Use in source · finite_core.tex, line 898
f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).
Exact mathematical passage
```
f_h(t,x)=\frac{e^{-it}}{\sqrt2}e_0
+\frac12e^{2\pi ix_1}u_A(t)
+\frac12e^{2\pi i(x_1+x_2+x_3)}u_D(t).
```

e^{2\phi}Use in source · 3 passages

Search aliases: e^2ϕ phi

Link to this entry ·

Use in source · core.tex, line 145
Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1
Exact mathematical passage
```
 e^0=e^\phi dt,\qquad e^1=e^\phi dx,
 \qquad ds^2=e^{2\phi}(-dt^2+dx^2).
```

e^3Use in source · 1 passage

Search aliases: e^3

Link to this entry ·

Use in source · constraints.tex, line 336
unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitution repairs the H…
Exact mathematical passage
```
\star F=-E e^2\wedge e^3
```

\mathcal E^aUse in source · 6 passages

Search aliases: E^a mathcal

Link to this entry ·

Use in source · constraints.tex, line 566
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
\mathcal E^a=\tfrac12\epsilon^{abc}\Pi_{bc}
```

e^aUse in source · 2 passages

Search aliases: e^a

Link to this entry ·

Use in source · spectral.tex, line 1015
∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subseries of this sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of …
Exact mathematical passage
```
 e^a\frac{a^{Q+1}}{(Q+1)!},\qquad a=|t|\norm{A_0}.
 \label{a-spectral-lane-tail-bound}
```

E^a_iUse in source · 5 passages

Search aliases: E^a_i

Link to this entry ·

Use in source · constraints.tex, line 203
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,
Exact mathematical passage
```
E^a_i
```

E^a_kUse in source · 1 passage

Search aliases: E^a_k

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

\mathcal E^bUse in source · 1 passage

Search aliases: E^b mathcal

Link to this entry ·

Use in source · constraints.tex, line 572
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
 \mathcal H_{\mathrm M}=\frac1{2e}q_{ab}
       (\mathcal E^a\mathcal E^b+\mathcal B^a\mathcal B^b),
 \qquad \mathcal B^a=\tfrac12\epsilon^{abc}\mathcal F_{bc}.
```

E^b_iUse in source · 1 passage

Search aliases: E^b_i

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

E^b_jUse in source · 2 passages

Search aliases: E^b_j

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

E^{bi}Use in source · 1 passage

Search aliases: E^bi

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

E^c_kUse in source · 1 passage

Search aliases: E^c_k

Link to this entry ·

Use in source · constraints.tex, line 213
H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
\det E=\tfrac16\epsilon_{abc}\epsilon^{ijk}E^a_iE^b_jE^c_k
```

e^{hA}Use in source · 1 passage

Search aliases: e^hA

Link to this entry ·

Use in source · core.tex, line 417
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
\label{g:commutator-loop}
 W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.
```

e^{hB}Use in source · 1 passage

Search aliases: e^hB

Link to this entry ·

Use in source · core.tex, line 417
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
\label{g:commutator-loop}
 W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.
```

E^IUse in source · 2 passages

Search aliases: E^I

Link to this entry ·

Use in source · constraints.tex, line 422
An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 e^I=e_\perp^I\mathrm dt+E^I,\qquad
 \omega^{IJ}=\omega_\perp^{IJ}\mathrm dt+\Omega^{IJ},\qquad
 a=a_\perp\mathrm dt+A.
```

E^iUse in source · 1 passage

Search aliases: E^i

Link to this entry ·

Use in source · constraints.tex, line 543
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 E^0=0,\qquad E^i=e_a{}^i\,\mathrm dx^a,
 \qquad e:=\det(e_a{}^i)>0.
 \label{c:constraints:regular-triad}
```

e^IUse in source · 5 passages

Search aliases: e^I

Link to this entry ·

Use in source · constraints.tex, line 287
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,
Exact mathematical passage
```
e^I
```

e^{i p(n)\cdot x}Use in source · 1 passage

Search aliases: e^i p(n)· x cdot

Link to this entry ·

Use in source · finite_core.tex, line 17
e_n(x)=e^2π i n· x=e^i p(n)· x.
Exact mathematical passage
```
e_n(x)=e^{2\pi i n\cdot x}=e^{i p(n)\cdot x}.
```

e^{i(s+t)E}Use in source · 1 passage

Search aliases: e^i(s+t)E

Link to this entry ·

Use in source · finite_core.tex, line 387
U_H(s)U_H(t) =e^-i(s+t)EP_++e^i(s+t)EP_-=U_H(s+t).
Exact mathematical passage
```
U_H(s)U_H(t)
=e^{-i(s+t)E}P_++e^{i(s+t)E}P_-=U_H(s+t).
```

e^{i(z-\lambda)\tau}Use in source · 1 passage

Search aliases: e^i(z-λ)τ lambda tau

Link to this entry ·

Use in source · spectral.tex, line 307
titution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D is self-adjoint, normK(τ)≤normB^2. The factor e^-Imz τ makes the transform absolutely convergent. On a spectral eigenvalue of D, ∫_0^∞ e^i(z-λ)τ dτ=i/(z-λ). The finite spectral sum then proves [source reference].
Exact mathematical passage
```
\int_0^\infty e^{i(z-\lambda)\tau}\,d\tau=i/(z-\lambda)
```

e^{ia}Use in source · 1 passage

Search aliases: e^ia

Link to this entry ·

Use in source · core.tex, line 319
A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).
Exact mathematical passage
```
|e^{ia}-e^{ib}|\le|a-b|
```

e^{ib}Use in source · 1 passage

Search aliases: e^ib

Link to this entry ·

Use in source · core.tex, line 319
A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).
Exact mathematical passage
```
|e^{ia}-e^{ib}|\le|a-b|
```

e^{ih^2F_{ij}}Use in source · 1 passage

Search aliases: e^ih^2F_ij

Link to this entry ·

Use in source · refinement.tex, line 353
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
\Omega_{ij}=e^{ih^2F_{ij}}
```

e^{ihp_j(n)}Use in source · 1 passage

Search aliases: e^ihp_j(n)

Link to this entry ·

Use in source · finite_core.tex, line 248
D_h,je_n =e_nfrace^ihp_j(n)-e^-ihp_j(n)2h =ifracsin(hp_j(n))he_n.
Exact mathematical passage
```
D_{h,j}e_n
=e_n\frac{e^{ihp_j(n)}-e^{-ihp_j(n)}}{2h}
=i\frac{\sin(hp_j(n))}{h}e_n.
```

e^{ik\cdot x}Use in source · 1 passage

Search aliases: e^ik· x cdot

Link to this entry ·

Use in source · constraints.tex, line 178
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K=\left\{\sum_{\|k\|_\infty\le K}
              \widehat f_k e^{ik\cdot x}\right\}.
```

e^{ip\alpha_q}Use in source · 1 passage

Search aliases: e^ipα_q alpha

Link to this entry ·

Use in source · spectral.tex, line 131
At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine Fourier index is uniquely k=j+qM. Its angle satisfies 2π k/(pM)=(θ_j+2π q)/p. Substitu…
Exact mathematical passage
```
e^{ip\alpha_q}=e^{i\theta}
```

e^{is}Use in source · 1 passage

Search aliases: e^is

Link to this entry ·

Use in source · refinement.tex, line 394
· -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancellation inside F_ij.
Exact mathematical passage
```
e^{it}-1=\int_0^t i e^{is}\,ds
```

e^{iS_C/\hbar}Use in source · 2 passages

Search aliases: e^iS_C/ℏ hbar

Link to this entry ·

Use in source · constraints.tex, line 816
regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution, Δ_C e^iS_C/ℏ =- N^-1∫_ L_DΔ_D e^iS_F/ℏ=0. Th…
Exact mathematical passage
```
 e^{iS_C/\hbar}=\mathcal N^{-1}\int_{\mathcal L_D}e^{iS_F/\hbar}.
```

e^{iS_F/\hbar}Use in source · 2 passages

Search aliases: e^iS_F/ℏ hbar

Link to this entry ·

Use in source · constraints.tex, line 816
regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution, Δ_C e^iS_C/ℏ =- N^-1∫_ L_DΔ_D e^iS_F/ℏ=0. Th…
Exact mathematical passage
```
 e^{iS_C/\hbar}=\mathcal N^{-1}\int_{\mathcal L_D}e^{iS_F/\hbar}.
```

e^{it}Use in source · 2 passages

Search aliases: e^it

Link to this entry ·

Use in source · refinement.tex, line 353
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
|e^{it}-1|\leq|t|
```

e^{itD}Use in source · 1 passage

Search aliases: e^itD

Link to this entry ·

Use in source · spectral.tex, line 299
together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D is self-adjoint, normK(τ)≤n…
Exact mathematical passage
```
e^{itD}v(t)
```

e^{itE}Use in source · 2 passages

Search aliases: e^itE

Link to this entry ·

Use in source · finite_core.tex, line 320
U_H(t)=e^-itEP_++e^itEP_- =cos(tE)I_4-ifracsin(tE)EH.
Exact mathematical passage
```
U_H(t)=e^{-itE}P_++e^{itE}P_-
=\cos(tE)I_4-i\frac{\sin(tE)}{E}H.
```

e^{iuA}Use in source · 2 passages

Search aliases: e^iuA

Link to this entry ·

Use in source · finite_core.tex, line 1010
For Hermitian matrices the two exponentials in the integrand are unitary. This does not rely on the special Clifford algebra: the series gives (e^-iuA)^*=e^iuA, and the Cauchy product of the two absolutely convergent series gives e^iuAe^-iuA=I. Indeed, the coefficient of A^k for k>0 is (iu)^k∑_r=0^kbinomkr(-1)^r/k!=0, and the constant coefficient is I. A unitary matrix on a nonzero space has norm one. Submultiplicativity therefore bounds the integrand norm by A-B . For t≥0, integration gives t A…
Exact mathematical passage
```
(e^{-iuA})^*=e^{iuA}
```

e^{iz\tau}Use in source · 1 passage

Search aliases: e^izτ tau

Link to this entry ·

Use in source · spectral.tex, line 293
K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D …
Exact mathematical passage
```
 \Sigma(z)=-i\int_0^\infty e^{iz\tau}K(\tau)\,d\tau.
 \label{a-spectral-memory-transform}
```

e^{i\alpha_qr}Use in source · 1 passage

Search aliases: e^iα_qr alpha

Link to this entry ·

Use in source · spectral.tex, line 129
p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine Fourier index is uniquely k=j+qM. Its angle sa…
Exact mathematical passage
```
p^{-1/2}e^{i\alpha_qr}
```

e^{i\theta}Use in source · 3 passages

Search aliases: e^iθ theta

Link to this entry ·

Use in source · spectral.tex, line 131
At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine Fourier index is uniquely k=j+qM. Its angle satisfies 2π k/(pM)=(θ_j+2π q)/p. Substitu…
Exact mathematical passage
```
e^{ip\alpha_q}=e^{i\theta}
```

e^{i\theta_f}Use in source · 1 passage

Search aliases: e^iθ_f theta

Link to this entry ·

Use in source · core.tex, line 173
six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a principal argument, it implies that the oriented phase sum lies in 2πZ. It does not force that sum to be zero as a real number. An unwrapped small-phase re…
Exact mathematical passage
```
W_f=e^{i\theta_f}
```

e^{i\theta_i(p,h)}Use in source · 1 passage

Search aliases: e^iθ_i(p,h) theta

Link to this entry ·

Use in source · refinement.tex, line 327
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
 \theta_i(p,h)=\int_0^h A_i(p+t e_i)\,dt,
 \qquad u_i(p,h)=e^{i\theta_i(p,h)}.
 \label{g:integrated-links}
```

e^{i\chi_j}Use in source · 1 passage

Search aliases: e^iχ_j chi

Link to this entry ·

Use in source · core.tex, line 277
For time-independent real χ_j, the transformation ψ_j↦ e^iχ_jψ_j, A_j↦ A_j+fracχ_j+1-χ_jh, E_j↦ E_j leaves the action and equations invariant. In fact U_j then has precisely the backward-link law [source reference], every hopping scalar z_j is unchanged, and the time derivatives of the gauge parameters vanish.
Exact mathematical passage
```
\label{g:residual-gauge}
 \psi_j\mapsto e^{i\chi_j}\psi_j,
 \qquad A_j\mapsto A_j+\frac{\chi_{j+1}-\chi_j}{h},
 \qquad E_j\mapsto E_j
```

E^JUse in source · 2 passages

Search aliases: E^J

Link to this entry ·

Use in source · constraints.tex, line 434
IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A
Exact mathematical passage
```
 B_{IJ}&=\tfrac1{4\kappa}\epsilon_{KLIJ}E^K\wedge E^L,\\
 \Pi&=\tfrac12\varphi_{IJ}E^I\wedge E^J,\qquad
 \sigma_I=\tfrac1{3!}\epsilon_{IJKL}E^J\wedge E^K\wedge E^L.
```

E^jUse in source · 1 passage

Search aliases: E^j

Link to this entry ·

Use in source · constraints.tex, line 556
c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ term. Those directions lie in the k…
Exact mathematical passage
```
 B_{ij}=0,\qquad
 B_{0i}=\tfrac1{4\kappa}\epsilon_{ijk}E^j\wedge E^k,
 \qquad
 \int_\Sigma B_{IJ}\wedge\delta\Omega^{IJ}
      =\frac1\kappa\int_\Sigma\widetilde E^a{}_i\delta K_a{}^i\,\mathrm d^3x.
 \label{c:constraints:simplicity}
```

e^JUse in source · 3 passages

Search aliases: e^J

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

E^KUse in source · 2 passages

Search aliases: E^K

Link to this entry ·

Use in source · constraints.tex, line 434
IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A
Exact mathematical passage
```
 B_{IJ}&=\tfrac1{4\kappa}\epsilon_{KLIJ}E^K\wedge E^L,\\
 \Pi&=\tfrac12\varphi_{IJ}E^I\wedge E^J,\qquad
 \sigma_I=\tfrac1{3!}\epsilon_{IJKL}E^J\wedge E^K\wedge E^L.
```

E^kUse in source · 2 passages

Search aliases: E^k

Link to this entry ·

Use in source · finite_core.tex, line 345
The identity I_4=P_++P_- and the two eigenvalue identities give, by induction, H^k=E^kP_++(-E)^kP_- for every k≥0. Substitute into the absolutely convergent series to obtain the spectral expression. Expanding its projectors gives the cosine and sine expression.
Exact mathematical passage
```
H^k=E^kP_++(-E)^kP_-
```

e^KUse in source · 2 passages

Search aliases: e^K

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

E^LUse in source · 2 passages

Search aliases: E^L

Link to this entry ·

Use in source · constraints.tex, line 434
IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A
Exact mathematical passage
```
 B_{IJ}&=\tfrac1{4\kappa}\epsilon_{KLIJ}E^K\wedge E^L,\\
 \Pi&=\tfrac12\varphi_{IJ}E^I\wedge E^J,\qquad
 \sigma_I=\tfrac1{3!}\epsilon_{IJKL}E^J\wedge E^K\wedge E^L.
```

e^LUse in source · 2 passages

Search aliases: e^L

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

e^{LT}Use in source · 1 passage

Search aliases: e^LT

Link to this entry ·

Use in source · core.tex, line 566
nd on g make the correction O(Δτ^3). Thus the corrected local error has the same order. Under the stated derivative bounds the corrected step map has Lipschitz constant at most 1+LΔτ for small steps. If e_n is its error against the exact solution, then e_n+1≤(1+LΔτ)e_n+MΔτ^3. Iterating this inequality for nΔτ≤ T bounds the error by MT e^LTΔτ^2 when e_0=0.
Exact mathematical passage
```
MT e^{LT}\Delta\tau^2
```

e^RUse in source · 1 passage

Search aliases: e^R

Link to this entry ·

Use in source · finite_core.tex, line 684
e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .
Exact mathematical passage
```
\|e^X-e^Y\|
\le\|X-Y\|\sum_{k=1}^{\infty}\frac{kR^{k-1}}{k!}
=e^R\|X-Y\|.
```

e^{R\|B\|}Use in source · 1 passage

Search aliases: e^R B

Link to this entry ·

Use in source · finite_core.tex, line 980
∑_k=1^∞frackR^k-1 B ^kk! = B e^R B .
Exact mathematical passage
```
\sum_{k=1}^{\infty}\frac{kR^{k-1}\|B\|^k}{k!}
=\|B\|e^{R\|B\|}.
```

e^{tA_0}Use in source · 1 passage

Search aliases: e^tA_0

Link to this entry ·

Use in source · spectral.tex, line 1003
[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber · =frac(tA_0)^jj! _0F_L-1 (- A_L,j; frac(tA_0)^LL^L),nonumber ∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subser…
Exact mathematical passage
```
 E_{L,j}(tA_0)
 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
                           e^{\omega_L^\ell tA_0},\nonumber\\
 &=\frac{(tA_0)^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                      \frac{(tA_0)^L}{L^L}\right),\nonumber\\
 \sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.
 \label{a-spectral-operator-lane-identities}
```

e^XUse in source · 1 passage

Search aliases: e^X

Link to this entry ·

Use in source · finite_core.tex, line 684
e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .
Exact mathematical passage
```
\|e^X-e^Y\|
\le\|X-Y\|\sum_{k=1}^{\infty}\frac{kR^{k-1}}{k!}
=e^R\|X-Y\|.
```

e^YUse in source · 1 passage

Search aliases: e^Y

Link to this entry ·

Use in source · finite_core.tex, line 684
e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .
Exact mathematical passage
```
\|e^X-e^Y\|
\le\|X-Y\|\sum_{k=1}^{\infty}\frac{kR^{k-1}}{k!}
=e^R\|X-Y\|.
```

e^zUse in source · 1 passage

Search aliases: e^z

Link to this entry ·

Use in source · spectral.tex, line 926
Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber
Exact mathematical passage
```
e^z
```

e^{zA}Use in source · 1 passage

Search aliases: e^zA

Link to this entry ·

Use in source · finite_core.tex, line 751
e^zA^ H=Te^zAT^-1.
Exact mathematical passage
```
e^{zA^{\mathcal H}}=Te^{zA}T^{-1}.
```

e^{zA^{\mathcal H}}Use in source · 1 passage

Search aliases: e^zA^ H mathcal

Link to this entry ·

Use in source · finite_core.tex, line 751
e^zA^ H=Te^zAT^-1.
Exact mathematical passage
```
e^{zA^{\mathcal H}}=Te^{zA}T^{-1}.
```

e^{zC_L}Use in source · 1 passage

Search aliases: e^zC_L

Link to this entry ·

Use in source · spectral.tex, line 970
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
 \mathbf E_L'=C_L\mathbf E_L,\qquad
 \mathbf E_L(0)=e_0,\qquad
 C_L^L=I,\qquad
 \mathbf E_L(z)=e^{zC_L}e_0.
 \label{a-spectral-companion-system}
```

e^{|t|E_n}Use in source · 1 passage

Search aliases: e^|t|E_n

Link to this entry ·

Use in source · finite_core.tex, line 694
U_H_h,n(t)-U_H_n(t) ≤ |t|e^|t|E_nh^2C(n).
Exact mathematical passage
```
\|U_{H_{h,n}}(t)-U_{H_n}(t)\|
\le |t|e^{|t|E_n}h^2C(n).
```

e^{|t||\lambda|}Use in source · 1 passage

Search aliases: e^|t||λ| lambda

Link to this entry ·

Use in source · spectral.tex, line 1141
A sufficient common-domain condition is ∫_R e^2|t||λ| d⟨ψ,E(λ)ψ⟩<∞, since |f_j(λ)|≤ e^|t||λ|. Spectral cutoffs P_R=E([-R,R]) supply a dense set satisfying this condition. On Ran P_R, the restricted Hamiltonian is bounded, the norm-series identities apply, and ∑_jf_j(H_∞)P_Rψ=e^-itH_∞P_Rψ ⟶ e^-itH_∞ψ. On the common domain of the individual lanes, each term also converges to f_j(H_∞)ψ by its spectral-domain integrability. For an arbitrary vector only convergence of the summed cutoff expression is …
Exact mathematical passage
```
|f_j(\lambda)|\leq e^{|t||\lambda|}
```

e^{\pm itE_{h,n}}Use in source · 1 passage

Search aliases: e^± itE_h,n pm

Link to this entry ·

Use in source · finite_core.tex, line 625
|e^± itE_h,n-e^± itE_n| ≤ |t| |E_h,n-E_n|,
Exact mathematical passage
```
|e^{\pm itE_{h,n}}-e^{\pm itE_n}|
\le |t|\,|E_{h,n}-E_n|,
```

e^{\pm itE_n}Use in source · 1 passage

Search aliases: e^± itE_n pm

Link to this entry ·

Use in source · finite_core.tex, line 625
|e^± itE_h,n-e^± itE_n| ≤ |t| |E_h,n-E_n|,
Exact mathematical passage
```
|e^{\pm itE_{h,n}}-e^{\pm itE_n}|
\le |t|\,|E_{h,n}-E_n|,
```

e^{\pm its}Use in source · 1 passage

Search aliases: e^± its pm

Link to this entry ·

Use in source · finite_core.tex, line 630
by integrating the derivative of s↦ e^± its between the two real energies. Phase comparison alone does not compare full spinor evolution: the eigenspaces also depend on the kinetic vector. The operator argument below controls both effects.
Exact mathematical passage
```
s\mapsto e^{\pm its}
```

e^{\mu(\operatorname{Re}\omega_L^\ell-1)}Use in source · 1 passage

Search aliases: e^μ(Reω_L^ℓ-1) mu operatorname omega ell

Link to this entry ·

Use in source · spectral.tex, line 1235
For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L), giving the second.
Exact mathematical passage
```
e^{\mu(\operatorname{Re}\omega_L^\ell-1)}/L
```

e^{\mu(\omega_L^\ell-1)}Use in source · 1 passage

Search aliases: e^μ(ω_L^ℓ-1) mu omega ell

Link to this entry ·

Use in source · spectral.tex, line 1207
For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series proves ∑_jp_L,j=1. Their equivalent hypergeometric form is e^-μ times [source reference] at z=μ.
Exact mathematical passage
```
 p_{L,j}(\mu)
 &=\langle\alpha,Q_j^{(L)}\alpha\rangle
   =e^{-\mu}E_{L,j}(\mu)\nonumber\\
 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
                      e^{\mu(\omega_L^\ell-1)}.
 \label{a-spectral-coherent-packets}
```

e^\phiUse in source · 1 passage

Search aliases: e^ϕ phi

Link to this entry ·

Use in source · core.tex, line 145
Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1
Exact mathematical passage
```
 e^0=e^\phi dt,\qquad e^1=e^\phi dx,
 \qquad ds^2=e^{2\phi}(-dt^2+dx^2).
```

e^{\chi}Use in source · 1 passage

Search aliases: e^χ chi

Link to this entry ·

Use in source · core.tex, line 185
If the integrated coefficient is χ, then e^χ K=coshχ I+sinhχ K. This preserves the Lorentz form diag(-1,1) and has positive eigenvalues e^χ and e^-χ. Its real principal matrix logarithm is χ K. These are Lorentz boosts. The positive Hilbert norm used for unitary color transport in the gauge monograph is a different structure.
Exact mathematical passage
```
e^{\chi}
```

e^{\chi K}Use in source · 1 passage

Search aliases: e^χ K chi

Link to this entry ·

Use in source · core.tex, line 181
If the integrated coefficient is χ, then e^χ K=coshχ I+sinhχ K. This preserves the Lorentz form diag(-1,1) and has positive eigenvalues e^χ and e^-χ. Its real principal matrix logarithm is χ K. These are Lorentz boosts. The positive Hilbert norm used for unitary color transport in the gauge monograph is a different structure.
Exact mathematical passage
```
 e^{\chi K}=\cosh\chi\,I+\sinh\chi\,K.
```

e^{\chi(t,x)K}Use in source · 1 passage

Search aliases: e^χ(t,x)K chi

Link to this entry ·

Use in source · core.tex, line 203
The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and parallel transport exp(-∫_eω). Its closed-loop rapidity is the negative of the rapidity of W_square defined here. The fixed-frame curvature identity [so…
Exact mathematical passage
```
 G(t,x)=e^{\chi(t,x)K},\qquad
 U_{a\to b}'=G(b)U_{a\to b}G(a)^{-1}.
```

e^{\omega_L^\ell tA_0}Use in source · 1 passage

Search aliases: e^ω_L^ℓ tA_0 omega ell

Link to this entry ·

Use in source · spectral.tex, line 1003
[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber · =frac(tA_0)^jj! _0F_L-1 (- A_L,j; frac(tA_0)^LL^L),nonumber ∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subser…
Exact mathematical passage
```
 E_{L,j}(tA_0)
 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
                           e^{\omega_L^\ell tA_0},\nonumber\\
 &=\frac{(tA_0)^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                      \frac{(tA_0)^L}{L^L}\right),\nonumber\\
 \sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.
 \label{a-spectral-operator-lane-identities}
```

e^{\omega_L^\ell z}Use in source · 1 passage

Search aliases: e^ω_L^ℓ z omega ell

Link to this entry ·

Use in source · spectral.tex, line 980
j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=frac1L∑_ℓ=0^L-1 ω_L^-jℓe^ω_L^ℓ z. Thus factorial, hypergeometric, companion-state, and root-filter forms describe the same analytic function. Each form has a different numerical conditioning: a root sum can contain cancellations even when the defining series ha…
Exact mathematical passage
```
 E_{L,j}(z)=\frac1L\sum_{\ell=0}^{L-1}
               \omega_L^{-j\ell}e^{\omega_L^\ell z}.
 \label{a-spectral-root-exponential}
```

\mathsf E_{\mathbf h}Use in source · 6 passages

Search aliases: E_ h mathsf mathbf

Link to this entry ·

Use in source · limit.tex, line 545
imensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and the identity is a finite telescoping sum, one factor o…
Exact mathematical passage
```
 \mathsf E_{\mathbf h}
 =I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathsf E_{p,h_p}.
```

\mathsf E_{\mathbf k}Use in source · 1 passage

Search aliases: E_ k mathsf mathbf

Link to this entry ·

Use in source · limit.tex, line 550
⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and the identity is a finite telescoping sum, one factor of which tends strongly to zero in each term, while all other projection factors have norm at mos…
Exact mathematical passage
```
 \mathsf E_{\mathbf h}\mathsf E_{\mathbf k}
 =\mathsf E_{\min(\mathbf h,\mathbf k)},
 \qquad \mathsf E_{\mathbf h}f\to f
 \quad\text{as every }h_p\to\infty.
 \label{a-limit-rectangular-projections}
```

E_+Use in source · 2 passages

Search aliases: E_+

Link to this entry ·

Use in source · finite.tex, line 226
0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though it no longer distinguishes the two rational eigenvalues. These E_± are complementary algebraic spectral idempotents. Since A is not self-adjoint in the…
Exact mathematical passage
```
E_+E_-=0
```

E_-Use in source · 2 passages

Search aliases: E_-

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Use in source · finite.tex, line 226
0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though it no longer distinguishes the two rational eigenvalues. These E_± are complementary algebraic spectral idempotents. Since A is not self-adjoint in the…
Exact mathematical passage
```
E_+E_-=0
```

E_0Use in source · 1 passage

Search aliases: E_0

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Use in source · finite.tex, line 301
For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.
Exact mathematical passage
```
E_0
```

\mathsf E_0Use in source · 2 passages

Search aliases: E_0 mathsf

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Use in source · limit.tex, line 146
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involvin
Exact mathematical passage
```
\mathsf F_0=\mathsf E_0
```

E_0^{(h)}Use in source · 1 passage

Search aliases: E_0^(h)

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Use in source · finite.tex, line 274
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 D_0^{(h)}=E_0^{(h)},\qquad
 D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\quad(1\leq r\leq h).
```

e_1Use in source · 4 passages

Search aliases: e_1

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Use in source · core.tex, line 283
[A stationary vector viewed through a rotating split] Let ψ(t)=e_1, K=0, and U(t)=cosθ(t) · -sinθ(t)
Exact mathematical passage
```
\psi(t)=e_1
```

e_2Use in source · 1 passage

Search aliases: e_2

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Use in source · core.tex, line 892
[An exact bracket failure] Use N=5, h=1, and (Df)_a=(f_a+1-f_a-1)/2. Take q=e_0, p=ξ=e_1, and η=e_2. Then (Dq)_1=-1/2, (Dη)_1=1/2, and η Dq=0. The direct bracket in the preceding proof is zero, while D_h[ξ Dη-η Dξ]=-tfrac14. The defect is therefore exactly +1/4. This finite witness shows that the naive centered scalar constraint does fail to close universally in the proposed undeformed form.
Exact mathematical passage
```
\eta=e_2
```

E_{2,0}Use in source · 1 passage

Search aliases: E_2,0

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Use in source · spectral.tex, line 1046
The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lane is an analytic contribution to it.
Exact mathematical passage
```
E_{2,0}(tA_0)=\cosh(tA_0)
```

E_{2,1}Use in source · 1 passage

Search aliases: E_2,1

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Use in source · spectral.tex, line 1047
The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lane is an analytic contribution to it.
Exact mathematical passage
```
E_{2,1}(tA_0)=\sinh(tA_0)
```

e_aUse in source · 19 passages

Search aliases: e_a

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Use in source · limit.tex, line 24
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after t
Exact mathematical passage
```
e_a
```

e_{a+d p^m}Use in source · 1 passage

Search aliases: e_a+d p^m

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Use in source · limit.tex, line 46
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
 V_{p,m}e_a=\frac1{\sqrt p}
 \sum_{d=0}^{p-1}e_{a+d p^m}.
```

e_a^2Use in source · 2 passages

Search aliases: e_a^2

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Use in source · extensions.tex, line 431
[No nonzero derivation of the finite pointwise algebra] Let D:C^N→C^N be complex linear and satisfy D(fg)=D(f)g+fD(g) for every pair of vectors, where multiplication is coordinatewise. Then D=0. For the coordinate idempotent e_a, the identity e_a^2=e_a gives D(e_a)=2e_aD(e_a). At a coordinate b≠ a, the right side is zero. At b=a, the equality reads (D(e_a))_a=2(D(e_a))_a, again forcing zero. Thus D(e_a)=0 for every basis vector, and linearity gives D=0.
Exact mathematical passage
```
e_a^2=e_a
```

E_a^IUse in source · 3 passages

Search aliases: E_a^I

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Use in source · constraints.tex, line 521
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
q_{ab}=\eta_{IJ}E_a^IE_b^J
```

E_a^JUse in source · 1 passage

Search aliases: E_a^J

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Use in source · constraints.tex, line 522
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
 \widetilde n_I=\tfrac1{3!}\epsilon_{IJKL}\epsilon^{abc}E_a^JE_b^KE_c^L.
```

E_A^{\rm con}Use in source · 1 passage

Search aliases: E_A^rm con rm

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Use in source · core.tex, line 737
For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.
Exact mathematical passage
```
 E_A^{\rm con}=\sum_{x\in A}h^3
 \left(|C_x|^2+\sum_i|C_{i,x}|^2\right)
```

e_bUse in source · 3 passages

Search aliases: e_b

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Use in source · limit.tex, line 32
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \inner{\mathcal U_{p,m}e_a}{\mathcal U_{p,m}e_b}
 =p^m\mu_p(B_{m,a}\cap B_{m,b})=\delta_{ab}.
```

E_b^JUse in source · 1 passage

Search aliases: E_b^J

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Use in source · constraints.tex, line 521
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
q_{ab}=\eta_{IJ}E_a^IE_b^J
```

E_b^KUse in source · 1 passage

Search aliases: E_b^K

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Use in source · constraints.tex, line 522
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
 \widetilde n_I=\tfrac1{3!}\epsilon_{IJKL}\epsilon^{abc}E_a^JE_b^KE_c^L.
```

e_cUse in source · 1 passage

Search aliases: e_c

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Use in source · constraints.tex, line 549
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 \widetilde E^a{}_i=\tfrac12\epsilon^{abc}\epsilon_{ijk}e_b{}^je_c{}^k,
 \qquad \widetilde E^a{}_ie_a{}^j=e\delta_i{}^j.
```

E_c^LUse in source · 1 passage

Search aliases: E_c^L

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Use in source · constraints.tex, line 522
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
 \widetilde n_I=\tfrac1{3!}\epsilon_{IJKL}\epsilon^{abc}E_a^JE_b^KE_c^L.
```

E_hUse in source · 1 passage

Search aliases: E_h

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Use in source · core.tex, line 70
s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.
Exact mathematical passage
```
E_h=\sum_je_j
```

\mathcal E_hUse in source · 3 passages

Search aliases: E_h mathcal

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Use in source · finite_core.tex, line 1111
with centered expectation E_h(a) obtained by replacing H_n with H_h,n. Hermiticity makes each expectation real: its complex conjugate is ⟨ H_na_n,a_n⟩=⟨ a_n,H_na_n⟩.
Exact mathematical passage
```
\mathcal E_h(a)
```

E_{h,n}Use in source · 11 passages

Search aliases: E_h,n

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Use in source · finite_core.tex, line 274
H_h,n^2=E_h,n^2I_4, E_h,n=sqrtm^2+|q_h(n)|^2.
Exact mathematical passage
```
H_{h,n}^2=E_{h,n}^2I_4,\quad
E_{h,n}=\sqrt{m^2+|q_h(n)|^2}.
```

E_{h,n}^2Use in source · 2 passages

Search aliases: E_h,n^2

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Use in source · finite_core.tex, line 274
H_h,n^2=E_h,n^2I_4, E_h,n=sqrtm^2+|q_h(n)|^2.
Exact mathematical passage
```
H_{h,n}^2=E_{h,n}^2I_4,\quad
E_{h,n}=\sqrt{m^2+|q_h(n)|^2}.
```

E_h^{(h)}Use in source · 1 passage

Search aliases: E_h^(h)

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Use in source · finite.tex, line 270
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 E_r^{(h)}E_s^{(h)}=E_{\min(r,s)}^{(h)},\qquad E_h^{(h)}=I.
```

E_h^{(h+1)}Use in source · 1 passage

Search aliases: E_h^(h+1)

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Use in source · finite.tex, line 391
is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an identity on the coarse space. The second is a projector on the fine space, not a fine-space identity.
Exact mathematical passage
```
 V_{p,h}^*V_{p,h}=I,\qquad
 V_{p,h}V_{p,h}^*=E_h^{(h+1)},\qquad
 I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.
 \label{a-finite-refinement-projectors}
```

e_h^2Use in source · 1 passage

Search aliases: e_h^2

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Use in source · strong_fields.tex, line 281
y≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonlinear difference satisfies |g(R_j)-g(widetilde R_j)|≤ L_g|e_j|, where g'(R)=fracMR-Q^2R^2sqrt2MR-Q^2. Thus tfrac12∂_tnorm e_h^2≤(L_g+S_1/2)norm e_h^2, and integration gives the claim.
Exact mathematical passage
```
\tfrac12\partial_t\norm e_h^2\le(L_g+S_1/2)\norm e_h^2
```

e_iUse in source · 6 passages

Search aliases: e_i

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Use in source · refinement.tex, line 327
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
 \theta_i(p,h)=\int_0^h A_i(p+t e_i)\,dt,
 \qquad u_i(p,h)=e^{i\theta_i(p,h)}.
 \label{g:integrated-links}
```

E_jUse in source · 9 passages

Search aliases: E_j

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Use in source · core.tex, line 196
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,
Exact mathematical passage
```
A_j,E_j\in\R
```

E_{j-1}Use in source · 1 passage

Search aliases: E_j-1

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Use in source · core.tex, line 227
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-def}
 \rho_j=\psi_j^*\psi_j,\qquad
 \bar\rho=\frac1N\sum_j\rho_j,\qquad
 \mathcal G_j=\frac{E_j-E_{j-1}}h+\rho_j-\bar\rho.
```

E_j^2Use in source · 1 passage

Search aliases: E_j^2

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Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

e_kUse in source · 2 passages

Search aliases: e_k

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Use in source · core.tex, line 388
l p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, with (e_k)_k indexing the basis of HH_B. In the resulting product basis the partial trace has entries (ρ_A)_ij=∑_k(ρ_AB)_(i,k),(j,k). This explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on x and summing t…
Exact mathematical passage
```
(e_k)_k
```

\mathbf E_LUse in source · 2 passages

Search aliases: E_L mathbf

Link to this entry ·

Use in source · spectral.tex, line 967
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
\mathbf E_L=(E_{L,0},\ldots,E_{L,L-1})^{\mathsf T}
```

\mathbf E_L'Use in source · 1 passage

Search aliases: E_L' mathbf

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Use in source · spectral.tex, line 970
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
 \mathbf E_L'=C_L\mathbf E_L,\qquad
 \mathbf E_L(0)=e_0,\qquad
 C_L^L=I,\qquad
 \mathbf E_L(z)=e^{zC_L}e_0.
 \label{a-spectral-companion-system}
```

E_{L,0}Use in source · 1 passage

Search aliases: E_L,0

Link to this entry ·

Use in source · spectral.tex, line 967
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
\mathbf E_L=(E_{L,0},\ldots,E_{L,L-1})^{\mathsf T}
```

E_{L,j-1}Use in source · 1 passage

Search aliases: E_L,j-1

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Use in source · spectral.tex, line 964
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
E'_{L,j}=E_{L,j-1}
```

E_{L,L-1}Use in source · 2 passages

Search aliases: E_L,L-1

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Use in source · spectral.tex, line 966
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
E'_{L,0}=E_{L,L-1}
```

\mathsf E_MUse in source · 16 passages

Search aliases: E_M mathsf

Link to this entry ·

Use in source · limit.tex, line 151
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
 \mathsf F_m\mathsf F_n=\delta_{mn}\mathsf F_m,
 \qquad \sum_{m=0}^M\mathsf F_m=\mathsf E_M.
 \label{a-limit-detail-algebra}
```

\mathsf E_{m-1}Use in source · 5 passages

Search aliases: E_m-1 mathsf

Link to this entry ·

Use in source · limit.tex, line 147
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_
Exact mathematical passage
```
\mathsf F_m=\mathsf E_m-\mathsf E_{m-1}
```

E_m^{(h)}Use in source · 2 passages

Search aliases: E_m^(h)

Link to this entry ·

Use in source · finite.tex, line 310
The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an additional bound on the detail weights of the state.
Exact mathematical passage
```
E_m^{(h)}=\sum_{r=0}^mD_r^{(h)}
```

\mathsf E_m^*Use in source · 1 passage

Search aliases: E_m^* mathsf

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Use in source · limit.tex, line 88
[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows innergf- E_mf=0. The image is in V_m, and averaging a function in V_m returns the function. These statements characterize the orthogonal projection and est…
Exact mathematical passage
```
 \mathsf E_m^*=\mathsf E_m,\qquad
 \mathsf E_m^2=\mathsf E_m,\qquad
 \mathsf E_m\mathsf E_n=\mathsf E_{\min(m,n)}.
 \label{a-limit-projection-product}
```

\mathsf E_m^2Use in source · 1 passage

Search aliases: E_m^2 mathsf

Link to this entry ·

Use in source · limit.tex, line 88
[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows innergf- E_mf=0. The image is in V_m, and averaging a function in V_m returns the function. These statements characterize the orthogonal projection and est…
Exact mathematical passage
```
 \mathsf E_m^*=\mathsf E_m,\qquad
 \mathsf E_m^2=\mathsf E_m,\qquad
 \mathsf E_m\mathsf E_n=\mathsf E_{\min(m,n)}.
 \label{a-limit-projection-product}
```

\mathsf E_{\min(\mathbf h,\mathbf k)}Use in source · 1 passage

Search aliases: E_min( h, k) mathsf min mathbf mathbf

Link to this entry ·

Use in source · limit.tex, line 550
⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and the identity is a finite telescoping sum, one factor of which tends strongly to zero in each term, while all other projection factors have norm at mos…
Exact mathematical passage
```
 \mathsf E_{\mathbf h}\mathsf E_{\mathbf k}
 =\mathsf E_{\min(\mathbf h,\mathbf k)},
 \qquad \mathsf E_{\mathbf h}f\to f
 \quad\text{as every }h_p\to\infty.
 \label{a-limit-rectangular-projections}
```

\mathsf E_{\min(m,n)}Use in source · 1 passage

Search aliases: E_min(m,n) mathsf min

Link to this entry ·

Use in source · limit.tex, line 88
[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows innergf- E_mf=0. The image is in V_m, and averaging a function in V_m returns the function. These statements characterize the orthogonal projection and est…
Exact mathematical passage
```
 \mathsf E_m^*=\mathsf E_m,\qquad
 \mathsf E_m^2=\mathsf E_m,\qquad
 \mathsf E_m\mathsf E_n=\mathsf E_{\min(m,n)}.
 \label{a-limit-projection-product}
```

E_{\min(r,s)}^{(h)}Use in source · 1 passage

Search aliases: E_min(r,s)^(h) min

Link to this entry ·

Use in source · finite.tex, line 270
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 E_r^{(h)}E_s^{(h)}=E_{\min(r,s)}^{(h)},\qquad E_h^{(h)}=I.
```

E_nUse in source · 15 passages

Search aliases: E_n

Link to this entry ·

Use in source · finite_core.tex, line 270
H_n^2=E_n^2I_4, E_n=sqrtm^2+|p(n)|^2,
Exact mathematical passage
```
H_n^2=E_n^2I_4,\quad E_n=\sqrt{m^2+|p(n)|^2},
```

\mathsf E_nUse in source · 3 passages

Search aliases: E_n mathsf

Link to this entry ·

Use in source · limit.tex, line 88
[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows innergf- E_mf=0. The image is in V_m, and averaging a function in V_m returns the function. These statements characterize the orthogonal projection and est…
Exact mathematical passage
```
 \mathsf E_m^*=\mathsf E_m,\qquad
 \mathsf E_m^2=\mathsf E_m,\qquad
 \mathsf E_m\mathsf E_n=\mathsf E_{\min(m,n)}.
 \label{a-limit-projection-product}
```

e_NUse in source · 2 passages

Search aliases: e_N

Link to this entry ·

Use in source · extensions.tex, line 145
[A small state can have a large generator image] On ℓ^2(N_≥1), let He_n=ne_n on its maximal domain. The finitely supported vectors x_N=e_N/N satisfy normx_N=1/N→0, while normHx_N=1. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because Hx_N=e_N has no norm limit.
Exact mathematical passage
```
x_N=e_N/N
```

e_nUse in source · 20 passages

Search aliases: e_n

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Use in source · continuum.tex, line 89
Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation, normF f_KK=normf_HH, F^-1F f=f, FF^-1a=a. The inverse is understood as …
Exact mathematical passage
```
e_n(x)=e^{2\pi i n\cdot x}
```

e_{N+1}Use in source · 1 passage

Search aliases: e_N+1

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Use in source · extensions.tex, line 350
[The identity never has a small omitted tail in operator norm] On an infinite-dimensional coordinate space, let S_N retain the first N coordinates. For X=I, normS_NIS_N-I=normS_N-I=1. A unit vector e_N+1 witnesses the lower bound. Nevertheless, S_Nx→ x for every fixed x∈ℓ^2. This is the difference between pointwise control over states and uniform control over the full unit ball.
Exact mathematical passage
```
e_{N+1}
```

e_{n+1}Use in source · 1 passage

Search aliases: e_n+1

Link to this entry ·

Use in source · core.tex, line 562
atives, the provisional constraint defect is also O(Δτ^3). The lower bound on the denominator and the upper bound on g make the correction O(Δτ^3). Thus the corrected local error has the same order. Under the stated derivative bounds the corrected step map has Lipschitz constant at most 1+LΔτ for small steps. If e_n is its error against the exact solution, then e_n+1≤(1+LΔτ)e_n+MΔτ^3. Iterating this inequality for nΔτ≤ T bounds the error by MT e^LTΔτ^2 when e_0=0.
Exact mathematical passage
```
 e_{n+1}\leq(1+L\Delta\tau)e_n+M\Delta\tau^3.
```

\mathsf E_{n-1}Use in source · 1 passage

Search aliases: E_n-1 mathsf

Link to this entry ·

Use in source · limit.tex, line 157
depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
(\mathsf E_m-\mathsf E_{m-1})(\mathsf E_n-\mathsf E_{n-1})
```

E_n^2Use in source · 3 passages

Search aliases: E_n^2

Link to this entry ·

Use in source · finite_core.tex, line 270
H_n^2=E_n^2I_4, E_n=sqrtm^2+|p(n)|^2,
Exact mathematical passage
```
H_n^2=E_n^2I_4,\quad E_n=\sqrt{m^2+|p(n)|^2},
```

e_{n\bmod L}Use in source · 1 passage

Search aliases: e_nbmod L bmod

Link to this entry ·

Use in source · spectral.tex, line 977
gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=frac1L∑_ℓ=0^L-1 ω_L^-jℓe^ω_L^ℓ z. Thus factorial, hypergeometric, companion-state, and root-filter forms describe the same analytic function. Each f…
Exact mathematical passage
```
C_L^ne_0=e_{n\bmod L}
```

\mathcal E_PUse in source · 3 passages

Search aliases: E_P mathcal

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

\mathcal E_{P,h}Use in source · 2 passages

Search aliases: E_P,h mathcal

Link to this entry ·

Use in source · core.tex, line 578
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
\mathcal E_{P,h}=\dot U-\dot U_{\rm raw}
```

\mathsf E_{p,h_p}Use in source · 1 passage

Search aliases: E_p,h_p mathsf

Link to this entry ·

Use in source · limit.tex, line 545
imensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and the identity is a finite telescoping sum, one factor o…
Exact mathematical passage
```
 \mathsf E_{\mathbf h}
 =I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathsf E_{p,h_p}.
```

E_{pL,j+aL}Use in source · 1 passage

Search aliases: E_pL,j+aL

Link to this entry ·

Use in source · spectral.tex, line 1175
Analytic refinement of a factorial lane is independent of this unit condition. Splitting q=pr+a, 0≤ a<p, gives the exact identity E_L,j(z)=∑_a=0^p-1E_pL,j+aL(z), 0≤ j<L. Absolute convergence permits this partition of the summation index. The labels j+aL all lie in [0,pL), but need not lie in the parent's short unit window. Exact complex analytic refinement and the local p-adic unit assertion therefore have separate hypotheses.
Exact mathematical passage
```
 E_{L,j}(z)=\sum_{a=0}^{p-1}E_{pL,j+aL}(z),\qquad 0\leq j<L.
 \label{a-spectral-analytic-refinement}
```

E_qUse in source · 1 passage

Search aliases: E_q

Link to this entry ·

Use in source · spectral.tex, line 119
[Exact folded cyclic spectrum] For q=0,…,p-1, the eigenvalues of H_p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source re…
Exact mathematical passage
```
 E_q(\theta)=\mu+2\kappa
     \cos\!\left(\frac{\theta+2\pi q}{p}\right).
 \label{a-spectral-band-formula}
```

\mathcal E_QUse in source · 3 passages

Search aliases: E_Q mathcal

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

\mathcal E_{Q,h}Use in source · 2 passages

Search aliases: E_Q,h mathcal

Link to this entry ·

Use in source · core.tex, line 579
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
\mathcal E_{Q,h}=\dot V-\dot V_{\rm raw}
```

E_rUse in source · 10 passages

Search aliases: E_r

Link to this entry ·

Use in source · finite.tex, line 292
If r≤ s, the level-s cylinders partition each level-r cylinder into equally sized groups. Averaging over the smaller groups and then over the whole group returns the full average, giving E_rE_s=E_r. The self-adjointness of both sides also gives E_sE_r=E_r. This proves the minimum rule.
Exact mathematical passage
```
E_rE_s=E_r
```

e_rUse in source · 3 passages

Search aliases: e_r

Link to this entry ·

Use in source · spectral.tex, line 102
M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1, e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
 P_p(\theta)e_r=
 \begin{cases}
 e_{r+1},&r<p-1,\\
 e^{-i\theta}e_0,&r=p-1.
 \end{cases}
 \qquad
 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
 \label{a-spectral-bloch-symbol}
```

e_{r+1}Use in source · 1 passage

Search aliases: e_r+1

Link to this entry ·

Use in source · spectral.tex, line 102
M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1, e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
 P_p(\theta)e_r=
 \begin{cases}
 e_{r+1},&r<p-1,\\
 e^{-i\theta}e_0,&r=p-1.
 \end{cases}
 \qquad
 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
 \label{a-spectral-bloch-symbol}
```

E_{r-1}Use in source · 3 passages

Search aliases: E_r-1

Link to this entry ·

Use in source · finite.tex, line 296
For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.
Exact mathematical passage
```
(E_r-E_{r-1})^2=E_r-2E_{r-1}+E_{r-1}=D_r
```

E_{r-1}^{(h)}Use in source · 1 passage

Search aliases: E_r-1^(h)

Link to this entry ·

Use in source · finite.tex, line 274
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 D_0^{(h)}=E_0^{(h)},\qquad
 D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\quad(1\leq r\leq h).
```

E_r^{(h)}Use in source · 7 passages

Search aliases: E_r^(h)

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Use in source · finite.tex, line 256
For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.
Exact mathematical passage
```
 (E_r^{(h)}f)(x)=p^{-(h-r)}
       \sum_{y\equiv x\, (\mathrm{mod}\,p^r)}f(y).
 \label{a-finite-cylinder-average}
```

E_r^{(h+1)}Use in source · 1 passage

Search aliases: E_r^(h+1)

Link to this entry ·

Use in source · finite.tex, line 402
Replicated vectors are constant in the new digit. All old cylinder averages commute with replication, giving E_r^(h+1)V_p,h · =V_p,hE_r^(h) (0≤ r≤ h), nonumber D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 E_r^{(h+1)}V_{p,h}&=V_{p,h}E_r^{(h)}\quad(0\leq r\leq h),
 \nonumber\\
 D_r^{(h+1)}V_{p,h}&=V_{p,h}D_r^{(h)}\quad(0\leq r\leq h),
 &D_{h+1}^{(h+1)}V_{p,h}&=0.
 \label{a-finite-detail-intertwining}
```

E_sUse in source · 3 passages

Search aliases: E_s

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Use in source · finite.tex, line 292
If r≤ s, the level-s cylinders partition each level-r cylinder into equally sized groups. Averaging over the smaller groups and then over the whole group returns the full average, giving E_rE_s=E_r. The self-adjointness of both sides also gives E_sE_r=E_r. This proves the minimum rule.
Exact mathematical passage
```
E_rE_s=E_r
```

E_{s-1}Use in source · 1 passage

Search aliases: E_s-1

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Use in source · finite.tex, line 298
For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.
Exact mathematical passage
```
(E_r-E_{r-1})(E_s-E_{s-1})
```

E_s^{(h)}Use in source · 1 passage

Search aliases: E_s^(h)

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Use in source · finite.tex, line 270
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 E_r^{(h)}E_s^{(h)}=E_{\min(r,s)}^{(h)},\qquad E_h^{(h)}=I.
```

e_yUse in source · 1 passage

Search aliases: e_y

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Use in source · refinement.tex, line 381
For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancella…
Exact mathematical passage
```
 \Theta_{xy}(p,h)
 ={}&\theta_y(p+h e_x,h)-\theta_y(p,h)\\
    &-\bigl(\theta_x(p+h e_y,h)-\theta_x(p,h)\bigr).
```

E_\pmUse in source · 3 passages

Search aliases: E_± pm

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Use in source · finite.tex, line 222
A separate arithmetic example shows why integral residue selectors and rational spectral idempotents should be distinguished. Let A=1 · 1 0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though it no longer …
Exact mathematical passage
```
 A=\begin{pmatrix}1&1\\0&-1\end{pmatrix},\qquad A^2=I,
 \qquad E_\pm=\frac{I\pm A}{2}.
```

E_\pm^2Use in source · 1 passage

Search aliases: E_±^2 pm

Link to this entry ·

Use in source · finite.tex, line 226
0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though it no longer distinguishes the two rational eigenvalues. These E_± are complementary algebraic spectral idempotents. Since A is not self-adjoint in the…
Exact mathematical passage
```
E_\pm^2=E_\pm
```

E_\lambdaUse in source · 3 passages

Search aliases: E_λ lambda

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Use in source · spectral.tex, line 222
Let D=∑_λλ E_λ be its spectral resolution. Then Σ(z)=∑_λfracB E_λ B^*z-λ. Every residue B E_λ B^* is positive semidefinite. Its rank records which directions of the hidden eigenspace reach the visible space. A vanishing residue removes the corresponding pole of Σ, even though z-D is still singular there.
Exact mathematical passage
```
D=\sum_\lambda\lambda E_\lambda
```

e_\perp^IUse in source · 1 passage

Search aliases: e_⊥^I perp

Link to this entry ·

Use in source · constraints.tex, line 422
An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 e^I=e_\perp^I\mathrm dt+E^I,\qquad
 \omega^{IJ}=\omega_\perp^{IJ}\mathrm dt+\Omega^{IJ},\qquad
 a=a_\perp\mathrm dt+A.
```

\mathrm{eff}Use in source · 5 passages

Search aliases: eff mathrm

Link to this entry ·

Use in source · core.tex, line 260
[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.
Exact mathematical passage
```
\label{f:effective-generator}
 \dot\phi=K_{\mathrm{eff}}\phi,\qquad
 K_{\mathrm{eff}}=U^*KU-\Gamma,\qquad
 \Gamma^*=-\Gamma,\quad K_{\mathrm{eff}}^*=-K_{\mathrm{eff}}.
```

\mathrm{emb}Use in source · 5 passages

Search aliases: emb mathrm

Link to this entry ·

Use in source · limit.tex, line 346
Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.
Exact mathematical passage
```
 U_M^{\mathrm{emb}}(t)
 =\sum_{m=0}^{M}e^{-it\epsilon_m}\mathsf F_m
 =U(t)\mathsf E_M.
 \label{a-limit-embedded-propagator}
```

\operatorname{erf}Use in source · 1 passage

Search aliases: erf operatorname

Link to this entry ·

Use in source · core.tex, line 288
For the normalized Gaussian density ρ(r)=fracM_0(2π)^3/2σ^3e^-r^2/(2σ^2), σ>0, direct differentiation verifies M_rm enc(r) · =M_0[ erf(frac rsqrt2σ) -sqrtfrac2πfrac rσ e^-r^2/(2σ^2)], Φ(r) · =-fracGM_0r erf(frac rsqrt2σ). The potential tends to zero at infinity and has the finite central value -sqrt2/π GM_0/σ. This removes the point-source singularity while preserving the attractive sign.
Exact mathematical passage
```
 M_{\rm enc}(r)&=M_0\left[
 \operatorname{erf}\left(\frac r{\sqrt2\sigma}\right)
 -\sqrt{\frac2\pi}\frac r\sigma e^{-r^2/(2\sigma^2)}\right],\\
 \Phi(r)&=-\frac{GM_0}{r}
 \operatorname{erf}\left(\frac r{\sqrt2\sigma}\right).
```

\mathcal FUse in source · 12 passages

Search aliases: F mathcal

Link to this entry ·

Use in source · finite_core.tex, line 56
The packet space F_S={f_a:a∈ K_S} will be identified isometrically with K_S. Its elements are smooth periodic functions, so the differential and centered-translation calculations below raise no regularity question. Empty support is allowed: both spaces then contain only zero and all packet identities reduce to 0=0.
Exact mathematical passage
```
\mathcal F_S=\{f_a:a\in\mathcal K_S\}
```

\mathrm fUse in source · 1 passage

Search aliases: f mathrm

Link to this entry ·

Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

\mathsf FUse in source · 27 passages

Search aliases: F mathsf

Link to this entry ·

Use in source · finite.tex, line 365
adicfigure02_haar_channelsFor p=3, the normalized child characters contain one average and two orthogonal complex details. Solid curves show real parts and dashed curves show imaginary parts. The level rank (p-1)p^r-1 grows by the branching factor. The atlas uses F_r for the corresponding infinite cylinder detail; its finite restriction is D_r^(h) for r≤ h.
Exact mathematical passage
```
\mathsf F_r
```

F'Use in source · 1 passage

Search aliases: F'

Link to this entry ·

Use in source · finite_core.tex, line 995
F'(s)=i e^-i(t-s)A(A-B)e^-isB.
Exact mathematical passage
```
F'(s)=i e^{-i(t-s)A}(A-B)e^{-isB}.
```

f'Use in source · 4 passages

Search aliases: f'

Link to this entry ·

Use in source · core.tex, line 469
An antisymmetric current also organizes derivatives of scalar functionals of the packet distribution. Let p_a(t)>0 be differentiable, normalized, and obey dot p_b=∑_aJ_b← a with J antisymmetric. For a differentiable real function f on the positive half-line, fracddt∑_b f(p_b) =frac12∑_b,aJ_b← a (f'(p_b)-f'(p_a)). To prove it, apply the ordinary chain rule and substitute the balance equation. Exchange the two indices in a second copy of the sum and use antisymmetry. Averaging the two expressions …
Exact mathematical passage
```
\label{f:functional-current}
 \frac{d}{dt}\sum_b f(p_b)
 =\frac12\sum_{b,a}J_{b\leftarrow a}
       \bigl(f'(p_b)-f'(p_a)\bigr).
```

f''Use in source · 1 passage

Search aliases: f''

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Use in source · core.tex, line 912
For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to control D_hq, the contraction in [source reference] is O(h^2) on a fixed periodic interval. Each summand is uniformly O(h^2) and hN is the interval leng…
Exact mathematical passage
```
 \mathfrak L_{D_h}(f,g)
 =\frac{h^2}{2}(f''g'+f'g'')+O(h^4).
 \label{c:core:smooth-defect}
```

f'''Use in source · 1 passage

Search aliases: f'''

Link to this entry ·

Use in source · core.tex, line 907
For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to control D_hq, the contraction in [source reference] is O(h^2) on a fixed periodic interval. Each summand is uniformly O(h^2) and hN is the interval leng…
Exact mathematical passage
```
 D_hf=f'+\frac{h^2}{6}f'''+O(h^4).
```

f^{-1}Use in source · 1 passage

Search aliases: f^-1

Link to this entry ·

Use in source · core.tex, line 189
Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a different physical system.
Exact mathematical passage
```
C_A=f^{-1}(A)
```

F^2Use in source · 1 passage

Search aliases: F^2

Link to this entry ·

Use in source · strong_fields.tex, line 142
ambda^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_trp(F) -fracMF_rF_tp(F)^3], which are the direct time derivatives of the two momenta. Finally, Φ_e'=QF_r/F^2 cancels NΛ Q/R^2 in dot A_r=NΛ Q/R^2-Φ_e'.
Exact mathematical passage
```
\Phi_e'=QF_r/F^2
```

f^2Use in source · 2 passages

Search aliases: f^2

Link to this entry ·

Use in source · finite.tex, line 305
The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an additional bound on the detail weights of the state.
Exact mathematical passage
```
 \mathcal H_{p,h}=\bigoplus_{r=0}^h\Ran D_r^{(h)},\qquad
 \norm f^2=\sum_{r=0}^h\norm{D_r^{(h)}f}^2.
 \label{a-finite-parseval}
```

F^{IJ}Use in source · 1 passage

Search aliases: F^IJ

Link to this entry ·

Use in source · constraints.tex, line 337
ta⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.
Exact mathematical passage
```
 S_{\mathrm M}^{(1)}\big|_{\varphi}
      =-\tfrac14\int F_{IJ}F^{IJ}\operatorname{vol}_e.
```

f_{\mathfrak H_p^s}Use in source · 1 passage

Search aliases: f_ H_p^s mathfrak

Link to this entry ·

Use in source · limit.tex, line 366
[Exact state error for a compatible cutoff] For every f∈HH_p, normU(t)f-U_M^emb(t)f_2 =norm(I- E_M)f_2. This equality is uniform for all real t. For f∈ H_p^s, the right side is bounded by p^-s(M+1)norm f_ H_p^s. For f∈Dom(H_ϵ) there is also the exact graph-norm identity · normU(t)f-U_M^emb(t)f_2^2 +normH_ϵ(U(t)f-U_M^emb(t)f)_2^2 nonumber
Exact mathematical passage
```
p^{-s(M+1)}\norm f_{\mathfrak H_p^s}
```

\mathsf F_{\mathbf r}Use in source · 5 passages

Search aliases: F_ r mathsf mathbf

Link to this entry ·

Use in source · limit.tex, line 575
Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.
Exact mathematical passage
```
 \mathsf F_{\mathbf r}
 =I_{\KK_{\mathrm{tor}}}\otimes
 \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
 \qquad \mathbf r\in\N^{\mathcal P}.
 \label{a-limit-joint-details}
```

f_{\mathbf r}Use in source · 2 passages

Search aliases: f_ r mathbf

Link to this entry ·

Use in source · limit.tex, line 633
domain and are dense. Symmetry holds by symmetry on every block. If g belongs to the adjoint domain and k=H_A^*g, testing against an arbitrary vector in KK_ r gives F_ rk=A_ r F_ rg. Square summability of k is precisely the domain condition in [source reference]. Thus H_A^*=H_A. Truncating the two square-summable sequences f_ r and A_ rf_ r proves the graph-core assertion.
Exact mathematical passage
```
f_{\mathbf r}
```

F_+Use in source · 4 passages

Search aliases: F_+

Link to this entry ·

Use in source · extensions.tex, line 20
[Overlapping two-channel weights] On C^2, take F_+=4/5 · 0 0 · 1/5, F_-=1/5 · 0 0 · 4/5, ψ=binom10. The true weights are 4/5 and 1/5. Squared effect-vector norms would instead give 16/25 and 1/25, whose sum is 17/25. The state still satisfies F_+ψ+F_-ψ=ψ, but its two summands are not orthogonal. Reconstruction and an orthogonal decomposition are different assertions.
Exact mathematical passage
```
 F_+=\begin{pmatrix}4/5&0\\0&1/5\end{pmatrix},\qquad
 F_-=\begin{pmatrix}1/5&0\\0&4/5\end{pmatrix},\qquad \psi=\binom10.
```

F_-Use in source · 3 passages

Search aliases: F_-

Link to this entry ·

Use in source · extensions.tex, line 20
[Overlapping two-channel weights] On C^2, take F_+=4/5 · 0 0 · 1/5, F_-=1/5 · 0 0 · 4/5, ψ=binom10. The true weights are 4/5 and 1/5. Squared effect-vector norms would instead give 16/25 and 1/25, whose sum is 17/25. The state still satisfies F_+ψ+F_-ψ=ψ, but its two summands are not orthogonal. Reconstruction and an orthogonal decomposition are different assertions.
Exact mathematical passage
```
 F_+=\begin{pmatrix}4/5&0\\0&1/5\end{pmatrix},\qquad
 F_-=\begin{pmatrix}1/5&0\\0&4/5\end{pmatrix},\qquad \psi=\binom10.
```

\mathsf F_0Use in source · 1 passage

Search aliases: F_0 mathsf

Link to this entry ·

Use in source · limit.tex, line 146
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involvin
Exact mathematical passage
```
\mathsf F_0=\mathsf E_0
```

f_0Use in source · 1 passage

Search aliases: f_0

Link to this entry ·

Use in source · spectral.tex, line 1122
For an explicit example, take L=4. The scalar formula gives f_0(λ)=tfrac12(cosh(tλ)+cos(tλ)), f_2(λ)=tfrac12(cos(tλ)-cosh(tλ)). If H_∞ is multiplication by λ on L^2(R) and t≠0, these functions define unbounded operators. Their sum cancels the growing hyperbolic contribution. The full propagator e^-itH_∞ is nevertheless unitary on all of L^2(R). For L=2, by contrast, the two Schr odinger lanes are cos(tH_∞) and -isin(tH_∞), both bounded. A larger lane count must not inherit this special dyadic bo…
Exact mathematical passage
```
 f_0(\lambda)=\tfrac12\bigl(\cosh(t\lambda)+\cos(t\lambda)\bigr),
 \quad
 f_2(\lambda)=\tfrac12\bigl(\cos(t\lambda)-\cosh(t\lambda)\bigr).
```

F_1Use in source · 4 passages

Search aliases: F_1

Link to this entry ·

Use in source · spectral.tex, line 1055
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -
Exact mathematical passage
```
 B(y)=\sum_{n=0}^\infty\frac{(-1)^ny^n}{(n!)^2}
     ={}_0F_1(;1;-y).
```

f_{1/12}Use in source · 1 passage

Search aliases: f_1/12

Link to this entry ·

Use in source · finite_core.tex, line 1086
f_1/12(t)-f(t) _L^2 ≤(2π-6)|t|≤fracπ^3108|t|.
Exact mathematical passage
```
\|f_{1/12}(t)-f(t)\|_{L^2}
\le(2\pi-6)|t|\le\frac{\pi^3}{108}|t|.
```

f_2Use in source · 3 passages

Search aliases: f_2

Link to this entry ·

Use in source · limit.tex, line 282
For f∈Dom(H_ϵ), the cylinder truncation E_M f satisfies both normf- E_M f_2→0 and normH_ϵ(f- E_M f)_2→0, because the two squared norms are the tails of convergent nonnegative series. This is precisely density in the graph norm and proves the core assertion. If the energies have supremum C<∞, Parseval gives normH_ϵ f_2≤ Cnorm f_2. Conversely, a unit vector on level m has image norm |ϵ_m|, so boundedness forces a finite supremum.
Exact mathematical passage
```
\norm{H_\epsilon f}_2\leq C\norm f_2
```

F_{2^{h_2}}Use in source · 1 passage

Search aliases: F_2^h_2

Link to this entry ·

Use in source · finite.tex, line 149
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
 \left[F_2\otimes F_{2^{h_2}}\otimes
 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
 \right]\mathcal U_{\mathrm{CRT}}.
 \label{a-finite-unit-fourier}
```

F_{2L-1}Use in source · 1 passage

Search aliases: F_2L-1

Link to this entry ·

Use in source · spectral.tex, line 1063
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters ar…
Exact mathematical passage
```
 B_{L,j}(y)
 =\frac{(-1)^jy^j}{(j!)^2}\,
 {}_0F_{2L-1}\!\left(
 \begin{matrix}-\\1,\mathcal A_{L,j},\mathcal A_{L,j}\end{matrix};
 \frac{(-1)^Ly^L}{L^{2L}}\right).
 \label{a-spectral-bessel-lane}
```

f_aUse in source · 4 passages

Search aliases: f_a

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Use in source · finite_core.tex, line 44
f_a(x)=∑_n∈ Se_n(x)a_n.
Exact mathematical passage
```
f_a(x)=\sum_{n\in S}e_n(x)a_n.
```

f_{a+1}Use in source · 1 passage

Search aliases: f_a+1

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Use in source · core.tex, line 891
[An exact bracket failure] Use N=5, h=1, and (Df)_a=(f_a+1-f_a-1)/2. Take q=e_0, p=ξ=e_1, and η=e_2. Then (Dq)_1=-1/2, (Dη)_1=1/2, and η Dq=0. The direct bracket in the preceding proof is zero, while D_h[ξ Dη-η Dξ]=-tfrac14. The defect is therefore exactly +1/4. This finite witness shows that the naive centered scalar constraint does fail to close universally in the proposed undeformed form.
Exact mathematical passage
```
(Df)_a=(f_{a+1}-f_{a-1})/2
```

f_{a-1}Use in source · 1 passage

Search aliases: f_a-1

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Use in source · core.tex, line 891
[An exact bracket failure] Use N=5, h=1, and (Df)_a=(f_a+1-f_a-1)/2. Take q=e_0, p=ξ=e_1, and η=e_2. Then (Dq)_1=-1/2, (Dη)_1=1/2, and η Dq=0. The direct bracket in the preceding proof is zero, while D_h[ξ Dη-η Dξ]=-tfrac14. The defect is therefore exactly +1/4. This finite witness shows that the naive centered scalar constraint does fail to close universally in the proposed undeformed form.
Exact mathematical passage
```
(Df)_a=(f_{a+1}-f_{a-1})/2
```

F_a^{1/2}Use in source · 1 passage

Search aliases: F_a^1/2

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Use in source · extensions.tex, line 88
*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_a=F_a^1/2. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measurement state is being made by this algebraic dilation.
Exact mathematical passage
```
V_a=F_a^{1/2}
```

F_a^2Use in source · 4 passages

Search aliases: F_a^2

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Use in source · extensions.tex, line 16
For a sharp atlas, F_a=P_a and p_a=normP_aψ^2. In general, p_a is not normF_aψ^2. The latter quantity is innerψF_a^2ψ and therefore loses precisely the positive defect F_a-F_a^2.
Exact mathematical passage
```
\inner{\psi}{F_a^2\psi}
```

F_{ab}^iUse in source · 1 passage

Search aliases: F_ab^i

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

F_{ab}^kUse in source · 1 passage

Search aliases: F_ab^k

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

F_{a\leftarrow b}Use in source · 1 passage

Search aliases: F_a← b leftarrow

Link to this entry ·

Use in source · core.tex, line 420
[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.
Exact mathematical passage
```
\label{f:jump-current}
 \dot p_b=\sum_aF_{b\leftarrow a},\qquad
 F_{b\leftarrow a}=\gamma_{ba}p_a-\gamma_{ab}p_b,
 \qquad F_{b\leftarrow a}=-F_{a\leftarrow b}.
```

F_bUse in source · 8 passages

Search aliases: F_b

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Use in source · finite.tex, line 171
P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permut…
Exact mathematical passage
```
 F_N=\mathcal P_{\mathrm{out}}^*(F_a\otimes F_b)
       \mathcal P_{\mathrm{in}}.
 \label{a-finite-additive-fourier}
```

\mathcal F_{bc}Use in source · 1 passage

Search aliases: F_bc mathcal

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Use in source · constraints.tex, line 572
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
 \mathcal H_{\mathrm M}=\frac1{2e}q_{ab}
       (\mathcal E^a\mathcal E^b+\mathcal B^a\mathcal B^b),
 \qquad \mathcal B^a=\tfrac12\epsilon^{abc}\mathcal F_{bc}.
```

F_{b\leftarrow a}Use in source · 3 passages

Search aliases: F_b← a leftarrow

Link to this entry ·

Use in source · core.tex, line 420
[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.
Exact mathematical passage
```
\label{f:jump-current}
 \dot p_b=\sum_aF_{b\leftarrow a},\qquad
 F_{b\leftarrow a}=\gamma_{ba}p_a-\gamma_{ab}p_b,
 \qquad F_{b\leftarrow a}=-F_{a\leftarrow b}.
```

F_hUse in source · 4 passages

Search aliases: F_h

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Use in source · core.tex, line 421
ations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connection has additional derivative terms and must be analyzed with those terms present.
Exact mathematical passage
```
\label{g:log-curvature}
 F_h=h^{-2}\log W_h,\qquad h\ne0.
```

f_hUse in source · 16 passages

Search aliases: f_h

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Use in source · finite_core.tex, line 453
f(t,x)=∑_n∈ Se_n(x)U_H_n(t)a_n, f_h(t,x)=∑_n∈ Se_n(x)U_H_h,n(t)a_n.
Exact mathematical passage
```
f(t,x)=\sum_{n\in S}e_n(x)U_{H_n}(t)a_n,\qquad
f_h(t,x)=\sum_{n\in S}e_n(x)U_{H_{h,n}}(t)a_n.
```

f_{h,a}Use in source · 1 passage

Search aliases: f_h,a

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Use in source · finite_core.tex, line 487
⟨ f_h,a(t),f_h,b(t)⟩_L^2 =∑_n∈ S ⟨ U_H_h,n(t)a_n,U_H_h,n(t)b_n⟩ =∑_n∈ S⟨ a_n,b_n⟩.
Exact mathematical passage
```
\langle f_{h,a}(t),f_{h,b}(t)\rangle_{L^2}
=\sum_{n\in S}
\langle U_{H_{h,n}}(t)a_n,U_{H_{h,n}}(t)b_n\rangle
=\sum_{n\in S}\langle a_n,b_n\rangle.
```

f_{h,b}Use in source · 1 passage

Search aliases: f_h,b

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Use in source · finite_core.tex, line 487
⟨ f_h,a(t),f_h,b(t)⟩_L^2 =∑_n∈ S ⟨ U_H_h,n(t)a_n,U_H_h,n(t)b_n⟩ =∑_n∈ S⟨ a_n,b_n⟩.
Exact mathematical passage
```
\langle f_{h,a}(t),f_{h,b}(t)\rangle_{L^2}
=\sum_{n\in S}
\langle U_{H_{h,n}}(t)a_n,U_{H_{h,n}}(t)b_n\rangle
=\sum_{n\in S}\langle a_n,b_n\rangle.
```

F_{h/2}Use in source · 2 passages

Search aliases: F_h/2

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Use in source · core.tex, line 430
[Local logarithmic refinement] On the common analytic logarithm branch, in any fixed matrix norm, F_h · =[A,B]+hC_3(A,B)+O(h^2), C_3(A,B) · =tfrac12[A+B,[A,B]], F_h^[1] · :=2F_h/2-F_h=[A,B]+O(h^2). For constant G∈ SU(2), simultaneous conjugation of A,B conjugates both F_h and F_h^[1] by G.
Exact mathematical passage
```
 F_h&=[A,B]+hC_3(A,B)+O(h^2),\label{g:log-series}\\
 C_3(A,B)&=\tfrac12[A+B,[A,B]],\label{g:cubic-coefficient}\\
 F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).
 \label{g:richardson-formula}
```

F_h^{[1]}Use in source · 2 passages

Search aliases: F_h^[1]

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Use in source · core.tex, line 430
[Local logarithmic refinement] On the common analytic logarithm branch, in any fixed matrix norm, F_h · =[A,B]+hC_3(A,B)+O(h^2), C_3(A,B) · =tfrac12[A+B,[A,B]], F_h^[1] · :=2F_h/2-F_h=[A,B]+O(h^2). For constant G∈ SU(2), simultaneous conjugation of A,B conjugates both F_h and F_h^[1] by G.
Exact mathematical passage
```
 F_h&=[A,B]+hC_3(A,B)+O(h^2),\label{g:log-series}\\
 C_3(A,B)&=\tfrac12[A+B,[A,B]],\label{g:cubic-coefficient}\\
 F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).
 \label{g:richardson-formula}
```

F_{IJ}Use in source · 1 passage

Search aliases: F_IJ

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Use in source · constraints.tex, line 337
ta⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.
Exact mathematical passage
```
 S_{\mathrm M}^{(1)}\big|_{\varphi}
      =-\tfrac14\int F_{IJ}F^{IJ}\operatorname{vol}_e.
```

F_{ij}Use in source · 4 passages

Search aliases: F_ij

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Use in source · refinement.tex, line 353
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
h^2F_{ij}
```

\mathcal F_{ij}Use in source · 2 passages

Search aliases: F_ij mathcal

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Use in source · extensions.tex, line 479
For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely when that X commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additiona…
Exact mathematical passage
```
\mathcal F_{ij}=i[B_i,B_j]
```

F_jUse in source · 3 passages

Search aliases: F_j

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Use in source · refinement.tex, line 248
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
F_j:Y\to Y
```

f_jUse in source · 13 passages

Search aliases: f_j

Link to this entry ·

Use in source · spectral.tex, line 1103
Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say that each lane acts on every vector.
Exact mathematical passage
```
f_j(\lambda)=E_{L,j}(-it\lambda)
```

F_{j+1}Use in source · 1 passage

Search aliases: F_j+1

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Use in source · refinement.tex, line 253
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_N\cdots F_1I-IC_N\cdots C_1
 =\sum_{j=1}^N F_N\cdots F_{j+1}D_jC_{j-1}\cdots C_1.
 \label{g:telescoping}
```

f_{j+1}Use in source · 2 passages

Search aliases: f_j+1

Link to this entry ·

Use in source · extensions.tex, line 437
[The precise defect of a forward difference] On a cyclic grid, let D_hf_j=(f_j+1-f_j)/h, with h≠0. Expanding the product gives D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j =h(D_hf)_j(D_hg)_j. Thus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go theorem, because that theorem concerns an exact identity for every vector at fixed finite resolution.
Exact mathematical passage
```
D_hf_j=(f_{j+1}-f_j)/h
```

F_{j+1/2}Use in source · 2 passages

Search aliases: F_j+1/2

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

f_{j-1}Use in source · 1 passage

Search aliases: f_j-1

Link to this entry ·

Use in source · core.tex, line 470
Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber
Exact mathematical passage
```
 (D_hf)_j=\frac{f_{j+1}-f_{j-1}}{2h},\qquad
 (\Delta_hf)_j=\frac{f_{j+1}-2f_j+f_{j-1}}{h^2}.
```

F_{j-1/2}Use in source · 1 passage

Search aliases: F_j-1/2

Link to this entry ·

Use in source · core.tex, line 66
s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.
Exact mathematical passage
```
 \dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.
 \label{c:core:scalar-balance}
```

F_MUse in source · 1 passage

Search aliases: F_M

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Use in source · spectral.tex, line 95
Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1,
Exact mathematical passage
```
(F_M)_{jk}=M^{-1/2}e^{-2\pi ijk/M}
```

\mathsf F_mUse in source · 19 passages

Search aliases: F_m mathsf

Link to this entry ·

Use in source · limit.tex, line 147
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_
Exact mathematical passage
```
\mathsf F_m=\mathsf E_m-\mathsf E_{m-1}
```

f_mUse in source · 2 passages

Search aliases: f_m

Link to this entry ·

Use in source · limit.tex, line 265
losed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on…
Exact mathematical passage
```
f_m=\mathsf F_mf
```

\mathsf F_{M+1}Use in source · 1 passage

Search aliases: F_M+1 mathsf

Link to this entry ·

Use in source · limit.tex, line 224
ence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted summands by this lower bound proves the squared estimate, and taking square roots gives [source reference]. At every finite M there is a unit vector in Ran F_M+1, on which I- E_M acts as the identity. Thus normI- E_M=1 for every M.
Exact mathematical passage
```
\operatorname{Ran}\mathsf F_{M+1}
```

F_NUse in source · 3 passages

Search aliases: F_N

Link to this entry ·

Use in source · finite.tex, line 171
P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permut…
Exact mathematical passage
```
 F_N=\mathcal P_{\mathrm{out}}^*(F_a\otimes F_b)
       \mathcal P_{\mathrm{in}}.
 \label{a-finite-additive-fourier}
```

F_nUse in source · 1 passage

Search aliases: F_n

Link to this entry ·

Use in source · finite.tex, line 144
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 (F_n)_{k x}=n^{-1/2}\exp(2\pi\mathrm{i}kx/n).
 \label{a-finite-fourier-convention}
```

\mathsf F_nUse in source · 1 passage

Search aliases: F_n mathsf

Link to this entry ·

Use in source · limit.tex, line 151
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
 \mathsf F_m\mathsf F_n=\delta_{mn}\mathsf F_m,
 \qquad \sum_{m=0}^M\mathsf F_m=\mathsf E_M.
 \label{a-limit-detail-algebra}
```

f_nUse in source · 6 passages

Search aliases: f_n

Link to this entry ·

Use in source · refinement.tex, line 438
[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence is Cauchy. Completeness provides a limit. Passing m→∞ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric ser…
Exact mathematical passage
```
f_n
```

f_{n+1}Use in source · 1 passage

Search aliases: f_n+1

Link to this entry ·

Use in source · refinement.tex, line 439
[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence is Cauchy. Completeness provides a limit. Passing m→∞ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric ser…
Exact mathematical passage
```
 d(f_n,f_{n+1})\leq C4^{-n}\qquad(n\geq0),\quad C\geq0.
```

F_pUse in source · 2 passages

Search aliases: F_p

Link to this entry ·

Use in source · finite.tex, line 464
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,
Exact mathematical passage
```
 \mathcal W_{p,h}=(F_p\otimes I_m)\mathcal A^{\mathrm{ad}}_{p,h}.
 \label{a-finite-child-fourier}
```

\mathsf F_{p,r_p}Use in source · 1 passage

Search aliases: F_p,r_p mathsf

Link to this entry ·

Use in source · limit.tex, line 575
Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.
Exact mathematical passage
```
 \mathsf F_{\mathbf r}
 =I_{\KK_{\mathrm{tor}}}\otimes
 \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
 \qquad \mathbf r\in\N^{\mathcal P}.
 \label{a-limit-joint-details}
```

F_{p-1}Use in source · 1 passage

Search aliases: F_p-1

Link to this entry ·

Use in source · finite.tex, line 149
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
 \left[F_2\otimes F_{2^{h_2}}\otimes
 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
 \right]\mathcal U_{\mathrm{CRT}}.
 \label{a-finite-unit-fourier}
```

F_{p^{h_p}}Use in source · 1 passage

Search aliases: F_p^h_p

Link to this entry ·

Use in source · finite.tex, line 149
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
 \left[F_2\otimes F_{2^{h_2}}\otimes
 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
 \right]\mathcal U_{\mathrm{CRT}}.
 \label{a-finite-unit-fourier}
```

F_{p_a}Use in source · 1 passage

Search aliases: F_p_a

Link to this entry ·

Use in source · orientation.tex, line 24
midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.
Exact mathematical passage
```
\{F,G\}=\sum_a(F_{q_a}G_{p_a}-F_{p_a}G_{q_a})
```

f_qUse in source · 5 passages

Search aliases: f_q

Link to this entry ·

Use in source · limit.tex, line 433
[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For q=1/8, the state norm tail decays faster than the graph norm tail. For q=1/2, the normalized state still has a convergent L^2 cylinder approximation a…
Exact mathematical passage
```
 f_q=\sum_{m\geq0}\sqrt{(1-q)q^m}\,u_m,
 \qquad \epsilon_m=2^m.
```

F_{q_a}Use in source · 1 passage

Search aliases: F_q_a

Link to this entry ·

Use in source · orientation.tex, line 24
midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.
Exact mathematical passage
```
\{F,G\}=\sum_a(F_{q_a}G_{p_a}-F_{p_a}G_{q_a})
```

F_rUse in source · 11 passages

Search aliases: F_r

Link to this entry ·

Use in source · strong_fields.tex, line 94
Take a smooth radial map R=F(t,r) with F_r>0 and identify t_P=t. Expanding dR=F_tdt+F_rdr identifies N=1, Λ=F_r, N^r=fracF_t+v(F)F_r. The corresponding canonical data are P_Λ=p(F), P_R=fracMF_rp(F), A_r=0, Φ_e=-frac QF, Q'=0. The sign in P_R is contained in p(F).
Exact mathematical passage
```
F_r>0
```

\mathsf F_rUse in source · 1 passage

Search aliases: F_r mathsf

Link to this entry ·

Use in source · finite.tex, line 365
adicfigure02_haar_channelsFor p=3, the normalized child characters contain one average and two orthogonal complex details. Solid curves show real parts and dashed curves show imaginary parts. The level rank (p-1)p^r-1 grows by the branching factor. The atlas uses F_r for the corresponding infinite cylinder detail; its finite restriction is D_r^(h) for r≤ h.
Exact mathematical passage
```
\mathsf F_r
```

F_{r-1/2}Use in source · 1 passage

Search aliases: F_r-1/2

Link to this entry ·

Use in source · core.tex, line 99
Every e_j is nonnegative under the stated coefficient assumptions. An interval of cells r,…,s obeys the exact coarse balance fracddt∑_j=r^se_j +F_s+1/2-F_r-1/2=∑_j=r^ss_j. Thus coarse energy is obtained by adding fine cell energies and retaining only the exposed boundary fluxes. This is the same cancellation mechanism as projector transfer, applied here to a classical field energy. Its physical units and constitutive coefficients come from the action.
Exact mathematical passage
```
 \frac{d}{dt}\sum_{j=r}^se_j
 +F_{s+1/2}-F_{r-1/2}=\sum_{j=r}^ss_j.
```

\mathcal F_SUse in source · 5 passages

Search aliases: F_S mathcal

Link to this entry ·

Use in source · finite_core.tex, line 56
The packet space F_S={f_a:a∈ K_S} will be identified isometrically with K_S. Its elements are smooth periodic functions, so the differential and centered-translation calculations below raise no regularity question. Empty support is allowed: both spaces then contain only zero and all packet identities reduce to 0=0.
Exact mathematical passage
```
\mathcal F_S=\{f_a:a\in\mathcal K_S\}
```

F_{s+1/2}Use in source · 1 passage

Search aliases: F_s+1/2

Link to this entry ·

Use in source · core.tex, line 99
Every e_j is nonnegative under the stated coefficient assumptions. An interval of cells r,…,s obeys the exact coarse balance fracddt∑_j=r^se_j +F_s+1/2-F_r-1/2=∑_j=r^ss_j. Thus coarse energy is obtained by adding fine cell energies and retaining only the exposed boundary fluxes. This is the same cancellation mechanism as projector transfer, applied here to a classical field energy. Its physical units and constitutive coefficients come from the action.
Exact mathematical passage
```
 \frac{d}{dt}\sum_{j=r}^se_j
 +F_{s+1/2}-F_{r-1/2}=\sum_{j=r}^ss_j.
```

F_tUse in source · 4 passages

Search aliases: F_t

Link to this entry ·

Use in source · strong_fields.tex, line 94
Take a smooth radial map R=F(t,r) with F_r>0 and identify t_P=t. Expanding dR=F_tdt+F_rdr identifies N=1, Λ=F_r, N^r=fracF_t+v(F)F_r. The corresponding canonical data are P_Λ=p(F), P_R=fracMF_rp(F), A_r=0, Φ_e=-frac QF, Q'=0. The sign in P_R is contained in p(F).
Exact mathematical passage
```
dR=F_tdt+F_rdr
```

F_{tr}Use in source · 3 passages

Search aliases: F_tr

Link to this entry ·

Use in source · strong_fields.tex, line 126
For evolution, the radius equation gives dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2],
Exact mathematical passage
```
F_{tr}
```

F_{U_M}Use in source · 1 passage

Search aliases: F_U_M

Link to this entry ·

Use in source · finite.tex, line 149
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
 \left[F_2\otimes F_{2^{h_2}}\otimes
 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
 \right]\mathcal U_{\mathrm{CRT}}.
 \label{a-finite-unit-fourier}
```

f_xUse in source · 5 passages

Search aliases: f_x

Link to this entry ·

Use in source · core.tex, line 311
For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bounds for A and ψ give the required uniform bound on f_x'''.
Exact mathematical passage
```
f_x(s)=\exp(-i\int_x^{x+s}A(r)\,dr)\psi(x+s)
```

f_x'Use in source · 2 passages

Search aliases: f_x'

Link to this entry ·

Use in source · core.tex, line 313
For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bounds for A and ψ give the required uniform bound on f_x'''.
Exact mathematical passage
```
f_x'(0)=\psi'(x)-iA(x)\psi(x)
```

f_x'''Use in source · 1 passage

Search aliases: f_x'''

Link to this entry ·

Use in source · core.tex, line 316
) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bounds for A and ψ give the required uniform bound on f_x'''.
Exact mathematical passage
```
f_x'''
```

F_{xy}Use in source · 2 passages

Search aliases: F_xy

Link to this entry ·

Use in source · refinement.tex, line 354
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
 \norm{D_{\rm total}(h)}
 \leq h^2\bigl(|F_{xy}|+2|F_{xz}|+2|F_{yz}|\bigr).
 \label{g:affine-bound}
```

F_{xz}Use in source · 1 passage

Search aliases: F_xz

Link to this entry ·

Use in source · refinement.tex, line 354
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
 \norm{D_{\rm total}(h)}
 \leq h^2\bigl(|F_{xy}|+2|F_{xz}|+2|F_{yz}|\bigr).
 \label{g:affine-bound}
```

F_{yz}Use in source · 1 passage

Search aliases: F_yz

Link to this entry ·

Use in source · refinement.tex, line 354
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
 \norm{D_{\rm total}(h)}
 \leq h^2\bigl(|F_{xy}|+2|F_{xz}|+2|F_{yz}|\bigr).
 \label{g:affine-bound}
```

F_{\mu\nu}Use in source · 4 passages

Search aliases: F_μν mu nu

Link to this entry ·

Use in source · core.tex, line 179
Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Conse…
Exact mathematical passage
```
\theta=-q\int_{\square}F_{\mu\nu}\,dS
```

F_\inftyUse in source · 1 passage

Search aliases: F_∞ infty

Link to this entry ·

Use in source · limit.tex, line 642
rectangular projection selects a collection of whole blocks, so it commutes with H_A on the domain and reduces the operator. The restriction to each retained block is literally the same matrix at every subsequent depth. This proves the finite intertwining, and finite-dimensional spectral calculus on each block gives [source reference]. Boundedness of F bounds its omitted image by norm F_∞norm(I- E_ h)f, proving strong convergence. Finally, each block exponential is unitary, so orthogonality of t…
Exact mathematical passage
```
\norm F_\infty\norm{(I-\mathsf E_{\mathbf h})f}
```

f_\inftyUse in source · 2 passages

Search aliases: f_∞ infty

Link to this entry ·

Use in source · refinement.tex, line 442
[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence is Cauchy. Completeness provides a limit. Passing m→∞ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric ser…
Exact mathematical passage
```
f_\infty
```

\mathrm{fine}Use in source · 1 passage

Search aliases: fine mathrm

Link to this entry ·

Use in source · refinement.tex, line 186
lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass substeps also intertwine, the identities compose through the entire execution order M-X-Y-Z-Y-X-M. The refined walk here replaces each coarse spatial fac…
Exact mathematical passage
```
 S_{i,\mathrm{fine}}^2 I_r=I_rS_{i,\mathrm{coarse}}.
```

\mathrm{fld}Use in source · 1 passage

Search aliases: fld mathrm

Link to this entry ·

Use in source · constraints.tex, line 350
The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.
Exact mathematical passage
```
 [\delta_U,\delta_V]_{\mathrm{fld}}
       =-\delta_{\left[U,V\right]_{\mathrm{par}}},
```

\mathcal GUse in source · 23 passages

Search aliases: G mathcal

Link to this entry ·

Use in source · limit.tex, line 819
Let N≥3 and h≥0. Let C_N be the physical cycle and let T_p,h be the rooted finite p-ary tree of prefixes of length at most h. The product graph is G=C_Nsquare T_p,h. A vertex is v=(x, d), with x∈Z/NZ and d a prefix. An edge changes either x by one on the cycle or the prefix by one parent-child step. The graph has N∑_r=0^h p^r vertices. It includes all tree depths, unlike a cylinder register which contains only the residues at its last depth.
Exact mathematical passage
```
 \mathcal G=C_N\square T_{p,h}.
```

\mathrm GUse in source · 4 passages

Search aliases: G mathrm

Link to this entry ·

Use in source · constraints.tex, line 303
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,
Exact mathematical passage
```
S_0=S_{\mathrm G}+S_{\mathrm M}^{(1)}+S_{\mathrm D}
```

g'Use in source · 3 passages

Search aliases: g'

Link to this entry ·

Use in source · core.tex, line 912
For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to control D_hq, the contraction in [source reference] is O(h^2) on a fixed periodic interval. Each summand is uniformly O(h^2) and hN is the interval leng…
Exact mathematical passage
```
 \mathfrak L_{D_h}(f,g)
 =\frac{h^2}{2}(f''g'+f'g'')+O(h^4).
 \label{c:core:smooth-defect}
```

g''Use in source · 1 passage

Search aliases: g''

Link to this entry ·

Use in source · core.tex, line 912
For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to control D_hq, the contraction in [source reference] is O(h^2) on a fixed periodic interval. Each summand is uniformly O(h^2) and hN is the interval leng…
Exact mathematical passage
```
 \mathfrak L_{D_h}(f,g)
 =\frac{h^2}{2}(f''g'+f'g'')+O(h^4).
 \label{c:core:smooth-defect}
```

g^2Use in source · 4 passages

Search aliases: g^2

Link to this entry ·

Use in source · constraints.tex, line 378
Nilpotency follows from the graded Jacobi identity for the parameter algebra and its action on each representation. A simple internal check makes the sign visible. Write the Lorentz representation-valued ghost as g=tfrac14χ^IJγ_IJ and suppress diffeomorphisms and the Abelian factor. Then sg=g^2, sψ=gψ, and the dual rule is sbarψ=barψ g when the odd ghost has been moved to the right. Thus s^2ψ=g^2ψ-g(gψ)=0, s^2barψ=(barψ g)g-barψ g^2=0. Using sbarψ=-barψ g with the same left-derivation convention…
Exact mathematical passage
```
sg=g^2
```

G^nUse in source · 1 passage

Search aliases: G^n

Link to this entry ·

Use in source · core.tex, line 80
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\label{g:walk-covariance}
 \mathcal W_n[U^g,T^g]G^n=G^{n+1}\mathcal W_n[U,T],
 \qquad (G^n\psi)_j=g_j^n\psi_j.
```

G^{n+1}Use in source · 1 passage

Search aliases: G^n+1

Link to this entry ·

Use in source · core.tex, line 80
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\label{g:walk-covariance}
 \mathcal W_n[U^g,T^g]G^n=G^{n+1}\mathcal W_n[U,T],
 \qquad (G^n\psi)_j=g_j^n\psi_j.
```

G^TUse in source · 2 passages

Search aliases: G^T

Link to this entry ·

Use in source · constraints.tex, line 685
equation overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse are skew, transposition proves (J_ D^Δ)^T=-J_ D^Δ. The endpoint tensor is a projection of the canonical direction into the kernel…
Exact mathematical passage
```
\Delta_\Delta=GJG^T
```

G_{\mathrm L}Use in source · 3 passages

Search aliases: G_ L mathrm

Link to this entry ·

Use in source · constraints.tex, line 478
What is explicit in the internal BFV sector. On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus {Ω_int,Ω_int}=0 under that regular moment-map hypot…
Exact mathematical passage
```
 \{G_{\mathrm L}[\lambda],G_{\mathrm L}[\mu]\}
       =G_{\mathrm L}[[\lambda,\mu]],\qquad
 \{G_{\mathrm L},G_{\mathrm U}\}=\{G_{\mathrm U},G_{\mathrm U}\}=0.
```

G_{\mathrm U}Use in source · 3 passages

Search aliases: G_ U mathrm

Link to this entry ·

Use in source · constraints.tex, line 478
What is explicit in the internal BFV sector. On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus {Ω_int,Ω_int}=0 under that regular moment-map hypot…
Exact mathematical passage
```
 \{G_{\mathrm L}[\lambda],G_{\mathrm L}[\mu]\}
       =G_{\mathrm L}[[\lambda,\mu]],\qquad
 \{G_{\mathrm L},G_{\mathrm U}\}=\{G_{\mathrm U},G_{\mathrm U}\}=0.
```

\mathcal G_{\mathrm U}Use in source · 2 passages

Search aliases: G_ U mathcal mathrm

Link to this entry ·

Use in source · constraints.tex, line 465
C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.
Exact mathematical passage
```
 \mathcal G_{IJ}
 &=D_\Omega B_{IJ}
       -\tfrac i8\sigma_K\bar\psi\{\gamma^K,\gamma_{IJ}\}\psi,\\
 \mathcal G_{\mathrm U}
 &=\mathrm d_\Sigma\Pi-q\sigma_I\bar\psi\gamma^I\psi.
```

\mathcal G_{\mathrm U}^{\mathrm{red}}Use in source · 1 passage

Search aliases: G_ U^red mathcal mathrm mathrm

Link to this entry ·

Use in source · constraints.tex, line 645
For example, the residual electric constraint is G_ U^red=∂_a E^a+qΨ^†Ψ, Q[u]=-∫_Σ u G_ U^red d^3x. It generates δ_uA_a=∂_au and δ_uΨ=iquΨ. The half-density Lie derivative in the spatial generator is L_η^(1/2)Ψ =η^a∂_aΨ+tfrac12(∂_aη^a)Ψ. This term is forced by the density weight. Treating Ψ as a scalar would give a different moment map.
Exact mathematical passage
```
 \mathcal G_{\mathrm U}^{\mathrm{red}}=\partial_a\mathcal E^a+q\Psi^\dagger\Psi,
 \qquad Q[u]=-\int_\Sigma u\mathcal G_{\mathrm U}^{\mathrm{red}}\,\mathrm d^3x.
```

g_0Use in source · 1 passage

Search aliases: g_0

Link to this entry ·

Use in source · spectral.tex, line 441
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
 \langle\mathrm{root},(z-H)^{-1}\mathrm{root}\rangle
   =g_0(z)=\frac1{z-\epsilon_0-\Sigma_0(z)}.
 \label{a-spectral-root-green}
```

G_gUse in source · 1 passage

Search aliases: G_g

Link to this entry ·

Use in source · core.tex, line 137
tution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing x through τ in each channel, gives ∑_xinnerSψ(x)Sϕ(x) =∑_xinnerψ(x)ϕ(x). For u^g(x)=g(τ x)u(x)g(x)^-1, direct endpoint cancellation gives S_A,τ,u^gG_g=G_gS_A,τ,u. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.
Exact mathematical passage
```
S_{A,\tau,u^g}G_g=G_gS_{A,\tau,u}
```

G_hUse in source · 2 passages

Search aliases: G_h

Link to this entry ·

Use in source · core.tex, line 498
g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
G_h=(0,0,g_U,g_V)
```

g_hUse in source · 1 passage

Search aliases: g_h

Link to this entry ·

Use in source · spectral.tex, line 433
For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rationa…
Exact mathematical passage
```
 g_h(z)&=(z-\epsilon_h)^{-1},\nonumber\\
 g_r(z)&=\frac1{z-\epsilon_r-\gamma_r^2g_{r+1}(z)},
          \qquad r<h.
 \label{a-spectral-continued-fraction}
```

\mathcal G_iUse in source · 2 passages

Search aliases: G_i mathcal

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

\mathcal G_{IJ}Use in source · 2 passages

Search aliases: G_IJ mathcal

Link to this entry ·

Use in source · constraints.tex, line 465
C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.
Exact mathematical passage
```
 \mathcal G_{IJ}
 &=D_\Omega B_{IJ}
       -\tfrac i8\sigma_K\bar\psi\{\gamma^K,\gamma_{IJ}\}\psi,\\
 \mathcal G_{\mathrm U}
 &=\mathrm d_\Sigma\Pi-q\sigma_I\bar\psi\gamma^I\psi.
```

\mathcal G_{ij}Use in source · 5 passages

Search aliases: G_ij mathcal

Link to this entry ·

Use in source · core.tex, line 618
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].
Exact mathematical passage
```
 C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
 &C_i&=-2\delta_j\pi_{ij},\nonumber\\
 \mathcal G_{ij}(h)&=\frac12\big[
 \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
 -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
 -\delta_{ij}C\big].
 \label{c:core:adm-operators}
```

\mathcal G_jUse in source · 3 passages

Search aliases: G_j mathcal

Link to this entry ·

Use in source · core.tex, line 227
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-def}
 \rho_j=\psi_j^*\psi_j,\qquad
 \bar\rho=\frac1N\sum_j\rho_j,\qquad
 \mathcal G_j=\frac{E_j-E_{j-1}}h+\rho_j-\bar\rho.
```

g_jUse in source · 2 passages

Search aliases: g_j

Link to this entry ·

Use in source · extensions.tex, line 438
[The precise defect of a forward difference] On a cyclic grid, let D_hf_j=(f_j+1-f_j)/h, with h≠0. Expanding the product gives D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j =h(D_hf)_j(D_hg)_j. Thus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go theorem, because that theorem concerns an exact identity for every vector at fixed finite resolution.
Exact mathematical passage
```
 D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j
 =h(D_hf)_j(D_hg)_j.
```

g_{j+1}^{-1}Use in source · 1 passage

Search aliases: g_j+1^-1

Link to this entry ·

Use in source · orientation.tex, line 27
Linked shifts · A coefficient U_j in U_jψ_j+1 transports from j+1 back to j. Thus U_j'=g_jU_jg_j+1^-1. An outward link uses the opposite endpoint convention.
Exact mathematical passage
```
U_j'=g_jU_jg_{j+1}^{-1}
```

g_{j+1}^nUse in source · 1 passage

Search aliases: g_j+1^n

Link to this entry ·

Use in source · core.tex, line 75
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\label{g:temporal-link}
 (U_j^n)^g=g_j^nU_j^n(g_{j+1}^n)^{-1},
 \qquad (T_j^n)^g=g_j^{n+1}T_j^n(g_j^n)^{-1}.
```

g_j^nUse in source · 4 passages

Search aliases: g_j^n

Link to this entry ·

Use in source · core.tex, line 74
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
g_j^n
```

g_j^{n+1}Use in source · 1 passage

Search aliases: g_j^n+1

Link to this entry ·

Use in source · core.tex, line 75
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\label{g:temporal-link}
 (U_j^n)^g=g_j^nU_j^n(g_{j+1}^n)^{-1},
 \qquad (T_j^n)^g=g_j^{n+1}T_j^n(g_j^n)^{-1}.
```

g_mUse in source · 2 passages

Search aliases: g_m

Link to this entry ·

Use in source · limit.tex, line 269
ginproof The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m gives k_m=ϵ_m g_m. Since k is square summable, ∑_m|ϵ_m|^2normg_m^2<∞. Thus g∈Dom(H_ϵ) and the adjoint agrees with H_ϵ. The reverse inclusion is symme…
Exact mathematical passage
```
k_m=\epsilon_m g_m
```

G_{p,h}Use in source · 1 passage

Search aliases: G_p,h

Link to this entry ·

Use in source · finite.tex, line 245
The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least significant base-p digits.
Exact mathematical passage
```
 G_{p,h}=\Z/p^h\Z,\qquad\mathcal H_{p,h}=\ell^2(G_{p,h};\C).
```

G_{p_a}Use in source · 1 passage

Search aliases: G_p_a

Link to this entry ·

Use in source · orientation.tex, line 24
midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.
Exact mathematical passage
```
\{F,G\}=\sum_a(F_{q_a}G_{p_a}-F_{p_a}G_{q_a})
```

g_qUse in source · 1 passage

Search aliases: g_q

Link to this entry ·

Use in source · spectral.tex, line 1159
Let p be prime, let L=mp for a positive integer m, and take 0≤ j<p. For every q≥0, g_q=fracj!(Lq)!(Lq+j)! =binomLq+jj^-1∈Z_p^×. Indeed binomLq+jj=∏_r=1^j(Lq+r)/r. Each numerator and denominator is a p-adic unit because Lq≡0pmod p and 1≤ r<p. For j=0 the empty product is one. The proof also covers p=2 in its short window. Outside this window, a factor can cross a multiple of p and the same proof gives no unit guarantee. This rational coefficient statement supplies no claim about the p-adic conver…
Exact mathematical passage
```
 g_q=\frac{j!(Lq)!}{(Lq+j)!}
       =\binom{Lq+j}{j}^{-1}\in\Z_p^\times.
 \label{a-spectral-unit-window}
```

G_{q_a}Use in source · 1 passage

Search aliases: G_q_a

Link to this entry ·

Use in source · orientation.tex, line 24
midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.
Exact mathematical passage
```
\{F,G\}=\sum_a(F_{q_a}G_{p_a}-F_{p_a}G_{q_a})
```

g_rUse in source · 4 passages

Search aliases: g_r

Link to this entry ·

Use in source · spectral.tex, line 431
For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber
Exact mathematical passage
```
g_r(z)=\langle r,(z-J_{r:h})^{-1}r\rangle
```

g_{r+1}Use in source · 1 passage

Search aliases: g_r+1

Link to this entry ·

Use in source · spectral.tex, line 433
For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rationa…
Exact mathematical passage
```
 g_h(z)&=(z-\epsilon_h)^{-1},\nonumber\\
 g_r(z)&=\frac1{z-\epsilon_r-\gamma_r^2g_{r+1}(z)},
          \qquad r<h.
 \label{a-spectral-continued-fraction}
```

g_UUse in source · 4 passages

Search aliases: g_U

Link to this entry ·

Use in source · core.tex, line 492
For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
 \nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\
 \nabla_QC_h&=-2hD_h(e^{2P}V),\\
 g_U:=\nabla_UC_h&=2hD_hP,
 &g_V:=\nabla_VC_h&=2h e^{2P}D_hQ.
```

g_VUse in source · 4 passages

Search aliases: g_V

Link to this entry ·

Use in source · core.tex, line 492
For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
 \nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\
 \nabla_QC_h&=-2hD_h(e^{2P}V),\\
 g_U:=\nabla_UC_h&=2hD_hP,
 &g_V:=\nabla_VC_h&=2h e^{2P}D_hQ.
```

g_xUse in source · 11 passages

Search aliases: g_x

Link to this entry ·

Use in source · core.tex, line 10
Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from ri…
Exact mathematical passage
```
g=(g_x)_{x\in V}
```

g_{x+\widehat\mu}Use in source · 1 passage

Search aliases: g_x+widehatμ widehat mu

Link to this entry ·

Use in source · core.tex, line 19
The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.
Exact mathematical passage
```
\label{g:forward-link}
 u_{x,\mu}=U_{x,\mu}^{-1}:\HH_x\longrightarrow\HH_{x+\widehat\mu},
 \qquad u_{x,\mu}^g=g_{x+\widehat\mu}u_{x,\mu}g_x^{-1}.
```

g_{x+\widehat\mu}^{-1}Use in source · 1 passage

Search aliases: g_x+widehatμ^-1 widehat mu

Link to this entry ·

Use in source · core.tex, line 11
ing on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the…
Exact mathematical passage
```
\label{g:link-transform}
 U_{x,\mu}^g=g_xU_{x,\mu}g_{x+\widehat\mu}^{-1},
 \qquad U_{x,\mu}^g\psi_{x+\widehat\mu}^g
 =g_xU_{x,\mu}\psi_{x+\widehat\mu}.
```

g_{x-\widehat\mu}Use in source · 1 passage

Search aliases: g_x-widehatμ widehat mu

Link to this entry ·

Use in source · core.tex, line 42
For the positive edge, substitute [source reference]. The two gauge factors at x+widehatμ cancel, leaving g_x on the left. For the negative edge, (U_x-widehatμ,μ^g)^-1g_x-widehatμ =g_xU_x-widehatμ,μ^-1. Every term in either difference therefore has the common factor g_x.
Exact mathematical passage
```
 (U_{x-\widehat\mu,\mu}^g)^{-1}g_{x-\widehat\mu}
 =g_xU_{x-\widehat\mu,\mu}^{-1}.
```

g_x^*Use in source · 2 passages

Search aliases: g_x^*

Link to this entry ·

Use in source · core.tex, line 340
Now take ψ_x∈C^2, U_x,μ∈ SU(2), and local gauges g_x∈ SU(2), as in the legacy non-Abelian chapter. The linked scalar z_x,μ=ψ_x^*U_x,μψ_x+widehatμ is gauge invariant: the adjoint factor g_x^* cancels g_x, and the factors at x+widehatμ also cancel. It follows that the source's scalar link observable j_x,μ=2Imz_x,μ is invariant. To obtain a continuity equation with its particular sign and normalization one must specify the hopping generator. Invariance of the observable alone is not an equation of …
Exact mathematical passage
```
g_x^*
```

g_x^{-1}Use in source · 4 passages

Search aliases: g_x^-1

Link to this entry ·

Use in source · core.tex, line 16
x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the input and output of every product.
Exact mathematical passage
```
g_yT_{y\leftarrow x}g_x^{-1}
```

g_yUse in source · 2 passages

Search aliases: g_y

Link to this entry ·

Use in source · core.tex, line 16
x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the input and output of every product.
Exact mathematical passage
```
g_yT_{y\leftarrow x}g_x^{-1}
```

g_{\mathrm{YM}}Use in source · 1 passage

Search aliases: g_YM mathrm

Link to this entry ·

Use in source · core.tex, line 160
For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the analogous term is 1-ReW. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interac…
Exact mathematical passage
```
\label{g:wilson-action}
 S_W[U]=\frac1{g_{\mathrm{YM}}^2}
 \sum_{x,\mu<\nu}\left(1-\tfrac12\operatorname{Re}\operatorname{Tr}W_{x,\mu\nu}\right),
 \qquad g_{\mathrm{YM}}\in\R\setminus\{0\}.
```

g_{\mathrm{YM}}^2Use in source · 1 passage

Search aliases: g_YM^2 mathrm

Link to this entry ·

Use in source · core.tex, line 160
For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the analogous term is 1-ReW. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interac…
Exact mathematical passage
```
\label{g:wilson-action}
 S_W[U]=\frac1{g_{\mathrm{YM}}^2}
 \sum_{x,\mu<\nu}\left(1-\tfrac12\operatorname{Re}\operatorname{Tr}W_{x,\mu\nu}\right),
 \qquad g_{\mathrm{YM}}\in\R\setminus\{0\}.
```

G_\betaUse in source · 1 passage

Search aliases: G_β beta

Link to this entry ·

Use in source · extensions.tex, line 65
If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β
Exact mathematical passage
```
G_\beta=\sum_{r(a)=\beta}F_a
```

\operatorname{gh}Use in source · 1 passage

Search aliases: gh operatorname

Link to this entry ·

Use in source · constraints.tex, line 386
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
-1-\operatorname{gh}(\Xi)
```

\mathrm{graph}Use in source · 2 passages

Search aliases: graph mathrm

Link to this entry ·

Use in source · continuum.tex, line 194
widehat H_ma are square summable. Given ε>0, choose finite sets that make each of their tails small and take their union. The resulting one finite set S satisfies norma-P_Sa<ε, normwidehat H_ma-widehat H_mP_Sa<ε. This graph-approximation statement is represented in the formal source. The finite Fourier fields are dense for the graph norm norma_graph^2 =norma^2+normwidehat H_ma^2. The term graph core describes this mathematical conclusion. The source proves the simultaneous approximation theorem …
Exact mathematical passage
```
\norm{a}_{\mathrm{graph}}^2
=\norm{a}^2+\norm{\widehat H_ma}^2.
```

\mathbf hUse in source · 8 passages

Search aliases: h mathbf

Link to this entry ·

Use in source · limit.tex, line 544
mathrmtor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and the identity is a finite telesc…
Exact mathematical passage
```
\mathbf h=(h_p)
```

\mathcal HUse in source · 19 passages

Search aliases: H mathcal

Link to this entry ·

Use in source · finite.tex, line 245
The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least significant base-p digits.
Exact mathematical passage
```
 G_{p,h}=\Z/p^h\Z,\qquad\mathcal H_{p,h}=\ell^2(G_{p,h};\C).
```

\mathfrak HUse in source · 5 passages

Search aliases: H mathfrak

Link to this entry ·

Use in source · limit.tex, line 193
For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.
Exact mathematical passage
```
 \norm{f}_{\mathfrak H_p^s}^2
 =\sum_{m\geq0}p^{2sm}\norm{\mathsf F_mf}_2^2,
 \qquad
 \mathfrak H_p^s=\{f\in\HH_p:\norm{f}_{\mathfrak H_p^s}<\infty\}.
```

\mathsf HUse in source · 3 passages

Search aliases: H mathsf

Link to this entry ·

Use in source · spectral.tex, line 102
M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1, e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
 P_p(\theta)e_r=
 \begin{cases}
 e_{r+1},&r<p-1,\\
 e^{-i\theta}e_0,&r=p-1.
 \end{cases}
 \qquad
 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
 \label{a-spectral-bloch-symbol}
```

H'Use in source · 1 passage

Search aliases: H'

Link to this entry ·

Use in source · orientation.tex, line 24
Refinement · An isometry V has V^*V=I. Its old-range projector is P=VV^* and its new-detail projector is Q=I-P. Exact dynamical compatibility is a separate identity, H'V=VH.
Exact mathematical passage
```
H'V=VH
```

H^*Use in source · 3 passages

Search aliases: H^*

Link to this entry ·

Use in source · spectral.tex, line 35
+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber
Exact mathematical passage
```
H=H^*
```

h^{-1}Use in source · 1 passage

Search aliases: h^-1

Link to this entry ·

Use in source · core.tex, line 298
real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription.
Exact mathematical passage
```
h^{-1}\int_{x_j}^{x_j+h}A(s)\,ds
```

h^{-2}Use in source · 3 passages

Search aliases: h^-2

Link to this entry ·

Use in source · core.tex, line 55
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
h^{-2}(\norm{\psi_{x+\widehat\mu}}^2+\norm{\psi_x}^2
-2\operatorname{Re}(\psi_x^*U_{x,\mu}\psi_{x+\widehat\mu}))
```

H^1Use in source · 1 passage

Search aliases: H^1

Link to this entry ·

Use in source · continuum.tex, line 233
The inhomogeneous weight retains the zero mode even when m=0. There is no loss of the ambient square-summability requirement in the massless case. The phrase first-order Fourier domain refers to this exact condition. It agrees with the conventional Fourier definition of periodic H^1 regularity; this manuscript does not claim a new checked equivalence with a separate weak-derivative implementation.
Exact mathematical passage
```
H^1
```

H^2Use in source · 1 passage

Search aliases: H^2

Link to this entry ·

Use in source · constraints.tex, line 96
A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0, PH=0, HI=0, H^2=0. Finally, require (s_0+δ)^2=0 and an inverse of id_F+Hδ. The inverse may exist in a declared operator topology, or in a complete filtration in which the Neumann series makes formal sense. A formal inverse is a formal c…
Exact mathematical passage
```
 \begin{gathered}
 PI=\mathrm{id}_C,\qquad
 \mathrm{id}_F-IP=s_0H+Hs_0,\\
 PH=0,\qquad HI=0,\qquad H^2=0.
 \end{gathered}
 \label{c:constraints:contraction}
```

h^2Use in source · 54 passages

Search aliases: h^2

Link to this entry ·

Use in source · continuum.tex, line 62
Take V=U_h(t), W=U(t), and b=P_Sa. For a fixed finite S, the compact-time packet estimate gives normU_h(t)P_Sa-U(t)P_Sa≤ K_S,a,Th^2 for all |t|≤ T and sufficiently small positive h. The constant is allowed to depend on the support, the amplitudes, and the time window. Combining the estimates gives normU_h(t)a-U(t)a≤ 2norma-P_Sa+K_S,a,Th^2.
Exact mathematical passage
```
\norm{U_h(t)P_Sa-U(t)P_Sa}\le K_{S,a,T}h^2
```

h^{2.084394965}Use in source · 1 passage

Search aliases: h^2.084394965

Link to this entry ·

Use in source · strong_fields.tex, line 529
The coarsest two event coordinates are not monotone with refinement. On the six finest resolutions the retained nonlinear fits are t_H(h) · ≃4.6893522370+33.88577338 h^2.173853282, r_H(h) · ≃3.1204965744-65.59310603 h^2.084394965. These fit exponents are consistent with second-order event convergence on this bounded family. They are not a convergence proof, a reliable error bar for every observable, or a critical-collapse exponent.
Exact mathematical passage
```
 t_H(h)&\simeq4.6893522370+33.88577338\,h^{2.173853282},\\
 r_H(h)&\simeq3.1204965744-65.59310603\,h^{2.084394965}.
```

h^{2.173853282}Use in source · 1 passage

Search aliases: h^2.173853282

Link to this entry ·

Use in source · strong_fields.tex, line 529
The coarsest two event coordinates are not monotone with refinement. On the six finest resolutions the retained nonlinear fits are t_H(h) · ≃4.6893522370+33.88577338 h^2.173853282, r_H(h) · ≃3.1204965744-65.59310603 h^2.084394965. These fit exponents are consistent with second-order event convergence on this bounded family. They are not a convergence proof, a reliable error bar for every observable, or a critical-collapse exponent.
Exact mathematical passage
```
 t_H(h)&\simeq4.6893522370+33.88577338\,h^{2.173853282},\\
 r_H(h)&\simeq3.1204965744-65.59310603\,h^{2.084394965}.
```

h^3Use in source · 8 passages

Search aliases: h^3

Link to this entry ·

Use in source · refinement.tex, line 77
T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for every real h. If each of the three samples used by T_2 has absolute error at most ε, its output error is at most frac8+6+13ε=5ε. This bound is attained by errors with signs matching the three weights. Thus exact removal of the first two powers does not imply a contraction on arbitrar…
Exact mathematical passage
```
f(h)=1+2h-3h^2+4h^3
```

h^4Use in source · 10 passages

Search aliases: h^4

Link to this entry ·

Use in source · finite_core.tex, line 602
q_j(h,p(n))=p_j(n)-frach^2p_j(n)^36+O(h^4),
Exact mathematical passage
```
q_j(h,p(n))=p_j(n)-\frac{h^2p_j(n)^3}{6}+O(h^4),
```

H^kUse in source · 1 passage

Search aliases: H^k

Link to this entry ·

Use in source · finite_core.tex, line 345
The identity I_4=P_++P_- and the two eigenvalue identities give, by induction, H^k=E^kP_++(-E)^kP_- for every k≥0. Substitute into the absolutely convergent series to obtain the spectral expression. Expanding its projectors gives the cosine and sine expression.
Exact mathematical passage
```
H^k=E^kP_++(-E)^kP_-
```

H^nUse in source · 1 passage

Search aliases: H^n

Link to this entry ·

Use in source · spectral.tex, line 49
For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
(\mathcal A H\mathcal A^*)^n=\mathcal A H^n\mathcal A^*
```

h^rUse in source · 4 passages

Search aliases: h^r

Link to this entry ·

Use in source · refinement.tex, line 18
[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
 T_m(h^rv)=c_{m,r}h^rv,\qquad
 c_{m,r}=\prod_{k=1}^{m}\frac{b^{k-r}-1}{b^k-1}.
```

H^TUse in source · 2 passages

Search aliases: H^T

Link to this entry ·

Use in source · constraints.tex, line 712
overline∇ H(z_0,z_1) with invertible Δ_Δ satisfies Φ(z_1)=Φ(z_0), H(z_1)=H(z_0). The componentwise chain rule gives Φ(z_1)-Φ(z_0)=G(z_1-z_0), which vanishes because GJ_ D^Δ=0. For the Hamiltonian, H(z_1)-H(z_0) =Δ t overline∇ H^T J_ D^Δoverline∇ H=0 by skewness.
Exact mathematical passage
```
 H(z_1)-H(z_0)
 =\Delta t\,\overline\nabla H^T J_{\mathrm D}^{\Delta}\overline\nabla H=0
```

H_{\mathrm m}Use in source · 3 passages

Search aliases: H_ m mathrm

Link to this entry ·

Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

\mathcal H_{\mathrm M}Use in source · 1 passage

Search aliases: H_ M mathcal mathrm

Link to this entry ·

Use in source · constraints.tex, line 572
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
 \mathcal H_{\mathrm M}=\frac1{2e}q_{ab}
       (\mathcal E^a\mathcal E^b+\mathcal B^a\mathcal B^b),
 \qquad \mathcal B^a=\tfrac12\epsilon^{abc}\mathcal F_{bc}.
```

H_0Use in source · 12 passages

Search aliases: H_0

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Use in source · limit.tex, line 729
Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.
Exact mathematical passage
```
H_0=H_0^*
```

H_0^*Use in source · 1 passage

Search aliases: H_0^*

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Use in source · limit.tex, line 729
Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.
Exact mathematical passage
```
H_0=H_0^*
```

h_2Use in source · 5 passages

Search aliases: h_2

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Use in source · finite.tex, line 120
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 M=2^{h_2+2}\prod_{p\in\mathcal P_{\mathrm o}}p^{h_p+1}.
```

H_AUse in source · 7 passages

Search aliases: H_A

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Use in source · limit.tex, line 600
[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber H_A f · =∑_ rA_ r F_ rf. Then H_A is self-adjoint, and finite rectangular packet sums form a core. Every rectangular range reduces H_A, and its finite restriction H_A^( h) satisfies exact embedding intertwining with every larger rectangular restriction. For bounded…
Exact mathematical passage
```
 \Dom(H_A)&=\left\{f:\sum_{\mathbf r}
 \norm{A_{\mathbf r}\mathsf F_{\mathbf r}f}^2<\infty\right\},
 \nonumber\\
 H_A f&=\sum_{\mathbf r}A_{\mathbf r}\mathsf F_{\mathbf r}f.
 \label{a-limit-block-domain}
```

H_A^{(\mathbf h)}Use in source · 1 passage

Search aliases: H_A^( h) mathbf

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Use in source · limit.tex, line 609
H_A f · =∑_ rA_ r F_ rf. Then H_A is self-adjoint, and finite rectangular packet sums form a core. Every rectangular range reduces H_A, and its finite restriction H_A^( h) satisfies exact embedding intertwining with every larger rectangular restriction. For bounded Borel F, F(H_A)f=∑_ rF(A_ r) F_ rf, F(H_A) E_ h→ F(H_A) strongly. In particular, norme^-itH_A(I- E_ h)f =norm(I- E_ h)f. Finite rectangular sums belong to the domain and are dense. Symmetry holds by symmetry on every block. If g belon…
Exact mathematical passage
```
H_A^{(\mathbf h)}
```

H_A^*Use in source · 2 passages

Search aliases: H_A^*

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Use in source · limit.tex, line 628
xtstrongly. In particular, norme^-itH_A(I- E_ h)f =norm(I- E_ h)f. Finite rectangular sums belong to the domain and are dense. Symmetry holds by symmetry on every block. If g belongs to the adjoint domain and k=H_A^*g, testing against an arbitrary vector in KK_ r gives F_ rk=A_ r F_ rg. Square summability of k is precisely the domain condition in [source reference]. Thus H_A^*=H_A. Truncating the two square-summable sequences f_ r and A_ rf_ r proves the graph-core assertion.
Exact mathematical passage
```
k=H_A^*g
```

H_{AA}Use in source · 1 passage

Search aliases: H_AA

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Use in source · core.tex, line 92
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.
Exact mathematical passage
```
 H_{AA}=H_{BB}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 H_{BA}=\begin{pmatrix}0&-1\\2&0\end{pmatrix},\qquad
 H_{AB}=H_{BA}^*.
```

H_{AB}Use in source · 1 passage

Search aliases: H_AB

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Use in source · core.tex, line 92
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.
Exact mathematical passage
```
 H_{AA}=H_{BB}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 H_{BA}=\begin{pmatrix}0&-1\\2&0\end{pmatrix},\qquad
 H_{AB}=H_{BA}^*.
```

H_{BA}Use in source · 1 passage

Search aliases: H_BA

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Use in source · core.tex, line 92
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.
Exact mathematical passage
```
 H_{AA}=H_{BB}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 H_{BA}=\begin{pmatrix}0&-1\\2&0\end{pmatrix},\qquad
 H_{AB}=H_{BA}^*.
```

H_{ba}Use in source · 1 passage

Search aliases: H_ba

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Use in source · core.tex, line 358
In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i
Exact mathematical passage
```
2\operatorname{Im}(H_{ba}\rho_{ab})
```

H_{BA}^*Use in source · 1 passage

Search aliases: H_BA^*

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Use in source · core.tex, line 92
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.
Exact mathematical passage
```
 H_{AA}=H_{BB}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 H_{BA}=\begin{pmatrix}0&-1\\2&0\end{pmatrix},\qquad
 H_{AB}=H_{BA}^*.
```

H_{BB}Use in source · 1 passage

Search aliases: H_BB

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Use in source · core.tex, line 92
2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.
Exact mathematical passage
```
 H_{AA}=H_{BB}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 H_{BA}=\begin{pmatrix}0&-1\\2&0\end{pmatrix},\qquad
 H_{AB}=H_{BA}^*.
```

H_cUse in source · 11 passages

Search aliases: H_c

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Use in source · finite.tex, line 514
[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,
Exact mathematical passage
```
H_c,H_f
```

\mathcal H_cUse in source · 1 passage

Search aliases: H_c mathcal

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Use in source · finite.tex, line 512
[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,
Exact mathematical passage
```
V:\mathcal H_c\to\mathcal H_f
```

H_DUse in source · 1 passage

Search aliases: H_D

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Use in source · spectral.tex, line 465
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation
Exact mathematical passage
```
A=H_D(k)=k\sigma_z+m_0\sigma_x
```

H_EUse in source · 1 passage

Search aliases: H_E

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Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

H_{\mathrm{eff}}Use in source · 1 passage

Search aliases: H_eff mathrm

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Use in source · core.tex, line 280
The connection Γ records the velocity of the frame. Omitting it would interpret a change of packet coordinates as if it were entirely a change of the physical state. In Hamiltonian notation the corresponding effective Hamiltonian is H_eff=U^*HU-iΓ, which is self-adjoint because Γ is skew-adjoint.
Exact mathematical passage
```
H_{\mathrm{eff}}=U^*HU-i\Gamma
```

H_fUse in source · 18 passages

Search aliases: H_f

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Use in source · finite.tex, line 514
[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,
Exact mathematical passage
```
H_c,H_f
```

\mathcal H_fUse in source · 1 passage

Search aliases: H_f mathcal

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Use in source · finite.tex, line 512
[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,
Exact mathematical passage
```
V:\mathcal H_c\to\mathcal H_f
```

H_GUse in source · 6 passages

Search aliases: H_G

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Use in source · core.tex, line 364
Define the densities H_G · =P_τ^2+γ P_θ^2 +e^2P(Q_τ^2+γ Q_θ^2),nonumber M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.
Exact mathematical passage
```
 H_G&=P_\tau^2+\gamma P_\theta^2
 +e^{2P}(Q_\tau^2+\gamma Q_\theta^2),\nonumber\\
 M_G&=2(P_\tau P_\theta+e^{2P}Q_\tau Q_\theta).
 \label{c:core:gowdy-densities}
```

H_{G,h}Use in source · 2 passages

Search aliases: H_G,h

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Use in source · core.tex, line 576
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
H_{G,h},M_{G,h}
```

H_hUse in source · 5 passages

Search aliases: H_h

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Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

H_{h,n}^2Use in source · 1 passage

Search aliases: H_h,n^2

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Use in source · finite_core.tex, line 274
H_h,n^2=E_h,n^2I_4, E_h,n=sqrtm^2+|q_h(n)|^2.
Exact mathematical passage
```
H_{h,n}^2=E_{h,n}^2I_4,\quad
E_{h,n}=\sqrt{m^2+|q_h(n)|^2}.
```

h_{ij}Use in source · 6 passages

Search aliases: h_ij

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Use in source · core.tex, line 615
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
h_{ij}=h_{ji}
```

h_{ik}Use in source · 2 passages

Search aliases: h_ik

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Use in source · core.tex, line 618
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].
Exact mathematical passage
```
 C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
 &C_i&=-2\delta_j\pi_{ij},\nonumber\\
 \mathcal G_{ij}(h)&=\frac12\big[
 \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
 -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
 -\delta_{ij}C\big].
 \label{c:core:adm-operators}
```

H_{\mathrm{int}}Use in source · 6 passages

Search aliases: H_int mathrm

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Use in source · finite.tex, line 740
Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_…
Exact mathematical passage
```
 H_{\mathrm{int}}=
 \sum_{\mathbf d}
 \left(\gamma_{p,w}|\mathbf d\rangle\langle c|
       +\overline{\gamma_{p,w}}|c\rangle\langle\mathbf d|\right).
 \label{a-finite-leaf-interaction}
```

h_{ji}Use in source · 1 passage

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Use in source · core.tex, line 615
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
h_{ij}=h_{ji}
```

h_{jk}Use in source · 2 passages

Search aliases: h_jk

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Use in source · core.tex, line 618
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].
Exact mathematical passage
```
 C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
 &C_i&=-2\delta_j\pi_{ij},\nonumber\\
 \mathcal G_{ij}(h)&=\frac12\big[
 \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
 -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
 -\delta_{ij}C\big].
 \label{c:core:adm-operators}
```

h_{kk}Use in source · 1 passage

Search aliases: h_kk

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Use in source · core.tex, line 616
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
h_{\rm tr}=h_{kk}
```

h_{kl}Use in source · 1 passage

Search aliases: h_kl

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Use in source · core.tex, line 636
Twice its divergence expands as 2δ_j G_ij = · δ_iδ_jδ_kh_jk +δ^2δ_kh_ik-δ^2δ_jh_ij -δ_iδ^2h_rm tr · -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h
Exact mathematical passage
```
 2\delta_j\mathcal G_{ij}
 ={}&\delta_i\delta_j\delta_kh_{jk}
 +\delta^2\delta_kh_{ik}-\delta^2\delta_jh_{ij}
 -\delta_i\delta^2h_{\rm tr}\\
 &-\delta_i\delta_k\delta_lh_{kl}
 +\delta_i\delta^2h_{\rm tr}.
```

H_m^{\mathrm{phys}}Use in source · 5 passages

Search aliases: H_m^phys mathrm

Link to this entry ·

Use in source · boundary.tex, line 24
Connecting the group to its generator. The next chapter proves that the constructed group is the spectral evolution of this Hamiltonian: U(t)=e^-itH_m^phys. The proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal generator has exactly that domain. The equality is therefore a conclusion about the complete Hilbert-space group, not an inference from agreement on finitel…
Exact mathematical passage
```
\mathcal U(t)=e^{-itH_m^{\mathrm{phys}}}.
```

h_nUse in source · 1 passage

Search aliases: h_n

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Use in source · refinement.tex, line 430
Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero need not converge; summability controls the accumulated tail.
Exact mathematical passage
```
h_n=a_0 2^{-n}
```

H_n^*Use in source · 2 passages

Search aliases: H_n^*

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Use in source · continuum.tex, line 205
The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass term is already summable because a∈KK. Membership is therefore equivalent to summability of ∑_n|p(n)|^2norma_n^2.
Exact mathematical passage
```
H_n^*=H_n
```

H_n^2Use in source · 2 passages

Search aliases: H_n^2

Link to this entry ·

Use in source · continuum.tex, line 205
The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass term is already summable because a∈KK. Membership is therefore equivalent to summability of ∑_n|p(n)|^2norma_n^2.
Exact mathematical passage
```
H_n^2=(m^2+|p(n)|^2)I_4
```

\mathsf H_pUse in source · 3 passages

Search aliases: H_p mathsf

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Use in source · spectral.tex, line 102
M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1, e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
 P_p(\theta)e_r=
 \begin{cases}
 e_{r+1},&r<p-1,\\
 e^{-i\theta}e_0,&r=p-1.
 \end{cases}
 \qquad
 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
 \label{a-spectral-bloch-symbol}
```

h_pUse in source · 8 passages

Search aliases: h_p

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Use in source · finite.tex, line 120
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 M=2^{h_2+2}\prod_{p\in\mathcal P_{\mathrm o}}p^{h_p+1}.
```

H_{p,h}Use in source · 1 passage

Search aliases: H_p,h

Link to this entry ·

Use in source · finite.tex, line 617
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
 H_{p,h}=H_{p,h}^{\mathrm{Haar}}+\kappa_p L_{p,h}
 \label{a-finite-compatible-laplacian}
```

\mathcal H_{p,h}Use in source · 2 passages

Search aliases: H_p,h mathcal

Link to this entry ·

Use in source · finite.tex, line 245
The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least significant base-p digits.
Exact mathematical passage
```
 G_{p,h}=\Z/p^h\Z,\qquad\mathcal H_{p,h}=\ell^2(G_{p,h};\C).
```

H_{p,h+1}^{\mathrm{Haar}}Use in source · 1 passage

Search aliases: H_p,h+1^Haar mathrm

Link to this entry ·

Use in source · finite.tex, line 596
Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonalizes the level blocks: T_p,hH_p,h^Haar T_p,h^* =∑_r=0^hϵ_p,rΠ_r. This is exact for each finite h and every finite real choice of the energies. Their gro…
Exact mathematical passage
```
H_{p,h+1}^{\mathrm{Haar}}V_{p,h}
=V_{p,h}H_{p,h}^{\mathrm{Haar}}
```

H_{p,h}^{\mathrm{Haar}}Use in source · 5 passages

Search aliases: H_p,h^Haar mathrm

Link to this entry ·

Use in source · finite.tex, line 589
Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonalizes the level blocks: T_p,hH_p,h^Haar T_p,h^* =∑_r=0^hϵ_p,rΠ_r. This is exact for each finite h and every finite real choice of the energies. Their gro…
Exact mathematical passage
```
 H_{p,h}^{\mathrm{Haar}}=\sum_{r=0}^h
          \epsilon_{p,r}D_r^{(h)}.
 \label{a-finite-hierarchical-hamiltonian}
```

\mathfrak H_p^sUse in source · 5 passages

Search aliases: H_p^s mathfrak

Link to this entry ·

Use in source · limit.tex, line 193
For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.
Exact mathematical passage
```
 \norm{f}_{\mathfrak H_p^s}^2
 =\sum_{m\geq0}p^{2sm}\norm{\mathsf F_mf}_2^2,
 \qquad
 \mathfrak H_p^s=\{f\in\HH_p:\norm{f}_{\mathfrak H_p^s}<\infty\}.
```

H_QUse in source · 4 passages

Search aliases: H_Q

Link to this entry ·

Use in source · finite.tex, line 521
QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.
Exact mathematical passage
```
 H_f=(VH_cV^*)|_{\Ran P}\ \oplus\ H_Q,
 \qquad H_Q=(QH_fQ)|_{\Ran Q}.
 \label{a-finite-hamiltonian-direct-sum}
```

H_qUse in source · 1 passage

Search aliases: H_q

Link to this entry ·

Use in source · core.tex, line 485
)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with zero populations require…
Exact mathematical passage
```
H_q=\log M_q/(1-q)
```

H_{\rm eff}^{(2)}Use in source · 1 passage

Search aliases: H_rm eff^(2) rm

Link to this entry ·

Use in source · spectral.tex, line 611
The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation retains every other site. It does not retain the average of each neighboring pair.
Exact mathematical passage
```
 H_{\rm eff}^{(2)}(z)
 =\mu I+\frac{\kappa^2}{z-\mu}(2I+T+T^*),\qquad z\ne\mu.
 \label{a-spectral-dyadic-effective}
```

H_{\rm eff}^{(p)}Use in source · 2 passages

Search aliases: H_rm eff^(p) rm

Link to this entry ·

Use in source · spectral.tex, line 573
[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber
Exact mathematical passage
```
\mathcal S_p(z)=zI-H_{\rm eff}^{(p)}(z)
```

H_{\rm full}Use in source · 1 passage

Search aliases: H_rm full rm

Link to this entry ·

Use in source · spectral.tex, line 474
=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this specific model. Its mass, couplings, and hi…
Exact mathematical passage
```
 \det(z-H_{\rm full})
 =\det(z-D)\bigl((z-\sigma(z))^2-k^2-m_0^2\bigr).
 \label{a-spectral-visible-determinant}
```

\mathcal H_{\rm matt}Use in source · 1 passage

Search aliases: H_rm matt mathcal rm

Link to this entry ·

Use in source · strong_fields.tex, line 344
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
 \mathcal H_{\rm matt}=\frac{P_\varphi^2}{2\Lambda R^2}
                         +\frac{R^2\varphi'^2}{2\Lambda},
 \qquad \mathcal D_{\rm matt}=P_\varphi\varphi'.
```

h_{\rm tr}Use in source · 4 passages

Search aliases: h_rm tr rm

Link to this entry ·

Use in source · core.tex, line 616
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
h_{\rm tr}=h_{kk}
```

H_{\rm TT}Use in source · 1 passage

Search aliases: H_rm TT rm

Link to this entry ·

Use in source · core.tex, line 761
Choose n=n_i=0 and impose the transverse and traceless conditions δ_jh_ij=0, h_rm tr=0, δ_jπ_ij=0, π_rm tr=0. They are preserved by the evolution because G_ij(h)=-tfrac12δ^2h_ij, dot h=2π, dotπ=tfrac12δ^2h, ddot h=δ^2h. On this reduced sector the energy is H_rm TT=normπ_h^2 +frac14∑_knormδ_kh_h^2≥0. Its derivative is zero: substituting the two first-order equations reduces it to ⟨π,δ^2h⟩_h +∑_k⟨δ_kh,δ_kπ⟩_h=0.
Exact mathematical passage
```
 H_{\rm TT}=\norm{\pi}_h^2
 +\frac14\sum_k\norm{\delta_kh}_h^2\geq0.
```

H_{rs}Use in source · 3 passages

Search aliases: H_rs

Link to this entry ·

Use in source · spectral.tex, line 64
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. endpr
Exact mathematical passage
```
i\dot\psi_r=\sum_sH_{rs}\psi_s
```

H_{rs}^*Use in source · 2 passages

Search aliases: H_rs^*

Link to this entry ·

Use in source · spectral.tex, line 35
mes|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber
Exact mathematical passage
```
H_{rs}^*=H_{sr}
```

H_{sr}Use in source · 3 passages

Search aliases: H_sr

Link to this entry ·

Use in source · spectral.tex, line 35
mes|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber
Exact mathematical passage
```
H_{rs}^*=H_{sr}
```

\mathcal H_tUse in source · 2 passages

Search aliases: H_t mathcal

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

H_\epsilonUse in source · 28 passages

Search aliases: H_ϵ epsilon

Link to this entry ·

Use in source · limit.tex, line 241
Let (ϵ_m)_m≥0 be any sequence of real numbers. Define Dom(H_ϵ) · ={f∈HH_p: ∑_m≥0|ϵ_m|^2norm F_mf_2^2<∞}, nonumber H_ϵ f · =∑_m≥0ϵ_m F_mf. The series converges in L^2 exactly on this domain. It is not necessary to bound the energies uniformly, but it is necessary to state the domain.
Exact mathematical passage
```
 \Dom(H_\epsilon)
 &=\left\{f\in\HH_p:
 \sum_{m\geq0}|\epsilon_m|^2\norm{\mathsf F_mf}_2^2<\infty\right\},
 \nonumber\\
 H_\epsilon f&=\sum_{m\geq0}\epsilon_m\mathsf F_mf.
 \label{a-limit-hamiltonian-domain}
```

H_\epsilon^{(M)}Use in source · 1 passage

Search aliases: H_ϵ^(M) epsilon

Link to this entry ·

Use in source · limit.tex, line 343
Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.
Exact mathematical passage
```
H_\epsilon^{(M)}
```

H_\epsilon^*Use in source · 2 passages

Search aliases: H_ϵ^* epsilon

Link to this entry ·

Use in source · limit.tex, line 267
pectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m gives k_m=ϵ_m g_m. Since k is square summable, ∑_m|ϵ_m|^2normg_m^2<∞. Thus g∈Dom(H_ϵ) and the adjoint agrees with H_ϵ. Th…
Exact mathematical passage
```
g\in\Dom(H_\epsilon^*)
```

H_\lambdaUse in source · 33 passages

Search aliases: H_λ lambda

Link to this entry ·

Use in source · limit.tex, line 734
Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.
Exact mathematical passage
```
H_\lambda=H_0+\lambda B
```

H_\lambda^*Use in source · 2 passages

Search aliases: H_λ^* lambda

Link to this entry ·

Use in source · extensions.tex, line 162
[Self-adjointness of the diagonal realization] The operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.
Exact mathematical passage
```
y\in D(H_\lambda^*)
```

H_\lambda^{1/2}Use in source · 2 passages

Search aliases: H_λ^1/2 lambda

Link to this entry ·

Use in source · extensions.tex, line 169
For positive λ_n, the quadratic-form domain instead uses ∑_nλ_n|c_n|^2<∞. This is the domain of H_λ^1/2, not generally the domain of H_λ. With λ_n=n and c_n=n^-s, the three thresholds are c|c membership · condition
Exact mathematical passage
```
H_\lambda^{1/2}
```

H_\inftyUse in source · 10 passages

Search aliases: H_∞ infty

Link to this entry ·

Use in source · spectral.tex, line 1101
Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It d
Exact mathematical passage
```
H_\infty
```

\mathrm{Haar}Use in source · 6 passages

Search aliases: Haar mathrm

Link to this entry ·

Use in source · finite.tex, line 589
Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonalizes the level blocks: T_p,hH_p,h^Haar T_p,h^* =∑_r=0^hϵ_p,rΠ_r. This is exact for each finite h and every finite real choice of the energies. Their gro…
Exact mathematical passage
```
 H_{p,h}^{\mathrm{Haar}}=\sum_{r=0}^h
          \epsilon_{p,r}D_r^{(h)}.
 \label{a-finite-hierarchical-hamiltonian}
```

\hboxUse in source · 1 passage

Search aliases: hbox

Link to this entry ·

Use in source · finite_core.tex, line 652
f_h(t)-f(t)=O(h^2) hboxin L^2( T^3; C^4),
Exact mathematical passage
```
f_h(t)-f(t)=O(h^2)
\quad\hbox{in }L^2(\mathbb T^3;\mathbb C^4),
```

\HH_{\mathcal P}^{\infty}Use in source · 2 passages

Search aliases: HH_ P^∞ HH mathcal infty

Link to this entry ·

Use in source · limit.tex, line 536
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections an…
Exact mathematical passage
```
 \HH_{\mathcal P}^{\infty}
 =\KK_{\mathrm{tor}}\otimes
 \bigotimes_{p\in\mathcal P}L^2(\Z_p,\mu_p).
 \label{a-limit-multiprime-space}
```

\HH_AUse in source · 1 passage

Search aliases: HH_A HH

Link to this entry ·

Use in source · core.tex, line 372
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr
Exact mathematical passage
```
\HH_A
```

\HH_BUse in source · 2 passages

Search aliases: HH_B HH

Link to this entry ·

Use in source · core.tex, line 372
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB
Exact mathematical passage
```
\HH_B
```

\HH_DUse in source · 2 passages

Search aliases: HH_D HH

Link to this entry ·

Use in source · spectral.tex, line 157
Let a finite Hilbert space be the orthogonal sum HH_L⊕HH_D, and write a self-adjoint Hamiltonian as H=A · B
Exact mathematical passage
```
\HH_L\oplus\HH_D
```

\HH_LUse in source · 4 passages

Search aliases: HH_L HH

Link to this entry ·

Use in source · spectral.tex, line 157
Let a finite Hilbert space be the orthogonal sum HH_L⊕HH_D, and write a self-adjoint Hamiltonian as H=A · B
Exact mathematical passage
```
\HH_L\oplus\HH_D
```

\HH_{p,m}Use in source · 2 passages

Search aliases: HH_p,m HH

Link to this entry ·

Use in source · limit.tex, line 23
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. end
Exact mathematical passage
```
\HH_{p,m}=\ell^2(\Z/p^m\Z)
```

\HH_SUse in source · 1 passage

Search aliases: HH_S HH

Link to this entry ·

Use in source · refinement.tex, line 217
Hilbert estimates with explicit constants. Finite-dimensional continuity is not enough to identify an operator norm with physical ℓ^2 energy. The earlier continuous-linear packaging retains the ambient norm on a function space. The Hilbert scalar layer instead realizes a finite four-spinor field as HH_S=ℓ^2(S×{0,1,2,3};C), normψ^2=∑_x∈ Snormψ(x)_C^4^2. The formal realization is codeEuclideanSpace; its norm is identified explicitly with the real part of the previously defined state inner product.…
Exact mathematical passage
```
 \HH_S=\ell^2(S\times\{0,1,2,3\};\C),\qquad
 \norm{\psi}^2=\sum_{x\in S}\norm{\psi(x)}_{\C^4}^2.
```

\HH_xUse in source · 2 passages

Search aliases: HH_x HH

Link to this entry ·

Use in source · core.tex, line 6
Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from ri…
Exact mathematical passage
```
\label{g:backward-link}
 U_{x,\mu}:\HH_{x+\widehat\mu}\longrightarrow\HH_x,
 \qquad U_{x,\mu}^*U_{x,\mu}=U_{x,\mu}U_{x,\mu}^*=I.
```

\HH_{x+\widehat\mu}Use in source · 2 passages

Search aliases: HH_x+widehatμ HH widehat mu

Link to this entry ·

Use in source · core.tex, line 6
Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from ri…
Exact mathematical passage
```
\label{g:backward-link}
 U_{x,\mu}:\HH_{x+\widehat\mu}\longrightarrow\HH_x,
 \qquad U_{x,\mu}^*U_{x,\mu}=U_{x,\mu}U_{x,\mu}^*=I.
```

\hlineUse in source · 1 passage

Search aliases: hline

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Use in source · extensions.tex, line 170
For positive λ_n, the quadratic-form domain instead uses ∑_nλ_n|c_n|^2<∞. This is the domain of H_λ^1/2, not generally the domain of H_λ. With λ_n=n and c_n=n^-s, the three thresholds are c|c membership · condition ℓ^2 · s>1/2 D(H_λ^1/2) · s>1 D(H_λ) · s>3/2. Each equality case diverges harmonically. Thus a state with finite mean positive energy can still lack the Hilbert-space derivative required by the Schr odinger equation. The harmonic oscillator example has λ_n=n+1/2 for n≥0 and the same as…
Exact mathematical passage
```
 \begin{array}{c|c}
 \text{membership}&\text{condition}\\ \hline
 \ell^2&s>1/2\\
 D(H_\lambda^{1/2})&s>1\\
 D(H_\lambda)&s>3/2.
 \end{array}
```

\mathcal IUse in source · 2 passages

Search aliases: I mathcal

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Use in source · core.tex, line 580
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
 \mathcal I_h=\partial_\tau M_{G,h}-D_hH_{G,h}
 -2(D_hP)\mathcal E_{P,h}
 -2e^{2P}(D_hQ)\mathcal E_{Q,h}
```

I_2Use in source · 6 passages

Search aliases: I_2

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Use in source · spectral.tex, line 466
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this…
Exact mathematical passage
```
D=J_{1:h}\otimes I_2
```

I_4Use in source · 25 passages

Search aliases: I_4

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Use in source · continuum.tex, line 205
The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass term is already summable because a∈KK. Membership is therefore equivalent to summability of ∑_n|p(n)|^2norma_n^2.
Exact mathematical passage
```
H_n^2=(m^2+|p(n)|^2)I_4
```

I_aUse in source · 10 passages

Search aliases: I_a

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Use in source · refinement.tex, line 31
Choose unit links a(x) and b(x) on the first and second half-edge with b(x)a(x)=u(x). The associated fine link is v(e_x)=a(x), v(m_x)=b(x). The interpolation and reconstruction maps are (I_aψ)(e_x) · =ψ(x), · (I_aψ)(m_x) · =a(x)ψ(x), (R_aΨ)(x) · =tfrac12(Ψ(e_x)+a(x)^-1Ψ(m_x)). The midpoint is a transported value, not an independently sampled field value. This identifies exactly the subspace on which the refinement theorem acts.
Exact mathematical passage
```
 (I_a\psi)(e_x)&=\psi(x),&
 (I_a\psi)(m_x)&=a(x)\psi(x),\label{g:edge-interpolation}\\
 (R_a\Psi)(x)&=\tfrac12\bigl(\Psi(e_x)+a(x)^{-1}\Psi(m_x)\bigr).
 \label{g:edge-restriction}
```

\mathcal I_hUse in source · 2 passages

Search aliases: I_h mathcal

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Use in source · core.tex, line 580
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
 \mathcal I_h=\partial_\tau M_{G,h}-D_hH_{G,h}
 -2(D_hP)\mathcal E_{P,h}
 -2e^{2P}(D_hQ)\mathcal E_{Q,h}
```

I_{\KK}Use in source · 1 passage

Search aliases: I_KK KK

Link to this entry ·

Use in source · extensions.tex, line 95
For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV. This is the exact missing-range term. Setting A=B=Π_a gives innerψ(F_a-F_a^2)ψ =norm(I-R)Π_aVψ^2. Thus the failure of an effect to be shar…
Exact mathematical passage
```
I_{\KK}
```

I_{\KK_{\mathrm{tor}}}Use in source · 3 passages

Search aliases: I_KK_tor KK mathrm

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Use in source · limit.tex, line 543
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections an…
Exact mathematical passage
```
I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathcal U_{p,h_p}
```

I_{K\to L}Use in source · 3 passages

Search aliases: I_K→ L to

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Use in source · constraints.tex, line 262
What the closure tower does and does not preserve. At a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space T_K is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies I_L→ MI_K→ L=I_K→ M. When the larger band and its grid resolve every generated product, the embedded constraints and their bracket values agree. This …
Exact mathematical passage
```
 I_{L\to M}I_{K\to L}=I_{K\to M}.
```

I_{K\to M}Use in source · 1 passage

Search aliases: I_K→ M to

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Use in source · constraints.tex, line 262
What the closure tower does and does not preserve. At a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space T_K is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies I_L→ MI_K→ L=I_K→ M. When the larger band and its grid resolve every generated product, the embedded constraints and their bracket values agree. This …
Exact mathematical passage
```
 I_{L\to M}I_{K\to L}=I_{K\to M}.
```

I_{L\to M}Use in source · 1 passage

Search aliases: I_L→ M to

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Use in source · constraints.tex, line 262
What the closure tower does and does not preserve. At a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space T_K is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies I_L→ MI_K→ L=I_K→ M. When the larger band and its grid resolve every generated product, the embedded constraints and their bracket values agree. This …
Exact mathematical passage
```
 I_{L\to M}I_{K\to L}=I_{K\to M}.
```

I_MUse in source · 3 passages

Search aliases: I_M

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Use in source · spectral.tex, line 25
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
J_r^*J_s=\delta_{rs}I_M
```

I_mUse in source · 2 passages

Search aliases: I_m

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Use in source · finite.tex, line 464
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,
Exact mathematical passage
```
 \mathcal W_{p,h}=(F_p\otimes I_m)\mathcal A^{\mathrm{ad}}_{p,h}.
 \label{a-finite-child-fourier}
```

I_NUse in source · 1 passage

Search aliases: I_N

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Use in source · spectral.tex, line 25
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
\sum_rJ_rJ_r^*=I_N
```

I_pUse in source · 4 passages

Search aliases: I_p

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Use in source · finite.tex, line 438
p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_…
Exact mathematical passage
```
 \mathcal A^{\mathrm{ad}}_{p,h}=
 (I_p\otimes\mathcal R_{p,h-1})
 \mathcal A^{\mathrm{std}}_{p,h}\mathcal R_{p,h}.
 \label{a-finite-digit-reversal}
```

I_{\mathrm{phase},\mathbf r}Use in source · 1 passage

Search aliases: I_phase, r mathrm mathbf

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Use in source · limit.tex, line 649
For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting wholly inside a new detail complement has no matrix element between that complement and the old range. A cross term between them would be a different…
Exact mathematical passage
```
 A_{\mathbf r}
 =\left(\sum_a\epsilon_{a,\mathbf r}T_a\right)
 \otimes I_{\mathrm{phase},\mathbf r},
 \qquad \epsilon_{a,\mathbf r}\in\R,\quad T_a=T_a^*.
```

I_rUse in source · 2 passages

Search aliases: I_r

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Use in source · refinement.tex, line 169
ndent The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phase frame r(s,c)∈ U(1) defines (I_rψ)(s,c)=r(s,c)ψ(s). With one-eighth fine weights this is again an isometry.
Exact mathematical passage
```
 (I_r\psi)(s,c)=r(s,c)\psi(s).
```

I_\deltaUse in source · 8 passages

Search aliases: I_δ delta

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Use in source · constraints.tex, line 108
[Transfer of a differential] Under these assumptions, define I_δ=(id_F+Hδ)^-1I, s_C=s_0,C+Pδ I_δ. Then PI_δ=id_C, (s_0+δ)I_δ=I_δ s_C, and s_C^2=0. The inverse equation is I_δ=I-Hδ I_δ. Application of P proves PI_δ=PI=id_C. Let K=(s_0+δ)I_δ-I_δ s_C. Using the chain-map identity and id_F-IP=s_0H+Hs_0 in this expression gives K=H(s_0δ I_δ+ δ I_δ s_0,C+δ I_δ Pδ I_δ). Nilpotency of the fine differential gives s_0δ+δ s_0+δ^2=0. The parenthesis therefore equals -δ K. Hence (id_F+Hδ)K=0, and invertibil
Exact mathematical passage
```
 I_\delta=(\mathrm{id}_F+H\delta)^{-1}I,
 \qquad
 s_C=s_{0,C}+P\delta I_\delta.
 \label{c:constraints:transfer-formula}
```

I_\ellUse in source · 1 passage

Search aliases: I_ℓ ell

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Use in source · refinement.tex, line 136
Interpolation is possible even when it depends on a path. Choose one reference path from the origin to each vertex and let r(v) be its transport. Then (I_ℓψ)(v)=r(v)ψ, innerΨΦ_rm cube =tfrac18∑_v∈{0,1}^3innerΨ(v)Φ(v). Every r(v) is unitary, so this map is a Hilbert isometry from one spinor to the eight-child cell. Its gauge covariance follows from telescoping along the selected paths. Neither property requires flatness. What curvature obstructs is independence from the reference-path choice.
Exact mathematical passage
```
 (I_\ell\psi)(v)=r(v)\psi,
 \qquad
 \inner{\Psi}{\Phi}_{\rm cube}
 =\tfrac18\sum_{v\in\{0,1\}^3}\inner{\Psi(v)}{\Phi(v)}.
```

\id_AUse in source · 2 passages

Search aliases: id_A id

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Use in source · core.tex, line 378
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
\id_A\otimes\id_B
```

\id_BUse in source · 3 passages

Search aliases: id_B id

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Use in source · core.tex, line 378
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
\id_A\otimes\id_B
```

\operatorname{Im}Use in source · 19 passages

Search aliases: Im operatorname

Link to this entry ·

Use in source · limit.tex, line 403
athsf F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber norm((z-widetilde H_M)^-1-(z-H_ϵ)^-1)f_2 · ≤(|z|^-1+|Imz|^-1) norm(I- E_M)f_2. Thus widetilde H_M converges to H_ϵ in the strong resolvent sense. These statements specify operators on one…
Exact mathematical passage
```
 (z-\widetilde H_M)^{-1}
 &=\mathsf E_M(z-H_\epsilon)^{-1}
   +z^{-1}(I-\mathsf E_M),\nonumber\\
 \norm{((z-\widetilde H_M)^{-1}-(z-H_\epsilon)^{-1})f}_2
 &\leq\left(|z|^{-1}+|\operatorname{Im}z|^{-1}\right)
 \norm{(I-\mathsf E_M)f}_2.
 \label{a-limit-strong-resolvent}
```

\mathrm{in}Use in source · 2 passages

Search aliases: in mathrm

Link to this entry ·

Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

\mathrm{int}Use in source · 8 passages

Search aliases: int mathrm

Link to this entry ·

Use in source · finite.tex, line 740
Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_…
Exact mathematical passage
```
 H_{\mathrm{int}}=
 \sum_{\mathbf d}
 \left(\gamma_{p,w}|\mathbf d\rangle\langle c|
       +\overline{\gamma_{p,w}}|c\rangle\langle\mathbf d|\right).
 \label{a-finite-leaf-interaction}
```

\mathcal JUse in source · 6 passages

Search aliases: J mathcal

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Use in source · spectral.tex, line 66
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
 \dot w_r=\sum_s\mathcal J_{r\leftarrow s},\qquad
 \mathcal J_{r\leftarrow s}
   =2\operatorname{Im}\langle\psi_r,H_{rs}\psi_s\rangle,
 \qquad
 \mathcal J_{r\leftarrow s}=-\mathcal J_{s\leftarrow r}.
```

J^{\mathrm c}Use in source · 2 passages

Search aliases: J^ c mathrm

Link to this entry ·

Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

J^{\mathrm c}_{B\leftarrow A}Use in source · 1 passage

Search aliases: J^ c_B← A mathrm leftarrow

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Use in source · core.tex, line 231
_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence dot p_A^ c=-3/2 and dot p_B^ c=3/2.
Exact mathematical passage
```
J^{\mathrm c}_{B\leftarrow A}=2-1/2=3/2
```

J^{(\rho)}_{a\leftarrow b}Use in source · 1 passage

Search aliases: J^(ρ)_a← b rho leftarrow

Link to this entry ·

Use in source · core.tex, line 334
[Trace-current balance] For a Hermitian ρ satisfying [source reference], dot p_b=∑_aJ^(ρ)_b← a, J^(ρ)_b← a=-J^(ρ)_a← b. The subset boundary formula and exact coarse-current summation remain valid.
Exact mathematical passage
```
 \dot p_b=\sum_aJ^{(\rho)}_{b\leftarrow a},\qquad
 J^{(\rho)}_{b\leftarrow a}=-J^{(\rho)}_{a\leftarrow b}.
```

J^{(\rho)}_{b\leftarrow a}Use in source · 2 passages

Search aliases: J^(ρ)_b← a rho leftarrow

Link to this entry ·

Use in source · core.tex, line 327
Suppose a differentiable matrix trajectory obeys the von Neumann equation dotρ=-i[H,ρ], H=H^*. For its coherent transfer define J^(ρ)_b← a =2ImTr(P_bHP_aρ).
Exact mathematical passage
```
\label{f:mixed-current}
 J^{(\rho)}_{b\leftarrow a}
 =2\operatorname{Im}\operatorname{Tr}(P_bHP_a\rho).
```

J^*Use in source · 1 passage

Search aliases: J^*

Link to this entry ·

Use in source · limit.tex, line 942
ds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its internal norm. The subsequent reversal shift sends these amplitudes to new arcs at the children, outside the old arc range. Therefore, for a unit internal state χ, norm(I-JJ^*)U_p,h+1,NJ(e_j⊗χ) =frac2sqrt p3+p>0. The state JU_p,h,N(e_j⊗χ) has zero amplitude there, so U_p,h+1,NJ≠ JU_p,h,N.
Exact mathematical passage
```
 \norm{(I-JJ^*)U_{p,h+1,N}J(e_j\otimes\chi)}
 =\frac{2\sqrt p}{3+p}>0.
 \label{a-limit-walk-refinement-leakage}
```

J^FUse in source · 1 passage

Search aliases: J^F

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Use in source · extensions.tex, line 71
r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.
Exact mathematical passage
```
J^F
```

J^F_{ba}Use in source · 6 passages

Search aliases: J^F_ba

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Use in source · extensions.tex, line 36
Suppose that a differentiable state curve satisfies dotψ=Kψ. Define J^F_ba=ReinnerF_bψKF_aψ -ReinnerF_aψKF_bψ. This current is antisymmetric directly from its definition. Skew-adjointness also gives J^F_ba=2ReinnerF_bψKF_aψ. The same bilinear expression occurs for sharp packets, but its interpretation now concerns expectation weights.
Exact mathematical passage
```
 J^F_{ba}=2\operatorname{Re}\inner{F_b\psi}{KF_a\psi}.
```

J^{F}_{ba}Use in source · 1 passage

Search aliases: J^F_ba

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Use in source · extensions.tex, line 30
Suppose that a differentiable state curve satisfies dotψ=Kψ. Define J^F_ba=ReinnerF_bψKF_aψ -ReinnerF_aψKF_bψ. This current is antisymmetric directly from its definition. Skew-adjointness also gives J^F_ba=2ReinnerF_bψKF_aψ. The same bilinear expression occurs for sharp packets, but its interpretation now concerns expectation weights.
Exact mathematical passage
```
 J^{F}_{ba}=\operatorname{Re}\inner{F_b\psi}{KF_a\psi}
             -\operatorname{Re}\inner{F_a\psi}{KF_b\psi}.
 \label{eq:foundation-effect-current}
```

J^{\rm c}_{\beta\alpha}Use in source · 2 passages

Search aliases: J^rm c_βα rm beta alpha

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Use in source · extensions.tex, line 66
If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.
Exact mathematical passage
```
 p^{\rm c}_\beta=\sum_{r(a)=\beta}p_a,\qquad
 J^{\rm c}_{\beta\alpha}
 =\sum_{\substack{r(b)=\beta\\r(a)=\alpha}}J^F_{ba}.
```

J_{\mathrm D}^{\Delta}Use in source · 7 passages

Search aliases: J_ D^Δ mathrm Delta

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Use in source · constraints.tex, line 686
labelc:constraints:discrete-gradient The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse are skew, transposition proves (J_ D^Δ)^T=-J_ D^Δ. The endpoint tensor is a projection of the canonical direction into the kernel of all discrete cons…
Exact mathematical passage
```
 J_{\mathrm D}^{\Delta}
       =J-JG^T\Delta_\Delta^{-1}GJ.
 \label{c:constraints:endpoint-tensor}
```

J_0Use in source · 6 passages

Search aliases: J_0

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Use in source · spectral.tex, line 1059
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -
Exact mathematical passage
```
J_0(2\sqrt y)
```

J_0'Use in source · 2 passages

Search aliases: J_0'

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Use in source · core.tex, line 433
Set Q=0 and take an integer n≥1. Define the entire function J_0(x)=∑_m=0^∞frac(-1)^m(x^2/4)^m(m!)^2. The series and its differentiated series converge uniformly on compact sets. Coefficient comparison gives x^2J_0''+xJ_0'+x^2J_0=0. Thus P(τ,θ)=A J_0(ne^-τ)cos(nθ) is an exact polarized solution. Indeed, for x=ne^-τ, ∂_τ^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x), whereas P_θθ=-n^2P. Also M_G=2P_τ P_θ has zero spatial integral, so periodic λ can be reconstructed.
Exact mathematical passage
```
x^2J_0''+xJ_0'+x^2J_0=0
```

J_0''Use in source · 2 passages

Search aliases: J_0''

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Use in source · core.tex, line 433
Set Q=0 and take an integer n≥1. Define the entire function J_0(x)=∑_m=0^∞frac(-1)^m(x^2/4)^m(m!)^2. The series and its differentiated series converge uniformly on compact sets. Coefficient comparison gives x^2J_0''+xJ_0'+x^2J_0=0. Thus P(τ,θ)=A J_0(ne^-τ)cos(nθ) is an exact polarized solution. Indeed, for x=ne^-τ, ∂_τ^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x), whereas P_θθ=-n^2P. Also M_G=2P_τ P_θ has zero spatial integral, so periodic λ can be reconstructed.
Exact mathematical passage
```
x^2J_0''+xJ_0'+x^2J_0=0
```

J_{1:h}Use in source · 3 passages

Search aliases: J_1:h

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Use in source · spectral.tex, line 466
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this…
Exact mathematical passage
```
D=J_{1:h}\otimes I_2
```

J_{1\leftarrow4}Use in source · 1 passage

Search aliases: J_1←4 leftarrow

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Use in source · core.tex, line 225
[Internal exchange disappears from the coarse boundary] Consider four labels with nonzero currents J_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence dot p_A^ c=-3/2 and dot p_B^ c=3/2.
Exact mathematical passage
```
 J_{2\leftarrow1}=3,\quad J_{3\leftarrow2}=2,\quad
 J_{4\leftarrow3}=1,\quad J_{1\leftarrow4}=\tfrac12,
```

J_{2\leftarrow1}Use in source · 4 passages

Search aliases: J_2←1 leftarrow

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Use in source · core.tex, line 173
-isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instantaneous flow must be kept separate.
Exact mathematical passage
```
 J_{2\leftarrow1}
 =2\operatorname{Im}\bigl(i g\sin(gt)\cos(gt)\bigr)
 =g\sin(2gt)=\dot p_2.
```

J_{3\leftarrow2}Use in source · 1 passage

Search aliases: J_3←2 leftarrow

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Use in source · core.tex, line 225
[Internal exchange disappears from the coarse boundary] Consider four labels with nonzero currents J_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence dot p_A^ c=-3/2 and dot p_B^ c=3/2.
Exact mathematical passage
```
 J_{2\leftarrow1}=3,\quad J_{3\leftarrow2}=2,\quad
 J_{4\leftarrow3}=1,\quad J_{1\leftarrow4}=\tfrac12,
```

J_{4\leftarrow3}Use in source · 1 passage

Search aliases: J_4←3 leftarrow

Link to this entry ·

Use in source · core.tex, line 225
[Internal exchange disappears from the coarse boundary] Consider four labels with nonzero currents J_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence dot p_A^ c=-3/2 and dot p_B^ c=3/2.
Exact mathematical passage
```
 J_{2\leftarrow1}=3,\quad J_{3\leftarrow2}=2,\quad
 J_{4\leftarrow3}=1,\quad J_{1\leftarrow4}=\tfrac12,
```

J_5^0Use in source · 1 passage

Search aliases: J_5^0

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Use in source · constraints.tex, line 621
a^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter…
Exact mathematical passage
```
 \Gamma_a{}^i=\mathring\Gamma_a{}^i(E)
                     -\tfrac\kappa4J_5^0e_a{}^i.
```

J_5^IUse in source · 2 passages

Search aliases: J_5^I

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Use in source · constraints.tex, line 612
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
J_5^I=\bar\psi\gamma^I\gamma_5\psi
```

J_5^JUse in source · 1 passage

Search aliases: J_5^J

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Use in source · constraints.tex, line 614
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
 T^I=\tfrac\kappa4\epsilon^I{}_{JKL}J_5^Je^K\wedge e^L,
 \qquad
 C^{IJ}=\tfrac\kappa4\epsilon^{IJ}{}_{KL}J_5^Ke^L,
```

J_5^KUse in source · 1 passage

Search aliases: J_5^K

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Use in source · constraints.tex, line 614
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
 T^I=\tfrac\kappa4\epsilon^I{}_{JKL}J_5^Je^K\wedge e^L,
 \qquad
 C^{IJ}=\tfrac\kappa4\epsilon^{IJ}{}_{KL}J_5^Ke^L,
```

J_{5I}Use in source · 1 passage

Search aliases: J_5I

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Use in source · constraints.tex, line 626
tion is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is -tfrac3κ16∫ J_5IJ_5^Ivol_e. Its sign belongs to that convention; changing the Clifford, orientation, or Dirac-action conventions requires translating it.
Exact mathematical passage
```
-\tfrac{3\kappa}{16}\int J_{5I}J_5^I\operatorname{vol}_e
```

J_{a-1}Use in source · 1 passage

Search aliases: J_a-1

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Use in source · core.tex, line 251
rge without compensating background would be inconsistent. Equation [source reference] supplies that neutralizing mean. Conservation of G_j is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over a≤ j≤ b gives the physical-length-weighted charge balance fracddt(h∑_j=a^bρ_j)=J_a-1-J_b.
Exact mathematical passage
```
 \frac{d}{dt}\left(h\sum_{j=a}^{b}\rho_j\right)=J_{a-1}-J_b.
```

J_{ab}Use in source · 1 passage

Search aliases: J_ab

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Use in source · extensions.tex, line 189
Suppose a supplied curve ψ(t)∈ D(H) is differentiable as a curve in HH and satisfies dotψ=-iHψ. For a domain-preserving atlas, put p_b=normP_bψ^2, J_ba=2IminnerP_bψHP_aψ. Symmetry gives J_ba=-J_ab. Differentiating the bounded projected norm gives dot p_b=2IminnerP_bψHψ. Graph compatibility permits direct insertion of HS_Fψ→ Hψ. The formal development also proves a stronger statement for one receiving packet: symmetry moves H onto that packet, so only the strong state sum is needed. Indeed, inner…
Exact mathematical passage
```
J_{ba}=-J_{ab}
```

J_{a\leftarrow a}Use in source · 1 passage

Search aliases: J_a← a leftarrow

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Use in source · core.tex, line 122
[Finite packet transfer] For the trajectory in [source reference], dot p_b · =∑_aJ_b← a, J_b← a · =-J_a← b, J_a← a=0, fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.
Exact mathematical passage
```
 \dot p_b&=\sum_aJ_{b\leftarrow a},\label{f:continuity}\\
 J_{b\leftarrow a}&=-J_{a\leftarrow b},\qquad
 J_{a\leftarrow a}=0,\label{f:antisymmetry}\\
 \frac{d}{dt}\sum_ap_a&=0.
```

J_{a\leftarrow b}Use in source · 1 passage

Search aliases: J_a← b leftarrow

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Use in source · core.tex, line 122
[Finite packet transfer] For the trajectory in [source reference], dot p_b · =∑_aJ_b← a, J_b← a · =-J_a← b, J_a← a=0, fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.
Exact mathematical passage
```
 \dot p_b&=\sum_aJ_{b\leftarrow a},\label{f:continuity}\\
 J_{b\leftarrow a}&=-J_{a\leftarrow b},\qquad
 J_{a\leftarrow a}=0,\label{f:antisymmetry}\\
 \frac{d}{dt}\sum_ap_a&=0.
```

J_{A\leftarrow B}^{\mathrm c}Use in source · 1 passage

Search aliases: J_A← B^ c leftarrow mathrm

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Use in source · core.tex, line 204
J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
J_{B\leftarrow A}^{\mathrm c}=-J_{A\leftarrow B}^{\mathrm c}
```

J_bUse in source · 1 passage

Search aliases: J_b

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Use in source · core.tex, line 251
rge without compensating background would be inconsistent. Equation [source reference] supplies that neutralizing mean. Conservation of G_j is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over a≤ j≤ b gives the physical-length-weighted charge balance fracddt(h∑_j=a^bρ_j)=J_a-1-J_b.
Exact mathematical passage
```
 \frac{d}{dt}\left(h\sum_{j=a}^{b}\rho_j\right)=J_{a-1}-J_b.
```

J_{ba}Use in source · 7 passages

Search aliases: J_ba

Link to this entry ·

Use in source · extensions.tex, line 6
Throughout this chapter, inner products are conjugate linear in the first variable. A bounded state generator is denoted by K, with K^*=-K. When a Hamiltonian is present, K=-iH. The receiving label is written first in J_ba.
Exact mathematical passage
```
J_{ba}
```

J_{B\leftarrow A}^{\mathrm c}Use in source · 3 passages

Search aliases: J_B← A^ c leftarrow mathrm

Link to this entry ·

Use in source · core.tex, line 197
[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
 p_A^{\mathrm c}&=\norm{Q_A\psi}^2=\sum_{a\in C_A}p_a,\\
 J_{B\leftarrow A}^{\mathrm c}
 &=2\operatorname{Re}\inner{Q_B\psi}{KQ_A\psi}
 =\sum_{b\in C_B}\sum_{a\in C_A}J_{b\leftarrow a}.
```

J_hUse in source · 2 passages

Search aliases: J_h

Link to this entry ·

Use in source · spectral.tex, line 353
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · ·
Exact mathematical passage
```
HR=RJ_h
```

J_jUse in source · 8 passages

Search aliases: J_j

Link to this entry ·

Use in source · core.tex, line 211
sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1, dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1). For example, if z_j=ψ_j^*α U_jψ_j+1, then ∂_A_jImz_j =Im(-ihz_j)=-hRez_j. This fixes the positive sign of dot E_j=J_j for the conventions in the act…
Exact mathematical passage
```
 \dot\psi_j&=-im\beta\psi_j
 -\frac1{2h}\alpha U_j\psi_{j+1}
 +\frac1{2h}U_{j-1}^*\alpha\psi_{j-1},\label{g:matter-ode}\\
 \dot A_j&=E_j,
 &\dot E_j&=J_j,
 &J_j&=\operatorname{Re}(\psi_j^*\alpha U_j\psi_{j+1}).
 \label{g:field-ode}
```

J_{j+1/2}^nUse in source · 2 passages

Search aliases: J_j+1/2^n

Link to this entry ·

Use in source · core.tex, line 88
[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
\label{g:walk-current}
 J_{j+1/2}^n=|\eta_j^+|^2-|\eta_{j+1}^-|^2.
```

J_{j-1}Use in source · 3 passages

Search aliases: J_j-1

Link to this entry ·

Use in source · core.tex, line 233
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-balance}
 \dot\rho_j+\frac{J_j-J_{j-1}}h=0,
 \qquad\dot{\bar\rho}=0,
 \qquad\dot{\mathcal G}_j=0.
```

J_{j-1/2}^nUse in source · 1 passage

Search aliases: J_j-1/2^n

Link to this entry ·

Use in source · core.tex, line 92
[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
\label{g:walk-balance}
 \rho_j^{n+1}-\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.
```

J_kUse in source · 2 passages

Search aliases: J_k

Link to this entry ·

Use in source · finite_core.tex, line 423
⟨ J_nv,J_kw⟩_L^2 =⟨ v,w⟩, · n=k, 0, · n≠ k.
Exact mathematical passage
```
\langle J_nv,J_kw\rangle_{L^2}
=\begin{cases}\langle v,w\rangle,&n=k,\\0,&n\ne k.\end{cases}
```

J_nUse in source · 6 passages

Search aliases: J_n

Link to this entry ·

Use in source · finite_core.tex, line 416
Let J_n: C^4→ L^2( T^3; C^4) be J_nv=e_nv.
Exact mathematical passage
```
J_n:\mathbb C^4\to L^2(\mathbb T^3;\mathbb C^4)
```

J_rUse in source · 5 passages

Search aliases: J_r

Link to this entry ·

Use in source · spectral.tex, line 21
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
J_r:\C^M\to\C^N
```

J_{r:h}Use in source · 2 passages

Search aliases: J_r:h

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Use in source · spectral.tex, line 430
For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber
Exact mathematical passage
```
J_{r:h}
```

J_r^*Use in source · 5 passages

Search aliases: J_r^*

Link to this entry ·

Use in source · spectral.tex, line 25
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
J_r^*J_s=\delta_{rs}I_M
```

\mathcal J_{r\leftarrow s}Use in source · 1 passage

Search aliases: J_r← s mathcal leftarrow

Link to this entry ·

Use in source · spectral.tex, line 66
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
 \dot w_r=\sum_s\mathcal J_{r\leftarrow s},\qquad
 \mathcal J_{r\leftarrow s}
   =2\operatorname{Im}\langle\psi_r,H_{rs}\psi_s\rangle,
 \qquad
 \mathcal J_{r\leftarrow s}=-\mathcal J_{s\leftarrow r}.
```

J_sUse in source · 2 passages

Search aliases: J_s

Link to this entry ·

Use in source · spectral.tex, line 25
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
J_r^*J_s=\delta_{rs}I_M
```

J_s^*Use in source · 2 passages

Search aliases: J_s^*

Link to this entry ·

Use in source · spectral.tex, line 29
section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt …
Exact mathematical passage
```
 X=\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\qquad
 (\mathcal A X\mathcal A^*)_{rs}=X_{rs}.
 \label{a-spectral-block-reconstruction}
```

\mathcal J_{s\leftarrow r}Use in source · 1 passage

Search aliases: J_s← r mathcal leftarrow

Link to this entry ·

Use in source · spectral.tex, line 66
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
 \dot w_r=\sum_s\mathcal J_{r\leftarrow s},\qquad
 \mathcal J_{r\leftarrow s}
   =2\operatorname{Im}\langle\psi_r,H_{rs}\psi_s\rangle,
 \qquad
 \mathcal J_{r\leftarrow s}=-\mathcal J_{s\leftarrow r}.
```

J_tUse in source · 1 passage

Search aliases: J_t

Link to this entry ·

Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

j_{x,\mu}Use in source · 1 passage

Search aliases: j_x,μ mu

Link to this entry ·

Use in source · core.tex, line 341
C^2, U_x,μ∈ SU(2), and local gauges g_x∈ SU(2), as in the legacy non-Abelian chapter. The linked scalar z_x,μ=ψ_x^*U_x,μψ_x+widehatμ is gauge invariant: the adjoint factor g_x^* cancels g_x, and the factors at x+widehatμ also cancel. It follows that the source's scalar link observable j_x,μ=2Imz_x,μ is invariant. To obtain a continuity equation with its particular sign and normalization one must specify the hopping generator. Invariance of the observable alone is not an equation of motion for co…
Exact mathematical passage
```
\label{g:su2-current}
 j_{x,\mu}=2\operatorname{Im}z_{x,\mu}
```

J_\SigmaUse in source · 2 passages

Search aliases: J_Σ Sigma

Link to this entry ·

Use in source · constraints.tex, line 499
, the Cartan identities give L_ηΩ=ι_η R+D_Ω(ι_ηΩ), L_η A=ι_η F+ d_Σ(ι_η A). Accordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression J_Σ[c,χ,u]=C[ζ(c)] +G_ L[χ-ι_cΩ] +G_ U[u+ι_cA]. For tangential c, ζ(c)=ι_cE is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that contraction. One must specify the embedding variables and conjugate momenta,…
Exact mathematical passage
```
 J_\Sigma[c,\chi,u]=C[\zeta(c)]
             +G_{\mathrm L}[\chi-\iota_c\Omega]
             +G_{\mathrm U}[u+\iota_cA].
 \label{c:constraints:boundary-candidate}
```

J_{\partial_+A}^nUse in source · 1 passage

Search aliases: J_∂_+A^n partial

Link to this entry ·

Use in source · core.tex, line 96
i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n
```

J_{\partial_-A}^nUse in source · 1 passage

Search aliases: J_∂_-A^n partial

Link to this entry ·

Use in source · core.tex, line 96
i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n
```

\mathbf kUse in source · 4 passages

Search aliases: k mathbf

Link to this entry ·

Use in source · limit.tex, line 550
⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and the identity is a finite telescoping sum, one factor of which tends strongly to zero in each term, while all other projection factors have norm at mos…
Exact mathematical passage
```
 \mathsf E_{\mathbf h}\mathsf E_{\mathbf k}
 =\mathsf E_{\min(\mathbf h,\mathbf k)},
 \qquad \mathsf E_{\mathbf h}f\to f
 \quad\text{as every }h_p\to\infty.
 \label{a-limit-rectangular-projections}
```

\mathcal KUse in source · 13 passages

Search aliases: K mathcal

Link to this entry ·

Use in source · spectral.tex, line 818
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,
Exact mathematical passage
```
\mathcal K_pf=f\circ R_p
```

kUse in source · 109 passages

Search aliases: k

Link to this entry ·

Use in source · finite.tex, line 46
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
 v_p\bigl((1+p)^{p^k}-1\bigr)=k+1
 \qquad(k\geq0)
```

k^{-2s}Use in source · 1 passage

Search aliases: k^-2s

Link to this entry ·

Use in source · continuum.tex, line 237
A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly when s>3/2. Thus the choice s=1 gives an L^2 field outside the Hamiltonian domain, while s=2 gives one inside it. These thresholds use axis-supported da…
Exact mathematical passage
```
\sum_n\norm{a_n}^2=\sum_{k\ge1}k^{-2s},
```

k^{-s}Use in source · 1 passage

Search aliases: k^-s

Link to this entry ·

Use in source · continuum.tex, line 236
A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly when s>3/2. Thus the choice s=1 gives an L^2 field outside the Hamiltonian domain, while s=2 gives one inside it. These thresholds use axis-supported da…
Exact mathematical passage
```
a_n=k^{-s}v
```

K^2Use in source · 3 passages

Search aliases: K^2

Link to this entry ·

Use in source · finite_core.tex, line 330
Proof. For E>0, the Hermitian matrix K=H/E satisfies K^2=I_4. Thus
Exact mathematical passage
```
K^2=I_4
```

k^2Use in source · 2 passages

Search aliases: k^2

Link to this entry ·

Use in source · spectral.tex, line 474
=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this specific model. Its mass, couplings, and hi…
Exact mathematical passage
```
 \det(z-H_{\rm full})
 =\det(z-D)\bigl((z-\sigma(z))^2-k^2-m_0^2\bigr).
 \label{a-spectral-visible-determinant}
```

k^{2-2s}Use in source · 1 passage

Search aliases: k^2-2s

Link to this entry ·

Use in source · continuum.tex, line 241
A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly when s>3/2. Thus the choice s=1 gives an L^2 field outside the Hamiltonian domain, while s=2 gives one inside it. These thresholds use axis-supported da…
Exact mathematical passage
```
4\pi^2\sum_{k\ge1}k^{2-2s},
```

K^iUse in source · 1 passage

Search aliases: K^i

Link to this entry ·

Use in source · constraints.tex, line 553
ented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ …
Exact mathematical passage
```
K^i=\Omega^{0i}
```

K_aUse in source · 3 passages

Search aliases: K_a

Link to this entry ·

Use in source · constraints.tex, line 556
c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ term. Those directions lie in the k…
Exact mathematical passage
```
 B_{ij}=0,\qquad
 B_{0i}=\tfrac1{4\kappa}\epsilon_{ijk}E^j\wedge E^k,
 \qquad
 \int_\Sigma B_{IJ}\wedge\delta\Omega^{IJ}
      =\frac1\kappa\int_\Sigma\widetilde E^a{}_i\delta K_a{}^i\,\mathrm d^3x.
 \label{c:constraints:simplicity}
```

k_aUse in source · 3 passages

Search aliases: k_a

Link to this entry ·

Use in source · finite.tex, line 177
al P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output per…
Exact mathematical passage
```
k_a=kb^{-1}\bmod a
```

k_bUse in source · 3 passages

Search aliases: k_b

Link to this entry ·

Use in source · finite.tex, line 177
gle · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permutation…
Exact mathematical passage
```
k_b=ka^{-1}\bmod b
```

K_{ba}Use in source · 1 passage

Search aliases: K_ba

Link to this entry ·

Use in source · core.tex, line 113
Define the receiving-first current by J_b← a(t) =2Reinnerψ_b(t)K(t)ψ_a(t) =2Reinnerψ_b(t)K_ba(t)ψ_a(t). The second equality follows because P_bψ_b=ψ_b. A positive value means that the contribution assigned to the pair increases the weight in b. The current is a real number, and its sign convention is fixed by the derivative calculation below.
Exact mathematical passage
```
\label{f:current}
 J_{b\leftarrow a}(t)
 =2\operatorname{Re}\inner{\psi_b(t)}{K(t)\psi_a(t)}
 =2\operatorname{Re}\inner{\psi_b(t)}{K_{ba}(t)\psi_a(t)}.
```

K_{\mathrm{eff}}Use in source · 3 passages

Search aliases: K_eff mathrm

Link to this entry ·

Use in source · core.tex, line 260
[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.
Exact mathematical passage
```
\label{f:effective-generator}
 \dot\phi=K_{\mathrm{eff}}\phi,\qquad
 K_{\mathrm{eff}}=U^*KU-\Gamma,\qquad
 \Gamma^*=-\Gamma,\quad K_{\mathrm{eff}}^*=-K_{\mathrm{eff}}.
```

K_{\mathrm{eff}}^*Use in source · 1 passage

Search aliases: K_eff^* mathrm

Link to this entry ·

Use in source · core.tex, line 260
[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.
Exact mathematical passage
```
\label{f:effective-generator}
 \dot\phi=K_{\mathrm{eff}}\phi,\qquad
 K_{\mathrm{eff}}=U^*KU-\Gamma,\qquad
 \Gamma^*=-\Gamma,\quad K_{\mathrm{eff}}^*=-K_{\mathrm{eff}}.
```

k_iUse in source · 2 passages

Search aliases: k_i

Link to this entry ·

Use in source · core.tex, line 771
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
\widetilde k_i=\sin(k_ih)/h
```

k_i^4Use in source · 1 passage

Search aliases: k_i^4

Link to this entry ·

Use in source · core.tex, line 793
The resulting frequency obeys ω_h^2=∑_ifracsin^2(k_ih)h^2. For fixed k≠0 as h→0, ω_h^2=| k|^2-frach^23∑_i k_i^4+O(h^4), ω_h=| k| -frach^26| k|∑_i k_i^4+O(h^4). These follow by expanding the sine and then the positive square root. The regime keeps k fixed. It is not uniform up to the Nyquist boundary.
Exact mathematical passage
```
 \omega_h^2=|\mathbf k|^2-\frac{h^2}{3}\sum_i k_i^4+O(h^4),
 \qquad
 \omega_h=|\mathbf k|
 -\frac{h^2}{6|\mathbf k|}\sum_i k_i^4+O(h^4).
```

K_jUse in source · 2 passages

Search aliases: K_j

Link to this entry ·

Use in source · constraints.tex, line 192
If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j),
Exact mathematical passage
```
f_j\in\mathcal T_{K_j}
```

K_MUse in source · 3 passages

Search aliases: K_M

Link to this entry ·

Use in source · limit.tex, line 665
There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0. The initial space V_0 and the successive difference spaces therefore have stable self-adjoint finite blocks. Their orthogonal direct sum with the ma…
Exact mathematical passage
```
K_M
```

k_mUse in source · 1 passage

Search aliases: k_m

Link to this entry ·

Use in source · limit.tex, line 269
ginproof The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m gives k_m=ϵ_m g_m. Since k is square summable, ∑_m|ϵ_m|^2normg_m^2<∞. Thus g∈Dom(H_ϵ) and the adjoint agrees with H_ϵ. The reverse inclusion is symme…
Exact mathematical passage
```
k_m=\epsilon_m g_m
```

K_{M+1}Use in source · 3 passages

Search aliases: K_M+1

Link to this entry ·

Use in source · limit.tex, line 666
There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0. The initial space V_0 and the successive difference spaces therefore have stable self-adjoint finite blocks. Their orthogonal direct sum with the ma…
Exact mathematical passage
```
K_{M+1}f=K_Mf
```

\mathcal K_pUse in source · 4 passages

Search aliases: K_p mathcal

Link to this entry ·

Use in source · spectral.tex, line 818
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,
Exact mathematical passage
```
\mathcal K_pf=f\circ R_p
```

K_{p-1}Use in source · 1 passage

Search aliases: K_p-1

Link to this entry ·

Use in source · spectral.tex, line 540
The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.
Exact mathematical passage
```
\kappa K_{p-1}(x)
```

\mathcal K_p^*Use in source · 3 passages

Search aliases: K_p^* mathcal

Link to this entry ·

Use in source · spectral.tex, line 823
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ
Exact mathematical passage
```
 \mathcal K_p\phi_n=\phi_{pn},\qquad
 \mathcal K_p^*\phi_m=
 \begin{cases}\phi_{m/p},&p\mid m,\\0,&p\nmid m.\end{cases}
 \label{a-spectral-koopman-routing}
```

\mathcal K_SUse in source · 4 passages

Search aliases: K_S mathcal

Link to this entry ·

Use in source · finite_core.tex, line 50
K_S=⊕_n∈ S C^4, a _ K_S^2=∑_n∈ S a_n _2^2.
Exact mathematical passage
```
\mathcal K_S=\bigoplus_{n\in S}\mathbb C^4,
\qquad
\|a\|_{\mathcal K_S}^2=\sum_{n\in S}\|a_n\|_2^2.
```

K_{S,a,T}Use in source · 2 passages

Search aliases: K_S,a,T

Link to this entry ·

Use in source · continuum.tex, line 62
Take V=U_h(t), W=U(t), and b=P_Sa. For a fixed finite S, the compact-time packet estimate gives normU_h(t)P_Sa-U(t)P_Sa≤ K_S,a,Th^2 for all |t|≤ T and sufficiently small positive h. The constant is allowed to depend on the support, the amplitudes, and the time window. Combining the estimates gives normU_h(t)a-U(t)a≤ 2norma-P_Sa+K_S,a,Th^2.
Exact mathematical passage
```
\norm{U_h(t)P_Sa-U(t)P_Sa}\le K_{S,a,T}h^2
```

K_\muUse in source · 3 passages

Search aliases: K_μ mu

Link to this entry ·

Use in source · spectral.tex, line 52
For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
\Phi(\rho)=\sum_\mu K_\mu\rho K_\mu^*
```

K_\mu^*Use in source · 2 passages

Search aliases: K_μ^* mu

Link to this entry ·

Use in source · spectral.tex, line 52
For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
\Phi(\rho)=\sum_\mu K_\mu\rho K_\mu^*
```

K_\mu^{\rm poly}Use in source · 1 passage

Search aliases: K_μ^rm poly mu rm

Link to this entry ·

Use in source · spectral.tex, line 54
xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
 K_\mu^{\rm poly}=\mathcal A K_\mu\mathcal A^*,\qquad
 \sum_\mu(K_\mu^{\rm poly})^*K_\mu^{\rm poly}=I.
```

\kerUse in source · 1 passage

Search aliases: ker

Link to this entry ·

Use in source · finite.tex, line 199
The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel that discards a subsystem. For example it sends the normalized state (|0,a⟩+|1,a⟩)/sqrt2 to sqrt2|a⟩. Its adjoint product therefore does not equal th…
Exact mathematical passage
```
 \operatorname{rank}C_{\mathrm{phase}}=2^h,
 \qquad\dim\ker C_{\mathrm{phase}}=2^h.
```

\KK_{\mathbf r}Use in source · 3 passages

Search aliases: KK_ r KK mathbf

Link to this entry ·

Use in source · limit.tex, line 598
[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber
Exact mathematical passage
```
\KK_{\mathbf r}=\operatorname{Ran}\mathsf F_{\mathbf r}
```

\KK_aUse in source · 4 passages

Search aliases: KK_a KK

Link to this entry ·

Use in source · extensions.tex, line 75
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
\KK_a
```

\KK_{\mathrm{tor}}Use in source · 6 passages

Search aliases: KK_tor KK mathrm

Link to this entry ·

Use in source · limit.tex, line 533
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every
Exact mathematical passage
```
\KK_{\mathrm{tor}}
```

\mathcal LUse in source · 13 passages

Search aliases: L mathcal

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\mathfrak LUse in source · 4 passages

Search aliases: L mathfrak

Link to this entry ·

Use in source · core.tex, line 831
Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.
Exact mathematical passage
```
 \mathcal D_h[\xi]=h\sum_a\xi_ap_a(Dq)_a,
 \qquad
 \mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).
```

\mathrm LUse in source · 3 passages

Search aliases: L mathrm

Link to this entry ·

Use in source · constraints.tex, line 478
What is explicit in the internal BFV sector. On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus {Ω_int,Ω_int}=0 under that regular moment-map hypot…
Exact mathematical passage
```
 \{G_{\mathrm L}[\lambda],G_{\mathrm L}[\mu]\}
       =G_{\mathrm L}[[\lambda,\mu]],\qquad
 \{G_{\mathrm L},G_{\mathrm U}\}=\{G_{\mathrm U},G_{\mathrm U}\}=0.
```

lUse in source · 9 passages

Search aliases: l

Link to this entry ·

Use in source · spectral.tex, line 54
xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
 K_\mu^{\rm poly}=\mathcal A K_\mu\mathcal A^*,\qquad
 \sum_\mu(K_\mu^{\rm poly})^*K_\mu^{\rm poly}=I.
```

L^*Use in source · 1 passage

Search aliases: L^*

Link to this entry ·

Use in source · core.tex, line 407
Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected bas…
Exact mathematical passage
```
\label{f:jump-operators}
 L_{ba}=\sqrt{\gamma_{ba}}\,|e_b\rangle\langle e_a|,
 \qquad
 \mathcal D_L(\rho)=L\rho L^*-	frac12(L^*L\rho+\rho L^*L).
```

L^{-1}Use in source · 1 passage

Search aliases: L^-1

Link to this entry ·

Use in source · spectral.tex, line 919
vergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follow by filtering these coefficients again. For an entire F the formulas hold everywhere. For a function continued outside its disk, the domain and any chosen branches must also support all r…
Exact mathematical passage
```
L^{-1}\sum_\ell\omega_L^{(n-j)\ell}
```

L^{2L}Use in source · 1 passage

Search aliases: L^2L

Link to this entry ·

Use in source · spectral.tex, line 1063
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters ar…
Exact mathematical passage
```
 B_{L,j}(y)
 =\frac{(-1)^jy^j}{(j!)^2}\,
 {}_0F_{2L-1}\!\left(
 \begin{matrix}-\\1,\mathcal A_{L,j},\mathcal A_{L,j}\end{matrix};
 \frac{(-1)^Ly^L}{L^{2L}}\right).
 \label{a-spectral-bessel-lane}
```

L^{Lq}Use in source · 1 passage

Search aliases: L^Lq

Link to this entry ·

Use in source · spectral.tex, line 934
Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber · =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (-
Exact mathematical passage
```
 \frac{(Lq+j)!}{j!}
 &=\prod_{r=1}^{L}\prod_{k=0}^{q-1}(j+r+Lk)\nonumber\\
 &=L^{Lq}\prod_{r=1}^{L}\left(\frac{j+r}{L}\right)_q.
 \label{a-spectral-factorial-split}
```

L_2Use in source · 1 passage

Search aliases: L_2

Link to this entry ·

Use in source · spectral.tex, line 770
Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_n(trM_0). The coefficient identity is the same xU_n-1-2U_n-2=2T_n used above.
Exact mathematical passage
```
M_0\in SL_2(\C)
```

L_{ba}^*Use in source · 1 passage

Search aliases: L_ba^*

Link to this entry ·

Use in source · core.tex, line 430
For the one transition a→ b, L_baρ L_ba^*=γ_bap_aP_b, L_ba^*L_ba=γ_baP_a. Its diagonal gain is γ_bap_a at b, and the two halves of the anticommutator remove that same amount at a. No other diagonal entry changes from this one jump. Sum over ordered transitions to obtain [source reference]. Antisymmetry is immediate, and the finite boundary cancellation applies.
Exact mathematical passage
```
 L_{ba}\rho L_{ba}^*=\gamma_{ba}p_aP_b,\qquad
 L_{ba}^*L_{ba}=\gamma_{ba}P_a.
```

\mathcal L_cUse in source · 2 passages

Search aliases: L_c mathcal

Link to this entry ·

Use in source · constraints.tex, line 363
Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,
Exact mathematical passage
```
 sc^\mu=c^\nu\partial_\nu c^\mu,\qquad
 s\chi=\mathcal L_c\chi+\chi^2,\qquad
 s\upsilon=\mathcal L_c\upsilon.
 \label{c:constraints:ghost-differential}
```

L_{CC}Use in source · 1 passage

Search aliases: L_CC

Link to this entry ·

Use in source · constraints.tex, line 827
Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local coarse action. Exact detail elimination can create long-range couplings.
Exact mathematical passage
```
 L_{\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},
```

L_{CD}Use in source · 1 passage

Search aliases: L_CD

Link to this entry ·

Use in source · constraints.tex, line 827
Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local coarse action. Exact detail elimination can create long-range couplings.
Exact mathematical passage
```
 L_{\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},
```

\mathcal L_DUse in source · 3 passages

Search aliases: L_D mathcal

Link to this entry ·

Use in source · constraints.tex, line 815
ectionThe conditional BV pushforward statement Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary c…
Exact mathematical passage
```
\mathcal L_D
```

\mathfrak L_DUse in source · 3 passages

Search aliases: L_D mathfrak

Link to this entry ·

Use in source · core.tex, line 831
Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.
Exact mathematical passage
```
 \mathcal D_h[\xi]=h\sum_a\xi_ap_a(Dq)_a,
 \qquad
 \mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).
```

\mathfrak L_{D_h}Use in source · 1 passage

Search aliases: L_D_h mathfrak

Link to this entry ·

Use in source · core.tex, line 912
For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to control D_hq, the contraction in [source reference] is O(h^2) on a fixed periodic interval. Each summand is uniformly O(h^2) and hN is the interval leng…
Exact mathematical passage
```
 \mathfrak L_{D_h}(f,g)
 =\frac{h^2}{2}(f''g'+f'g'')+O(h^4).
 \label{c:core:smooth-defect}
```

L_{DC}Use in source · 1 passage

Search aliases: L_DC

Link to this entry ·

Use in source · constraints.tex, line 827
Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local coarse action. Exact detail elimination can create long-range couplings.
Exact mathematical passage
```
 L_{\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},
```

L_{DD}^{-1}Use in source · 1 passage

Search aliases: L_DD^-1

Link to this entry ·

Use in source · constraints.tex, line 827
Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local coarse action. Exact detail elimination can create long-range couplings.
Exact mathematical passage
```
 L_{\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},
```

L_{\mathrm{eff}}Use in source · 1 passage

Search aliases: L_eff mathrm

Link to this entry ·

Use in source · constraints.tex, line 827
Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local coarse action. Exact detail elimination can create long-range couplings.
Exact mathematical passage
```
 L_{\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},
```

L_gUse in source · 4 passages

Search aliases: L_g

Link to this entry ·

Use in source · strong_fields.tex, line 267
it-form energy identity] Let D be skew adjoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R(0)_h, normDb_∞≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source ref…
Exact mathematical passage
```
|g'|\le L_g
```

L_hUse in source · 3 passages

Search aliases: L_h

Link to this entry ·

Use in source · core.tex, line 31
w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring…
Exact mathematical passage
```
 L_h=\frac h2\sum_jw_j\dot\varphi_j^2
 -\frac1{2h}\sum_jc_{j+1/2}\Delta_j^2
 -\frac h2\sum_jv_j\varphi_j^2.
 \label{c:core:scalar-action}
```

L_jUse in source · 1 passage

Search aliases: L_j

Link to this entry ·

Use in source · core.tex, line 23
The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.
Exact mathematical passage
```
L_j
```

L_{p,h}Use in source · 2 passages

Search aliases: L_p,h

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Use in source · finite.tex, line 615
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
L_{p,h}=2I-S_{p,h}-S_{p,h}^*
```

\mathcal L_\etaUse in source · 2 passages

Search aliases: L_η mathcal eta

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Use in source · constraints.tex, line 355
The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.
Exact mathematical passage
```
 \left[(\xi,\lambda,u),(\eta,\mu,v)\right]_{\mathrm{par}}
 =\left([\xi,\eta],\mathcal L_\xi\mu-\mathcal L_\eta\lambda+[\lambda,\mu],
                  \mathcal L_\xi v-\mathcal L_\eta u\right).
```

\mathcal L_\eta^{(1/2)}Use in source · 1 passage

Search aliases: L_η^(1/2) mathcal eta

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Use in source · constraints.tex, line 650
For example, the residual electric constraint is G_ U^red=∂_a E^a+qΨ^†Ψ, Q[u]=-∫_Σ u G_ U^red d^3x. It generates δ_uA_a=∂_au and δ_uΨ=iquΨ. The half-density Lie derivative in the spatial generator is L_η^(1/2)Ψ =η^a∂_aΨ+tfrac12(∂_aη^a)Ψ. This term is forced by the density weight. Treating Ψ as a scalar would give a different moment map.
Exact mathematical passage
```
 \mathcal L_\eta^{(1/2)}\Psi
       =\eta^a\partial_a\Psi+\tfrac12(\partial_a\eta^a)\Psi.
```

\mathcal L_\xiUse in source · 4 passages

Search aliases: L_ξ mathcal xi

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Use in source · constraints.tex, line 345
Gauge transformations and graded conventions. For even parameters (ξ,λ,u), the coframe transforms by its Lie derivative and Lorentz rotation, the connection by L_ξω-D_ωλ, and the Abelian potential by L_ξ a+ du. The spinor transforms by L_ξψ+tfrac14λ^IJγ_IJψ+iquψ, with the dual action on barψ. All internal contractions in~[source reference] are invariant. A diffeomorphism changes its four-form density by a Lie derivative, which integrates to zero on a periodic domain or under stated boundary cond…
Exact mathematical passage
```
\mathcal L_\xi\omega-D_\omega\lambda
```

\mathcal L_\xi^{(-1)}Use in source · 2 passages

Search aliases: L_ξ^(-1) mathcal xi

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Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\operatorname{lcp}Use in source · 2 passages

Search aliases: lcp operatorname

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Use in source · limit.tex, line 888
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
\operatorname{lcp}
```

\limUse in source · 3 passages

Search aliases: lim

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Use in source · limit.tex, line 335
ive strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint operator is the generator. A general L^2 state evolves unitarily even if this derivative does not exist.
Exact mathematical passage
```
 \lim_{t\to0}\frac{U(t)f-f}{t}=-iH_\epsilon f.
```

\lim_rUse in source · 1 passage

Search aliases: lim_r lim

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Use in source · continuum.tex, line 177
Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If a^(r)→ a and widehat H_ma^(r)→ b in KK, then convergence holds at each coordinate. Since the finite matrix H_n is continuous, b_n=lim_rH_na_n^(r)=H_na_n. The output sequence equals b∈KK, so a belongs to the maximal domain and widehat H_ma=b. This direct argument is included as an ordinary explanatory deduction.
Exact mathematical passage
```
b_n=\lim_rH_na_n^{(r)}=H_na_n.
```

\lim_{r\downarrow0}Use in source · 1 passage

Search aliases: lim_rdownarrow0 lim downarrow

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Use in source · core.tex, line 230
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
\lim_{r\downarrow0}r^2\Phi'(r)=0
```

\lim_{t\to0}Use in source · 1 passage

Search aliases: lim_t→0 lim to

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Use in source · limit.tex, line 335
ive strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint operator is the generator. A general L^2 state evolves unitarily even if this derivative does not exist.
Exact mathematical passage
```
 \lim_{t\to0}\frac{U(t)f-f}{t}=-iH_\epsilon f.
```

\llUse in source · 1 passage

Search aliases: ll

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Use in source · core.tex, line 302
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 ds^2=-\left(1+\frac{2\Phi}{c_0^2}\right)c_0^2dt^2
       +\left(1-\frac{2\Phi}{c_0^2}\right)d\mathbf x^2,
 \qquad \left|\frac{\Phi}{c_0^2}\right|\ll1,
```

\logUse in source · 20 passages

Search aliases: log

Link to this entry ·

Use in source · limit.tex, line 520
tor has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It is trace class exactly when this series converges. For ϵ_m=c m with c>0, the condition is β c>log p, by the geometric-series test. For ϵ_m=p^α m with α>0, it converges for every β>0, since the exponential energy term eventually dominates the linear-in-m logarithm of the multiplicity. A thermodynamic partition function therefore ne…
Exact mathematical passage
```
\beta c>\log p
```

\log_2Use in source · 2 passages

Search aliases: log_2 log

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Use in source · limit.tex, line 706
Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors with equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bits is S_sign(t) =h_2(frac1+|cos(gt)|2), h_2(u)=-ulog_2u-(1-u)log_2(1-u), with 0log_2 0=0. At t_* =π/(2|g|) the two sign states are orthogonal and the …
Exact mathematical passage
```
 S_{\mathrm{sign}}(t)
 =h_2\left(\frac{1+|\cos(gt)|}{2}\right),
 \quad h_2(u)=-u\log_2u-(1-u)\log_2(1-u),
 \label{a-limit-entanglement-entropy}
```

\LongleftrightarrowUse in source · 1 passage

Search aliases: Longleftrightarrow

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Use in source · continuum.tex, line 212
The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass term is already summable because a∈KK. Membership is therefore equivalent to summability of ∑_n|p(n)|^2norma_n^2.
Exact mathematical passage
```
a\in\Dom(\widehat H_m)
\quad\Longleftrightarrow\quad
\sum_n(m^2+|p(n)|^2)\norm{a_n}^2<\infty.
```

\LongrightarrowUse in source · 2 passages

Search aliases: Longrightarrow

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Use in source · limit.tex, line 883
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
 d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
 \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
 \label{a-limit-product-cone}
```

\lVertUse in source · 1 passage

Search aliases: lVert

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Use in source · core.tex, line 50
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
\label{g:laplace-form}
 -\inner{\psi}{\Delta_U\psi}_V
 =\sum_{x,\mu}\left\lVert(D^+_{\mu,U}\psi)_x\right\rVert^2.
```

\mathrm MUse in source · 6 passages

Search aliases: M mathrm

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Use in source · constraints.tex, line 303
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,
Exact mathematical passage
```
S_0=S_{\mathrm G}+S_{\mathrm M}^{(1)}+S_{\mathrm D}
```

\mathrm mUse in source · 4 passages

Search aliases: m mathrm

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Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

\mathcal M'Use in source · 3 passages

Search aliases: M' mathcal

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Use in source · strong_fields.tex, line 55
[Exact triangular mass identity] For differentiable canonical data in the regular chart, M'=-fracR'Λ H -fracP_ΛRΛ D +frac QR G. Consequently, on a patch where R'/Λ≠0, the systems ( H, D, G)=0 and ( M', D, G)=0 have the same zero set. Differentiating [source reference] gives M'= · fracR'2-fracR'^32Λ^2 -fracRR'R''Λ^2+fracRR'^2Λ'Λ^3 +fracP_Λ P_Λ'R-fracP_Λ^2R'2R^2
Exact mathematical passage
```
 \mathcal M'=-\frac{R'}\Lambda\mathcal H
             -\frac{P_\Lambda}{R\Lambda}\mathcal D
             +\frac QR\mathcal G.
\label{c:strong:mass-identity}
```

m'Use in source · 4 passages

Search aliases: m'

Link to this entry ·

Use in source · strong_fields.tex, line 371
The momentum constraint removes P_R algebraically: P_R=P_Λ'-P_φΦ. Since m'=P_Λ P_Λ'/r-P_Λ^2/(2r^2), inserting this relation into the Hamiltonian constraint gives m'=r^2 E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ),
Exact mathematical passage
```
m'=P_\Lambda P_\Lambda'/r-P_\Lambda^2/(2r^2)
```

M^{-1/2}Use in source · 1 passage

Search aliases: M^-1/2

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Use in source · spectral.tex, line 95
Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1,
Exact mathematical passage
```
(F_M)_{jk}=M^{-1/2}e^{-2\pi ijk/M}
```

M^2Use in source · 5 passages

Search aliases: M^2

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Use in source · strong_fields.tex, line 154
Charged horizons and the turning surface. Write f(R)=1-frac2MR+fracQ^2R^2. For 0<|Q|<M, its two positive roots are r_±=M±sqrtM^2-Q^2. At either root p(R)^2=R^2, so the canonical quantities in [source reference] remain finite if F_r does. For |Q|=M, the two roots coincide at R=M. For |Q|>M there is no real root. At Q=0, the outer root is 2M and the formal inner root 0 is the singular endpoint, not a second regular horizon.
Exact mathematical passage
```
 r_\pm=M\pm\sqrt{M^2-Q^2}.
```

m^2Use in source · 20 passages

Search aliases: m^2

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Use in source · continuum.tex, line 205
The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass term is already summable because a∈KK. Membership is therefore equivalent to summability of ∑_n|p(n)|^2norma_n^2.
Exact mathematical passage
```
H_n^2=(m^2+|p(n)|^2)I_4
```

M_0Use in source · 20 passages

Search aliases: M_0

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Use in source · spectral.tex, line 770
Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_n(trM_0). The coefficient identity is the same xU_n-1-2U_n-2=2T_n used above.
Exact mathematical passage
```
M_0\in SL_2(\C)
```

m_0Use in source · 3 passages

Search aliases: m_0

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Use in source · spectral.tex, line 465
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation
Exact mathematical passage
```
A=H_D(k)=k\sigma_z+m_0\sigma_x
```

M_0^2Use in source · 2 passages

Search aliases: M_0^2

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Use in source · spectral.tex, line 772
Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_n(trM_0). The coefficient identity is the same xU_n-1-2U_n-2=2T_n used above.
Exact mathematical passage
```
M_0^2-xM_0+I=0
```

m_0^2Use in source · 2 passages

Search aliases: m_0^2

Link to this entry ·

Use in source · spectral.tex, line 474
=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this specific model. Its mass, couplings, and hi…
Exact mathematical passage
```
 \det(z-H_{\rm full})
 =\det(z-D)\bigl((z-\sigma(z))^2-k^2-m_0^2\bigr).
 \label{a-spectral-visible-determinant}
```

M_0^jUse in source · 1 passage

Search aliases: M_0^j

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Use in source · spectral.tex, line 788
If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the required nearest-neighbor realization. For a general transfer problem only the matrix power and trace statements have been proved.
Exact mathematical passage
```
 w_{nq+j}=M_0^j(M_0^n)^q w_0,\qquad 0\leq j<n.
```

M_0^nUse in source · 3 passages

Search aliases: M_0^n

Link to this entry ·

Use in source · spectral.tex, line 773
Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_n(trM_0). The coefficient identity is the same xU_n-1-2U_n-2=2T_n used above.
Exact mathematical passage
```
 M_0^n=U_{n-1}(x/2)M_0-U_{n-2}(x/2)I,\qquad n\geq1,
 \label{a-spectral-sl2-power}
```

M_1Use in source · 3 passages

Search aliases: M_1

Link to this entry ·

Use in source · spectral.tex, line 990
For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.
Exact mathematical passage
```
M_1
```

m_1Use in source · 1 passage

Search aliases: m_1

Link to this entry ·

Use in source · continuum.tex, line 226
[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exactly when the momentum series is finite. Adding the already finite unweighted series gives precisely the displayed inhomogeneous first-order condition.…
Exact mathematical passage
```
\Dom(\widehat H_{m_1})=\Dom(\widehat H_{m_2})
```

M_1^jUse in source · 1 passage

Search aliases: M_1^j

Link to this entry ·

Use in source · spectral.tex, line 991
For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.
Exact mathematical passage
```
v_{Lq+j}=M_1^j(M_1^L)^qv_0
```

M_1^LUse in source · 1 passage

Search aliases: M_1^L

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Use in source · spectral.tex, line 991
For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.
Exact mathematical passage
```
v_{Lq+j}=M_1^j(M_1^L)^qv_0
```

m_2Use in source · 1 passage

Search aliases: m_2

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Use in source · continuum.tex, line 226
[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exactly when the momentum series is finite. Adding the already finite unweighted series gives precisely the displayed inhomogeneous first-order condition.…
Exact mathematical passage
```
\Dom(\widehat H_{m_1})=\Dom(\widehat H_{m_2})
```

M_DUse in source · 1 passage

Search aliases: M_D

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Use in source · constraints.tex, line 788
nsfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m -m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_…
Exact mathematical passage
```
 M_D=\begin{pmatrix}x_1^2+m&-m\\-m&m+n\end{pmatrix},
 \qquad D=(m+n)x_1^2+mn>0.
```

M_GUse in source · 10 passages

Search aliases: M_G

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Use in source · core.tex, line 364
Define the densities H_G · =P_τ^2+γ P_θ^2 +e^2P(Q_τ^2+γ Q_θ^2),nonumber M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.
Exact mathematical passage
```
 H_G&=P_\tau^2+\gamma P_\theta^2
 +e^{2P}(Q_\tau^2+\gamma Q_\theta^2),\nonumber\\
 M_G&=2(P_\tau P_\theta+e^{2P}Q_\tau Q_\theta).
 \label{c:core:gowdy-densities}
```

M_{G,h}Use in source · 2 passages

Search aliases: M_G,h

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Use in source · core.tex, line 576
The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservation does not force I_h to vanish at each site. Its refinement behavior tests whether the modified evolution approaches the local continuum reconstructio…
Exact mathematical passage
```
H_{G,h},M_{G,h}
```

m_HUse in source · 2 passages

Search aliases: m_H

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Use in source · strong_fields.tex, line 536
For N_r=2400, the interpolated event has t_H=4.69182094580, r_H=3.11341154797, m_H=r_H/2=1.55670577399. The relation m_H=r_H/2 is imposed by the interpolated marginal-sphere criterion. It is not an independent numerical conservation test.
Exact mathematical passage
```
 t_H=4.69182094580,\quad r_H=3.11341154797,
 \quad m_H=r_H/2=1.55670577399.
```

m_jUse in source · 2 passages

Search aliases: m_j

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Use in source · limit.tex, line 490
limit-strong-resolvent. If the absolute energies tend to infinity, the right side of [source reference] tends to zero. The retained part of the resolvent has finite rank, so the resolvent is a norm limit of finite-rank operators and is compact. Conversely, if the absolute energies do not tend to infinity, there are infinitely many distinct levels m_j with |ϵ_m_j|≤ C for some C. Unit vectors u_j on these levels are orthogonal. For a fixed nonreal z, their resolvent images are also orthogonal and …
Exact mathematical passage
```
m_j
```

M_{p,h}Use in source · 1 passage

Search aliases: M_p,h

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Use in source · finite.tex, line 451
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
M_{p,h}|a+d p^{h-1}\rangle=\omega_p^d|a+d p^{h-1}\rangle
```

M_{p,h}^{\ell}Use in source · 1 passage

Search aliases: M_p,h^ℓ ell

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Use in source · finite.tex, line 453
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
 R_d=\frac1p\sum_{\ell=0}^{p-1}\omega_p^{-d\ell}M_{p,h}^{\ell}.
 \label{a-finite-root-selector}
```

M_qUse in source · 4 passages

Search aliases: M_q

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Use in source · core.tex, line 480
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
M_q=\sum_ap_a^q
```

M_{\rm enc}Use in source · 2 passages

Search aliases: M_rm enc rm

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Use in source · core.tex, line 232
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
 (r^2\Phi')'=4\pi G r^2\rho,
 \qquad M_{\rm enc}(r)=4\pi\int_0^rs^2\rho(s)\,ds,
 \qquad \Phi'(r)=\frac{GM_{\rm enc}(r)}{r^2}.
```

\mathringUse in source · 2 passages

Search aliases: mathring

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Use in source · constraints.tex, line 619
ebraic torsion and the reduced phase space For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representation…
Exact mathematical passage
```
\omega=\mathring\omega(e)+C
```

\maxUse in source · 6 passages

Search aliases: max

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Use in source · finite_core.tex, line 726
Let A be the coordinate space {0,1,2,3}→ C with maximum norm v _∞=max_r|v_r|, and let H be the same coordinates with the Euclidean inner product. The map
Exact mathematical passage
```
\|v\|_\infty=\max_r|v_r|
```

\max_rUse in source · 1 passage

Search aliases: max_r max

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Use in source · finite_core.tex, line 726
Let A be the coordinate space {0,1,2,3}→ C with maximum norm v _∞=max_r|v_r|, and let H be the same coordinates with the Euclidean inner product. The map
Exact mathematical passage
```
\|v\|_\infty=\max_r|v_r|
```

\max_{x\in S}Use in source · 1 passage

Search aliases: max_x∈ S max in

Link to this entry ·

Use in source · refinement.tex, line 225
[Constant-one directional mismatch bound] Let A be a Hermitian Clifford involution, τ a site bijection, and u,v:S→ U(1). Then normS_A,τ,v-S_A,τ,u ≤δ(u,v):=max_x∈ S|v(x)-u(x)| =max_x∈ S|v(x)u(x)^-1-1|. The constant is independent of the number of sites. The actual half-grid defect in [source reference], with half-cell Hilbert measure, obeys the same bound. At each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy…
Exact mathematical passage
```
 \norm{S_{A,\tau,v}-S_{A,\tau,u}}
 \leq\delta(u,v):=\max_{x\in S}|v(x)-u(x)|
 =\max_{x\in S}|v(x)u(x)^{-1}-1|.
 \label{g:constant-one}
```

\midUse in source · 9 passages

Search aliases: mid

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Use in source · spectral.tex, line 823
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ
Exact mathematical passage
```
 \mathcal K_p\phi_n=\phi_{pn},\qquad
 \mathcal K_p^*\phi_m=
 \begin{cases}\phi_{m/p},&p\mid m,\\0,&p\nmid m.\end{cases}
 \label{a-spectral-koopman-routing}
```

\minUse in source · 5 passages

Search aliases: min

Link to this entry ·

Use in source · finite.tex, line 270
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber
Exact mathematical passage
```
 E_r^{(h)}E_s^{(h)}=E_{\min(r,s)}^{(h)},\qquad E_h^{(h)}=I.
```

\mathrm{mod}Use in source · 3 passages

Search aliases: mod mathrm

Link to this entry ·

Use in source · finite.tex, line 256
For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.
Exact mathematical passage
```
 (E_r^{(h)}f)(x)=p^{-(h-r)}
       \sum_{y\equiv x\, (\mathrm{mod}\,p^r)}f(y).
 \label{a-finite-cylinder-average}
```

\NUse in source · 3 passages

Search aliases: N

Link to this entry ·

Use in source · limit.tex, line 575
Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.
Exact mathematical passage
```
 \mathsf F_{\mathbf r}
 =I_{\KK_{\mathrm{tor}}}\otimes
 \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
 \qquad \mathbf r\in\N^{\mathcal P}.
 \label{a-limit-joint-details}
```

\mathcal NUse in source · 4 passages

Search aliases: N mathcal

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Use in source · constraints.tex, line 816
regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution, Δ_C e^iS_C/ℏ =- N^-1∫_ L_DΔ_D e^iS_F/ℏ=0. Th…
Exact mathematical passage
```
 e^{iS_C/\hbar}=\mathcal N^{-1}\int_{\mathcal L_D}e^{iS_F/\hbar}.
```

\N^{\mathcal P}Use in source · 1 passage

Search aliases: N^ P N mathcal

Link to this entry ·

Use in source · limit.tex, line 575
Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.
Exact mathematical passage
```
 \mathsf F_{\mathbf r}
 =I_{\KK_{\mathrm{tor}}}\otimes
 \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
 \qquad \mathbf r\in\N^{\mathcal P}.
 \label{a-limit-joint-details}
```

\mathcal N^{-1}Use in source · 2 passages

Search aliases: N^-1 mathcal

Link to this entry ·

Use in source · constraints.tex, line 816
regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution, Δ_C e^iS_C/ℏ =- N^-1∫_ L_DΔ_D e^iS_F/ℏ=0. Th…
Exact mathematical passage
```
 e^{iS_C/\hbar}=\mathcal N^{-1}\int_{\mathcal L_D}e^{iS_F/\hbar}.
```

n^{-1}Use in source · 1 passage

Search aliases: n^-1

Link to this entry ·

Use in source · finite.tex, line 729
After w refinements a packet has n=p^w orthonormal descendants. Its normalized radial vector is |R_p,w⟩=n^-1/2∑_ d| d⟩. The norm is n^-1∑_ d1=1, so each descendant has amplitude modulus p^-w/2 and Born probability p^-w. This reproduces [source reference]. The words ``radial'' and ``angular'' here refer to the uniform vector and its orthogonal complement in a finite descendant space. They do not introduce a spacetime radius or a rotation group.
Exact mathematical passage
```
n^{-1}\sum_{\mathbf d}1=1
```

N^{-1/2}Use in source · 3 passages

Search aliases: N^-1/2

Link to this entry ·

Use in source · finite.tex, line 179
bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permutation would generally give a diffe…
Exact mathematical passage
```
 N^{-1/2}\exp\!\left[2\pi\mathrm{i}
      \left(\frac{k_a n_a}{a}+\frac{k_b n_b}{b}\right)\right].
```

n^{-1/2}Use in source · 2 passages

Search aliases: n^-1/2

Link to this entry ·

Use in source · finite.tex, line 144
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 (F_n)_{k x}=n^{-1/2}\exp(2\pi\mathrm{i}kx/n).
 \label{a-finite-fourier-convention}
```

n^{-2}Use in source · 3 passages

Search aliases: n^-2

Link to this entry ·

Use in source · extensions.tex, line 279
foundationfigure[H]11_graph_tailsFor the normalized state proportional to n^-2 and He_n=ne_n, the state tail and generator tail decay at different rates. Graph control retains both contributions.
Exact mathematical passage
```
n^{-2}
```

N^{-3}Use in source · 1 passage

Search aliases: N^-3

Link to this entry ·

Use in source · constraints.tex, line 190
e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
\tau_N=N^{-3}\sum_x
```

N^{-3/2}Use in source · 1 passage

Search aliases: N^-3/2

Link to this entry ·

Use in source · extensions.tex, line 306
For the faster normalized tail c_n=C/n^2, both errors are controlled: normc-S_Nc^2=C^2∑_n>Nn^-4, normH_λ c-S_NH_λ c^2=C^2∑_n>Nn^-2. The state tail is O(N^-3/2) in norm, while the velocity tail is O(N^-1/2). A small state error can therefore coexist with a much larger derivative error even inside the true domain.
Exact mathematical passage
```
O(N^{-3/2})
```

n^{-4}Use in source · 1 passage

Search aliases: n^-4

Link to this entry ·

Use in source · extensions.tex, line 302
For the faster normalized tail c_n=C/n^2, both errors are controlled: normc-S_Nc^2=C^2∑_n>Nn^-4, normH_λ c-S_NH_λ c^2=C^2∑_n>Nn^-2. The state tail is O(N^-3/2) in norm, while the velocity tail is O(N^-1/2). A small state error can therefore coexist with a much larger derivative error even inside the true domain.
Exact mathematical passage
```
 \norm{c-S_Nc}^2=C^2\sum_{n>N}n^{-4},\qquad
 \norm{H_\lambda c-S_NH_\lambda c}^2=C^2\sum_{n>N}n^{-2}.
```

n^{-s}Use in source · 2 passages

Search aliases: n^-s

Link to this entry ·

Use in source · extensions.tex, line 169
For positive λ_n, the quadratic-form domain instead uses ∑_nλ_n|c_n|^2<∞. This is the domain of H_λ^1/2, not generally the domain of H_λ. With λ_n=n and c_n=n^-s, the three thresholds are c|c membership · condition
Exact mathematical passage
```
c_n=n^{-s}
```

n^2Use in source · 7 passages

Search aliases: n^2

Link to this entry ·

Use in source · spectral.tex, line 1082
bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^2B_L,j(y)=-yB_L,j-1bmod L(y). For a term of positive degree n, its new coefficient is n^2(-1)^n/(n!)^2=-(-1)^n-1/[(n-1)!]^2. The degree-zero term disappears under Θ^2, which also verifies the wraparound case j=0. The scalar Gowdy Bessel mode therefore admits these analytic coordin…
Exact mathematical passage
```
n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2
```

N^3Use in source · 3 passages

Search aliases: N^3

Link to this entry ·

Use in source · constraints.tex, line 189
T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
N^3
```

n_0Use in source · 5 passages

Search aliases: n_0

Link to this entry ·

Use in source · finite_core.tex, line 803
S={n_0,n_A,n_D}, n_0=(0,0,0), n_A=(1,0,0), n_D=(1,1,1).
Exact mathematical passage
```
S=\{n_0,n_A,n_D\},\qquad
n_0=(0,0,0),\quad n_A=(1,0,0),\quad n_D=(1,1,1).
```

n_1Use in source · 1 passage

Search aliases: n_1

Link to this entry ·

Use in source · finite_core.tex, line 14
equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is
Exact mathematical passage
```
n=(n_1,n_2,n_3)\in\mathbb Z^3
```

n_2Use in source · 1 passage

Search aliases: n_2

Link to this entry ·

Use in source · finite_core.tex, line 14
equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is
Exact mathematical passage
```
n=(n_1,n_2,n_3)\in\mathbb Z^3
```

n_3Use in source · 1 passage

Search aliases: n_3

Link to this entry ·

Use in source · finite_core.tex, line 14
equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is
Exact mathematical passage
```
n=(n_1,n_2,n_3)\in\mathbb Z^3
```

n_AUse in source · 5 passages

Search aliases: n_A

Link to this entry ·

Use in source · finite_core.tex, line 803
S={n_0,n_A,n_D}, n_0=(0,0,0), n_A=(1,0,0), n_D=(1,1,1).
Exact mathematical passage
```
S=\{n_0,n_A,n_D\},\qquad
n_0=(0,0,0),\quad n_A=(1,0,0),\quad n_D=(1,1,1).
```

n_aUse in source · 2 passages

Search aliases: n_a

Link to this entry ·

Use in source · finite.tex, line 176
P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permut…
Exact mathematical passage
```
n_a=n\bmod a
```

n_bUse in source · 2 passages

Search aliases: n_b

Link to this entry ·

Use in source · finite.tex, line 176
P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permut…
Exact mathematical passage
```
n_b=n\bmod b
```

n_DUse in source · 5 passages

Search aliases: n_D

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Use in source · finite_core.tex, line 803
S={n_0,n_A,n_D}, n_0=(0,0,0), n_A=(1,0,0), n_D=(1,1,1).
Exact mathematical passage
```
S=\{n_0,n_A,n_D\},\qquad
n_0=(0,0,0),\quad n_A=(1,0,0),\quad n_D=(1,1,1).
```

n_iUse in source · 7 passages

Search aliases: n_i

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Use in source · core.tex, line 663
Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber
Exact mathematical passage
```
n_i
```

N_jUse in source · 1 passage

Search aliases: N_j

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Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

n_jUse in source · 3 passages

Search aliases: n_j

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Use in source · core.tex, line 671
innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic gradient is 2π_ij-δ_ijπ_rm tr. Lemma~[source reference] makes the potential gradient G(h). The lapse gradient follows by two spatial integrations by pa…
Exact mathematical passage
```
 \dot h_{ij}&=2\pi_{ij}-\delta_{ij}\pi_{\rm tr}
             +\delta_in_j+\delta_jn_i,\nonumber\\
 \dot\pi_{ij}&=-\mathcal G_{ij}(h)
              +\delta_i\delta_jn-\delta_{ij}\delta^2n.
 \label{c:core:adm-evolution}
```

N_{j+1/2}Use in source · 1 passage

Search aliases: N_j+1/2

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Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

n_kUse in source · 1 passage

Search aliases: n_k

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Use in source · core.tex, line 706
The trace of the momentum part of dot h is -π_rm tr. Its contribution to dot C is therefore 2δ_iδ_jπ_ij=-δ_iC_i. The shift contribution is δ_iδ_j(δ_in_j+δ_jn_i) -δ^2(2δ_kn_k)=0. For dot C_i, the divergence of G vanishes by Lemma~[source reference], and δ_j(δ_iδ_jn-δ_ijδ^2n)=0. This proves the propagation equations and their stated implication.
Exact mathematical passage
```
 \delta_i\delta_j(\delta_in_j+\delta_jn_i)
 -\delta^2(2\delta_kn_k)=0.
```

N_rUse in source · 4 passages

Search aliases: N_r

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Use in source · strong_fields.tex, line 516
Recorded first marginally trapped sphere. The event finder takes a quadratic interpolation of the spatial maximum of 2m/r and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is small rrrrr N_r · h · t_H · r_H · Δ M/M_0
Exact mathematical passage
```
N_r
```

\mathcal N_{\rm emit}Use in source · 1 passage

Search aliases: N_rm emit mathcal rm

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\mathcal N_{\rm obs}Use in source · 1 passage

Search aliases: N_rm obs mathcal rm

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\N_{\geq1}Use in source · 1 passage

Search aliases: N_≥1 N geq

Link to this entry ·

Use in source · extensions.tex, line 145
[A small state can have a large generator image] On ℓ^2(N_≥1), let He_n=ne_n on its maximal domain. The finitely supported vectors x_N=e_N/N satisfy normx_N=1/N→0, while normHx_N=1. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because Hx_N=e_N has no norm limit.
Exact mathematical passage
```
\ell^2(\N_{\geq1})
```

\nmidUse in source · 2 passages

Search aliases: nmid

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Use in source · finite.tex, line 52
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
p\nmid c
```

OUse in source · 48 passages

Search aliases: O

Link to this entry ·

Use in source · finite.tex, line 676
For a self-adjoint fine Hamiltonian with coarse projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional example H_f=0 · overlineγ
Exact mathematical passage
```
 Qe^{-\mathrm{i}tH_f}\psi=-\mathrm{i}tB\psi+O(t^2),
 \qquad
 \norm{Qe^{-\mathrm{i}tH_f}\psi}^2
 =t^2\norm{B\psi}^2+O(|t|^3).
```

\mathrm oUse in source · 5 passages

Search aliases: o mathrm

Link to this entry ·

Use in source · finite.tex, line 119
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique
Exact mathematical passage
```
\mathcal P_{\mathrm o}
```

oUse in source · 12 passages

Search aliases: o

Link to this entry ·

Use in source · spectral.tex, line 54
xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
 K_\mu^{\rm poly}=\mathcal A K_\mu\mathcal A^*,\qquad
 \sum_\mu(K_\mu^{\rm poly})^*K_\mu^{\rm poly}=I.
```

\ointUse in source · 1 passage

Search aliases: oint

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Use in source · core.tex, line 168
We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt along the two horizontal endpoints and ϕ_xx along the two vertical endpoints, with the boundary orientation from dtwedge dx. This is the rectangular funda…
Exact mathematical passage
```
 W_\square
 =\exp\left(K\oint_{\partial\square}(\phi_xdt+\phi_tdx)\right)
 =\exp\left(\frac K2\int_\square e^{2\phi}R\,dt\,dx\right).
 \label{c:core:conformal-holonomy}
```

\oint_{\partial\square}Use in source · 1 passage

Search aliases: oint_∂square oint partial square

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Use in source · core.tex, line 168
We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt along the two horizontal endpoints and ϕ_xx along the two vertical endpoints, with the boundary orientation from dtwedge dx. This is the rectangular funda…
Exact mathematical passage
```
 W_\square
 =\exp\left(K\oint_{\partial\square}(\phi_xdt+\phi_tdx)\right)
 =\exp\left(\frac K2\int_\square e^{2\phi}R\,dt\,dx\right).
 \label{c:core:conformal-holonomy}
```

\operatorname{or}Use in source · 2 passages

Search aliases: or operatorname

Link to this entry ·

Use in source · core.tex, line 170
For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a principal argument, it implies that the oriented phase sum lies in 2πZ. It does not force that su…
Exact mathematical passage
```
\label{g:abelian-cube}
 \prod_{f\subset\partial c}W_f^{\operatorname{or}(f,c)}=1.
```

\mathrm{out}Use in source · 2 passages

Search aliases: out mathrm

Link to this entry ·

Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

\overline AUse in source · 1 passage

Search aliases: overline A overline

Link to this entry ·

Use in source · constraints.tex, line 270
The Lorentzian limitation of this self-dual chart is also explicit. A single Lorentzian self-dual connection is complex, with a physical sector requiring real E and A+overline A=2Γ(E). The source chapter does not construct a finite step preserving that sector. The real parent below addresses reality by changing variables and retaining both chiral sectors, rather than by declaring this missing step solved inside the single complex chart.
Exact mathematical passage
```
A+\overline A=2\Gamma(E)
```

\overline fUse in source · 1 passage

Search aliases: overline f overline

Link to this entry ·

Use in source · limit.tex, line 17
Fix a prime p. The compact additive group Z_p has normalized Haar measure μ_p, with B_m,a=a+p^mZ_p, 0≤ a<p^m, μ_p(B_m,a)=p^-m. Write HH_p=L^2(Z_p,μ_p;C). We use the complex inner product innerfg=∫overline f g dμ_p, linear in its second argument. The p-adic coordinate belongs to the measurable space. Amplitudes and the Born probabilities use ordinary complex Hilbert space.
Exact mathematical passage
```
\inner{f}{g}=\int\overline f g\,d\mu_p
```

\overline GUse in source · 1 passage

Search aliases: overline G overline

Link to this entry ·

Use in source · finite.tex, line 774
Assume those invariance conditions and order the basis as (|c⟩,|R_p,w⟩). With visible energy E_c and radial energy ϵ, the exact bright block is H_bright= E_c · overline G_p,w G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t…
Exact mathematical passage
```
 H_{\mathrm{bright}}=
 \begin{pmatrix}E_c&\overline G_{p,w}\\G_{p,w}&\epsilon\end{pmatrix}.
 \label{a-finite-bright-block}
```

\overline gUse in source · 1 passage

Search aliases: overline g overline

Link to this entry ·

Use in source · limit.tex, line 99
E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows innergf- E_mf=0. The image is in V_m, and averaging a function in V_m returns the function. These statements characterize the orthogonal projection and establish the first two identities. The spaces are nested, because every cylinder partitions into it…
Exact mathematical passage
```
\overline g
```

\overline G_{p,w}Use in source · 1 passage

Search aliases: overline G_p,w overline

Link to this entry ·

Use in source · finite.tex, line 774
Assume those invariance conditions and order the basis as (|c⟩,|R_p,w⟩). With visible energy E_c and radial energy ϵ, the exact bright block is H_bright= E_c · overline G_p,w G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t…
Exact mathematical passage
```
 H_{\mathrm{bright}}=
 \begin{pmatrix}E_c&\overline G_{p,w}\\G_{p,w}&\epsilon\end{pmatrix}.
 \label{a-finite-bright-block}
```

\overline{G_{p,w}}Use in source · 1 passage

Search aliases: overlineG_p,w overline

Link to this entry ·

Use in source · finite.tex, line 754
c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_ d=0. Hence H_int|A⟩=0. Only the radial vector is bright under this equal-leaf interaction.
Exact mathematical passage
```
H_{\mathrm{int}}|R_{p,w}\rangle=
\overline{G_{p,w}}|c\rangle
```

\overline{\inner{\psi_b}{K\psi_a}}Use in source · 1 passage

Search aliases: overlineinnerψ_bKψ_a overline inner psi psi

Link to this entry ·

Use in source · core.tex, line 144
· =∑_a2Reinnerψ_bKψ_a. The reconstruction of ψ justifies the last finite sum. Skew-adjointness gives innerψ_aKψ_b =-innerKψ_aψ_b =-overlineinnerψ_bKψ_a. Taking real parts proves antisymmetry. Putting a=b gives zero diagonal current. Every internal ordered pair in C× C cancels with its reverse. Splitting the sum over all sources into C and its complement proves [source reference]. Taking C=I proves total conservation.
Exact mathematical passage
```
 \inner{\psi_a}{K\psi_b}
 =-\inner{K\psi_a}{\psi_b}
 =-\overline{\inner{\psi_b}{K\psi_a}}.
```

\overline{P_+X}Use in source · 1 passage

Search aliases: overlineP_+X overline

Link to this entry ·

Use in source · constraints.tex, line 415
Reality is built into e and ω. For real Lorentz bivectors, the internal dual obeys ⋆^2=-id and the projectors P_±=tfrac12(id∓ i⋆) satisfy overlineP_+X=P_-X. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical starting point. It does not prove a missing finite reality-preserving step for the earlier isolated self-dual coordinates.
Exact mathematical passage
```
\overline{P_+X}=P_-X
```

\overline{v_r}Use in source · 1 passage

Search aliases: overlinev_r overline

Link to this entry ·

Use in source · finite_core.tex, line 24
⟨ v,w⟩=∑_r=0^3overlinev_rw_r, v _2^2=∑_r=0^3|v_r|^2.
Exact mathematical passage
```
\langle v,w\rangle=\sum_{r=0}^{3}\overline{v_r}w_r,
\qquad
\|v\|_2^2=\sum_{r=0}^{3}|v_r|^2.
```

\overline{z_{ba}}Use in source · 1 passage

Search aliases: overlinez_ba overline

Link to this entry ·

Use in source · core.tex, line 343
Let z_ba=Tr(P_bHP_aρ). Taking the conjugate trace and cyclically moving the first factor to the end yields overlinez_ba=Tr(ρ P_aHP_b) =Tr(P_aHP_bρ)=z_ab. Completeness and cyclicity resolve the derivative as dot p_b=-i∑_a(z_ba-z_ab) =∑_a2Imz_ba. The conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exactly as before.
Exact mathematical passage
```
 \overline{z_{ba}}=\operatorname{Tr}(\rho P_aHP_b)
 =\operatorname{Tr}(P_aHP_b\rho)=z_{ab}.
```

\overline{\{\epsilon_m:m\geq0\}}Use in source · 1 passage

Search aliases: overline{ϵ_m:m≥0} overline epsilon geq

Link to this entry ·

Use in source · limit.tex, line 258
[Real level energies define a self-adjoint operator] The operator in [source reference] is densely defined, closed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, s…
Exact mathematical passage
```
 \sigma(H_\epsilon)=\overline{\{\epsilon_m:m\geq0\}}.
 \label{a-limit-spectrum}
```

\overline\gammaUse in source · 1 passage

Search aliases: overlineγ overline gamma

Link to this entry ·

Use in source · finite.tex, line 685
se projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional example H_f=0 · overlineγ γ · 0, P=1 · 0 0 · 0 has compressed Hamiltonian zero but detail probability sin^2(|γ|t) from the first basis state. It fails exact refinement compatibility whenever γ≠0.
Exact mathematical passage
```
 H_f=\begin{pmatrix}0&\overline\gamma\\\gamma&0\end{pmatrix},
 \qquad P=\begin{pmatrix}1&0\\0&0\end{pmatrix}
```

\overline{\gamma_{p,w}}Use in source · 1 passage

Search aliases: overlineγ_p,w overline gamma

Link to this entry ·

Use in source · finite.tex, line 740
Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_…
Exact mathematical passage
```
 H_{\mathrm{int}}=
 \sum_{\mathbf d}
 \left(\gamma_{p,w}|\mathbf d\rangle\langle c|
       +\overline{\gamma_{p,w}}|c\rangle\langle\mathbf d|\right).
 \label{a-finite-leaf-interaction}
```

\overline\psiUse in source · 2 passages

Search aliases: overlineψ overline psi

Link to this entry ·

Use in source · core.tex, line 210
al Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1,
Exact mathematical passage
```
ih\dot\psi_j=\partial H_h/\partial\overline\psi_j
```

\overline\psi_jUse in source · 2 passages

Search aliases: overlineψ_j overline psi

Link to this entry ·

Use in source · core.tex, line 210
al Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1,
Exact mathematical passage
```
ih\dot\psi_j=\partial H_h/\partial\overline\psi_j
```

\overline\nablaUse in source · 9 passages

Search aliases: overline∇ overline nabla

Link to this entry ·

Use in source · constraints.tex, line 676
hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let …
Exact mathematical passage
```
 \overline\nabla F(z_0,z_1)=\int_0^1\nabla F((1-s)z_0+sz_1)\,\mathrm ds.
 \label{c:constraints:discrete-gradient}
```

\overline{\langle\psi_r,H_{rs}\psi_s\rangle}Use in source · 1 passage

Search aliases: overline⟨ψ_r,H_rsψ_s⟩ overline langle psi psi rangle

Link to this entry ·

Use in source · spectral.tex, line 80
al J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a probability or squared amplitude weight. Calling it an energy would require an additional energy observable and its balance law.
Exact mathematical passage
```
\langle\psi_s,H_{sr}\psi_r\rangle
=\overline{\langle\psi_r,H_{rs}\psi_s\rangle}
```

\mathcal PUse in source · 11 passages

Search aliases: P mathcal

Link to this entry ·

Use in source · finite.tex, line 119
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique
Exact mathematical passage
```
\mathcal P_{\mathrm o}
```

\mathrm PUse in source · 1 passage

Search aliases: P mathrm

Link to this entry ·

Use in source · constraints.tex, line 586
the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
 \{\chi_A(x),\bar\chi_B(y)\}_{\mathrm P}
            =-ie(x)\delta_{AB}\delta^3(x-y).
```

p^{\mathrm c}Use in source · 2 passages

Search aliases: p^ c mathrm

Link to this entry ·

Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

p^{(1-2\alpha)w}Use in source · 1 passage

Search aliases: p^(1-2α)w alpha

Link to this entry ·

Use in source · finite.tex, line 820
Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf interaction: its two radial eigenvalues are ±|G_p,w|, so normH_int=|G_p,w|. They do not determine the boundedness of a different infinite operator with man…
Exact mathematical passage
```
 G_{p,w}=g p^{(1/2-\alpha)w},\qquad
 |G_{p,w}|^2=|g|^2p^{(1-2\alpha)w}.
 \label{a-finite-coupling-exponent}
```

p^{(1/2-\alpha)w}Use in source · 1 passage

Search aliases: p^(1/2-α)w alpha

Link to this entry ·

Use in source · finite.tex, line 820
Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf interaction: its two radial eigenvalues are ±|G_p,w|, so normH_int=|G_p,w|. They do not determine the boundedness of a different infinite operator with man…
Exact mathematical passage
```
 G_{p,w}=g p^{(1/2-\alpha)w},\qquad
 |G_{p,w}|^2=|g|^2p^{(1-2\alpha)w}.
 \label{a-finite-coupling-exponent}
```

p^{(m+1)/2}Use in source · 1 passage

Search aliases: p^(m+1)/2

Link to this entry ·

Use in source · limit.tex, line 56
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
p^{-1/2}p^{(m+1)/2}=p^{m/2}
```

p^{(m-1)/2}Use in source · 1 passage

Search aliases: p^(m-1)/2

Link to this entry ·

Use in source · limit.tex, line 174
For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω_p^(q'-q)d=δ_qq'. The number of these orthonormal functions is exactly d_m, so they span that level. Together with ψ_0=1, they form a complete orthono…
Exact mathematical passage
```
 \psi_{m,a,q}=p^{(m-1)/2}\sum_{d=0}^{p-1}
 \omega_p^{qd}\mathbf1_{B_{m,a+d p^{m-1}}},
 \quad
 \begin{matrix}m\geq1,\quad 0\leq a<p^{m-1},\\1\leq q<p.\end{matrix}
 \label{a-limit-haar-wavelets}
```

P^+_AUse in source · 1 passage

Search aliases: P^+_A

Link to this entry ·

Use in source · core.tex, line 115
The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness follows from A=A^*. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coord…
Exact mathematical passage
```
\label{g:clifford-projections}
 P^+_A=\tfrac12(I+A),\qquad P^-_A=\tfrac12(I-A)
```

p^{-(h-r)}Use in source · 3 passages

Search aliases: p^-(h-r)

Link to this entry ·

Use in source · finite.tex, line 256
For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.
Exact mathematical passage
```
 (E_r^{(h)}f)(x)=p^{-(h-r)}
       \sum_{y\equiv x\, (\mathrm{mod}\,p^r)}f(y).
 \label{a-finite-cylinder-average}
```

p^{-(h-r+1)}Use in source · 1 passage

Search aliases: p^-(h-r+1)

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

p^{-(r+1)/2}Use in source · 1 passage

Search aliases: p^-(r+1)/2

Link to this entry ·

Use in source · spectral.tex, line 347
dots,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributions instead give p p^-r/2γ_r-1/sqrt p =γ_r-1p^-(r-1)/2.
Exact mathematical passage
```
p^{-r/2}\gamma_r/\sqrt p=\gamma_r p^{-(r+1)/2}
```

p^{-(r-1)/2}Use in source · 1 passage

Search aliases: p^-(r-1)/2

Link to this entry ·

Use in source · spectral.tex, line 349
on calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributions instead give p p^-r/2γ_r-1/sqrt p =γ_r-1p^-(r-1)/2.
Exact mathematical passage
```
p\,p^{-r/2}\gamma_{r-1}/\sqrt p
=\gamma_{r-1}p^{-(r-1)/2}
```

p^{-1}Use in source · 4 passages

Search aliases: p^-1

Link to this entry ·

Use in source · finite.tex, line 343
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
 p^{-1}\sum_{d=0}^{p-1}\omega_p^{(q'-q)d}=\delta_{qq'}.
```

p^{-1/2}Use in source · 7 passages

Search aliases: p^-1/2

Link to this entry ·

Use in source · finite.tex, line 374
One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an id…
Exact mathematical passage
```
 (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),
 \quad 0\leq a<p^h,\quad0\leq d<p.
 \label{a-finite-refinement}
```

p^{-2w}Use in source · 1 passage

Search aliases: p^-2w

Link to this entry ·

Use in source · finite.tex, line 862
A model that directly couples |c⟩ to one normalized hidden packet with matrix element g p^-w instead has a collective coefficient of that same size and a self-energy numerator |g|^2p^-2w. It is a valid two-level model, but it is not the equal-leaf model [source reference]. The number of descendant addresses alone does not select between these interaction geometries. A physical refinement law must specify the coupling, the retained hidden modes, their energies, and the observation scale.
Exact mathematical passage
```
|g|^2p^{-2w}
```

P^-_AUse in source · 1 passage

Search aliases: P^-_A

Link to this entry ·

Use in source · core.tex, line 115
The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness follows from A=A^*. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coord…
Exact mathematical passage
```
\label{g:clifford-projections}
 P^+_A=\tfrac12(I+A),\qquad P^-_A=\tfrac12(I-A)
```

p^{-h}Use in source · 1 passage

Search aliases: p^-h

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

p^{-h/2}Use in source · 1 passage

Search aliases: p^-h/2

Link to this entry ·

Use in source · finite.tex, line 335
0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
\psi_0^{(h)}(x)=p^{-h/2}
```

p^{-m}Use in source · 4 passages

Search aliases: p^-m

Link to this entry ·

Use in source · limit.tex, line 12
Fix a prime p. The compact additive group Z_p has normalized Haar measure μ_p, with B_m,a=a+p^mZ_p, 0≤ a<p^m, μ_p(B_m,a)=p^-m. Write HH_p=L^2(Z_p,μ_p;C). We use the complex inner product innerfg=∫overline f g dμ_p, linear in its second argument. The p-adic coordinate belongs to the measurable space. Amplitudes and the Born probabilities use ordinary complex Hilbert space.
Exact mathematical passage
```
 B_{m,a}=a+p^m\Z_p,\qquad 0\leq a<p^m,\qquad
 \mu_p(B_{m,a})=p^{-m}.
```

p^{-r/2}Use in source · 3 passages

Search aliases: p^-r/2

Link to this entry ·

Use in source · spectral.tex, line 334
Define normalized level states |r⟩=p^-r/2∑_|v|=r|v⟩, r=0,…,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributions instead give p p^-r/2γ_r-1/sqrt p =γ_r-1p^-(r-1)/2.
Exact mathematical passage
```
 |r\rangle=p^{-r/2}\sum_{|v|=r}|v\rangle,
 \qquad r=0,\ldots,h.
```

p^{-s(M+1)}Use in source · 2 passages

Search aliases: p^-s(M+1)

Link to this entry ·

Use in source · limit.tex, line 211
[Exact tail and regularity bound] For f∈HH_p and every M≥0, norm(I- E_M)f_2^2 =∑_m>Mnorm F_mf_2^2. If f∈ H_p^s, then norm(I- E_M)f_2 ≤ p^-s(M+1)normf_ H_p^s. The convergence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted summands by this lower bound proves the squared estimate, and taking square roots gives [source reference]. At every finite M there is a unit vector …
Exact mathematical passage
```
 \norm{(I-\mathsf E_M)f}_2
 \leq p^{-s(M+1)}\norm{f}_{\mathfrak H_p^s}.
 \label{a-limit-tail-bound}
```

p^{-s/2}Use in source · 1 passage

Search aliases: p^-s/2

Link to this entry ·

Use in source · spectral.tex, line 376
There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-c…
Exact mathematical passage
```
 |u,c;s\rangle
 =p^{-s/2}\sum_{d=0}^{p-1}c_d
       \sum_{w\in\{0,\ldots,p-1\}^{s}}|u d w\rangle.
 \label{a-spectral-angular-chain}
```

p^{-w}Use in source · 11 passages

Search aliases: p^-w

Link to this entry ·

Use in source · finite.tex, line 730
After w refinements a packet has n=p^w orthonormal descendants. Its normalized radial vector is |R_p,w⟩=n^-1/2∑_ d| d⟩. The norm is n^-1∑_ d1=1, so each descendant has amplitude modulus p^-w/2 and Born probability p^-w. This reproduces [source reference]. The words ``radial'' and ``angular'' here refer to the uniform vector and its orthogonal complement in a finite descendant space. They do not introduce a spacetime radius or a rotation group.
Exact mathematical passage
```
p^{-w}
```

p^{-w/2}Use in source · 5 passages

Search aliases: p^-w/2

Link to this entry ·

Use in source · finite.tex, line 411
D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 V_{p,h+w-1}\cdots V_{p,h}|a\rangle
 =p^{-w/2}\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}|a,\mathbf d\rangle,
 \label{a-finite-nested-refinement}
```

p^{-\alpha w}Use in source · 1 passage

Search aliases: p^-α w alpha

Link to this entry ·

Use in source · finite.tex, line 818
Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf interaction: its two radial eigenvalues are ±|G_p,w|, so normH_int=|G_p,w|. They do not determine the boundedness of a different infinite operator with man…
Exact mathematical passage
```
\gamma_{p,w}=g p^{-\alpha w}
```

p^{2s(M+1)}Use in source · 1 passage

Search aliases: p^2s(M+1)

Link to this entry ·

Use in source · limit.tex, line 221
a-limit-tail-identity If f∈ H_p^s, then norm(I- E_M)f_2 ≤ p^-s(M+1)normf_ H_p^s. The convergence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted summands by this lower bound proves the squared estimate, and taking square roots gives [source reference]. At every finite M there is a unit vector in Ran F_M+1, on which I- E_M acts as the identity. Thus normI- E_M=1 for eve…
Exact mathematical passage
```
p^{2sm}\geq p^{2s(M+1)}
```

p^{2sm}Use in source · 2 passages

Search aliases: p^2sm

Link to this entry ·

Use in source · limit.tex, line 193
For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.
Exact mathematical passage
```
 \norm{f}_{\mathfrak H_p^s}^2
 =\sum_{m\geq0}p^{2sm}\norm{\mathsf F_mf}_2^2,
 \qquad
 \mathfrak H_p^s=\{f\in\HH_p:\norm{f}_{\mathfrak H_p^s}<\infty\}.
```

p^AUse in source · 1 passage

Search aliases: p^A

Link to this entry ·

Use in source · core.tex, line 394
For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0 0 · 1/2, p^A=p^B=(1/2,1/2). The product density matrix (id_A/2)⊗(id_B/2) has the same marginals but every joint entry is 1/4. Marginals therefore fail to reconstruct even the joint measurement distribution. Moreover, the classical mixture of e_0⊗ e_0 and e_1⊗ e_1 has the same joint table as…
Exact mathematical passage
```
 (p_{a\mu})=\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix},
 \qquad p^A=p^B=(1/2,1/2).
```

p^BUse in source · 1 passage

Search aliases: p^B

Link to this entry ·

Use in source · core.tex, line 394
For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0 0 · 1/2, p^A=p^B=(1/2,1/2). The product density matrix (id_A/2)⊗(id_B/2) has the same marginals but every joint entry is 1/4. Marginals therefore fail to reconstruct even the joint measurement distribution. Moreover, the classical mixture of e_0⊗ e_0 and e_1⊗ e_1 has the same joint table as…
Exact mathematical passage
```
 (p_{a\mu})=\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix},
 \qquad p^A=p^B=(1/2,1/2).
```

p^bUse in source · 3 passages

Search aliases: p^b

Link to this entry ·

Use in source · spectral.tex, line 409
its fixed ancestral coefficient, on all children below the lower vertex at a common level. The lower chain has zero sum over its own child contrasts, so their inner product is zero. The same zero-sum calculation makes every angular vector orthogonal to the uniform radial level vector. At a fixed depth b, the radial vector contributes one dimension. The angular vectors present there contribute ∑_r=1^b(p-1)p^r-1=p^b-1 dimensions. They are orthogonal and therefore form a basis of that p^b-dimension…
Exact mathematical passage
```
\sum_{r=1}^b(p-1)p^{r-1}=p^b-1
```

p^hUse in source · 15 passages

Search aliases: p^h

Link to this entry ·

Use in source · finite.tex, line 32
[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ, (s,a) · ⟼(-1)^s5^apmod2^h+2, · (s,a) · ∈Z/2Z×Z/2^hZ are group isomorphisms onto U_p,h and U_2,h respectively.
Exact mathematical passage
```
 (t,a)&\longmapsto\zeta_{p,h}^{\,t}(1+p)^a
       \pmod{p^{h+1}},
 & (t,a)&\in\Z/(p-1)\Z\times\Z/p^h\Z,
 \label{a-finite-odd-coordinate}\\
 (s,a)&\longmapsto(-1)^s5^a\pmod{2^{h+2}},
 & (s,a)&\in\Z/2\Z\times\Z/2^h\Z
 \label{a-finite-dyadic-coordinate}
```

p^{h+1}Use in source · 4 passages

Search aliases: p^h+1

Link to this entry ·

Use in source · finite.tex, line 15
The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into one phase coordinate.
Exact mathematical passage
```
 U_{p,h}=(\Z/p^{h+1}\Z)^\times.
```

p^{h-1}Use in source · 4 passages

Search aliases: p^h-1

Link to this entry ·

Use in source · finite.tex, line 423
For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber
Exact mathematical passage
```
m=p^{h-1}
```

p^{h-2}Use in source · 1 passage

Search aliases: p^h-2

Link to this entry ·

Use in source · finite.tex, line 447
-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_0+⋯+d_h-2p^h-2. This proves the identity on an orthonormal basis.
Exact mathematical passage
```
a=d_0+\cdots+d_{h-2}p^{h-2}
```

p^{h-k-1}Use in source · 1 passage

Search aliases: p^h-k-1

Link to this entry ·

Use in source · finite.tex, line 663
x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born probability, is constant on each such valuation shell. At a fixed initial address y, the shell of valuation k<h contains (p-1)p^h-k-1 final addresses. Its total probability includes this multiplicity. The diagonal has one address. Uniform probability per address on a shell does not mean equal total pr…
Exact mathematical passage
```
(p-1)p^{h-k-1}
```

p^{h-r}Use in source · 1 passage

Search aliases: p^h-r

Link to this entry ·

Use in source · finite.tex, line 250
The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least significant base-p digits.
Exact mathematical passage
```
p^{h-r}
```

p^{h-r+1}Use in source · 1 passage

Search aliases: p^h-r+1

Link to this entry ·

Use in source · finite.tex, line 340
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
p^{h-r+1}
```

p^{h_p}Use in source · 3 passages

Search aliases: p^h_p

Link to this entry ·

Use in source · finite.tex, line 126
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 \mathcal U_{\mathrm{CRT}}:
 \ell^2\bigl((\Z/M\Z)^\times\bigr)
 \longrightarrow
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^{h_2}\Z)
 \otimes\bigotimes_{p\in\mathcal P_{\mathrm o}}
 \left(\C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^{h_p}\Z)\right).
 \label{a-finite-crt}
```

p^{h_p+1}Use in source · 1 passage

Search aliases: p^h_p+1

Link to this entry ·

Use in source · finite.tex, line 120
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 M=2^{h_2+2}\prod_{p\in\mathcal P_{\mathrm o}}p^{h_p+1}.
```

p^kUse in source · 1 passage

Search aliases: p^k

Link to this entry ·

Use in source · finite.tex, line 46
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
 v_p\bigl((1+p)^{p^k}-1\bigr)=k+1
 \qquad(k\geq0)
```

p^{k+1}Use in source · 1 passage

Search aliases: p^k+1

Link to this entry ·

Use in source · finite.tex, line 51
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
u=1+c p^{k+1}
```

p^{k+2}Use in source · 1 passage

Search aliases: p^k+2

Link to this entry ·

Use in source · finite.tex, line 53
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
c p^{k+2}
```

p^{m-1}Use in source · 5 passages

Search aliases: p^m-1

Link to this entry ·

Use in source · limit.tex, line 170
Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.
Exact mathematical passage
```
d_m=(p-1)p^{m-1}
```

p^{m/2}Use in source · 5 passages

Search aliases: p^m/2

Link to this entry ·

Use in source · limit.tex, line 25
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
 \qquad \mathcal V_m=\operatorname{span}
 \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
 \label{a-limit-cylinder-embedding}
```

p^nUse in source · 2 passages

Search aliases: p^n

Link to this entry ·

Use in source · finite.tex, line 60
For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the required order. The cyclicity of this residue group follows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finit…
Exact mathematical passage
```
p^n
```

p^{n+1}Use in source · 1 passage

Search aliases: p^n+1

Link to this entry ·

Use in source · finite.tex, line 61
For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the required order. The cyclicity of this residue group follows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finit…
Exact mathematical passage
```
p^{n+1}
```

p^rUse in source · 8 passages

Search aliases: p^r

Link to this entry ·

Use in source · finite.tex, line 249
The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least significant base-p digits.
Exact mathematical passage
```
p^r
```

p^{\rm c}_\betaUse in source · 1 passage

Search aliases: p^rm c_β rm beta

Link to this entry ·

Use in source · extensions.tex, line 66
If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.
Exact mathematical passage
```
 p^{\rm c}_\beta=\sum_{r(a)=\beta}p_a,\qquad
 J^{\rm c}_{\beta\alpha}
 =\sum_{\substack{r(b)=\beta\\r(a)=\alpha}}J^F_{ba}.
```

p^{w/2}Use in source · 3 passages

Search aliases: p^w/2

Link to this entry ·

Use in source · finite.tex, line 825
Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf interaction: its two radial eigenvalues are ±|G_p,w|, so normH_int=|G_p,w|. They do not determine the boundedness of a different infinite operator with man…
Exact mathematical passage
```
p^{w/2}
```

p^{\alpha m}Use in source · 1 passage

Search aliases: p^α m alpha

Link to this entry ·

Use in source · limit.tex, line 521
-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It is trace class exactly when this series converges. For ϵ_m=c m with c>0, the condition is β c>log p, by the geometric-series test. For ϵ_m=p^α m with α>0, it converges for every β>0, since the exponential energy term eventually dominates the linear-in-m logarithm of the multiplicity. A thermodynamic partition function therefore needs an energy choic…
Exact mathematical passage
```
\epsilon_m=p^{\alpha m}
```

\mathcal P_{\mathrm o}Use in source · 5 passages

Search aliases: P_ o mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 119
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique
Exact mathematical passage
```
\mathcal P_{\mathrm o}
```

p_*Use in source · 2 passages

Search aliases: p_*

Link to this entry ·

Use in source · strong_fields.tex, line 232
A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞, normδ P_Λ_∞ · ≤frac Mp_*normδ R_∞, normδ P_R_∞ · ≤frac Mp_*normD_4δ R_∞ +fracM^2d^*p_*^3normδ R_∞. For example, ∂_Rp=M/p and the two partial derivatives of MΛ/p have magnitudes at most M/p_* and M^2d^*/p_*^3. The admissible segment preserves th…
Exact mathematical passage
```
 \norm{\delta\Lambda}_\infty&\le\norm{D_4\delta R}_\infty,\\
 \norm{\delta P_\Lambda}_\infty&\le\frac M{p_*}\norm{\delta R}_\infty,\\
 \norm{\delta P_R}_\infty&\le\frac M{p_*}\norm{D_4\delta R}_\infty
                          +\frac{M^2d^*}{p_*^3}\norm{\delta R}_\infty.
\label{c:strong:projection-bound}
```

p_*^2Use in source · 1 passage

Search aliases: p_*^2

Link to this entry ·

Use in source · strong_fields.tex, line 227
A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,
Exact mathematical passage
```
 R_j\ge R_*>0,\quad 2MR_j-Q^2\ge p_*^2>0,
 \quad 0<d_*\le D_4R_j\le d^*.
```

p_*^3Use in source · 2 passages

Search aliases: p_*^3

Link to this entry ·

Use in source · strong_fields.tex, line 232
A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞, normδ P_Λ_∞ · ≤frac Mp_*normδ R_∞, normδ P_R_∞ · ≤frac Mp_*normD_4δ R_∞ +fracM^2d^*p_*^3normδ R_∞. For example, ∂_Rp=M/p and the two partial derivatives of MΛ/p have magnitudes at most M/p_* and M^2d^*/p_*^3. The admissible segment preserves th…
Exact mathematical passage
```
 \norm{\delta\Lambda}_\infty&\le\norm{D_4\delta R}_\infty,\\
 \norm{\delta P_\Lambda}_\infty&\le\frac M{p_*}\norm{\delta R}_\infty,\\
 \norm{\delta P_R}_\infty&\le\frac M{p_*}\norm{D_4\delta R}_\infty
                          +\frac{M^2d^*}{p_*^3}\norm{\delta R}_\infty.
\label{c:strong:projection-bound}
```

p_0Use in source · 1 passage

Search aliases: p_0

Link to this entry ·

Use in source · constraints.tex, line 731
-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpoints on the unit sphere can have bar q=0 if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an a…
Exact mathematical passage
```
\bar p=(p_0+p_1)/2
```

p_1Use in source · 5 passages

Search aliases: p_1

Link to this entry ·

Use in source · core.tex, line 168
0. Direct differentiation verifies the solution ψ(t)=cos(gt) -isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instantaneous flow must be kept separate.
Exact mathematical passage
```
 \psi(t)=\begin{pmatrix}\cos(gt)\\-i\sin(gt)\end{pmatrix},
 \quad p_1=\cos^2(gt),\quad p_2=\sin^2(gt).
```

p_2Use in source · 4 passages

Search aliases: p_2

Link to this entry ·

Use in source · core.tex, line 168
0. Direct differentiation verifies the solution ψ(t)=cos(gt) -isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instantaneous flow must be kept separate.
Exact mathematical passage
```
 \psi(t)=\begin{pmatrix}\cos(gt)\\-i\sin(gt)\end{pmatrix},
 \quad p_1=\cos^2(gt),\quad p_2=\sin^2(gt).
```

P_AUse in source · 7 passages

Search aliases: P_A

Link to this entry ·

Use in source · orientation.tex, line 58
Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.
Exact mathematical passage
```
P_A=\sum_{a\in C(A)}P_a
```

p_AUse in source · 1 passage

Search aliases: p_A

Link to this entry ·

Use in source · core.tex, line 47
[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights …
Exact mathematical passage
```
p_A=2/7
```

p_A^{\mathrm c}Use in source · 3 passages

Search aliases: p_A^ c mathrm

Link to this entry ·

Use in source · core.tex, line 197
[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
 p_A^{\mathrm c}&=\norm{Q_A\psi}^2=\sum_{a\in C_A}p_a,\\
 J_{B\leftarrow A}^{\mathrm c}
 &=2\operatorname{Re}\inner{Q_B\psi}{KQ_A\psi}
 =\sum_{b\in C_B}\sum_{a\in C_A}J_{b\leftarrow a}.
```

P_a^{(0)}Use in source · 4 passages

Search aliases: P_a^(0)

Link to this entry ·

Use in source · core.tex, line 248
A resolution can change with time even when the state is stationary. To retain orthogonality, transport the entire atlas by a unitary family. Let U(t) be differentiable in operator norm and satisfy both U^*U=id and UU^*=id. Set P_a(t)=U(t)P_a^(0)U(t)^*, ϕ(t)=U(t)^*ψ(t), Γ(t)=U(t)^*dot U(t). The coordinates ϕ see fixed projectors P_a^(0). The physical weight in the moving packet is the same scalar: normP_a(t)ψ(t)^2 =normP_a^(0)ϕ(t)^2.
Exact mathematical passage
```
\label{f:moving-defs}
 P_a(t)=U(t)P_a^{(0)}U(t)^*,\qquad
 \phi(t)=U(t)^*\psi(t),\qquad \Gamma(t)=U(t)^*\dot U(t).
```

P_A^+Use in source · 3 passages

Search aliases: P_A^+

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Use in source · core.tex, line 118
The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness follows from A=A^*. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coord…
Exact mathematical passage
```
P_A^+P_A^-=0
```

P_A^-Use in source · 3 passages

Search aliases: P_A^-

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Use in source · core.tex, line 118
The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness follows from A=A^*. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coord…
Exact mathematical passage
```
P_A^+P_A^-=0
```

p_a^AUse in source · 3 passages

Search aliases: p_a^A

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Use in source · core.tex, line 379
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
\label{f:joint-marginal}
 p_{a\mu}=\operatorname{Tr}(R_{a\mu}\rho_{AB}),\qquad
 p_a^A=\sum_\mu p_{a\mu},\quad p_\mu^B=\sum_a p_{a\mu}.
```

p_a^qUse in source · 1 passage

Search aliases: p_a^q

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Use in source · core.tex, line 480
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
M_q=\sum_ap_a^q
```

p_a^{q-1}Use in source · 1 passage

Search aliases: p_a^q-1

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Use in source · core.tex, line 481
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
 \dot M_q=\frac q2\sum_{b,a}J_{b\leftarrow a}
 (p_b^{q-1}-p_a^{q-1}).
```

p_{a\mu}Use in source · 3 passages

Search aliases: p_aμ mu

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Use in source · core.tex, line 379
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
\label{f:joint-marginal}
 p_{a\mu}=\operatorname{Tr}(R_{a\mu}\rho_{AB}),\qquad
 p_a^A=\sum_\mu p_{a\mu},\quad p_\mu^B=\sum_a p_{a\mu}.
```

P_BUse in source · 3 passages

Search aliases: P_B

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Use in source · orientation.tex, line 58
Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.
Exact mathematical passage
```
P_B=\sum_{b\in C(B)}P_b
```

p_BUse in source · 1 passage

Search aliases: p_B

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Use in source · core.tex, line 47
[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights …
Exact mathematical passage
```
p_B=5/7
```

p_bUse in source · 7 passages

Search aliases: p_b

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Use in source · core.tex, line 129
fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.
Exact mathematical passage
```
\label{f:boundary-current}
 \frac{d}{dt}\sum_{b\in C}p_b
 =\sum_{b\in C}\sum_{a\notin C}J_{b\leftarrow a}.
```

p_B^{\mathrm c}Use in source · 1 passage

Search aliases: p_B^ c mathrm

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Use in source · core.tex, line 459
The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equality of outgoing rates to each destination cell, the fine population distribution inside a source cell matters. This direct consequence of the gain sum …
Exact mathematical passage
```
\dot p_B^{\mathrm c}=\sum_{A\ne B}(q_{BA}p_A^{\mathrm c}-q_{AB}p_B^{\mathrm c})
```

P_b^2Use in source · 1 passage

Search aliases: P_b^2

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Use in source · core.tex, line 78
Insert ∑_bP_b=id to the left of X and ∑_aP_a=id to its right. This gives the first formula. Evaluate the right resolution on ψ for the second. For the product, P_cXYP_a=P_cX(∑_bP_b)YP_a =∑_b(P_cXP_b)(P_bYP_a), where P_b^2=P_b supplies the repeated middle projector. Finally, (P_bXP_a)^*=P_aX^*P_b.
Exact mathematical passage
```
P_b^2=P_b
```

p_b^{q-1}Use in source · 1 passage

Search aliases: p_b^q-1

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Use in source · core.tex, line 481
For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers and logarithmic derivatives valid, and M_q>0 when the finite index set is nonempty. Boundary points with z…
Exact mathematical passage
```
 \dot M_q=\frac q2\sum_{b,a}J_{b\leftarrow a}
 (p_b^{q-1}-p_a^{q-1}).
```

P_cUse in source · 3 passages

Search aliases: P_c

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Use in source · orientation.tex, line 53
For two operators L and M, insert the resolution between them: P_cMLP_a=∑_b∈ A(P_cMP_b)(P_bLP_a). This is block multiplication written without a choice of basis inside each packet. The intermediate index records the interface through which a composition passes. It is also the reason that a change of resolution must specify how fine packets assemble into coarse packets.
Exact mathematical passage
```
P_cMLP_a=\sum_{b\in A}(P_cMP_b)(P_bLP_a).
```

\mathcal P_{\mathrm{in}}Use in source · 2 passages

Search aliases: P_in mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

P_jUse in source · 1 passage

Search aliases: P_j

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Use in source · core.tex, line 484
dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.
Exact mathematical passage
```
 C_h(z)=2h\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].
 \label{c:core:gowdy-momentum}
```

p_j^2Use in source · 2 passages

Search aliases: p_j^2

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Use in source · finite_core.tex, line 281
Proof. Adjoint is additive, and a real multiple of a Hermitian matrix is Hermitian. PH01 therefore proves the first identity. Expanding the square gives the square terms m^2β^2+∑_jp_j^2α_j^2, the terms mp_j(βα_j+α_jβ), and the terms p_jp_k(α_jα_k+α_kα_j) for j<k. The anticommutators vanish and all four squares are I_4, giving (m^2+∑_jp_j^2)I_4. Substitution proves both mode statements. square
Exact mathematical passage
```
m^2\beta^2+\sum_jp_j^2\alpha_j^2
```

p_kUse in source · 1 passage

Search aliases: p_k

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Use in source · finite_core.tex, line 283
Proof. Adjoint is additive, and a real multiple of a Hermitian matrix is Hermitian. PH01 therefore proves the first identity. Expanding the square gives the square terms m^2β^2+∑_jp_j^2α_j^2, the terms mp_j(βα_j+α_jβ), and the terms p_jp_k(α_jα_k+α_kα_j) for j<k. The anticommutators vanish and all four squares are I_4, giving (m^2+∑_jp_j^2)I_4. Substitution proves both mode statements. square
Exact mathematical passage
```
p_jp_k(\alpha_j\alpha_k+\alpha_k\alpha_j)
```

P_LUse in source · 2 passages

Search aliases: P_L

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Use in source · spectral.tex, line 204
If P_L=ι_Lι_L^*, the full space compression is P_L(z-H)^-1P_L=ι_L S(z)^-1ι_L^*. When a Haar transform W first exposes low and detail channels, the embedding in the original coordinates is ι_L= W^*(|0⟩⊗ I). The inverse on HH_L must be embedded in this way before it is compared with a full space matrix.
Exact mathematical passage
```
P_L=\iota_L\iota_L^*
```

p_{L,0}Use in source · 1 passage

Search aliases: p_L,0

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Use in source · spectral.tex, line 1216
· =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series proves ∑_jp_L,j=1. Their equivalent hypergeometric form is e^-μ times [source reference] at z=μ.
Exact mathematical passage
```
p_{L,0}(0)=1
```

p_{L,j}Use in source · 4 passages

Search aliases: p_L,j

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Use in source · spectral.tex, line 1207
For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series proves ∑_jp_L,j=1. Their equivalent hypergeometric form is e^-μ times [source reference] at z=μ.
Exact mathematical passage
```
 p_{L,j}(\mu)
 &=\langle\alpha,Q_j^{(L)}\alpha\rangle
   =e^{-\mu}E_{L,j}(\mu)\nonumber\\
 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
                      e^{\mu(\omega_L^\ell-1)}.
 \label{a-spectral-coherent-packets}
```

P_nUse in source · 2 passages

Search aliases: P_n

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Use in source · refinement.tex, line 493
The transported and concrete closed levels are then proved equal. Their union is dense, and the orthogonal projections P_n satisfy normP_nf-f_L^2⟶0 for every f∈ L^2([0,1]^3;C^4). This is a state-space approximation theorem. Strong convergence of projections does not control an unbounded Dirac operator, establish graph convergence, or identify the limit of gauge-coupled propagators. Those statements need consistency on suitable domains and stability estimates for the transported operators.
Exact mathematical passage
```
P_n
```

\mathcal P_{\mathrm{out}}Use in source · 1 passage

Search aliases: P_out mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 164
The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n…
Exact mathematical passage
```
 \mathcal P_{\mathrm{in}}|n\rangle
 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
 \mathcal P_{\mathrm{out}}|k\rangle
 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
```

\mathcal P_{\mathrm{out}}^*Use in source · 1 passage

Search aliases: P_out^* mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 171
P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b, hence modulo ab. This kernel equals N^-1/2exp(2πikn/N), proving [source reference]. Omitting the dual-output permut…
Exact mathematical passage
```
 F_N=\mathcal P_{\mathrm{out}}^*(F_a\otimes F_b)
       \mathcal P_{\mathrm{in}}.
 \label{a-finite-additive-fourier}
```

P_pUse in source · 6 passages

Search aliases: P_p

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Use in source · spectral.tex, line 102
M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1, e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
 P_p(\theta)e_r=
 \begin{cases}
 e_{r+1},&r<p-1,\\
 e^{-i\theta}e_0,&r=p-1.
 \end{cases}
 \qquad
 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
 \label{a-spectral-bloch-symbol}
```

P_p^*Use in source · 1 passage

Search aliases: P_p^*

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Use in source · spectral.tex, line 133
y, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine Fourier index is uniquely k=j+qM. Its angle satisfies 2π k/(pM)=(θ_j+2π q)/p. Substitution into the fine cyclic eigenvalue μ+2κcos(2π k/N) proves the m…
Exact mathematical passage
```
\mu I+\kappa(P_p+P_p^*)
```

P_qUse in source · 1 passage

Search aliases: P_q

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Use in source · refinement.tex, line 280
A two-stage edge and path estimate. The cube phase discrepancies can be made into bounded operators on the same finite Hilbert state space. For z∈ U(1), let P_zψ=zψ uniformly at every site. If p=Ω q, then P_p-P_q=q(Ω-1)id, normP_p-P_q=|Ω-1|. The norm equality uses a nonzero Hilbert space. It is the operator form of [source reference].
Exact mathematical passage
```
 P_p-P_q=q(\Omega-1)\id,\qquad
 \norm{P_p-P_q}=|\Omega-1|.
```

P_RUse in source · 20 passages

Search aliases: P_R

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Use in source · spectral.tex, line 1142
A sufficient common-domain condition is ∫_R e^2|t||λ| d⟨ψ,E(λ)ψ⟩<∞, since |f_j(λ)|≤ e^|t||λ|. Spectral cutoffs P_R=E([-R,R]) supply a dense set satisfying this condition. On Ran P_R, the restricted Hamiltonian is bounded, the norm-series identities apply, and ∑_jf_j(H_∞)P_Rψ=e^-itH_∞P_Rψ ⟶ e^-itH_∞ψ. On the common domain of the individual lanes, each term also converges to f_j(H_∞)ψ by its spectral-domain integrability. For an arbitrary vector only convergence of the summed cutoff expression is …
Exact mathematical passage
```
P_R=E([-R,R])
```

p_RUse in source · 1 passage

Search aliases: p_R

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Use in source · strong_fields.tex, line 126
For evolution, the radius equation gives dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2],
Exact mathematical passage
```
p_R=M/p
```

P_{R,j}Use in source · 1 passage

Search aliases: P_R,j

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Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

P_SUse in source · 9 passages

Search aliases: P_S

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Use in source · continuum.tex, line 11
For a finite set S⊂Z^3, define the truncation P_Sa by retaining a_n when n∈ S and assigning zero otherwise. Then norma-P_Sa^2=∑_n∉ Snorma_n^2. The sum on the right can be made arbitrarily small because the full nonnegative series converges. Thus finite packets are dense in KK. The statement concerns the norm of the omitted amplitudes and does not depend on a particular ordering of the integer lattice.
Exact mathematical passage
```
P_Sa
```

p_xUse in source · 2 passages

Search aliases: p_x

Link to this entry ·

Use in source · finite_core.tex, line 200
H(m,p)=mβ+p_xα_x+p_yα_y+p_zα_z.
Exact mathematical passage
```
H(m,p)=m\beta+p_x\alpha_x+p_y\alpha_y+p_z\alpha_z.
```

p_{xy}Use in source · 2 passages

Search aliases: p_xy

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Use in source · refinement.tex, line 100
For example, the two positive paths across an xy face at height z are p_xy(z)=ℓ_y(1,z)ℓ_x(0,z), p_yx(z)=ℓ_x(1,z)ℓ_y(0,z). Define Ω_xy(z)=p_xy(z)p_yx(z)^-1, and define the xz and yz face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each Ω is gauge invariant, and Ω=1 precisely when the two path phases agree.
Exact mathematical passage
```
 p_{xy}(z)=\ell_y(1,z)\ell_x(0,z),\qquad
 p_{yx}(z)=\ell_x(1,z)\ell_y(0,z).
```

p_{XYZ}Use in source · 5 passages

Search aliases: p_XYZ

Link to this entry ·

Use in source · refinement.tex, line 107
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,
Exact mathematical passage
```
p_{XYZ}
```

p_{XZY}Use in source · 3 passages

Search aliases: p_XZY

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Use in source · refinement.tex, line 107
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,
Exact mathematical passage
```
p_{XZY}
```

p_yUse in source · 2 passages

Search aliases: p_y

Link to this entry ·

Use in source · finite_core.tex, line 200
H(m,p)=mβ+p_xα_x+p_yα_y+p_zα_z.
Exact mathematical passage
```
H(m,p)=m\beta+p_x\alpha_x+p_y\alpha_y+p_z\alpha_z.
```

p_{yx}Use in source · 2 passages

Search aliases: p_yx

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Use in source · refinement.tex, line 100
For example, the two positive paths across an xy face at height z are p_xy(z)=ℓ_y(1,z)ℓ_x(0,z), p_yx(z)=ℓ_x(1,z)ℓ_y(0,z). Define Ω_xy(z)=p_xy(z)p_yx(z)^-1, and define the xz and yz face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each Ω is gauge invariant, and Ω=1 precisely when the two path phases agree.
Exact mathematical passage
```
 p_{xy}(z)=\ell_y(1,z)\ell_x(0,z),\qquad
 p_{yx}(z)=\ell_x(1,z)\ell_y(0,z).
```

p_{YXZ}Use in source · 4 passages

Search aliases: p_YXZ

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Use in source · refinement.tex, line 107
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,
Exact mathematical passage
```
p_{YXZ}
```

p_{YZX}Use in source · 3 passages

Search aliases: p_YZX

Link to this entry ·

Use in source · refinement.tex, line 107
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,
Exact mathematical passage
```
p_{YZX}
```

P_zUse in source · 1 passage

Search aliases: P_z

Link to this entry ·

Use in source · refinement.tex, line 279
A two-stage edge and path estimate. The cube phase discrepancies can be made into bounded operators on the same finite Hilbert state space. For z∈ U(1), let P_zψ=zψ uniformly at every site. If p=Ω q, then P_p-P_q=q(Ω-1)id, normP_p-P_q=|Ω-1|. The norm equality uses a nonzero Hilbert space. It is the operator form of [source reference].
Exact mathematical passage
```
P_z\psi=z\psi
```

p_zUse in source · 2 passages

Search aliases: p_z

Link to this entry ·

Use in source · finite_core.tex, line 200
H(m,p)=mβ+p_xα_x+p_yα_y+p_zα_z.
Exact mathematical passage
```
H(m,p)=m\beta+p_x\alpha_x+p_y\alpha_y+p_z\alpha_z.
```

p_{ZXY}Use in source · 3 passages

Search aliases: p_ZXY

Link to this entry ·

Use in source · refinement.tex, line 107
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,
Exact mathematical passage
```
p_{ZXY}
```

p_{ZYX}Use in source · 4 passages

Search aliases: p_ZYX

Link to this entry ·

Use in source · refinement.tex, line 107
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,
Exact mathematical passage
```
p_{ZYX}
```

P_\pm^*Use in source · 1 passage

Search aliases: P_±^* pm

Link to this entry ·

Use in source · finite_core.tex, line 316
Then P_±^*=P_±, P_±^2=P_±, P_+P_-=P_-P_+=0, P_++P_-=I_4, and HP_±=± EP_±. Both projectors have rank two, and
Exact mathematical passage
```
P_\pm^*=P_\pm
```

P_\pm^2Use in source · 2 passages

Search aliases: P_±^2 pm

Link to this entry ·

Use in source · finite_core.tex, line 316
Then P_±^*=P_±, P_±^2=P_±, P_+P_-=P_-P_+=0, P_++P_-=I_4, and HP_±=± EP_±. Both projectors have rank two, and
Exact mathematical passage
```
P_\pm^2=P_\pm
```

P_\thetaUse in source · 7 passages

Search aliases: P_θ theta

Link to this entry ·

Use in source · constraints.tex, line 279
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
\mathcal D=\Pi_PP_\theta+\Pi_QQ_\theta
```

P_\theta^2Use in source · 3 passages

Search aliases: P_θ^2 theta

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

P_{\theta\theta}Use in source · 3 passages

Search aliases: P_θθ theta theta

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

P_\Lambda'Use in source · 7 passages

Search aliases: P_Λ' Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 20
The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),
Exact mathematical passage
```
 \mathcal D=P_RR'-\Lambda P_\Lambda',\qquad \mathcal G=Q',
```

P_{\Lambda,j}Use in source · 1 passage

Search aliases: P_Λ,j Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

P_{\Lambda,j}^2Use in source · 1 passage

Search aliases: P_Λ,j^2 Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

P_\Lambda^2Use in source · 5 passages

Search aliases: P_Λ^2 Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 24
The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),
Exact mathematical passage
```
\begin{aligned}
 \mathcal H={}&-\frac{P_RP_\Lambda}{R}
 +\frac{\Lambda P_\Lambda^2}{2R^2}
 +\frac{RR''}{\Lambda}-\frac{RR'\Lambda'}{\Lambda^2}
 +\frac{R'^2}{2\Lambda}-\frac\Lambda2
 +\frac{\Lambda Q^2}{2R^2}.
\end{aligned}
\label{c:strong:electrovac-H}
```

p_\mu^BUse in source · 2 passages

Search aliases: p_μ^B mu

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Use in source · core.tex, line 379
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
\label{f:joint-marginal}
 p_{a\mu}=\operatorname{Tr}(R_{a\mu}\rho_{AB}),\qquad
 p_a^A=\sum_\mu p_{a\mu},\quad p_\mu^B=\sum_a p_{a\mu}.
```

P_\tauUse in source · 5 passages

Search aliases: P_τ tau

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

P_\tau^2Use in source · 2 passages

Search aliases: P_τ^2 tau

Link to this entry ·

Use in source · core.tex, line 343
Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber
Exact mathematical passage
```
 S_G=\frac12\int_I\int_0^{2\pi}
 \left[P_\tau^2-\gamma P_\theta^2
 +e^{2P}(Q_\tau^2-\gamma Q_\theta^2)\right]d\theta\,d\tau.
 \label{c:core:gowdy-action}
```

P_{\tau\theta}Use in source · 1 passage

Search aliases: P_τθ tau theta

Link to this entry ·

Use in source · core.tex, line 389
Differentiate the two densities. The terms with P_τ P_τθ and Q_τ Q_τθ cancel, leaving ∂_τ M_G-∂_θ H_G = · 2P_θ(P_ττ-γ P_θθ)
Exact mathematical passage
```
P_\tau
P_{\tau\theta}
```

P_{\tau\tau}Use in source · 2 passages

Search aliases: P_ττ tau tau

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

P_\varphiUse in source · 4 passages

Search aliases: P_φ varphi

Link to this entry ·

Use in source · strong_fields.tex, line 343
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
(\varphi,P_\varphi)
```

P_\varphi^2Use in source · 1 passage

Search aliases: P_φ^2 varphi

Link to this entry ·

Use in source · strong_fields.tex, line 344
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
 \mathcal H_{\rm matt}=\frac{P_\varphi^2}{2\Lambda R^2}
                         +\frac{R^2\varphi'^2}{2\Lambda},
 \qquad \mathcal D_{\rm matt}=P_\varphi\varphi'.
```

\mathrm{par}Use in source · 2 passages

Search aliases: par mathrm

Link to this entry ·

Use in source · constraints.tex, line 350
The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.
Exact mathematical passage
```
 [\delta_U,\delta_V]_{\mathrm{fld}}
       =-\delta_{\left[U,V\right]_{\mathrm{par}}},
```

\mathrm{phase}Use in source · 3 passages

Search aliases: phase mathrm

Link to this entry ·

Use in source · finite.tex, line 192
The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel that discards a subsystem. For example it sends the normalized state (|0,a⟩+|1,a⟩)/sqrt2 to sqrt2|a⟩. Its adjoint product therefore does not equal th…
Exact mathematical passage
```
 C_{\mathrm{phase}}|s,a\rangle=|a\rangle
 \label{a-finite-forgetting}
```

\mathrm{phys}Use in source · 5 passages

Search aliases: phys mathrm

Link to this entry ·

Use in source · boundary.tex, line 24
Connecting the group to its generator. The next chapter proves that the constructed group is the spectral evolution of this Hamiltonian: U(t)=e^-itH_m^phys. The proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal generator has exactly that domain. The equality is therefore a conclusion about the complete Hilbert-space group, not an inference from agreement on finitel…
Exact mathematical passage
```
\mathcal U(t)=e^{-itH_m^{\mathrm{phys}}}.
```

\proptoUse in source · 1 passage

Search aliases: propto

Link to this entry ·

Use in source · constraints.tex, line 798
x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gaug…
Exact mathematical passage
```
\rho_C\propto D^{-1/2}
```

\mathbb QUse in source · 1 passage

Search aliases: Q mathbb

Link to this entry ·

Use in source · primary_context.tex, line 23
item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is proved separately.
Exact mathematical passage
```
\mathbb Q_p
```

Q'Use in source · 4 passages

Search aliases: Q'

Link to this entry ·

Use in source · strong_fields.tex, line 20
The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),
Exact mathematical passage
```
 \mathcal D=P_RR'-\Lambda P_\Lambda',\qquad \mathcal G=Q',
```

q'Use in source · 3 passages

Search aliases: q'

Link to this entry ·

Use in source · finite.tex, line 342
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
q,q'
```

q^2Use in source · 1 passage

Search aliases: q^2

Link to this entry ·

Use in source · core.tex, line 184
u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluat…
Exact mathematical passage
```
\label{g:wilson-expansion}
 1-\operatorname{Re}W
 =\tfrac12q^2a_\mu^2a_\nu^2 F_{\mu\nu}(x_c)^2+O(a^6).
```

Q^4Use in source · 1 passage

Search aliases: Q^4

Link to this entry ·

Use in source · strong_fields.tex, line 166
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
 \mathcal K=\frac{48M^2}{R^6}-\frac{96MQ^2}{R^7}
                            +\frac{56Q^4}{R^8}.
```

q^{ab}Use in source · 1 passage

Search aliases: q^ab

Link to this entry ·

Use in source · constraints.tex, line 528
nequation widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket …
Exact mathematical passage
```
 S^a(\nu,\mu)=\det(q_{cd})q^{ab}
                          (\nu\partial_b\mu-\mu\partial_b\nu).
```

q^mUse in source · 1 passage

Search aliases: q^m

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Use in source · limit.tex, line 433
[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For q=1/8, the state norm tail decays faster than the graph norm tail. For q=1/2, the normalized state still has a convergent L^2 cylinder approximation a…
Exact mathematical passage
```
 f_q=\sum_{m\geq0}\sqrt{(1-q)q^m}\,u_m,
 \qquad \epsilon_m=2^m.
```

q^{M+1}Use in source · 2 passages

Search aliases: q^M+1

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Use in source · limit.tex, line 438
[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For q=1/8, the state norm tail decays faster than the graph norm tail. For q=1/2, the normalized state still has a convergent L^2 cylinder approximation a…
Exact mathematical passage
```
 \norm{(I-\mathsf E_M)f_q}_2^2=q^{M+1}.
 \label{a-limit-geometric-state-tail}
```

q_0Use in source · 1 passage

Search aliases: q_0

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Use in source · constraints.tex, line 731
-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpoints on the unit sphere can have bar q=0 if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an a…
Exact mathematical passage
```
\bar q=(q_0+q_1)/2
```

q_1Use in source · 1 passage

Search aliases: q_1

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Use in source · constraints.tex, line 731
-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpoints on the unit sphere can have bar q=0 if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an a…
Exact mathematical passage
```
\bar q=(q_0+q_1)/2
```

Q_AUse in source · 3 passages

Search aliases: Q_A

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Use in source · core.tex, line 190
Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a different physical system.
Exact mathematical passage
```
\label{f:coarse-projector}
 Q_A=\sum_{a\in C_A}P_a.
```

q_aUse in source · 5 passages

Search aliases: q_a

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Use in source · refinement.tex, line 472
Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2 =∑_a∫_C_av^-1normψ_a^2 =∑_anormψ_a^2. Equivalently, physical cell values carry the weighted norm v∑_anormq_a^2. The adjoint averaging map yields the …
Exact mathematical passage
```
q_a\in\C^4
```

Q_A^nUse in source · 1 passage

Search aliases: Q_A^n

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Use in source · core.tex, line 96
i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n
```

Q_A^{n+1}Use in source · 1 passage

Search aliases: Q_A^n+1

Link to this entry ·

Use in source · core.tex, line 96
i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n
```

q_{AB}Use in source · 1 passage

Search aliases: q_AB

Link to this entry ·

Use in source · core.tex, line 459
The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equality of outgoing rates to each destination cell, the fine population distribution inside a source cell matters. This direct consequence of the gain sum …
Exact mathematical passage
```
\dot p_B^{\mathrm c}=\sum_{A\ne B}(q_{BA}p_A^{\mathrm c}-q_{AB}p_B^{\mathrm c})
```

q_{ab}Use in source · 4 passages

Search aliases: q_ab

Link to this entry ·

Use in source · constraints.tex, line 301
Dbarψ · = dbarψ-tfrac14barψω^IJγ_IJ+iq abarψ. The parameter q here is electric charge. The determinant of a spatial metric below will be written det(q_ab) to avoid confusing the two uses.
Exact mathematical passage
```
\det(q_{ab})
```

Q_BUse in source · 1 passage

Search aliases: Q_B

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Use in source · core.tex, line 197
[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
 p_A^{\mathrm c}&=\norm{Q_A\psi}^2=\sum_{a\in C_A}p_a,\\
 J_{B\leftarrow A}^{\mathrm c}
 &=2\operatorname{Re}\inner{Q_B\psi}{KQ_A\psi}
 =\sum_{b\in C_B}\sum_{a\in C_A}J_{b\leftarrow a}.
```

q_{BA}Use in source · 3 passages

Search aliases: q_BA

Link to this entry ·

Use in source · core.tex, line 454
The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equality of outgoing rates to each destination cell, the fine population distribution inside a source cell matters. This direct consequence of the gain sum …
Exact mathematical passage
```
 \sum_{b\in C_B}\gamma_{ba}=q_{BA}
 \quad\text{for every }a\in C_A.
```

Q_cUse in source · 4 passages

Search aliases: Q_c

Link to this entry ·

Use in source · refinement.tex, line 292
The formal construction selects five ordered phase pairs, in this order: (p_XYZ,p_YXZ), (p_YXZ,p_YZX), (p_YZX,p_ZYX), (p_XYZ,p_XZY), (p_XZY,p_ZXY). Let Q_f be the execution-order product of the first phase operators in these pairs, and Q_c the product of the second ones. Identity factors can be placed at the ends to fit the seven-factor interface. Applying [source reference] yields normQ_f-Q_c≤κ(ℓ). These are explicitly constructed products of uniform phase operators. Calling them a path-reorder…
Exact mathematical passage
```
Q_c
```

q_{cd}Use in source · 1 passage

Search aliases: q_cd

Link to this entry ·

Use in source · constraints.tex, line 528
nequation widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket …
Exact mathematical passage
```
 S^a(\nu,\mu)=\det(q_{cd})q^{ab}
                          (\nu\partial_b\mu-\mu\partial_b\nu).
```

Q_fUse in source · 5 passages

Search aliases: Q_f

Link to this entry ·

Use in source · refinement.tex, line 292
The formal construction selects five ordered phase pairs, in this order: (p_XYZ,p_YXZ), (p_YXZ,p_YZX), (p_YZX,p_ZYX), (p_XYZ,p_XZY), (p_XZY,p_ZXY). Let Q_f be the execution-order product of the first phase operators in these pairs, and Q_c the product of the second ones. Identity factors can be placed at the ends to fit the seven-factor interface. Applying [source reference] yields normQ_f-Q_c≤κ(ℓ). These are explicitly constructed products of uniform phase operators. Calling them a path-reorder…
Exact mathematical passage
```
Q_f
```

q_hUse in source · 7 passages

Search aliases: q_h

Link to this entry ·

Use in source · finite_core.tex, line 235
Write q_h(n)=q(h,p(n)) and H_h,n=H(m,q_h(n)). The function q_j is even in h, and its value at zero is justified by sin z/z→1. For h≠0, a centered translation difference on a periodic smooth function is
Exact mathematical passage
```
q_h(n)=q(h,p(n))
```

Q_jUse in source · 3 passages

Search aliases: Q_j

Link to this entry ·

Use in source · spectral.tex, line 1196
unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
Q_jQ_k=\delta_{jk}Q_j
```

q_jUse in source · 6 passages

Search aliases: q_j

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Use in source · finite_core.tex, line 227
q_j(h,p)= sin(hp_j)/h, · h≠0, p_j, · h=0.
Exact mathematical passage
```
q_j(h,p)=
\begin{cases}
\sin(hp_j)/h,&h\ne0,\\
p_j,&h=0.
\end{cases}
```

Q_j^{(L)}Use in source · 3 passages

Search aliases: Q_j^(L)

Link to this entry ·

Use in source · spectral.tex, line 1189
On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
 Q_j^{(L)}=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}V^\ell
          =\sum_{n\equiv j\;({\rm mod}\;L)}|n\rangle\langle n|.
 \label{a-spectral-number-projectors}
```

q_j^*Use in source · 2 passages

Search aliases: q_j^*

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Use in source · core.tex, line 266
The first formula follows from dot E_j=J_j. For the matter contribution, let q_j=∂ H_ m/∂overlineψ_j=ihdotψ_j. The state-dependent part of the real derivative is 2Re∑_jq_j^*dotψ_j=0, because q_j^*dotψ_j=-ihnormdotψ_j^2 is purely imaginary. The remaining explicit link derivative is ∑_jfrac∂ H_ m∂ A_jdot A_j =-h∑_jJ_jE_j. Adding the two sectors proves total conservation.
Exact mathematical passage
```
2\operatorname{Re}\sum_jq_j^*\dot\psi_j=0
```

Q_kUse in source · 1 passage

Search aliases: Q_k

Link to this entry ·

Use in source · spectral.tex, line 1196
unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
Q_jQ_k=\delta_{jk}Q_j
```

\mathbb Q_pUse in source · 1 passage

Search aliases: Q_p mathbb

Link to this entry ·

Use in source · primary_context.tex, line 23
item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is proved separately.
Exact mathematical passage
```
\mathbb Q_p
```

Q_r^{(L)}Use in source · 1 passage

Search aliases: Q_r^(L)

Link to this entry ·

Use in source · spectral.tex, line 1246
Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number norm. Truncation of a vector in either domain converges in the corresponding graph norm, so the identities extend to those domains. For the isolated …
Exact mathematical passage
```
 Q_r^{(L)}aQ_s^{(L)}=0\quad\text{unless }r\equiv s-1\pmod L,
 \qquad
 Q_r^{(L)}a^\dagger Q_s^{(L)}=0\quad\text{unless }r\equiv s+1\pmod L.
 \label{a-spectral-ladder-routing}
```

Q_s^{(L)}Use in source · 1 passage

Search aliases: Q_s^(L)

Link to this entry ·

Use in source · spectral.tex, line 1246
Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number norm. Truncation of a vector in either domain converges in the corresponding graph norm, so the identities extend to those domains. For the isolated …
Exact mathematical passage
```
 Q_r^{(L)}aQ_s^{(L)}=0\quad\text{unless }r\equiv s-1\pmod L,
 \qquad
 Q_r^{(L)}a^\dagger Q_s^{(L)}=0\quad\text{unless }r\equiv s+1\pmod L.
 \label{a-spectral-ladder-routing}
```

q_x^aUse in source · 1 passage

Search aliases: q_x^a

Link to this entry ·

Use in source · core.tex, line 346
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
q_x^a=\psi_x^*\sigma^a\psi_x
```

q_x^bUse in source · 1 passage

Search aliases: q_x^b

Link to this entry ·

Use in source · core.tex, line 347
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
\label{g:adjoint-color}
 g_x^*\sigma^a g_x=\sum_b R_x{}^a{}_b\sigma^b,
 \qquad q_x^{g,a}=\sum_bR_x{}^a{}_bq_x^b.
```

q_x^{g,a}Use in source · 1 passage

Search aliases: q_x^g,a

Link to this entry ·

Use in source · core.tex, line 347
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
\label{g:adjoint-color}
 g_x^*\sigma^a g_x=\sum_b R_x{}^a{}_b\sigma^b,
 \qquad q_x^{g,a}=\sum_bR_x{}^a{}_bq_x^b.
```

Q_\alphaUse in source · 8 passages

Search aliases: Q_α alpha

Link to this entry ·

Use in source · extensions.tex, line 216
For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.
Exact mathematical passage
```
Q_\alpha=\sum_{a\in C_\alpha}P_a
```

Q_\betaUse in source · 3 passages

Search aliases: Q_β beta

Link to this entry ·

Use in source · extensions.tex, line 223
Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately: J^rm c_βα · =2IminnerQ_βψHQ_αψ · =∑_b∈ C_β(∑_a∈ C_αJ_ba) =∑_a∈ C_α(∑_b∈ C_βJ_ba). For example, first summing a moves H onto P_bψ, then the outer sum pairs ∑_bP_bψ with the fixed vector HQ_αψ. The reverse order has its own symmetric argument. Flattening these expressions to a single unordered sum on C_β× C_α requires joint summability. Equality of two justified iterated sums does not supp…
Exact mathematical passage
```
 J^{\rm c}_{\beta\alpha}
 &=2\operatorname{Im}\inner{Q_\beta\psi}{HQ_\alpha\psi}\\
 &=\sum_{b\in C_\beta}\left(\sum_{a\in C_\alpha}J_{ba}\right)
 =\sum_{a\in C_\alpha}\left(\sum_{b\in C_\beta}J_{ba}\right).
```

Q_\thetaUse in source · 6 passages

Search aliases: Q_θ theta

Link to this entry ·

Use in source · constraints.tex, line 279
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
\mathcal D=\Pi_PP_\theta+\Pi_QQ_\theta
```

Q_\theta^2Use in source · 6 passages

Search aliases: Q_θ^2 theta

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

Q_{\theta\theta}Use in source · 2 passages

Search aliases: Q_θθ theta theta

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

Q_\muUse in source · 2 passages

Search aliases: Q_μ mu

Link to this entry ·

Use in source · core.tex, line 372
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
(Q_\mu)
```

Q_\tauUse in source · 5 passages

Search aliases: Q_τ tau

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Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

Q_\tau^2Use in source · 4 passages

Search aliases: Q_τ^2 tau

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Use in source · core.tex, line 343
Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber
Exact mathematical passage
```
 S_G=\frac12\int_I\int_0^{2\pi}
 \left[P_\tau^2-\gamma P_\theta^2
 +e^{2P}(Q_\tau^2-\gamma Q_\theta^2)\right]d\theta\,d\tau.
 \label{c:core:gowdy-action}
```

Q_{\tau\theta}Use in source · 1 passage

Search aliases: Q_τθ tau theta

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Use in source · core.tex, line 390
Differentiate the two densities. The terms with P_τ P_τθ and Q_τ Q_τθ cancel, leaving ∂_τ M_G-∂_θ H_G = · 2P_θ(P_ττ-γ P_θθ)
Exact mathematical passage
```
Q_\tau Q_{\tau\theta}
```

Q_{\tau\tau}Use in source · 2 passages

Search aliases: Q_ττ tau tau

Link to this entry ·

Use in source · core.tex, line 352
∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
 \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
 -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
 \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
 +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
 \label{c:core:gowdy-euler}
```

\RUse in source · 16 passages

Search aliases: R

Link to this entry ·

Use in source · limit.tex, line 303
For bounded Borel F:R→C, define F(H_ϵ)f=∑_m≥0F(ϵ_m) F_mf. The series converges by Parseval and normF(H_ϵ)f_2≤sup_m|F(ϵ_m)|norm f_2. The same level decomposition verifies sums, products, and adjoints of these multipliers directly. In particular, U(t) · =e^-itH_ϵ =∑_m≥0e^-itϵ_m F_m,
Exact mathematical passage
```
F:\R\to\C
```

\mathbb RUse in source · 7 passages

Search aliases: R mathbb

Link to this entry ·

Use in source · finite_core.tex, line 7
T^3=( R/ Z)^3,
Exact mathematical passage
```
\mathbb T^3=(\mathbb R/\mathbb Z)^3,
```

\mathbf rUse in source · 12 passages

Search aliases: r mathbf

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Use in source · limit.tex, line 575
Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.
Exact mathematical passage
```
 \mathsf F_{\mathbf r}
 =I_{\KK_{\mathrm{tor}}}\otimes
 \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
 \qquad \mathbf r\in\N^{\mathcal P}.
 \label{a-limit-joint-details}
```

\mathcal RUse in source · 8 passages

Search aliases: R mathcal

Link to this entry ·

Use in source · finite.tex, line 436
A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores th…
Exact mathematical passage
```
\mathcal R_{p,h}
```

\mathfrak RUse in source · 3 passages

Search aliases: R mathfrak

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Use in source · spectral.tex, line 902
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The id…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)
 =\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}F(\omega_L^\ell z),
 \quad 0\leq j<L.
 \label{a-spectral-analytic-filter}
```

R''Use in source · 3 passages

Search aliases: R''

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Use in source · strong_fields.tex, line 24
The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),
Exact mathematical passage
```
\begin{aligned}
 \mathcal H={}&-\frac{P_RP_\Lambda}{R}
 +\frac{\Lambda P_\Lambda^2}{2R^2}
 +\frac{RR''}{\Lambda}-\frac{RR'\Lambda'}{\Lambda^2}
 +\frac{R'^2}{2\Lambda}-\frac\Lambda2
 +\frac{\Lambda Q^2}{2R^2}.
\end{aligned}
\label{c:strong:electrovac-H}
```

R'^2Use in source · 2 passages

Search aliases: R'^2

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Use in source · strong_fields.tex, line 24
The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),
Exact mathematical passage
```
\begin{aligned}
 \mathcal H={}&-\frac{P_RP_\Lambda}{R}
 +\frac{\Lambda P_\Lambda^2}{2R^2}
 +\frac{RR''}{\Lambda}-\frac{RR'\Lambda'}{\Lambda^2}
 +\frac{R'^2}{2\Lambda}-\frac\Lambda2
 +\frac{\Lambda Q^2}{2R^2}.
\end{aligned}
\label{c:strong:electrovac-H}
```

R'^3Use in source · 1 passage

Search aliases: R'^3

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Use in source · strong_fields.tex, line 67
-fracP_ΛRΛ D +frac QR G. Consequently, on a patch where R'/Λ≠0, the systems ( H, D, G)=0 and ( M', D, G)=0 have the same zero set. Differentiating [source reference] gives M'= · fracR'2-fracR'^32Λ^2 -fracRR'R''Λ^2+fracRR'^2Λ'Λ^3 +fracP_Λ P_Λ'R-fracP_Λ^2R'2R^2 · +fracQQ'R-fracQ^2R'2R^2. Expanding the right side of [source reference] cancels the two P_R terms and produces precisely this expression. The constraint transformation is upper triangular with diagonal (-R'/Λ,1,1), so it is invertible und…
Exact mathematical passage
```
\begin{aligned}
 \mathcal M'={}&\frac{R'}2-\frac{R'^3}{2\Lambda^2}
 -\frac{RR'R''}{\Lambda^2}+\frac{RR'^2\Lambda'}{\Lambda^3}
 +\frac{P_\Lambda P_\Lambda'}R-\frac{P_\Lambda^2R'}{2R^2}\\
 &+\frac{QQ'}R-\frac{Q^2R'}{2R^2}.
\end{aligned}
```

R^{\mathsf T}Use in source · 2 passages

Search aliases: R^ T mathsf

Link to this entry ·

Use in source · core.tex, line 212
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
\label{f:incidence}
 p^{\mathrm c}=Rp,\qquad J^{\mathrm c}=RJR^{\mathsf T},\qquad
 R^{\mathsf T}\boldsymbol1_{\mathrm c}=\boldsymbol1_{\mathrm f}.
```

R^*Use in source · 1 passage

Search aliases: R^*

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Use in source · extensions.tex, line 80
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
 V^*V=I,\qquad R=VV^*,\qquad R^*=R^2=R.
```

r^{-2}Use in source · 1 passage

Search aliases: r^-2

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Use in source · strong_fields.tex, line 388
E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.
Exact mathematical passage
```
 \dot\varphi&=\sigma(\Pi+v\Phi),\\
 \dot\Phi&=\partial_r[\sigma(\Pi+v\Phi)],\\
 \dot\Pi&=r^{-2}\partial_r[r^2\sigma(\Phi+v\Pi)].
\label{c:strong:scalar-evolution}
```

r^2Use in source · 19 passages

Search aliases: r^2

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Use in source · core.tex, line 230
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
\lim_{r\downarrow0}r^2\Phi'(r)=0
```

\mathbb R^{2n}Use in source · 1 passage

Search aliases: R^2n mathbb

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Use in source · constraints.tex, line 675
Endpoint Dirac projection. Consider a finite-dimensional real canonical chart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theo…
Exact mathematical passage
```
z\in\mathbb R^{2n}
```

R^3Use in source · 1 passage

Search aliases: R^3

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Use in source · strong_fields.tex, line 127
ginequation dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2], dot P_R · =fracΛ(P_Λ^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_…
Exact mathematical passage
```
 \dot P_\Lambda&=N^rP_\Lambda'
 +\frac12\left[1-\frac{P_\Lambda^2+Q^2}{R^2}
                 -\left(\frac{R'}\Lambda\right)^2\right],\\
 \dot P_R&=\frac{\Lambda(P_\Lambda^2+Q^2)}{R^3}
 -\frac{P_\Lambda P_R}{R^2}+(N^rP_R)'
 -\frac{R''}\Lambda+\frac{\Lambda'R'}{\Lambda^2}.
```

\mathbb R^3Use in source · 2 passages

Search aliases: R^3 mathbb

Link to this entry ·

Use in source · finite_core.tex, line 198
For p∈ R^3, define
Exact mathematical passage
```
p\in\mathbb R^3
```

\mathbb R^4Use in source · 1 passage

Search aliases: R^4 mathbb

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Use in source · finite_core.tex, line 591
The reverse triangle inequality in R^4, applied to (m,q_h(n)) and (m,p(n)), gives
Exact mathematical passage
```
\mathbb R^4
```

R^6Use in source · 1 passage

Search aliases: R^6

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Use in source · strong_fields.tex, line 166
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
 \mathcal K=\frac{48M^2}{R^6}-\frac{96MQ^2}{R^7}
                            +\frac{56Q^4}{R^8}.
```

\mathbb R^6Use in source · 1 passage

Search aliases: R^6 mathbb

Link to this entry ·

Use in source · constraints.tex, line 722
A worked particle constraint. In R^6 with z=(q,p) and canonical J, set Φ_1=q· q-1, Φ_2=q· p. At one point the ordinary Dirac matrix is Δ(z)=0 · 2 q ^2
Exact mathematical passage
```
\mathbb R^6
```

R^7Use in source · 1 passage

Search aliases: R^7

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Use in source · strong_fields.tex, line 166
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
 \mathcal K=\frac{48M^2}{R^6}-\frac{96MQ^2}{R^7}
                            +\frac{56Q^4}{R^8}.
```

R^8Use in source · 1 passage

Search aliases: R^8

Link to this entry ·

Use in source · strong_fields.tex, line 166
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
 \mathcal K=\frac{48M^2}{R^6}-\frac{96MQ^2}{R^7}
                            +\frac{56Q^4}{R^8}.
```

R^{IJ}Use in source · 1 passage

Search aliases: R^IJ

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

R^{k-1}Use in source · 3 passages

Search aliases: R^k-1

Link to this entry ·

Use in source · finite_core.tex, line 678
X^k-Y^k ≤ kR^k-1 X-Y .
Exact mathematical passage
```
\|X^k-Y^k\|\le kR^{k-1}\|X-Y\|.
```

R^{KL}Use in source · 1 passage

Search aliases: R^KL

Link to this entry ·

Use in source · constraints.tex, line 304
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e, S_ M^(1) · =∫(tfrac12φ_IJFwedge e^Iwedge e^J -tfrac14φ_IJφ^IJvol_e), S_ D · =frac i2∫Σ_Iwedge (barψγ^I Dψ -( Dbarψ)γ^Iψ) -m∫vol_ebarψψ. This formulation uses no inverse tetrad in its defining action. Its largest monomials have six field factors. Nondegeneracy becomes necessary when the auxiliary variables are eliminated or a metric interpretation is imposed.
Exact mathematical passage
```
 S_{\mathrm G}
 &=\frac1{4\kappa}\int\epsilon_{IJKL}e^I\wedge e^J\wedge R^{KL}
                  -\frac\Lambda\kappa\int\operatorname{vol}_e,\\
 S_{\mathrm M}^{(1)}
 &=\int\left(\tfrac12\varphi_{IJ}F\wedge e^I\wedge e^J
                    -\tfrac14\varphi_{IJ}\varphi^{IJ}\operatorname{vol}_e\right),\\
 S_{\mathrm D}
 &=\frac i2\int\Sigma_I\wedge
       \left(\bar\psi\gamma^I\mathcal D\psi
               -(\mathcal D\bar\psi)\gamma^I\psi\right)
       -m\int\operatorname{vol}_e\bar\psi\psi.
 \label{c:constraints:parent-action}
```

\mathcal R^{KL}Use in source · 1 passage

Search aliases: R^KL mathcal

Link to this entry ·

Use in source · constraints.tex, line 452
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ,
Exact mathematical passage
```
 \mathcal C_I^{\mathrm G}
 &=\tfrac1{2\kappa}\epsilon_{IJKL}E^J\wedge\mathcal R^{KL}
                  -\tfrac\Lambda\kappa\sigma_I,\\
 \mathcal C_I^{\mathrm M}
 &=\varphi_{IJ}\mathcal F\wedge E^J-\tfrac14\varphi_{KL}\varphi^{KL}\sigma_I,\\
 \mathcal C_I^{\mathrm D}
 &=\tfrac i4\epsilon_{JIKL}E^K\wedge E^L\wedge
       \bigl(\bar\psi\gamma^J\mathcal D_\Sigma\psi
                  -(\mathcal D_\Sigma\bar\psi)\gamma^J\psi\bigr)
                      -m\sigma_I\bar\psi\psi.
```

R_*Use in source · 1 passage

Search aliases: R_*

Link to this entry ·

Use in source · strong_fields.tex, line 227
A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,
Exact mathematical passage
```
 R_j\ge R_*>0,\quad 2MR_j-Q^2\ge p_*^2>0,
 \quad 0<d_*\le D_4R_j\le d^*.
```

r_+Use in source · 4 passages

Search aliases: r_+

Link to this entry ·

Use in source · strong_fields.tex, line 170
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
r_+=1.6
```

r_{+,h}Use in source · 1 passage

Search aliases: r_+,h

Link to this entry ·

Use in source · strong_fields.tex, line 336
The outer horizon remains at the fixed areal radius 1.6. Its computational coordinate moves because F changes. The retained final coordinate comparison at N=256 gives |r_+,h-r_+^rm exact|=1.89262 10^-11. This measures tracking of a known horizon under coordinate deformation. It does not measure the formation or growth of a black hole.
Exact mathematical passage
```
|r_{+,h}-r_+^{\rm exact}|=1.89262\,10^{-11}
```

r_+^2Use in source · 2 passages

Search aliases: r_+^2

Link to this entry ·

Use in source · strong_fields.tex, line 176
Horizon conditioning. For a subextremal outer horizon, put d=sqrtM^2-Q^2>0. Then f'(r_+)=frac2dr_+^2. If a nearby continuous numerical profile has a selected root near this simple root, the local inverse estimate gives |δ r_+|≤fracr_+^22d normδ f_∞ +O(normδ f_∞^2). The estimate describes the nearby root and presupposes its localization to a region where the exact derivative is nonzero. It does not make every zero of an arbitrarily oscillatory perturbation physically relevant. The factor is 2.133…
Exact mathematical passage
```
 f'(r_+)=\frac{2d}{r_+^2}.
```

r_+^{\rm exact}Use in source · 1 passage

Search aliases: r_+^rm exact rm

Link to this entry ·

Use in source · strong_fields.tex, line 336
The outer horizon remains at the fixed areal radius 1.6. Its computational coordinate moves because F changes. The retained final coordinate comparison at N=256 gives |r_+,h-r_+^rm exact|=1.89262 10^-11. This measures tracking of a known horizon under coordinate deformation. It does not measure the formation or growth of a black hole.
Exact mathematical passage
```
|r_{+,h}-r_+^{\rm exact}|=1.89262\,10^{-11}
```

r_-Use in source · 1 passage

Search aliases: r_-

Link to this entry ·

Use in source · strong_fields.tex, line 170
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
r_-=0.4
```

R_0Use in source · 4 passages

Search aliases: R_0

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

R_1Use in source · 2 passages

Search aliases: R_1

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

R_2Use in source · 2 passages

Search aliases: R_2

Link to this entry ·

Use in source · spectral.tex, line 610
The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation retains every other site. It does not retain the average of each neighboring pair.
Exact mathematical passage
```
R_2(x)=x^2-2
```

R_3Use in source · 1 passage

Search aliases: R_3

Link to this entry ·

Use in source · spectral.tex, line 883
adicfigure11_prime_renormalizationThe normalized maps R_2, R_3, and R_5 preserve the same arcsine spectral measure. The accompanying bounded moment checks illustrate the exact measure identity, while each Schur realization retains its own excluded poles.
Exact mathematical passage
```
R_3
```

R_5Use in source · 1 passage

Search aliases: R_5

Link to this entry ·

Use in source · spectral.tex, line 883
adicfigure11_prime_renormalizationThe normalized maps R_2, R_3, and R_5 preserve the same arcsine spectral measure. The accompanying bounded moment checks illustrate the exact measure identity, while each Schur realization retains its own excluded poles.
Exact mathematical passage
```
R_5
```

R_aUse in source · 4 passages

Search aliases: R_a

Link to this entry ·

Use in source · refinement.tex, line 31
Choose unit links a(x) and b(x) on the first and second half-edge with b(x)a(x)=u(x). The associated fine link is v(e_x)=a(x), v(m_x)=b(x). The interpolation and reconstruction maps are (I_aψ)(e_x) · =ψ(x), · (I_aψ)(m_x) · =a(x)ψ(x), (R_aΨ)(x) · =tfrac12(Ψ(e_x)+a(x)^-1Ψ(m_x)). The midpoint is a transported value, not an independently sampled field value. This identifies exactly the subspace on which the refinement theorem acts.
Exact mathematical passage
```
 (I_a\psi)(e_x)&=\psi(x),&
 (I_a\psi)(m_x)&=a(x)\psi(x),\label{g:edge-interpolation}\\
 (R_a\Psi)(x)&=\tfrac12\bigl(\Psi(e_x)+a(x)^{-1}\Psi(m_x)\bigr).
 \label{g:edge-restriction}
```

R_{Aa}Use in source · 1 passage

Search aliases: R_Aa

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Use in source · core.tex, line 211
For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
R_{Aa}=1
```

R_{a\mu}Use in source · 3 passages

Search aliases: R_aμ mu

Link to this entry ·

Use in source · core.tex, line 373
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
 R_{a\mu}=P_a\otimes Q_\mu.
```

R_{b\nu}Use in source · 1 passage

Search aliases: R_bν nu

Link to this entry ·

Use in source · core.tex, line 377
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
R_{a\mu}R_{b\nu}=\delta_{ab}\delta_{\mu\nu}R_{a\mu}
```

R_CUse in source · 3 passages

Search aliases: R_C

Link to this entry ·

Use in source · constraints.tex, line 794
-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonli…
Exact mathematical passage
```
 S_C=\frac{\mu x_2^2}{2d},\qquad
 R_C=\partial_{x_1}+\frac{x_1x_2}{d}\partial_{x_2}.
```

R_dUse in source · 4 passages

Search aliases: R_d

Link to this entry ·

Use in source · finite.tex, line 453
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
 R_d=\frac1p\sum_{\ell=0}^{p-1}\omega_p^{-d\ell}M_{p,h}^{\ell}.
 \label{a-finite-root-selector}
```

R_d^*Use in source · 1 passage

Search aliases: R_d^*

Link to this entry ·

Use in source · finite.tex, line 459
dulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
R_d^*=R_d
```

R_eUse in source · 1 passage

Search aliases: R_e

Link to this entry ·

Use in source · finite.tex, line 459
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
R_dR_e=\delta_{de}R_d
```

R_FUse in source · 1 passage

Search aliases: R_F

Link to this entry ·

Use in source · constraints.tex, line 787
A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m
Exact mathematical passage
```
R_F=\partial_{x_1}+y\partial_{x_2}
```

r_HUse in source · 4 passages

Search aliases: r_H

Link to this entry ·

Use in source · strong_fields.tex, line 516
Recorded first marginally trapped sphere. The event finder takes a quadratic interpolation of the spatial maximum of 2m/r and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is small rrrrr N_r · h · t_H · r_H · Δ M/M_0
Exact mathematical passage
```
r_H
```

r_iUse in source · 9 passages

Search aliases: r_i

Link to this entry ·

Use in source · core.tex, line 437
In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable …
Exact mathematical passage
```
r_i=\sum_{b\ne i}\gamma_{bi}
```

r_i^2Use in source · 1 passage

Search aliases: r_i^2

Link to this entry ·

Use in source · strong_fields.tex, line 436
On a fixed radial interval [r_i,r_o], the exact balance is fracddt(m(t,r_o)-m(t,r_i)) =r_o^2 J(t,r_o)-r_i^2 J(t,r_i). Boundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid and time step.
Exact mathematical passage
```
 \frac{d}{dt}\bigl(m(t,r_o)-m(t,r_i)\bigr)
 =r_o^2\mathcal J(t,r_o)-r_i^2\mathcal J(t,r_i).
```

R_{i\eta}Use in source · 1 passage

Search aliases: R_iη eta

Link to this entry ·

Use in source · extensions.tex, line 407
projection preserving D(H), and [H,P] on D(H) extends to a bounded operator B. For nonreal z, let R_z=(H-z)^-1. It maps HH into D(H), so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives [R_z,P]=-R_zBR_z. Indeed, for x∈HH, apply H-z to PR_zx to obtain Px+BR_zx, then apply R_z. Hence norm[R_z,P]≤normR_z^2norm B, norm[R_iη,P]≤fracnorm Bη^2 (η≠0). The estimate converts a domain-sensitive commutator into a bounded one. In the…
Exact mathematical passage
```
 \norm{[R_z,P]}\leq\norm{R_z}^2\norm B,
 \qquad
 \norm{[R_{i\eta},P]}\leq\frac{\norm B}{\eta^2}
 \quad(\eta\ne0).
```

r_jUse in source · 3 passages

Search aliases: r_j

Link to this entry ·

Use in source · finite_core.tex, line 937
u_q(t)-u_r(t) _2^2 =(c_q-c_r)^2+(s_q-s_r)^2 +∑_j=x,y,z(q_js_q-r_js_r)^2.
Exact mathematical passage
```
\|u_q(t)-u_r(t)\|_2^2
=(c_q-c_r)^2+(s_q-s_r)^2
+\sum_{j=x,y,z}(q_js_q-r_js_r)^2.
```

\mathfrak R_{L,j}Use in source · 3 passages

Search aliases: R_L,j mathfrak

Link to this entry ·

Use in source · spectral.tex, line 902
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The id…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)
 =\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}F(\omega_L^\ell z),
 \quad 0\leq j<L.
 \label{a-spectral-analytic-filter}
```

\mathfrak R_{L,k}Use in source · 1 passage

Search aliases: R_L,k mathfrak

Link to this entry ·

Use in source · spectral.tex, line 909
, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follo…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
 \quad
 \mathfrak R_{L,j}\mathfrak R_{L,k}
   =\delta_{jk}\mathfrak R_{L,j},\quad
 \sum_j\mathfrak R_{L,j}=I.
 \label{a-spectral-analytic-lanes}
```

R_mUse in source · 4 passages

Search aliases: R_m

Link to this entry ·

Use in source · spectral.tex, line 709
[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R_n(…
Exact mathematical passage
```
 R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\nonumber\\
 U_{mn-1}(t)&=U_{n-1}(T_m(t))\,U_{m-1}(t)
 \label{a-spectral-polynomial-semigroup}
```

r_{\max}Use in source · 3 passages

Search aliases: r_max max

Link to this entry ·

Use in source · strong_fields.tex, line 485
What is local in the algorithm. At each RK4 stage the matter channels determine the geometry through two radial solves. The mass constraint is integrated outward with m(r_rm exc)=0. The lapse derivative is integrated and the result normalized by σ(r_max)=1. These are triangular prefix operations. A matter value near the inner end can affect the reconstructed mass and lapse far along the interval in the same stage. The full stage map is therefore globally coupled.
Exact mathematical passage
```
\sigma(r_{\max})=1
```

R_{mn}Use in source · 3 passages

Search aliases: R_mn

Link to this entry ·

Use in source · spectral.tex, line 709
[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R_n(…
Exact mathematical passage
```
 R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\nonumber\\
 U_{mn-1}(t)&=U_{n-1}(T_m(t))\,U_{m-1}(t)
 \label{a-spectral-polynomial-semigroup}
```

R_nUse in source · 6 passages

Search aliases: R_n

Link to this entry ·

Use in source · spectral.tex, line 527
We now eliminate intermediate residue sites in [source reference], assuming κ≠0. The retained sites are C={0,p,…,(M-1)p} and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).
Exact mathematical passage
```
 x=\frac{z-\mu}{\kappa},\qquad
 R_n(x)=2T_n(x/2),
 \label{a-spectral-normalized-energy}
```

r_oUse in source · 2 passages

Search aliases: r_o

Link to this entry ·

Use in source · strong_fields.tex, line 435
On a fixed radial interval [r_i,r_o], the exact balance is fracddt(m(t,r_o)-m(t,r_i)) =r_o^2 J(t,r_o)-r_i^2 J(t,r_i). Boundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid and time step.
Exact mathematical passage
```
[r_i,r_o]
```

r_o^2Use in source · 1 passage

Search aliases: r_o^2

Link to this entry ·

Use in source · strong_fields.tex, line 436
On a fixed radial interval [r_i,r_o], the exact balance is fracddt(m(t,r_o)-m(t,r_i)) =r_o^2 J(t,r_o)-r_i^2 J(t,r_i). Boundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid and time step.
Exact mathematical passage
```
 \frac{d}{dt}\bigl(m(t,r_o)-m(t,r_i)\bigr)
 =r_o^2\mathcal J(t,r_o)-r_i^2\mathcal J(t,r_i).
```

r_pUse in source · 1 passage

Search aliases: r_p

Link to this entry ·

Use in source · limit.tex, line 575
Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.
Exact mathematical passage
```
 \mathsf F_{\mathbf r}
 =I_{\KK_{\mathrm{tor}}}\otimes
 \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
 \qquad \mathbf r\in\N^{\mathcal P}.
 \label{a-limit-joint-details}
```

R_p'Use in source · 2 passages

Search aliases: R_p'

Link to this entry ·

Use in source · spectral.tex, line 868
The mean logarithmic expansion of this map is also exact: ∫_-2^2log|R_p'(x)| dν(x)=log p. Indeed for x=2cosθ, R_p'(x)=psin(pθ)/sinθ. Logarithms at its finitely many zeros are integrable, since they have the local form log|θ-θ_0| plus a bounded term. Hence the integral is log p plus the difference of the integrals of log|sin(pθ)| and logsinθ over [0,π]. The substitution u=pθ and π-periodicity of log|sin u| make these two integrals equal. This calculation does not assert the entropy of a different…
Exact mathematical passage
```
 \int_{-2}^2\log|R_p'(x)|\,d\nu(x)=\log p.
 \label{a-spectral-log-expansion}
```

\mathcal R_{p,0}Use in source · 1 passage

Search aliases: R_p,0 mathcal

Link to this entry ·

Use in source · finite.tex, line 437
A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores th…
Exact mathematical passage
```
\mathcal R_{p,0}=I
```

\mathcal R_{p,h}Use in source · 2 passages

Search aliases: R_p,h mathcal

Link to this entry ·

Use in source · finite.tex, line 436
A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores th…
Exact mathematical passage
```
\mathcal R_{p,h}
```

\mathcal R_{p,h-1}Use in source · 1 passage

Search aliases: R_p,h-1 mathcal

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Use in source · finite.tex, line 438
p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_…
Exact mathematical passage
```
 \mathcal A^{\mathrm{ad}}_{p,h}=
 (I_p\otimes\mathcal R_{p,h-1})
 \mathcal A^{\mathrm{std}}_{p,h}\mathcal R_{p,h}.
 \label{a-finite-digit-reversal}
```

r_{\rm exc}Use in source · 3 passages

Search aliases: r_rm exc rm

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Use in source · strong_fields.tex, line 485
What is local in the algorithm. At each RK4 stage the matter channels determine the geometry through two radial solves. The mass constraint is integrated outward with m(r_rm exc)=0. The lapse derivative is integrated and the result normalized by σ(r_max)=1. These are triangular prefix operations. A matter value near the inner end can affect the reconstructed mass and lapse far along the interval in the same stage. The full stage map is therefore globally coupled.
Exact mathematical passage
```
m(r_{\rm exc})=0
```

R_{\rm turn}Use in source · 2 passages

Search aliases: R_rm turn rm

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Use in source · strong_fields.tex, line 160
The square-root chart instead meets a turning surface at R_rm turn=fracQ^22M. There p=0, and P_R=MF_r/p becomes unbounded for nonzero F_r. This is a limitation of the chosen flat spatial slice and canonical chart. Horizon penetration does not remove the need for a positive distance from this turning surface. It also does not remove the physical curvature singularity at R=0.
Exact mathematical passage
```
 R_{\rm turn}=\frac{Q^2}{2M}.
```

R_vUse in source · 2 passages

Search aliases: R_v

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Use in source · limit.tex, line 837
directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The on…
Exact mathematical passage
```
R_v
```

R_xUse in source · 2 passages

Search aliases: R_x

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Use in source · core.tex, line 347
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
\label{g:adjoint-color}
 g_x^*\sigma^a g_x=\sum_b R_x{}^a{}_b\sigma^b,
 \qquad q_x^{g,a}=\sum_bR_x{}^a{}_bq_x^b.
```

r_xUse in source · 3 passages

Search aliases: r_x

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Use in source · finite_core.tex, line 856
H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
Exact mathematical passage
```
H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
```

R_x^{\mathsf T}Use in source · 1 passage

Search aliases: R_x^ T mathsf

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Use in source · core.tex, line 351
pand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x is separately invariant. A color component relative to a selected Pauli axis is not general…
Exact mathematical passage
```
R_xR_x^{\mathsf T}=I
```

r_x^2Use in source · 2 passages

Search aliases: r_x^2

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Use in source · finite_core.tex, line 914
u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.
Exact mathematical passage
```
\|u_r(t)\|_2^2
=c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)
=\cos^2(E_rt)+\sin^2(E_rt)=1.
```

r_yUse in source · 3 passages

Search aliases: r_y

Link to this entry ·

Use in source · finite_core.tex, line 856
H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
Exact mathematical passage
```
H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
```

r_y^2Use in source · 2 passages

Search aliases: r_y^2

Link to this entry ·

Use in source · finite_core.tex, line 914
u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.
Exact mathematical passage
```
\|u_r(t)\|_2^2
=c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)
=\cos^2(E_rt)+\sin^2(E_rt)=1.
```

R_zUse in source · 6 passages

Search aliases: R_z

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Use in source · extensions.tex, line 402
A packet need not commute with an unbounded Hamiltonian. Suppose now that H is self-adjoint, P is a bounded orthogonal projection preserving D(H), and [H,P] on D(H) extends to a bounded operator B. For nonreal z, let R_z=(H-z)^-1. It maps HH into D(H), so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives [R_z,P]=-R_zBR_z. Indeed, for x∈HH, apply H-z to PR_zx to obtain Px+BR_zx, then apply R_z. Hence norm[R_z,P]≤normR_z^2n…
Exact mathematical passage
```
R_z=(H-z)^{-1}
```

r_zUse in source · 2 passages

Search aliases: r_z

Link to this entry ·

Use in source · finite_core.tex, line 856
H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
Exact mathematical passage
```
H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
```

r_z^2Use in source · 1 passage

Search aliases: r_z^2

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Use in source · finite_core.tex, line 914
u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.
Exact mathematical passage
```
\|u_r(t)\|_2^2
=c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)
=\cos^2(E_rt)+\sin^2(E_rt)=1.
```

r_\pmUse in source · 1 passage

Search aliases: r_± pm

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Use in source · strong_fields.tex, line 154
Charged horizons and the turning surface. Write f(R)=1-frac2MR+fracQ^2R^2. For 0<|Q|<M, its two positive roots are r_±=M±sqrtM^2-Q^2. At either root p(R)^2=R^2, so the canonical quantities in [source reference] remain finite if F_r does. For |Q|=M, the two roots coincide at R=M. For |Q|>M there is no real root. At Q=0, the outer root is 2M and the formal inner root 0 is the singular endpoint, not a second regular horizon.
Exact mathematical passage
```
 r_\pm=M\pm\sqrt{M^2-Q^2}.
```

\RanUse in source · 15 passages

Search aliases: Ran

Link to this entry ·

Use in source · finite.tex, line 300
For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.
Exact mathematical passage
```
\Ran E_{r-1}\subset\Ran E_r
```

\operatorname{Ran}Use in source · 5 passages

Search aliases: Ran operatorname

Link to this entry ·

Use in source · limit.tex, line 164
Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.
Exact mathematical passage
```
 \HH_p=\bigoplus_{m=0}^{\infty}\operatorname{Ran}\mathsf F_m,
 \qquad \norm{f}_2^2=\sum_{m=0}^{\infty}\norm{\mathsf F_mf}_2^2.
 \label{a-limit-parseval}
```

\operatorname{rank}Use in source · 2 passages

Search aliases: rank operatorname

Link to this entry ·

Use in source · finite.tex, line 199
The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel that discards a subsystem. For example it sends the normalized state (|0,a⟩+|1,a⟩)/sqrt2 to sqrt2|a⟩. Its adjoint product therefore does not equal th…
Exact mathematical passage
```
 \operatorname{rank}C_{\mathrm{phase}}=2^h,
 \qquad\dim\ker C_{\mathrm{phase}}=2^h.
```

\operatorname{Re}Use in source · 20 passages

Search aliases: Re operatorname

Link to this entry ·

Use in source · spectral.tex, line 77
um_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a probability or squared amplitude weight. Calling it an energy w…
Exact mathematical passage
```
\dot w_r=2\operatorname{Re}\langle\psi_r,-i\sum_sH_{rs}\psi_s\rangle
```

\rmUse in source · 43 passages

Search aliases: rm

Link to this entry ·

Use in source · spectral.tex, line 54
xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
 K_\mu^{\rm poly}=\mathcal A K_\mu\mathcal A^*,\qquad
 \sum_\mu(K_\mu^{\rm poly})^*K_\mu^{\rm poly}=I.
```

\mathrm{root}Use in source · 1 passage

Search aliases: root mathrm

Link to this entry ·

Use in source · spectral.tex, line 441
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
 \langle\mathrm{root},(z-H)^{-1}\mathrm{root}\rangle
   =g_0(z)=\frac1{z-\epsilon_0-\Sigma_0(z)}.
 \label{a-spectral-root-green}
```

\rVertUse in source · 1 passage

Search aliases: rVert

Link to this entry ·

Use in source · core.tex, line 50
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
\label{g:laplace-form}
 -\inner{\psi}{\Delta_U\psi}_V
 =\sum_{x,\mu}\left\lVert(D^+_{\mu,U}\psi)_x\right\rVert^2.
```

\rVert^2Use in source · 1 passage

Search aliases: rVert^2 rVert

Link to this entry ·

Use in source · core.tex, line 50
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
\label{g:laplace-form}
 -\inner{\psi}{\Delta_U\psi}_V
 =\sum_{x,\mu}\left\lVert(D^+_{\mu,U}\psi)_x\right\rVert^2.
```

\mathcal SUse in source · 18 passages

Search aliases: S mathcal

Link to this entry ·

Use in source · spectral.tex, line 170
[Finite Feshbach identity] If z∉σ(D), define Σ(z)=B(z-D)^-1B^*, S(z)=z-A-Σ(z). Then z-H is invertible exactly when S(z) is invertible. On this common regular set, ι_L^*(z-H)^-1ι_L · = S(z)^-1,nonumber
Exact mathematical passage
```
 \Sigma(z)=B(z-D)^{-1}B^*,\qquad
 \mathcal S(z)=z-A-\Sigma(z).
 \label{a-spectral-schur-definition}
```

S^{\mathsf T}Use in source · 1 passage

Search aliases: S^ T mathsf

Link to this entry ·

Use in source · core.tex, line 217
Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.
Exact mathematical passage
```
 S(Rp)=(SR)p,\qquad S(RJR^{\mathsf T})S^{\mathsf T}
 =(SR)J(SR)^{\mathsf T}.
```

S^*Use in source · 3 passages

Search aliases: S^*

Link to this entry ·

Use in source · limit.tex, line 842
malized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph edges, with no continuum limit assumed.
Exact mathematical passage
```
S^*=S
```

S^2Use in source · 1 passage

Search aliases: S^2

Link to this entry ·

Use in source · limit.tex, line 842
oing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph edges, with no continuum limit assumed.
Exact mathematical passage
```
S^2=I
```

S^aUse in source · 4 passages

Search aliases: S^a

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

S^\ellUse in source · 1 passage

Search aliases: S^ℓ ell

Link to this entry ·

Use in source · spectral.tex, line 140
The same construction works for any circulant polynomial H=∑_ℓ c_ℓ S^ℓ, with symbol ∑_ℓ c_ℓ P_p(θ)^ℓ. It preserves all bands, whether or not their energies coincide. For N=p^h, digit reversal relates this standard residue split to the newest-digit adic split developed earlier. It is still necessary to distinguish the retained residue-zero sites from the retained Haar average channel. They are different subspaces.
Exact mathematical passage
```
H=\sum_\ell c_\ell S^\ell
```

S_{\mathrm D}Use in source · 2 passages

Search aliases: S_ D mathrm

Link to this entry ·

Use in source · constraints.tex, line 303
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,
Exact mathematical passage
```
S_0=S_{\mathrm G}+S_{\mathrm M}^{(1)}+S_{\mathrm D}
```

S_{\mathrm G}Use in source · 2 passages

Search aliases: S_ G mathrm

Link to this entry ·

Use in source · constraints.tex, line 303
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,
Exact mathematical passage
```
S_0=S_{\mathrm G}+S_{\mathrm M}^{(1)}+S_{\mathrm D}
```

S_{\mathrm M}^{(1)}Use in source · 3 passages

Search aliases: S_ M^(1) mathrm

Link to this entry ·

Use in source · constraints.tex, line 303
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,
Exact mathematical passage
```
S_0=S_{\mathrm G}+S_{\mathrm M}^{(1)}+S_{\mathrm D}
```

S_0Use in source · 5 passages

Search aliases: S_0

Link to this entry ·

Use in source · constraints.tex, line 303
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,
Exact mathematical passage
```
S_0=S_{\mathrm G}+S_{\mathrm M}^{(1)}+S_{\mathrm D}
```

s_0Use in source · 9 passages

Search aliases: s_0

Link to this entry ·

Use in source · constraints.tex, line 91
A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0,
Exact mathematical passage
```
s_F=s_0+\delta
```

s_{0,C}Use in source · 6 passages

Search aliases: s_0,C

Link to this entry ·

Use in source · constraints.tex, line 91
A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0,
Exact mathematical passage
```
(\mathcal C_C,s_{0,C})
```

s_0^2Use in source · 1 passage

Search aliases: s_0^2

Link to this entry ·

Use in source · constraints.tex, line 91
A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0,
Exact mathematical passage
```
s_0^2=0
```

S_1Use in source · 2 passages

Search aliases: S_1

Link to this entry ·

Use in source · strong_fields.tex, line 268
djoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R(0)_h, normDb_∞≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonli…
Exact mathematical passage
```
 \norm{R(t)-\widetilde R(t)}_h
 \le e^{(L_g+S_1/2)t}\norm{R(0)-\widetilde R(0)}_h,
 \qquad \norm{Db}_\infty\le S_1.
\label{c:strong:semidiscrete-bound}
```

S_{A,1,-1}Use in source · 1 passage

Search aliases: S_A,1,-1

Link to this entry ·

Use in source · refinement.tex, line 204
A concrete counterexample removes any possible inference from flatness alone. Let both half-links be i, and prescribe u=1. Their product is v=-1. A cube whose twelve positive links are all i has every face plaquette equal to one. On a one-site periodic lattice, however, τ=1 and S_A,1,1=P_++P_-=1, S_A,1,-1=-P_+-P_-=-1. Thus the coarse mismatch is -2ψ, and the normalized fine defect has norm 2normψ. The same link data are perfectly flat in the path-comparison sense and maximally wrong relative to …
Exact mathematical passage
```
 S_{A,1,1}=P_++P_-=1,\qquad
 S_{A,1,-1}=-P_+-P_-=-1.
```

S_{A,1,1}Use in source · 1 passage

Search aliases: S_A,1,1

Link to this entry ·

Use in source · refinement.tex, line 204
A concrete counterexample removes any possible inference from flatness alone. Let both half-links be i, and prescribe u=1. Their product is v=-1. A cube whose twelve positive links are all i has every face plaquette equal to one. On a one-site periodic lattice, however, τ=1 and S_A,1,1=P_++P_-=1, S_A,1,-1=-P_+-P_-=-1. Thus the coarse mismatch is -2ψ, and the normalized fine defect has norm 2normψ. The same link data are perfectly flat in the path-comparison sense and maximally wrong relative to …
Exact mathematical passage
```
 S_{A,1,1}=P_++P_-=1,\qquad
 S_{A,1,-1}=-P_+-P_-=-1.
```

S_{A,\sigma,(a,b)}^2Use in source · 1 passage

Search aliases: S_A,σ,(a,b)^2 sigma

Link to this entry ·

Use in source · refinement.tex, line 195
A flat cube need not reconstruct the prescribed edge. Conversely, take an arbitrary ordered half-link pair (a,b) and call its actual product v=ba. It is always a subdivision of v, even when the separately prescribed coarse link is u≠ v. The exact directional defect is D_rm fineψ · :=S_A,σ,(a,b)^2I_aψ-I_aS_A,τ,uψnotag · =I_a(S_A,τ,v-S_A,τ,u)ψ. The equality follows by applying the exact theorem to the product v and using linearity of I_a. Endpoint restriction makes I_a injective, so the fine defec…
Exact mathematical passage
```
 D_{\rm fine}\psi
  &:=S_{A,\sigma,(a,b)}^2I_a\psi-I_aS_{A,\tau,u}\psi\notag\\
  &=I_a(S_{A,\tau,v}-S_{A,\tau,u})\psi.
 \label{g:exact-mismatch}
```

S_{A,\sigma,v}Use in source · 1 passage

Search aliases: S_A,σ,v sigma

Link to this entry ·

Use in source · refinement.tex, line 89
s step can be refined factor by factor. If a local linear map N satisfies N^2=M, then N commutes with the scalar interpolation. Consequently N^2S_A,σ,v^ 2N^2I_a =I_aMS_A,τ,uM. Unitary spectral phases provide explicit choices of N. The refined expression replaces each coarse factor by its two half factors. It does not assert that two repetitions of a complete fine N S_A,σ,vN walk equal the coarse walk: the intervening order would be different.
Exact mathematical passage
```
N S_{A,\sigma,v}N
```

S_{A,\sigma,v}^{\,2}Use in source · 2 passages

Search aliases: S_A,σ,v^ 2 sigma

Link to this entry ·

Use in source · refinement.tex, line 52
[Exact two-half-step refinement] For every coarse state ψ and subdivision satisfying [source reference], S_A,σ,v^ 2I_aψ=I_aS_A,τ,uψ. No flatness hypothesis is needed. When the fine shift is applied twice, the mixed projector terms vanish because P_+P_-=P_-P_+=0. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At e_x, the positive coefficient is b(τ^-1x)a(τ^-1x)=u(τ^-1x); the negative coefficient is a(x)^-1b(x)^-1=u(x)^-1. Th…
Exact mathematical passage
```
 S_{A,\sigma,v}^{\,2}I_a\psi=I_aS_{A,\tau,u}\psi.
 \label{g:half-intertwining}
```

S_{A,\tau,u}^{-1}Use in source · 1 passage

Search aliases: S_A,τ,u^-1 tau

Link to this entry ·

Use in source · core.tex, line 127
For a bijection τ:V→ V and a forward scalar link u:V→ U(1), define (S_A,τ,uψ)(x) =u(τ^-1x)P_A^+ψ(τ^-1x) +u(x)^-1P_A^-ψ(τ x). This agrees with the orientation dictionary [source reference]. Its inverse is (S_A,τ,u^-1ϕ)(x) =u(x)^-1P_A^+ϕ(τ x) +u(τ^-1x)P_A^-ϕ(τ^-1x). Substitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing x through τ in each channel, gives ∑_xinnerSψ(x)Sϕ(x) =∑_xinnerψ(x)ϕ…
Exact mathematical passage
```
\label{g:clifford-inverse}
 (S_{A,\tau,u}^{-1}\phi)(x)
 =u(x)^{-1}P_A^+\phi(\tau x)
 +u(\tau^{-1}x)P_A^-\phi(\tau^{-1}x).
```

S_{A,\tau,u^g}Use in source · 1 passage

Search aliases: S_A,τ,u^g tau

Link to this entry ·

Use in source · core.tex, line 137
tution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing x through τ in each channel, gives ∑_xinnerSψ(x)Sϕ(x) =∑_xinnerψ(x)ϕ(x). For u^g(x)=g(τ x)u(x)g(x)^-1, direct endpoint cancellation gives S_A,τ,u^gG_g=G_gS_A,τ,u. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.
Exact mathematical passage
```
S_{A,\tau,u^g}G_g=G_gS_{A,\tau,u}
```

S_{A,\tau,v}Use in source · 2 passages

Search aliases: S_A,τ,v tau

Link to this entry ·

Use in source · refinement.tex, line 195
A flat cube need not reconstruct the prescribed edge. Conversely, take an arbitrary ordered half-link pair (a,b) and call its actual product v=ba. It is always a subdivision of v, even when the separately prescribed coarse link is u≠ v. The exact directional defect is D_rm fineψ · :=S_A,σ,(a,b)^2I_aψ-I_aS_A,τ,uψnotag · =I_a(S_A,τ,v-S_A,τ,u)ψ. The equality follows by applying the exact theorem to the product v and using linearity of I_a. Endpoint restriction makes I_a injective, so the fine defec…
Exact mathematical passage
```
 D_{\rm fine}\psi
  &:=S_{A,\sigma,(a,b)}^2I_a\psi-I_aS_{A,\tau,u}\psi\notag\\
  &=I_a(S_{A,\tau,v}-S_{A,\tau,u})\psi.
 \label{g:exact-mismatch}
```

s_bUse in source · 2 passages

Search aliases: s_b

Link to this entry ·

Use in source · extensions.tex, line 47
[Effect continuity] Under these hypotheses, dot p_b=∑_a J^F_ba, ∑_bdot p_b=0. If F_b=F_b(t) is differentiable in operator norm, the first identity becomes dot p_b=s_b+∑_a J^F_ba, s_b=innerψdot F_bψ, ∑_b s_b=0. Self-adjointness of F_b and differentiation give dot p_b=2ReinnerF_bψKψ for a fixed effect. Summing the two terms of [source reference] over a and using ∑_aF_a=I gives ∑_aJ^F_ba=ReinnerF_bψKψ -ReinnerψKF_bψ. The second real part is the negative of the first. The moving case adds the produc…
Exact mathematical passage
```
 \dot p_b=s_b+\sum_a J^F_{ba},\qquad
 s_b=\inner{\psi}{\dot F_b\psi},\qquad \sum_b s_b=0.
```

S_{\mathrm{BV}}Use in source · 2 passages

Search aliases: S_BV mathrm

Link to this entry ·

Use in source · constraints.tex, line 387
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
 S_{\mathrm{BV}}=S_0+
     \int\sum_\Xi\langle\Xi^\ddagger,s\Xi\rangle.
 \label{c:constraints:bv-action}
```

S_CUse in source · 5 passages

Search aliases: S_C

Link to this entry ·

Use in source · constraints.tex, line 794
-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonli…
Exact mathematical passage
```
 S_C=\frac{\mu x_2^2}{2d},\qquad
 R_C=\partial_{x_1}+\frac{x_1x_2}{d}\partial_{x_2}.
```

s_CUse in source · 6 passages

Search aliases: s_C

Link to this entry ·

Use in source · constraints.tex, line 84
0 · 1/25 · 0 . Its square is nonzero, as its (2,2) entry is -1/25. The transferred block in that same test is s_C=0 · 0 · 0 1 · 0 · 1/25 0 · 0 · 0, s_C^2=0. This small rational example illustrates the distinction between a projection and a transfer. The matrices by themselves are an algebraic diagnostic. They do not define gravitational constraints, a symplectic BFV theory, or a continuum model.
Exact mathematical passage
```
 s_C=\begin{pmatrix}0&0&0\\1&0&1/25\\0&0&0\end{pmatrix},
 \qquad s_C^2=0.
```

S_{C,\hbar}Use in source · 1 passage

Search aliases: S_C,ℏ hbar

Link to this entry ·

Use in source · constraints.tex, line 806
Eliminating z first gives the effective coefficient μ for y^2 and the factor (m+n)^-1/2. Eliminating y next gives d^-1/2. Their product is D^-1/2, exactly as in direct elimination. For the Euclidean weight e^-S/ℏ, using a flat coarse measure instead records this determinant as S_C,ℏ=S_C+tfracℏ2log D, up to an additive constant. A nonlinear symmetry that preserves the classical action can still require a nonconstant induced measure.
Exact mathematical passage
```
S_{C,\hbar}=S_C+\tfrac\hbar2\log D
```

s_C^2Use in source · 4 passages

Search aliases: s_C^2

Link to this entry ·

Use in source · constraints.tex, line 84
0 · 1/25 · 0 . Its square is nonzero, as its (2,2) entry is -1/25. The transferred block in that same test is s_C=0 · 0 · 0 1 · 0 · 1/25 0 · 0 · 0, s_C^2=0. This small rational example illustrates the distinction between a projection and a transfer. The matrices by themselves are an algebraic diagnostic. They do not define gravitational constraints, a symplectic BFV theory, or a continuum model.
Exact mathematical passage
```
 s_C=\begin{pmatrix}0&0&0\\1&0&1/25\\0&0&0\end{pmatrix},
 \qquad s_C^2=0.
```

S_GUse in source · 1 passage

Search aliases: S_G

Link to this entry ·

Use in source · core.tex, line 343
Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber
Exact mathematical passage
```
 S_G=\frac12\int_I\int_0^{2\pi}
 \left[P_\tau^2-\gamma P_\theta^2
 +e^{2P}(Q_\tau^2-\gamma Q_\theta^2)\right]d\theta\,d\tau.
 \label{c:core:gowdy-action}
```

S_hUse in source · 2 passages

Search aliases: S_h

Link to this entry ·

Use in source · core.tex, line 250
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 S_h[u]=\sum_{i=0}^{M-1}\frac{(u_{i+1}-u_i)^2}{2Gh}
 +4\pi h\sum_{i=1}^{M-1}r_i\rho_i u_i.
 \label{c:core:poisson-action}
```

\mathcal S_hUse in source · 1 passage

Search aliases: S_h mathcal

Link to this entry ·

Use in source · core.tex, line 205
βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out…
Exact mathematical passage
```
\label{g:action}
 \mathcal S_h=\int\left[
 \frac{ih}{2}\sum_j(\psi_j^*\dot\psi_j-\dot\psi_j^*\psi_j)
 +h\sum_jE_j\dot A_j-H_h\right]dt.
```

S_h^{(2)}Use in source · 1 passage

Search aliases: S_h^(2)

Link to this entry ·

Use in source · core.tex, line 664
Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber
Exact mathematical passage
```
 S_h^{(2)}=\int\left[
 \inner{\pi}{\dot h}_h-T_h-U_h
 +\inner{n}{C}_h-\inner{n_i}{C_i}_h\right]dt.
 \label{c:core:adm-action}
```

S_{i,\mathrm{coarse}}Use in source · 1 passage

Search aliases: S_i,coarse mathrm

Link to this entry ·

Use in source · refinement.tex, line 186
lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass substeps also intertwine, the identities compose through the entire execution order M-X-Y-Z-Y-X-M. The refined walk here replaces each coarse spatial fac…
Exact mathematical passage
```
 S_{i,\mathrm{fine}}^2 I_r=I_rS_{i,\mathrm{coarse}}.
```

S_{i,\mathrm{fine}}^2Use in source · 1 passage

Search aliases: S_i,fine^2 mathrm

Link to this entry ·

Use in source · refinement.tex, line 186
lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass substeps also intertwine, the identities compose through the entire execution order M-X-Y-Z-Y-X-M. The refined walk here replaces each coarse spatial fac…
Exact mathematical passage
```
 S_{i,\mathrm{fine}}^2 I_r=I_rS_{i,\mathrm{coarse}}.
```

s_jUse in source · 5 passages

Search aliases: s_j

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

S_NUse in source · 26 passages

Search aliases: S_N

Link to this entry ·

Use in source · extensions.tex, line 162
operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.
Exact mathematical passage
```
S_Nc=\sum_{n\leq N}c_ne_n
```

S_N^*Use in source · 1 passage

Search aliases: S_N^*

Link to this entry ·

Use in source · extensions.tex, line 340
For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anorm Y_2 and the displayed decomposition finish the proof.
Exact mathematical passage
```
S_N^*=S_N
```

\mathcal S_pUse in source · 4 passages

Search aliases: S_p mathcal

Link to this entry ·

Use in source · spectral.tex, line 568
[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber
Exact mathematical passage
```
 \mathcal S_p(z)=\frac{\kappa}{U_{p-1}(x/2)}
          \bigl[R_p(x)I_M-(T+T^*)\bigr].
 \label{a-spectral-decimation-formula}
```

S_{p,h}Use in source · 3 passages

Search aliases: S_p,h

Link to this entry ·

Use in source · finite.tex, line 608
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
(S_{p,h}f)(x)=f(x-1)
```

S_{p,h+1}Use in source · 1 passage

Search aliases: S_p,h+1

Link to this entry ·

Use in source · finite.tex, line 611
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
 S_{p,h+1}V_{p,h}=V_{p,h}S_{p,h}.
```

S_{p,h}^*Use in source · 1 passage

Search aliases: S_p,h^*

Link to this entry ·

Use in source · finite.tex, line 615
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
L_{p,h}=2I-S_{p,h}-S_{p,h}^*
```

s_qUse in source · 2 passages

Search aliases: s_q

Link to this entry ·

Use in source · finite_core.tex, line 934
The error also has an exact expression. For fixed mode, let r=p(n), q=q_h(n), with the corresponding c_r,s_r,c_q,s_q. Coordinate subtraction gives
Exact mathematical passage
```
c_r,s_r,c_q,s_q
```

s_rUse in source · 5 passages

Search aliases: s_r

Link to this entry ·

Use in source · finite_core.tex, line 861
Set E_r=sqrt1+|r|^2, c_r(t)=cos(E_rt), and s_r(t)=sin(E_rt)/E_r. PH03 gives
Exact mathematical passage
```
s_r(t)=\sin(E_rt)/E_r
```

S_{\mathrm{sign}}Use in source · 1 passage

Search aliases: S_sign mathrm

Link to this entry ·

Use in source · limit.tex, line 706
Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors with equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bits is S_sign(t) =h_2(frac1+|cos(gt)|2), h_2(u)=-ulog_2u-(1-u)log_2(1-u), with 0log_2 0=0. At t_* =π/(2|g|) the two sign states are orthogonal and the …
Exact mathematical passage
```
 S_{\mathrm{sign}}(t)
 =h_2\left(\frac{1+|\cos(gt)|}{2}\right),
 \quad h_2(u)=-u\log_2u-(1-u)\log_2(1-u),
 \label{a-limit-entanglement-entropy}
```

S_uUse in source · 1 passage

Search aliases: S_u

Link to this entry ·

Use in source · refinement.tex, line 235
1|. The constant is independent of the number of sites. The actual half-grid defect in [source reference], with half-cell Hilbert measure, obeys the same bound. At each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity norm(S_v-S_u)ψ^2 =∑_x[ · |v(τ^-1x)-u(τ^-1x)|^2 normP_+ψ(τ^-1x)^2 · +|v(x)^-1-u(x)^-1|^2 normP_-ψ(τ x)^2]. There is no mixed term, even though the two spinors come from different sites: th…
Exact mathematical passage
```
 \norm{(S_v-S_u)\psi}^2
 =\sum_x\bigl[&|v(\tau^{-1}x)-u(\tau^{-1}x)|^2
                   \norm{P_+\psi(\tau^{-1}x)}^2\\
 &+|v(x)^{-1}-u(x)^{-1}|^2
                   \norm{P_-\psi(\tau x)}^2\bigr].
```

S_vUse in source · 1 passage

Search aliases: S_v

Link to this entry ·

Use in source · refinement.tex, line 235
1|. The constant is independent of the number of sites. The actual half-grid defect in [source reference], with half-cell Hilbert measure, obeys the same bound. At each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity norm(S_v-S_u)ψ^2 =∑_x[ · |v(τ^-1x)-u(τ^-1x)|^2 normP_+ψ(τ^-1x)^2 · +|v(x)^-1-u(x)^-1|^2 normP_-ψ(τ x)^2]. There is no mixed term, even though the two spinors come from different sites: th…
Exact mathematical passage
```
 \norm{(S_v-S_u)\psi}^2
 =\sum_x\bigl[&|v(\tau^{-1}x)-u(\tau^{-1}x)|^2
                   \norm{P_+\psi(\tau^{-1}x)}^2\\
 &+|v(x)^{-1}-u(x)^{-1}|^2
                   \norm{P_-\psi(\tau x)}^2\bigr].
```

S_WUse in source · 2 passages

Search aliases: S_W

Link to this entry ·

Use in source · core.tex, line 160
For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the analogous term is 1-ReW. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interac…
Exact mathematical passage
```
\label{g:wilson-action}
 S_W[U]=\frac1{g_{\mathrm{YM}}^2}
 \sum_{x,\mu<\nu}\left(1-\tfrac12\operatorname{Re}\operatorname{Tr}W_{x,\mu\nu}\right),
 \qquad g_{\mathrm{YM}}\in\R\setminus\{0\}.
```

\mathrm{sign}Use in source · 8 passages

Search aliases: sign mathrm

Link to this entry ·

Use in source · finite.tex, line 94
In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They do not assert that a generic state is a product state, or that a generic Hamiltonian is a sum of independent register Hamiltonians. Entanglement and in…
Exact mathematical passage
```
 \ell^2(U_{p,h})&\cong
 \C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^h\Z),
 &&p\text{ odd},\nonumber\\
 \ell^2(U_{2,h})&\cong
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^h\Z).
 \label{a-finite-unit-hilbert}
```

\sinhUse in source · 2 passages

Search aliases: sinh

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Use in source · spectral.tex, line 1047
The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lane is an analytic contribution to it.
Exact mathematical passage
```
E_{2,1}(tA_0)=\sinh(tA_0)
```

\mathrm{Site}Use in source · 1 passage

Search aliases: Site mathrm

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Use in source · refinement.tex, line 168
Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phase frame r(s,c)∈ U(1) defines (I_rψ)(s,c)=r(s,c)ψ(s). With one-eighth fine weights this is again an isometry.
Exact mathematical passage
```
\mathrm{Site}\times\{0,1\}^3
```

\operatorname{span}Use in source · 1 passage

Search aliases: span operatorname

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Use in source · limit.tex, line 25
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
 \qquad \mathcal V_m=\operatorname{span}
 \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
 \label{a-limit-cylinder-embedding}
```

\mathrm{std}Use in source · 2 passages

Search aliases: std mathrm

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Use in source · finite.tex, line 425
For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+…
Exact mathematical passage
```
 \mathcal A^{\mathrm{std}}_{p,h}|pk+r\rangle
 &=|r\rangle\otimes|k\rangle,
 &0\leq r<p,\quad0\leq k<m,\nonumber\\
 \mathcal A^{\mathrm{ad}}_{p,h}|a+d p^{h-1}\rangle
 &=|d\rangle\otimes|a\rangle,
 &0\leq d<p,\quad0\leq a<m.
 \label{a-finite-digit-splits}
```

\mathfrak{su}Use in source · 1 passage

Search aliases: su mathfrak

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Use in source · core.tex, line 416
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
A,B\in\mathfrak{su}(2)
```

\supUse in source · 8 passages

Search aliases: sup

Link to this entry ·

Use in source · limit.tex, line 256
[Real level energies define a self-adjoint operator] The operator in [source reference] is densely defined, closed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, s…
Exact mathematical passage
```
\sup_m|\epsilon_m|<\infty
```

\sup_mUse in source · 2 passages

Search aliases: sup_m sup

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Use in source · limit.tex, line 256
[Real level energies define a self-adjoint operator] The operator in [source reference] is densely defined, closed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, s…
Exact mathematical passage
```
\sup_m|\epsilon_m|<\infty
```

\sup_{m>M}Use in source · 1 passage

Search aliases: sup_m>M sup

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Use in source · limit.tex, line 468
The omitted embedded resolvent has the exact norm norm(I- E_M)(z-H_ϵ)^-1 =sup_m>M|z-ϵ_m|^-1. The upper bound follows by Parseval. For the lower bound, test on a unit vector at any omitted level and take the supremum. Since every level is nonzero and finite dimensional, this proves the following equivalence.
Exact mathematical passage
```
 \norm{(I-\mathsf E_M)(z-H_\epsilon)^{-1}}
 =\sup_{m>M}|z-\epsilon_m|^{-1}.
 \label{a-limit-resolvent-tail-norm}
```

\sup_{n>N}Use in source · 4 passages

Search aliases: sup_n>N sup

Link to this entry ·

Use in source · extensions.tex, line 356
For a diagonal bounded multiplier f(H_λ), zero padding gives the exact formula normf(H_λ)-S_Nf(H_λ)S_N =sup_n>N|f(λ_n)|. For the unitary multiplier f(λ)=e^-itλ, this norm is one at every finite cutoff. Equation [source reference] remains valid. Uniformity in time for a fixed state and failure of operator-norm convergence are therefore compatible.
Exact mathematical passage
```
 \norm{f(H_\lambda)-S_Nf(H_\lambda)S_N}
 =\sup_{n>N}|f(\lambda_n)|.
```

\sup_{|t|\le T}Use in source · 1 passage

Search aliases: sup_|t|≤ T sup le

Link to this entry ·

Use in source · finite_core.tex, line 1039
sup_|t|≤ T f_h(t)-f(t) _L^2 ≤ Th^2(∑_n∈ SC(n)^2 a_n _2^2)^1/2,
Exact mathematical passage
```
\sup_{|t|\le T}\|f_h(t)-f(t)\|_{L^2}
\le Th^2\left(\sum_{n\in S}C(n)^2\|a_n\|_2^2\right)^{1/2},
```

\TUse in source · 4 passages

Search aliases: T

Link to this entry ·

Use in source · continuum.tex, line 90
Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation, normF f_KK=normf_HH, F^-1F f=f, FF^-1a=a. The inverse is understood as …
Exact mathematical passage
```
\F:\HH\longrightarrow\KK,
\qquad \HH=L^2(\T^3;\C^4).
```

\mathbb TUse in source · 9 passages

Search aliases: T mathbb

Link to this entry ·

Use in source · finite_core.tex, line 7
T^3=( R/ Z)^3,
Exact mathematical passage
```
\mathbb T^3=(\mathbb R/\mathbb Z)^3,
```

\mathcal TUse in source · 12 passages

Search aliases: T mathcal

Link to this entry ·

Use in source · finite.tex, line 482
W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the n…
Exact mathematical passage
```
\mathcal T_{p,h}
```

\mathsf TUse in source · 10 passages

Search aliases: T mathsf

Link to this entry ·

Use in source · spectral.tex, line 197
-C^-1B^* · I. Both outer factors are invertible and have determinant one. This proves the invertibility equivalence and determinant formula. Inverting the factors gives the stated upper left block. Alternatively, in (z-H)(u,v)^ T=(f,0)^ T, the lower equation gives v=C^-1B^*u and the upper equation becomes S(z)u=f. The same elimination with f=0 proves the eigenvector statement. If u=0 at such an E, invertibility of E-D forces v=0, so a nonzero full eigenvector indeed has u≠0.
Exact mathematical passage
```
(z-H)(u,v)^{\mathsf T}=(f,0)^{\mathsf T}
```

T^*Use in source · 5 passages

Search aliases: T^*

Link to this entry ·

Use in source · spectral.tex, line 568
[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber
Exact mathematical passage
```
 \mathcal S_p(z)=\frac{\kappa}{U_{p-1}(x/2)}
          \bigl[R_p(x)I_M-(T+T^*)\bigr].
 \label{a-spectral-decimation-formula}
```

\T^3Use in source · 2 passages

Search aliases: T^3 T

Link to this entry ·

Use in source · continuum.tex, line 90
Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation, normF f_KK=normf_HH, F^-1F f=f, FF^-1a=a. The inverse is understood as …
Exact mathematical passage
```
\F:\HH\longrightarrow\KK,
\qquad \HH=L^2(\T^3;\C^4).
```

\mathbb T^3Use in source · 9 passages

Search aliases: T^3 mathbb

Link to this entry ·

Use in source · finite_core.tex, line 7
T^3=( R/ Z)^3,
Exact mathematical passage
```
\mathbb T^3=(\mathbb R/\mathbb Z)^3,
```

T^gUse in source · 1 passage

Search aliases: T^g

Link to this entry ·

Use in source · core.tex, line 80
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\label{g:walk-covariance}
 \mathcal W_n[U^g,T^g]G^n=G^{n+1}\mathcal W_n[U,T],
 \qquad (G^n\psi)_j=g_j^n\psi_j.
```

T^IUse in source · 1 passage

Search aliases: T^I

Link to this entry ·

Use in source · constraints.tex, line 614
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
 T^I=\tfrac\kappa4\epsilon^I{}_{JKL}J_5^Je^K\wedge e^L,
 \qquad
 C^{IJ}=\tfrac\kappa4\epsilon^{IJ}{}_{KL}J_5^Ke^L,
```

t_*Use in source · 1 passage

Search aliases: t_*

Link to this entry ·

Use in source · limit.tex, line 712
th equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bits is S_sign(t) =h_2(frac1+|cos(gt)|2), h_2(u)=-ulog_2u-(1-u)log_2(1-u), with 0log_2 0=0. At t_* =π/(2|g|) the two sign states are orthogonal and the entropy is one bit. A fixed level projector is compatible under refinement, so this interaction can entangle the prime factors while preserving the cylinder embeddi…
Exact mathematical passage
```
t_* =\pi/(2|g|)
```

t_0Use in source · 1 passage

Search aliases: t_0

Link to this entry ·

Use in source · limit.tex, line 326
The modulus-one multipliers give U(t)^*U(t)=I and U(t+s)=U(t)U(s). They also give strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint operator is the generator. A general L^2 state evolves unitarily even i…
Exact mathematical passage
```
 \norm{(U(t)-U(t_0))f}_2^2
 =\sum_m|e^{-it\epsilon_m}-e^{-it_0\epsilon_m}|^2
 \norm{\mathsf F_mf}_2^2\longrightarrow0.
```

T_1Use in source · 2 passages

Search aliases: T_1

Link to this entry ·

Use in source · refinement.tex, line 72
A complete three-level example. For b=2, direct substitution gives T_1f(h) · =2f(h/2)-f(h), T_2f(h) · =frac8f(h/4)-6f(h/2)+f(h)3, T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for every real h. If each of the three samples used by T_2 has absolute error at most ε, its output error is at most frac8+6+13ε=5ε. This bound is attained by…
Exact mathematical passage
```
 T_1f(h)&=2f(h/2)-f(h),\\
 T_2f(h)&=\frac{8f(h/4)-6f(h/2)+f(h)}{3},\\
 T_3f(h)&=\frac{64f(h/8)-56f(h/4)+14f(h/2)-f(h)}{21}.
```

T_2Use in source · 3 passages

Search aliases: T_2

Link to this entry ·

Use in source · refinement.tex, line 72
A complete three-level example. For b=2, direct substitution gives T_1f(h) · =2f(h/2)-f(h), T_2f(h) · =frac8f(h/4)-6f(h/2)+f(h)3, T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for every real h. If each of the three samples used by T_2 has absolute error at most ε, its output error is at most frac8+6+13ε=5ε. This bound is attained by…
Exact mathematical passage
```
 T_1f(h)&=2f(h/2)-f(h),\\
 T_2f(h)&=\frac{8f(h/4)-6f(h/2)+f(h)}{3},\\
 T_3f(h)&=\frac{64f(h/8)-56f(h/4)+14f(h/2)-f(h)}{21}.
```

T_3Use in source · 2 passages

Search aliases: T_3

Link to this entry ·

Use in source · refinement.tex, line 72
A complete three-level example. For b=2, direct substitution gives T_1f(h) · =2f(h/2)-f(h), T_2f(h) · =frac8f(h/4)-6f(h/2)+f(h)3, T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for every real h. If each of the three samples used by T_2 has absolute error at most ε, its output error is at most frac8+6+13ε=5ε. This bound is attained by…
Exact mathematical passage
```
 T_1f(h)&=2f(h/2)-f(h),\\
 T_2f(h)&=\frac{8f(h/4)-6f(h/2)+f(h)}{3},\\
 T_3f(h)&=\frac{64f(h/8)-56f(h/4)+14f(h/2)-f(h)}{21}.
```

T_aUse in source · 2 passages

Search aliases: T_a

Link to this entry ·

Use in source · limit.tex, line 649
For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting wholly inside a new detail complement has no matrix element between that complement and the old range. A cross term between them would be a different…
Exact mathematical passage
```
 A_{\mathbf r}
 =\left(\sum_a\epsilon_{a,\mathbf r}T_a\right)
 \otimes I_{\mathrm{phase},\mathbf r},
 \qquad \epsilon_{a,\mathbf r}\in\R,\quad T_a=T_a^*.
```

T_a^*Use in source · 1 passage

Search aliases: T_a^*

Link to this entry ·

Use in source · limit.tex, line 649
For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting wholly inside a new detail complement has no matrix element between that complement and the old range. A cross term between them would be a different…
Exact mathematical passage
```
 A_{\mathbf r}
 =\left(\sum_a\epsilon_{a,\mathbf r}T_a\right)
 \otimes I_{\mathrm{phase},\mathbf r},
 \qquad \epsilon_{a,\mathbf r}\in\R,\quad T_a=T_a^*.
```

T_hUse in source · 2 passages

Search aliases: T_h

Link to this entry ·

Use in source · core.tex, line 659
Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber
Exact mathematical passage
```
 T_h=\inner{\pi}{\pi}_h-\tfrac12\inner{\pi_{\rm tr}}{\pi_{\rm tr}}_h,
 \qquad U_h=\tfrac12\inner{h}{\mathcal G(h)}_h.
```

t_HUse in source · 4 passages

Search aliases: t_H

Link to this entry ·

Use in source · strong_fields.tex, line 516
Recorded first marginally trapped sphere. The event finder takes a quadratic interpolation of the spatial maximum of 2m/r and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is small rrrrr N_r · h · t_H · r_H · Δ M/M_0
Exact mathematical passage
```
t_H
```

T_j^nUse in source · 3 passages

Search aliases: T_j^n

Link to this entry ·

Use in source · core.tex, line 65
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,
Exact mathematical passage
```
T_j^n\in U(1)
```

\mathcal T_KUse in source · 7 passages

Search aliases: T_K mathcal

Link to this entry ·

Use in source · constraints.tex, line 178
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K=\left\{\sum_{\|k\|_\infty\le K}
              \widehat f_k e^{ik\cdot x}\right\}.
```

\mathcal T_{K+L}Use in source · 1 passage

Search aliases: T_K+L mathcal

Link to this entry ·

Use in source · constraints.tex, line 183
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K\mathcal T_L\subset\mathcal T_{K+L},\qquad
 \partial_a\mathcal T_K\subset\mathcal T_K.
```

\mathcal T_K^{(4)}Use in source · 1 passage

Search aliases: T_K^(4) mathcal

Link to this entry ·

Use in source · constraints.tex, line 409
For coefficients in T_K^(4) and a finite Grassmann algebra, the parent action has degree at most six, and the displayed antifield couplings have degree at most three. The master-equation and invariance coefficients have at most seven band-K factors. The sufficient rule 16K<N places them below Nyquist and transfers the classical identities coefficientwise. The rule concerns the polynomial parent. It does not automatically survive elimination of inverse tetrads or density square roots.
Exact mathematical passage
```
\mathcal T_K^{(4)}
```

\mathcal T_{K_j}Use in source · 1 passage

Search aliases: T_K_j mathcal

Link to this entry ·

Use in source · constraints.tex, line 192
If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j),
Exact mathematical passage
```
f_j\in\mathcal T_{K_j}
```

\mathcal T_LUse in source · 1 passage

Search aliases: T_L mathcal

Link to this entry ·

Use in source · constraints.tex, line 183
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K\mathcal T_L\subset\mathcal T_{K+L},\qquad
 \partial_a\mathcal T_K\subset\mathcal T_K.
```

T_nUse in source · 4 passages

Search aliases: T_n

Link to this entry ·

Use in source · spectral.tex, line 527
We now eliminate intermediate residue sites in [source reference], assuming κ≠0. The retained sites are C={0,p,…,(M-1)p} and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).
Exact mathematical passage
```
 x=\frac{z-\mu}{\kappa},\qquad
 R_n(x)=2T_n(x/2),
 \label{a-spectral-normalized-energy}
```

T_pUse in source · 3 passages

Search aliases: T_p

Link to this entry ·

Use in source · spectral.tex, line 596
eighboring paths. Each path contributes the onsite self-energy κ U_p-2/U_p-1 by Lemma~[source reference]; the two contributions add. The path joining two adjacent retained sites contributes off-diagonal self-energy κ/U_p-1. Thus S_p(z) =κ(x-frac2U_p-2(x/2)U_p-1(x/2))I -fracκU_p-1(x/2)(T+T^*). The identity xU_p-1(x/2)-2U_p-2(x/2)=2T_p(x/2) follows from the three-term recurrence and U_p-U_p-2=2T_p. For the latter identity, substituting x/2=cosθ gives [sin((p+1)θ)-sin((p-1)θ)]/sinθ =2cos(pθ), and p…
Exact mathematical passage
```
xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)
```

t_PUse in source · 2 passages

Search aliases: t_P

Link to this entry ·

Use in source · strong_fields.tex, line 88
Fix M>0, a real charge Q, and a branch ε∈{+1,-1}. On 2MR-Q^2>0, define p(R)=εsqrt2MR-Q^2, v(R)=fracp(R)R. The Painlev e--Gullstrand metric takes the form ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2dΩ^2. No inverse factor of 1-2M/R+Q^2/R^2 appears. The metric can therefore cross a horizon while the lapse and radial spatial metric remain regular.
Exact mathematical passage
```
 ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2d\Omega^2.
\label{c:strong:RN-PG}
```

\mathcal T_{p,h}Use in source · 3 passages

Search aliases: T_p,h mathcal

Link to this entry ·

Use in source · finite.tex, line 482
W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the n…
Exact mathematical passage
```
\mathcal T_{p,h}
```

\mathcal T_{p,h}^*Use in source · 2 passages

Search aliases: T_p,h^* mathcal

Link to this entry ·

Use in source · finite.tex, line 484
d The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the nonzero character labels q↔ p-q and leaves th…
Exact mathematical passage
```
 \mathcal T_{p,h}^*\Pi_r\mathcal T_{p,h}=D_r^{(h)}.
 \label{a-finite-haar-analysis}
```

t_P^2Use in source · 1 passage

Search aliases: t_P^2

Link to this entry ·

Use in source · strong_fields.tex, line 88
Fix M>0, a real charge Q, and a branch ε∈{+1,-1}. On 2MR-Q^2>0, define p(R)=εsqrt2MR-Q^2, v(R)=fracp(R)R. The Painlev e--Gullstrand metric takes the form ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2dΩ^2. No inverse factor of 1-2M/R+Q^2/R^2 appears. The metric can therefore cross a horizon while the lapse and radial spatial metric remain regular.
Exact mathematical passage
```
 ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2d\Omega^2.
\label{c:strong:RN-PG}
```

t_{\rm Shapiro}Use in source · 1 passage

Search aliases: t_rm Shapiro rm

Link to this entry ·

Use in source · core.tex, line 317
N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b) · =frac2c_0^2|∫_-L^L∂_bΦ(x,b) dx| =frac4GM_0Lc_0^2bsqrtL^2+b^2. Here α is the magnitude of the deflection toward the source. The limit L→∞ gives 4GM…
Exact mathematical passage
```
 \Delta t_{\rm Shapiro}
 &=-\frac2{c_0^3}\int_{-L}^L\Phi(x,b)\,dx
 =\frac{4GM_0}{c_0^3}\operatorname{asinh}\left(\frac Lb\right),\\
 \alpha(L,b)
 &=\frac2{c_0^2}\left|\int_{-L}^L\partial_b\Phi(x,b)\,dx\right|
 =\frac{4GM_0L}{c_0^2b\sqrt{L^2+b^2}}.
```

T_{y\leftarrow x}Use in source · 5 passages

Search aliases: T_y← x leftarrow

Link to this entry ·

Use in source · core.tex, line 16
_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the input and output of every product.
Exact mathematical passage
```
T_{y\leftarrow x}
```

T_{z\leftarrow y}Use in source · 1 passage

Search aliases: T_z← y leftarrow

Link to this entry ·

Use in source · orientation.tex, line 26
Matrix order · T_z← yT_y← x transports first from x to y, then from y to z. The rightmost factor acts first.
Exact mathematical passage
```
T_{z\leftarrow y}T_{y\leftarrow x}
```

\mathrm{tor}Use in source · 6 passages

Search aliases: tor mathrm

Link to this entry ·

Use in source · limit.tex, line 533
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every
Exact mathematical passage
```
\KK_{\mathrm{tor}}
```

\mathrm{tors}Use in source · 2 passages

Search aliases: tors mathrm

Link to this entry ·

Use in source · finite.tex, line 94
In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They do not assert that a generic state is a product state, or that a generic Hamiltonian is a sum of independent register Hamiltonians. Entanglement and in…
Exact mathematical passage
```
 \ell^2(U_{p,h})&\cong
 \C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^h\Z),
 &&p\text{ odd},\nonumber\\
 \ell^2(U_{2,h})&\cong
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^h\Z).
 \label{a-finite-unit-hilbert}
```

\operatorname{Tr}_BUse in source · 1 passage

Search aliases: Tr_B operatorname

Link to this entry ·

Use in source · core.tex, line 384
al because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, with (e_k)_k indexing the basis of HH_B. In the resulting product basis the partial trace has entries (ρ_A)_ij=∑_k(ρ_AB)_(i,k),(j,k). This explicit formula s…
Exact mathematical passage
```
 p_a^A=\operatorname{Tr}\bigl((P_a\otimes\id_B)\rho_{AB}\bigr)
 =\operatorname{Tr}(P_a\rho_A),\qquad \rho_A=\operatorname{Tr}_B\rho_{AB}.
```

\operatorname{Tr}_{\mathrm{sign}}Use in source · 1 passage

Search aliases: Tr_sign operatorname mathrm

Link to this entry ·

Use in source · finite.tex, line 210
Discarding the sign as a quantum operation instead takes the partial trace ρ↦Tr_signρ. Both pure states |0,a⟩ and |1,a⟩ then produce the same reduced phase state |a⟩⟨ a|. Yet the sign observable Z_sign=(|0⟩⟨0|-|1⟩⟨1|)⊗ I has expectations +1 and -1 on those states. No measurement on the retained phase state can reconstruct this difference. The obstruction is information loss in a stated map, not an alternative probability rule.
Exact mathematical passage
```
\rho\mapsto\operatorname{Tr}_{\mathrm{sign}}\rho
```

\mathrm UUse in source · 6 passages

Search aliases: U mathrm

Link to this entry ·

Use in source · constraints.tex, line 465
C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.
Exact mathematical passage
```
 \mathcal G_{IJ}
 &=D_\Omega B_{IJ}
       -\tfrac i8\sigma_K\bar\psi\{\gamma^K,\gamma_{IJ}\}\psi,\\
 \mathcal G_{\mathrm U}
 &=\mathrm d_\Sigma\Pi-q\sigma_I\bar\psi\gamma^I\psi.
```

u'Use in source · 2 passages

Search aliases: u'

Link to this entry ·

Use in source · core.tex, line 241
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 r^2\Phi'=ru'-u,\qquad
 (r^2\Phi')'=ru'',\qquad u''=4\pi G r\rho.
```

u''Use in source · 1 passage

Search aliases: u''

Link to this entry ·

Use in source · core.tex, line 241
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 r^2\Phi'=ru'-u,\qquad
 (r^2\Phi')'=ru'',\qquad u''=4\pi G r\rho.
```

u^{-1}Use in source · 1 passage

Search aliases: u^-1

Link to this entry ·

Use in source · refinement.tex, line 242
· +|v(x)^-1-u(x)^-1|^2 normP_-ψ(τ x)^2]. There is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies |v^-1-u^-1|=|v-u|. Bound every coefficient by δ^2, reindex the two sums through τ, and use normP_+η^2+normP_-η^2=normη^2. The result is at most δ^2normψ^2. Taking square roots and then the operator supremum proves the first inequality. Multiplication by u(x)^-1 proves the equality of scalar maxima. Finally [sourc…
Exact mathematical passage
```
|v^{-1}-u^{-1}|=|v-u|
```

u^2Use in source · 1 passage

Search aliases: u^2

Link to this entry ·

Use in source · finite_core.tex, line 553
|1-cos u|=2sin^2(u/2)≤ u^2/2.
Exact mathematical passage
```
|1-\cos u|=2\sin^2(u/2)\le u^2/2.
```

U^gUse in source · 4 passages

Search aliases: U^g

Link to this entry ·

Use in source · core.tex, line 36
(Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
D^+_{\mu,U^g}(g\psi)=gD^+_{\mu,U}\psi
```

u^pUse in source · 1 passage

Search aliases: u^p

Link to this entry ·

Use in source · finite.tex, line 52
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
u^p-1
```

u^TUse in source · 1 passage

Search aliases: u^T

Link to this entry ·

Use in source · strong_fields.tex, line 287
For an interval summation-by-parts derivative with a diagonal positive norm matrix H and HD+D^TH=B, multiplication by b commutes with H. The uncancelled term is instead inneruA_h[b]u_H =frac12u^TB diag(b)u -frac12inneru(Db)u_H. Its boundary flux must be controlled using an actual boundary closure and appropriate boundary data. Saying that a stencil is centered in the interior does not establish this property for the complete interval operator.
Exact mathematical passage
```
 \inner{u}{A_h[b]u}_H
 =\frac12u^TB\,\operatorname{diag}(b)u
  -\frac12\inner{u}{(Db)u}_H.
```

u_{(2\pi,0,0)}Use in source · 2 passages

Search aliases: u_(2π,0,0) pi

Link to this entry ·

Use in source · finite_core.tex, line 905
The continuum packet has the same form with u_A,u_D replaced by u_(2π,0,0) and u_(2π,2π,2π). At t=0, both equal
Exact mathematical passage
```
u_{(2\pi,0,0)}
```

u_{(2\pi,2\pi,2\pi)}Use in source · 2 passages

Search aliases: u_(2π,2π,2π) pi pi pi

Link to this entry ·

Use in source · finite_core.tex, line 905
The continuum packet has the same form with u_A,u_D replaced by u_(2π,0,0) and u_(2π,2π,2π). At t=0, both equal
Exact mathematical passage
```
u_{(2\pi,2\pi,2\pi)}
```

u_{(6,0,0)}Use in source · 1 passage

Search aliases: u_(6,0,0)

Link to this entry ·

Use in source · finite_core.tex, line 945
f_h(t)-f(t) _L^2^2 = · frac14 u_(6,0,0)(t)-u_(2π,0,0)(t) _2^2 · +frac14 u_(6,6,6)(t)-u_(2π,2π,2π)(t) _2^2.
Exact mathematical passage
```
\begin{aligned}
\|f_h(t)-f(t)\|_{L^2}^2
={}&\frac14\|u_{(6,0,0)}(t)-u_{(2\pi,0,0)}(t)\|_2^2\\
&+\frac14\|u_{(6,6,6)}(t)-u_{(2\pi,2\pi,2\pi)}(t)\|_2^2.
\end{aligned}
```

u_{(6,6,6)}Use in source · 1 passage

Search aliases: u_(6,6,6)

Link to this entry ·

Use in source · finite_core.tex, line 945
f_h(t)-f(t) _L^2^2 = · frac14 u_(6,0,0)(t)-u_(2π,0,0)(t) _2^2 · +frac14 u_(6,6,6)(t)-u_(2π,2π,2π)(t) _2^2.
Exact mathematical passage
```
\begin{aligned}
\|f_h(t)-f(t)\|_{L^2}^2
={}&\frac14\|u_{(6,0,0)}(t)-u_{(2\pi,0,0)}(t)\|_2^2\\
&+\frac14\|u_{(6,6,6)}(t)-u_{(2\pi,2\pi,2\pi)}(t)\|_2^2.
\end{aligned}
```

u_*Use in source · 1 passage

Search aliases: u_*

Link to this entry ·

Use in source · spectral.tex, line 489
For a nondegenerate visible spin direction with normalized low-space direction u_*, the visible weight of its normalized full eigenstate is [1-σ'(E)]^-1. If k=m_0=0, the spin eigenspace is degenerate, but the same normalization relation holds for each chosen unit vector in that eigenspace.
Exact mathematical passage
```
u_*
```

U_{-1}Use in source · 3 passages

Search aliases: U_-1

Link to this entry ·

Use in source · spectral.tex, line 533
dots,(M-1)p} and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).
Exact mathematical passage
```
U_{-1}=0
```

U_0Use in source · 1 passage

Search aliases: U_0

Link to this entry ·

Use in source · spectral.tex, line 533
and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).
Exact mathematical passage
```
U_0=1
```

u_0Use in source · 4 passages

Search aliases: u_0

Link to this entry ·

Use in source · spectral.tex, line 279
[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber
Exact mathematical passage
```
(u_0,v_0)
```

U_1Use in source · 1 passage

Search aliases: U_1

Link to this entry ·

Use in source · spectral.tex, line 610
The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation retains every other site. It does not retain the average of each neighboring pair.
Exact mathematical passage
```
U_1(x/2)=x
```

U_{2,h}Use in source · 3 passages

Search aliases: U_2,h

Link to this entry ·

Use in source · finite.tex, line 19
The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into one phase coordinate.
Exact mathematical passage
```
 U_{2,h}=(\Z/2^{h+2}\Z)^\times.
```

u_AUse in source · 3 passages

Search aliases: u_A

Link to this entry ·

Use in source · finite_core.tex, line 876
u_A(t)= cos(sqrt37t)-isin(sqrt37t)/sqrt37 0 0 -6isin(sqrt37t)/sqrt37 ,
Exact mathematical passage
```
u_A(t)=
\begin{pmatrix}
\cos(\sqrt{37}t)-i\sin(\sqrt{37}t)/\sqrt{37}\\
0\\
0\\
-6i\sin(\sqrt{37}t)/\sqrt{37}
\end{pmatrix},
```

u_aUse in source · 6 passages

Search aliases: u_a

Link to this entry ·

Use in source · core.tex, line 494
For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).
Exact mathematical passage
```
u_a=p_a/\pi_a
```

U_{AB}Use in source · 1 passage

Search aliases: U_AB

Link to this entry ·

Use in source · constraints.tex, line 147
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
 \Omega_F=c^AC_A-\tfrac12c^Ac^BU_{AB}{}^C\varpi_C
             +\Omega_F^{(\ge3)}.
 \label{c:constraints:bfv-charge}
```

U_{a\to b}Use in source · 1 passage

Search aliases: U_a→ b to

Link to this entry ·

Use in source · core.tex, line 203
The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and parallel transport exp(-∫_eω). Its closed-loop rapidity is the negative of the rapidity of W_square defined here. The fixed-frame curvature identity [so…
Exact mathematical passage
```
 G(t,x)=e^{\chi(t,x)K},\qquad
 U_{a\to b}'=G(b)U_{a\to b}G(a)^{-1}.
```

U_{a\to b}'Use in source · 1 passage

Search aliases: U_a→ b' to

Link to this entry ·

Use in source · core.tex, line 203
The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and parallel transport exp(-∫_eω). Its closed-loop rapidity is the negative of the rapidity of W_square defined here. The fixed-frame curvature identity [so…
Exact mathematical passage
```
 G(t,x)=e^{\chi(t,x)K},\qquad
 U_{a\to b}'=G(b)U_{a\to b}G(a)^{-1}.
```

u_bUse in source · 5 passages

Search aliases: u_b

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Use in source · core.tex, line 495
For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).
Exact mathematical passage
```
F_{b\leftarrow a}=c_{ba}(u_a-u_b)
```

\mathcal U_{\mathrm{CRT}}Use in source · 3 passages

Search aliases: U_CRT mathcal mathrm

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Use in source · finite.tex, line 126
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 \mathcal U_{\mathrm{CRT}}:
 \ell^2\bigl((\Z/M\Z)^\times\bigr)
 \longrightarrow
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^{h_2}\Z)
 \otimes\bigotimes_{p\in\mathcal P_{\mathrm o}}
 \left(\C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^{h_p}\Z)\right).
 \label{a-finite-crt}
```

\mathcal U_{\mathrm{CRT}}^*Use in source · 1 passage

Search aliases: U_CRT^* mathcal mathrm

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Use in source · finite.tex, line 149
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
 \left[F_2\otimes F_{2^{h_2}}\otimes
 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
 \right]\mathcal U_{\mathrm{CRT}}.
 \label{a-finite-unit-fourier}
```

u_DUse in source · 3 passages

Search aliases: u_D

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Use in source · finite_core.tex, line 886
u_D(t)= cos(sqrt109t)-isin(sqrt109t)/sqrt109 0 -6isin(sqrt109t)/sqrt109 6(1-i)sin(sqrt109t)/sqrt109 .
Exact mathematical passage
```
u_D(t)=
\begin{pmatrix}
\cos(\sqrt{109}t)-i\sin(\sqrt{109}t)/\sqrt{109}\\
0\\
-6i\sin(\sqrt{109}t)/\sqrt{109}\\
6(1-i)\sin(\sqrt{109}t)/\sqrt{109}
\end{pmatrix}.
```

U_eUse in source · 1 passage

Search aliases: U_e

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Use in source · core.tex, line 165
We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt along the two horizontal endpoints and ϕ_xx along the two vertical endpoints, with the boundary orientation from dtwedge dx. This is the rectangular funda…
Exact mathematical passage
```
U_e=\exp(+\int_e\omega)
```

U_HUse in source · 11 passages

Search aliases: U_H

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Use in source · finite_core.tex, line 299
U_H(t)=e^-itH=∑_k=0^∞frac(-itH)^kk!.
Exact mathematical passage
```
U_H(t)=e^{-itH}=\sum_{k=0}^{\infty}\frac{(-itH)^k}{k!}.
```

U_{H(1,r)}Use in source · 1 passage

Search aliases: U_H(1,r)

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Use in source · finite_core.tex, line 863
u_r(t)=U_H(1,r)(t)e_0= c_r(t)-is_r(t) 0 -ir_zs_r(t) (r_y-ir_x)s_r(t) .
Exact mathematical passage
```
u_r(t)=U_{H(1,r)}(t)e_0=
\begin{pmatrix}
c_r(t)-is_r(t)\\
0\\
-ir_zs_r(t)\\
(r_y-ir_x)s_r(t)
\end{pmatrix}.
```

U_{h,S}Use in source · 2 passages

Search aliases: U_h,S

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Use in source · finite_core.tex, line 458
Both remain in F_S. Their operators U_S(t),U_h,S(t) have the finite block diagonal coefficient actions with blocks U_H_n(t),U_H_h,n(t).
Exact mathematical passage
```
U_S(t),U_{h,S}(t)
```

U_{H_{h,n}}Use in source · 9 passages

Search aliases: U_H_h,n

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Use in source · finite_core.tex, line 453
f(t,x)=∑_n∈ Se_n(x)U_H_n(t)a_n, f_h(t,x)=∑_n∈ Se_n(x)U_H_h,n(t)a_n.
Exact mathematical passage
```
f(t,x)=\sum_{n\in S}e_n(x)U_{H_n}(t)a_n,\qquad
f_h(t,x)=\sum_{n\in S}e_n(x)U_{H_{h,n}}(t)a_n.
```

U_{H_n}Use in source · 7 passages

Search aliases: U_H_n

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Use in source · finite_core.tex, line 453
f(t,x)=∑_n∈ Se_n(x)U_H_n(t)a_n, f_h(t,x)=∑_n∈ Se_n(x)U_H_h,n(t)a_n.
Exact mathematical passage
```
f(t,x)=\sum_{n\in S}e_n(x)U_{H_n}(t)a_n,\qquad
f_h(t,x)=\sum_{n\in S}e_n(x)U_{H_{h,n}}(t)a_n.
```

u_iUse in source · 10 passages

Search aliases: u_i

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Use in source · refinement.tex, line 179
The straight compatibility contract. For each direction, canonical fine links can be defined by v_i(p)=r(σ_i p) γ_i(p) r(p)^-1, where γ_i(p)=1 on a noncarrying half step and γ_i(p)=u_i(s) on a carrying half step. Cancellation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument…
Exact mathematical passage
```
\gamma_i(p)=u_i(s)
```

u_{i+1}Use in source · 3 passages

Search aliases: u_i+1

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Use in source · core.tex, line 250
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 S_h[u]=\sum_{i=0}^{M-1}\frac{(u_{i+1}-u_i)^2}{2Gh}
 +4\pi h\sum_{i=1}^{M-1}r_i\rho_i u_i.
 \label{c:core:poisson-action}
```

u_{i-1}Use in source · 2 passages

Search aliases: u_i-1

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Use in source · core.tex, line 260
[Unique finite radial solution] With u_0,u_M fixed, S_h has a unique minimizer. Its interior values satisfy fracu_i+1-2u_i+u_i-1h^2=4π G r_iρ_i.
Exact mathematical passage
```
 \frac{u_{i+1}-2u_i+u_{i-1}}{h^2}=4\pi G r_i\rho_i.
```

U_jUse in source · 10 passages

Search aliases: U_j

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Use in source · core.tex, line 196
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,
Exact mathematical passage
```
U_j=e^{-ihA_j}
```

u_jUse in source · 4 passages

Search aliases: u_j

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Use in source · limit.tex, line 292
If z is outside the closed energy set, the sequence (ϵ_m-z)^-1 is bounded and defines an inverse for H_ϵ-z. Its image lies in the domain because ϵ_m/(ϵ_m-z)=1+z/(ϵ_m-z) is bounded as well. This proves that such z is in the resolvent set. An energy ϵ_m is an eigenvalue, while a limit point z has unit level vectors u_j with norm(H_ϵ-z)u_j_2→0. A bounded inverse would contradict this limit. Hence the closed energy set is exactly the spectrum.
Exact mathematical passage
```
u_j
```

U_j'Use in source · 1 passage

Search aliases: U_j'

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Use in source · orientation.tex, line 27
Linked shifts · A coefficient U_j in U_jψ_j+1 transports from j+1 back to j. Thus U_j'=g_jU_jg_j+1^-1. An outward link uses the opposite endpoint convention.
Exact mathematical passage
```
U_j'=g_jU_jg_{j+1}^{-1}
```

u_{j+1}Use in source · 2 passages

Search aliases: u_j+1

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Use in source · strong_fields.tex, line 193
On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,Λ,P_Λ,P_R,Q), rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining R, …
Exact mathematical passage
```
 (D_4u)_j=\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.
```

u_{j+2}Use in source · 2 passages

Search aliases: u_j+2

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Use in source · strong_fields.tex, line 193
On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,Λ,P_Λ,P_R,Q), rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining R, …
Exact mathematical passage
```
 (D_4u)_j=\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.
```

U_{j-1}Use in source · 1 passage

Search aliases: U_j-1

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Use in source · core.tex, line 242
Differentiate ρ_j and use dotρ_j=2Re(ψ_j^*dotψ_j). The mass term contributes zero, since ψ_j^*βψ_j is real. The forward hopping term contributes -J_j/h. The backward scalar is the conjugate of ψ_j-1^*α U_j-1ψ_j, using α=α^* and the scalar nature of U. Its contribution is J_j-1/h. This proves continuity. Periodic summation gives dotbarρ=0. Finally, dot G_j =fracdot E_j-dot E_j-1h+dotρ_j-dotbarρ =fracJ_j-J_j-1h-fracJ_j-J_j-1h=0.
Exact mathematical passage
```
\psi_{j-1}^*\alpha U_{j-1}\psi_j
```

u_{j-1}Use in source · 2 passages

Search aliases: u_j-1

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Use in source · strong_fields.tex, line 193
On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,Λ,P_Λ,P_R,Q), rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining R, …
Exact mathematical passage
```
 (D_4u)_j=\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.
```

U_{j-1}^*Use in source · 1 passage

Search aliases: U_j-1^*

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Use in source · core.tex, line 211
sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1, dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1). For example, if z_j=ψ_j^*α U_jψ_j+1, then ∂_A_jImz_j =Im(-ihz_j)=-hRez_j. This fixes the positive sign of dot E_j=J_j for the conventions in the act…
Exact mathematical passage
```
 \dot\psi_j&=-im\beta\psi_j
 -\frac1{2h}\alpha U_j\psi_{j+1}
 +\frac1{2h}U_{j-1}^*\alpha\psi_{j-1},\label{g:matter-ode}\\
 \dot A_j&=E_j,
 &\dot E_j&=J_j,
 &J_j&=\operatorname{Re}(\psi_j^*\alpha U_j\psi_{j+1}).
 \label{g:field-ode}
```

U_{j-1}^nUse in source · 1 passage

Search aliases: U_j-1^n

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Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

u_{j-2}Use in source · 2 passages

Search aliases: u_j-2

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Use in source · strong_fields.tex, line 193
On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,Λ,P_Λ,P_R,Q), rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining R, …
Exact mathematical passage
```
 (D_4u)_j=\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.
```

U_j^nUse in source · 3 passages

Search aliases: U_j^n

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Use in source · core.tex, line 65
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,
Exact mathematical passage
```
U_j^n\in U(1)
```

U_LUse in source · 2 passages

Search aliases: U_L

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Use in source · constraints.tex, line 774
For an embedding I_K→ L and a level-L step U_L, the adaptive map is U_L∘ I_K→ L. The level-L step preserves the invariants of its embedded initial state. To preserve the intended old data, the embedding must also satisfy the appropriate pullback relations for the Hamiltonian, constraints, gauge conditions, and symplectic form. A generic zero padding or a generic change of gauge does not prove all those relations.
Exact mathematical passage
```
U_L
```

U_MUse in source · 1 passage

Search aliases: U_M

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Use in source · finite.tex, line 149
We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit group, rather than the additive Fourier transform on all residues modulo M.
Exact mathematical passage
```
 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
 \left[F_2\otimes F_{2^{h_2}}\otimes
 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
 \right]\mathcal U_{\mathrm{CRT}}.
 \label{a-finite-unit-fourier}
```

u_MUse in source · 1 passage

Search aliases: u_M

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Use in source · core.tex, line 258
[Unique finite radial solution] With u_0,u_M fixed, S_h has a unique minimizer. Its interior values satisfy fracu_i+1-2u_i+u_i-1h^2=4π G r_iρ_i.
Exact mathematical passage
```
u_0,u_M
```

u_mUse in source · 2 passages

Search aliases: u_m

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Use in source · limit.tex, line 431
[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For q=1/8, the state norm tail decays faster than the graph norm tail. For q=1/2, the normalized state still has a convergent L^2 cylinder approximation a…
Exact mathematical passage
```
u_m\in\operatorname{Ran}\mathsf F_m
```

U_{m-1}Use in source · 3 passages

Search aliases: U_m-1

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Use in source · spectral.tex, line 709
[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R_n(…
Exact mathematical passage
```
 R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\nonumber\\
 U_{mn-1}(t)&=U_{n-1}(T_m(t))\,U_{m-1}(t)
 \label{a-spectral-polynomial-semigroup}
```

U_M^{\mathrm{emb}}Use in source · 5 passages

Search aliases: U_M^emb mathrm

Link to this entry ·

Use in source · limit.tex, line 346
Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.
Exact mathematical passage
```
 U_M^{\mathrm{emb}}(t)
 =\sum_{m=0}^{M}e^{-it\epsilon_m}\mathsf F_m
 =U(t)\mathsf E_M.
 \label{a-limit-embedded-propagator}
```

U_{mn-1}Use in source · 4 passages

Search aliases: U_mn-1

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Use in source · spectral.tex, line 709
[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R_n(…
Exact mathematical passage
```
 R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\nonumber\\
 U_{mn-1}(t)&=U_{n-1}(T_m(t))\,U_{m-1}(t)
 \label{a-spectral-polynomial-semigroup}
```

U_{n+1}Use in source · 1 passage

Search aliases: U_n+1

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Use in source · spectral.tex, line 534
liminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).
Exact mathematical passage
```
U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)
```

U_{n-2}Use in source · 3 passages

Search aliases: U_n-2

Link to this entry ·

Use in source · spectral.tex, line 655
1 · 0. Its determinant is one and induction gives A(x)^n= U_n(x/2) · -U_n-1(x/2) U_n-1(x/2) · -U_n-2(x/2) , n≥1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j+χ_j-1=0 for χ_j=ψ_pj. It explains the n…
Exact mathematical passage
```
 A(x)^n=\begin{pmatrix}
 U_n(x/2)&-U_{n-1}(x/2)\\
 U_{n-1}(x/2)&-U_{n-2}(x/2)
 \end{pmatrix},\quad n\geq1.
 \label{a-spectral-transfer-power}
```

U_pUse in source · 1 passage

Search aliases: U_p

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Use in source · spectral.tex, line 597
-1 by Lemma~[source reference]; the two contributions add. The path joining two adjacent retained sites contributes off-diagonal self-energy κ/U_p-1. Thus S_p(z) =κ(x-frac2U_p-2(x/2)U_p-1(x/2))I -fracκU_p-1(x/2)(T+T^*). The identity xU_p-1(x/2)-2U_p-2(x/2)=2T_p(x/2) follows from the three-term recurrence and U_p-U_p-2=2T_p. For the latter identity, substituting x/2=cosθ gives [sin((p+1)θ)-sin((p-1)θ)]/sinθ =2cos(pθ), and polynomial equality extends it to all x. This proves [source reference] inc…
Exact mathematical passage
```
U_p-U_{p-2}=2T_p
```

U_{p,h}Use in source · 6 passages

Search aliases: U_p,h

Link to this entry ·

Use in source · finite.tex, line 15
The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into one phase coordinate.
Exact mathematical passage
```
 U_{p,h}=(\Z/p^{h+1}\Z)^\times.
```

U_{p,h+1,N}Use in source · 2 passages

Search aliases: U_p,h+1,N

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Use in source · limit.tex, line 942
ds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its internal norm. The subsequent reversal shift sends these amplitudes to new arcs at the children, outside the old arc range. Therefore, for a unit internal state χ, norm(I-JJ^*)U_p,h+1,NJ(e_j⊗χ) =frac2sqrt p3+p>0. The state JU_p,h,N(e_j⊗χ) has zero amplitude there, so U_p,h+1,NJ≠ JU_p,h,N.
Exact mathematical passage
```
 \norm{(I-JJ^*)U_{p,h+1,N}J(e_j\otimes\chi)}
 =\frac{2\sqrt p}{3+p}>0.
 \label{a-limit-walk-refinement-leakage}
```

U_{p,h,N}Use in source · 4 passages

Search aliases: U_p,h,N

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Use in source · limit.tex, line 844
nd the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph edges, with no continuum limit assumed.
Exact mathematical passage
```
 U_{p,h,N}=SC
 \label{a-limit-walk-unitary}
```

U_{p,h,N}^*Use in source · 1 passage

Search aliases: U_p,h,N^*

Link to this entry ·

Use in source · limit.tex, line 848
|s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph edges, with no continuum limit assumed.
Exact mathematical passage
```
U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I
```

\mathcal U_{p,h_p}Use in source · 1 passage

Search aliases: U_p,h_p mathcal

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Use in source · limit.tex, line 543
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections an…
Exact mathematical passage
```
I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathcal U_{p,h_p}
```

\mathcal U_{p,M}Use in source · 2 passages

Search aliases: U_p,M mathcal

Link to this entry ·

Use in source · limit.tex, line 134
The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projection in V_M onto V_m. This is the exact relation between the finite matrices and the common infinite Hilbert space.
Exact mathematical passage
```
 \mathsf E_m\big|_{\mathcal V_M}
 =\mathcal U_{p,M}V_{p,m\to M}V_{p,m\to M}^*
 \mathcal U_{p,M}^*.
 \label{a-limit-finite-projection}
```

\mathcal U_{p,m}Use in source · 5 passages

Search aliases: U_p,m mathcal

Link to this entry ·

Use in source · limit.tex, line 25
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
 \qquad \mathcal V_m=\operatorname{span}
 \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
 \label{a-limit-cylinder-embedding}
```

\mathcal U_{p,m+1}Use in source · 1 passage

Search aliases: U_p,m+1 mathcal

Link to this entry ·

Use in source · limit.tex, line 51
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
 \mathcal U_{p,m+1}V_{p,m}=\mathcal U_{p,m}.
 \label{a-limit-common-embedding}
```

\mathcal U_{p,M}^*Use in source · 1 passage

Search aliases: U_p,M^* mathcal

Link to this entry ·

Use in source · limit.tex, line 134
The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projection in V_M onto V_m. This is the exact relation between the finite matrices and the common infinite Hilbert space.
Exact mathematical passage
```
 \mathsf E_m\big|_{\mathcal V_M}
 =\mathcal U_{p,M}V_{p,m\to M}V_{p,m\to M}^*
 \mathcal U_{p,M}^*.
 \label{a-limit-finite-projection}
```

U_{p-1}Use in source · 10 passages

Search aliases: U_p-1

Link to this entry ·

Use in source · spectral.tex, line 567
[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber
Exact mathematical passage
```
U_{p-1}(x/2)\ne0
```

U_{p-2}Use in source · 4 passages

Search aliases: U_p-2

Link to this entry ·

Use in source · spectral.tex, line 586
, μ_p(z)=z-fracκ R_p(x)U_p-1(x/2). The eliminated sites split into M disjoint Dirichlet paths of length p-1. Hence their resolvent exists exactly when U_p-1(x/2)≠0. A retained site touches one endpoint on each of its two neighboring paths. Each path contributes the onsite self-energy κ U_p-2/U_p-1 by Lemma~[source reference]; the two contributions add. The path joining two adjacent retained sites contributes off-diagonal self-energy κ/U_p-1. Thus S_p(z) =κ(x-frac2U_p-2(x/2)U_p-1(x/2))I -fracκU_p…
Exact mathematical passage
```
\kappa U_{p-2}/U_{p-1}
```

u_qUse in source · 1 passage

Search aliases: u_q

Link to this entry ·

Use in source · finite_core.tex, line 937
u_q(t)-u_r(t) _2^2 =(c_q-c_r)^2+(s_q-s_r)^2 +∑_j=x,y,z(q_js_q-r_js_r)^2.
Exact mathematical passage
```
\|u_q(t)-u_r(t)\|_2^2
=(c_q-c_r)^2+(s_q-s_r)^2
+\sum_{j=x,y,z}(q_js_q-r_js_r)^2.
```

u_rUse in source · 3 passages

Search aliases: u_r

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Use in source · finite_core.tex, line 863
u_r(t)=U_H(1,r)(t)e_0= c_r(t)-is_r(t) 0 -ir_zs_r(t) (r_y-ir_x)s_r(t) .
Exact mathematical passage
```
u_r(t)=U_{H(1,r)}(t)e_0=
\begin{pmatrix}
c_r(t)-is_r(t)\\
0\\
-ir_zs_r(t)\\
(r_y-ir_x)s_r(t)
\end{pmatrix}.
```

U_{rs}Use in source · 1 passage

Search aliases: U_rs

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Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

U_{rs}^*Use in source · 1 passage

Search aliases: U_rs^*

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Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

U_{rt}Use in source · 1 passage

Search aliases: U_rt

Link to this entry ·

Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

U_SUse in source · 2 passages

Search aliases: U_S

Link to this entry ·

Use in source · finite_core.tex, line 458
Both remain in F_S. Their operators U_S(t),U_h,S(t) have the finite block diagonal coefficient actions with blocks U_H_n(t),U_H_h,n(t).
Exact mathematical passage
```
U_S(t),U_{h,S}(t)
```

U_{ts}^*Use in source · 1 passage

Search aliases: U_ts^*

Link to this entry ·

Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

u_xUse in source · 1 passage

Search aliases: u_x

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Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
u_x,u_y,u_z
```

U_{x+\widehat\mu,\nu}Use in source · 1 passage

Search aliases: U_x+widehatμ,ν widehat mu nu

Link to this entry ·

Use in source · core.tex, line 148
On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x^-1. For U(1) the plaquette itself is invariant. For a nonco…
Exact mathematical passage
```
\label{g:plaquette}
 W_{x,\mu\nu}
 =U_{x,\mu}U_{x+\widehat\mu,\nu}
 U_{x+\widehat\nu,\mu}^{-1}U_{x,\nu}^{-1}.
```

U_{x+\widehat\nu,\mu}^{-1}Use in source · 1 passage

Search aliases: U_x+widehatν,μ^-1 widehat nu mu

Link to this entry ·

Use in source · core.tex, line 148
On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x^-1. For U(1) the plaquette itself is invariant. For a nonco…
Exact mathematical passage
```
\label{g:plaquette}
 W_{x,\mu\nu}
 =U_{x,\mu}U_{x+\widehat\mu,\nu}
 U_{x+\widehat\nu,\mu}^{-1}U_{x,\nu}^{-1}.
```

U_{x,\mu}Use in source · 7 passages

Search aliases: U_x,μ mu

Link to this entry ·

Use in source · core.tex, line 6
Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from ri…
Exact mathematical passage
```
\label{g:backward-link}
 U_{x,\mu}:\HH_{x+\widehat\mu}\longrightarrow\HH_x,
 \qquad U_{x,\mu}^*U_{x,\mu}=U_{x,\mu}U_{x,\mu}^*=I.
```

u_{x,\mu}Use in source · 1 passage

Search aliases: u_x,μ mu

Link to this entry ·

Use in source · core.tex, line 19
The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.
Exact mathematical passage
```
\label{g:forward-link}
 u_{x,\mu}=U_{x,\mu}^{-1}:\HH_x\longrightarrow\HH_{x+\widehat\mu},
 \qquad u_{x,\mu}^g=g_{x+\widehat\mu}u_{x,\mu}g_x^{-1}.
```

U_{x,\mu}^*Use in source · 1 passage

Search aliases: U_x,μ^* mu

Link to this entry ·

Use in source · core.tex, line 6
Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from ri…
Exact mathematical passage
```
\label{g:backward-link}
 U_{x,\mu}:\HH_{x+\widehat\mu}\longrightarrow\HH_x,
 \qquad U_{x,\mu}^*U_{x,\mu}=U_{x,\mu}U_{x,\mu}^*=I.
```

U_{x,\mu}^{-1}Use in source · 1 passage

Search aliases: U_x,μ^-1 mu

Link to this entry ·

Use in source · core.tex, line 19
The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.
Exact mathematical passage
```
\label{g:forward-link}
 u_{x,\mu}=U_{x,\mu}^{-1}:\HH_x\longrightarrow\HH_{x+\widehat\mu},
 \qquad u_{x,\mu}^g=g_{x+\widehat\mu}u_{x,\mu}g_x^{-1}.
```

U_{x,\mu}^gUse in source · 1 passage

Search aliases: U_x,μ^g mu

Link to this entry ·

Use in source · core.tex, line 11
ing on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the…
Exact mathematical passage
```
\label{g:link-transform}
 U_{x,\mu}^g=g_xU_{x,\mu}g_{x+\widehat\mu}^{-1},
 \qquad U_{x,\mu}^g\psi_{x+\widehat\mu}^g
 =g_xU_{x,\mu}\psi_{x+\widehat\mu}.
```

u_{x,\mu}^gUse in source · 1 passage

Search aliases: u_x,μ^g mu

Link to this entry ·

Use in source · core.tex, line 19
The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.
Exact mathematical passage
```
\label{g:forward-link}
 u_{x,\mu}=U_{x,\mu}^{-1}:\HH_x\longrightarrow\HH_{x+\widehat\mu},
 \qquad u_{x,\mu}^g=g_{x+\widehat\mu}u_{x,\mu}g_x^{-1}.
```

U_{x,\nu}^{-1}Use in source · 1 passage

Search aliases: U_x,ν^-1 nu

Link to this entry ·

Use in source · core.tex, line 148
On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x^-1. For U(1) the plaquette itself is invariant. For a nonco…
Exact mathematical passage
```
\label{g:plaquette}
 W_{x,\mu\nu}
 =U_{x,\mu}U_{x+\widehat\mu,\nu}
 U_{x+\widehat\nu,\mu}^{-1}U_{x,\nu}^{-1}.
```

U_{x-\widehat\mu,\mu}^{-1}Use in source · 2 passages

Search aliases: U_x-widehatμ,μ^-1 widehat mu mu

Link to this entry ·

Use in source · core.tex, line 27
[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
 (D^+_{\mu,U}\psi)_x
 &=\frac{U_{x,\mu}\psi_{x+\widehat\mu}-\psi_x}{h},\label{g:forward-difference}\\
 (\Delta_U\psi)_x
 &=\frac1{h^2}\sum_\mu\left(
 U_{x,\mu}\psi_{x+\widehat\mu}
 +U_{x-\widehat\mu,\mu}^{-1}\psi_{x-\widehat\mu}-2\psi_x\right).
 \label{g:laplacian}
```

U_{x-\widehat\mu,\mu}^gUse in source · 1 passage

Search aliases: U_x-widehatμ,μ^g widehat mu mu

Link to this entry ·

Use in source · core.tex, line 42
For the positive edge, substitute [source reference]. The two gauge factors at x+widehatμ cancel, leaving g_x on the left. For the negative edge, (U_x-widehatμ,μ^g)^-1g_x-widehatμ =g_xU_x-widehatμ,μ^-1. Every term in either difference therefore has the common factor g_x.
Exact mathematical passage
```
 (U_{x-\widehat\mu,\mu}^g)^{-1}g_{x-\widehat\mu}
 =g_xU_{x-\widehat\mu,\mu}^{-1}.
```

u_yUse in source · 1 passage

Search aliases: u_y

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Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
u_x,u_y,u_z
```

u_zUse in source · 1 passage

Search aliases: u_z

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Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
u_x,u_y,u_z
```

\mathcal VUse in source · 23 passages

Search aliases: V mathcal

Link to this entry ·

Use in source · limit.tex, line 25
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
 \qquad \mathcal V_m=\operatorname{span}
 \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
 \label{a-limit-cylinder-embedding}
```

v'Use in source · 3 passages

Search aliases: v'

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Use in source · strong_fields.tex, line 423
partial_t(m'-r^2 E)=0. A radial mass constraint imposed initially propagates under this continuum system. Expanding [source reference] gives dotΦ=(σ A)' and dotΠ=(σ B)'+2σ B/r. Differentiate E and subtract the radial derivative of the flux. Terms involving Π', Φ', and v' cancel, leaving ∂_t(r^2 E)-∂_r(r^2 J) =r^2[σ'AB+ΠΦdot v]. Since m=rv^2/2, the flux equation implies dot v=rσ AB/v. The remaining expression is r^2AB[σ'+fracσ rΠΦv]=0 by [source reference]. This establishes the identity without a…
Exact mathematical passage
```
v'
```

V^*Use in source · 18 passages

Search aliases: V^*

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Use in source · finite.tex, line 513
[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,
Exact mathematical passage
```
P=VV^*
```

v^{-1}Use in source · 2 passages

Search aliases: v^-1

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Use in source · refinement.tex, line 242
· +|v(x)^-1-u(x)^-1|^2 normP_-ψ(τ x)^2]. There is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies |v^-1-u^-1|=|v-u|. Bound every coefficient by δ^2, reindex the two sums through τ, and use normP_+η^2+normP_-η^2=normη^2. The result is at most δ^2normψ^2. Taking square roots and then the operator supremum proves the first inequality. Multiplication by u(x)^-1 proves the equality of scalar maxima. Finally [sourc…
Exact mathematical passage
```
|v^{-1}-u^{-1}|=|v-u|
```

v^{-1/2}Use in source · 1 passage

Search aliases: v^-1/2

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Use in source · refinement.tex, line 473
Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2 =∑_a∫_C_av^-1normψ_a^2 =∑_anormψ_a^2. Equivalently, physical cell values carry the weighted norm v∑_anormq_a^2. The adjoint averaging map yields the …
Exact mathematical passage
```
 (J\psi)(x)=v^{-1/2}\psi_a\quad(x\in C_a),
```

V^2Use in source · 1 passage

Search aliases: V^2

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Use in source · core.tex, line 476
Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber dot U · =γΔ_hP+e^2P[V^2-γ(D_hQ)^2],nonumber dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity va…
Exact mathematical passage
```
 \dot P&=U,&\dot Q&=V,\nonumber\\
 \dot U&=\gamma\Delta_hP+e^{2P}[V^2-\gamma(D_hQ)^2],\nonumber\\
 \dot V&=\gamma\Delta_hQ-2[UV-\gamma(D_hP)(D_hQ)].
 \label{c:core:gowdy-raw}
```

v^2Use in source · 4 passages

Search aliases: v^2

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Use in source · strong_fields.tex, line 357
Impose R=r and Λ=1, and set N=σ>0, N^r=σ v, P_Λ=rv, P_φ=r^2Π, Φ=φ'. Here Π is the scalar momentum per areal area, not the reconstruction map Π_h. Define m=frac r2v^2, v=sqrtfrac2mr. The inward branch gives ds^2=-σ^2dt^2+(dr+σ vdt)^2+r^2dΩ^2. The following derivation applies on smooth patches with r>0, σ>0, and v>0. These hypotheses keep every division legitimate.
Exact mathematical passage
```
 m=\frac r2v^2,\qquad v=\sqrt{\frac{2m}{r}}.
```

V^\ellUse in source · 1 passage

Search aliases: V^ℓ ell

Link to this entry ·

Use in source · spectral.tex, line 1189
On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
 Q_j^{(L)}=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}V^\ell
          =\sum_{n\equiv j\;({\rm mod}\;L)}|n\rangle\langle n|.
 \label{a-spectral-number-projectors}
```

\mathcal V_0Use in source · 2 passages

Search aliases: V_0 mathcal

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Use in source · limit.tex, line 664
There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0. The initial space V_0 and the successive difference spaces therefore have stable self-adjoint finite blocks. Their orthogonal direct sum with the ma…
Exact mathematical passage
```
\mathcal V_0\subset\mathcal V_1
\subset\cdots
```

v_0Use in source · 8 passages

Search aliases: v_0

Link to this entry ·

Use in source · limit.tex, line 882
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
v_0
```

\mathcal V_1Use in source · 1 passage

Search aliases: V_1 mathcal

Link to this entry ·

Use in source · limit.tex, line 664
There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0. The initial space V_0 and the successive difference spaces therefore have stable self-adjoint finite blocks. Their orthogonal direct sum with the ma…
Exact mathematical passage
```
\mathcal V_0\subset\mathcal V_1
\subset\cdots
```

v_1Use in source · 1 passage

Search aliases: v_1

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Use in source · finite_core.tex, line 112
In particular, α_yv=(-iv_3,iv_2,-iv_1,iv_0); this fixes a sign that cannot be changed silently in a worked example. The matrices act on column spinors. Grouping the first two and last two components gives their block form without any coordinate permutation:
Exact mathematical passage
```
\alpha_yv=(-iv_3,iv_2,-iv_1,iv_0)
```

v_2Use in source · 2 passages

Search aliases: v_2

Link to this entry ·

Use in source · finite.tex, line 79
For the dyadic case, successive squaring gives v_2(5^2^k-1)=k+2. Indeed 5-1 has valuation two, and for every such power the factor 5^2^k+1 has valuation one. Thus 5 has order 2^h modulo 2^h+2. Its powers are exactly the 2^h units congruent to one modulo four. Multiplication by -1 supplies the units congruent to three modulo four. These two classes are disjoint, giving the claimed unique coordinates.
Exact mathematical passage
```
 v_2\bigl(5^{2^k}-1\bigr)=k+2.
```

v_3Use in source · 1 passage

Search aliases: v_3

Link to this entry ·

Use in source · finite_core.tex, line 112
In particular, α_yv=(-iv_3,iv_2,-iv_1,iv_0); this fixes a sign that cannot be changed silently in a worked example. The matrices act on column spinors. Grouping the first two and last two components gives their block form without any coordinate permutation:
Exact mathematical passage
```
\alpha_yv=(-iv_3,iv_2,-iv_1,iv_0)
```

V_aUse in source · 6 passages

Search aliases: V_a

Link to this entry ·

Use in source · extensions.tex, line 75
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
V_a:\HH\to\KK_a
```

\mathcal V_aUse in source · 2 passages

Search aliases: V_a mathcal

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

V_a^*Use in source · 4 passages

Search aliases: V_a^*

Link to this entry ·

Use in source · extensions.tex, line 76
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
 \sum_aV_a^*V_a=I.
```

V_bUse in source · 1 passage

Search aliases: V_b

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Use in source · extensions.tex, line 90
For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV. This is the exact missing-range term. Setting A=B=Π_a gives innerψ(F_a-F_a^2)ψ =norm(I-R)Π_aVψ^2. Thus the failure of an
Exact mathematical passage
```
X_{ba}^{V}=V_bXV_a^*
```

V_b^*Use in source · 1 passage

Search aliases: V_b^*

Link to this entry ·

Use in source · extensions.tex, line 91
For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV. This is the exact missing-range term. Setting A=B=Π_a gives innerψ(F_a-F_a^2)ψ =norm(I-R)Π_aVψ^2. Thus the failure of an effect to be shar…
Exact mathematical passage
```
 X=\sum_{a,b}V_b^*X_{ba}^{V}V_a.
 \label{eq:foundation-channel-reconstruction}
```

V_hUse in source · 2 passages

Search aliases: V_h

Link to this entry ·

Use in source · core.tex, line 604
Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric tensors.
Exact mathematical passage
```
V_h
```

v_iUse in source · 5 passages

Search aliases: v_i

Link to this entry ·

Use in source · refinement.tex, line 176
The straight compatibility contract. For each direction, canonical fine links can be defined by v_i(p)=r(σ_i p) γ_i(p) r(p)^-1, where γ_i(p)=1 on a noncarrying half step and γ_i(p)=u_i(s) on a carrying half step. Cancellation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument…
Exact mathematical passage
```
 v_i(p)=r(\sigma_i p)\,\gamma_i(p)\,r(p)^{-1},
```

v_{i+1}Use in source · 1 passage

Search aliases: v_i+1

Link to this entry ·

Use in source · core.tex, line 270
Differentiation with respect to u_i gives (2u_i-u_i-1-u_i+1)/(Gh)+4π h r_iρ_i. Setting it to zero gives the equation. For a variation v with v_0=v_M=0, the Hessian quadratic form is frac1Gh∑_i=0^M-1(v_i+1-v_i)^2. Its vanishing forces v to be constant and hence zero. The finite Hessian is positive definite, so the quadratic action has one stationary point and that point is its unique minimizer.
Exact mathematical passage
```
 \frac1{Gh}\sum_{i=0}^{M-1}(v_{i+1}-v_i)^2.
```

V_jUse in source · 1 passage

Search aliases: V_j

Link to this entry ·

Use in source · core.tex, line 484
dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.
Exact mathematical passage
```
 C_h(z)=2h\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].
 \label{c:core:gowdy-momentum}
```

v_jUse in source · 4 passages

Search aliases: v_j

Link to this entry ·

Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

v_{j-1}Use in source · 1 passage

Search aliases: v_j-1

Link to this entry ·

Use in source · spectral.tex, line 969
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
(C_Lv)_j=v_{j-1}
```

\mathcal V_LUse in source · 1 passage

Search aliases: V_L mathcal

Link to this entry ·

Use in source · limit.tex, line 231
The rate is a bound on states in the displayed regularity class. For an arbitrary L^2 state the tail still tends to zero, but no common geometric rate follows. A finite array already in V_L has zero tail once M≥ L, which is a different statement from testing an infinite tail.
Exact mathematical passage
```
\mathcal V_L
```

v_{L-1}Use in source · 1 passage

Search aliases: v_L-1

Link to this entry ·

Use in source · spectral.tex, line 968
Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=f…
Exact mathematical passage
```
(C_Lv)_0=v_{L-1}
```

v_{Lq+j}Use in source · 1 passage

Search aliases: v_Lq+j

Link to this entry ·

Use in source · spectral.tex, line 991
For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.
Exact mathematical passage
```
v_{Lq+j}=M_1^j(M_1^L)^qv_0
```

\mathcal V_MUse in source · 8 passages

Search aliases: V_M mathcal

Link to this entry ·

Use in source · limit.tex, line 120
Given ε>0, choose g∈ V_M with normf-g_2<ε. For m≥ M, the projection fixes g, so normf- E_m f_2 =norm(I- E_m)(f-g)_2≤normf-g_2<ε. This proves strong convergence without exchanging an infinite integral with a limit.
Exact mathematical passage
```
g\in\mathcal V_M
```

\mathcal V_mUse in source · 8 passages

Search aliases: V_m mathcal

Link to this entry ·

Use in source · limit.tex, line 25
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
 \qquad \mathcal V_m=\operatorname{span}
 \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
 \label{a-limit-cylinder-embedding}
```

v_MUse in source · 1 passage

Search aliases: v_M

Link to this entry ·

Use in source · core.tex, line 268
Differentiation with respect to u_i gives (2u_i-u_i-1-u_i+1)/(Gh)+4π h r_iρ_i. Setting it to zero gives the equation. For a variation v with v_0=v_M=0, the Hessian quadratic form is frac1Gh∑_i=0^M-1(v_i+1-v_i)^2. Its vanishing forces v to be constant and hence zero. The finite Hessian is positive definite, so the quadratic action has one stationary point and that point is its unique minimizer.
Exact mathematical passage
```
v_0=v_M=0
```

\mathcal V_{M+1}Use in source · 1 passage

Search aliases: V_M+1 mathcal

Link to this entry ·

Use in source · limit.tex, line 667
There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0. The initial space V_0 and the successive difference spaces therefore have stable self-adjoint finite blocks. Their orthogonal direct sum with the ma…
Exact mathematical passage
```
\mathcal V_M^\perp\cap\mathcal V_{M+1}
```

\mathcal V_M^\perpUse in source · 2 passages

Search aliases: V_M^⊥ mathcal perp

Link to this entry ·

Use in source · limit.tex, line 353
Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.
Exact mathematical passage
```
\mathcal V_M^\perp
```

V_pUse in source · 1 passage

Search aliases: V_p

Link to this entry ·

Use in source · limit.tex, line 565
Refinement in distinct coordinates is path independent because (V_p⊗ I)(I⊗ V_q) =(I⊗ V_q)(V_p⊗ I)=V_p⊗ V_q. The identity on KK_tor leaves the separately retained sign channel untouched by the averaging. This is a finite set of primes. An infinite tensor product, a restricted adelic product, and a model including the Archimedean place each require further choices.
Exact mathematical passage
```
 (V_p\otimes I)(I\otimes V_q)
 =(I\otimes V_q)(V_p\otimes I)=V_p\otimes V_q.
```

v_pUse in source · 2 passages

Search aliases: v_p

Link to this entry ·

Use in source · finite.tex, line 46
For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup generated by 1+p is contained in the kernel of reduction modulo p and has the same cardinality p^h, so it is that entire kernel.
Exact mathematical passage
```
 v_p\bigl((1+p)^{p^k}-1\bigr)=k+1
 \qquad(k\geq0)
```

V_{p,h}Use in source · 7 passages

Search aliases: V_p,h

Link to this entry ·

Use in source · finite.tex, line 374
One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an id…
Exact mathematical passage
```
 (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),
 \quad 0\leq a<p^h,\quad0\leq d<p.
 \label{a-finite-refinement}
```

V_{p,h+w-1}Use in source · 1 passage

Search aliases: V_p,h+w-1

Link to this entry ·

Use in source · finite.tex, line 411
D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 V_{p,h+w-1}\cdots V_{p,h}|a\rangle
 =p^{-w/2}\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}|a,\mathbf d\rangle,
 \label{a-finite-nested-refinement}
```

V_{p,h-1}^*Use in source · 1 passage

Search aliases: V_p,h-1^*

Link to this entry ·

Use in source · finite.tex, line 470
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*, W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] an…
Exact mathematical passage
```
\begin{aligned}
 (\langle0|\otimes I_m)\mathcal W_{p,h}&=V_{p,h-1}^*,\\
 \mathcal W_{p,h}^*
 \bigl[(I_p-|0\rangle\langle0|)\otimes I_m\bigr]
 \mathcal W_{p,h}&=D_h^{(h)}.
\end{aligned}
\label{a-finite-child-detail}
```

V_{p,h}^*Use in source · 2 passages

Search aliases: V_p,h^*

Link to this entry ·

Use in source · finite.tex, line 386
_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an identity on the coarse space. The second is a projector on the fine space, not a fine-sp…
Exact mathematical passage
```
 (V_{p,h}^*g)(a)=p^{-1/2}\sum_{d=0}^{p-1}g(a+d p^h).
 \label{a-finite-refinement-adjoint}
```

V_{p,m}Use in source · 4 passages

Search aliases: V_p,m

Link to this entry ·

Use in source · limit.tex, line 46
The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the function in HH_p.
Exact mathematical passage
```
 V_{p,m}e_a=\frac1{\sqrt p}
 \sum_{d=0}^{p-1}e_{a+d p^m}.
```

V_{p,M-1}Use in source · 1 passage

Search aliases: V_p,M-1

Link to this entry ·

Use in source · limit.tex, line 133
The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projection in V_M onto V_m. This is the exact relation between the finite matrices and the common infinite Hilbert space.
Exact mathematical passage
```
V_{p,m\to M}=V_{p,M-1}\cdots V_{p,m}
```

V_{p,m\to M}Use in source · 2 passages

Search aliases: V_p,m→ M to

Link to this entry ·

Use in source · limit.tex, line 133
The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projection in V_M onto V_m. This is the exact relation between the finite matrices and the common infinite Hilbert space.
Exact mathematical passage
```
V_{p,m\to M}=V_{p,M-1}\cdots V_{p,m}
```

V_{p,m\to M}^*Use in source · 1 passage

Search aliases: V_p,m→ M^* to

Link to this entry ·

Use in source · limit.tex, line 134
The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projection in V_M onto V_m. This is the exact relation between the finite matrices and the common infinite Hilbert space.
Exact mathematical passage
```
 \mathsf E_m\big|_{\mathcal V_M}
 =\mathcal U_{p,M}V_{p,m\to M}V_{p,m\to M}^*
 \mathcal U_{p,M}^*.
 \label{a-limit-finite-projection}
```

V_qUse in source · 1 passage

Search aliases: V_q

Link to this entry ·

Use in source · limit.tex, line 565
Refinement in distinct coordinates is path independent because (V_p⊗ I)(I⊗ V_q) =(I⊗ V_q)(V_p⊗ I)=V_p⊗ V_q. The identity on KK_tor leaves the separately retained sign channel untouched by the averaging. This is a finite set of primes. An infinite tensor product, a restricted adelic product, and a model including the Archimedean place each require further choices.
Exact mathematical passage
```
 (V_p\otimes I)(I\otimes V_q)
 =(I\otimes V_q)(V_p\otimes I)=V_p\otimes V_q.
```

v_qUse in source · 3 passages

Search aliases: v_q

Link to this entry ·

Use in source · spectral.tex, line 129
mathsf H_p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine Fourier index is uniquely k=j+qM. Its…
Exact mathematical passage
```
v_q
```

v_RUse in source · 1 passage

Search aliases: v_R

Link to this entry ·

Use in source · strong_fields.tex, line 126
For evolution, the radius equation gives dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2],
Exact mathematical passage
```
v_R=M/(pR)-p/R^2
```

v_rUse in source · 4 passages

Search aliases: v_r

Link to this entry ·

Use in source · finite_core.tex, line 24
⟨ v,w⟩=∑_r=0^3overlinev_rw_r, v _2^2=∑_r=0^3|v_r|^2.
Exact mathematical passage
```
\langle v,w\rangle=\sum_{r=0}^{3}\overline{v_r}w_r,
\qquad
\|v\|_2^2=\sum_{r=0}^{3}|v_r|^2.
```

v_xUse in source · 1 passage

Search aliases: v_x

Link to this entry ·

Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
v_x,v_y,v_z
```

v_yUse in source · 1 passage

Search aliases: v_y

Link to this entry ·

Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
v_x,v_y,v_z
```

v_zUse in source · 1 passage

Search aliases: v_z

Link to this entry ·

Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
v_x,v_y,v_z
```

\VertUse in source · 1 passage

Search aliases: Vert

Link to this entry ·

Use in source · core.tex, line 500
[Relative-entropy dissipation] For a strictly positive solution of the finite jump balance and a fixed positive reference satisfying [source reference], D(pVertπ) · =∑_ap_alogfracp_aπ_a, fracddtD(pVertπ) · =-frac12∑_b,ac_ba(u_b-u_a)(log u_b-log u_a)≤0. notag
Exact mathematical passage
```
\label{f:dissipation}
 D(p\Vert\pi)&=\sum_ap_a\log\frac{p_a}{\pi_a},\\
 \frac{d}{dt}D(p\Vert\pi)
 &=-\frac12\sum_{b,a}c_{ba}(u_b-u_a)(\log u_b-\log u_a)\le0.
 \notag
```

\operatorname{vol}Use in source · 6 passages

Search aliases: vol operatorname

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\operatorname{vol}_eUse in source · 6 passages

Search aliases: vol_e operatorname

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\mathcal WUse in source · 6 passages

Search aliases: W mathcal

Link to this entry ·

Use in source · finite.tex, line 464
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,
Exact mathematical passage
```
 \mathcal W_{p,h}=(F_p\otimes I_m)\mathcal A^{\mathrm{ad}}_{p,h}.
 \label{a-finite-child-fourier}
```

w'Use in source · 1 passage

Search aliases: w'

Link to this entry ·

Use in source · strong_fields.tex, line 452
dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
\dot w+c\,w'=\text{source}
```

W^{(2)}Use in source · 1 passage

Search aliases: W^(2)

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

\mathcal W^*Use in source · 1 passage

Search aliases: W^* mathcal

Link to this entry ·

Use in source · spectral.tex, line 208
If P_L=ι_Lι_L^*, the full space compression is P_L(z-H)^-1P_L=ι_L S(z)^-1ι_L^*. When a Haar transform W first exposes low and detail channels, the embedding in the original coordinates is ι_L= W^*(|0⟩⊗ I). The inverse on HH_L must be embedded in this way before it is compared with a full space matrix.
Exact mathematical passage
```
\iota_L=\mathcal W^*(|0\rangle\otimes I)
```

W^{1,\infty}Use in source · 2 passages

Search aliases: W^1,∞ infty

Link to this entry ·

Use in source · strong_fields.tex, line 241
The input norm contains a radial derivative. A small pointwise radius error alone therefore does not establish a small error in every reconstructed canonical field. This estimate transfers an already controlled discrete W^1,∞ radius error to the reconstructed state. It does not create that derivative control from an L^2 estimate.
Exact mathematical passage
```
W^{1,\infty}
```

w^2Use in source · 1 passage

Search aliases: w^2

Link to this entry ·

Use in source · extensions.tex, line 267
The constructed orbit is the unique global domain-valued solution with a given domain initial state. If two supplied solutions have the same initial value, their difference w satisfies dot w=-iH_λ w. Symmetry gives fracddtnorm w^2 =2Reinnerw-iH_λ w=0. The initial zero norm remains zero. This uniqueness proof extends to supplied solutions of a symmetric operator, but the proof of existence above uses the explicit real diagonal construction.
Exact mathematical passage
```
 \frac{d}{dt}\norm w^2
 =2\operatorname{Re}\inner{w}{-iH_\lambda w}=0.
```

w_+Use in source · 3 passages

Search aliases: w_+

Link to this entry ·

Use in source · strong_fields.tex, line 443
Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-],
Exact mathematical passage
```
w_+=\Pi+\Phi
```

w_+'Use in source · 1 passage

Search aliases: w_+'

Link to this entry ·

Use in source · strong_fields.tex, line 444
Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
 \dot w_+={}&\sigma(v+1)w_+'
 +[\sigma v'+\sigma'(v+1)]w_+
 +\frac\sigma r[(v+1)w_++(v-1)w_-],\\
 \dot w_-={}&\sigma(v-1)w_-'
 +[\sigma v'+\sigma'(v-1)]w_-
 +\frac\sigma r[(v-1)w_-+(v+1)w_+].
```

w_+^2Use in source · 2 passages

Search aliases: w_+^2

Link to this entry ·

Use in source · strong_fields.tex, line 460
The same channels give the exact algebraic identities E · =frac14[(1+v)w_+^2+(1-v)w_-^2], J · =fracσ4[(1+v)^2w_+^2-(1-v)^2w_-^2]. For 0≤ v<1, E≥(1-v)(w_+^2+w_-^2)/4. This lower bound degenerates at a horizon. For v>1, the outward-channel coefficient in this coordinate mass density is negative. The scalar's local normal-frame quadratic density (Π^2+Φ^2)/2 remains nonnegative. The loss of coercivity in E therefore diagnoses this slicing's mass-coordinate energy estimate, rather than negative physi…
Exact mathematical passage
```
 \mathcal E&=\frac14[(1+v)w_+^2+(1-v)w_-^2],\\
 \mathcal J&=\frac\sigma4[(1+v)^2w_+^2-(1-v)^2w_-^2].
\label{c:strong:null-energy}
```

w_-Use in source · 3 passages

Search aliases: w_-

Link to this entry ·

Use in source · strong_fields.tex, line 443
Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-],
Exact mathematical passage
```
w_-=\Pi-\Phi
```

w_-'Use in source · 1 passage

Search aliases: w_-'

Link to this entry ·

Use in source · strong_fields.tex, line 444
Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.
Exact mathematical passage
```
 \dot w_+={}&\sigma(v+1)w_+'
 +[\sigma v'+\sigma'(v+1)]w_+
 +\frac\sigma r[(v+1)w_++(v-1)w_-],\\
 \dot w_-={}&\sigma(v-1)w_-'
 +[\sigma v'+\sigma'(v-1)]w_-
 +\frac\sigma r[(v-1)w_-+(v+1)w_+].
```

w_-^2Use in source · 2 passages

Search aliases: w_-^2

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Use in source · strong_fields.tex, line 460
The same channels give the exact algebraic identities E · =frac14[(1+v)w_+^2+(1-v)w_-^2], J · =fracσ4[(1+v)^2w_+^2-(1-v)^2w_-^2]. For 0≤ v<1, E≥(1-v)(w_+^2+w_-^2)/4. This lower bound degenerates at a horizon. For v>1, the outward-channel coefficient in this coordinate mass density is negative. The scalar's local normal-frame quadratic density (Π^2+Φ^2)/2 remains nonnegative. The loss of coercivity in E therefore diagnoses this slicing's mass-coordinate energy estimate, rather than negative physi…
Exact mathematical passage
```
 \mathcal E&=\frac14[(1+v)w_+^2+(1-v)w_-^2],\\
 \mathcal J&=\frac\sigma4[(1+v)^2w_+^2-(1-v)^2w_-^2].
\label{c:strong:null-energy}
```

W_0Use in source · 1 passage

Search aliases: W_0

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Use in source · core.tex, line 420
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
W_0=I
```

w_0Use in source · 1 passage

Search aliases: w_0

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Use in source · spectral.tex, line 788
If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the required nearest-neighbor realization. For a general transfer problem only the matrix power and trace statements have been proved.
Exact mathematical passage
```
 w_{nq+j}=M_0^j(M_0^n)^q w_0,\qquad 0\leq j<n.
```

W_fUse in source · 1 passage

Search aliases: W_f

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Use in source · core.tex, line 173
six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a principal argument, it implies that the oriented phase sum lies in 2πZ. It does not force that sum to be zero as a real number. An unwrapped small-phase re…
Exact mathematical passage
```
W_f=e^{i\theta_f}
```

W_f^{\operatorname{or}(f,c)}Use in source · 1 passage

Search aliases: W_f^or(f,c) operatorname

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Use in source · core.tex, line 170
For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a principal argument, it implies that the oriented phase sum lies in 2πZ. It does not force that su…
Exact mathematical passage
```
\label{g:abelian-cube}
 \prod_{f\subset\partial c}W_f^{\operatorname{or}(f,c)}=1.
```

W_hUse in source · 4 passages

Search aliases: W_h

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Use in source · core.tex, line 417
Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local constant-matrix model of noncommuting transport. A varying connect…
Exact mathematical passage
```
\label{g:commutator-loop}
 W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.
```

w_jUse in source · 5 passages

Search aliases: w_j

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Use in source · core.tex, line 22
Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation…
Exact mathematical passage
```
 w_j=\frac{a_j}{N_j}>0,\qquad
 c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
 v_j=N_ja_jm^2\geq0.
```

w_kUse in source · 1 passage

Search aliases: w_k

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Use in source · spectral.tex, line 787
If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the required nearest-neighbor realization. For a general transfer problem only the matrix power and trace statements have been proved.
Exact mathematical passage
```
w_{k+1}=M_0w_k
```

w_{k+1}Use in source · 1 passage

Search aliases: w_k+1

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Use in source · spectral.tex, line 787
If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the required nearest-neighbor realization. For a general transfer problem only the matrix power and trace statements have been proved.
Exact mathematical passage
```
w_{k+1}=M_0w_k
```

W_nUse in source · 1 passage

Search aliases: W_n

Link to this entry ·

Use in source · continuum.tex, line 18
[Completion of a modewise isometry] Suppose that for every mode n, V_n:C^4→C^4 is a complex linear isometry. The coordinate rule (Va)_n=V_na_n defines a complex linear isometry on KK. If each V_n has an inverse isometry W_n, the induced V and W are inverse maps on KK. For each n, normV_na_n^2=norma_n^2. Summability of the latter terms gives membership of Va in KK, and summing the equalities gives normVa=norma. Linearity holds coordinatewise. The two inverse identities hold on each coordinate and…
Exact mathematical passage
```
W_n
```

\mathcal W_nUse in source · 2 passages

Search aliases: W_n mathcal

Link to this entry ·

Use in source · core.tex, line 79
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\mathcal W_n
```

w_{nq+j}Use in source · 1 passage

Search aliases: w_nq+j

Link to this entry ·

Use in source · spectral.tex, line 788
If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the required nearest-neighbor realization. For a general transfer problem only the matrix power and trace statements have been proved.
Exact mathematical passage
```
 w_{nq+j}=M_0^j(M_0^n)^q w_0,\qquad 0\leq j<n.
```

\mathcal W_{p,h}Use in source · 2 passages

Search aliases: W_p,h mathcal

Link to this entry ·

Use in source · finite.tex, line 464
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,
Exact mathematical passage
```
 \mathcal W_{p,h}=(F_p\otimes I_m)\mathcal A^{\mathrm{ad}}_{p,h}.
 \label{a-finite-child-fourier}
```

\mathcal W_{p,h}^*Use in source · 1 passage

Search aliases: W_p,h^* mathcal

Link to this entry ·

Use in source · finite.tex, line 470
Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*, W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] an…
Exact mathematical passage
```
\begin{aligned}
 (\langle0|\otimes I_m)\mathcal W_{p,h}&=V_{p,h-1}^*,\\
 \mathcal W_{p,h}^*
 \bigl[(I_p-|0\rangle\langle0|)\otimes I_m\bigr]
 \mathcal W_{p,h}&=D_h^{(h)}.
\end{aligned}
\label{a-finite-child-detail}
```

w_rUse in source · 4 passages

Search aliases: w_r

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Use in source · spectral.tex, line 65
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
w_r=\norm{\psi_r}^2
```

W_\squareUse in source · 3 passages

Search aliases: W_square square

Link to this entry ·

Use in source · core.tex, line 168
We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt along the two horizontal endpoints and ϕ_xx along the two vertical endpoints, with the boundary orientation from dtwedge dx. This is the rectangular funda…
Exact mathematical passage
```
 W_\square
 =\exp\left(K\oint_{\partial\square}(\phi_xdt+\phi_tdx)\right)
 =\exp\left(\frac K2\int_\square e^{2\phi}R\,dt\,dx\right).
 \label{c:core:conformal-holonomy}
```

W_uUse in source · 5 passages

Search aliases: W_u

Link to this entry ·

Use in source · refinement.tex, line 266
both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need not be attained by the complete walk: …
Exact mathematical passage
```
 \norm{W_v-W_u}\leq 2\delta_x+2\delta_y+\delta_z.
 \label{g:sevenbudget}
```

W_vUse in source · 3 passages

Search aliases: W_v

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Use in source · refinement.tex, line 266
both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need not be attained by the complete walk: …
Exact mathematical passage
```
 \norm{W_v-W_u}\leq 2\delta_x+2\delta_y+\delta_z.
 \label{g:sevenbudget}
```

W_{x,\mu\nu}Use in source · 3 passages

Search aliases: W_x,μν mu nu

Link to this entry ·

Use in source · core.tex, line 148
On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x^-1. For U(1) the plaquette itself is invariant. For a nonco…
Exact mathematical passage
```
\label{g:plaquette}
 W_{x,\mu\nu}
 =U_{x,\mu}U_{x+\widehat\mu,\nu}
 U_{x+\widehat\nu,\mu}^{-1}U_{x,\nu}^{-1}.
```

W_{x,\mu\nu}^gUse in source · 1 passage

Search aliases: W_x,μν^g mu nu

Link to this entry ·

Use in source · core.tex, line 154
ν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x^-1. For U(1) the plaquette itself is invariant. For a noncommutative group it is its conjugacy class that is invariant; the matrix entries depend on …
Exact mathematical passage
```
\label{g:plaquette-covariance}
 W_{x,\mu\nu}^g=g_xW_{x,\mu\nu}g_x^{-1}.
```

\wedgeUse in source · 18 passages

Search aliases: wedge

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\widehat DUse in source · 1 passage

Search aliases: widehat D widehat

Link to this entry ·

Use in source · strong_fields.tex, line 295
The periodic Fourier symbol is widehat D_4(θ)=fracihfrac8sinθ-sin2θ6. Its maximum dimensionless magnitude is 1.3722219798…. Applying the RK4 imaginary-axis stability interval to frozen constant advection gives frac|b|Δ th≤2.0612023174…. This is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requires a uniform stability argument for the implemented boundary closure and a suitable stronger norm. Fourth-order consistency, the semidiscrete L^2 es…
Exact mathematical passage
```
 \widehat D_4(\theta)=\frac{i}{h}\frac{8\sin\theta-\sin2\theta}{6}.
```

\widehat D_4Use in source · 1 passage

Search aliases: widehat D_4 widehat

Link to this entry ·

Use in source · strong_fields.tex, line 295
The periodic Fourier symbol is widehat D_4(θ)=fracihfrac8sinθ-sin2θ6. Its maximum dimensionless magnitude is 1.3722219798…. Applying the RK4 imaginary-axis stability interval to frozen constant advection gives frac|b|Δ th≤2.0612023174…. This is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requires a uniform stability argument for the implemented boundary closure and a suitable stronger norm. Fourth-order consistency, the semidiscrete L^2 es…
Exact mathematical passage
```
 \widehat D_4(\theta)=\frac{i}{h}\frac{8\sin\theta-\sin2\theta}{6}.
```

\widehat fUse in source · 7 passages

Search aliases: widehat f widehat

Link to this entry ·

Use in source · continuum.tex, line 281
[Physical torus Hamiltonian] The operator H_m^phys is densely defined, symmetric, self-adjoint, and closed on HH=L^2(T^3;C^4). Its domain consists exactly of fields satisfying ∑_n∈Z^3(1+|2π n|^2)normwidehat f(n)^2<∞, and for every domain vector, normH_m^physf_HH^2 =∑_n(m^2+|2π n|^2)normwidehat f(n)^2. The domain characterization is the coefficient characterization applied to F f. Density transports through the isometric equivalence. For two domain vectors, transporting the inner product and usin…
Exact mathematical passage
```
\sum_{n\in\Z^3}(1+|2\pi n|^2)\norm{\widehat f(n)}^2<\infty,
```

\widehat f_hUse in source · 4 passages

Search aliases: widehat f_h widehat

Link to this entry ·

Use in source · core.tex, line 500
g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
 \Lambda_h=\frac{\nabla C_h\cdot f_h}{d_h},\qquad
 \widehat f_h=f_h-\Lambda_hG_h.
```

\widehat f_kUse in source · 1 passage

Search aliases: widehat f_k widehat

Link to this entry ·

Use in source · constraints.tex, line 178
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K=\left\{\sum_{\|k\|_\infty\le K}
              \widehat f_k e^{ik\cdot x}\right\}.
```

\widehat HUse in source · 25 passages

Search aliases: widehat H widehat

Link to this entry ·

Use in source · continuum.tex, line 133
Define Dom(widehat H_m) ={a∈KK:∑_nnormH_na_n^2<∞}, (widehat H_ma)_n=H_na_n. The domain is maximal for this coefficient action: it contains precisely those coefficient fields for which the stated output belongs to the ambient Hilbert space. It is a vector subspace. For example, normx+y^2≤2normx^2+2normy^2 and linearity of each H_n prove closure under addition. Scalar multiplication preserves summability as well.
Exact mathematical passage
```
\Dom(\widehat H_m)
=\left\{a\in\KK:\sum_n\norm{H_na_n}^2<\infty\right\},
\qquad
(\widehat H_ma)_n=H_na_n.
```

\widehat H_mUse in source · 22 passages

Search aliases: widehat H_m widehat

Link to this entry ·

Use in source · continuum.tex, line 133
Define Dom(widehat H_m) ={a∈KK:∑_nnormH_na_n^2<∞}, (widehat H_ma)_n=H_na_n. The domain is maximal for this coefficient action: it contains precisely those coefficient fields for which the stated output belongs to the ambient Hilbert space. It is a vector subspace. For example, normx+y^2≤2normx^2+2normy^2 and linearity of each H_n prove closure under addition. Scalar multiplication preserves summability as well.
Exact mathematical passage
```
\Dom(\widehat H_m)
=\left\{a\in\KK:\sum_n\norm{H_na_n}^2<\infty\right\},
\qquad
(\widehat H_ma)_n=H_na_n.
```

\widehat H_m^*Use in source · 4 passages

Search aliases: widehat H_m^* widehat

Link to this entry ·

Use in source · continuum.tex, line 154
The adjoint domain and self-adjointness. Modewise Hermiticity gives symmetry on the maximal domain. If a,b belong to that domain, the two series in innerwidehat H_mab =∑_ninnerH_na_nb_n =∑_ninnera_nH_nb_n =innerawidehat H_mb converge absolutely by Cauchy--Schwarz. This establishes the inclusion widehat H_m⊆widehat H_m^*, including the corresponding domain inclusion.
Exact mathematical passage
```
\widehat H_m\subseteq\widehat H_m^*
```

\widehat H_{m_1}Use in source · 1 passage

Search aliases: widehat H_m_1 widehat

Link to this entry ·

Use in source · continuum.tex, line 226
[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exactly when the momentum series is finite. Adding the already finite unweighted series gives precisely the displayed inhomogeneous first-order condition.…
Exact mathematical passage
```
\Dom(\widehat H_{m_1})=\Dom(\widehat H_{m_2})
```

\widehat H_{m_2}Use in source · 1 passage

Search aliases: widehat H_m_2 widehat

Link to this entry ·

Use in source · continuum.tex, line 226
[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exactly when the momentum series is finite. Adding the already finite unweighted series gives precisely the displayed inhomogeneous first-order condition.…
Exact mathematical passage
```
\Dom(\widehat H_{m_1})=\Dom(\widehat H_{m_2})
```

\widehat NUse in source · 3 passages

Search aliases: widehat N widehat

Link to this entry ·

Use in source · spectral.tex, line 1187
On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
\widehat N|n\rangle=n|n\rangle
```

\widehat UUse in source · 2 passages

Search aliases: widehat U widehat

Link to this entry ·

Use in source · core.tex, line 322
[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The formula U_j=e^-ihA_j by itself does not identify a second-order sampling scheme.
Exact mathematical passage
```
\widehat U_h(x)=e^{-ihA(x)}
```

\widehat U_hUse in source · 2 passages

Search aliases: widehat U_h widehat

Link to this entry ·

Use in source · core.tex, line 322
[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The formula U_j=e^-ihA_j by itself does not identify a second-order sampling scheme.
Exact mathematical passage
```
\widehat U_h(x)=e^{-ihA(x)}
```

\widehat\muUse in source · 11 passages

Search aliases: widehatμ widehat mu

Link to this entry ·

Use in source · core.tex, line 3
A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as ψ_x+widehatμ-ψ_x therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.
Exact mathematical passage
```
\psi_{x+\widehat\mu}-\psi_x
```

\widehat\nuUse in source · 2 passages

Search aliases: widehatν widehat nu

Link to this entry ·

Use in source · core.tex, line 148
On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x^-1. For U(1) the plaquette itself is invariant. For a nonco…
Exact mathematical passage
```
\label{g:plaquette}
 W_{x,\mu\nu}
 =U_{x,\mu}U_{x+\widehat\mu,\nu}
 U_{x+\widehat\nu,\mu}^{-1}U_{x,\nu}^{-1}.
```

\widetilde dUse in source · 3 passages

Search aliases: widetilde d widetilde

Link to this entry ·

Use in source · core.tex, line 517
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde d_h>0
```

\widetilde d_hUse in source · 3 passages

Search aliases: widetilde d_h widetilde

Link to this entry ·

Use in source · core.tex, line 517
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde d_h>0
```

\widetilde EUse in source · 4 passages

Search aliases: widetilde E widetilde

Link to this entry ·

Use in source · constraints.tex, line 549
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 \widetilde E^a{}_i=\tfrac12\epsilon^{abc}\epsilon_{ijk}e_b{}^je_c{}^k,
 \qquad \widetilde E^a{}_ie_a{}^j=e\delta_i{}^j.
```

\widetilde E^aUse in source · 4 passages

Search aliases: widetilde E^a widetilde

Link to this entry ·

Use in source · constraints.tex, line 549
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 \widetilde E^a{}_i=\tfrac12\epsilon^{abc}\epsilon_{ijk}e_b{}^je_c{}^k,
 \qquad \widetilde E^a{}_ie_a{}^j=e\delta_i{}^j.
```

\widetilde gUse in source · 3 passages

Search aliases: widetilde g widetilde

Link to this entry ·

Use in source · core.tex, line 516
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde g_U,\widetilde g_V
```

\widetilde g_UUse in source · 3 passages

Search aliases: widetilde g_U widetilde

Link to this entry ·

Use in source · core.tex, line 516
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde g_U,\widetilde g_V
```

\widetilde g_VUse in source · 3 passages

Search aliases: widetilde g_V widetilde

Link to this entry ·

Use in source · core.tex, line 516
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde g_U,\widetilde g_V
```

\widetilde HUse in source · 5 passages

Search aliases: widetilde H widetilde

Link to this entry ·

Use in source · limit.tex, line 388
An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber
Exact mathematical passage
```
 \widetilde H_M=H_\epsilon\mathsf E_M
 =\sum_{m\leq M}\epsilon_m\mathsf F_m.
```

\widetilde H_MUse in source · 5 passages

Search aliases: widetilde H_M widetilde

Link to this entry ·

Use in source · limit.tex, line 388
An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber
Exact mathematical passage
```
 \widetilde H_M=H_\epsilon\mathsf E_M
 =\sum_{m\leq M}\epsilon_m\mathsf F_m.
```

\widetilde KUse in source · 3 passages

Search aliases: widetilde K widetilde

Link to this entry ·

Use in source · extensions.tex, line 108
The dynamics respects the same construction. Put widetilde K=VKV^*. It is skew-adjoint, and fracddtVψ=widetilde K Vψ. Moving adjoints across the inner product shows 2ReinnerΠ_bVψwidetilde KΠ_aVψ =2ReinnerF_bψKF_aψ=J^F_ba. The positive-effect current is therefore exactly the sharp current of this lifted flow evaluated on the embedded state. It is not obtained by treating arbitrary compression as an algebra homomorphism.
Exact mathematical passage
```
\widetilde K=VKV^*
```

\widetilde kUse in source · 2 passages

Search aliases: widetilde k widetilde

Link to this entry ·

Use in source · core.tex, line 771
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
\widetilde k_i=\sin(k_ih)/h
```

\widetilde{\mathbf k}Use in source · 2 passages

Search aliases: widetilde k widetilde mathbf

Link to this entry ·

Use in source · core.tex, line 771
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
\widetilde{\mathbf k}\neq0
```

\widetilde k_iUse in source · 2 passages

Search aliases: widetilde k_i widetilde

Link to this entry ·

Use in source · core.tex, line 771
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
\widetilde k_i=\sin(k_ih)/h
```

\widetilde k_jUse in source · 1 passage

Search aliases: widetilde k_j widetilde

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\widetilde nUse in source · 4 passages

Search aliases: widetilde n widetilde

Link to this entry ·

Use in source · constraints.tex, line 522
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
 \widetilde n_I=\tfrac1{3!}\epsilon_{IJKL}\epsilon^{abc}E_a^JE_b^KE_c^L.
```

\widetilde n^IUse in source · 1 passage

Search aliases: widetilde n^I widetilde

Link to this entry ·

Use in source · constraints.tex, line 527
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
\zeta^I=\nu\widetilde n^I+\eta^aE_a^I
```

\widetilde n_IUse in source · 3 passages

Search aliases: widetilde n_I widetilde

Link to this entry ·

Use in source · constraints.tex, line 522
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
 \widetilde n_I=\tfrac1{3!}\epsilon_{IJKL}\epsilon^{abc}E_a^JE_b^KE_c^L.
```

\widetilde n_JUse in source · 1 passage

Search aliases: widetilde n_J widetilde

Link to this entry ·

Use in source · constraints.tex, line 526
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
-\eta^{IJ}\widetilde n_I\widetilde n_J=\det(q_{ab})
```

\widetilde PUse in source · 2 passages

Search aliases: widetilde P widetilde

Link to this entry ·

Use in source · core.tex, line 515
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde z=(\widetilde P,\widetilde Q,\widetilde U,\widetilde V)
```

\widetilde QUse in source · 2 passages

Search aliases: widetilde Q widetilde

Link to this entry ·

Use in source · core.tex, line 515
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde z=(\widetilde P,\widetilde Q,\widetilde U,\widetilde V)
```

\widetilde RUse in source · 3 passages

Search aliases: widetilde R widetilde

Link to this entry ·

Use in source · strong_fields.tex, line 268
djoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R(0)_h, normDb_∞≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonli…
Exact mathematical passage
```
 \norm{R(t)-\widetilde R(t)}_h
 \le e^{(L_g+S_1/2)t}\norm{R(0)-\widetilde R(0)}_h,
 \qquad \norm{Db}_\infty\le S_1.
\label{c:strong:semidiscrete-bound}
```

\widetilde R_jUse in source · 1 passage

Search aliases: widetilde R_j widetilde

Link to this entry ·

Use in source · strong_fields.tex, line 277
ns normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R(0)_h, normDb_∞≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonlinear difference satisfies |g(R_j)-g(widetilde R_j)|≤ L_g|e_j|, where g'(R)=fracMR-Q^2R^2sqrt2MR-Q^2. Thus tfrac12∂_tnorm e_h^2≤(L_g+S_1/2)norm e_h^2, and integration gives the claim.
Exact mathematical passage
```
|g(R_j)-g(\widetilde R_j)|\le L_g|e_j|
```

\widetilde UUse in source · 3 passages

Search aliases: widetilde U widetilde

Link to this entry ·

Use in source · core.tex, line 319
A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).
Exact mathematical passage
```
\widetilde U_h(x)=e^{-ihA(x+h/2)}
```

\widetilde U_hUse in source · 1 passage

Search aliases: widetilde U_h widetilde

Link to this entry ·

Use in source · core.tex, line 319
A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).
Exact mathematical passage
```
\widetilde U_h(x)=e^{-ihA(x+h/2)}
```

\widetilde VUse in source · 2 passages

Search aliases: widetilde V widetilde

Link to this entry ·

Use in source · core.tex, line 515
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde z=(\widetilde P,\widetilde Q,\widetilde U,\widetilde V)
```

\widetilde XUse in source · 1 passage

Search aliases: widetilde X widetilde

Link to this entry ·

Use in source · extensions.tex, line 95
For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV. This is the exact missing-range term. Setting A=B=Π_a gives innerψ(F_a-F_a^2)ψ =norm(I-R)Π_aVψ^2. Thus the failure of an effect to be shar…
Exact mathematical passage
```
\widetilde X=VXV^*
```

\widetilde zUse in source · 4 passages

Search aliases: widetilde z widetilde

Link to this entry ·

Use in source · core.tex, line 515
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
\widetilde z=(\widetilde P,\widetilde Q,\widetilde U,\widetilde V)
```

\mathbf xUse in source · 1 passage

Search aliases: x mathbf

Link to this entry ·

Use in source · core.tex, line 302
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 ds^2=-\left(1+\frac{2\Phi}{c_0^2}\right)c_0^2dt^2
       +\left(1-\frac{2\Phi}{c_0^2}\right)d\mathbf x^2,
 \qquad \left|\frac{\Phi}{c_0^2}\right|\ll1,
```

\mathbf x^2Use in source · 1 passage

Search aliases: x^2 mathbf

Link to this entry ·

Use in source · core.tex, line 302
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 ds^2=-\left(1+\frac{2\Phi}{c_0^2}\right)c_0^2dt^2
       +\left(1-\frac{2\Phi}{c_0^2}\right)d\mathbf x^2,
 \qquad \left|\frac{\Phi}{c_0^2}\right|\ll1,
```

x^aUse in source · 1 passage

Search aliases: x^a

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Use in source · constraints.tex, line 543
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 E^0=0,\qquad E^i=e_a{}^i\,\mathrm dx^a,
 \qquad e:=\det(e_a{}^i)>0.
 \label{c:constraints:regular-triad}
```

X^kUse in source · 2 passages

Search aliases: X^k

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Use in source · finite_core.tex, line 669
X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).
Exact mathematical passage
```
X^k-Y^k=\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r
\quad(k\ge1).
```

X^{k-1-r}Use in source · 2 passages

Search aliases: X^k-1-r

Link to this entry ·

Use in source · finite_core.tex, line 669
X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).
Exact mathematical passage
```
X^k-Y^k=\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r
\quad(k\ge1).
```

X^{k-r}Use in source · 1 passage

Search aliases: X^k-r

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Use in source · finite_core.tex, line 675
No commutation is assumed. Expanding the summands gives successive differences X^k-rY^r-X^k-1-rY^r+1, which cancel. If X , Y ≤ R, then
Exact mathematical passage
```
X^{k-r}Y^r-X^{k-1-r}Y^{r+1}
```

x^mUse in source · 1 passage

Search aliases: x^m

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Use in source · finite.tex, line 70
lows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finite subgroup of the multiplicative group, the least common multiple m of the element orders is realized by an element. To see this, take for each prime divisor of m a component having the maximal prime-power order and multiply those commuting components of coprime orders. All elements solve x^m=1. Consequently the subgroup has size at most m, and Lagrange's theorem gives the reverse bound. The subg…
Exact mathematical passage
```
x^m=1
```

x^{p-1}Use in source · 1 passage

Search aliases: x^p-1

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Use in source · finite.tex, line 59
For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the required order. The cyclicity of this residue group follows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finit…
Exact mathematical passage
```
x^{p-1}=1
```

x^{p-2}Use in source · 1 passage

Search aliases: x^p-2

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Use in source · finite.tex, line 60
For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the required order. The cyclicity of this residue group follows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finit…
Exact mathematical passage
```
(p-1)x^{p-2}
```

x_0Use in source · 1 passage

Search aliases: x_0

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Use in source · limit.tex, line 883
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
 d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
 \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
 \label{a-limit-product-cone}
```

X_{0,j+1}^{-1}Use in source · 1 passage

Search aliases: X_0,j+1^-1

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

X_{00}Use in source · 2 passages

Search aliases: X_00

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

X_{01}Use in source · 2 passages

Search aliases: X_01

Link to this entry ·

Use in source · core.tex, line 399
· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source reference].
Exact mathematical passage
```
 R_0Y_{00}R_1Y_{00}^{-1}
 ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}
 \underbrace{Y_{00}^{-1}Y_{00}}_{I}\\
 &\quad\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.
```

X_{01}^{-1}Use in source · 2 passages

Search aliases: X_01^-1

Link to this entry ·

Use in source · core.tex, line 399
· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source reference].
Exact mathematical passage
```
 R_0Y_{00}R_1Y_{00}^{-1}
 ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}
 \underbrace{Y_{00}^{-1}Y_{00}}_{I}\\
 &\quad\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.
```

X_{02}^{-1}Use in source · 2 passages

Search aliases: X_02^-1

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

x_1Use in source · 7 passages

Search aliases: x_1

Link to this entry ·

Use in source · finite_core.tex, line 898
f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).
Exact mathematical passage
```
f_h(t,x)=\frac{e^{-it}}{\sqrt2}e_0
+\frac12e^{2\pi ix_1}u_A(t)
+\frac12e^{2\pi i(x_1+x_2+x_3)}u_D(t).
```

X_{1,j+1}^{-1}Use in source · 1 passage

Search aliases: X_1,j+1^-1

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

X_{10}Use in source · 2 passages

Search aliases: X_10

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

X_{11}Use in source · 2 passages

Search aliases: X_11

Link to this entry ·

Use in source · core.tex, line 399
· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source reference].
Exact mathematical passage
```
 R_0Y_{00}R_1Y_{00}^{-1}
 ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}
 \underbrace{Y_{00}^{-1}Y_{00}}_{I}\\
 &\quad\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.
```

X_{11}^{-1}Use in source · 2 passages

Search aliases: X_11^-1

Link to this entry ·

Use in source · core.tex, line 399
· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source reference].
Exact mathematical passage
```
 R_0Y_{00}R_1Y_{00}^{-1}
 ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}
 \underbrace{Y_{00}^{-1}Y_{00}}_{I}\\
 &\quad\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.
```

X_{12}^{-1}Use in source · 2 passages

Search aliases: X_12^-1

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

x_1^2Use in source · 2 passages

Search aliases: x_1^2

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Use in source · constraints.tex, line 788
nsfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m -m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_…
Exact mathematical passage
```
 M_D=\begin{pmatrix}x_1^2+m&-m\\-m&m+n\end{pmatrix},
 \qquad D=(m+n)x_1^2+mn>0.
```

X_{1j}Use in source · 1 passage

Search aliases: X_1j

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

X_2Use in source · 1 passage

Search aliases: X_2

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Use in source · extensions.tex, line 340
For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anorm Y_2 and the displayed decomposition finish the proof.
Exact mathematical passage
```
\norm{X^*}_2=\norm X_2
```

x_2Use in source · 6 passages

Search aliases: x_2

Link to this entry ·

Use in source · finite_core.tex, line 898
f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).
Exact mathematical passage
```
f_h(t,x)=\frac{e^{-it}}{\sqrt2}e_0
+\frac12e^{2\pi ix_1}u_A(t)
+\frac12e^{2\pi i(x_1+x_2+x_3)}u_D(t).
```

X_2^2Use in source · 1 passage

Search aliases: X_2^2

Link to this entry ·

Use in source · extensions.tex, line 337
For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anorm Y_2 and the displayed decomposition finish the proof.
Exact mathematical passage
```
 \norm X_2^2=\sum_j\norm{Xe_j}^2<\infty.
```

x_2^2Use in source · 1 passage

Search aliases: x_2^2

Link to this entry ·

Use in source · constraints.tex, line 794
-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonli…
Exact mathematical passage
```
 S_C=\frac{\mu x_2^2}{2d},\qquad
 R_C=\partial_{x_1}+\frac{x_1x_2}{d}\partial_{x_2}.
```

x_3Use in source · 2 passages

Search aliases: x_3

Link to this entry ·

Use in source · finite_core.tex, line 898
f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).
Exact mathematical passage
```
f_h(t,x)=\frac{e^{-it}}{\sqrt2}e_0
+\frac12e^{2\pi ix_1}u_A(t)
+\frac12e^{2\pi i(x_1+x_2+x_3)}u_D(t).
```

X_{ba}Use in source · 4 passages

Search aliases: X_ba

Link to this entry ·

Use in source · core.tex, line 57
The same atlas resolves an operator on both its input and output sides. For X∈ B(HH), define X_ba=P_bXP_a. The first index names the destination and the second names the source. As a map between packet spaces, this is X_ba:Ran P_a→Ran P_b. As a bounded operator on HH, it is extended by zero on the orthogonal complement of Ran P_a. Keeping these two descriptions aligned prevents a block from being composed with an incompatible input space.
Exact mathematical passage
```
\label{f:corner-def}
 X_{ba}=P_bXP_a.
```

X_{ba}^{V}Use in source · 2 passages

Search aliases: X_ba^V

Link to this entry ·

Use in source · extensions.tex, line 90
For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV. This is the exact missing-range term. Setting A=B=Π_a gives innerψ(F_a-F_a^2)ψ =norm(I-R)Π_aVψ^2. Thus the failure of an
Exact mathematical passage
```
X_{ba}^{V}=V_bXV_a^*
```

X_{bc}Use in source · 1 passage

Search aliases: X_bc

Link to this entry ·

Use in source · extensions.tex, line 493
Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.
Exact mathematical passage
```
 P_b\delta_B(X)P_a
 =i\sum_c\left(B_{bc}X_{ca}-X_{bc}B_{ca}\right).
 \label{eq:foundation-corner-path}
```

x_cUse in source · 3 passages

Search aliases: x_c

Link to this entry ·

Use in source · core.tex, line 179
ampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μ…
Exact mathematical passage
```
x_c
```

X_{ca}Use in source · 1 passage

Search aliases: X_ca

Link to this entry ·

Use in source · extensions.tex, line 493
Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.
Exact mathematical passage
```
 P_b\delta_B(X)P_a
 =i\sum_c\left(B_{bc}X_{ca}-X_{bc}B_{ca}\right).
 \label{eq:foundation-corner-path}
```

X_{cb}Use in source · 1 passage

Search aliases: X_cb

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Use in source · core.tex, line 64
[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.
Exact mathematical passage
```
 X&=\sum_{b,a}X_{ba}, & P_bX\psi&=\sum_aX_{ba}\psi_a,\label{f:block-reconstruction}\\
 (XY)_{ca}&=\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.
 \label{f:block-product}
```

x_kUse in source · 1 passage

Search aliases: x_k

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Use in source · spectral.tex, line 687
For the fine cycle, x_k=2cos(2π k/(pM)) maps to 2cos(2π k/M). Each coarse Fourier label has exactly p fine preimages. If two labels already have the same numerical energy, their multiplicities add as usual. Band folding retains the branch label and its vector, whereas normalized decimation identifies all of these labels at the same coarse energy. Their spectral relation does not make elimination an invertible operator change of basis.
Exact mathematical passage
```
x_k=2\cos(2\pi k/(pM))
```

x_mUse in source · 2 passages

Search aliases: x_m

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Use in source · spectral.tex, line 738
an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =fracκ U_m-1(x/2)U_n-1(T_m(x/2)) =fracκU_mn-1(x/2). Similarly μ_m,n=z-κ_m,nR_n(R_m(x)), which gives…
Exact mathematical passage
```
x_m=(z-\mu_m)/\kappa_m=R_m(x)
```

x_NUse in source · 4 passages

Search aliases: x_N

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Use in source · extensions.tex, line 145
[A small state can have a large generator image] On ℓ^2(N_≥1), let He_n=ne_n on its maximal domain. The finitely supported vectors x_N=e_N/N satisfy normx_N=1/N→0, while normHx_N=1. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because Hx_N=e_N has no norm limit.
Exact mathematical passage
```
x_N=e_N/N
```

x_nUse in source · 2 passages

Search aliases: x_n

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Use in source · finite.tex, line 60
For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the required order. The cyclicity of this residue group follows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finit…
Exact mathematical passage
```
x_n
```

x_{p,q}Use in source · 1 passage

Search aliases: x_p,q

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Use in source · spectral.tex, line 674
The folded branches from Proposition~[source reference], in normalized coordinates, satisfy x_p,q(θ)=2cos (fracθ+2π qp), R_p(x_p,q(θ))=2cosθ. The second equality is just T_p(cosα)=cos(pα). For a coarse normalized energy in the open interval (-2,2), choose its angle in (0,π). These formulas supply the p distinct real roots of R_p(x)=2cosθ. At the endpoints, some roots coincide and algebraic multiplicities must be retained. Globally the branches exchange labels when the angle crosses a cell bounda…
Exact mathematical passage
```
 x_{p,q}(\theta)=2\cos\!\left(\frac{\theta+2\pi q}{p}\right),
 \qquad R_p(x_{p,q}(\theta))=2\cos\theta.
 \label{a-spectral-branch-relation}
```

X_{rs}Use in source · 2 passages

Search aliases: X_rs

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Use in source · spectral.tex, line 28
rimality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U i…
Exact mathematical passage
```
X_{rs}=J_r^*XJ_s
```

Y^*Use in source · 1 passage

Search aliases: Y^*

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Use in source · extensions.tex, line 334
For compact X, choose finite-rank Y with normX-Y<ε. Strong convergence is uniform on each fixed finite-dimensional subspace, and self-adjointness of S_N controls the right error by the corresponding left error for Y^*. Thus normS_NYS_N-Y→0. The difference between the two reconstruction errors is at most 2ε, proving the claim.
Exact mathematical passage
```
Y^*
```

y^{-1}Use in source · 1 passage

Search aliases: y^-1

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Use in source · spectral.tex, line 242
[Upper half-plane sign and the exact visible norm] For z=x+iy with y>0, ImΣ(z) =-yB((x-D)^2+y^2I)^-1B^*≤0, Im S(z)≥ yI. In particular S(z) is invertible and norm S(z)^-1≤ y^-1. For real E∉σ(D), Σ'(E)=-B(E-D)^-2B^*. If (u,v) is a normalized full eigenvector at E, then 1=⟨ u,(I-Σ'(E))u⟩. For real λ, Im(z-λ)^-1=-y/((x-λ)^2+y^2). Applying this identity to the spectral sum gives the first formula and then the bound on Im S. For any u, Cauchy--Schwarz implies ynormu^2≤|⟨ u, S(z)u⟩| ≤normunorm S(z)u. T…
Exact mathematical passage
```
\norm{\mathcal S(z)^{-1}}\leq y^{-1}
```

y^2Use in source · 3 passages

Search aliases: y^2

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Use in source · spectral.tex, line 235
[Upper half-plane sign and the exact visible norm] For z=x+iy with y>0, ImΣ(z) =-yB((x-D)^2+y^2I)^-1B^*≤0, Im S(z)≥ yI. In particular S(z) is invertible and norm S(z)^-1≤ y^-1. For real E∉σ(D), Σ'(E)=-B(E-D)^-2B^*. If (u,v) is a normalized full eigenvector at E, then 1=⟨ u,(I-Σ'(E))u⟩. For real λ, Im(z-λ)^-1=-y/((x-λ)^2+y^2). Applying this identity to the spectral sum gives the first formula and then the bound on Im S. For any u, Cauchy--Schwarz implies ynormu^2≤|⟨ u, S(z)u⟩| ≤normunorm S(z)u. T…
Exact mathematical passage
```
 \operatorname{Im}\Sigma(z)
  =-yB\bigl((x-D)^2+y^2I\bigr)^{-1}B^*\leq0,
 \qquad
 \operatorname{Im}\mathcal S(z)\geq yI.
```

y^jUse in source · 1 passage

Search aliases: y^j

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Use in source · spectral.tex, line 1063
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters ar…
Exact mathematical passage
```
 B_{L,j}(y)
 =\frac{(-1)^jy^j}{(j!)^2}\,
 {}_0F_{2L-1}\!\left(
 \begin{matrix}-\\1,\mathcal A_{L,j},\mathcal A_{L,j}\end{matrix};
 \frac{(-1)^Ly^L}{L^{2L}}\right).
 \label{a-spectral-bessel-lane}
```

Y^kUse in source · 2 passages

Search aliases: Y^k

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Use in source · finite_core.tex, line 669
X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).
Exact mathematical passage
```
X^k-Y^k=\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r
\quad(k\ge1).
```

y^LUse in source · 1 passage

Search aliases: y^L

Link to this entry ·

Use in source · spectral.tex, line 1063
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters ar…
Exact mathematical passage
```
 B_{L,j}(y)
 =\frac{(-1)^jy^j}{(j!)^2}\,
 {}_0F_{2L-1}\!\left(
 \begin{matrix}-\\1,\mathcal A_{L,j},\mathcal A_{L,j}\end{matrix};
 \frac{(-1)^Ly^L}{L^{2L}}\right).
 \label{a-spectral-bessel-lane}
```

y^{Lq+j}Use in source · 1 passage

Search aliases: y^Lq+j

Link to this entry ·

Use in source · spectral.tex, line 1061
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -
Exact mathematical passage
```
B_{L,j}(y)=\sum_{q\geq0}(-1)^{Lq+j}y^{Lq+j}/[(Lq+j)!]^2
```

y^nUse in source · 1 passage

Search aliases: y^n

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Use in source · spectral.tex, line 1055
The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -
Exact mathematical passage
```
 B(y)=\sum_{n=0}^\infty\frac{(-1)^ny^n}{(n!)^2}
     ={}_0F_1(;1;-y).
```

Y^rUse in source · 2 passages

Search aliases: Y^r

Link to this entry ·

Use in source · finite_core.tex, line 669
X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).
Exact mathematical passage
```
X^k-Y^k=\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r
\quad(k\ge1).
```

Y^{r+1}Use in source · 1 passage

Search aliases: Y^r+1

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Use in source · finite_core.tex, line 675
No commutation is assumed. Expanding the summands gives successive differences X^k-rY^r-X^k-1-rY^r+1, which cancel. If X , Y ≤ R, then
Exact mathematical passage
```
X^{k-r}Y^r-X^{k-1-r}Y^{r+1}
```

y^{rq+j}Use in source · 1 passage

Search aliases: y^rq+j

Link to this entry ·

Use in source · core.tex, line 447
The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the separated Taylor channels are coupled. Their sum satisfies the Bessel equation. This decomposition resolves series order and does not introduce addition…
Exact mathematical passage
```
 \mathcal B_{r,j}(y)=\sum_{q=0}^{\infty}
 \frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},
 \qquad j=0,\ldots,r-1.
```

y_*Use in source · 2 passages

Search aliases: y_*

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Use in source · constraints.tex, line 793
-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonli…
Exact mathematical passage
```
y_*=x_1x_2/d
```

Y_{00}Use in source · 2 passages

Search aliases: Y_00

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

Y_{00}^{-1}Use in source · 2 passages

Search aliases: Y_00^-1

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

Y_{01}^{-1}Use in source · 2 passages

Search aliases: Y_01^-1

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

Y_{0j}^{-1}Use in source · 1 passage

Search aliases: Y_0j^-1

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

Y_{1j}Use in source · 2 passages

Search aliases: Y_1j

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

Y_{1j}^{-1}Use in source · 2 passages

Search aliases: Y_1j^-1

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

Y_2Use in source · 1 passage

Search aliases: Y_2

Link to this entry ·

Use in source · extensions.tex, line 340
For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anorm Y_2 and the displayed decomposition finish the proof.
Exact mathematical passage
```
\norm{AY}_2\leq\norm A\norm Y_2
```

Y_{20}Use in source · 2 passages

Search aliases: Y_20

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

Y_{21}Use in source · 2 passages

Search aliases: Y_21

Link to this entry ·

Use in source · core.tex, line 379
[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.
Exact mathematical passage
```
 W^{(2)}
 &:=X_{00}X_{10}Y_{20}Y_{21}
 X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
 \label{g:coarse-boundary}\\
 &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
```

Y_{2j}Use in source · 1 passage

Search aliases: Y_2j

Link to this entry ·

Use in source · core.tex, line 392
First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I
Exact mathematical passage
```
 R_j
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
 X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
 &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
```

y_aUse in source · 1 passage

Search aliases: y_a

Link to this entry ·

Use in source · extensions.tex, line 79
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
V^*(y_a)_a=\sum_aV_a^*y_a
```

Y_{ba}Use in source · 1 passage

Search aliases: Y_ba

Link to this entry ·

Use in source · core.tex, line 64
[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.
Exact mathematical passage
```
 X&=\sum_{b,a}X_{ba}, & P_bX\psi&=\sum_aX_{ba}\psi_a,\label{f:block-reconstruction}\\
 (XY)_{ca}&=\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.
 \label{f:block-product}
```

y_nUse in source · 1 passage

Search aliases: y_n

Link to this entry ·

Use in source · extensions.tex, line 162
[Self-adjointness of the diagonal realization] The operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.
Exact mathematical passage
```
z_n=\lambda_ny_n
```

\mathrm{YM}Use in source · 1 passage

Search aliases: YM mathrm

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Use in source · core.tex, line 160
For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the analogous term is 1-ReW. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interac…
Exact mathematical passage
```
\label{g:wilson-action}
 S_W[U]=\frac1{g_{\mathrm{YM}}^2}
 \sum_{x,\mu<\nu}\left(1-\tfrac12\operatorname{Re}\operatorname{Tr}W_{x,\mu\nu}\right),
 \qquad g_{\mathrm{YM}}\in\R\setminus\{0\}.
```

ZUse in source · 31 passages

Search aliases: Z

Link to this entry ·

Use in source · finite.tex, line 213
Discarding the sign as a quantum operation instead takes the partial trace ρ↦Tr_signρ. Both pure states |0,a⟩ and |1,a⟩ then produce the same reduced phase state |a⟩⟨ a|. Yet the sign observable Z_sign=(|0⟩⟨0|-|1⟩⟨1|)⊗ I has expectations +1 and -1 on those states. No measurement on the retained phase state can reconstruct this difference. The obstruction is information loss in a stated map, not an alternative probability rule.
Exact mathematical passage
```
 Z_{\mathrm{sign}}=(|0\rangle\langle0|-|1\rangle\langle1|)\otimes I
```

\mathbb ZUse in source · 5 passages

Search aliases: Z mathbb

Link to this entry ·

Use in source · primary_context.tex, line 24
item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is proved separately.
Exact mathematical passage
```
\mathbb Z_p
```

z^{-1}Use in source · 2 passages

Search aliases: z^-1

Link to this entry ·

Use in source · limit.tex, line 403
athsf F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber norm((z-widetilde H_M)^-1-(z-H_ϵ)^-1)f_2 · ≤(|z|^-1+|Imz|^-1) norm(I- E_M)f_2. Thus widetilde H_M converges to H_ϵ in the strong resolvent sense. These statements specify operators on one…
Exact mathematical passage
```
 (z-\widetilde H_M)^{-1}
 &=\mathsf E_M(z-H_\epsilon)^{-1}
   +z^{-1}(I-\mathsf E_M),\nonumber\\
 \norm{((z-\widetilde H_M)^{-1}-(z-H_\epsilon)^{-1})f}_2
 &\leq\left(|z|^{-1}+|\operatorname{Im}z|^{-1}\right)
 \norm{(I-\mathsf E_M)f}_2.
 \label{a-limit-strong-resolvent}
```

Z^2Use in source · 1 passage

Search aliases: Z^2

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Use in source · core.tex, line 444
Multiplying the four exponential power series through degree three gives W_h=I+h^2[A,B]+frach^32[A+B,[A,B]]+O(h^4). The analytic logarithm has log(I+Z)=Z+O(norm Z^2) near zero. Since Z=W_h-I=O(h^2), its correction begins at order h^4. Dividing by h^2 proves [source reference]. At half spacing the linear term is hC_3/2, so 2F_h/2-F_h removes it exactly and leaves an O(h^2) remainder. Finally, the power series for both the exponential and the local logarithm commute with conjugation. Applying them…
Exact mathematical passage
```
\log(I+Z)=Z+O(\norm Z^2)
```

z^2Use in source · 1 passage

Search aliases: z^2

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Use in source · constraints.tex, line 783
A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m
Exact mathematical passage
```
 S_F(x_1,x_2,y,z)=\tfrac12(x_2-x_1y)^2
                         +\tfrac m2(y-z)^2+\tfrac n2z^2.
```

\Z^3Use in source · 3 passages

Search aliases: Z^3 Z

Link to this entry ·

Use in source · continuum.tex, line 5
Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of amplitudes embeds by assigning zero to every absent mode.
Exact mathematical passage
```
\KK=\ell^2(\Z^3;\C^4)
=\left\{a=(a_n):\sum_{n\in\Z^3}\norm{a_n}^2<\infty\right\}.
```

\mathbb Z^3Use in source · 2 passages

Search aliases: Z^3 mathbb

Link to this entry ·

Use in source · finite_core.tex, line 14
equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is
Exact mathematical passage
```
n=(n_1,n_2,n_3)\in\mathbb Z^3
```

Z^AUse in source · 4 passages

Search aliases: Z^A

Link to this entry ·

Use in source · constraints.tex, line 7
An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.
Exact mathematical passage
```
(Z^A,P_A)
```

z^kUse in source · 1 passage

Search aliases: z^k

Link to this entry ·

Use in source · finite_core.tex, line 766
∑_k=0^Nfracz^k(A^ H)^kk! =T(∑_k=0^Nfracz^kA^kk!)T^-1.
Exact mathematical passage
```
\sum_{k=0}^{N}\frac{z^k(A^{\mathcal H})^k}{k!}
=T\left(\sum_{k=0}^{N}\frac{z^kA^k}{k!}\right)T^{-1}.
```

z^{Lq+j}Use in source · 2 passages

Search aliases: z^Lq+j

Link to this entry ·

Use in source · spectral.tex, line 909
, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follo…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
 \quad
 \mathfrak R_{L,j}\mathfrak R_{L,k}
   =\delta_{jk}\mathfrak R_{L,j},\quad
 \sum_j\mathfrak R_{L,j}=I.
 \label{a-spectral-analytic-lanes}
```

z^nUse in source · 2 passages

Search aliases: z^n

Link to this entry ·

Use in source · spectral.tex, line 900
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓ
Exact mathematical passage
```
F(z)=\sum_{n=0}^\infty a_nz^n
```

z_*Use in source · 1 passage

Search aliases: z_*

Link to this entry ·

Use in source · constraints.tex, line 793
-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonli…
Exact mathematical passage
```
z_*=my_* /(m+n)
```

z_0Use in source · 12 passages

Search aliases: z_0

Link to this entry ·

Use in source · constraints.tex, line 676
hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let …
Exact mathematical passage
```
 \overline\nabla F(z_0,z_1)=\int_0^1\nabla F((1-s)z_0+sz_1)\,\mathrm ds.
 \label{c:constraints:discrete-gradient}
```

z_1Use in source · 12 passages

Search aliases: z_1

Link to this entry ·

Use in source · constraints.tex, line 676
hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let …
Exact mathematical passage
```
 \overline\nabla F(z_0,z_1)=\int_0^1\nabla F((1-s)z_0+sz_1)\,\mathrm ds.
 \label{c:constraints:discrete-gradient}
```

z_{ab}Use in source · 2 passages

Search aliases: z_ab

Link to this entry ·

Use in source · core.tex, line 343
Let z_ba=Tr(P_bHP_aρ). Taking the conjugate trace and cyclically moving the first factor to the end yields overlinez_ba=Tr(ρ P_aHP_b) =Tr(P_aHP_bρ)=z_ab. Completeness and cyclicity resolve the derivative as dot p_b=-i∑_a(z_ba-z_ab) =∑_a2Imz_ba. The conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exactly as before.
Exact mathematical passage
```
 \overline{z_{ba}}=\operatorname{Tr}(\rho P_aHP_b)
 =\operatorname{Tr}(P_aHP_b\rho)=z_{ab}.
```

z_{ba}Use in source · 2 passages

Search aliases: z_ba

Link to this entry ·

Use in source · core.tex, line 342
Let z_ba=Tr(P_bHP_aρ). Taking the conjugate trace and cyclically moving the first factor to the end yields overlinez_ba=Tr(ρ P_aHP_b) =Tr(P_aHP_bρ)=z_ab. Completeness and cyclicity resolve the derivative as dot p_b=-i∑_a(z_ba-z_ab) =∑_a2Imz_ba. The conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exactly as before.
Exact mathematical passage
```
z_{ba}=\operatorname{Tr}(P_bHP_a\rho)
```

z_jUse in source · 5 passages

Search aliases: z_j

Link to this entry ·

Use in source · spectral.tex, line 623
The sine formula U_p-1(cosθ)=sin(pθ)/sinθ shows that the excluded normalized energies are x_j=2cos(jπ/p), z_j=μ+2κcos(jπ/p), 1≤ j<p. These are the energies of one eliminated path. Indeed its jth eigenvector has interior components sin(jπ r/p), r=1,…,p-1, and the nearest-neighbor recurrence gives precisely z_j. Both boundary values at r=0,p vanish. The eigenvectors are orthogonal and there are p-1 of them, so these exhaust the path spectrum.
Exact mathematical passage
```
 x_j=2\cos(j\pi/p),\qquad
 z_j=\mu+2\kappa\cos(j\pi/p),\quad 1\leq j<p.
 \label{a-spectral-dirichlet-resonances}
```

z_nUse in source · 6 passages

Search aliases: z_n

Link to this entry ·

Use in source · extensions.tex, line 162
[Self-adjointness of the diagonal realization] The operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.
Exact mathematical passage
```
z_n=\lambda_ny_n
```

z_{n+1}Use in source · 3 passages

Search aliases: z_n+1

Link to this entry ·

Use in source · core.tex, line 518
[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetilde V-μwidetilde g_V) satisfies C_h(z_n+1)=C_h(z_n) exactly. It is the correction of smallest Euclidean velocity norm at the fixed configuration.
Exact mathematical passage
```
 \mu=\frac{C_h(\widetilde z)-C_h(z_n)}{\widetilde d_h},\qquad
 z_{n+1}=(\widetilde P,\widetilde Q,
 \widetilde U-\mu\widetilde g_U,
 \widetilde V-\mu\widetilde g_V)
```

\mathbb Z_pUse in source · 1 passage

Search aliases: Z_p mathbb

Link to this entry ·

Use in source · primary_context.tex, line 24
item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is proved separately.
Exact mathematical passage
```
\mathbb Z_p
```

\Z_p^\timesUse in source · 1 passage

Search aliases: Z_p^× Z times

Link to this entry ·

Use in source · spectral.tex, line 1159
Let p be prime, let L=mp for a positive integer m, and take 0≤ j<p. For every q≥0, g_q=fracj!(Lq)!(Lq+j)! =binomLq+jj^-1∈Z_p^×. Indeed binomLq+jj=∏_r=1^j(Lq+r)/r. Each numerator and denominator is a p-adic unit because Lq≡0pmod p and 1≤ r<p. For j=0 the empty product is one. The proof also covers p=2 in its short window. Outside this window, a factor can cross a multiple of p and the same proof gives no unit guarantee. This rational coefficient statement supplies no claim about the p-adic conver…
Exact mathematical passage
```
 g_q=\frac{j!(Lq)!}{(Lq+j)!}
       =\binom{Lq+j}{j}^{-1}\in\Z_p^\times.
 \label{a-spectral-unit-window}
```

Z_{\mathrm{sign}}Use in source · 4 passages

Search aliases: Z_sign mathrm

Link to this entry ·

Use in source · finite.tex, line 213
Discarding the sign as a quantum operation instead takes the partial trace ρ↦Tr_signρ. Both pure states |0,a⟩ and |1,a⟩ then produce the same reduced phase state |a⟩⟨ a|. Yet the sign observable Z_sign=(|0⟩⟨0|-|1⟩⟨1|)⊗ I has expectations +1 and -1 on those states. No measurement on the retained phase state can reconstruct this difference. The obstruction is information loss in a stated map, not an alternative probability rule.
Exact mathematical passage
```
 Z_{\mathrm{sign}}=(|0\rangle\langle0|-|1\rangle\langle1|)\otimes I
```

z_{x,\mu}Use in source · 2 passages

Search aliases: z_x,μ mu

Link to this entry ·

Use in source · core.tex, line 337
Now take ψ_x∈C^2, U_x,μ∈ SU(2), and local gauges g_x∈ SU(2), as in the legacy non-Abelian chapter. The linked scalar z_x,μ=ψ_x^*U_x,μψ_x+widehatμ is gauge invariant: the adjoint factor g_x^* cancels g_x, and the factors at x+widehatμ also cancel. It follows that the source's scalar link observable j_x,μ=2Imz_x,μ is invariant. To obtain a continuity equation with its particular sign and normalization one must specify the hopping generator. Invariance of the observable alone is not an equation of …
Exact mathematical passage
```
\label{g:linked-scalar}
 z_{x,\mu}=\psi_x^*U_{x,\mu}\psi_{x+\widehat\mu}
```

|^{-1}Use in source · 2 passages

Search aliases: |^-1

Link to this entry ·

Use in source · limit.tex, line 403
athsf F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber norm((z-widetilde H_M)^-1-(z-H_ϵ)^-1)f_2 · ≤(|z|^-1+|Imz|^-1) norm(I- E_M)f_2. Thus widetilde H_M converges to H_ϵ in the strong resolvent sense. These statements specify operators on one…
Exact mathematical passage
```
 (z-\widetilde H_M)^{-1}
 &=\mathsf E_M(z-H_\epsilon)^{-1}
   +z^{-1}(I-\mathsf E_M),\nonumber\\
 \norm{((z-\widetilde H_M)^{-1}-(z-H_\epsilon)^{-1})f}_2
 &\leq\left(|z|^{-1}+|\operatorname{Im}z|^{-1}\right)
 \norm{(I-\mathsf E_M)f}_2.
 \label{a-limit-strong-resolvent}
```

|^3Use in source · 4 passages

Search aliases: |^3

Link to this entry ·

Use in source · finite.tex, line 676
For a self-adjoint fine Hamiltonian with coarse projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional example H_f=0 · overlineγ
Exact mathematical passage
```
 Qe^{-\mathrm{i}tH_f}\psi=-\mathrm{i}tB\psi+O(t^2),
 \qquad
 \norm{Qe^{-\mathrm{i}tH_f}\psi}^2
 =t^2\norm{B\psi}^2+O(|t|^3).
```

|^5Use in source · 1 passage

Search aliases: |^5

Link to this entry ·

Use in source · finite_core.tex, line 612
For clarity, Taylor's theorem through degree four has zero even coefficients for sine, and its fifth-derivative remainder is at most |hp_j|^5/120. Division by |h| gives O(h^4); squaring the three expansions and summing preserves that order for a fixed mode. If E_n>0, the square root is smooth near E_n^2, so
Exact mathematical passage
```
|hp_j|^5/120
```

|^6Use in source · 1 passage

Search aliases: |^6

Link to this entry ·

Use in source · finite_core.tex, line 519
C(n)=frac16(∑_j=x,y,z|p_j(n)|^6)^1/2.
Exact mathematical passage
```
C(n)=\frac16\left(\sum_{j=x,y,z}|p_j(n)|^6\right)^{1/2}.
```

|^pUse in source · 3 passages

Search aliases: |^p

Link to this entry ·

Use in source · refinement.tex, line 45
[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower step gives normT_m+1R(h)≤ fracb^m+1normT_mR(h/b)+normT_mR(h)b^m+1-1. Starting with the stated remainder bound and proceeding by induction multiplies i…
Exact mathematical passage
```
 \norm{R(h)}\leq C|h|^p,\qquad
 \norm{g(h)-f(h)}\leq\varepsilon
 \quad(0<|h|<\delta),
```

|_{\mathcal V_M}Use in source · 2 passages

Search aliases: |_ V_M mathcal

Link to this entry ·

Use in source · limit.tex, line 134
The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projection in V_M onto V_m. This is the exact relation between the finite matrices and the common infinite Hilbert space.
Exact mathematical passage
```
 \mathsf E_m\big|_{\mathcal V_M}
 =\mathcal U_{p,M}V_{p,m\to M}V_{p,m\to M}^*
 \mathcal U_{p,M}^*.
 \label{a-limit-finite-projection}
```

|_{Q\HH}Use in source · 3 passages

Search aliases: |_QHH HH

Link to this entry ·

Use in source · limit.tex, line 747
normQe^-itH_λP · ≤min{1,|tλ|normQBP}. For z∉σ(QH_λ Q|_QHH), the coarse self-energy is Σ_P(z)=PH_λ Q (z-QH_λ Q|_QHH)^-1QH_λ P, where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-
Exact mathematical passage
```
z\notin\sigma(QH_\lambda Q|_{Q\HH})
```

|_{\Ran P}Use in source · 1 passage

Search aliases: |_Ran P Ran

Link to this entry ·

Use in source · finite.tex, line 521
QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.
Exact mathematical passage
```
 H_f=(VH_cV^*)|_{\Ran P}\ \oplus\ H_Q,
 \qquad H_Q=(QH_fQ)|_{\Ran Q}.
 \label{a-finite-hamiltonian-direct-sum}
```

|_{\Ran Q}Use in source · 2 passages

Search aliases: |_Ran Q Ran

Link to this entry ·

Use in source · finite.tex, line 521
QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.
Exact mathematical passage
```
 H_f=(VH_cV^*)|_{\Ran P}\ \oplus\ H_Q,
 \qquad H_Q=(QH_fQ)|_{\Ran Q}.
 \label{a-finite-hamiltonian-direct-sum}
```

|_{\varphi}Use in source · 1 passage

Search aliases: |_φ varphi

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Use in source · constraints.tex, line 337
ta⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.
Exact mathematical passage
```
 S_{\mathrm M}^{(1)}\big|_{\varphi}
      =-\tfrac14\int F_{IJ}F^{IJ}\operatorname{vol}_e.
```

\timesUse in source · 24 passages

Search aliases: × times

Link to this entry ·

Use in source · finite.tex, line 15
The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into one phase coordinate.
Exact mathematical passage
```
 U_{p,h}=(\Z/p^{h+1}\Z)^\times.
```

\alphaUse in source · 59 passages

Search aliases: α alpha

Link to this entry ·

Use in source · finite.tex, line 818
Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf interaction: its two radial eigenvalues are ±|G_p,w|, so normH_int=|G_p,w|. They do not determine the boundedness of a different infinite operator with man…
Exact mathematical passage
```
\gamma_{p,w}=g p^{-\alpha w}
```

\alpha^*Use in source · 2 passages

Search aliases: α^* alpha

Link to this entry ·

Use in source · core.tex, line 196
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,
Exact mathematical passage
```
\alpha=\alpha^*
```

\alpha^nUse in source · 1 passage

Search aliases: α^n alpha

Link to this entry ·

Use in source · spectral.tex, line 1201
For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber
Exact mathematical passage
```
 |\alpha\rangle=e^{-|\alpha|^2/2}
        \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}|n\rangle,
 \qquad \mu=|\alpha|^2\geq0,
```

\alpha_jUse in source · 11 passages

Search aliases: α_j alpha

Link to this entry ·

Use in source · continuum.tex, line 301
On finite Fourier polynomials, the operator agrees with the matrix differential expression -i∑_jα_j∂_j+mβ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every L^2 representative. Equality with a separately constructed weak differential operator is a further interface statement if that alternate realization is introduced.
Exact mathematical passage
```
-i\sum_j\alpha_j\partial_j+m\beta
```

\alpha_j^*Use in source · 2 passages

Search aliases: α_j^* alpha

Link to this entry ·

Use in source · finite_core.tex, line 140
α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,
Exact mathematical passage
```
\alpha_j^*=\alpha_j,\quad \beta^*=\beta,\quad
\alpha_j^2=\beta^2=I_4,
```

\alpha_j^2Use in source · 3 passages

Search aliases: α_j^2 alpha

Link to this entry ·

Use in source · finite_core.tex, line 140
α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,
Exact mathematical passage
```
\alpha_j^*=\alpha_j,\quad \beta^*=\beta,\quad
\alpha_j^2=\beta^2=I_4,
```

\alpha_kUse in source · 3 passages

Search aliases: α_k alpha

Link to this entry ·

Use in source · finite_core.tex, line 145
α_jα_k+α_kα_j=0 (j≠ k), α_jβ+βα_j=0.
Exact mathematical passage
```
\alpha_j\alpha_k+\alpha_k\alpha_j=0\quad(j\ne k),
\qquad
\alpha_j\beta+\beta\alpha_j=0.
```

\alpha_qUse in source · 4 passages

Search aliases: α_q alpha

Link to this entry ·

Use in source · spectral.tex, line 128
al-folded-bands For q=0,…,p-1, the eigenvalues of H_p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every…
Exact mathematical passage
```
\alpha_q=(\theta+2\pi q)/p
```

\alpha_xUse in source · 2 passages

Search aliases: α_x alpha

Link to this entry ·

Use in source · finite_core.tex, line 76
α_x= 0 · 0 · 0 · 1 0 · 0 · 1 · 0 0 · 1 · 0 · 0 1 · 0 · 0 · 0 , α_y= 0 · 0 · 0 · -i 0 · 0 · i · 0 0 · -i · 0 · 0 i · 0 · 0 · 0 ,
Exact mathematical passage
```
\alpha_x=
\begin{pmatrix}
0&0&0&1\\
0&0&1&0\\
0&1&0&0\\
1&0&0&0
\end{pmatrix},
\qquad
\alpha_y=
\begin{pmatrix}
0&0&0&-i\\
0&0&i&0\\
0&-i&0&0\\
i&0&0&0
\end{pmatrix},
```

\alpha_yUse in source · 3 passages

Search aliases: α_y alpha

Link to this entry ·

Use in source · finite_core.tex, line 76
α_x= 0 · 0 · 0 · 1 0 · 0 · 1 · 0 0 · 1 · 0 · 0 1 · 0 · 0 · 0 , α_y= 0 · 0 · 0 · -i 0 · 0 · i · 0 0 · -i · 0 · 0 i · 0 · 0 · 0 ,
Exact mathematical passage
```
\alpha_x=
\begin{pmatrix}
0&0&0&1\\
0&0&1&0\\
0&1&0&0\\
1&0&0&0
\end{pmatrix},
\qquad
\alpha_y=
\begin{pmatrix}
0&0&0&-i\\
0&0&i&0\\
0&-i&0&0\\
i&0&0&0
\end{pmatrix},
```

\alpha_zUse in source · 2 passages

Search aliases: α_z alpha

Link to this entry ·

Use in source · finite_core.tex, line 94
α_z= 0 · 0 · 1 · 0 0 · 0 · 0 · -1 1 · 0 · 0 · 0 0 · -1 · 0 · 0 , β= 1 · 0 · 0 · 0 0 · 1 · 0 · 0 0 · 0 · -1 · 0 0 · 0 · 0 · -1 .
Exact mathematical passage
```
\alpha_z=
\begin{pmatrix}
0&0&1&0\\
0&0&0&-1\\
1&0&0&0\\
0&-1&0&0
\end{pmatrix},
\qquad
\beta=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&-1
\end{pmatrix}.
```

\betaUse in source · 33 passages

Search aliases: β beta

Link to this entry ·

Use in source · finite.tex, line 559
[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber
Exact mathematical passage
```
\beta
```

\beta^*Use in source · 2 passages

Search aliases: β^* beta

Link to this entry ·

Use in source · finite_core.tex, line 140
α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,
Exact mathematical passage
```
\alpha_j^*=\alpha_j,\quad \beta^*=\beta,\quad
\alpha_j^2=\beta^2=I_4,
```

\beta^2Use in source · 2 passages

Search aliases: β^2 beta

Link to this entry ·

Use in source · finite_core.tex, line 140
α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,
Exact mathematical passage
```
\alpha_j^*=\alpha_j,\quad \beta^*=\beta,\quad
\alpha_j^2=\beta^2=I_4,
```

\GammaUse in source · 15 passages

Search aliases: Γ Gamma

Link to this entry ·

Use in source · core.tex, line 248
A resolution can change with time even when the state is stationary. To retain orthogonality, transport the entire atlas by a unitary family. Let U(t) be differentiable in operator norm and satisfy both U^*U=id and UU^*=id. Set P_a(t)=U(t)P_a^(0)U(t)^*, ϕ(t)=U(t)^*ψ(t), Γ(t)=U(t)^*dot U(t). The coordinates ϕ see fixed projectors P_a^(0). The physical weight in the moving packet is the same scalar: normP_a(t)ψ(t)^2 =normP_a^(0)ϕ(t)^2.
Exact mathematical passage
```
\label{f:moving-defs}
 P_a(t)=U(t)P_a^{(0)}U(t)^*,\qquad
 \phi(t)=U(t)^*\psi(t),\qquad \Gamma(t)=U(t)^*\dot U(t).
```

\gammaUse in source · 59 passages

Search aliases: γ gamma

Link to this entry ·

Use in source · finite.tex, line 685
se projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional example H_f=0 · overlineγ γ · 0, P=1 · 0 0 · 0 has compressed Hamiltonian zero but detail probability sin^2(|γ|t) from the first basis state. It fails exact refinement compatibility whenever γ≠0.
Exact mathematical passage
```
 H_f=\begin{pmatrix}0&\overline\gamma\\\gamma&0\end{pmatrix},
 \qquad P=\begin{pmatrix}1&0\\0&0\end{pmatrix}
```

\Gamma^*Use in source · 2 passages

Search aliases: Γ^* Gamma

Link to this entry ·

Use in source · core.tex, line 260
[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.
Exact mathematical passage
```
\label{f:effective-generator}
 \dot\phi=K_{\mathrm{eff}}\phi,\qquad
 K_{\mathrm{eff}}=U^*KU-\Gamma,\qquad
 \Gamma^*=-\Gamma,\quad K_{\mathrm{eff}}^*=-K_{\mathrm{eff}}.
```

\gamma^0Use in source · 3 passages

Search aliases: γ^0 gamma

Link to this entry ·

Use in source · constraints.tex, line 580
Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†…
Exact mathematical passage
```
\bar\psi=\psi^\dagger\gamma^0
```

\gamma^1Use in source · 1 passage

Search aliases: γ^1 gamma

Link to this entry ·

Use in source · constraints.tex, line 611
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3
```

\gamma^2Use in source · 1 passage

Search aliases: γ^2 gamma

Link to this entry ·

Use in source · constraints.tex, line 611
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3
```

\gamma^3Use in source · 1 passage

Search aliases: γ^3 gamma

Link to this entry ·

Use in source · constraints.tex, line 611
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3
```

\Gamma^iUse in source · 1 passage

Search aliases: Γ^i Gamma

Link to this entry ·

Use in source · constraints.tex, line 554
E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ term. Those directions …
Exact mathematical passage
```
\Gamma^i=-\tfrac12\epsilon^i{}_{jk}\Omega^{jk}
```

\gamma^IUse in source · 7 passages

Search aliases: γ^I gamma

Link to this entry ·

Use in source · constraints.tex, line 295
vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,
Exact mathematical passage
```
\{\gamma^I,\gamma^J\}=2\eta^{IJ}
```

\gamma^JUse in source · 3 passages

Search aliases: γ^J gamma

Link to this entry ·

Use in source · constraints.tex, line 295
vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,
Exact mathematical passage
```
\{\gamma^I,\gamma^J\}=2\eta^{IJ}
```

\gamma^KUse in source · 1 passage

Search aliases: γ^K gamma

Link to this entry ·

Use in source · constraints.tex, line 465
C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.
Exact mathematical passage
```
 \mathcal G_{IJ}
 &=D_\Omega B_{IJ}
       -\tfrac i8\sigma_K\bar\psi\{\gamma^K,\gamma_{IJ}\}\psi,\\
 \mathcal G_{\mathrm U}
 &=\mathrm d_\Sigma\Pi-q\sigma_I\bar\psi\gamma^I\psi.
```

\gamma_0Use in source · 3 passages

Search aliases: γ_0 gamma

Link to this entry ·

Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\gamma_0^2Use in source · 2 passages

Search aliases: γ_0^2 gamma

Link to this entry ·

Use in source · spectral.tex, line 469
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this…
Exact mathematical passage
```
 \Sigma_D(z)=\sigma(z)I_2,\qquad
 \sigma(z)=\gamma_0^2\langle1,(z-J_{1:h})^{-1}1\rangle.
```

\gamma_1Use in source · 1 passage

Search aliases: γ_1 gamma

Link to this entry ·

Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\gamma_{12}Use in source · 1 passage

Search aliases: γ_12 gamma

Link to this entry ·

Use in source · core.tex, line 444
[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding -i[H,ρ] to [source reference] adds the coherent current to the jump current. Th…
Exact mathematical passage
```
\gamma_{21}=\gamma_{12}=k>0
```

\gamma_{21}Use in source · 1 passage

Search aliases: γ_21 gamma

Link to this entry ·

Use in source · core.tex, line 444
[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding -i[H,ρ] to [source reference] adds the coherent current to the jump current. Th…
Exact mathematical passage
```
\gamma_{21}=\gamma_{12}=k>0
```

\gamma_5Use in source · 2 passages

Search aliases: γ_5 gamma

Link to this entry ·

Use in source · constraints.tex, line 611
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3
```

\Gamma_aUse in source · 1 passage

Search aliases: Γ_a Gamma

Link to this entry ·

Use in source · constraints.tex, line 621
a^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter…
Exact mathematical passage
```
 \Gamma_a{}^i=\mathring\Gamma_a{}^i(E)
                     -\tfrac\kappa4J_5^0e_a{}^i.
```

\gamma_{a+1}Use in source · 1 passage

Search aliases: γ_a+1 gamma

Link to this entry ·

Use in source · spectral.tex, line 385
ldots,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-counting calculation as in [source reference] produces the neighboring chain vectors.
Exact mathematical passage
```
\gamma_{a+1},\ldots,\gamma_{h-1}
```

\gamma_{ab}Use in source · 2 passages

Search aliases: γ_ab gamma

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Use in source · core.tex, line 420
[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.
Exact mathematical passage
```
\label{f:jump-current}
 \dot p_b=\sum_aF_{b\leftarrow a},\qquad
 F_{b\leftarrow a}=\gamma_{ba}p_a-\gamma_{ab}p_b,
 \qquad F_{b\leftarrow a}=-F_{a\leftarrow b}.
```

\gamma_{ba}Use in source · 8 passages

Search aliases: γ_ba gamma

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Use in source · core.tex, line 406
Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected bas…
Exact mathematical passage
```
\gamma_{ba}\ge0
```

\gamma_{bi}Use in source · 1 passage

Search aliases: γ_bi gamma

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Use in source · core.tex, line 437
In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable …
Exact mathematical passage
```
r_i=\sum_{b\ne i}\gamma_{bi}
```

\gamma_{h-1}Use in source · 2 passages

Search aliases: γ_h-1 gamma

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Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\gamma_IUse in source · 1 passage

Search aliases: γ_I gamma

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Use in source · constraints.tex, line 296
vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,
Exact mathematical passage
```
\gamma_{IJ}=\tfrac12[\gamma_I,\gamma_J]
```

\gamma_iUse in source · 3 passages

Search aliases: γ_i gamma

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Use in source · refinement.tex, line 176
The straight compatibility contract. For each direction, canonical fine links can be defined by v_i(p)=r(σ_i p) γ_i(p) r(p)^-1, where γ_i(p)=1 on a noncarrying half step and γ_i(p)=u_i(s) on a carrying half step. Cancellation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument…
Exact mathematical passage
```
 v_i(p)=r(\sigma_i p)\,\gamma_i(p)\,r(p)^{-1},
```

\gamma_{IJ}Use in source · 6 passages

Search aliases: γ_IJ gamma

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Use in source · constraints.tex, line 296
vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,
Exact mathematical passage
```
\gamma_{IJ}=\tfrac12[\gamma_I,\gamma_J]
```

\gamma_JUse in source · 1 passage

Search aliases: γ_J gamma

Link to this entry ·

Use in source · constraints.tex, line 296
vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,
Exact mathematical passage
```
\gamma_{IJ}=\tfrac12[\gamma_I,\gamma_J]
```

\gamma_{p,w}Use in source · 5 passages

Search aliases: γ_p,w gamma

Link to this entry ·

Use in source · finite.tex, line 738
Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coeffi
Exact mathematical passage
```
\gamma_{p,w}
```

\gamma_rUse in source · 5 passages

Search aliases: γ_r gamma

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Use in source · spectral.tex, line 326
Let T_p,h be a rooted p-ary tree of depth h≥1. Vertices at depth r have onsite energy ϵ_r∈R. Every parent-to-child edge from level r to r+1 has real coupling γ_r/sqrt p, where γ_r∈R. The Hamiltonian is the corresponding real symmetric adjacency matrix with onsite diagonal. The level symmetry and this branch normalization are structural assumptions of the model.
Exact mathematical passage
```
\gamma_r/\sqrt p
```

\gamma_{r-1}Use in source · 2 passages

Search aliases: γ_r-1 gamma

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Use in source · spectral.tex, line 340
Define normalized level states |r⟩=p^-r/2∑_|v|=r|v⟩, r=0,…,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributions instead give p p^-r/2γ_r-1/sqrt p =γ_r-1p^-(r-1)/2.
Exact mathematical passage
```
 H|r\rangle=\gamma_{r-1}|r-1\rangle
              +\epsilon_r|r\rangle+\gamma_r|r+1\rangle,
 \label{a-spectral-radial-action}
```

\gamma_r^2Use in source · 2 passages

Search aliases: γ_r^2 gamma

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Use in source · spectral.tex, line 433
For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rationa…
Exact mathematical passage
```
 g_h(z)&=(z-\epsilon_h)^{-1},\nonumber\\
 g_r(z)&=\frac1{z-\epsilon_r-\gamma_r^2g_{r+1}(z)},
          \qquad r<h.
 \label{a-spectral-continued-fraction}
```

\DeltaUse in source · 53 passages

Search aliases: Δ Delta

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Use in source · finite_core.tex, line 510
Δ_h,n:=H_h,n-H_n =∑_j=x,y,zδ_j(h,n)α_j, δ_j(h,n)=q_j(h,p(n))-p_j(n).
Exact mathematical passage
```
\Delta_{h,n}:=H_{h,n}-H_n
=\sum_{j=x,y,z}\delta_j(h,n)\alpha_j,
\qquad
\delta_j(h,n)=q_j(h,p(n))-p_j(n).
```

\delta^3Use in source · 2 passages

Search aliases: δ^3 delta

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Use in source · constraints.tex, line 586
the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
 \{\chi_A(x),\bar\chi_B(y)\}_{\mathrm P}
            =-ie(x)\delta_{AB}\delta^3(x-y).
```

\delta_{\mathcal F_{ij}}Use in source · 1 passage

Search aliases: δ_ F_ij delta mathcal

Link to this entry ·

Use in source · extensions.tex, line 480
For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely when that X commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additiona…
Exact mathematical passage
```
 [\delta_{B_i},\delta_{B_j}]X
 =-[[B_i,B_j],X]
 =i[\mathcal F_{ij},X]=\delta_{\mathcal F_{ij}}X.
 \label{eq:foundation-corner-curvature}
```

\delta_1Use in source · 1 passage

Search aliases: δ_1 delta

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Use in source · core.tex, line 605
Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric tensors.
Exact mathematical passage
```
\delta_1,\delta_2,\delta_3
```

\delta_2Use in source · 1 passage

Search aliases: δ_2 delta

Link to this entry ·

Use in source · core.tex, line 605
Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric tensors.
Exact mathematical passage
```
\delta_1,\delta_2,\delta_3
```

\delta_3Use in source · 1 passage

Search aliases: δ_3 delta

Link to this entry ·

Use in source · core.tex, line 605
Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric tensors.
Exact mathematical passage
```
\delta_1,\delta_2,\delta_3
```

\delta_{\left[U,V\right]_{\mathrm{par}}}Use in source · 1 passage

Search aliases: δ_[U,V]_par delta left right mathrm

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Use in source · constraints.tex, line 350
The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.
Exact mathematical passage
```
 [\delta_U,\delta_V]_{\mathrm{fld}}
       =-\delta_{\left[U,V\right]_{\mathrm{par}}},
```

\delta_aUse in source · 5 passages

Search aliases: δ_a delta

Link to this entry ·

Use in source · constraints.tex, line 189
|_∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
\delta_a
```

\delta_{AB}Use in source · 2 passages

Search aliases: δ_AB delta

Link to this entry ·

Use in source · constraints.tex, line 586
the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
 \{\chi_A(x),\bar\chi_B(y)\}_{\mathrm P}
            =-ie(x)\delta_{AB}\delta^3(x-y).
```

\delta_bUse in source · 2 passages

Search aliases: δ_b delta

Link to this entry ·

Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

\delta_{B_i}Use in source · 1 passage

Search aliases: δ_B_i delta

Link to this entry ·

Use in source · extensions.tex, line 480
For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely when that X commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additiona…
Exact mathematical passage
```
 [\delta_{B_i},\delta_{B_j}]X
 =-[[B_i,B_j],X]
 =i[\mathcal F_{ij},X]=\delta_{\mathcal F_{ij}}X.
 \label{eq:foundation-corner-curvature}
```

\delta_{B_j}Use in source · 1 passage

Search aliases: δ_B_j delta

Link to this entry ·

Use in source · extensions.tex, line 480
For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely when that X commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additiona…
Exact mathematical passage
```
 [\delta_{B_i},\delta_{B_j}]X
 =-[[B_i,B_j],X]
 =i[\mathcal F_{ij},X]=\delta_{\mathcal F_{ij}}X.
 \label{eq:foundation-corner-curvature}
```

\Delta_CUse in source · 2 passages

Search aliases: Δ_C Delta

Link to this entry ·

Use in source · constraints.tex, line 815
The conditional BV pushforward statement. Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contri…
Exact mathematical passage
```
\Delta_F=\Delta_C+\Delta_D
```

\Delta_DUse in source · 2 passages

Search aliases: Δ_D Delta

Link to this entry ·

Use in source · constraints.tex, line 815
The conditional BV pushforward statement. Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contri…
Exact mathematical passage
```
\Delta_F=\Delta_C+\Delta_D
```

\delta_{de}Use in source · 2 passages

Search aliases: δ_de delta

Link to this entry ·

Use in source · finite.tex, line 458
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
p^{-1}\sum_\ell\omega_p^{(e-d)\ell}=\delta_{de}
```

\Delta_FUse in source · 2 passages

Search aliases: Δ_F Delta

Link to this entry ·

Use in source · constraints.tex, line 810
The conditional BV pushforward statement. Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contri…
Exact mathematical passage
```
 \tfrac12(S_F,S_F)-i\hbar\Delta_FS_F=0.
 \label{c:constraints:qme}
```

\Delta_hUse in source · 2 passages

Search aliases: Δ_h Delta

Link to this entry ·

Use in source · core.tex, line 470
Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber
Exact mathematical passage
```
 (D_hf)_j=\frac{f_{j+1}-f_{j-1}}{2h},\qquad
 (\Delta_hf)_j=\frac{f_{j+1}-2f_j+f_{j-1}}{h^2}.
```

\Delta_{h,n}Use in source · 8 passages

Search aliases: Δ_h,n Delta

Link to this entry ·

Use in source · finite_core.tex, line 510
Δ_h,n:=H_h,n-H_n =∑_j=x,y,zδ_j(h,n)α_j, δ_j(h,n)=q_j(h,p(n))-p_j(n).
Exact mathematical passage
```
\Delta_{h,n}:=H_{h,n}-H_n
=\sum_{j=x,y,z}\delta_j(h,n)\alpha_j,
\qquad
\delta_j(h,n)=q_j(h,p(n))-p_j(n).
```

\Delta_{h,n}^*Use in source · 2 passages

Search aliases: Δ_h,n^* Delta

Link to this entry ·

Use in source · finite_core.tex, line 572
Δ_h,nv _2^2 =⟨ v,Δ_h,n^*Δ_h,nv⟩ =|δ(h,n)|^2 v _2^2.
Exact mathematical passage
```
\|\Delta_{h,n}v\|_2^2
=\langle v,\Delta_{h,n}^*\Delta_{h,n}v\rangle
=|\delta(h,n)|^2\|v\|_2^2.
```

\Delta_{h,n}^2Use in source · 1 passage

Search aliases: Δ_h,n^2 Delta

Link to this entry ·

Use in source · finite_core.tex, line 570
The Hermitian matrix Δ_h,n satisfies Δ_h,n^2=|δ(h,n)|^2I_4 by PH01. Hence
Exact mathematical passage
```
\Delta_{h,n}^2=|\delta(h,n)|^2I_4
```

\delta_iUse in source · 14 passages

Search aliases: δ_i delta

Link to this entry ·

Use in source · refinement.tex, line 265
For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need …
Exact mathematical passage
```
\delta_i=\delta(u_i,v_i)
```

\delta_i^2Use in source · 1 passage

Search aliases: δ_i^2 delta

Link to this entry ·

Use in source · core.tex, line 606
Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric tensors.
Exact mathematical passage
```
 \delta_i\delta_j=\delta_j\delta_i,
 \qquad\inner{\delta_iX}{Y}_h=-\inner{X}{\delta_iY}_h,
 \qquad\delta^2=\sum_i\delta_i^2.
```

\delta_{ij}Use in source · 6 passages

Search aliases: δ_ij delta

Link to this entry ·

Use in source · core.tex, line 618
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].
Exact mathematical passage
```
 C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
 &C_i&=-2\delta_j\pi_{ij},\nonumber\\
 \mathcal G_{ij}(h)&=\frac12\big[
 \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
 -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
 -\delta_{ij}C\big].
 \label{c:core:adm-operators}
```

\Delta_jUse in source · 4 passages

Search aliases: Δ_j Delta

Link to this entry ·

Use in source · core.tex, line 30
ites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jd…
Exact mathematical passage
```
\Delta_j=\varphi_{j+1}-\varphi_j
```

\delta_jUse in source · 16 passages

Search aliases: δ_j delta

Link to this entry ·

Use in source · finite_core.tex, line 510
Δ_h,n:=H_h,n-H_n =∑_j=x,y,zδ_j(h,n)α_j, δ_j(h,n)=q_j(h,p(n))-p_j(n).
Exact mathematical passage
```
\Delta_{h,n}:=H_{h,n}-H_n
=\sum_{j=x,y,z}\delta_j(h,n)\alpha_j,
\qquad
\delta_j(h,n)=q_j(h,p(n))-p_j(n).
```

\Delta_{j-1}Use in source · 1 passage

Search aliases: Δ_j-1 Delta

Link to this entry ·

Use in source · core.tex, line 38
equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms in the action produce the two terms in the spatial numerator. The time integration by parts differentiates w_jdotφ_j, including the dot w_jdotφ_j term.
Exact mathematical passage
```
 w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
 =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
 -v_j\varphi_j.
 \label{c:core:scalar-evolution}
```

\Delta_{j-1}^2Use in source · 1 passage

Search aliases: Δ_j-1^2 Delta

Link to this entry ·

Use in source · core.tex, line 53
[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and do…
Exact mathematical passage
```
 e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
 +\frac{c_{j+1/2}}{4h}\Delta_j^2
 +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
 F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
             (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
 s_j&=-\frac h2\dot w_j\dot\varphi_j^2
 +\frac h2\dot v_j\varphi_j^2
 +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
 +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
 \label{c:core:scalar-densities}
```

\Delta_j^2Use in source · 4 passages

Search aliases: Δ_j^2 Delta

Link to this entry ·

Use in source · core.tex, line 31
w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring…
Exact mathematical passage
```
 L_h=\frac h2\sum_jw_j\dot\varphi_j^2
 -\frac1{2h}\sum_jc_{j+1/2}\Delta_j^2
 -\frac h2\sum_jv_j\varphi_j^2.
 \label{c:core:scalar-action}
```

\delta_{jk}Use in source · 2 passages

Search aliases: δ_jk delta

Link to this entry ·

Use in source · spectral.tex, line 909
, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follo…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
 \quad
 \mathfrak R_{L,j}\mathfrak R_{L,k}
   =\delta_{jk}\mathfrak R_{L,j},\quad
 \sum_j\mathfrak R_{L,j}=I.
 \label{a-spectral-analytic-lanes}
```

\delta_KUse in source · 4 passages

Search aliases: δ_K delta

Link to this entry ·

Use in source · core.tex, line 931
For the full matrix algebra, choose a Hermitian matrix K and set δ_KX=i[K,X]. Direct multiplication gives δ_K(XY)=(δ_KX)Y+X(δ_KY). It also obeys δ_K(X^*)=(δ_KX)^*, Trδ_KX=0, and is skew-adjoint for the Hilbert--Schmidt inner product. The last identity follows by moving K through the trace. These properties supply an exact finite product calculus on a noncommutative algebra.
Exact mathematical passage
```
\delta_KX=i[K,X]
```

\delta_kUse in source · 7 passages

Search aliases: δ_k delta

Link to this entry ·

Use in source · core.tex, line 618
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].
Exact mathematical passage
```
 C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
 &C_i&=-2\delta_j\pi_{ij},\nonumber\\
 \mathcal G_{ij}(h)&=\frac12\big[
 \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
 -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
 -\delta_{ij}C\big].
 \label{c:core:adm-operators}
```

\delta_lUse in source · 1 passage

Search aliases: δ_l delta

Link to this entry ·

Use in source · core.tex, line 636
Twice its divergence expands as 2δ_j G_ij = · δ_iδ_jδ_kh_jk +δ^2δ_kh_ik-δ^2δ_jh_ij -δ_iδ^2h_rm tr · -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h
Exact mathematical passage
```
 2\delta_j\mathcal G_{ij}
 ={}&\delta_i\delta_j\delta_kh_{jk}
 +\delta^2\delta_kh_{ik}-\delta^2\delta_jh_{ij}
 -\delta_i\delta^2h_{\rm tr}\\
 &-\delta_i\delta_k\delta_lh_{kl}
 +\delta_i\delta^2h_{\rm tr}.
```

\delta_{mn}Use in source · 1 passage

Search aliases: δ_mn delta

Link to this entry ·

Use in source · limit.tex, line 151
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
 \mathsf F_m\mathsf F_n=\delta_{mn}\mathsf F_m,
 \qquad \sum_{m=0}^M\mathsf F_m=\mathsf E_M.
 \label{a-limit-detail-algebra}
```

\delta_{pn,m}Use in source · 1 passage

Search aliases: δ_pn,m delta

Link to this entry ·

Use in source · spectral.tex, line 854
ty and the binomial variance bound, by separating |k/n-t|≤δ from its complement and then letting n→∞, δ→0. Continuous functions are dense for this finite measure, as one sees by approximating interval steps in the angle coordinate by continuous ramps. This supplies completeness. Finally, Chebyshev composition gives K_pϕ_n=ϕ_pn. For every n,m≥0, ⟨ϕ_n, K_p^*ϕ_m⟩ =⟨ϕ_pn,ϕ_m⟩=δ_pn,m. Completeness yields the adjoint formula.
Exact mathematical passage
```
\langle\phi_n,\mathcal K_p^*\phi_m\rangle
=\langle\phi_{pn},\phi_m\rangle=\delta_{pn,m}
```

\delta_{qq'}Use in source · 2 passages

Search aliases: δ_qq' delta

Link to this entry ·

Use in source · finite.tex, line 343
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
 p^{-1}\sum_{d=0}^{p-1}\omega_p^{(q'-q)d}=\delta_{qq'}.
```

\delta_{rs}Use in source · 2 passages

Search aliases: δ_rs delta

Link to this entry ·

Use in source · finite.tex, line 279
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber rankD_0^(h) · =1, · rankD_r^(h) · =(p-1)p^r-1 (r≥1).
Exact mathematical passage
```
 \sum_{r=0}^hD_r^{(h)}&=I,
 &D_r^{(h)}D_s^{(h)}&=\delta_{rs}D_r^{(h)},\nonumber\\
 \operatorname{rank}D_0^{(h)}&=1,
 &\operatorname{rank}D_r^{(h)}&=(p-1)p^{r-1}\quad(r\geq1).
 \label{a-finite-detail-ranks}
```

\delta_{rt}Use in source · 1 passage

Search aliases: δ_rt delta

Link to this entry ·

Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

\delta_{st}Use in source · 1 passage

Search aliases: δ_st delta

Link to this entry ·

Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

\Delta_UUse in source · 4 passages

Search aliases: Δ_U Delta

Link to this entry ·

Use in source · core.tex, line 27
[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
 (D^+_{\mu,U}\psi)_x
 &=\frac{U_{x,\mu}\psi_{x+\widehat\mu}-\psi_x}{h},\label{g:forward-difference}\\
 (\Delta_U\psi)_x
 &=\frac1{h^2}\sum_\mu\left(
 U_{x,\mu}\psi_{x+\widehat\mu}
 +U_{x-\widehat\mu,\mu}^{-1}\psi_{x-\widehat\mu}-2\psi_x\right).
 \label{g:laplacian}
```

\delta_UUse in source · 1 passage

Search aliases: δ_U delta

Link to this entry ·

Use in source · constraints.tex, line 350
The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.
Exact mathematical passage
```
 [\delta_U,\delta_V]_{\mathrm{fld}}
       =-\delta_{\left[U,V\right]_{\mathrm{par}}},
```

\delta_uUse in source · 2 passages

Search aliases: δ_u delta

Link to this entry ·

Use in source · constraints.tex, line 649
For example, the residual electric constraint is G_ U^red=∂_a E^a+qΨ^†Ψ, Q[u]=-∫_Σ u G_ U^red d^3x. It generates δ_uA_a=∂_au and δ_uΨ=iquΨ. The half-density Lie derivative in the spatial generator is L_η^(1/2)Ψ =η^a∂_aΨ+tfrac12(∂_aη^a)Ψ. This term is forced by the density weight. Treating Ψ as a scalar would give a different moment map.
Exact mathematical passage
```
\delta_uA_a=\partial_au
```

\Delta_{U^g}Use in source · 1 passage

Search aliases: Δ_U^g Delta

Link to this entry ·

Use in source · core.tex, line 37
(Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
\Delta_{U^g}(g\psi)=g\Delta_U\psi
```

\delta_VUse in source · 1 passage

Search aliases: δ_V delta

Link to this entry ·

Use in source · constraints.tex, line 350
The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.
Exact mathematical passage
```
 [\delta_U,\delta_V]_{\mathrm{fld}}
       =-\delta_{\left[U,V\right]_{\mathrm{par}}},
```

\delta_xUse in source · 3 passages

Search aliases: δ_x delta

Link to this entry ·

Use in source · refinement.tex, line 266
both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need not be attained by the complete walk: …
Exact mathematical passage
```
 \norm{W_v-W_u}\leq 2\delta_x+2\delta_y+\delta_z.
 \label{g:sevenbudget}
```

\delta_yUse in source · 3 passages

Search aliases: δ_y delta

Link to this entry ·

Use in source · refinement.tex, line 266
both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need not be attained by the complete walk: …
Exact mathematical passage
```
 \norm{W_v-W_u}\leq 2\delta_x+2\delta_y+\delta_z.
 \label{g:sevenbudget}
```

\delta_zUse in source · 3 passages

Search aliases: δ_z delta

Link to this entry ·

Use in source · refinement.tex, line 266
both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] prove normW_v-W_u≤ 2δ_x+2δ_y+δ_z. The repeated X and Y factors explain the coefficients two. The upper bound need not be attained by the complete walk: …
Exact mathematical passage
```
 \norm{W_v-W_u}\leq 2\delta_x+2\delta_y+\delta_z.
 \label{g:sevenbudget}
```

\Delta_\DeltaUse in source · 4 passages

Search aliases: Δ_Δ Delta Delta

Link to this entry ·

Use in source · constraints.tex, line 685
equation overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse are skew, transposition proves (J_ D^Δ)^T=-J_ D^Δ. The endpoint tensor is a projection of the canonical direction into the kernel…
Exact mathematical passage
```
\Delta_\Delta=GJG^T
```

\Delta_\Delta^{-1}Use in source · 1 passage

Search aliases: Δ_Δ^-1 Delta Delta

Link to this entry ·

Use in source · constraints.tex, line 686
labelc:constraints:discrete-gradient The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse are skew, transposition proves (J_ D^Δ)^T=-J_ D^Δ. The endpoint tensor is a projection of the canonical direction into the kernel of all discrete cons…
Exact mathematical passage
```
 J_{\mathrm D}^{\Delta}
       =J-JG^T\Delta_\Delta^{-1}GJ.
 \label{c:constraints:endpoint-tensor}
```

\delta_{\mu\nu}Use in source · 1 passage

Search aliases: δ_μν delta mu nu

Link to this entry ·

Use in source · core.tex, line 377
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
R_{a\mu}R_{b\nu}=\delta_{ab}\delta_{\mu\nu}R_{a\mu}
```

\varepsilonUse in source · 29 passages

Search aliases: ε varepsilon

Link to this entry ·

Use in source · limit.tex, line 120
Given ε>0, choose g∈ V_M with normf-g_2<ε. For m≥ M, the projection fixes g, so normf- E_m f_2 =norm(I- E_m)(f-g)_2≤normf-g_2<ε. This proves strong convergence without exchanging an infinite integral with a limit.
Exact mathematical passage
```
\varepsilon>0
```

\varepsilon^2Use in source · 1 passage

Search aliases: ε^2 varepsilon

Link to this entry ·

Use in source · extensions.tex, line 434
The theorem concerns the semisimple pointwise algebra C^N. It does not assert that every finite-dimensional commutative complex algebra has only the zero derivation. For example, on C[ε]/(ε^2), the map D(a+bε)=bε is a nonzero derivation. The nilpotent element changes the algebraic hypothesis.
Exact mathematical passage
```
\C[\varepsilon]/(\varepsilon^2)
```

\epsilon^{abc}Use in source · 4 passages

Search aliases: ϵ^abc epsilon

Link to this entry ·

Use in source · constraints.tex, line 522
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
 \widetilde n_I=\tfrac1{3!}\epsilon_{IJKL}\epsilon^{abc}E_a^JE_b^KE_c^L.
```

\epsilon^IUse in source · 1 passage

Search aliases: ϵ^I epsilon

Link to this entry ·

Use in source · constraints.tex, line 614
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
 T^I=\tfrac\kappa4\epsilon^I{}_{JKL}J_5^Je^K\wedge e^L,
 \qquad
 C^{IJ}=\tfrac\kappa4\epsilon^{IJ}{}_{KL}J_5^Ke^L,
```

\epsilon^iUse in source · 3 passages

Search aliases: ϵ^i epsilon

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

\epsilon^{IJ}Use in source · 1 passage

Search aliases: ϵ^IJ epsilon

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Use in source · constraints.tex, line 614
Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representa…
Exact mathematical passage
```
 T^I=\tfrac\kappa4\epsilon^I{}_{JKL}J_5^Je^K\wedge e^L,
 \qquad
 C^{IJ}=\tfrac\kappa4\epsilon^{IJ}{}_{KL}J_5^Ke^L,
```

\epsilon^{ij}Use in source · 1 passage

Search aliases: ϵ^ij epsilon

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

\epsilon^{ijk}Use in source · 1 passage

Search aliases: ϵ^ijk epsilon

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Use in source · constraints.tex, line 213
H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
\det E=\tfrac16\epsilon_{abc}\epsilon^{ijk}E^a_iE^b_jE^c_k
```

\epsilon_{0123}Use in source · 1 passage

Search aliases: ϵ_0123 epsilon

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Use in source · constraints.tex, line 287
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,
Exact mathematical passage
```
\epsilon_{0123}=1
```

\epsilon_1Use in source · 1 passage

Search aliases: ϵ_1 epsilon

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Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\epsilon_2Use in source · 1 passage

Search aliases: ϵ_2 epsilon

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Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\epsilon_{a+1}Use in source · 1 passage

Search aliases: ϵ_a+1 epsilon

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Use in source · spectral.tex, line 384
choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-counting calculation as in [source reference] produces the neighboring chain vectors.
Exact mathematical passage
```
\epsilon_{a+1},\ldots,\epsilon_h
```

\epsilon_{a,\mathbf r}Use in source · 1 passage

Search aliases: ϵ_a, r epsilon mathbf

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Use in source · limit.tex, line 649
For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting wholly inside a new detail complement has no matrix element between that complement and the old range. A cross term between them would be a different…
Exact mathematical passage
```
 A_{\mathbf r}
 =\left(\sum_a\epsilon_{a,\mathbf r}T_a\right)
 \otimes I_{\mathrm{phase},\mathbf r},
 \qquad \epsilon_{a,\mathbf r}\in\R,\quad T_a=T_a^*.
```

\epsilon_{abc}Use in source · 1 passage

Search aliases: ϵ_abc epsilon

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Use in source · constraints.tex, line 213
H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
\det E=\tfrac16\epsilon_{abc}\epsilon^{ijk}E^a_iE^b_jE^c_k
```

\epsilon_hUse in source · 3 passages

Search aliases: ϵ_h epsilon

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Use in source · spectral.tex, line 354
Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decomposition, rather than only a compressed approximation. The root belongs to the radial block.
Exact mathematical passage
```
 J_h=\begin{pmatrix}
 \epsilon_0&\gamma_0&&&\\
 \gamma_0&\epsilon_1&\gamma_1&&\\
 &\gamma_1&\epsilon_2&\ddots&\\
 &&\ddots&\ddots&\gamma_{h-1}\\
 &&&\gamma_{h-1}&\epsilon_h
 \end{pmatrix}.
 \label{a-spectral-jacobi-matrix}
```

\epsilon_{ij}Use in source · 1 passage

Search aliases: ϵ_ij epsilon

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Use in source · constraints.tex, line 204
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),
Exact mathematical passage
```
 F_{ab}^i&=\delta_aA_b^i-\delta_bA_a^i
                    +\epsilon^i{}_{jk}A_a^jA_b^k,\\
 \mathcal G_i&=\delta_aE^a_i+\epsilon_{ij}{}^kA_a^jE^a_k,\\
 \mathcal V_a&=F_{ab}^iE^b_i,\qquad
 \mathcal D_a=\mathcal V_a-A_a^i\mathcal G_i,\\
 \mathcal H&=\tfrac12\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k
                                  +\Lambda\det E.
```

\epsilon_{ijk}Use in source · 2 passages

Search aliases: ϵ_ijk epsilon

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Use in source · constraints.tex, line 549
Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for…
Exact mathematical passage
```
 \widetilde E^a{}_i=\tfrac12\epsilon^{abc}\epsilon_{ijk}e_b{}^je_c{}^k,
 \qquad \widetilde E^a{}_ie_a{}^j=e\delta_i{}^j.
```

\epsilon_{IJKL}Use in source · 5 passages

Search aliases: ϵ_IJKL epsilon

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Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\epsilon_{JIKL}Use in source · 1 passage

Search aliases: ϵ_JIKL epsilon

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Use in source · constraints.tex, line 452
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ,
Exact mathematical passage
```
 \mathcal C_I^{\mathrm G}
 &=\tfrac1{2\kappa}\epsilon_{IJKL}E^J\wedge\mathcal R^{KL}
                  -\tfrac\Lambda\kappa\sigma_I,\\
 \mathcal C_I^{\mathrm M}
 &=\varphi_{IJ}\mathcal F\wedge E^J-\tfrac14\varphi_{KL}\varphi^{KL}\sigma_I,\\
 \mathcal C_I^{\mathrm D}
 &=\tfrac i4\epsilon_{JIKL}E^K\wedge E^L\wedge
       \bigl(\bar\psi\gamma^J\mathcal D_\Sigma\psi
                  -(\mathcal D_\Sigma\bar\psi)\gamma^J\psi\bigr)
                      -m\sigma_I\bar\psi\psi.
```

\epsilon_{KLIJ}Use in source · 1 passage

Search aliases: ϵ_KLIJ epsilon

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Use in source · constraints.tex, line 434
IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A
Exact mathematical passage
```
 B_{IJ}&=\tfrac1{4\kappa}\epsilon_{KLIJ}E^K\wedge E^L,\\
 \Pi&=\tfrac12\varphi_{IJ}E^I\wedge E^J,\qquad
 \sigma_I=\tfrac1{3!}\epsilon_{IJKL}E^J\wedge E^K\wedge E^L.
```

\epsilon_mUse in source · 26 passages

Search aliases: ϵ_m epsilon

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Use in source · limit.tex, line 240
Let (ϵ_m)_m≥0 be any sequence of real numbers. Define Dom(H_ϵ) · ={f∈HH_p: ∑_m≥0|ϵ_m|^2norm F_mf_2^2<∞}, nonumber
Exact mathematical passage
```
(\epsilon_m)_{m\geq0}
```

\epsilon_{m_j}Use in source · 1 passage

Search aliases: ϵ_m_j epsilon

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Use in source · limit.tex, line 491
g-resolvent. If the absolute energies tend to infinity, the right side of [source reference] tends to zero. The retained part of the resolvent has finite rank, so the resolvent is a norm limit of finite-rank operators and is compact. Conversely, if the absolute energies do not tend to infinity, there are infinitely many distinct levels m_j with |ϵ_m_j|≤ C for some C. Unit vectors u_j on these levels are orthogonal. For a fixed nonreal z, their resolvent images are also orthogonal and have norm a…
Exact mathematical passage
```
|\epsilon_{m_j}|\leq C
```

\varepsilon_nUse in source · 2 passages

Search aliases: ε_n varepsilon

Link to this entry ·

Use in source · refinement.tex, line 431
Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero need not converge; summability controls the accumulated tail.
Exact mathematical passage
```
 \varepsilon_n=C4^{-n},\qquad C=a_0^2\mathcal A_{\partial}.
```

\epsilon_{p,0}Use in source · 1 passage

Search aliases: ϵ_p,0 epsilon

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

\epsilon_{p,h+1}Use in source · 1 passage

Search aliases: ϵ_p,h+1 epsilon

Link to this entry ·

Use in source · finite.tex, line 598
silon_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonalizes the level blocks: T_p,hH_p,h^Haar T_p,h^* =∑_r=0^hϵ_p,rΠ_r. This is exact for each finite h and every finite real choice of the energies. Their growth becomes a domain question only wh…
Exact mathematical passage
```
\epsilon_{p,h+1}
```

\epsilon_{p,r}Use in source · 6 passages

Search aliases: ϵ_p,r epsilon

Link to this entry ·

Use in source · finite.tex, line 588
Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonalizes the level blocks: T_p,hH_p,h^Haar T_p,h^* =∑_r=0^hϵ_p,rΠ_r. This is exact for each finite h and every finite real choice of the energies. Th
Exact mathematical passage
```
\epsilon_{p,r}
```

\epsilon_rUse in source · 4 passages

Search aliases: ϵ_r epsilon

Link to this entry ·

Use in source · spectral.tex, line 325
Let T_p,h be a rooted p-ary tree of depth h≥1. Vertices at depth r have onsite energy ϵ_r∈R. Every parent-to-child edge from level r to r+1 has real coupling γ_r/sqrt p, where γ_r∈R. The Hamiltonian is the corresponding real symmetric adjacency matrix with onsite diagonal. The level symmetry and this branch normalization are structural assumptions of the model.
Exact mathematical passage
```
\epsilon_r\in\R
```

\epsilon_{r+1}Use in source · 1 passage

Search aliases: ϵ_r+1 epsilon

Link to this entry ·

Use in source · spectral.tex, line 440
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
\Sigma_r=\gamma_r^2/(z-\epsilon_{r+1}-\Sigma_{r+1})
```

\zetaUse in source · 8 passages

Search aliases: ζ zeta

Link to this entry ·

Use in source · finite.tex, line 30
[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ,
Exact mathematical passage
```
\zeta_{p,h}
```

\zeta^IUse in source · 1 passage

Search aliases: ζ^I zeta

Link to this entry ·

Use in source · constraints.tex, line 527
ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differ…
Exact mathematical passage
```
\zeta^I=\nu\widetilde n^I+\eta^aE_a^I
```

\zeta_{p,h}Use in source · 3 passages

Search aliases: ζ_p,h zeta

Link to this entry ·

Use in source · finite.tex, line 30
[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ,
Exact mathematical passage
```
\zeta_{p,h}
```

\zeta_{p,h+1}Use in source · 1 passage

Search aliases: ζ_p,h+1 zeta

Link to this entry ·

Use in source · finite.tex, line 108
The lifting construction in the proof permits the coherent choice ζ_p,h+1≡ζ_p,hpmodp^h+1. With that choice, reduction of unit addresses retains t and reduces a modulo p^h. If the torsion lifts are instead chosen independently at each depth, the finite factorizations remain exact, but a comparison between levels must include the resulting coordinate changes. A levelwise permutation is not by itself a specified refinement tower.
Exact mathematical passage
```
\zeta_{p,h+1}\equiv\zeta_{p,h}\pmod{p^{h+1}}
```

\zeta_{p,h}^{\,t}Use in source · 2 passages

Search aliases: ζ_p,h^ t zeta

Link to this entry ·

Use in source · finite.tex, line 32
[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ, (s,a) · ⟼(-1)^s5^apmod2^h+2, · (s,a) · ∈Z/2Z×Z/2^hZ are group isomorphisms onto U_p,h and U_2,h respectively.
Exact mathematical passage
```
 (t,a)&\longmapsto\zeta_{p,h}^{\,t}(1+p)^a
       \pmod{p^{h+1}},
 & (t,a)&\in\Z/(p-1)\Z\times\Z/p^h\Z,
 \label{a-finite-odd-coordinate}\\
 (s,a)&\longmapsto(-1)^s5^a\pmod{2^{h+2}},
 & (s,a)&\in\Z/2\Z\times\Z/2^h\Z
 \label{a-finite-dyadic-coordinate}
```

\etaUse in source · 35 passages

Search aliases: η eta

Link to this entry ·

Use in source · extensions.tex, line 362
For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.
Exact mathematical passage
```
i\eta
```

\eta^2Use in source · 2 passages

Search aliases: η^2 eta

Link to this entry ·

Use in source · extensions.tex, line 363
For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.
Exact mathematical passage
```
 \sup_{n>N}\frac1{\sqrt{n^2+\eta^2}}
 =\frac1{\sqrt{(N+1)^2+\eta^2}}\longrightarrow0.
```

\eta^aUse in source · 3 passages

Search aliases: η^a eta

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\eta^bUse in source · 1 passage

Search aliases: η^b eta

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\eta^{IJ}Use in source · 3 passages

Search aliases: η^IJ eta

Link to this entry ·

Use in source · constraints.tex, line 295
vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,
Exact mathematical passage
```
\{\gamma^I,\gamma^J\}=2\eta^{IJ}
```

\eta_aUse in source · 2 passages

Search aliases: η_a eta

Link to this entry ·

Use in source · core.tex, line 841
[Exact nonlinear defect identity] For arbitrary finite arrays q,p,ξ,η, { D_h[ξ], D_h[η]}_h = · D_h[ξ Dη-η Dξ]nonumber · +h∑_ap_a[ ξ_a L_D(η,Dq)_a -η_a L_D(ξ,Dq)_a].
Exact mathematical passage
```
 \{\mathcal D_h[\xi],\mathcal D_h[\eta]\}_h
 ={}&\mathcal D_h[\xi D\eta-\eta D\xi]\nonumber\\
 &+h\sum_ap_a\big[
 \xi_a\mathfrak L_D(\eta,Dq)_a
 -\eta_a\mathfrak L_D(\xi,Dq)_a\big].
 \label{c:core:nonlinear-defect-identity}
```

\eta_{IJ}Use in source · 2 passages

Search aliases: η_IJ eta

Link to this entry ·

Use in source · constraints.tex, line 287
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,
Exact mathematical passage
```
\eta_{IJ}=\operatorname{diag}(-1,1,1,1)
```

\eta_jUse in source · 3 passages

Search aliases: η_j eta

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\eta_{j+1}^-Use in source · 3 passages

Search aliases: η_j+1^- eta

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\eta_{j-1}^+Use in source · 2 passages

Search aliases: η_j-1^+ eta

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\eta_j^+Use in source · 2 passages

Search aliases: η_j^+ eta

Link to this entry ·

Use in source · core.tex, line 88
[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
\label{g:walk-current}
 J_{j+1/2}^n=|\eta_j^+|^2-|\eta_{j+1}^-|^2.
```

\eta_j^-Use in source · 1 passage

Search aliases: η_j^- eta

Link to this entry ·

Use in source · core.tex, line 101
The first coin preserves the local norm, so ρ_j^n=|η_j^+|^2+|η_j^-|^2. The second coin and temporal phase preserve the norm after the shift, giving ρ_j^n+1=|η_j-1^+|^2+|η_j+1^-|^2. Subtract these expressions and group the two terms associated with each boundary edge. This is [source reference]. The transformed η_j is g_j^nη_j, so its channel norms are unchanged. Summing the balance over the full cycle or the chosen arc cancels its internal currents.
Exact mathematical passage
```
\rho_j^n=|\eta_j^+|^2+|\eta_j^-|^2
```

\ThetaUse in source · 9 passages

Search aliases: Θ Theta

Link to this entry ·

Use in source · spectral.tex, line 1075
l A_L,j; frac(-1)^Ly^LL^2L). One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^2B_L,j(y)=-yB_L,j-1bmod L(y). For a term of positive degree n, its new coefficient is n^2(-1)^n/(n!)^2=-(-1)^n-1/[(n-1)!]…
Exact mathematical passage
```
\Theta=y\,d/dy
```

\thetaUse in source · 74 passages

Search aliases: θ theta

Link to this entry ·

Use in source · spectral.tex, line 96
Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1,
Exact mathematical passage
```
\theta_j=2\pi j/M
```

\varthetaUse in source · 2 passages

Search aliases: ϑ vartheta

Link to this entry ·

Use in source · core.tex, line 453
The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the separated Taylor channels are coupled. Their sum satisfies the Bessel equation. This decomposition resolves series order and does not introduce addition…
Exact mathematical passage
```
\vartheta=y\,d/dy
```

\Theta^2Use in source · 2 passages

Search aliases: Θ^2 Theta

Link to this entry ·

Use in source · spectral.tex, line 1077
One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^2B_L,j(y)=-yB_L,j-1bmod L(y). For a term of positive degree n, its new coefficient is n^2(-1)^n/(n!)^2=-(-1)^n-1/[(n-1)!]^2. The degree-zero term disa…
Exact mathematical passage
```
 \Theta^2B_{L,j}(y)=-yB_{L,j-1\bmod L}(y).
 \label{a-spectral-bessel-euler}
```

\theta^2Use in source · 1 passage

Search aliases: θ^2 theta

Link to this entry ·

Use in source · core.tex, line 188
of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These orders conce…
Exact mathematical passage
```
1-\cos\theta=\theta^2/2+O(\theta^4)
```

\vartheta^2Use in source · 1 passage

Search aliases: ϑ^2 vartheta

Link to this entry ·

Use in source · core.tex, line 454
The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the separated Taylor channels are coupled. Their sum satisfies the Bessel equation. This decomposition resolves series order and does not introduce addition…
Exact mathematical passage
```
 \vartheta^2\mathcal B_{r,j}+y\mathcal B_{r,j-1}=0,
 \qquad j\pmod r,
```

\theta^4Use in source · 1 passage

Search aliases: θ^4 theta

Link to this entry ·

Use in source · core.tex, line 188
of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These orders conce…
Exact mathematical passage
```
1-\cos\theta=\theta^2/2+O(\theta^4)
```

\Theta_{\mathrm D}Use in source · 2 passages

Search aliases: Θ_ D Theta mathrm

Link to this entry ·

Use in source · constraints.tex, line 581
Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†…
Exact mathematical passage
```
 \Theta_{\mathrm D}=\frac i2\int_\Sigma e
        (\psi^\dagger\delta\psi-\delta\psi^\dagger\psi)\,\mathrm d^3x.
```

\theta_+Use in source · 1 passage

Search aliases: θ_+ theta

Link to this entry ·

Use in source · strong_fields.tex, line 477
For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric di…
Exact mathematical passage
```
 \theta_+=\frac2r(1-v),\qquad
 \theta_-=-\frac2r(1+v).
\label{c:strong:expansions}
```

\theta_-Use in source · 1 passage

Search aliases: θ_- theta

Link to this entry ·

Use in source · strong_fields.tex, line 477
For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric di…
Exact mathematical passage
```
 \theta_+=\frac2r(1-v),\qquad
 \theta_-=-\frac2r(1+v).
\label{c:strong:expansions}
```

\theta_0Use in source · 1 passage

Search aliases: θ_0 theta

Link to this entry ·

Use in source · spectral.tex, line 875
The mean logarithmic expansion of this map is also exact: ∫_-2^2log|R_p'(x)| dν(x)=log p. Indeed for x=2cosθ, R_p'(x)=psin(pθ)/sinθ. Logarithms at its finitely many zeros are integrable, since they have the local form log|θ-θ_0| plus a bounded term. Hence the integral is log p plus the difference of the integrals of log|sin(pθ)| and logsinθ over [0,π]. The substitution u=pθ and π-periodicity of log|sin u| make these two integrals equal. This calculation does not assert the entropy of a different…
Exact mathematical passage
```
\log|\theta-\theta_0|
```

\theta_fUse in source · 2 passages

Search aliases: θ_f theta

Link to this entry ·

Use in source · core.tex, line 173
six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a principal argument, it implies that the oriented phase sum lies in 2πZ. It does not force that sum to be zero as a real number. An unwrapped small-phase re…
Exact mathematical passage
```
W_f=e^{i\theta_f}
```

\theta_iUse in source · 3 passages

Search aliases: θ_i theta

Link to this entry ·

Use in source · refinement.tex, line 327
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
 \theta_i(p,h)=\int_0^h A_i(p+t e_i)\,dt,
 \qquad u_i(p,h)=e^{i\theta_i(p,h)}.
 \label{g:integrated-links}
```

\theta_jUse in source · 6 passages

Search aliases: θ_j theta

Link to this entry ·

Use in source · spectral.tex, line 96
Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, · r<p-1,
Exact mathematical passage
```
\theta_j=2\pi j/M
```

\Theta_{\mathrm{red}}Use in source · 1 passage

Search aliases: Θ_red Theta mathrm

Link to this entry ·

Use in source · constraints.tex, line 635
The resulting real canonical superphase space has coordinates (K_a^i,widetilde E^a_i), (A_a, E^a), (Ψ,Ψ^†), and one-form Θ_red=∫_Σ d^3x [frac1κwidetilde E^a_iδ K_a^i + E^aδ A_a +frac i2(Ψ^†δΨ-δΨ^†Ψ)]. Its even canonical dimension is 24 per spatial point: eighteen gravitational and six electromagnetic coordinates. The odd primary block has eight components in the stated complexified graded presentation, with the real spinor relation imposed. These counts are local phase-space counts, not a count …
Exact mathematical passage
```
 \Theta_{\mathrm{red}}=\int_\Sigma\mathrm d^3x\,
 \left[\frac1\kappa\widetilde E^a{}_i\delta K_a{}^i
       +\mathcal E^a\delta A_a
       +\frac i2(\Psi^\dagger\delta\Psi-\delta\Psi^\dagger\Psi)\right].
 \label{c:constraints:reduced-one-form}
```

\theta_xUse in source · 1 passage

Search aliases: θ_x theta

Link to this entry ·

Use in source · refinement.tex, line 381
For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancella…
Exact mathematical passage
```
 \Theta_{xy}(p,h)
 ={}&\theta_y(p+h e_x,h)-\theta_y(p,h)\\
    &-\bigl(\theta_x(p+h e_y,h)-\theta_x(p,h)\bigr).
```

\Theta_{xy}Use in source · 2 passages

Search aliases: Θ_xy Theta

Link to this entry ·

Use in source · refinement.tex, line 381
For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancella…
Exact mathematical passage
```
 \Theta_{xy}(p,h)
 ={}&\theta_y(p+h e_x,h)-\theta_y(p,h)\\
    &-\bigl(\theta_x(p+h e_y,h)-\theta_x(p,h)\bigr).
```

\theta_yUse in source · 2 passages

Search aliases: θ_y theta

Link to this entry ·

Use in source · refinement.tex, line 381
For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancella…
Exact mathematical passage
```
 \Theta_{xy}(p,h)
 ={}&\theta_y(p+h e_x,h)-\theta_y(p,h)\\
    &-\bigl(\theta_x(p+h e_y,h)-\theta_x(p,h)\bigr).
```

\Theta_\SigmaUse in source · 1 passage

Search aliases: Θ_Σ Theta Sigma

Link to this entry ·

Use in source · constraints.tex, line 440
Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A · -tfrac i2σ_I(barψγ^Iδψ -δbarψγ^Iψ)]. The minus sign in the spinor term is fixed by the time--space orientation and by moving dt past the spatial three-form. It must be retained when comparing the spacetime action and its canonical chart.
Exact mathematical passage
```
 \begin{aligned}
 \Theta_\Sigma=\int_\Sigma\bigl[&B_{IJ}\wedge\delta\Omega^{IJ}+\Pi\wedge\delta A\\
 &-\tfrac i2\sigma_I(\bar\psi\gamma^I\delta\psi
                         -\delta\bar\psi\gamma^I\psi)\bigr].
 \end{aligned}
 \label{c:constraints:boundary-potential}
```

\iota_cUse in source · 2 passages

Search aliases: ι_c iota

Link to this entry ·

Use in source · constraints.tex, line 499
, the Cartan identities give L_ηΩ=ι_η R+D_Ω(ι_ηΩ), L_η A=ι_η F+ d_Σ(ι_η A). Accordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression J_Σ[c,χ,u]=C[ζ(c)] +G_ L[χ-ι_cΩ] +G_ U[u+ι_cA]. For tangential c, ζ(c)=ι_cE is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that contraction. One must specify the embedding variables and conjugate momenta,…
Exact mathematical passage
```
 J_\Sigma[c,\chi,u]=C[\zeta(c)]
             +G_{\mathrm L}[\chi-\iota_c\Omega]
             +G_{\mathrm U}[u+\iota_cA].
 \label{c:constraints:boundary-candidate}
```

\iota_LUse in source · 5 passages

Search aliases: ι_L iota

Link to this entry ·

Use in source · spectral.tex, line 162
B^* · D, A=A^*, D=D^*. Here B:HH_D→HH_L. Let ι_L embed HH_L into the full space. This notation fixes the coordinate type of every compression.
Exact mathematical passage
```
\iota_L
```

\iota_L^*Use in source · 3 passages

Search aliases: ι_L^* iota

Link to this entry ·

Use in source · spectral.tex, line 177
[Finite Feshbach identity] If z∉σ(D), define Σ(z)=B(z-D)^-1B^*, S(z)=z-A-Σ(z). Then z-H is invertible exactly when S(z) is invertible. On this common regular set, ι_L^*(z-H)^-1ι_L · = S(z)^-1,nonumber det(z-H) · =det(z-D)det S(z). For E∈R∖σ(D), an eigenvector of H is equivalent to a nonzero solution of S(E)u=0, with hidden component v=(E-D)^-1B^*u. Put C=z-D. Direct block multiplication gives z-H= I · -BC^-1
Exact mathematical passage
```
 \iota_L^*(z-H)^{-1}\iota_L&=\mathcal S(z)^{-1},\nonumber\\
 \det(z-H)&=\det(z-D)\det\mathcal S(z).
 \label{a-spectral-schur-identities}
```

\iota_\etaUse in source · 1 passage

Search aliases: ι_η iota eta

Link to this entry ·

Use in source · constraints.tex, line 493
Normal deformations require more boundary data. For a spatial vector field η, the Cartan identities give L_ηΩ=ι_η R+D_Ω(ι_ηΩ), L_η A=ι_η F+ d_Σ(ι_η A). Accordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression J_Σ[c,χ,u]=C[ζ(c)] +G_ L[χ-ι_cΩ] +G_ U[u+ι_cA]. For tangential c, ζ(c)=ι_cE is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that …
Exact mathematical passage
```
 \mathcal L_\eta\Omega=\iota_\eta\mathcal R+D_\Omega(\iota_\eta\Omega),
 \qquad
 \mathcal L_\eta A=\iota_\eta\mathcal F+\mathrm d_\Sigma(\iota_\eta A).
```

\kappaUse in source · 36 passages

Search aliases: κ kappa

Link to this entry ·

Use in source · finite.tex, line 616
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
\kappa_p
```

\kappa^2Use in source · 1 passage

Search aliases: κ^2 kappa

Link to this entry ·

Use in source · spectral.tex, line 611
The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation retains every other site. It does not retain the average of each neighboring pair.
Exact mathematical passage
```
 H_{\rm eff}^{(2)}(z)
 =\mu I+\frac{\kappa^2}{z-\mu}(2I+T+T^*),\qquad z\ne\mu.
 \label{a-spectral-dyadic-effective}
```

\kappa^{p-1}Use in source · 1 passage

Search aliases: κ^p-1 kappa

Link to this entry ·

Use in source · spectral.tex, line 643
Let Q project onto the eliminated sites. On the regular chart, det(z-H)=det(z-QHQ)det S_p(z), det(z-QHQ)=[κ^p-1U_p-1(x/2)]^M. Their product is a polynomial identity after cancellation. It records both the hidden path factor and the coarse Schur factor, including their cancellations at the resonant energies.
Exact mathematical passage
```
 \det(z-H)=\det(z-QHQ)\det\mathcal S_p(z),\qquad
 \det(z-QHQ)=\bigl[\kappa^{p-1}U_{p-1}(x/2)\bigr]^M.
```

\kappa_mUse in source · 2 passages

Search aliases: κ_m kappa

Link to this entry ·

Use in source · spectral.tex, line 738
an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =fracκ U_m-1(x/2)U_n-1(T_m(x/2)) =fracκU_mn-1(x/2). Similarly μ_m,n=z-κ_m,nR_n(R_m(x)), which gives…
Exact mathematical passage
```
x_m=(z-\mu_m)/\kappa_m=R_m(x)
```

\kappa_{m,n}Use in source · 2 passages

Search aliases: κ_m,n kappa

Link to this entry ·

Use in source · spectral.tex, line 739
ls, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =fracκ U_m-1(x/2)U_n-1(T_m(x/2)) =fracκU_mn-1(x/2). Similarly μ_m,n=z-κ_m,nR_n(R_m(x)), which gives [source reference]. At each stage the matrices remai…
Exact mathematical passage
```
 \kappa_{m,n}
 =\frac{\kappa_m}{U_{n-1}(x_m/2)}
 =\frac{\kappa}
   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}
 =\frac{\kappa}{U_{mn-1}(x/2)}.
```

\kappa_{mn}Use in source · 1 passage

Search aliases: κ_mn kappa

Link to this entry ·

Use in source · spectral.tex, line 718
U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R_n(R_m(x))=2cos(mnθ)=R_mn(x). It holds on an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of …
Exact mathematical passage
```
 \kappa_{mn}(z)=\frac{\kappa}{U_{mn-1}(x/2)},\qquad
 \mu_{mn}(z)=z-\frac{\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.
 \label{a-spectral-iterated-coefficients}
```

\kappa_pUse in source · 5 passages

Search aliases: κ_p kappa

Link to this entry ·

Use in source · finite.tex, line 616
Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is self-adjoint and refinement compatible. Translation permutes the level-r cylinders, so it commutes with each E_r^(h) and D_r^(h). The Laplacian thus …
Exact mathematical passage
```
\kappa_p
```

\lambdaUse in source · 92 passages

Search aliases: λ lambda

Link to this entry ·

Use in source · limit.tex, line 734
Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.
Exact mathematical passage
```
\lambda
```

\Lambda'Use in source · 3 passages

Search aliases: Λ' Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 24
The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),
Exact mathematical passage
```
\begin{aligned}
 \mathcal H={}&-\frac{P_RP_\Lambda}{R}
 +\frac{\Lambda P_\Lambda^2}{2R^2}
 +\frac{RR''}{\Lambda}-\frac{RR'\Lambda'}{\Lambda^2}
 +\frac{R'^2}{2\Lambda}-\frac\Lambda2
 +\frac{\Lambda Q^2}{2R^2}.
\end{aligned}
\label{c:strong:electrovac-H}
```

\lambda^2Use in source · 1 passage

Search aliases: λ^2 lambda

Link to this entry ·

Use in source · limit.tex, line 798
To verify this, subtract the constant integrand QBP in Duhamel's formula and use norme^-iuK-I≤|u|norm K for a finite self-adjoint K. The two deviations are bounded by (|t-s|normH_λ+|s|normH_0)normQBP on the oriented integration interval. Their integrals are both t^2/2. For |λ|≤1, this is a uniform O(t^2|λ|) remainder. A generic O(t^2λ^2) remainder would omit the first-order interaction with the background Hamiltonian.
Exact mathematical passage
```
O(t^2\lambda^2)
```

\Lambda^3Use in source · 1 passage

Search aliases: Λ^3 Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 67
-fracP_ΛRΛ D +frac QR G. Consequently, on a patch where R'/Λ≠0, the systems ( H, D, G)=0 and ( M', D, G)=0 have the same zero set. Differentiating [source reference] gives M'= · fracR'2-fracR'^32Λ^2 -fracRR'R''Λ^2+fracRR'^2Λ'Λ^3 +fracP_Λ P_Λ'R-fracP_Λ^2R'2R^2 · +fracQQ'R-fracQ^2R'2R^2. Expanding the right side of [source reference] cancels the two P_R terms and produces precisely this expression. The constraint transformation is upper triangular with diagonal (-R'/Λ,1,1), so it is invertible und…
Exact mathematical passage
```
\begin{aligned}
 \mathcal M'={}&\frac{R'}2-\frac{R'^3}{2\Lambda^2}
 -\frac{RR'R''}{\Lambda^2}+\frac{RR'^2\Lambda'}{\Lambda^3}
 +\frac{P_\Lambda P_\Lambda'}R-\frac{P_\Lambda^2R'}{2R^2}\\
 &+\frac{QQ'}R-\frac{Q^2R'}{2R^2}.
\end{aligned}
```

\lambda^iUse in source · 2 passages

Search aliases: λ^i lambda

Link to this entry ·

Use in source · constraints.tex, line 203
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,
Exact mathematical passage
```
\lambda^i
```

\lambda^{IJ}Use in source · 1 passage

Search aliases: λ^IJ lambda

Link to this entry ·

Use in source · constraints.tex, line 346
Gauge transformations and graded conventions. For even parameters (ξ,λ,u), the coframe transforms by its Lie derivative and Lorentz rotation, the connection by L_ξω-D_ωλ, and the Abelian potential by L_ξ a+ du. The spinor transforms by L_ξψ+tfrac14λ^IJγ_IJψ+iquψ, with the dual action on barψ. All internal contractions in~[source reference] are invariant. A diffeomorphism changes its four-form density by a Lie derivative, which integrates to zero on a periodic domain or under stated boundary cond…
Exact mathematical passage
```
\mathcal L_\xi\psi+\tfrac14\lambda^{IJ}\gamma_{IJ}\psi+iqu\psi
```

\lambda^jUse in source · 1 passage

Search aliases: λ^j lambda

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\Lambda^{\rm TT}_{ij,kl}Use in source · 1 passage

Search aliases: Λ^rm TT_ij,kl Lambda rm

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\lambda_0Use in source · 1 passage

Search aliases: λ_0 lambda

Link to this entry ·

Use in source · core.tex, line 409
If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The difference of two such functions has both derivatives zero, so is constant.
Exact mathematical passage
```
 \lambda(\tau,\theta)=\lambda_0
 +\int_{\tau_0}^{\tau}H_G(s,0)\,ds
 +\int_0^{\theta}M_G(\tau,\eta)\,d\eta.
```

\Lambda_hUse in source · 3 passages

Search aliases: Λ_h Lambda

Link to this entry ·

Use in source · constraints.tex, line 59
The closed finite object is the full inner algebra. To identify a continuum spatial diffeomorphism, one must additionally construct parameters Λ_h[ξ] whose action approximates the Lie derivative on resolved observables. An arbitrary subset of matrices chosen to resemble first derivatives need not be closed under commutators. Equation~[source reference] does not establish that extra geometric identification.
Exact mathematical passage
```
\Lambda_h[\xi]
```

\Lambda_jUse in source · 1 passage

Search aliases: Λ_j Lambda

Link to this entry ·

Use in source · strong_fields.tex, line 208
[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each constraint vanishes by direct substitution. In the discrete mass coordinate D_4R/Λ=1 and P_Λ^2=2MR-Q^2, leaving tfracR2(2M/R-Q^2/R^2)+Q^2/(2R)=M.
Exact mathematical passage
```
\begin{aligned}
 C_{\Lambda,j}&=\Lambda_j-D_4R_j,&
 C_{M,j}&=P_{\Lambda,j}^2-2MR_j+Q^2,\\
 C_{P,j}&=P_{R,j}P_{\Lambda,j}-M\Lambda_j,&
 C_{Q,j}&=Q_j-Q
\end{aligned}
```

\lambda_nUse in source · 15 passages

Search aliases: λ_n lambda

Link to this entry ·

Use in source · extensions.tex, line 150
Let (λ_n) be any real sequence and define D(H_λ)={c∈ℓ^2: ∑_nλ_n^2|c_n|^2<∞}, (H_λ c)_n=λ_nc_n. The operator is dense because it contains every finitely supported sequence. It is symmetric by the reality of the multipliers. Its domain is maximal for this coordinate action.
Exact mathematical passage
```
(\lambda_n)
```

\lambda_n^2Use in source · 4 passages

Search aliases: λ_n^2 lambda

Link to this entry ·

Use in source · extensions.tex, line 151
Let (λ_n) be any real sequence and define D(H_λ)={c∈ℓ^2: ∑_nλ_n^2|c_n|^2<∞}, (H_λ c)_n=λ_nc_n. The operator is dense because it contains every finitely supported sequence. It is symmetric by the reality of the multipliers. Its domain is maximal for this coordinate action.
Exact mathematical passage
```
 D(H_\lambda)=\left\{c\in\ell^2:
       \sum_n\lambda_n^2|c_n|^2<\infty\right\},\qquad
 (H_\lambda c)_n=\lambda_nc_n.
```

\lambda_\thetaUse in source · 2 passages

Search aliases: λ_θ lambda theta

Link to this entry ·

Use in source · core.tex, line 371
M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.
Exact mathematical passage
```
\lambda_\theta=M_G
```

\lambda_\tauUse in source · 1 passage

Search aliases: λ_τ lambda tau

Link to this entry ·

Use in source · core.tex, line 371
M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.
Exact mathematical passage
```
\lambda_\tau=H_G
```

\mu^kUse in source · 1 passage

Search aliases: μ^k mu

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\mu_mUse in source · 1 passage

Search aliases: μ_m mu

Link to this entry ·

Use in source · spectral.tex, line 738
an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =fracκ U_m-1(x/2)U_n-1(T_m(x/2)) =fracκU_mn-1(x/2). Similarly μ_m,n=z-κ_m,nR_n(R_m(x)), which gives…
Exact mathematical passage
```
x_m=(z-\mu_m)/\kappa_m=R_m(x)
```

\mu_{m,n}Use in source · 1 passage

Search aliases: μ_m,n mu

Link to this entry ·

Use in source · spectral.tex, line 747
(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =fracκ U_m-1(x/2)U_n-1(T_m(x/2)) =fracκU_mn-1(x/2). Similarly μ_m,n=z-κ_m,nR_n(R_m(x)), which gives [source reference]. At each stage the matrices remain scalar nearest-neighbor circulants with the displayed complex coefficients, so their algebraic Schur formula conti…
Exact mathematical passage
```
\mu_{m,n}=z-\kappa_{m,n}R_n(R_m(x))
```

\mu_{mn}Use in source · 1 passage

Search aliases: μ_mn mu

Link to this entry ·

Use in source · spectral.tex, line 718
U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R_n(R_m(x))=2cos(mnθ)=R_mn(x). It holds on an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of …
Exact mathematical passage
```
 \kappa_{mn}(z)=\frac{\kappa}{U_{mn-1}(x/2)},\qquad
 \mu_{mn}(z)=z-\frac{\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.
 \label{a-spectral-iterated-coefficients}
```

\nuUse in source · 24 passages

Search aliases: ν nu

Link to this entry ·

Use in source · limit.tex, line 111
For completeness, the union of these ranges is dense. Cylinders form a countable clopen base for the topology on Z_p and generate its Borel sigma-algebra. If f∈ L^2 is orthogonal to every cylinder indicator, the finite complex measure ν(A)=∫_A f dμ_p vanishes on cylinders and their finite unions. The class of sets on which ν is zero contains the whole space, is closed under complements, and is closed under countable disjoint unions. The elementary monotone-class extension from the cylinder algeb…
Exact mathematical passage
```
\nu(A)=\int_A f\,d\mu_p
```

\nu_{\rm emit}Use in source · 1 passage

Search aliases: ν_rm emit nu rm

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\nu_{\rm obs}Use in source · 1 passage

Search aliases: ν_rm obs nu rm

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\XiUse in source · 5 passages

Search aliases: Ξ Xi

Link to this entry ·

Use in source · constraints.tex, line 386
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
\Xi
```

\xiUse in source · 27 passages

Search aliases: ξ xi

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\xi^aUse in source · 4 passages

Search aliases: ξ^a xi

Link to this entry ·

Use in source · constraints.tex, line 203
Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,
Exact mathematical passage
```
\xi^a
```

\xi^bUse in source · 1 passage

Search aliases: ξ^b xi

Link to this entry ·

Use in source · constraints.tex, line 233
Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.
Exact mathematical passage
```
 (\lambda\times\mu)^i&=\epsilon^i{}_{jk}\lambda^j\mu^k,\\
 [\xi,\eta]^a&=\xi^b\delta_b\eta^a-\eta^b\delta_b\xi^a,\\
 \mathcal L_\xi^{(-1)}n&=\xi^a\delta_an-n\delta_a\xi^a,\\
 S^a(n,m)&=E^a_iE^{bi}(m\delta_bn-n\delta_bm).
```

\Xi^\ddaggerUse in source · 2 passages

Search aliases: Ξ^ddagger Xi ddagger

Link to this entry ·

Use in source · constraints.tex, line 386
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
\Xi^\ddagger
```

\xi_aUse in source · 3 passages

Search aliases: ξ_a xi

Link to this entry ·

Use in source · core.tex, line 831
Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.
Exact mathematical passage
```
 \mathcal D_h[\xi]=h\sum_a\xi_ap_a(Dq)_a,
 \qquad
 \mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).
```

\xi_iUse in source · 3 passages

Search aliases: ξ_i xi

Link to this entry ·

Use in source · core.tex, line 697
[Exact finite linearized closure] For every solution of [source reference], dot C=-δ_iC_i, dot C_i=0. Thus initially vanishing constraints remain zero. With the canonical Frobenius bracket and smearings C[n]=⟨ n,C⟩_h, D[ξ]=⟨ξ_i,C_i⟩_h, all three brackets {C[n],C[m]}, {D[ξ],D[η]}, and {C[n],D[ξ]} vanish identically.
Exact mathematical passage
```
D[\xi]=\langle\xi_i,C_i\rangle_h
```

\xi_jUse in source · 3 passages

Search aliases: ξ_j xi

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\xi_j^+Use in source · 1 passage

Search aliases: ξ_j^+ xi

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\xi_j^-Use in source · 1 passage

Search aliases: ξ_j^- xi

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\xi_kUse in source · 1 passage

Search aliases: ξ_k xi

Link to this entry ·

Use in source · core.tex, line 732
The momentum generator changes the metric perturbation by h_ij↦ h_ij+δ_iξ_j+δ_jξ_i. The corresponding change of the scalar constraint is 2δ^2δ_kξ_k-2δ^2δ_kξ_k=0. No product rule is used in these cancellations.
Exact mathematical passage
```
2\delta^2\delta_k\xi_k-2\delta^2\delta_k\xi_k=0
```

\PiUse in source · 43 passages

Search aliases: Π Pi

Link to this entry ·

Use in source · finite.tex, line 482
p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the nonzero chara…
Exact mathematical passage
```
\Pi_r
```

\varpiUse in source · 2 passages

Search aliases: ϖ varpi

Link to this entry ·

Use in source · constraints.tex, line 146
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
\varpi_A
```

\Pi'Use in source · 1 passage

Search aliases: Π' Pi

Link to this entry ·

Use in source · strong_fields.tex, line 423
endequation Thus ∂_t(m'-r^2 E)=0. A radial mass constraint imposed initially propagates under this continuum system. Expanding [source reference] gives dotΦ=(σ A)' and dotΠ=(σ B)'+2σ B/r. Differentiate E and subtract the radial derivative of the flux. Terms involving Π', Φ', and v' cancel, leaving ∂_t(r^2 E)-∂_r(r^2 J) =r^2[σ'AB+ΠΦdot v]. Since m=rv^2/2, the flux equation implies dot v=rσ AB/v. The remaining expression is r^2AB[σ'+fracσ rΠΦv]=0 by [source reference]. This establishes the identit…
Exact mathematical passage
```
\Pi'
```

\Pi^2Use in source · 7 passages

Search aliases: Π^2 Pi

Link to this entry ·

Use in source · core.tex, line 782
lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the projector has rank two.
Exact mathematical passage
```
\Pi^2=\Pi
```

\pi^2Use in source · 4 passages

Search aliases: π^2 pi

Link to this entry ·

Use in source · spectral.tex, line 1241
The exponent is positive for each fixed L, but decreases as 2π^2/L^2 when L grows. The conclusion is therefore uniform in the finite packet label j at fixed L; it is not a depth-uniform estimate at arbitrarily large residue modulus.
Exact mathematical passage
```
2\pi^2/L^2
```

\pi^3Use in source · 2 passages

Search aliases: π^3 pi

Link to this entry ·

Use in source · finite_core.tex, line 1070
(∑_n∈ SC(n)^2 a_n _2^2)^1/2 =frac(2π)^36=frac4π^33.
Exact mathematical passage
```
\left(\sum_{n\in S}C(n)^2\|a_n\|_2^2\right)^{1/2}
=\frac{(2\pi)^3}{6}=\frac{4\pi^3}{3}.
```

\Pi_aUse in source · 5 passages

Search aliases: Π_a Pi

Link to this entry ·

Use in source · extensions.tex, line 83
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
\Pi_a
```

\pi_aUse in source · 5 passages

Search aliases: π_a pi

Link to this entry ·

Use in source · core.tex, line 490
For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).
Exact mathematical passage
```
\pi_a>0
```

\varpi_AUse in source · 1 passage

Search aliases: ϖ_A varpi

Link to this entry ·

Use in source · constraints.tex, line 146
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
\varpi_A
```

\Pi_bUse in source · 1 passage

Search aliases: Π_b Pi

Link to this entry ·

Use in source · extensions.tex, line 109
The dynamics respects the same construction. Put widetilde K=VKV^*. It is skew-adjoint, and fracddtVψ=widetilde K Vψ. Moving adjoints across the inner product shows 2ReinnerΠ_bVψwidetilde KΠ_aVψ =2ReinnerF_bψKF_aψ=J^F_ba. The positive-effect current is therefore exactly the sharp current of this lifted flow evaluated on the embedded state. It is not obtained by treating arbitrary compression as an algebra homomorphism.
Exact mathematical passage
```
 2\operatorname{Re}\inner{\Pi_bV\psi}{\widetilde K\Pi_aV\psi}
 =2\operatorname{Re}\inner{F_b\psi}{KF_a\psi}=J^F_{ba}.
```

\pi_bUse in source · 1 passage

Search aliases: π_b pi

Link to this entry ·

Use in source · core.tex, line 491
For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).
Exact mathematical passage
```
\label{f:detailed-balance}
 \gamma_{ba}\pi_a=\gamma_{ab}\pi_b.
```

\Pi_{bc}Use in source · 1 passage

Search aliases: Π_bc Pi

Link to this entry ·

Use in source · constraints.tex, line 566
Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.
Exact mathematical passage
```
\mathcal E^a=\tfrac12\epsilon^{abc}\Pi_{bc}
```

\varpi_CUse in source · 1 passage

Search aliases: ϖ_C varpi

Link to this entry ·

Use in source · constraints.tex, line 147
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
 \Omega_F=c^AC_A-\tfrac12c^Ac^BU_{AB}{}^C\varpi_C
             +\Omega_F^{(\ge3)}.
 \label{c:constraints:bfv-charge}
```

\Pi_hUse in source · 3 passages

Search aliases: Π_h Pi

Link to this entry ·

Use in source · strong_fields.tex, line 197
On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,Λ,P_Λ,P_R,Q), rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining R, …
Exact mathematical passage
```
 \Pi_h[R]_j=\left(R_j,D_4R_j,
                 \varepsilon\sqrt{2MR_j-Q^2},
                 \frac{M D_4R_j}{\varepsilon\sqrt{2MR_j-Q^2}},Q\right).
\label{c:strong:reconstruct}
```

\Pi_{ij}Use in source · 1 passage

Search aliases: Π_ij Pi

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\pi_{ij}Use in source · 7 passages

Search aliases: π_ij pi

Link to this entry ·

Use in source · core.tex, line 615
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
\pi_{ij}=\pi_{ji}
```

\Pi_{ik}Use in source · 1 passage

Search aliases: Π_ik Pi

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\Pi_{il}Use in source · 1 passage

Search aliases: Π_il Pi

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\pi_{ji}Use in source · 1 passage

Search aliases: π_ji pi

Link to this entry ·

Use in source · core.tex, line 615
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
\pi_{ij}=\pi_{ji}
```

\Pi_{jk}Use in source · 1 passage

Search aliases: Π_jk Pi

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\Pi_{jl}Use in source · 1 passage

Search aliases: Π_jl Pi

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\pi_{kk}Use in source · 1 passage

Search aliases: π_kk pi

Link to this entry ·

Use in source · core.tex, line 617
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
\pi_{\rm tr}=\pi_{kk}
```

\Pi_{kl}Use in source · 1 passage

Search aliases: Π_kl Pi

Link to this entry ·

Use in source · core.tex, line 773
For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse two-plane, symmetric tensors have dimension three and the trace condition removes one dimension. Thus the p…
Exact mathematical passage
```
 \Pi_{ij}=\delta_{ij}
 -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
 \qquad
 \Lambda^{\rm TT}_{ij,kl}
 =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
```

\Pi_PUse in source · 1 passage

Search aliases: Π_P Pi

Link to this entry ·

Use in source · constraints.tex, line 279
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
\mathcal D=\Pi_PP_\theta+\Pi_QQ_\theta
```

\Pi_P^2Use in source · 1 passage

Search aliases: Π_P^2 Pi

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

\Pi_QUse in source · 1 passage

Search aliases: Π_Q Pi

Link to this entry ·

Use in source · constraints.tex, line 279
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
\mathcal D=\Pi_PP_\theta+\Pi_QQ_\theta
```

\Pi_Q^2Use in source · 1 passage

Search aliases: Π_Q^2 Pi

Link to this entry ·

Use in source · constraints.tex, line 274
The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make H_t a vanishing gravitational constraint. Areal time is already a gauge choice.
Exact mathematical passage
```
 \mathcal H_t=\frac{\Pi_P^2+e^{-2P}\Pi_Q^2}{2t}
                   +\frac t2(P_\theta^2+e^{2P}Q_\theta^2).
```

\Pi_rUse in source · 3 passages

Search aliases: Π_r Pi

Link to this entry ·

Use in source · finite.tex, line 482
p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the nonzero chara…
Exact mathematical passage
```
\Pi_r
```

\pi_{\rm tr}Use in source · 6 passages

Search aliases: π_rm tr pi rm

Link to this entry ·

Use in source · core.tex, line 617
Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber
Exact mathematical passage
```
\pi_{\rm tr}=\pi_{kk}
```

\pi_\chiUse in source · 1 passage

Search aliases: π_χ pi chi

Link to this entry ·

Use in source · constraints.tex, line 484
nal BFV sector On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus {Ω_int,Ω_int}=0 under that regular moment-map hypothesis. This finite Lie-algebra…
Exact mathematical passage
```
 \Omega_{\mathrm{int}}=G_{\mathrm L}[\chi]+G_{\mathrm U}[\upsilon]
                   -\tfrac12\langle\pi_\chi,[\chi,\chi]\rangle.
```

\rho_{(x,\mathbf d)}^nUse in source · 1 passage

Search aliases: ρ_(x, d)^n rho mathbf

Link to this entry ·

Use in source · limit.tex, line 883
[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-| d_0||≤ n. Every path in a Cartesian product graph uses a number of physical edges at least the physical distance and a number of tree edges at least…
Exact mathematical passage
```
 d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
 \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
 \label{a-limit-product-cone}
```

\rho_+Use in source · 2 passages

Search aliases: ρ_+ rho

Link to this entry ·

Use in source · core.tex, line 359
In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i -i · 1, ρ_-=frac121 · -i i · 1, ρ_0=frac121 · 0 0 · 1. All three are density matrices with packet probabilities (1/2,1/2). For H=g(0 · 1
Exact mathematical passage
```
 \rho_+=\frac12\begin{pmatrix}1&i\\-i&1\end{pmatrix},\qquad
 \rho_-=\frac12\begin{pmatrix}1&-i\\i&1\end{pmatrix},\qquad
 \rho_0=\frac12\begin{pmatrix}1&0\\0&1\end{pmatrix}.
```

\rho_-Use in source · 2 passages

Search aliases: ρ_- rho

Link to this entry ·

Use in source · core.tex, line 359
In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i -i · 1, ρ_-=frac121 · -i i · 1, ρ_0=frac121 · 0 0 · 1. All three are density matrices with packet probabilities (1/2,1/2). For H=g(0 · 1
Exact mathematical passage
```
 \rho_+=\frac12\begin{pmatrix}1&i\\-i&1\end{pmatrix},\qquad
 \rho_-=\frac12\begin{pmatrix}1&-i\\i&1\end{pmatrix},\qquad
 \rho_0=\frac12\begin{pmatrix}1&0\\0&1\end{pmatrix}.
```

\rho_0Use in source · 1 passage

Search aliases: ρ_0 rho

Link to this entry ·

Use in source · core.tex, line 359
In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i -i · 1, ρ_-=frac121 · -i i · 1, ρ_0=frac121 · 0 0 · 1. All three are density matrices with packet probabilities (1/2,1/2). For H=g(0 · 1
Exact mathematical passage
```
 \rho_+=\frac12\begin{pmatrix}1&i\\-i&1\end{pmatrix},\qquad
 \rho_-=\frac12\begin{pmatrix}1&-i\\i&1\end{pmatrix},\qquad
 \rho_0=\frac12\begin{pmatrix}1&0\\0&1\end{pmatrix}.
```

\rho_{12}Use in source · 1 passage

Search aliases: ρ_12 rho

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Use in source · core.tex, line 445
[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding -i[H,ρ] to [source reference] adds the coherent current to the jump current. Th…
Exact mathematical passage
```
 \dot p_1=k(1-2p_1),\qquad
 p_1(t)=\tfrac12+\bigl(p_1(0)-\tfrac12\bigr)e^{-2kt},
 \qquad \rho_{12}(t)=e^{-kt}\rho_{12}(0).
```

\rho_AUse in source · 2 passages

Search aliases: ρ_A rho

Link to this entry ·

Use in source · core.tex, line 384
al because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, with (e_k)_k indexing the basis of HH_B. In the resulting product basis the partial trace has entries (ρ_A)_ij=∑_k(ρ_AB)_(i,k),(j,k). This explicit formula s…
Exact mathematical passage
```
 p_a^A=\operatorname{Tr}\bigl((P_a\otimes\id_B)\rho_{AB}\bigr)
 =\operatorname{Tr}(P_a\rho_A),\qquad \rho_A=\operatorname{Tr}_B\rho_{AB}.
```

\rho_{AB}Use in source · 5 passages

Search aliases: ρ_AB rho

Link to this entry ·

Use in source · core.tex, line 378
Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, wit…
Exact mathematical passage
```
\rho_{AB}
```

\rho_{ab}Use in source · 1 passage

Search aliases: ρ_ab rho

Link to this entry ·

Use in source · core.tex, line 358
In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i
Exact mathematical passage
```
2\operatorname{Im}(H_{ba}\rho_{ab})
```

\rho_{bb}Use in source · 1 passage

Search aliases: ρ_bb rho

Link to this entry ·

Use in source · core.tex, line 419
[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.
Exact mathematical passage
```
p_b=\rho_{bb}
```

\rho_CUse in source · 2 passages

Search aliases: ρ_C rho

Link to this entry ·

Use in source · constraints.tex, line 798
x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gaug…
Exact mathematical passage
```
\rho_C\propto D^{-1/2}
```

\rho_iUse in source · 3 passages

Search aliases: ρ_i rho

Link to this entry ·

Use in source · core.tex, line 250
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 S_h[u]=\sum_{i=0}^{M-1}\frac{(u_{i+1}-u_i)^2}{2Gh}
 +4\pi h\sum_{i=1}^{M-1}r_i\rho_i u_i.
 \label{c:core:poisson-action}
```

\rho_{ij}Use in source · 2 passages

Search aliases: ρ_ij rho

Link to this entry ·

Use in source · core.tex, line 438
In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable …
Exact mathematical passage
```
\label{f:coherence-decay}
 \dot\rho_{ij}=-\tfrac12(r_i+r_j)\rho_{ij}.
```

\rho_jUse in source · 3 passages

Search aliases: ρ_j rho

Link to this entry ·

Use in source · core.tex, line 227
[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.
Exact mathematical passage
```
\label{g:gauss-def}
 \rho_j=\psi_j^*\psi_j,\qquad
 \bar\rho=\frac1N\sum_j\rho_j,\qquad
 \mathcal G_j=\frac{E_j-E_{j-1}}h+\rho_j-\bar\rho.
```

\rho_j^nUse in source · 3 passages

Search aliases: ρ_j^n rho

Link to this entry ·

Use in source · core.tex, line 87
[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
\rho_j^n=\norm{\psi_j^n}^2
```

\rho_j^{n+1}Use in source · 2 passages

Search aliases: ρ_j^n+1 rho

Link to this entry ·

Use in source · core.tex, line 92
[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
\label{g:walk-balance}
 \rho_j^{n+1}-\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.
```

\Sigma'Use in source · 4 passages

Search aliases: Σ' Sigma

Link to this entry ·

Use in source · spectral.tex, line 244
[Upper half-plane sign and the exact visible norm] For z=x+iy with y>0, ImΣ(z) =-yB((x-D)^2+y^2I)^-1B^*≤0, Im S(z)≥ yI. In particular S(z) is invertible and norm S(z)^-1≤ y^-1. For real E∉σ(D), Σ'(E)=-B(E-D)^-2B^*. If (u,v) is a normalized full eigenvector at E, then 1=⟨ u,(I-Σ'(E))u⟩. For real λ, Im(z-λ)^-1=-y/((x-λ)^2+y^2). Applying this identity to the spectral sum gives the first formula and then the bound on Im S. For any u, Cauchy--Schwarz implies ynormu^2≤|⟨ u, S(z)u⟩| ≤normunorm S(z)u. T…
Exact mathematical passage
```
 \Sigma'(E)=-B(E-D)^{-2}B^*.
```

\sigma'Use in source · 7 passages

Search aliases: σ' sigma

Link to this entry ·

Use in source · spectral.tex, line 490
For a nondegenerate visible spin direction with normalized low-space direction u_*, the visible weight of its normalized full eigenstate is [1-σ'(E)]^-1. If k=m_0=0, the spin eigenspace is degenerate, but the same normalization relation holds for each chosen unit vector in that eigenspace.
Exact mathematical passage
```
[1-\sigma'(E)]^{-1}
```

\sigma^{-1}Use in source · 1 passage

Search aliases: σ^-1 sigma

Link to this entry ·

Use in source · strong_fields.tex, line 473
For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric di…
Exact mathematical passage
```
 \ell_\pm=\sigma^{-1}\partial_t+(-v\pm1)\partial_r.
```

\sigma^1Use in source · 1 passage

Search aliases: σ^1 sigma

Link to this entry ·

Use in source · core.tex, line 346
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
\sigma^1,\sigma^2,\sigma^3
```

\sigma^3Use in source · 2 passages

Search aliases: σ^3 sigma

Link to this entry ·

Use in source · core.tex, line 346
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
\sigma^1,\sigma^2,\sigma^3
```

\sigma^aUse in source · 2 passages

Search aliases: σ^a sigma

Link to this entry ·

Use in source · core.tex, line 346
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
q_x^a=\psi_x^*\sigma^a\psi_x
```

\sigma^bUse in source · 1 passage

Search aliases: σ^b sigma

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Use in source · core.tex, line 347
Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge ψ_x^*ψ_x i…
Exact mathematical passage
```
\label{g:adjoint-color}
 g_x^*\sigma^a g_x=\sum_b R_x{}^a{}_b\sigma^b,
 \qquad q_x^{g,a}=\sum_bR_x{}^a{}_bq_x^b.
```

\Sigma_0Use in source · 1 passage

Search aliases: Σ_0 Sigma

Link to this entry ·

Use in source · spectral.tex, line 441
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
 \langle\mathrm{root},(z-H)^{-1}\mathrm{root}\rangle
   =g_0(z)=\frac1{z-\epsilon_0-\Sigma_0(z)}.
 \label{a-spectral-root-green}
```

\sigma_0Use in source · 1 passage

Search aliases: σ_0 sigma

Link to this entry ·

Use in source · constraints.tex, line 580
Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†…
Exact mathematical passage
```
\sigma_0=e\,\mathrm d^3x
```

\sigma_AUse in source · 1 passage

Search aliases: σ_A sigma

Link to this entry ·

Use in source · core.tex, line 392
For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0
Exact mathematical passage
```
\rho_{AB}=\sigma_A\otimes\tau_B
```

\Sigma_DUse in source · 1 passage

Search aliases: Σ_D Sigma

Link to this entry ·

Use in source · spectral.tex, line 469
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation of this…
Exact mathematical passage
```
 \Sigma_D(z)=\sigma(z)I_2,\qquad
 \sigma(z)=\gamma_0^2\langle1,(z-J_{1:h})^{-1}1\rangle.
```

\Sigma_hUse in source · 1 passage

Search aliases: Σ_h Sigma

Link to this entry ·

Use in source · spectral.tex, line 439
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
\Sigma_h=0
```

\Sigma_IUse in source · 2 passages

Search aliases: Σ_I Sigma

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\sigma_IUse in source · 4 passages

Search aliases: σ_I sigma

Link to this entry ·

Use in source · constraints.tex, line 434
IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A
Exact mathematical passage
```
 B_{IJ}&=\tfrac1{4\kappa}\epsilon_{KLIJ}E^K\wedge E^L,\\
 \Pi&=\tfrac12\varphi_{IJ}E^I\wedge E^J,\qquad
 \sigma_I=\tfrac1{3!}\epsilon_{IJKL}E^J\wedge E^K\wedge E^L.
```

\sigma_iUse in source · 4 passages

Search aliases: σ_i sigma

Link to this entry ·

Use in source · refinement.tex, line 168
Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phase frame r(s,c)∈ U(1) defines (I_rψ)(s,c)=r(s,c)ψ(s). With one-eighth fine weights this is again an isometry.
Exact mathematical passage
```
\sigma_i
```

\sigma_i^2Use in source · 2 passages

Search aliases: σ_i^2 sigma

Link to this entry ·

Use in source · refinement.tex, line 168
Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phase frame r(s,c)∈ U(1) defines (I_rψ)(s,c)=r(s,c)ψ(s). With one-eighth fine weights this is again an isometry.
Exact mathematical passage
```
\sigma_i^2
```

\sigma_jUse in source · 4 passages

Search aliases: σ_j sigma

Link to this entry ·

Use in source · finite_core.tex, line 117

α_j= 0 · σ_j σ_j · 0, β=I_2 · 0 0 · -I_2,

Exact mathematical passage


\alpha_j=
\begin{pmatrix}0&\sigma_j\\\sigma_j&0\end{pmatrix},
\qquad
\beta=\begin{pmatrix}I_2&0\\0&-I_2\end{pmatrix},

\sigma_j^*Use in source · 1 passage

Search aliases: σ_j^* sigma

Link to this entry ·

Use in source · finite_core.tex, line 173

α_j^*= 0 · σ_j^* σ_j^* · 0 =α_j, α_jα_k= σ_jσ_k · 0 0 · σ_jσ_k.

Exact mathematical passage


\alpha_j^*=
\begin{pmatrix}0&\sigma_j^*\\\sigma_j^*&0\end{pmatrix}
=\alpha_j,\qquad
\alpha_j\alpha_k=
\begin{pmatrix}\sigma_j\sigma_k&0\\0&\sigma_j\sigma_k\end{pmatrix}.

\sigma_KUse in source · 1 passage

Search aliases: σ_K sigma

Link to this entry ·

Use in source · constraints.tex, line 465
C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.
Exact mathematical passage
```
 \mathcal G_{IJ}
 &=D_\Omega B_{IJ}
       -\tfrac i8\sigma_K\bar\psi\{\gamma^K,\gamma_{IJ}\}\psi,\\
 \mathcal G_{\mathrm U}
 &=\mathrm d_\Sigma\Pi-q\sigma_I\bar\psi\gamma^I\psi.
```

\sigma_kUse in source · 1 passage

Search aliases: σ_k sigma

Link to this entry ·

Use in source · finite_core.tex, line 173

α_j^*= 0 · σ_j^* σ_j^* · 0 =α_j, α_jα_k= σ_jσ_k · 0 0 · σ_jσ_k.

Exact mathematical passage


\alpha_j^*=
\begin{pmatrix}0&\sigma_j^*\\\sigma_j^*&0\end{pmatrix}
=\alpha_j,\qquad
\alpha_j\alpha_k=
\begin{pmatrix}\sigma_j\sigma_k&0\\0&\sigma_j\sigma_k\end{pmatrix}.

\Sigma_PUse in source · 3 passages

Search aliases: Σ_P Sigma

Link to this entry ·

Use in source · finite.tex, line 695
For z outside the spectrum of H_Q=(QH_fQ)|_Ran Q, the coarse Schur self-energy is Σ_P(z)=PH_fQ(z-H_Q)^-1QH_fP. Compatibility makes it identically zero by setting B=0. This zero is an algebraic statement, not a uniform bound on a resolvent near its spectrum. A nonzero coupling requires the energy-dependent Schur term when the detail sector is eliminated. The spectral part of this monograph develops that elimination and its time-domain memory.
Exact mathematical passage
```
 \Sigma_P(z)=PH_fQ(z-H_Q)^{-1}QH_fP.
 \label{a-finite-self-energy}
```

\Sigma_{p,w}Use in source · 2 passages

Search aliases: Σ_p,w Sigma

Link to this entry ·

Use in source · finite.tex, line 805
Eliminating the hidden radial level gives Σ_p,w(z)=frac|G_p,w|^2z-ϵ, z≠ϵ. This is the exact finite Schur term. The full visible Green function, when its denominator is nonzero, is [z-E_c-Σ_p,w(z)]^-1. It is distinct from the self-energy itself, and the locations of its poles depend on both sectors.
Exact mathematical passage
```
 \Sigma_{p,w}(z)=\frac{|G_{p,w}|^2}{z-\epsilon},
 \qquad z\neq\epsilon.
 \label{a-finite-radial-self-energy}
```

\Sigma_rUse in source · 1 passage

Search aliases: Σ_r Sigma

Link to this entry ·

Use in source · spectral.tex, line 440
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
\Sigma_r=\gamma_r^2/(z-\epsilon_{r+1}-\Sigma_{r+1})
```

\Sigma_{r+1}Use in source · 1 passage

Search aliases: Σ_r+1 Sigma

Link to this entry ·

Use in source · spectral.tex, line 440
g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled final rational expression is finite. One should use the resolvent or the canceled rational function at such a point, rather than perform division by ze…
Exact mathematical passage
```
\Sigma_r=\gamma_r^2/(z-\epsilon_{r+1}-\Sigma_{r+1})
```

\sigma_xUse in source · 3 passages

Search aliases: σ_x sigma

Link to this entry ·

Use in source · spectral.tex, line 465
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation
Exact mathematical passage
```
A=H_D(k)=k\sigma_z+m_0\sigma_x
```

\sigma_yUse in source · 2 passages

Search aliases: σ_y sigma

Link to this entry ·

Use in source · finite_core.tex, line 126

σ_x=0 · 1 1 · 0, σ_y=0 · -i i · 0, σ_z=1 · 0 0 · -1.

Exact mathematical passage


\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\quad
\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},
\quad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.

\sigma_zUse in source · 4 passages

Search aliases: σ_z sigma

Link to this entry ·

Use in source · spectral.tex, line 465
As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenvalue in the regular chart obeys (E-σ(E))^2=k^2+m_0^2. The formula is a dressed dispersion relation
Exact mathematical passage
```
A=H_D(k)=k\sigma_z+m_0\sigma_x
```

\tau^2Use in source · 2 passages

Search aliases: τ^2 tau

Link to this entry ·

Use in source · core.tex, line 548
a compact regular neighborhood where d_h≥ d_0>0. Assume the mixed time and state derivatives of widehat f_h through order three and the state derivatives of C_h through order four are bounded there. Apply the endpoint correction of Proposition~[source reference] after explicit midpoint. Then its local error is O(Δτ^3) and its global error on a fixed time interval is O(Δτ^2).
Exact mathematical passage
```
O(\Delta\tau^2)
```

\tau^3Use in source · 5 passages

Search aliases: τ^3 tau

Link to this entry ·

Use in source · core.tex, line 547
of dot z=widehat f_h(τ,z) and its numerical steps remain in a compact regular neighborhood where d_h≥ d_0>0. Assume the mixed time and state derivatives of widehat f_h through order three and the state derivatives of C_h through order four are bounded there. Apply the endpoint correction of Proposition~[source reference] after explicit midpoint. Then its local error is O(Δτ^3) and its global error on a fixed time interval is O(Δτ^2).
Exact mathematical passage
```
O(\Delta\tau^3)
```

\tau_0Use in source · 2 passages

Search aliases: τ_0 tau

Link to this entry ·

Use in source · core.tex, line 407
If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The difference of two such functions has both derivatives zero, so is constant.
Exact mathematical passage
```
\tau_0
```

\tau_BUse in source · 1 passage

Search aliases: τ_B tau

Link to this entry ·

Use in source · core.tex, line 392
For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0
Exact mathematical passage
```
\rho_{AB}=\sigma_A\otimes\tau_B
```

\tau_iUse in source · 2 passages

Search aliases: τ_i tau

Link to this entry ·

Use in source · refinement.tex, line 168
Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phase frame r(s,c)∈ U(1) defines (I_rψ)(s,c)=r(s,c)ψ(s). With one-eighth fine weights this is again an isometry.
Exact mathematical passage
```
\tau_i
```

\tau_NUse in source · 3 passages

Search aliases: τ_N tau

Link to this entry ·

Use in source · constraints.tex, line 190
e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
\tau_N=N^{-3}\sum_x
```

\upsilonUse in source · 4 passages

Search aliases: υ upsilon

Link to this entry ·

Use in source · constraints.tex, line 362
Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,
Exact mathematical passage
```
c,\chi,\upsilon
```

\varphiUse in source · 28 passages

Search aliases: φ varphi

Link to this entry ·

Use in source · finite.tex, line 138
n⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are preserved. The dimensions also agree: φ(M)=2^h_2+1∏_p∈ P_ o(p-1)p^h_p. Multiplication of units becomes addition in each cyclic coordinate.
Exact mathematical passage
```
 \varphi(M)=2^{h_2+1}\prod_{p\in\mathcal P_{\mathrm o}}(p-1)p^{h_p}.
```

\Phi'Use in source · 5 passages

Search aliases: Φ' Phi

Link to this entry ·

Use in source · core.tex, line 230
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
\lim_{r\downarrow0}r^2\Phi'(r)=0
```

\varphi'Use in source · 3 passages

Search aliases: φ' varphi

Link to this entry ·

Use in source · strong_fields.tex, line 344
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
 \mathcal H_{\rm matt}=\frac{P_\varphi^2}{2\Lambda R^2}
                         +\frac{R^2\varphi'^2}{2\Lambda},
 \qquad \mathcal D_{\rm matt}=P_\varphi\varphi'.
```

\varphi'^2Use in source · 1 passage

Search aliases: φ'^2 varphi

Link to this entry ·

Use in source · strong_fields.tex, line 344
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
 \mathcal H_{\rm matt}=\frac{P_\varphi^2}{2\Lambda R^2}
                         +\frac{R^2\varphi'^2}{2\Lambda},
 \qquad \mathcal D_{\rm matt}=P_\varphi\varphi'.
```

\Phi^2Use in source · 7 passages

Search aliases: Φ^2 Phi

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\varphi^{23}Use in source · 1 passage

Search aliases: φ^23 varphi

Link to this entry ·

Use in source · constraints.tex, line 336
ge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The substitution repairs the Hodge shorthand while retaining the on-sh…
Exact mathematical passage
```
\varphi^{23}=E
```

\varphi^{IJ}Use in source · 2 passages

Search aliases: φ^IJ varphi

Link to this entry ·

Use in source · constraints.tex, line 304
The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e, S_ M^(1) · =∫(tfrac12φ_IJFwedge e^Iwedge e^J -tfrac14φ_IJφ^IJvol_e), S_ D · =frac i2∫Σ_Iwedge (barψγ^I Dψ -( Dbarψ)γ^Iψ) -m∫vol_ebarψψ. This formulation uses no inverse tetrad in its defining action. Its largest monomials have six field factors. Nondegeneracy becomes necessary when the auxiliary variables are eliminated or a metric interpretation is imposed.
Exact mathematical passage
```
 S_{\mathrm G}
 &=\frac1{4\kappa}\int\epsilon_{IJKL}e^I\wedge e^J\wedge R^{KL}
                  -\frac\Lambda\kappa\int\operatorname{vol}_e,\\
 S_{\mathrm M}^{(1)}
 &=\int\left(\tfrac12\varphi_{IJ}F\wedge e^I\wedge e^J
                    -\tfrac14\varphi_{IJ}\varphi^{IJ}\operatorname{vol}_e\right),\\
 S_{\mathrm D}
 &=\frac i2\int\Sigma_I\wedge
       \left(\bar\psi\gamma^I\mathcal D\psi
               -(\mathcal D\bar\psi)\gamma^I\psi\right)
       -m\int\operatorname{vol}_e\bar\psi\psi.
 \label{c:constraints:parent-action}
```

\varphi^{KL}Use in source · 1 passage

Search aliases: φ^KL varphi

Link to this entry ·

Use in source · constraints.tex, line 452
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ,
Exact mathematical passage
```
 \mathcal C_I^{\mathrm G}
 &=\tfrac1{2\kappa}\epsilon_{IJKL}E^J\wedge\mathcal R^{KL}
                  -\tfrac\Lambda\kappa\sigma_I,\\
 \mathcal C_I^{\mathrm M}
 &=\varphi_{IJ}\mathcal F\wedge E^J-\tfrac14\varphi_{KL}\varphi^{KL}\sigma_I,\\
 \mathcal C_I^{\mathrm D}
 &=\tfrac i4\epsilon_{JIKL}E^K\wedge E^L\wedge
       \bigl(\bar\psi\gamma^J\mathcal D_\Sigma\psi
                  -(\mathcal D_\Sigma\bar\psi)\gamma^J\psi\bigr)
                      -m\sigma_I\bar\psi\psi.
```

\phi_0Use in source · 2 passages

Search aliases: ϕ_0 phi

Link to this entry ·

Use in source · spectral.tex, line 819
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,
Exact mathematical passage
```
 \phi_0=1,\qquad \phi_n(x)=\sqrt2\,T_n(x/2),\quad n\geq1,
```

\Phi_1Use in source · 3 passages

Search aliases: Φ_1 Phi

Link to this entry ·

Use in source · constraints.tex, line 675
Endpoint Dirac projection. Consider a finite-dimensional real canonical chart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theo…
Exact mathematical passage
```
\Phi=(\Phi_1,\ldots,\Phi_{2m})
```

\phi_1Use in source · 1 passage

Search aliases: ϕ_1 phi

Link to this entry ·

Use in source · spectral.tex, line 860
K_p is an isometry with proper range, rather than a unitary: for example ϕ_1 is orthogonal to its range. Its adjoint discards nondivisible modes. Invariance of ν is a statement about the normalized free spectral distribution under R_p. It is not a proof that a sequence of growing Hilbert spaces has converged to an interacting Hamiltonian or that its quantum state converges. The direct-limit operator construction has separate hypotheses in the final part.
Exact mathematical passage
```
\phi_1
```

\Phi_2Use in source · 2 passages

Search aliases: Φ_2 Phi

Link to this entry ·

Use in source · constraints.tex, line 723
A worked particle constraint. In R^6 with z=(q,p) and canonical J, set Φ_1=q· q-1, Φ_2=q· p. At one point the ordinary Dirac matrix is Δ(z)=0 · 2 q ^2
Exact mathematical passage
```
 \Phi_1=q\cdot q-1,\qquad \Phi_2=q\cdot p.
```

\Phi_{2m}Use in source · 1 passage

Search aliases: Φ_2m Phi

Link to this entry ·

Use in source · constraints.tex, line 675
Endpoint Dirac projection. Consider a finite-dimensional real canonical chart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theo…
Exact mathematical passage
```
\Phi=(\Phi_1,\ldots,\Phi_{2m})
```

\Phi_e'Use in source · 2 passages

Search aliases: Φ_e' Phi

Link to this entry ·

Use in source · strong_fields.tex, line 142
ambda^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_trp(F) -fracMF_rF_tp(F)^3], which are the direct time derivatives of the two momenta. Finally, Φ_e'=QF_r/F^2 cancels NΛ Q/R^2 in dot A_r=NΛ Q/R^2-Φ_e'.
Exact mathematical passage
```
\Phi_e'=QF_r/F^2
```

\varphi_{IJ}Use in source · 4 passages

Search aliases: φ_IJ varphi

Link to this entry ·

Use in source · constraints.tex, line 287
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,
Exact mathematical passage
```
\varphi_{IJ}
```

\varphi_jUse in source · 3 passages

Search aliases: φ_j varphi

Link to this entry ·

Use in source · core.tex, line 29
N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w…
Exact mathematical passage
```
\varphi_j(t)
```

\varphi_{j+1}Use in source · 1 passage

Search aliases: φ_j+1 varphi

Link to this entry ·

Use in source · core.tex, line 30
ites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jd…
Exact mathematical passage
```
\Delta_j=\varphi_{j+1}-\varphi_j
```

\varphi_j^2Use in source · 3 passages

Search aliases: φ_j^2 varphi

Link to this entry ·

Use in source · core.tex, line 31
w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring…
Exact mathematical passage
```
 L_h=\frac h2\sum_jw_j\dot\varphi_j^2
 -\frac1{2h}\sum_jc_{j+1/2}\Delta_j^2
 -\frac h2\sum_jv_j\varphi_j^2.
 \label{c:core:scalar-action}
```

\Phi_j^TUse in source · 1 passage

Search aliases: Φ_j^T Phi

Link to this entry ·

Use in source · constraints.tex, line 685
ous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse are skew, transposition proves (J_ D^Δ)^T=-J_ D^Δ. The endpoint tensor is a projectio…
Exact mathematical passage
```
\overline\nabla\Phi_j^T
```

\varphi_{KL}Use in source · 1 passage

Search aliases: φ_KL varphi

Link to this entry ·

Use in source · constraints.tex, line 452
The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ,
Exact mathematical passage
```
 \mathcal C_I^{\mathrm G}
 &=\tfrac1{2\kappa}\epsilon_{IJKL}E^J\wedge\mathcal R^{KL}
                  -\tfrac\Lambda\kappa\sigma_I,\\
 \mathcal C_I^{\mathrm M}
 &=\varphi_{IJ}\mathcal F\wedge E^J-\tfrac14\varphi_{KL}\varphi^{KL}\sigma_I,\\
 \mathcal C_I^{\mathrm D}
 &=\tfrac i4\epsilon_{JIKL}E^K\wedge E^L\wedge
       \bigl(\bar\psi\gamma^J\mathcal D_\Sigma\psi
                  -(\mathcal D_\Sigma\bar\psi)\gamma^J\psi\bigr)
                      -m\sigma_I\bar\psi\psi.
```

\phi_mUse in source · 2 passages

Search aliases: ϕ_m phi

Link to this entry ·

Use in source · spectral.tex, line 823
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ
Exact mathematical passage
```
 \mathcal K_p\phi_n=\phi_{pn},\qquad
 \mathcal K_p^*\phi_m=
 \begin{cases}\phi_{m/p},&p\mid m,\\0,&p\nmid m.\end{cases}
 \label{a-spectral-koopman-routing}
```

\phi_{m/p}Use in source · 1 passage

Search aliases: ϕ_m/p phi

Link to this entry ·

Use in source · spectral.tex, line 823
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ
Exact mathematical passage
```
 \mathcal K_p\phi_n=\phi_{pn},\qquad
 \mathcal K_p^*\phi_m=
 \begin{cases}\phi_{m/p},&p\mid m,\\0,&p\nmid m.\end{cases}
 \label{a-spectral-koopman-routing}
```

\phi_nUse in source · 5 passages

Search aliases: ϕ_n phi

Link to this entry ·

Use in source · spectral.tex, line 819
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,
Exact mathematical passage
```
 \phi_0=1,\qquad \phi_n(x)=\sqrt2\,T_n(x/2),\quad n\geq1,
```

\phi_{pn}Use in source · 3 passages

Search aliases: ϕ_pn phi

Link to this entry ·

Use in source · spectral.tex, line 823
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ
Exact mathematical passage
```
 \mathcal K_p\phi_n=\phi_{pn},\qquad
 \mathcal K_p^*\phi_m=
 \begin{cases}\phi_{m/p},&p\mid m,\\0,&p\nmid m.\end{cases}
 \label{a-spectral-koopman-routing}
```

\Phi_{\rm emit}Use in source · 1 passage

Search aliases: Φ_rm emit Phi rm

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\Phi_{\rm obs}Use in source · 1 passage

Search aliases: Φ_rm obs Phi rm

Link to this entry ·

Use in source · core.tex, line 309
In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb),
Exact mathematical passage
```
 \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
 =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
 =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
 +O\left(\frac{\Phi^2}{c_0^4}\right).
```

\phi_{\rm phys}Use in source · 1 passage

Search aliases: ϕ_rm phys phi rm

Link to this entry ·

Use in source · strong_fields.tex, line 343
To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source reference] with Q=0.
Exact mathematical passage
```
\varphi=\sqrt{4\pi}\,\phi_{\rm phys}
```

\phi_tUse in source · 2 passages

Search aliases: ϕ_t phi

Link to this entry ·

Use in source · core.tex, line 150
Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1 1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.
Exact mathematical passage
```
 K=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 \omega=(\phi_xdt+\phi_tdx)K.
```

\phi_{tt}Use in source · 3 passages

Search aliases: ϕ_tt phi

Link to this entry ·

Use in source · core.tex, line 157
1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.
Exact mathematical passage
```
 \Omega=d\omega=(\phi_{tt}-\phi_{xx})K\,dt\wedge dx,
 \qquad R=2e^{-2\phi}(\phi_{tt}-\phi_{xx}).
 \label{c:core:conformal-curvature}
```

\phi_xUse in source · 2 passages

Search aliases: ϕ_x phi

Link to this entry ·

Use in source · core.tex, line 150
Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1 1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.
Exact mathematical passage
```
 K=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 \omega=(\phi_xdt+\phi_tdx)K.
```

\varphi_xUse in source · 1 passage

Search aliases: φ_x varphi

Link to this entry ·

Use in source · core.tex, line 124
For a smooth field sampled at sites and a smooth coefficient sampled at edge midpoints, the spatial expression is second-order consistent with ∂_x(c∂_xφ). Indeed, expanding at x_j gives c(x_j+h/2)[φ(x_j+h)-φ(x_j)] -c(x_j-h/2)[φ(x_j)-φ(x_j-h)] =h^2(cφ_x)_x(x_j)+O(h^4). Uniform bounds on the requisite fourth derivatives make the remainder uniform on a compact spatial region. The exact finite balance holds without any Taylor approximation.
Exact mathematical passage
```
 c(x_j+h/2)[\varphi(x_j+h)-\varphi(x_j)]
 -c(x_j-h/2)[\varphi(x_j)-\varphi(x_j-h)]
 =h^2(c\varphi_x)_x(x_j)+O(h^4).
```

\phi_x^*Use in source · 1 passage

Search aliases: ϕ_x^* phi

Link to this entry ·

Use in source · core.tex, line 49
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
\inner{\phi}{\psi}_V=\sum_x\phi_x^*\psi_x
```

\phi_{xx}Use in source · 3 passages

Search aliases: ϕ_xx phi

Link to this entry ·

Use in source · core.tex, line 157
1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.
Exact mathematical passage
```
 \Omega=d\omega=(\phi_{tt}-\phi_{xx})K\,dt\wedge dx,
 \qquad R=2e^{-2\phi}(\phi_{tt}-\phi_{xx}).
 \label{c:core:conformal-curvature}
```

\chiUse in source · 26 passages

Search aliases: χ chi

Link to this entry ·

Use in source · limit.tex, line 941
nt. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its internal norm. The subsequent reversal shift sends these amplitudes to new arcs at the children, outside the old arc range. Therefore, for a unit internal state χ, norm(I-JJ^*)U_p,h+1,NJ(e_j⊗χ) =frac2sqrt p3+p>0. The state JU_p,h,N(e_j⊗χ) has zero amplitude there, so U_p,…
Exact mathematical passage
```
\chi
```

\chi^2Use in source · 1 passage

Search aliases: χ^2 chi

Link to this entry ·

Use in source · constraints.tex, line 363
Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,
Exact mathematical passage
```
 sc^\mu=c^\nu\partial_\nu c^\mu,\qquad
 s\chi=\mathcal L_c\chi+\chi^2,\qquad
 s\upsilon=\mathcal L_c\upsilon.
 \label{c:constraints:ghost-differential}
```

\chi^{IJ}Use in source · 2 passages

Search aliases: χ^IJ chi

Link to this entry ·

Use in source · constraints.tex, line 370
ultiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ, sbarψ · = L_cbarψ -tfrac14χ^IJbarψγ_IJ-iqυbarψ. The ordering in the second line matters: both χ and barψ are odd. The expression -barψχ is not the same expression. Moving a ghost through an odd dual field changes its sign…
Exact mathematical passage
```
 s\psi&=\mathcal L_c\psi+\tfrac14\chi^{IJ}\gamma_{IJ}\psi+iq\upsilon\psi,\\
 s\bar\psi&=\mathcal L_c\bar\psi
                -\tfrac14\chi^{IJ}\bar\psi\gamma_{IJ}-iq\upsilon\bar\psi.
 \label{c:constraints:dual-ghost-action}
```

\chi_AUse in source · 1 passage

Search aliases: χ_A chi

Link to this entry ·

Use in source · constraints.tex, line 586
the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
 \{\chi_A(x),\bar\chi_B(y)\}_{\mathrm P}
            =-ie(x)\delta_{AB}\delta^3(x-y).
```

\chi_jUse in source · 5 passages

Search aliases: χ_j chi

Link to this entry ·

Use in source · spectral.tex, line 667
1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j+χ_j-1=0 for χ_j=ψ_pj. It explains the normalized spectral polynomial. The path-inverse proof above additionally supplies the prefactor needed for the…
Exact mathematical passage
```
\chi_{j+1}-R_p(x)\chi_j+\chi_{j-1}=0
```

\chi_{j+1}Use in source · 2 passages

Search aliases: χ_j+1 chi

Link to this entry ·

Use in source · spectral.tex, line 667
1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j+χ_j-1=0 for χ_j=ψ_pj. It explains the normalized spectral polynomial. The path-inverse proof above additionally supplies the prefactor needed for the…
Exact mathematical passage
```
\chi_{j+1}-R_p(x)\chi_j+\chi_{j-1}=0
```

\chi_{j-1}Use in source · 1 passage

Search aliases: χ_j-1 chi

Link to this entry ·

Use in source · spectral.tex, line 667
1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j+χ_j-1=0 for χ_j=ψ_pj. It explains the normalized spectral polynomial. The path-inverse proof above additionally supplies the prefactor needed for the…
Exact mathematical passage
```
\chi_{j+1}-R_p(x)\chi_j+\chi_{j-1}=0
```

\PsiUse in source · 18 passages

Search aliases: Ψ Psi

Link to this entry ·

Use in source · limit.tex, line 687
Z_sign have eigenvalues +1,-1 on a dyadic sign qubit, and let B=B^* act on the remaining factors. Assume there are orthonormal states |b_0⟩,|b_1⟩ with B|b_0⟩=0 and B|b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sign⊗ B, |Ψ(0)⟩=|+_x⟩⊗ frac|b_0⟩+|b_1⟩sqrt2. Evolution under this explicit interaction alone gives |Ψ(t)⟩=frac1sqrt2 (|+_x⟩⊗|b_0⟩ +e^-igtZ_sign|+_x⟩⊗|b_1…
Exact mathematical passage
```
 H_{\mathrm{int}}=gZ_{\mathrm{sign}}\otimes B,
 \qquad
 |\Psi(0)\rangle=|+_x\rangle\otimes
 \frac{|b_0\rangle+|b_1\rangle}{\sqrt2}.
```

\psi'Use in source · 1 passage

Search aliases: ψ' psi

Link to this entry ·

Use in source · core.tex, line 313
For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bounds for A and ψ give the required uniform bound on f_x'''.
Exact mathematical passage
```
f_x'(0)=\psi'(x)-iA(x)\psi(x)
```

\Psi^\daggerUse in source · 6 passages

Search aliases: Ψ^† Psi dagger

Link to this entry ·

Use in source · constraints.tex, line 591
tion Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,
Exact mathematical passage
```
\Psi^\dagger=e^{1/2}\psi^\dagger
```

\psi^\daggerUse in source · 4 passages

Search aliases: ψ^† psi dagger

Link to this entry ·

Use in source · constraints.tex, line 580
Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†…
Exact mathematical passage
```
\bar\psi=\psi^\dagger\gamma^0
```

\psi_0Use in source · 1 passage

Search aliases: ψ_0 psi

Link to this entry ·

Use in source · limit.tex, line 187
h has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω_p^(q'-q)d=δ_qq'. The number of these orthonormal functions is exactly d_m, so they span that level. Together with ψ_0=1, they form a complete orthonormal basis by [source reference].
Exact mathematical passage
```
\psi_0=1
```

\psi_0^{(h)}Use in source · 1 passage

Search aliases: ψ_0^(h) psi

Link to this entry ·

Use in source · finite.tex, line 335
0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.
Exact mathematical passage
```
\psi_0^{(h)}(x)=p^{-h/2}
```

\Psi_AUse in source · 1 passage

Search aliases: Ψ_A Psi

Link to this entry ·

Use in source · constraints.tex, line 600
δΨ^†Ψ · =eδψ^†ψ+tfrac12ψ^†ψ δ e. The determinant terms cancel, so Θ_ D=frac i2∫_Σ (Ψ^†δΨ-δΨ^†Ψ) d^3x, {Ψ_A(x),Ψ_B^†(y)}_ D =-iδ_ABδ^3(x-y). The half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of e, so it is a local coordinate choice on~[source reference].
Exact mathematical passage
```
 \Theta_{\mathrm D}=\frac i2\int_\Sigma
       (\Psi^\dagger\delta\Psi-\delta\Psi^\dagger\Psi)\,\mathrm d^3x,
 \qquad
 \{\Psi_A(x),\Psi_B^\dagger(y)\}_{\mathrm D}
             =-i\delta_{AB}\delta^3(x-y).
```

\psi_AUse in source · 1 passage

Search aliases: ψ_A psi

Link to this entry ·

Use in source · core.tex, line 46
[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights …
Exact mathematical passage
```
\psi_A=(1,i,0,0)^{\mathsf T}/\sqrt7
```

\psi_BUse in source · 1 passage

Search aliases: ψ_B psi

Link to this entry ·

Use in source · core.tex, line 47
[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights …
Exact mathematical passage
```
\psi_B=(0,0,2,-i)^{\mathsf T}/\sqrt7
```

\psi_bUse in source · 5 passages

Search aliases: ψ_b psi

Link to this entry ·

Use in source · core.tex, line 113
Define the receiving-first current by J_b← a(t) =2Reinnerψ_b(t)K(t)ψ_a(t) =2Reinnerψ_b(t)K_ba(t)ψ_a(t). The second equality follows because P_bψ_b=ψ_b. A positive value means that the contribution assigned to the pair increases the weight in b. The current is a real number, and its sign convention is fixed by the derivative calculation below.
Exact mathematical passage
```
\label{f:current}
 J_{b\leftarrow a}(t)
 =2\operatorname{Re}\inner{\psi_b(t)}{K(t)\psi_a(t)}
 =2\operatorname{Re}\inner{\psi_b(t)}{K_{ba}(t)\psi_a(t)}.
```

\Psi_B^\daggerUse in source · 1 passage

Search aliases: Ψ_B^† Psi dagger

Link to this entry ·

Use in source · constraints.tex, line 600
δΨ^†Ψ · =eδψ^†ψ+tfrac12ψ^†ψ δ e. The determinant terms cancel, so Θ_ D=frac i2∫_Σ (Ψ^†δΨ-δΨ^†Ψ) d^3x, {Ψ_A(x),Ψ_B^†(y)}_ D =-iδ_ABδ^3(x-y). The half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of e, so it is a local coordinate choice on~[source reference].
Exact mathematical passage
```
 \Theta_{\mathrm D}=\frac i2\int_\Sigma
       (\Psi^\dagger\delta\Psi-\delta\Psi^\dagger\Psi)\,\mathrm d^3x,
 \qquad
 \{\Psi_A(x),\Psi_B^\dagger(y)\}_{\mathrm D}
             =-i\delta_{AB}\delta^3(x-y).
```

\psi_jUse in source · 10 passages

Search aliases: ψ_j psi

Link to this entry ·

Use in source · core.tex, line 80
For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channe…
Exact mathematical passage
```
\label{g:walk-covariance}
 \mathcal W_n[U^g,T^g]G^n=G^{n+1}\mathcal W_n[U,T],
 \qquad (G^n\psi)_j=g_j^n\psi_j.
```

\psi_{j+1}Use in source · 4 passages

Search aliases: ψ_j+1 psi

Link to this entry ·

Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

\psi_{j-1}Use in source · 1 passage

Search aliases: ψ_j-1 psi

Link to this entry ·

Use in source · core.tex, line 211
sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1, dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1). For example, if z_j=ψ_j^*α U_jψ_j+1, then ∂_A_jImz_j =Im(-ihz_j)=-hRez_j. This fixes the positive sign of dot E_j=J_j for the conventions in the act…
Exact mathematical passage
```
 \dot\psi_j&=-im\beta\psi_j
 -\frac1{2h}\alpha U_j\psi_{j+1}
 +\frac1{2h}U_{j-1}^*\alpha\psi_{j-1},\label{g:matter-ode}\\
 \dot A_j&=E_j,
 &\dot E_j&=J_j,
 &J_j&=\operatorname{Re}(\psi_j^*\alpha U_j\psi_{j+1}).
 \label{g:field-ode}
```

\psi_{j-1}^*Use in source · 1 passage

Search aliases: ψ_j-1^* psi

Link to this entry ·

Use in source · core.tex, line 242
Differentiate ρ_j and use dotρ_j=2Re(ψ_j^*dotψ_j). The mass term contributes zero, since ψ_j^*βψ_j is real. The forward hopping term contributes -J_j/h. The backward scalar is the conjugate of ψ_j-1^*α U_j-1ψ_j, using α=α^* and the scalar nature of U. Its contribution is J_j-1/h. This proves continuity. Periodic summation gives dotbarρ=0. Finally, dot G_j =fracdot E_j-dot E_j-1h+dotρ_j-dotbarρ =fracJ_j-J_j-1h-fracJ_j-J_j-1h=0.
Exact mathematical passage
```
\psi_{j-1}^*\alpha U_{j-1}\psi_j
```

\psi_j^*Use in source · 7 passages

Search aliases: ψ_j^* psi

Link to this entry ·

Use in source · core.tex, line 197
Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign…
Exact mathematical passage
```
 H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
 H_{\mathrm m}
 &=hm\sum_j\psi_j^*\beta\psi_j
   +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
 &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
```

\psi_j^nUse in source · 3 passages

Search aliases: ψ_j^n psi

Link to this entry ·

Use in source · core.tex, line 65
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,
Exact mathematical passage
```
\psi_j^n\in\C^2
```

\psi_j^{n+1}Use in source · 1 passage

Search aliases: ψ_j^n+1 psi

Link to this entry ·

Use in source · core.tex, line 66
Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^+ · =(U_j-1^n)^-1η_j-1^+, · ξ_j^- · =U_j^nη_j+1^-, ψ_j^n+1 · =T_j^n C_2,j^nξ_j. Here the + channel moves toward increasing j, while the - channel move…
Exact mathematical passage
```
 \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
 \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
 &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
 \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
```

\psi_{m,a,q}Use in source · 1 passage

Search aliases: ψ_m,a,q psi

Link to this entry ·

Use in source · limit.tex, line 174
For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω_p^(q'-q)d=δ_qq'. The number of these orthonormal functions is exactly d_m, so they span that level. Together with ψ_0=1, they form a complete orthono…
Exact mathematical passage
```
 \psi_{m,a,q}=p^{(m-1)/2}\sum_{d=0}^{p-1}
 \omega_p^{qd}\mathbf1_{B_{m,a+d p^{m-1}}},
 \quad
 \begin{matrix}m\geq1,\quad 0\leq a<p^{m-1},\\1\leq q<p.\end{matrix}
 \label{a-limit-haar-wavelets}
```

\psi_nUse in source · 1 passage

Search aliases: ψ_n psi

Link to this entry ·

Use in source · spectral.tex, line 539
The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.
Exact mathematical passage
```
\psi_{n+1}=x\psi_n-\psi_{n-1}
```

\psi_{n+1}Use in source · 1 passage

Search aliases: ψ_n+1 psi

Link to this entry ·

Use in source · spectral.tex, line 539
The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.
Exact mathematical passage
```
\psi_{n+1}=x\psi_n-\psi_{n-1}
```

\psi_{n-1}Use in source · 1 passage

Search aliases: ψ_n-1 psi

Link to this entry ·

Use in source · spectral.tex, line 539
The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.
Exact mathematical passage
```
\psi_{n+1}=x\psi_n-\psi_{n-1}
```

\psi_{pj}Use in source · 1 passage

Search aliases: ψ_pj psi

Link to this entry ·

Use in source · spectral.tex, line 668
equation For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j+χ_j-1=0 for χ_j=ψ_pj. It explains the normalized spectral polynomial. The path-inverse proof above additionally supplies the prefactor needed f…
Exact mathematical passage
```
\chi_j=\psi_{pj}
```

\psi_rUse in source · 4 passages

Search aliases: ψ_r psi

Link to this entry ·

Use in source · spectral.tex, line 65
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a …
Exact mathematical passage
```
w_r=\norm{\psi_r}^2
```

\psi_sUse in source · 4 passages

Search aliases: ψ_s psi

Link to this entry ·

Use in source · spectral.tex, line 64
[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. endpr
Exact mathematical passage
```
i\dot\psi_r=\sum_sH_{rs}\psi_s
```

\psi_xUse in source · 8 passages

Search aliases: ψ_x psi

Link to this entry ·

Use in source · core.tex, line 3
A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as ψ_x+widehatμ-ψ_x therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.
Exact mathematical passage
```
\psi_{x+\widehat\mu}-\psi_x
```

\psi_{x+\widehat\mu}Use in source · 5 passages

Search aliases: ψ_x+widehatμ psi widehat mu

Link to this entry ·

Use in source · core.tex, line 3
A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as ψ_x+widehatμ-ψ_x therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.
Exact mathematical passage
```
\psi_{x+\widehat\mu}-\psi_x
```

\psi_{x+\widehat\mu}^gUse in source · 1 passage

Search aliases: ψ_x+widehatμ^g psi widehat mu

Link to this entry ·

Use in source · core.tex, line 11
ing on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the…
Exact mathematical passage
```
\label{g:link-transform}
 U_{x,\mu}^g=g_xU_{x,\mu}g_{x+\widehat\mu}^{-1},
 \qquad U_{x,\mu}^g\psi_{x+\widehat\mu}^g
 =g_xU_{x,\mu}\psi_{x+\widehat\mu}.
```

\psi_{x-\widehat\mu}Use in source · 1 passage

Search aliases: ψ_x-widehatμ psi widehat mu

Link to this entry ·

Use in source · core.tex, line 27
[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.
Exact mathematical passage
```
 (D^+_{\mu,U}\psi)_x
 &=\frac{U_{x,\mu}\psi_{x+\widehat\mu}-\psi_x}{h},\label{g:forward-difference}\\
 (\Delta_U\psi)_x
 &=\frac1{h^2}\sum_\mu\left(
 U_{x,\mu}\psi_{x+\widehat\mu}
 +U_{x-\widehat\mu,\mu}^{-1}\psi_{x-\widehat\mu}-2\psi_x\right).
 \label{g:laplacian}
```

\psi_x^*Use in source · 4 passages

Search aliases: ψ_x^* psi

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Use in source · core.tex, line 55
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
h^{-2}(\norm{\psi_{x+\widehat\mu}}^2+\norm{\psi_x}^2
-2\operatorname{Re}(\psi_x^*U_{x,\mu}\psi_{x+\widehat\mu}))
```

\psi_x^gUse in source · 1 passage

Search aliases: ψ_x^g psi

Link to this entry ·

Use in source · core.tex, line 10
a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to le…
Exact mathematical passage
```
\psi_x^g=g_x\psi_x
```

\omega'Use in source · 1 passage

Search aliases: ω' omega

Link to this entry ·

Use in source · core.tex, line 211
The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and parallel transport exp(-∫_eω). Its closed-loop rapidity is the negative of the rapidity of W_square defined here. The fixed-frame curvature identity [so…
Exact mathematical passage
```
\omega'=\omega-d\chi\,K
```

\Omega^{0i}Use in source · 1 passage

Search aliases: Ω^0i Omega

Link to this entry ·

Use in source · constraints.tex, line 553
ented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ …
Exact mathematical passage
```
K^i=\Omega^{0i}
```

\omega^IUse in source · 1 passage

Search aliases: ω^I omega

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\Omega^{IJ}Use in source · 3 passages

Search aliases: Ω^IJ Omega

Link to this entry ·

Use in source · constraints.tex, line 422
An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 e^I=e_\perp^I\mathrm dt+E^I,\qquad
 \omega^{IJ}=\omega_\perp^{IJ}\mathrm dt+\Omega^{IJ},\qquad
 a=a_\perp\mathrm dt+A.
```

\omega^{IJ}Use in source · 4 passages

Search aliases: ω^IJ omega

Link to this entry ·

Use in source · constraints.tex, line 287
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,
Exact mathematical passage
```
\omega^{IJ}
```

\Omega^{jk}Use in source · 1 passage

Search aliases: Ω^jk Omega

Link to this entry ·

Use in source · constraints.tex, line 554
E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x. The antisymmetric IJ sum includes both 0i and i0, accounting for the factor in the last identity. There is no δΓ term. Those directions …
Exact mathematical passage
```
\Gamma^i=-\tfrac12\epsilon^i{}_{jk}\Omega^{jk}
```

\omega^{KJ}Use in source · 1 passage

Search aliases: ω^KJ omega

Link to this entry ·

Use in source · constraints.tex, line 288
Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The cov…
Exact mathematical passage
```
 R^{IJ}&=\mathrm d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ},
 &F&=\mathrm da,\\
 \operatorname{vol}_e&=\tfrac1{4!}\epsilon_{IJKL}
                         e^I\wedge e^J\wedge e^K\wedge e^L,
 &\Sigma_I&=\tfrac1{3!}\epsilon_{IJKL}e^J\wedge e^K\wedge e^L.
```

\omega_+Use in source · 1 passage

Search aliases: ω_+ omega

Link to this entry ·

Use in source · core.tex, line 208
The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and parallel transport exp(-∫_eω). Its closed-loop rapidity is the negative of the rapidity of W_square defined here. The fixed-frame curvature identity [so…
Exact mathematical passage
```
\omega_+'=\omega_++d\chi\,K
```

\omega_+'Use in source · 1 passage

Search aliases: ω_+' omega

Link to this entry ·

Use in source · core.tex, line 208
The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and parallel transport exp(-∫_eω). Its closed-loop rapidity is the negative of the rapidity of W_square defined here. The fixed-frame curvature identity [so…
Exact mathematical passage
```
\omega_+'=\omega_++d\chi\,K
```

\Omega_CUse in source · 2 passages

Search aliases: Ω_C Omega

Link to this entry ·

Use in source · constraints.tex, line 155
If {Ω_F,Ω_F}=0, then s_F={Ω_F,·} is a differential. To transfer a nonlinear Hamiltonian gauge structure, the contraction must also lift compatibly to the completed nonlinear coalgebra and respect its cyclic or symplectic structure. On a local exact canonical chart, a resulting Hamiltonian vector field can have a charge Ω_C. Its square then has Hamiltonian tfrac12{Ω_C,Ω_C}, which vanishes when the canonical center has no ghost-number-two constants. These are additional assumptions. Connectedness …
Exact mathematical passage
```
\Omega_C
```

\Omega_FUse in source · 3 passages

Search aliases: Ω_F Omega

Link to this entry ·

Use in source · constraints.tex, line 147
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
 \Omega_F=c^AC_A-\tfrac12c^Ac^BU_{AB}{}^C\varpi_C
             +\Omega_F^{(\ge3)}.
 \label{c:constraints:bfv-charge}
```

\Omega_F^{(\ge3)}Use in source · 1 passage

Search aliases: Ω_F^(≥3) Omega ge

Link to this entry ·

Use in source · constraints.tex, line 147
There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-depende…
Exact mathematical passage
```
 \Omega_F=c^AC_A-\tfrac12c^Ac^BU_{AB}{}^C\varpi_C
             +\Omega_F^{(\ge3)}.
 \label{c:constraints:bfv-charge}
```

\omega_hUse in source · 1 passage

Search aliases: ω_h omega

Link to this entry ·

Use in source · core.tex, line 793
The resulting frequency obeys ω_h^2=∑_ifracsin^2(k_ih)h^2. For fixed k≠0 as h→0, ω_h^2=| k|^2-frach^23∑_i k_i^4+O(h^4), ω_h=| k| -frach^26| k|∑_i k_i^4+O(h^4). These follow by expanding the sine and then the positive square root. The regime keeps k fixed. It is not uniform up to the Nyquist boundary.
Exact mathematical passage
```
 \omega_h^2=|\mathbf k|^2-\frac{h^2}{3}\sum_i k_i^4+O(h^4),
 \qquad
 \omega_h=|\mathbf k|
 -\frac{h^2}{6|\mathbf k|}\sum_i k_i^4+O(h^4).
```

\omega_h^2Use in source · 2 passages

Search aliases: ω_h^2 omega

Link to this entry ·

Use in source · core.tex, line 788
The resulting frequency obeys ω_h^2=∑_ifracsin^2(k_ih)h^2. For fixed k≠0 as h→0, ω_h^2=| k|^2-frach^23∑_i k_i^4+O(h^4), ω_h=| k| -frach^26| k|∑_i k_i^4+O(h^4). These follow by expanding the sine and then the positive square root. The regime keeps k fixed. It is not uniform up to the Nyquist boundary.
Exact mathematical passage
```
 \omega_h^2=\sum_i\frac{\sin^2(k_ih)}{h^2}.
 \label{c:core:adm-dispersion}
```

\Omega_{ij}Use in source · 1 passage

Search aliases: Ω_ij Omega

Link to this entry ·

Use in source · refinement.tex, line 353
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
\Omega_{ij}=e^{ih^2F_{ij}}
```

\Omega_{\mathrm{int}}Use in source · 2 passages

Search aliases: Ω_int Omega mathrm

Link to this entry ·

Use in source · constraints.tex, line 484
nal BFV sector On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus {Ω_int,Ω_int}=0 under that regular moment-map hypothesis. This finite Lie-algebra…
Exact mathematical passage
```
 \Omega_{\mathrm{int}}=G_{\mathrm L}[\chi]+G_{\mathrm U}[\upsilon]
                   -\tfrac12\langle\pi_\chi,[\chi,\chi]\rangle.
```

\omega_LUse in source · 1 passage

Search aliases: ω_L omega

Link to this entry ·

Use in source · spectral.tex, line 899
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in a-spec
Exact mathematical passage
```
\omega_L=e^{2\pi i/L}
```

\omega_L^{(n-j)\ell}Use in source · 1 passage

Search aliases: ω_L^(n-j)ℓ omega ell

Link to this entry ·

Use in source · spectral.tex, line 919
vergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follow by filtering these coefficients again. For an entire F the formulas hold everywhere. For a function continued outside its disk, the domain and any chosen branches must also support all r…
Exact mathematical passage
```
L^{-1}\sum_\ell\omega_L^{(n-j)\ell}
```

\omega_L^{-j\ell}Use in source · 5 passages

Search aliases: ω_L^-jℓ omega ell

Link to this entry ·

Use in source · spectral.tex, line 902
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The id…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)
 =\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}F(\omega_L^\ell z),
 \quad 0\leq j<L.
 \label{a-spectral-analytic-filter}
```

\omega_L^\ellUse in source · 5 passages

Search aliases: ω_L^ℓ omega ell

Link to this entry ·

Use in source · spectral.tex, line 902
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The id…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)
 =\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}F(\omega_L^\ell z),
 \quad 0\leq j<L.
 \label{a-spectral-analytic-filter}
```

\omega_p^{(e-d)\ell}Use in source · 1 passage

Search aliases: ω_p^(e-d)ℓ omega ell

Link to this entry ·

Use in source · finite.tex, line 458
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
p^{-1}\sum_\ell\omega_p^{(e-d)\ell}=\delta_{de}
```

\omega_p^{(q'-q)d}Use in source · 2 passages

Search aliases: ω_p^(q'-q)d omega

Link to this entry ·

Use in source · finite.tex, line 343
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
 p^{-1}\sum_{d=0}^{p-1}\omega_p^{(q'-q)d}=\delta_{qq'}.
```

\omega_p^{-d\ell}Use in source · 1 passage

Search aliases: ω_p^-dℓ omega ell

Link to this entry ·

Use in source · finite.tex, line 453
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
 R_d=\frac1p\sum_{\ell=0}^{p-1}\omega_p^{-d\ell}M_{p,h}^{\ell}.
 \label{a-finite-root-selector}
```

\omega_p^dUse in source · 1 passage

Search aliases: ω_p^d omega

Link to this entry ·

Use in source · finite.tex, line 451
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
M_{p,h}|a+d p^{h-1}\rangle=\omega_p^d|a+d p^{h-1}\rangle
```

\omega_p^{qd}Use in source · 3 passages

Search aliases: ω_p^qd omega

Link to this entry ·

Use in source · finite.tex, line 339
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
\sum_{d=0}^{p-1}\omega_p^{qd}=0
```

\Omega_{xy}Use in source · 7 passages

Search aliases: Ω_xy Omega

Link to this entry ·

Use in source · refinement.tex, line 104
For example, the two positive paths across an xy face at height z are p_xy(z)=ℓ_y(1,z)ℓ_x(0,z), p_yx(z)=ℓ_x(1,z)ℓ_y(0,z). Define Ω_xy(z)=p_xy(z)p_yx(z)^-1, and define the xz and yz face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each Ω is gauge invariant, and Ω=1 precisely when the two path phases agree.
Exact mathematical passage
```
\Omega_{xy}(z)=p_{xy}(z)p_{yx}(z)^{-1}
```

\Omega_{xz}Use in source · 3 passages

Search aliases: Ω_xz Omega

Link to this entry ·

Use in source · refinement.tex, line 109
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY, p_YXZ · =Ω_xz(1)p_YZX, · p_XZY · =Ω_xz(0)p_ZXY, p_YZX · =Ω_yz(0)p_ZYX, · p_ZXY · =Ω_xy(1)p_ZYX. Each identity follows by replacing the two-edge portion being swapped by its alte…
Exact mathematical passage
```
 p_{XYZ}&=\Omega_{xy}(0)p_{YXZ},&
 p_{XYZ}&=\Omega_{yz}(1)p_{XZY},\\
 p_{YXZ}&=\Omega_{xz}(1)p_{YZX},&
 p_{XZY}&=\Omega_{xz}(0)p_{ZXY},\\
 p_{YZX}&=\Omega_{yz}(0)p_{ZYX},&
 p_{ZXY}&=\Omega_{xy}(1)p_{ZYX}.
```

\Omega_{yz}Use in source · 3 passages

Search aliases: Ω_yz Omega

Link to this entry ·

Use in source · refinement.tex, line 109
Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY, p_YXZ · =Ω_xz(1)p_YZX, · p_XZY · =Ω_xz(0)p_ZXY, p_YZX · =Ω_yz(0)p_ZYX, · p_ZXY · =Ω_xy(1)p_ZYX. Each identity follows by replacing the two-edge portion being swapped by its alte…
Exact mathematical passage
```
 p_{XYZ}&=\Omega_{xy}(0)p_{YXZ},&
 p_{XYZ}&=\Omega_{yz}(1)p_{XZY},\\
 p_{YXZ}&=\Omega_{xz}(1)p_{YZX},&
 p_{XZY}&=\Omega_{xz}(0)p_{ZXY},\\
 p_{YZX}&=\Omega_{yz}(0)p_{ZYX},&
 p_{ZXY}&=\Omega_{xy}(1)p_{ZYX}.
```

\omega_\perpUse in source · 1 passage

Search aliases: ω_⊥ omega perp

Link to this entry ·

Use in source · constraints.tex, line 428
lit On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 R=\mathrm dt\wedge(\dot\Omega-D_\Omega\omega_\perp)+\mathcal R,
 \qquad
 F=\mathrm dt\wedge(\dot A-\mathrm d_\Sigma a_\perp)+\mathcal F.
```

\omega_\perp^{IJ}Use in source · 1 passage

Search aliases: ω_⊥^IJ omega perp

Link to this entry ·

Use in source · constraints.tex, line 422
An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,
Exact mathematical passage
```
 e^I=e_\perp^I\mathrm dt+E^I,\qquad
 \omega^{IJ}=\omega_\perp^{IJ}\mathrm dt+\Omega^{IJ},\qquad
 a=a_\perp\mathrm dt+A.
```

\daggerUse in source · 11 passages

Search aliases: † dagger

Link to this entry ·

Use in source · spectral.tex, line 1246
Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number norm. Truncation of a vector in either domain converges in the corresponding graph norm, so the identities extend to those domains. For the isolated …
Exact mathematical passage
```
 Q_r^{(L)}aQ_s^{(L)}=0\quad\text{unless }r\equiv s-1\pmod L,
 \qquad
 Q_r^{(L)}a^\dagger Q_s^{(L)}=0\quad\text{unless }r\equiv s+1\pmod L.
 \label{a-spectral-ladder-routing}
```

\ell_xUse in source · 2 passages

Search aliases: ℓ_x ell

Link to this entry ·

Use in source · refinement.tex, line 97
Cubical transport and the curvature obstruction. Three one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one 2×2×2 vertex cube. Let the vertices be (x,y,z)∈{0,1}^3, with positive links ℓ_x(y,z), ℓ_y(x,z) and ℓ_z(x,y). There are twelve links and six faces. Under a vertex gauge, every link acquires its target gauge on the left and its source inverse on the right.
Exact mathematical passage
```
\ell_x(y,z)
```

\ell_yUse in source · 2 passages

Search aliases: ℓ_y ell

Link to this entry ·

Use in source · refinement.tex, line 97
Cubical transport and the curvature obstruction. Three one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one 2×2×2 vertex cube. Let the vertices be (x,y,z)∈{0,1}^3, with positive links ℓ_x(y,z), ℓ_y(x,z) and ℓ_z(x,y). There are twelve links and six faces. Under a vertex gauge, every link acquires its target gauge on the left and its source inverse on the right.
Exact mathematical passage
```
\ell_y(x,z)
```

\ell_zUse in source · 1 passage

Search aliases: ℓ_z ell

Link to this entry ·

Use in source · refinement.tex, line 97
Cubical transport and the curvature obstruction. Three one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one 2×2×2 vertex cube. Let the vertices be (x,y,z)∈{0,1}^3, with positive links ℓ_x(y,z), ℓ_y(x,z) and ℓ_z(x,y). There are twelve links and six faces. Under a vertex gauge, every link acquires its target gauge on the left and its source inverse on the right.
Exact mathematical passage
```
\ell_z(x,y)
```

\ell_\pmUse in source · 1 passage

Search aliases: ℓ_± ell pm

Link to this entry ·

Use in source · strong_fields.tex, line 473
For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric di…
Exact mathematical passage
```
 \ell_\pm=\sigma^{-1}\partial_t+(-v\pm1)\partial_r.
```

\leftrightarrowUse in source · 1 passage

Search aliases: ↔ leftrightarrow

Link to this entry ·

Use in source · finite.tex, line 490
sis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the nonzero character labels q↔ p-q and leaves the detail projectors unchanged.
Exact mathematical passage
```
q\leftrightarrow p-q
```

\partialUse in source · 57 passages

Search aliases: ∂ partial

Link to this entry ·

Use in source · continuum.tex, line 301
On finite Fourier polynomials, the operator agrees with the matrix differential expression -i∑_jα_j∂_j+mβ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every L^2 representative. Equality with a separately constructed weak differential operator is a further interface statement if that alternate realization is introduced.
Exact mathematical passage
```
-i\sum_j\alpha_j\partial_j+m\beta
```

\partial_+Use in source · 1 passage

Search aliases: ∂_+ partial

Link to this entry ·

Use in source · core.tex, line 96
i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n
```

\partial_-Use in source · 1 passage

Search aliases: ∂_- partial

Link to this entry ·

Use in source · core.tex, line 96
i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.
Exact mathematical passage
```
Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n
```

\partial_aUse in source · 5 passages

Search aliases: ∂_a partial

Link to this entry ·

Use in source · constraints.tex, line 183
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K\mathcal T_L\subset\mathcal T_{K+L},\qquad
 \partial_a\mathcal T_K\subset\mathcal T_K.
```

\partial_{A_j}Use in source · 1 passage

Search aliases: ∂_A_j partial

Link to this entry ·

Use in source · core.tex, line 221
dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1). For example, if z_j=ψ_j^*α U_jψ_j+1, then ∂_A_jImz_j =Im(-ihz_j)=-hRez_j. This fixes the positive sign of dot E_j=J_j for the conventions in the action.
Exact mathematical passage
```
\partial_{A_j}\operatorname{Im}z_j
=\operatorname{Im}(-ihz_j)=-h\operatorname{Re}z_j
```

\partial_bUse in source · 2 passages

Search aliases: ∂_b partial

Link to this entry ·

Use in source · constraints.tex, line 528
nequation widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket …
Exact mathematical passage
```
 S^a(\nu,\mu)=\det(q_{cd})q^{ab}
                          (\nu\partial_b\mu-\mu\partial_b\nu).
```

\partial_IUse in source · 2 passages

Search aliases: ∂_I partial

Link to this entry ·

Use in source · constraints.tex, line 320
The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor i, the flat-space operators iγ^I∂_I± m have product (iγ^I∂_I+m)(iγ^J∂_J-m) =-Box-m^2, Box=η^IJ∂_I∂_J. Thus its squared field equation is Box+m^2=0, whereas the conventional real massive dispersion in this mostly-plus metric uses Box-m^2=0. A consistent translation of the Clifford, k…
Exact mathematical passage
```
i\gamma^I\partial_I\pm m
```

\partial_iUse in source · 5 passages

Search aliases: ∂_i partial

Link to this entry ·

Use in source · refinement.tex, line 349
For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.
Exact mathematical passage
```
A_i(p+t e_i)=A_i(p)+t\,\partial_iA_i
```

\partial_JUse in source · 1 passage

Search aliases: ∂_J partial

Link to this entry ·

Use in source · constraints.tex, line 321
The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor i, the flat-space operators iγ^I∂_I± m have product (iγ^I∂_I+m)(iγ^J∂_J-m) =-Box-m^2, Box=η^IJ∂_I∂_J. Thus its squared field equation is Box+m^2=0, whereas the conventional real massive dispersion in this mostly-plus metric uses Box-m^2=0. A consistent translation of the Clifford, k…
Exact mathematical passage
```
 (i\gamma^I\partial_I+m)(i\gamma^J\partial_J-m)
       =-\Box-m^2,
 \qquad \Box=\eta^{IJ}\partial_I\partial_J.
```

\partial_jUse in source · 4 passages

Search aliases: ∂_j partial

Link to this entry ·

Use in source · continuum.tex, line 301
On finite Fourier polynomials, the operator agrees with the matrix differential expression -i∑_jα_j∂_j+mβ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every L^2 representative. Equality with a separately constructed weak differential operator is a further interface statement if that alternate realization is introduced.
Exact mathematical passage
```
-i\sum_j\alpha_j\partial_j+m\beta
```

\partial_pUse in source · 1 passage

Search aliases: ∂_p partial

Link to this entry ·

Use in source · core.tex, line 853
Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], and the two defect terms give the second.
Exact mathematical passage
```
\partial_p\mathcal D_h[\xi]=h\xi Dq
```

\partial_qUse in source · 1 passage

Search aliases: ∂_q partial

Link to this entry ·

Use in source · core.tex, line 854
Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], and the two defect terms give the second.
Exact mathematical passage
```
\partial_q\mathcal D_h[\xi]=-hD(\xi p)
```

\partial_RUse in source · 1 passage

Search aliases: ∂_R partial

Link to this entry ·

Use in source · strong_fields.tex, line 239
normδ P_R_∞ · ≤frac Mp_*normD_4δ R_∞ +fracM^2d^*p_*^3normδ R_∞. For example, ∂_Rp=M/p and the two partial derivatives of MΛ/p have magnitudes at most M/p_* and M^2d^*/p_*^3. The admissible segment preserves the displayed linear radius and derivative bounds. The constants are independent of h.
Exact mathematical passage
```
\partial_Rp=M/p
```

\partial_rUse in source · 5 passages

Search aliases: ∂_r partial

Link to this entry ·

Use in source · strong_fields.tex, line 388
E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.
Exact mathematical passage
```
 \dot\varphi&=\sigma(\Pi+v\Phi),\\
 \dot\Phi&=\partial_r[\sigma(\Pi+v\Phi)],\\
 \dot\Pi&=r^{-2}\partial_r[r^2\sigma(\Phi+v\Pi)].
\label{c:strong:scalar-evolution}
```

\partial_tUse in source · 7 passages

Search aliases: ∂_t partial

Link to this entry ·

Use in source · core.tex, line 71
s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.
Exact mathematical passage
```
\dot E_h=\sum_js_j=-\partial_tL_h
```

\partial_xUse in source · 4 passages

Search aliases: ∂_x partial

Link to this entry ·

Use in source · core.tex, line 302
[Local second-order consistency] With uniform bounds on A and its derivatives through order two and on ψ and its derivatives through order three on a fixed neighborhood, (D_h^Aψ)(x)=(∂_x-iA(x))ψ(x)+O(h^2) (h→0). The remainder is uniform over points whose neighborhoods have the same derivative bounds.
Exact mathematical passage
```
\label{g:stencil-limit}
 (D_h^A\psi)(x)=(\partial_x-iA(x))\psi(x)+O(h^2)
 \qquad(h\to0).
```

\partial_{x_1}Use in source · 2 passages

Search aliases: ∂_x_1 partial

Link to this entry ·

Use in source · constraints.tex, line 787
A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m
Exact mathematical passage
```
R_F=\partial_{x_1}+y\partial_{x_2}
```

\partial_{x_2}Use in source · 2 passages

Search aliases: ∂_x_2 partial

Link to this entry ·

Use in source · constraints.tex, line 787
A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m
Exact mathematical passage
```
R_F=\partial_{x_1}+y\partial_{x_2}
```

\partial_\thetaUse in source · 3 passages

Search aliases: ∂_θ partial theta

Link to this entry ·

Use in source · core.tex, line 360
E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
\partial_\tau(e^{2P}Q_\tau)-\partial_\theta(e^{2P}\gamma Q_\theta)=0
```

\partial_\nuUse in source · 1 passage

Search aliases: ∂_ν partial nu

Link to this entry ·

Use in source · constraints.tex, line 363
Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,
Exact mathematical passage
```
 sc^\mu=c^\nu\partial_\nu c^\mu,\qquad
 s\chi=\mathcal L_c\chi+\chi^2,\qquad
 s\upsilon=\mathcal L_c\upsilon.
 \label{c:constraints:ghost-differential}
```

\partial_\tauUse in source · 4 passages

Search aliases: ∂_τ partial tau

Link to this entry ·

Use in source · core.tex, line 360
E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.
Exact mathematical passage
```
\partial_\tau(e^{2P}Q_\tau)-\partial_\theta(e^{2P}\gamma Q_\theta)=0
```

\partial_\tau^2Use in source · 1 passage

Search aliases: ∂_τ^2 partial tau

Link to this entry ·

Use in source · core.tex, line 439
. Define the entire function J_0(x)=∑_m=0^∞frac(-1)^m(x^2/4)^m(m!)^2. The series and its differentiated series converge uniformly on compact sets. Coefficient comparison gives x^2J_0''+xJ_0'+x^2J_0=0. Thus P(τ,θ)=A J_0(ne^-τ)cos(nθ) is an exact polarized solution. Indeed, for x=ne^-τ, ∂_τ^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x), whereas P_θθ=-n^2P. Also M_G=2P_τ P_θ has zero spatial integral, so periodic λ can be reconstructed.
Exact mathematical passage
```
\partial_\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)
```

\nablaUse in source · 9 passages

Search aliases: ∇ nabla

Link to this entry ·

Use in source · constraints.tex, line 8
An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.
Exact mathematical passage
```
 \{F,G\}=\sum_A\operatorname{Tr}\left(
 \nabla_{Z^A}F\nabla_{P_A}G-
 \nabla_{P_A}F\nabla_{Z^A}G\right).
 \label{c:constraints:trace-bracket}
```

\nabla_hUse in source · 2 passages

Search aliases: ∇_h nabla

Link to this entry ·

Use in source · core.tex, line 716
For the bracket, use {F,G}=⟨∇_hF,∇_π G⟩_h -⟨∇_π F,∇_hG⟩_h. Two scalar constraints depend only on h and two momentum constraints depend only on π, so their brackets vanish. The gradients for the mixed bracket are (∇_hC[n])_ij=δ_iδ_jn-δ_ijδ^2n, (∇_π D[ξ])_ij=δ_iξ_j+δ_jξ_i. Their contraction vanishes after one integration by parts because the first tensor is divergence-free.
Exact mathematical passage
```
\{F,G\}=\langle\nabla_hF,\nabla_\pi G\rangle_h
-\langle\nabla_\pi F,\nabla_hG\rangle_h
```

\nabla_PUse in source · 2 passages

Search aliases: ∇_P nabla

Link to this entry ·

Use in source · constraints.tex, line 24
equation The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.
Exact mathematical passage
```
\nabla_PD[\Lambda]=i[\Lambda,Z]
```

\nabla_{P_A}Use in source · 1 passage

Search aliases: ∇_P_A nabla

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Use in source · constraints.tex, line 8
An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.
Exact mathematical passage
```
 \{F,G\}=\sum_A\operatorname{Tr}\left(
 \nabla_{Z^A}F\nabla_{P_A}G-
 \nabla_{P_A}F\nabla_{Z^A}G\right).
 \label{c:constraints:trace-bracket}
```

\nabla_QUse in source · 1 passage

Search aliases: ∇_Q nabla

Link to this entry ·

Use in source · core.tex, line 492
For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
 \nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\
 \nabla_QC_h&=-2hD_h(e^{2P}V),\\
 g_U:=\nabla_UC_h&=2hD_hP,
 &g_V:=\nabla_VC_h&=2h e^{2P}D_hQ.
```

\nabla_UUse in source · 1 passage

Search aliases: ∇_U nabla

Link to this entry ·

Use in source · core.tex, line 492
For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
 \nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\
 \nabla_QC_h&=-2hD_h(e^{2P}V),\\
 g_U:=\nabla_UC_h&=2hD_hP,
 &g_V:=\nabla_VC_h&=2h e^{2P}D_hQ.
```

\nabla_VUse in source · 1 passage

Search aliases: ∇_V nabla

Link to this entry ·

Use in source · core.tex, line 492
For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keeps C_h constant.
Exact mathematical passage
```
 \nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\
 \nabla_QC_h&=-2hD_h(e^{2P}V),\\
 g_U:=\nabla_UC_h&=2hD_hP,
 &g_V:=\nabla_VC_h&=2h e^{2P}D_hQ.
```

\nabla_ZUse in source · 1 passage

Search aliases: ∇_Z nabla

Link to this entry ·

Use in source · constraints.tex, line 25
ned by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.
Exact mathematical passage
```
\nabla_ZD[\Lambda]=i[P,\Lambda]
```

\nabla_{Z^A}Use in source · 1 passage

Search aliases: ∇_Z^A nabla

Link to this entry ·

Use in source · constraints.tex, line 8
An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.
Exact mathematical passage
```
 \{F,G\}=\sum_A\operatorname{Tr}\left(
 \nabla_{Z^A}F\nabla_{P_A}G-
 \nabla_{P_A}F\nabla_{Z^A}G\right).
 \label{c:constraints:trace-bracket}
```

\nabla_\piUse in source · 2 passages

Search aliases: ∇_π nabla pi

Link to this entry ·

Use in source · core.tex, line 716
For the bracket, use {F,G}=⟨∇_hF,∇_π G⟩_h -⟨∇_π F,∇_hG⟩_h. Two scalar constraints depend only on h and two momentum constraints depend only on π, so their brackets vanish. The gradients for the mixed bracket are (∇_hC[n])_ij=δ_iδ_jn-δ_ijδ^2n, (∇_π D[ξ])_ij=δ_iξ_j+δ_jξ_i. Their contraction vanishes after one integration by parts because the first tensor is divergence-free.
Exact mathematical passage
```
\{F,G\}=\langle\nabla_hF,\nabla_\pi G\rangle_h
-\langle\nabla_\pi F,\nabla_hG\rangle_h
```

\notinUse in source · 8 passages

Search aliases: ∉ notin

Link to this entry ·

Use in source · finite.tex, line 560
[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber
Exact mathematical passage
```
z\notin\sigma(H_f)
```

\prod_{f\subset\partial c}Use in source · 1 passage

Search aliases: ∏_f⊂∂ c prod subset partial

Link to this entry ·

Use in source · core.tex, line 170
For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a principal argument, it implies that the oriented phase sum lies in 2πZ. It does not force that su…
Exact mathematical passage
```
\label{g:abelian-cube}
 \prod_{f\subset\partial c}W_f^{\operatorname{or}(f,c)}=1.
```

\prod_jUse in source · 1 passage

Search aliases: ∏_j prod

Link to this entry ·

Use in source · constraints.tex, line 193
If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j), τ_N(∏_jS_Nf_j) · =(2π)^-3∫_ T^3∏_jf_j d^3x. The Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled pointwise products. This is how the resolved construction avoids the finite pointwise derivation obstruction: it retains enough detail instead of demandi…
Exact mathematical passage
```
 \delta_a\prod_jS_Nf_j
 &=S_N\partial_a\left(\prod_jf_j\right),\\
 \tau_N\left(\prod_jS_Nf_j\right)
 &=(2\pi)^{-3}\int_{\mathbb T^3}\prod_jf_j\,\mathrm d^3x.
 \label{c:constraints:alias-free}
```

\prod_{k=0}^{q-1}Use in source · 1 passage

Search aliases: ∏_k=0^q-1 prod

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Use in source · spectral.tex, line 934
Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber · =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (-
Exact mathematical passage
```
 \frac{(Lq+j)!}{j!}
 &=\prod_{r=1}^{L}\prod_{k=0}^{q-1}(j+r+Lk)\nonumber\\
 &=L^{Lq}\prod_{r=1}^{L}\left(\frac{j+r}{L}\right)_q.
 \label{a-spectral-factorial-split}
```

\prod_{k=1}^{m}Use in source · 2 passages

Search aliases: ∏_k=1^m prod

Link to this entry ·

Use in source · refinement.tex, line 18
[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next factor to the product. For r=0, each factor equals one. For 1≤ r≤ m, the factor at k=r vanishes.
Exact mathematical passage
```
 T_m(h^rv)=c_{m,r}h^rv,\qquad
 c_{m,r}=\prod_{k=1}^{m}\frac{b^{k-r}-1}{b^k-1}.
```

\prod_{p\in\mathcal P_{\mathrm o}}Use in source · 2 passages

Search aliases: ∏_p∈ P_ o prod in mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 120
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 M=2^{h_2+2}\prod_{p\in\mathcal P_{\mathrm o}}p^{h_p+1}.
```

\prod_{r=1}^jUse in source · 1 passage

Search aliases: ∏_r=1^j prod

Link to this entry ·

Use in source · spectral.tex, line 1165
Let p be prime, let L=mp for a positive integer m, and take 0≤ j<p. For every q≥0, g_q=fracj!(Lq)!(Lq+j)! =binomLq+jj^-1∈Z_p^×. Indeed binomLq+jj=∏_r=1^j(Lq+r)/r. Each numerator and denominator is a p-adic unit because Lq≡0pmod p and 1≤ r<p. For j=0 the empty product is one. The proof also covers p=2 in its short window. Outside this window, a factor can cross a multiple of p and the same proof gives no unit guarantee. This rational coefficient statement supplies no claim about the p-adic conver…
Exact mathematical passage
```
\binom{Lq+j}{j}=\prod_{r=1}^j(Lq+r)/r
```

\prod_{r=1}^{L}Use in source · 1 passage

Search aliases: ∏_r=1^L prod

Link to this entry ·

Use in source · spectral.tex, line 934
Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber · =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (-
Exact mathematical passage
```
 \frac{(Lq+j)!}{j!}
 &=\prod_{r=1}^{L}\prod_{k=0}^{q-1}(j+r+Lk)\nonumber\\
 &=L^{Lq}\prod_{r=1}^{L}\left(\frac{j+r}{L}\right)_q.
 \label{a-spectral-factorial-split}
```

\sum_{\mathbf d}Use in source · 5 passages

Search aliases: ∑_ d sum mathbf

Link to this entry ·

Use in source · finite.tex, line 725
After w refinements a packet has n=p^w orthonormal descendants. Its normalized radial vector is |R_p,w⟩=n^-1/2∑_ d| d⟩. The norm is n^-1∑_ d1=1, so each descendant has amplitude modulus p^-w/2 and Born probability p^-w. This reproduces [source reference]. The words ``radial'' and ``angular'' here refer to the uniform vector and its orthogonal complement in a finite descendant space. They do not introduce a spacetime radius or a rotation group.
Exact mathematical passage
```
 |R_{p,w}\rangle=n^{-1/2}\sum_{\mathbf d}|\mathbf d\rangle.
 \label{a-finite-radial-vector}
```

\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}Use in source · 1 passage

Search aliases: ∑_ d∈{0,…,p-1}^w sum mathbf in ldots

Link to this entry ·

Use in source · finite.tex, line 411
D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 V_{p,h+w-1}\cdots V_{p,h}|a\rangle
 =p^{-w/2}\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}|a,\mathbf d\rangle,
 \label{a-finite-nested-refinement}
```

\sum_{\|k\|_\infty\le K}Use in source · 1 passage

Search aliases: ∑_ k _∞≤ K sum infty le

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Use in source · constraints.tex, line 178
A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
 \mathcal T_K=\left\{\sum_{\|k\|_\infty\le K}
              \widehat f_k e^{ik\cdot x}\right\}.
```

\sum_{\mathbf r}Use in source · 2 passages

Search aliases: ∑_ r sum mathbf

Link to this entry ·

Use in source · limit.tex, line 600
[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber H_A f · =∑_ rA_ r F_ rf. Then H_A is self-adjoint, and finite rectangular packet sums form a core. Every rectangular range reduces H_A, and its finite restriction H_A^( h) satisfies exact embedding intertwining with every larger rectangular restriction. For bounded…
Exact mathematical passage
```
 \Dom(H_A)&=\left\{f:\sum_{\mathbf r}
 \norm{A_{\mathbf r}\mathsf F_{\mathbf r}f}^2<\infty\right\},
 \nonumber\\
 H_A f&=\sum_{\mathbf r}A_{\mathbf r}\mathsf F_{\mathbf r}f.
 \label{a-limit-block-domain}
```

\sum_AUse in source · 3 passages

Search aliases: ∑_A sum

Link to this entry ·

Use in source · core.tex, line 203
J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
\dot p_B^{\mathrm c}=\sum_AJ_{B\leftarrow A}^{\mathrm c}
```

\sum_{a,b}Use in source · 2 passages

Search aliases: ∑_a,b sum

Link to this entry ·

Use in source · orientation.tex, line 50
is a resolved operator rather than a preferred coordinate matrix. Let V be a complex vector space with a finite family of projectors (P_a)_a∈ A satisfying P_aP_b=δ_abP_a, ∑_a∈ AP_a=I. The component P_av is the packet of v selected by a. For a linear map L:V→ V, its transfer block from packet a to packet b is L_ba=P_bLP_a. Completeness gives L=∑_a,bL_ba. Both indices matter. Keeping only P_aLP_a would remove every transfer between distinct packets.
Exact mathematical passage
```
L=\sum_{a,b}L_{ba}
```

\sum_{a=0}^{p-1}Use in source · 1 passage

Search aliases: ∑_a=0^p-1 sum

Link to this entry ·

Use in source · spectral.tex, line 1175
Analytic refinement of a factorial lane is independent of this unit condition. Splitting q=pr+a, 0≤ a<p, gives the exact identity E_L,j(z)=∑_a=0^p-1E_pL,j+aL(z), 0≤ j<L. Absolute convergence permits this partition of the summation index. The labels j+aL all lie in [0,pL), but need not lie in the parent's short unit window. Exact complex analytic refinement and the local p-adic unit assertion therefore have separate hypotheses.
Exact mathematical passage
```
 E_{L,j}(z)=\sum_{a=0}^{p-1}E_{pL,j+aL}(z),\qquad 0\leq j<L.
 \label{a-spectral-analytic-refinement}
```

\sum_{a=0}^{p^{r-1}-1}Use in source · 1 passage

Search aliases: ∑_a=0^p^r-1-1 sum

Link to this entry ·

Use in source · finite.tex, line 350
joint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level r, equal to the rank in [source reference], so they span that detail subspace. Hence D_r^(h)=∑_a=0^p^r-1-1∑_q=1^p-1 |ψ_r,a,q^(h)⟩⟨ψ_r,a,q^(h)|. The omitted character q=0 is the parent average. For p=2 there is one average and one difference per paren…
Exact mathematical passage
```
 D_r^{(h)}=\sum_{a=0}^{p^{r-1}-1}\sum_{q=1}^{p-1}
       |\psi_{r,a,q}^{(h)}\rangle\langle\psi_{r,a,q}^{(h)}|.
 \label{a-finite-wavelet-projector}
```

\sum_{a\in C(A)}Use in source · 2 passages

Search aliases: ∑_a∈ C(A) sum in

Link to this entry ·

Use in source · orientation.tex, line 58
Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.
Exact mathematical passage
```
P_A=\sum_{a\in C(A)}P_a
```

\sum_{a\in C_A}Use in source · 2 passages

Search aliases: ∑_a∈ C_A sum in

Link to this entry ·

Use in source · core.tex, line 190
Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a different physical system.
Exact mathematical passage
```
\label{f:coarse-projector}
 Q_A=\sum_{a\in C_A}P_a.
```

\sum_{a\in C_\alpha}Use in source · 4 passages

Search aliases: ∑_a∈ C_α sum in alpha

Link to this entry ·

Use in source · extensions.tex, line 216
For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.
Exact mathematical passage
```
Q_\alpha=\sum_{a\in C_\alpha}P_a
```

\sum_{a\notin C}Use in source · 1 passage

Search aliases: ∑_a∉ C sum notin

Link to this entry ·

Use in source · core.tex, line 129
fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.
Exact mathematical passage
```
\label{f:boundary-current}
 \frac{d}{dt}\sum_{b\in C}p_b
 =\sum_{b\in C}\sum_{a\notin C}J_{b\leftarrow a}.
```

\sum_{A\ne B}Use in source · 1 passage

Search aliases: ∑_A≠ B sum ne

Link to this entry ·

Use in source · core.tex, line 459
The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equality of outgoing rates to each destination cell, the fine population distribution inside a source cell matters. This direct consequence of the gain sum …
Exact mathematical passage
```
\dot p_B^{\mathrm c}=\sum_{A\ne B}(q_{BA}p_A^{\mathrm c}-q_{AB}p_B^{\mathrm c})
```

\sum_bUse in source · 9 passages

Search aliases: ∑_b sum

Link to this entry ·

Use in source · core.tex, line 64
[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.
Exact mathematical passage
```
 X&=\sum_{b,a}X_{ba}, & P_bX\psi&=\sum_aX_{ba}\psi_a,\label{f:block-reconstruction}\\
 (XY)_{ca}&=\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.
 \label{f:block-product}
```

\sum_{b,a}Use in source · 6 passages

Search aliases: ∑_b,a sum

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Use in source · core.tex, line 64
[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.
Exact mathematical passage
```
 X&=\sum_{b,a}X_{ba}, & P_bX\psi&=\sum_aX_{ba}\psi_a,\label{f:block-reconstruction}\\
 (XY)_{ca}&=\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.
 \label{f:block-product}
```

\sum_{b=0}^hUse in source · 1 passage

Search aliases: ∑_b=0^h sum

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Use in source · spectral.tex, line 421
The rank (p-1)p^r-1 thus counts chains born at level r or new Haar directions on a level. A single level's detail space is usually transported to the next level; it is the entire angular chain that is invariant under the tree Hamiltonian. The full dark dimension is ∑_r=1^h(p-1)p^r-1(h-r+1) =∑_b=0^hp^b-(h+1). Every one of these states is invisible to a root-localized symmetric source because it is orthogonal to the radial reducing block.
Exact mathematical passage
```
 \sum_{r=1}^h(p-1)p^{r-1}(h-r+1)
 =\sum_{b=0}^hp^b-(h+1).
```

\sum_{b\in A}Use in source · 1 passage

Search aliases: ∑_b∈ A sum in

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Use in source · orientation.tex, line 53
For two operators L and M, insert the resolution between them: P_cMLP_a=∑_b∈ A(P_cMP_b)(P_bLP_a). This is block multiplication written without a choice of basis inside each packet. The intermediate index records the interface through which a composition passes. It is also the reason that a change of resolution must specify how fine packets assemble into coarse packets.
Exact mathematical passage
```
P_cMLP_a=\sum_{b\in A}(P_cMP_b)(P_bLP_a).
```

\sum_{b\in C}Use in source · 1 passage

Search aliases: ∑_b∈ C sum in

Link to this entry ·

Use in source · core.tex, line 129
fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.
Exact mathematical passage
```
\label{f:boundary-current}
 \frac{d}{dt}\sum_{b\in C}p_b
 =\sum_{b\in C}\sum_{a\notin C}J_{b\leftarrow a}.
```

\sum_{b\in C(B)}Use in source · 2 passages

Search aliases: ∑_b∈ C(B) sum in

Link to this entry ·

Use in source · orientation.tex, line 58
Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.
Exact mathematical passage
```
P_B=\sum_{b\in C(B)}P_b
```

\sum_{b\in C_B}Use in source · 2 passages

Search aliases: ∑_b∈ C_B sum in

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Use in source · core.tex, line 197
[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.
Exact mathematical passage
```
 p_A^{\mathrm c}&=\norm{Q_A\psi}^2=\sum_{a\in C_A}p_a,\\
 J_{B\leftarrow A}^{\mathrm c}
 &=2\operatorname{Re}\inner{Q_B\psi}{KQ_A\psi}
 =\sum_{b\in C_B}\sum_{a\in C_A}J_{b\leftarrow a}.
```

\sum_{b\in C_\beta}Use in source · 1 passage

Search aliases: ∑_b∈ C_β sum in beta

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Use in source · extensions.tex, line 223
Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately: J^rm c_βα · =2IminnerQ_βψHQ_αψ · =∑_b∈ C_β(∑_a∈ C_αJ_ba) =∑_a∈ C_α(∑_b∈ C_βJ_ba). For example, first summing a moves H onto P_bψ, then the outer sum pairs ∑_bP_bψ with the fixed vector HQ_αψ. The reverse order has its own symmetric argument. Flattening these expressions to a single unordered sum on C_β× C_α requires joint summability. Equality of two justified iterated sums does not supp…
Exact mathematical passage
```
 J^{\rm c}_{\beta\alpha}
 &=2\operatorname{Im}\inner{Q_\beta\psi}{HQ_\alpha\psi}\\
 &=\sum_{b\in C_\beta}\left(\sum_{a\in C_\alpha}J_{ba}\right)
 =\sum_{a\in C_\alpha}\left(\sum_{b\in C_\beta}J_{ba}\right).
```

\sum_{b\ne a}Use in source · 1 passage

Search aliases: ∑_b≠ a sum ne

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Use in source · core.tex, line 413
of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected basis. A general dissipative operator or a higher-rank atlas does not automatically produce the same closed rate equation.
Exact mathematical passage
```
\label{f:lindblad}
 \dot\rho=\sum_{b\ne a}\mathcal D_{L_{ba}}(\rho).
```

\sum_{b\ne i}Use in source · 1 passage

Search aliases: ∑_b≠ i sum ne

Link to this entry ·

Use in source · core.tex, line 437
In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable …
Exact mathematical passage
```
r_i=\sum_{b\ne i}\gamma_{bi}
```

\sum_cUse in source · 2 passages

Search aliases: ∑_c sum

Link to this entry ·

Use in source · extensions.tex, line 492
Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.
Exact mathematical passage
```
I=\sum_cP_c
```

\sum_dUse in source · 7 passages

Search aliases: ∑_d sum

Link to this entry ·

Use in source · finite.tex, line 380
One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an id…
Exact mathematical passage
```
|+_p\rangle=p^{-1/2}\sum_d|d\rangle
```

\sum_{d=0}^{p-1}Use in source · 6 passages

Search aliases: ∑_d=0^p-1 sum

Link to this entry ·

Use in source · finite.tex, line 339
Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level…
Exact mathematical passage
```
\sum_{d=0}^{p-1}\omega_p^{qd}=0
```

\sum_iUse in source · 4 passages

Search aliases: ∑_i sum

Link to this entry ·

Use in source · core.tex, line 606
Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric tensors.
Exact mathematical passage
```
 \delta_i\delta_j=\delta_j\delta_i,
 \qquad\inner{\delta_iX}{Y}_h=-\inner{X}{\delta_iY}_h,
 \qquad\delta^2=\sum_i\delta_i^2.
```

\sum_{i=0}^{M-1}Use in source · 2 passages

Search aliases: ∑_i=0^M-1 sum

Link to this entry ·

Use in source · core.tex, line 250
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 S_h[u]=\sum_{i=0}^{M-1}\frac{(u_{i+1}-u_i)^2}{2Gh}
 +4\pi h\sum_{i=1}^{M-1}r_i\rho_i u_i.
 \label{c:core:poisson-action}
```

\sum_{i=1}^{M-1}Use in source · 1 passage

Search aliases: ∑_i=1^M-1 sum

Link to this entry ·

Use in source · core.tex, line 250
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 S_h[u]=\sum_{i=0}^{M-1}\frac{(u_{i+1}-u_i)^2}{2Gh}
 +4\pi h\sum_{i=1}^{M-1}r_i\rho_i u_i.
 \label{c:core:poisson-action}
```

\sum_jUse in source · 27 passages

Search aliases: ∑_j sum

Link to this entry ·

Use in source · spectral.tex, line 909
, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follo…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
 \quad
 \mathfrak R_{L,j}\mathfrak R_{L,k}
   =\delta_{jk}\mathfrak R_{L,j},\quad
 \sum_j\mathfrak R_{L,j}=I.
 \label{a-spectral-analytic-lanes}
```

\sum_{j=0}^{L-1}Use in source · 1 passage

Search aliases: ∑_j=0^L-1 sum

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Use in source · spectral.tex, line 1003
[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber · =frac(tA_0)^jj! _0F_L-1 (- A_L,j; frac(tA_0)^LL^L),nonumber ∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subser…
Exact mathematical passage
```
 E_{L,j}(tA_0)
 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
                           e^{\omega_L^\ell tA_0},\nonumber\\
 &=\frac{(tA_0)^j}{j!}\,
 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
                      \frac{(tA_0)^L}{L^L}\right),\nonumber\\
 \sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.
 \label{a-spectral-operator-lane-identities}
```

\sum_{j=1}^NUse in source · 2 passages

Search aliases: ∑_j=1^N sum

Link to this entry ·

Use in source · refinement.tex, line 253
The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty products are identities. If every F_j,C_j is contractive, the triangle inequality and submultiplicativity yield normF_N⋯ F_1I-IC_N⋯ C_1 ≤∑_j=1^NnormD…
Exact mathematical passage
```
 F_N\cdots F_1I-IC_N\cdots C_1
 =\sum_{j=1}^N F_N\cdots F_{j+1}D_jC_{j-1}\cdots C_1.
 \label{g:telescoping}
```

\sum_{j=a}^{b}Use in source · 1 passage

Search aliases: ∑_j=a^b sum

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Use in source · core.tex, line 251
rge without compensating background would be inconsistent. Equation [source reference] supplies that neutralizing mean. Conservation of G_j is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over a≤ j≤ b gives the physical-length-weighted charge balance fracddt(h∑_j=a^bρ_j)=J_a-1-J_b.
Exact mathematical passage
```
 \frac{d}{dt}\left(h\sum_{j=a}^{b}\rho_j\right)=J_{a-1}-J_b.
```

\sum_{j=r}^sUse in source · 1 passage

Search aliases: ∑_j=r^s sum

Link to this entry ·

Use in source · core.tex, line 99
Every e_j is nonnegative under the stated coefficient assumptions. An interval of cells r,…,s obeys the exact coarse balance fracddt∑_j=r^se_j +F_s+1/2-F_r-1/2=∑_j=r^ss_j. Thus coarse energy is obtained by adding fine cell energies and retaining only the exposed boundary fluxes. This is the same cancellation mechanism as projector transfer, applied here to a classical field energy. Its physical units and constitutive coefficients come from the action.
Exact mathematical passage
```
 \frac{d}{dt}\sum_{j=r}^se_j
 +F_{s+1/2}-F_{r-1/2}=\sum_{j=r}^ss_j.
```

\sum_{j=x,y,z}Use in source · 4 passages

Search aliases: ∑_j=x,y,z sum

Link to this entry ·

Use in source · finite_core.tex, line 219
-i∑_j=x,y,zα_j∂_j+mβ
Exact mathematical passage
```
-i\sum_{j=x,y,z}\alpha_j\partial_j+m\beta
```

\sum_kUse in source · 5 passages

Search aliases: ∑_k sum

Link to this entry ·

Use in source · finite_core.tex, line 304
The series converges absolutely in operator norm, being dominated by ∑_k(|t| H )^k/k!. This is exact continuous-time evolution, with no time step or series truncation.
Exact mathematical passage
```
\sum_k(|t|\|H\|)^k/k!
```

\sum_{k=0}^{N}Use in source · 1 passage

Search aliases: ∑_k=0^N sum

Link to this entry ·

Use in source · finite_core.tex, line 766
∑_k=0^Nfracz^k(A^ H)^kk! =T(∑_k=0^Nfracz^kA^kk!)T^-1.
Exact mathematical passage
```
\sum_{k=0}^{N}\frac{z^k(A^{\mathcal H})^k}{k!}
=T\left(\sum_{k=0}^{N}\frac{z^kA^k}{k!}\right)T^{-1}.
```

\sum_{k=0}^{N-1}Use in source · 1 passage

Search aliases: ∑_k=0^N-1 sum

Link to this entry ·

Use in source · spectral.tex, line 808
On [-2,2] define the probability measure dν(x)=fracdxπsqrt4-x^2. It is the distribution of 2cosθ for a uniform angle on the circle, or on [0,π]. It is also the limiting normalized spectral measure of the free cycles: for a continuous test function f, frac1N∑_k=0^N-1f(2cos(2π k/N)) ⟶frac12π∫_0^2πf(2cosθ) dθ. This is the Riemann-sum limit of a continuous periodic function.
Exact mathematical passage
```
 \frac1N\sum_{k=0}^{N-1}f(2\cos(2\pi k/N))
 \longrightarrow\frac1{2\pi}\int_0^{2\pi}f(2\cos\theta)\,d\theta.
```

\sum_{k=0}^\inftyUse in source · 1 passage

Search aliases: ∑_k=0^∞ sum infty

Link to this entry ·

Use in source · spectral.tex, line 1029
is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane by the full tail. Since (Q+1+k)!≥(Q+1)!k!, ∑_n=Q+1^∞fraca^nn! ≤fraca^Q+1(Q+1)!∑_k=0^∞fraca^kk!, which proves [source reference].
Exact mathematical passage
```
 \sum_{n=Q+1}^\infty\frac{a^n}{n!}
 \leq\frac{a^{Q+1}}{(Q+1)!}\sum_{k=0}^\infty\frac{a^k}{k!},
```

\sum_{k=0}^{\infty}Use in source · 1 passage

Search aliases: ∑_k=0^∞ sum infty

Link to this entry ·

Use in source · finite_core.tex, line 299
U_H(t)=e^-itH=∑_k=0^∞frac(-itH)^kk!.
Exact mathematical passage
```
U_H(t)=e^{-itH}=\sum_{k=0}^{\infty}\frac{(-itH)^k}{k!}.
```

\sum_{k=1}^{\infty}Use in source · 2 passages

Search aliases: ∑_k=1^∞ sum infty

Link to this entry ·

Use in source · finite_core.tex, line 684
e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .
Exact mathematical passage
```
\|e^X-e^Y\|
\le\|X-Y\|\sum_{k=1}^{\infty}\frac{kR^{k-1}}{k!}
=e^R\|X-Y\|.
```

\sum_{k=n}^{m-1}Use in source · 1 passage

Search aliases: ∑_k=n^m-1 sum

Link to this entry ·

Use in source · refinement.tex, line 451
pace limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence is Cauchy. Completeness provides a limit. Passing m→∞ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric series.
Exact mathematical passage
```
 d(f_n,f_m)\leq\sum_{k=n}^{m-1}C4^{-k}
 \leq\tfrac43C4^{-n}.
```

\sum_{k=n}^{\infty}Use in source · 1 passage

Search aliases: ∑_k=n^∞ sum infty

Link to this entry ·

Use in source · refinement.tex, line 443
[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence is Cauchy. Completeness provides a limit. Passing m→∞ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric ser…
Exact mathematical passage
```
 d(f_n,f_\infty)\leq\sum_{k=n}^{\infty}C4^{-k}
 =\tfrac43C4^{-n}.
 \label{g:dyadic-tail}
```

\sum_{k\ge1}Use in source · 2 passages

Search aliases: ∑_k≥1 sum ge

Link to this entry ·

Use in source · continuum.tex, line 237
A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly when s>3/2. Thus the choice s=1 gives an L^2 field outside the Hamiltonian domain, while s=2 gives one inside it. These thresholds use axis-supported da…
Exact mathematical passage
```
\sum_n\norm{a_n}^2=\sum_{k\ge1}k^{-2s},
```

\sum_mUse in source · 3 passages

Search aliases: ∑_m sum

Link to this entry ·

Use in source · limit.tex, line 270
it-parseval identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m gives k_m=ϵ_m g_m. Since k is square summable, ∑_m|ϵ_m|^2normg_m^2<∞. Thus g∈Dom(H_ϵ) and the adjoint agrees with H_ϵ. The reverse inclusion is symmetry. This proves self-adjointness and therefo…
Exact mathematical passage
```
\sum_m|\epsilon_m|^2\norm{g_m}^2<\infty
```

\sum_{m=0}^MUse in source · 1 passage

Search aliases: ∑_m=0^M sum

Link to this entry ·

Use in source · limit.tex, line 151
Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases involving m=0 use E_0 E_n= E_0.
Exact mathematical passage
```
 \mathsf F_m\mathsf F_n=\delta_{mn}\mathsf F_m,
 \qquad \sum_{m=0}^M\mathsf F_m=\mathsf E_M.
 \label{a-limit-detail-algebra}
```

\sum_{m=0}^{M}Use in source · 1 passage

Search aliases: ∑_m=0^M sum

Link to this entry ·

Use in source · limit.tex, line 346
Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.
Exact mathematical passage
```
 U_M^{\mathrm{emb}}(t)
 =\sum_{m=0}^{M}e^{-it\epsilon_m}\mathsf F_m
 =U(t)\mathsf E_M.
 \label{a-limit-embedded-propagator}
```

\sum_{m=0}^{\infty}Use in source · 2 passages

Search aliases: ∑_m=0^∞ sum infty

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Use in source · limit.tex, line 164
Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.
Exact mathematical passage
```
 \HH_p=\bigoplus_{m=0}^{\infty}\operatorname{Ran}\mathsf F_m,
 \qquad \norm{f}_2^2=\sum_{m=0}^{\infty}\norm{\mathsf F_mf}_2^2.
 \label{a-limit-parseval}
```

\sum_{m>M}Use in source · 4 passages

Search aliases: ∑_m>M sum

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Use in source · limit.tex, line 205
[Exact tail and regularity bound] For f∈HH_p and every M≥0, norm(I- E_M)f_2^2 =∑_m>Mnorm F_mf_2^2. If f∈ H_p^s, then norm(I- E_M)f_2 ≤ p^-s(M+1)normf_ H_p^s. The convergence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted summands by this lower bound proves the squared estimate, and taking square roots gives [source reference]. At every finite M there is a unit vector …
Exact mathematical passage
```
 \norm{(I-\mathsf E_M)f}_2^2
 =\sum_{m>M}\norm{\mathsf F_mf}_2^2.
 \label{a-limit-tail-identity}
```

\sum_{m\leq M}Use in source · 1 passage

Search aliases: ∑_m≤ M sum leq

Link to this entry ·

Use in source · limit.tex, line 388
An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber
Exact mathematical passage
```
 \widetilde H_M=H_\epsilon\mathsf E_M
 =\sum_{m\leq M}\epsilon_m\mathsf F_m.
```

\sum_{m\geq0}Use in source · 6 passages

Search aliases: ∑_m≥0 sum geq

Link to this entry ·

Use in source · limit.tex, line 193
For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.
Exact mathematical passage
```
 \norm{f}_{\mathfrak H_p^s}^2
 =\sum_{m\geq0}p^{2sm}\norm{\mathsf F_mf}_2^2,
 \qquad
 \mathfrak H_p^s=\{f\in\HH_p:\norm{f}_{\mathfrak H_p^s}<\infty\}.
```

\sum_{m\geq1}Use in source · 1 passage

Search aliases: ∑_m≥1 sum geq

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Use in source · limit.tex, line 511
silon is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It…
Exact mathematical passage
```
 \operatorname{Tr}(e^{-\beta H_\epsilon})
 =e^{-\beta\epsilon_0}
 +\sum_{m\geq1}(p-1)p^{m-1}e^{-\beta\epsilon_m},
 \label{a-limit-thermal-trace}
```

\sum_nUse in source · 15 passages

Search aliases: ∑_n sum

Link to this entry ·

Use in source · continuum.tex, line 9
Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of amplitudes embeds by assigning zero to every absent mode.
Exact mathematical passage
```
\inner{a}{b}=\sum_n\inner{a_n}{b_n}
```

\sum_{n=0}^\inftyUse in source · 3 passages

Search aliases: ∑_n=0^∞ sum infty

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Use in source · spectral.tex, line 900
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓ
Exact mathematical passage
```
F(z)=\sum_{n=0}^\infty a_nz^n
```

\sum_{n=Q+1}^\inftyUse in source · 1 passage

Search aliases: ∑_n=Q+1^∞ sum infty

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Use in source · spectral.tex, line 1029
is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane by the full tail. Since (Q+1+k)!≥(Q+1)!k!, ∑_n=Q+1^∞fraca^nn! ≤fraca^Q+1(Q+1)!∑_k=0^∞fraca^kk!, which proves [source reference].
Exact mathematical passage
```
 \sum_{n=Q+1}^\infty\frac{a^n}{n!}
 \leq\frac{a^{Q+1}}{(Q+1)!}\sum_{k=0}^\infty\frac{a^k}{k!},
```

\sum_{n>N}Use in source · 2 passages

Search aliases: ∑_n>N sum

Link to this entry ·

Use in source · extensions.tex, line 163
adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.
Exact mathematical passage
```
 \norm{c-S_Nc}_{H_\lambda}^2
 =\sum_{n>N}(1+\lambda_n^2)|c_n|^2\longrightarrow0.
```

\sum_{n\in S}Use in source · 16 passages

Search aliases: ∑_n∈ S sum in

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Use in source · finite_core.tex, line 44
f_a(x)=∑_n∈ Se_n(x)a_n.
Exact mathematical passage
```
f_a(x)=\sum_{n\in S}e_n(x)a_n.
```

\sum_{n\in\Z^3}Use in source · 2 passages

Search aliases: ∑_n∈Z^3 sum in Z

Link to this entry ·

Use in source · continuum.tex, line 5
Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of amplitudes embeds by assigning zero to every absent mode.
Exact mathematical passage
```
\KK=\ell^2(\Z^3;\C^4)
=\left\{a=(a_n):\sum_{n\in\Z^3}\norm{a_n}^2<\infty\right\}.
```

\sum_{n\notin S}Use in source · 1 passage

Search aliases: ∑_n∉ S sum notin

Link to this entry ·

Use in source · continuum.tex, line 12
For a finite set S⊂Z^3, define the truncation P_Sa by retaining a_n when n∈ S and assigning zero otherwise. Then norma-P_Sa^2=∑_n∉ Snorma_n^2. The sum on the right can be made arbitrarily small because the full nonnegative series converges. Thus finite packets are dense in KK. The statement concerns the norm of the omitted amplitudes and does not depend on a particular ordering of the integer lattice.
Exact mathematical passage
```
\norm{a-P_Sa}^2=\sum_{n\notin S}\norm{a_n}^2.
```

\sum_{n\equiv j\;({\rm mod}\;L)}Use in source · 1 passage

Search aliases: ∑_n≡ j (rm mod L) sum equiv rm

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Use in source · spectral.tex, line 1189
On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is needed; its spectral calculus defines V.
Exact mathematical passage
```
 Q_j^{(L)}=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}V^\ell
          =\sum_{n\equiv j\;({\rm mod}\;L)}|n\rangle\langle n|.
 \label{a-spectral-number-projectors}
```

\sum_{n\leq N}Use in source · 1 passage

Search aliases: ∑_n≤ N sum leq

Link to this entry ·

Use in source · extensions.tex, line 162
operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.
Exact mathematical passage
```
S_Nc=\sum_{n\leq N}c_ne_n
```

\sum_{n\geq1}Use in source · 1 passage

Search aliases: ∑_n≥1 sum geq

Link to this entry ·

Use in source · extensions.tex, line 298
[Strong convergence with no differentiable limiting orbit] For λ_n=n, take c_n=C/n with C=(∑_n≥1n^-2)^-1/2. This is a normalized ℓ^2 state outside D(H_λ). Every finite cutoff orbit is differentiable, and [source reference] gives uniform convergence to the full orbit. Yet the full orbit is nowhere differentiable in ℓ^2, by the preceding proposition. In fact the norm of the cutoff velocity is Csqrt N, which diverges.
Exact mathematical passage
```
C=(\sum_{n\geq1}n^{-2})^{-1/2}
```

\sum_{q=0}^\inftyUse in source · 2 passages

Search aliases: ∑_q=0^∞ sum infty

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Use in source · spectral.tex, line 909
, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follo…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
 \quad
 \mathfrak R_{L,j}\mathfrak R_{L,k}
   =\delta_{jk}\mathfrak R_{L,j},\quad
 \sum_j\mathfrak R_{L,j}=I.
 \label{a-spectral-analytic-lanes}
```

\sum_{q=0}^{\infty}Use in source · 1 passage

Search aliases: ∑_q=0^∞ sum infty

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Use in source · core.tex, line 447
The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the separated Taylor channels are coupled. Their sum satisfies the Bessel equation. This decomposition resolves series order and does not introduce addition…
Exact mathematical passage
```
 \mathcal B_{r,j}(y)=\sum_{q=0}^{\infty}
 \frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},
 \qquad j=0,\ldots,r-1.
```

\sum_{q=1}^{p-1}Use in source · 1 passage

Search aliases: ∑_q=1^p-1 sum

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Use in source · finite.tex, line 350
joint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level r, equal to the rank in [source reference], so they span that detail subspace. Hence D_r^(h)=∑_a=0^p^r-1-1∑_q=1^p-1 |ψ_r,a,q^(h)⟩⟨ψ_r,a,q^(h)|. The omitted character q=0 is the parent average. For p=2 there is one average and one difference per paren…
Exact mathematical passage
```
 D_r^{(h)}=\sum_{a=0}^{p^{r-1}-1}\sum_{q=1}^{p-1}
       |\psi_{r,a,q}^{(h)}\rangle\langle\psi_{r,a,q}^{(h)}|.
 \label{a-finite-wavelet-projector}
```

\sum_rUse in source · 4 passages

Search aliases: ∑_r sum

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Use in source · spectral.tex, line 25
Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=…
Exact mathematical passage
```
\sum_rJ_rJ_r^*=I_N
```

\sum_{r(a)=\beta}Use in source · 2 passages

Search aliases: ∑_r(a)=β sum beta

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Use in source · extensions.tex, line 65
If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β
Exact mathematical passage
```
G_\beta=\sum_{r(a)=\beta}F_a
```

\sum_{r,s=0}^{p-1}Use in source · 1 passage

Search aliases: ∑_r,s=0^p-1 sum

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Use in source · spectral.tex, line 29
section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt …
Exact mathematical passage
```
 X=\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\qquad
 (\mathcal A X\mathcal A^*)_{rs}=X_{rs}.
 \label{a-spectral-block-reconstruction}
```

\sum_{r=0}^{3}Use in source · 1 passage

Search aliases: ∑_r=0^3 sum

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Use in source · finite_core.tex, line 24
⟨ v,w⟩=∑_r=0^3overlinev_rw_r, v _2^2=∑_r=0^3|v_r|^2.
Exact mathematical passage
```
\langle v,w\rangle=\sum_{r=0}^{3}\overline{v_r}w_r,
\qquad
\|v\|_2^2=\sum_{r=0}^{3}|v_r|^2.
```

\sum_{r=0}^hUse in source · 6 passages

Search aliases: ∑_r=0^h sum

Link to this entry ·

Use in source · finite.tex, line 279
[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber rankD_0^(h) · =1, · rankD_r^(h) · =(p-1)p^r-1 (r≥1).
Exact mathematical passage
```
 \sum_{r=0}^hD_r^{(h)}&=I,
 &D_r^{(h)}D_s^{(h)}&=\delta_{rs}D_r^{(h)},\nonumber\\
 \operatorname{rank}D_0^{(h)}&=1,
 &\operatorname{rank}D_r^{(h)}&=(p-1)p^{r-1}\quad(r\geq1).
 \label{a-finite-detail-ranks}
```

\sum_{r=0}^{k}Use in source · 1 passage

Search aliases: ∑_r=0^k sum

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Use in source · finite_core.tex, line 1013
For Hermitian matrices the two exponentials in the integrand are unitary. This does not rely on the special Clifford algebra: the series gives (e^-iuA)^*=e^iuA, and the Cauchy product of the two absolutely convergent series gives e^iuAe^-iuA=I. Indeed, the coefficient of A^k for k>0 is (iu)^k∑_r=0^kbinomkr(-1)^r/k!=0, and the constant coefficient is I. A unitary matrix on a nonzero space has norm one. Submultiplicativity therefore bounds the integrand norm by A-B . For t≥0, integration gives t A…
Exact mathematical passage
```
(iu)^k\sum_{r=0}^{k}\binom{k}{r}(-1)^r/k!=0
```

\sum_{r=0}^{k-1}Use in source · 1 passage

Search aliases: ∑_r=0^k-1 sum

Link to this entry ·

Use in source · finite_core.tex, line 669
X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).
Exact mathematical passage
```
X^k-Y^k=\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r
\quad(k\ge1).
```

\sum_{r=0}^mUse in source · 1 passage

Search aliases: ∑_r=0^m sum

Link to this entry ·

Use in source · finite.tex, line 310
The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an additional bound on the detail weights of the state.
Exact mathematical passage
```
E_m^{(h)}=\sum_{r=0}^mD_r^{(h)}
```

\sum_{r=1}^bUse in source · 1 passage

Search aliases: ∑_r=1^b sum

Link to this entry ·

Use in source · spectral.tex, line 409
its fixed ancestral coefficient, on all children below the lower vertex at a common level. The lower chain has zero sum over its own child contrasts, so their inner product is zero. The same zero-sum calculation makes every angular vector orthogonal to the uniform radial level vector. At a fixed depth b, the radial vector contributes one dimension. The angular vectors present there contribute ∑_r=1^b(p-1)p^r-1=p^b-1 dimensions. They are orthogonal and therefore form a basis of that p^b-dimension…
Exact mathematical passage
```
\sum_{r=1}^b(p-1)p^{r-1}=p^b-1
```

\sum_{r=1}^hUse in source · 2 passages

Search aliases: ∑_r=1^h sum

Link to this entry ·

Use in source · finite.tex, line 649
For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born …
Exact mathematical passage
```
 U_{p,h}(t;x,y)
 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
       -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
 \right].
 \label{a-finite-shell-kernel}
```

\sum_{r=1}^{m}Use in source · 1 passage

Search aliases: ∑_r=1^m sum

Link to this entry ·

Use in source · refinement.tex, line 33
Suppose that a finite expansion is available, f(h)=F+∑_r=1^mh^rv_r+R(h). Linearity and the proposition give the exact identity T_mf(h)-F=T_mR(h). Only a finite sum has been used. No convergence of an infinite formal series is needed. The size of the right-hand side remains an analytic question.
Exact mathematical passage
```
 f(h)=F+\sum_{r=1}^{m}h^rv_r+R(h).
```

\sum_{r=m+1}^hUse in source · 1 passage

Search aliases: ∑_r=m+1^h sum

Link to this entry ·

Use in source · finite.tex, line 312
The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an additional bound on the detail weights of the state.
Exact mathematical passage
```
 \norm{(I-E_m^{(h)})f}^2=
 \sum_{r=m+1}^h\norm{D_r^{(h)}f}^2.
```

\sum_sUse in source · 4 passages

Search aliases: ∑_s sum

Link to this entry ·

Use in source · spectral.tex, line 37
, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are the block unitary identities. Every coordinate is retained.
Exact mathematical passage
```
 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
 \label{a-spectral-paraunitarity}
```

\sum_{\substack{b\in C_\beta\\a\in C_\alpha}}Use in source · 1 passage

Search aliases: ∑_substackb∈ C_β a∈ C_α sum substack in beta a in alpha

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Use in source · extensions.tex, line 501
Summing the differentiated corners reconstructs δ_BX. For a coarse partition with Q_α=∑_a∈ C_αP_a, Q_βδ_B(X)Q_α =∑_substackb∈ C_β a∈ C_αP_bδ_B(X)P_a. This is exact aggregation of the already differentiated operator. It does not claim that discarding all internal matrix structure produces a new pointwise algebra with a nonzero exact derivation. The retained corners are the structure that makes the finite differential calculus work.
Exact mathematical passage
```
 Q_\beta\delta_B(X)Q_\alpha
 =\sum_{\substack{b\in C_\beta\\a\in C_\alpha}}P_b\delta_B(X)P_a.
```

\sum_{\substack{r(b)=\beta\\r(a)=\alpha}}Use in source · 1 passage

Search aliases: ∑_substackr(b)=β r(a)=α sum substack beta r alpha

Link to this entry ·

Use in source · extensions.tex, line 66
If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.
Exact mathematical passage
```
 p^{\rm c}_\beta=\sum_{r(a)=\beta}p_a,\qquad
 J^{\rm c}_{\beta\alpha}
 =\sum_{\substack{r(b)=\beta\\r(a)=\alpha}}J^F_{ba}.
```

\sum_{\substack{u\notin A,\ v\in A\\u\sim v}}Use in source · 1 passage

Search aliases: ∑_substacku∉ A, v∈ A u∼ v sum substack notin in u sim

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Use in source · limit.tex, line 870
in gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A, v∈ A u∼ vF_u→ v^n -∑_substackv∈ A, u∉ A u∼ vF_v→ u^n. Taking A to be all vertices gives conservation of total probability. The coarse identity uses a subset of vertices and does not presume amplit…
Exact mathematical passage
```
 \sum_{v\in A}(\rho_v^{n+1}-\rho_v^n)
 =\sum_{\substack{u\notin A,\ v\in A\\u\sim v}}F_{u\to v}^n
 -\sum_{\substack{v\in A,\ u\notin A\\u\sim v}}F_{v\to u}^n.
 \label{a-limit-walk-coarse-balance}
```

\sum_{\substack{v\in A,\ u\notin A\\u\sim v}}Use in source · 1 passage

Search aliases: ∑_substackv∈ A, u∉ A u∼ v sum substack in notin u sim

Link to this entry ·

Use in source · limit.tex, line 870
in gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A, v∈ A u∼ vF_u→ v^n -∑_substackv∈ A, u∉ A u∼ vF_v→ u^n. Taking A to be all vertices gives conservation of total probability. The coarse identity uses a subset of vertices and does not presume amplit…
Exact mathematical passage
```
 \sum_{v\in A}(\rho_v^{n+1}-\rho_v^n)
 =\sum_{\substack{u\notin A,\ v\in A\\u\sim v}}F_{u\to v}^n
 -\sum_{\substack{v\in A,\ u\notin A\\u\sim v}}F_{v\to u}^n.
 \label{a-limit-walk-coarse-balance}
```

\sum_{v\in A}Use in source · 1 passage

Search aliases: ∑_v∈ A sum in

Link to this entry ·

Use in source · limit.tex, line 870
in gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A, v∈ A u∼ vF_u→ v^n -∑_substackv∈ A, u∉ A u∼ vF_v→ u^n. Taking A to be all vertices gives conservation of total probability. The coarse identity uses a subset of vertices and does not presume amplit…
Exact mathematical passage
```
 \sum_{v\in A}(\rho_v^{n+1}-\rho_v^n)
 =\sum_{\substack{u\notin A,\ v\in A\\u\sim v}}F_{u\to v}^n
 -\sum_{\substack{v\in A,\ u\notin A\\u\sim v}}F_{v\to u}^n.
 \label{a-limit-walk-coarse-balance}
```

\sum_{v\in\{0,1\}^3}Use in source · 1 passage

Search aliases: ∑_v∈{0,1}^3 sum in

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Use in source · refinement.tex, line 136
Interpolation is possible even when it depends on a path. Choose one reference path from the origin to each vertex and let r(v) be its transport. Then (I_ℓψ)(v)=r(v)ψ, innerΨΦ_rm cube =tfrac18∑_v∈{0,1}^3innerΨ(v)Φ(v). Every r(v) is unitary, so this map is a Hilbert isometry from one spinor to the eight-child cell. Its gauge covariance follows from telescoping along the selected paths. Neither property requires flatness. What curvature obstructs is independence from the reference-path choice.
Exact mathematical passage
```
 (I_\ell\psi)(v)=r(v)\psi,
 \qquad
 \inner{\Psi}{\Phi}_{\rm cube}
 =\tfrac18\sum_{v\in\{0,1\}^3}\inner{\Psi(v)}{\Phi(v)}.
```

\sum_{w\in\{0,\ldots,p-1\}^{s}}Use in source · 1 passage

Search aliases: ∑_w∈{0,…,p-1}^s sum in ldots

Link to this entry ·

Use in source · spectral.tex, line 376
There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the same branch-c…
Exact mathematical passage
```
 |u,c;s\rangle
 =p^{-s/2}\sum_{d=0}^{p-1}c_d
       \sum_{w\in\{0,\ldots,p-1\}^{s}}|u d w\rangle.
 \label{a-spectral-angular-chain}
```

\sum_xUse in source · 5 passages

Search aliases: ∑_x sum

Link to this entry ·

Use in source · core.tex, line 49
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
\inner{\phi}{\psi}_V=\sum_x\phi_x^*\psi_x
```

\sum_{x,a,i}Use in source · 1 passage

Search aliases: ∑_x,a,i sum

Link to this entry ·

Use in source · constraints.tex, line 223
On the full grid canonical phase space the bracket is {F,G}_N=N^3∑_x,a,i (frac∂ F∂ A_a^i(x) frac∂ G∂ E^a_i(x) -frac∂ F∂ E^a_i(x) frac∂ G∂ A_a^i(x)). The factor N^3 matches the normalized trace in the action. The resolved identities below are evaluated at band-limited data in this phase space. Replacing this bracket by a bracket that first projects every gradient back to T_K is a different operation and can reintroduce the leakage of equation~[source reference].
Exact mathematical passage
```
 \{F,G\}_N=N^3\sum_{x,a,i}
 \left(\frac{\partial F}{\partial A_a^i(x)}
       \frac{\partial G}{\partial E^a_i(x)}
      -\frac{\partial F}{\partial E^a_i(x)}
       \frac{\partial G}{\partial A_a^i(x)}\right).
```

\sum_{x,\mu}Use in source · 1 passage

Search aliases: ∑_x,μ sum mu

Link to this entry ·

Use in source · core.tex, line 50
The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so Δ_U is self-adjoint and nonpositive. This finite identity uses the …
Exact mathematical passage
```
\label{g:laplace-form}
 -\inner{\psi}{\Delta_U\psi}_V
 =\sum_{x,\mu}\left\lVert(D^+_{\mu,U}\psi)_x\right\rVert^2.
```

\sum_{x,\mu<\nu}Use in source · 1 passage

Search aliases: ∑_x,μ<ν sum mu nu

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Use in source · core.tex, line 160
For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the analogous term is 1-ReW. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interac…
Exact mathematical passage
```
\label{g:wilson-action}
 S_W[U]=\frac1{g_{\mathrm{YM}}^2}
 \sum_{x,\mu<\nu}\left(1-\tfrac12\operatorname{Re}\operatorname{Tr}W_{x,\mu\nu}\right),
 \qquad g_{\mathrm{YM}}\in\R\setminus\{0\}.
```

\sum_{x\in A}Use in source · 1 passage

Search aliases: ∑_x∈ A sum in

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Use in source · core.tex, line 737
For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.
Exact mathematical passage
```
 E_A^{\rm con}=\sum_{x\in A}h^3
 \left(|C_x|^2+\sum_i|C_{i,x}|^2\right)
```

\sum_{x\in S}Use in source · 1 passage

Search aliases: ∑_x∈ S sum in

Link to this entry ·

Use in source · refinement.tex, line 217
Hilbert estimates with explicit constants. Finite-dimensional continuity is not enough to identify an operator norm with physical ℓ^2 energy. The earlier continuous-linear packaging retains the ambient norm on a function space. The Hilbert scalar layer instead realizes a finite four-spinor field as HH_S=ℓ^2(S×{0,1,2,3};C), normψ^2=∑_x∈ Snormψ(x)_C^4^2. The formal realization is codeEuclideanSpace; its norm is identified explicitly with the real part of the previously defined state inner product.…
Exact mathematical passage
```
 \HH_S=\ell^2(S\times\{0,1,2,3\};\C),\qquad
 \norm{\psi}^2=\sum_{x\in S}\norm{\psi(x)}_{\C^4}^2.
```

\sum_{y\equiv x\, (\mathrm{mod}\,p^r)}Use in source · 1 passage

Search aliases: ∑_y≡ x (mod p^r) sum equiv mathrm

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Use in source · finite.tex, line 256
For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.
Exact mathematical passage
```
 (E_r^{(h)}f)(x)=p^{-(h-r)}
       \sum_{y\equiv x\, (\mathrm{mod}\,p^r)}f(y).
 \label{a-finite-cylinder-average}
```

\sum_{|v|=r}Use in source · 1 passage

Search aliases: ∑_|v|=r sum

Link to this entry ·

Use in source · spectral.tex, line 334
Define normalized level states |r⟩=p^-r/2∑_|v|=r|v⟩, r=0,…,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributions instead give p p^-r/2γ_r-1/sqrt p =γ_r-1p^-(r-1)/2.
Exact mathematical passage
```
 |r\rangle=p^{-r/2}\sum_{|v|=r}|v\rangle,
 \qquad r=0,\ldots,h.
```

\sum_\lambdaUse in source · 2 passages

Search aliases: ∑_λ sum lambda

Link to this entry ·

Use in source · spectral.tex, line 222
Let D=∑_λλ E_λ be its spectral resolution. Then Σ(z)=∑_λfracB E_λ B^*z-λ. Every residue B E_λ B^* is positive semidefinite. Its rank records which directions of the hidden eigenspace reach the visible space. A vanishing residue removes the corresponding pole of Σ, even though z-D is still singular there.
Exact mathematical passage
```
D=\sum_\lambda\lambda E_\lambda
```

\sum_\muUse in source · 5 passages

Search aliases: ∑_μ sum mu

Link to this entry ·

Use in source · spectral.tex, line 52
For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This is a change of coordinates in a given channel. It does not supply a new dissipative dynamics.
Exact mathematical passage
```
\Phi(\rho)=\sum_\mu K_\mu\rho K_\mu^*
```

\sum_\XiUse in source · 1 passage

Search aliases: ∑_Ξ sum Xi

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Use in source · constraints.tex, line 387
The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor 1/2. No higher-antifield terms are needed for this off-shell closed representation algebra.
Exact mathematical passage
```
 S_{\mathrm{BV}}=S_0+
     \int\sum_\Xi\langle\Xi^\ddagger,s\Xi\rangle.
 \label{c:constraints:bv-action}
```

\sum_\ellUse in source · 4 passages

Search aliases: ∑_ℓ sum ell

Link to this entry ·

Use in source · finite.tex, line 458
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
p^{-1}\sum_\ell\omega_p^{(e-d)\ell}=\delta_{de}
```

\sum_{\ell=0}^{L-1}Use in source · 5 passages

Search aliases: ∑_ℓ=0^L-1 sum ell

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Use in source · spectral.tex, line 902
Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The id…
Exact mathematical passage
```
 (\mathfrak R_{L,j}F)(z)
 =\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}F(\omega_L^\ell z),
 \quad 0\leq j<L.
 \label{a-spectral-analytic-filter}
```

\sum_{\ell=0}^{p-1}Use in source · 1 passage

Search aliases: ∑_ℓ=0^p-1 sum ell

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Use in source · finite.tex, line 453
Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto Fourier characters of a child register.
Exact mathematical passage
```
 R_d=\frac1p\sum_{\ell=0}^{p-1}\omega_p^{-d\ell}M_{p,h}^{\ell}.
 \label{a-finite-root-selector}
```

\sum_{\ell=1}^{L-1}Use in source · 1 passage

Search aliases: ∑_ℓ=1^L-1 sum ell

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Use in source · spectral.tex, line 1223
[Approach to uniform residue probability] For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L), giving the second.
Exact mathematical passage
```
 \left|p_{L,j}(\mu)-\frac1L\right|
 \leq\frac1L\sum_{\ell=1}^{L-1}
       e^{-\mu(1-\cos(2\pi\ell/L))}
 \leq\frac{L-1}{L}
       e^{-\mu(1-\cos(2\pi/L))}.
 \label{a-spectral-packet-mixing-bound}
```

\mpUse in source · 1 passage

Search aliases: ∓ mp

Link to this entry ·

Use in source · constraints.tex, line 412
Reality is built into e and ω. For real Lorentz bivectors, the internal dual obeys ⋆^2=-id and the projectors P_±=tfrac12(id∓ i⋆) satisfy overlineP_+X=P_-X. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical starting point. It does not prove a missing finite reality-preserving step for the earlier isolated self-dual coordinates.
Exact mathematical passage
```
 P_\pm=\tfrac12(\mathrm{id}\mp i\star)
```

\setminusUse in source · 3 passages

Search aliases: ∖ setminus

Link to this entry ·

Use in source · spectral.tex, line 182
det(z-H) · =det(z-D)det S(z). For E∈R∖σ(D), an eigenvector of H is equivalent to a nonzero solution of S(E)u=0, with hidden component v=(E-D)^-1B^*u. Put C=z-D. Direct block multiplication gives z-H= I · -BC^-1
Exact mathematical passage
```
E\in\R\setminus\sigma(D)
```

\circUse in source · 2 passages

Search aliases: ∘ circ

Link to this entry ·

Use in source · spectral.tex, line 818
[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,
Exact mathematical passage
```
\mathcal K_pf=f\circ R_p
```

\inftyUse in source · 67 passages

Search aliases: ∞ infty

Link to this entry ·

Use in source · limit.tex, line 164
Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.
Exact mathematical passage
```
 \HH_p=\bigoplus_{m=0}^{\infty}\operatorname{Ran}\mathsf F_m,
 \qquad \norm{f}_2^2=\sum_{m=0}^{\infty}\norm{\mathsf F_mf}_2^2.
 \label{a-limit-parseval}
```

\infty^2Use in source · 2 passages

Search aliases: ∞^2 infty

Link to this entry ·

Use in source · finite_core.tex, line 743
show that both directions are continuous. The first follows because each coordinate square is bounded by the sum; the second because each of four squares is at most v _∞^2.
Exact mathematical passage
```
\|v\|_\infty^2
```

\capUse in source · 2 passages

Search aliases: ∩ cap

Link to this entry ·

Use in source · limit.tex, line 32
At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \inner{\mathcal U_{p,m}e_a}{\mathcal U_{p,m}e_b}
 =p^m\mu_p(B_{m,a}\cap B_{m,b})=\delta_{ab}.
```

\int_{\mathcal L_D}Use in source · 2 passages

Search aliases: ∫_ L_D int mathcal

Link to this entry ·

Use in source · constraints.tex, line 816
regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution, Δ_C e^iS_C/ℏ =- N^-1∫_ L_DΔ_D e^iS_F/ℏ=0. Th…
Exact mathematical passage
```
 e^{iS_C/\hbar}=\mathcal N^{-1}\int_{\mathcal L_D}e^{iS_F/\hbar}.
```

\int_{\mathbb T^3}Use in source · 4 passages

Search aliases: ∫_ T^3 int mathbb

Link to this entry ·

Use in source · finite_core.tex, line 12
equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is
Exact mathematical passage
```
\int_{\mathbb T^3}1\,dx=1
```

\int_{-2}^2Use in source · 1 passage

Search aliases: ∫_-2^2 int

Link to this entry ·

Use in source · spectral.tex, line 868
The mean logarithmic expansion of this map is also exact: ∫_-2^2log|R_p'(x)| dν(x)=log p. Indeed for x=2cosθ, R_p'(x)=psin(pθ)/sinθ. Logarithms at its finitely many zeros are integrable, since they have the local form log|θ-θ_0| plus a bounded term. Hence the integral is log p plus the difference of the integrals of log|sin(pθ)| and logsinθ over [0,π]. The substitution u=pθ and π-periodicity of log|sin u| make these two integrals equal. This calculation does not assert the entropy of a different…
Exact mathematical passage
```
 \int_{-2}^2\log|R_p'(x)|\,d\nu(x)=\log p.
 \label{a-spectral-log-expansion}
```

\int_{-L}^LUse in source · 1 passage

Search aliases: ∫_-L^L int

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Use in source · core.tex, line 317
N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b) · =frac2c_0^2|∫_-L^L∂_bΦ(x,b) dx| =frac4GM_0Lc_0^2bsqrtL^2+b^2. Here α is the magnitude of the deflection toward the source. The limit L→∞ gives 4GM…
Exact mathematical passage
```
 \Delta t_{\rm Shapiro}
 &=-\frac2{c_0^3}\int_{-L}^L\Phi(x,b)\,dx
 =\frac{4GM_0}{c_0^3}\operatorname{asinh}\left(\frac Lb\right),\\
 \alpha(L,b)
 &=\frac2{c_0^2}\left|\int_{-L}^L\partial_b\Phi(x,b)\,dx\right|
 =\frac{4GM_0L}{c_0^2b\sqrt{L^2+b^2}}.
```

\int_0^1Use in source · 1 passage

Search aliases: ∫_0^1 int

Link to this entry ·

Use in source · constraints.tex, line 676
hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let …
Exact mathematical passage
```
 \overline\nabla F(z_0,z_1)=\int_0^1\nabla F((1-s)z_0+sz_1)\,\mathrm ds.
 \label{c:constraints:discrete-gradient}
```

\int_0^{2\pi}Use in source · 4 passages

Search aliases: ∫_0^2π int pi

Link to this entry ·

Use in source · spectral.tex, line 808
On [-2,2] define the probability measure dν(x)=fracdxπsqrt4-x^2. It is the distribution of 2cosθ for a uniform angle on the circle, or on [0,π]. It is also the limiting normalized spectral measure of the free cycles: for a continuous test function f, frac1N∑_k=0^N-1f(2cos(2π k/N)) ⟶frac12π∫_0^2πf(2cosθ) dθ. This is the Riemann-sum limit of a continuous periodic function.
Exact mathematical passage
```
 \frac1N\sum_{k=0}^{N-1}f(2\cos(2\pi k/N))
 \longrightarrow\frac1{2\pi}\int_0^{2\pi}f(2\cos\theta)\,d\theta.
```

\int_0^{2\pi p}Use in source · 1 passage

Search aliases: ∫_0^2π p int pi

Link to this entry ·

Use in source · spectral.tex, line 833
0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ · =frac12π p∫_0^2π pf(2cos u) du =∫ f(x) dν(x). The last equality uses periodicity. Applying this equality to |f|^2 proves the isometry, first on bounded functions and then by truncation on all of L^2(ν). Pulling back to [0,π] makes the ϕ_n the normalized cosine functions, and elementary cosine orthogonality proves th…
Exact mathematical passage
```
 \int f(R_p(x))\,d\nu(x)
 &=\frac1{2\pi}\int_0^{2\pi}f(2\cos(p\theta))\,d\theta\\
 &=\frac1{2\pi p}\int_0^{2\pi p}f(2\cos u)\,du
 =\int f(x)\,d\nu(x).
```

\int_0^hUse in source · 1 passage

Search aliases: ∫_0^h int

Link to this entry ·

Use in source · refinement.tex, line 327
Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,h). The coupling is absorbed into A_i. These links are automatically unitary, and the interval integral is oriented when h<0.
Exact mathematical passage
```
 \theta_i(p,h)=\int_0^h A_i(p+t e_i)\,dt,
 \qquad u_i(p,h)=e^{i\theta_i(p,h)}.
 \label{g:integrated-links}
```

\int_0^{h/2}Use in source · 1 passage

Search aliases: ∫_0^h/2 int

Link to this entry ·

Use in source · refinement.tex, line 336
Exact composition precedes every estimate. Continuity of A_i along its own coordinate lines permits interval-integral additivity and a change of variable: θ_i(p,h) · =∫_0^h/2A_i(p+t e_i) dt +∫_h/2^hA_i(p+t e_i) dt · =θ_i(p,h/2)+θ_i(p+(h/2)e_i,h/2). Exponentiation gives the ordered identity u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h). Thus the reconstructed and prescribed links agree exactly and all three straight-edge errors vanish. This is an additivity theorem for exact line integrals. Independentl…
Exact mathematical passage
```
 \theta_i(p,h)
 &=\int_0^{h/2}A_i(p+t e_i)\,dt
   +\int_{h/2}^{h}A_i(p+t e_i)\,dt\\
 &=\theta_i(p,h/2)+\theta_i(p+(h/2)e_i,h/2).
```

\int_0^rUse in source · 1 passage

Search aliases: ∫_0^r int

Link to this entry ·

Use in source · core.tex, line 232
Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.
Exact mathematical passage
```
 (r^2\Phi')'=4\pi G r^2\rho,
 \qquad M_{\rm enc}(r)=4\pi\int_0^rs^2\rho(s)\,ds,
 \qquad \Phi'(r)=\frac{GM_{\rm enc}(r)}{r^2}.
```

\int_0^{R_0}Use in source · 1 passage

Search aliases: ∫_0^R_0 int

Link to this entry ·

Use in source · core.tex, line 246
Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.
Exact mathematical passage
```
 S[u]=\int_0^{R_0}\left[\frac{(u')^2}{2G}+4\pi r\rho u\right]dr.
```

\int_0^tUse in source · 5 passages

Search aliases: ∫_0^t int

Link to this entry ·

Use in source · limit.tex, line 765
and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP e^-isH_0P. The two propagators and the exterior projections are contractions. Integrating the norm bound over an interval of length |t| gives the sec…
Exact mathematical passage
```
 Qe^{-itH_\lambda}P
 =-i\lambda\int_0^t
 Qe^{-i(t-s)H_\lambda}B e^{-isH_0}P\,ds.
```

\int_0^uUse in source · 1 passage

Search aliases: ∫_0^u int

Link to this entry ·

Use in source · finite_core.tex, line 550
Proof. First prove the scalar estimate without a sign restriction on h. The fundamental theorem of calculus gives |sin u|≤|u|, since sin u=∫_0^ucos v dv and |cos v|≤1. Therefore
Exact mathematical passage
```
\sin u=\int_0^u\cos v\,dv
```

\int_0^zUse in source · 1 passage

Search aliases: ∫_0^z int

Link to this entry ·

Use in source · finite_core.tex, line 558
For z≥0, integrate this bound in sin z-z=∫_0^z(cos u-1) du to obtain |sin z-z|≤ z^3/6. Oddness of the residual proves |sin z-z|≤|z|^3/6 for negative z. For h≠0, substitute z=hp_j(n) and divide by |h|:
Exact mathematical passage
```
\sin z-z=\int_0^z(\cos u-1)\,du
```

\int_0^{\theta}Use in source · 1 passage

Search aliases: ∫_0^θ int theta

Link to this entry ·

Use in source · core.tex, line 409
If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The difference of two such functions has both derivatives zero, so is constant.
Exact mathematical passage
```
 \lambda(\tau,\theta)=\lambda_0
 +\int_{\tau_0}^{\tau}H_G(s,0)\,ds
 +\int_0^{\theta}M_G(\tau,\eta)\,d\eta.
```

\int_0^\inftyUse in source · 2 passages

Search aliases: ∫_0^∞ int infty

Link to this entry ·

Use in source · spectral.tex, line 293
K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D …
Exact mathematical passage
```
 \Sigma(z)=-i\int_0^\infty e^{iz\tau}K(\tau)\,d\tau.
 \label{a-spectral-memory-transform}
```

\int_AUse in source · 1 passage

Search aliases: ∫_A int

Link to this entry ·

Use in source · limit.tex, line 111
For completeness, the union of these ranges is dense. Cylinders form a countable clopen base for the topology on Z_p and generate its Borel sigma-algebra. If f∈ L^2 is orthogonal to every cylinder indicator, the finite complex measure ν(A)=∫_A f dμ_p vanishes on cylinders and their finite unions. The class of sets on which ν is zero contains the whole space, is closed under complements, and is closed under countable disjoint unions. The elementary monotone-class extension from the cylinder algeb…
Exact mathematical passage
```
\nu(A)=\int_A f\,d\mu_p
```

\int_{B_{m,a}}Use in source · 1 passage

Search aliases: ∫_B_m,a int

Link to this entry ·

Use in source · limit.tex, line 39
. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.
Exact mathematical passage
```
 \int_{B_{m,a}}|\mathcal U_{p,m}c|^2\,d\mu_p=|c_a|^2.
 \label{a-limit-born-normalization}
```

\int_{C_a}Use in source · 1 passage

Search aliases: ∫_C_a int

Link to this entry ·

Use in source · refinement.tex, line 477
Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2 =∑_a∫_C_av^-1normψ_a^2 =∑_anormψ_a^2. Equivalently, physical cell values carry the weighted norm v∑_anormq_a^2. The adjoint averaging map yields the …
Exact mathematical passage
```
 \norm{J\psi}_{L^2}^2
 =\sum_a\int_{C_a}v^{-1}\norm{\psi_a}^2
 =\sum_a\norm{\psi_a}^2.
```

\int_eUse in source · 2 passages

Search aliases: ∫_e int

Link to this entry ·

Use in source · core.tex, line 165
We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt along the two horizontal endpoints and ϕ_xx along the two vertical endpoints, with the boundary orientation from dtwedge dx. This is the rectangular funda…
Exact mathematical passage
```
U_e=\exp(+\int_e\omega)
```

\int_{h/2}^{h}Use in source · 1 passage

Search aliases: ∫_h/2^h int

Link to this entry ·

Use in source · refinement.tex, line 336
Exact composition precedes every estimate. Continuity of A_i along its own coordinate lines permits interval-integral additivity and a change of variable: θ_i(p,h) · =∫_0^h/2A_i(p+t e_i) dt +∫_h/2^hA_i(p+t e_i) dt · =θ_i(p,h/2)+θ_i(p+(h/2)e_i,h/2). Exponentiation gives the ordered identity u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h). Thus the reconstructed and prescribed links agree exactly and all three straight-edge errors vanish. This is an additivity theorem for exact line integrals. Independentl…
Exact mathematical passage
```
 \theta_i(p,h)
 &=\int_0^{h/2}A_i(p+t e_i)\,dt
   +\int_{h/2}^{h}A_i(p+t e_i)\,dt\\
 &=\theta_i(p,h/2)+\theta_i(p+(h/2)e_i,h/2).
```

\int_IUse in source · 1 passage

Search aliases: ∫_I int

Link to this entry ·

Use in source · core.tex, line 343
Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber
Exact mathematical passage
```
 S_G=\frac12\int_I\int_0^{2\pi}
 \left[P_\tau^2-\gamma P_\theta^2
 +e^{2P}(Q_\tau^2-\gamma Q_\theta^2)\right]d\theta\,d\tau.
 \label{c:core:gowdy-action}
```

\int_{\R}Use in source · 2 passages

Search aliases: ∫_R int R

Link to this entry ·

Use in source · spectral.tex, line 1105
Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say that each lane acts on every vector.
Exact mathematical passage
```
 \Dom(f_j(H_\infty))
 =\left\{\psi:\int_{\R}|f_j(\lambda)|^2
                       \,d\langle\psi,E(\lambda)\psi\rangle<\infty\right\}.
 \label{a-spectral-lane-domain}
```

\int_\squareUse in source · 1 passage

Search aliases: ∫_square int square

Link to this entry ·

Use in source · core.tex, line 168
We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt along the two horizontal endpoints and ϕ_xx along the two vertical endpoints, with the boundary orientation from dtwedge dx. This is the rectangular funda…
Exact mathematical passage
```
 W_\square
 =\exp\left(K\oint_{\partial\square}(\phi_xdt+\phi_tdx)\right)
 =\exp\left(\frac K2\int_\square e^{2\phi}R\,dt\,dx\right).
 \label{c:core:conformal-holonomy}
```

\int_{\square}Use in source · 1 passage

Search aliases: ∫_square int square

Link to this entry ·

Use in source · core.tex, line 179
Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Conse…
Exact mathematical passage
```
\theta=-q\int_{\square}F_{\mu\nu}\,dS
```

\int_x^{x+h}Use in source · 1 passage

Search aliases: ∫_x^x+h int

Link to this entry ·

Use in source · core.tex, line 293
The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let A be a real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription.
Exact mathematical passage
```
\label{g:integrated-link}
 U_h(x)=\exp\left(-i\int_x^{x+h}A(s)\,ds\right),\qquad
 (D_h^A\psi)(x)
 =\frac{U_h(x)\psi(x+h)-U_h(x-h)^{-1}\psi(x-h)}{2h}.
```

\int_x^{x+s}Use in source · 1 passage

Search aliases: ∫_x^x+s int

Link to this entry ·

Use in source · core.tex, line 311
For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bounds for A and ψ give the required uniform bound on f_x'''.
Exact mathematical passage
```
f_x(s)=\exp(-i\int_x^{x+s}A(r)\,dr)\psi(x+s)
```

\int_{x_j}^{x_j+h}Use in source · 1 passage

Search aliases: ∫_x_j^x_j+h int

Link to this entry ·

Use in source · core.tex, line 298
real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription.
Exact mathematical passage
```
h^{-1}\int_{x_j}^{x_j+h}A(s)\,ds
```

\int_\SigmaUse in source · 7 passages

Search aliases: ∫_Σ int Sigma

Link to this entry ·

Use in source · constraints.tex, line 440
Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A · -tfrac i2σ_I(barψγ^Iδψ -δbarψγ^Iψ)]. The minus sign in the spinor term is fixed by the time--space orientation and by moving dt past the spatial three-form. It must be retained when comparing the spacetime action and its canonical chart.
Exact mathematical passage
```
 \begin{aligned}
 \Theta_\Sigma=\int_\Sigma\bigl[&B_{IJ}\wedge\delta\Omega^{IJ}+\Pi\wedge\delta A\\
 &-\tfrac i2\sigma_I(\bar\psi\gamma^I\delta\psi
                         -\delta\bar\psi\gamma^I\psi)\bigr].
 \end{aligned}
 \label{c:constraints:boundary-potential}
```

\int_{\tau_0}^{\tau}Use in source · 1 passage

Search aliases: ∫_τ_0^τ int tau tau

Link to this entry ·

Use in source · core.tex, line 409
If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The difference of two such functions has both derivatives zero, so is constant.
Exact mathematical passage
```
 \lambda(\tau,\theta)=\lambda_0
 +\int_{\tau_0}^{\tau}H_G(s,0)\,ds
 +\int_0^{\theta}M_G(\tau,\eta)\,d\eta.
```

\simeqUse in source · 3 passages

Search aliases: ≃ simeq

Link to this entry ·

Use in source · strong_fields.tex, line 170
The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this interval.
Exact mathematical passage
```
\mathcal K\simeq199.6646853
```

\congUse in source · 2 passages

Search aliases: ≅ cong

Link to this entry ·

Use in source · finite.tex, line 94
In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They do not assert that a generic state is a product state, or that a generic Hamiltonian is a sum of independent register Hamiltonians. Entanglement and in…
Exact mathematical passage
```
 \ell^2(U_{p,h})&\cong
 \C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^h\Z),
 &&p\text{ odd},\nonumber\\
 \ell^2(U_{2,h})&\cong
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^h\Z).
 \label{a-finite-unit-hilbert}
```

\neUse in source · 33 passages

Search aliases: ≠ ne

Link to this entry ·

Use in source · spectral.tex, line 201
eterminant formula. Inverting the factors gives the stated upper left block. Alternatively, in (z-H)(u,v)^ T=(f,0)^ T, the lower equation gives v=C^-1B^*u and the upper equation becomes S(z)u=f. The same elimination with f=0 proves the eigenvector statement. If u=0 at such an E, invertibility of E-D forces v=0, so a nonzero full eigenvector indeed has u≠0.
Exact mathematical passage
```
u\ne0
```

\neqUse in source · 6 passages

Search aliases: ≠ neq

Link to this entry ·

Use in source · finite.tex, line 691
0 · 0 has compressed Hamiltonian zero but detail probability sin^2(|γ|t) from the first basis state. It fails exact refinement compatibility whenever γ≠0.
Exact mathematical passage
```
\gamma\neq0
```

\leUse in source · 59 passages

Search aliases: ≤ le

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Use in source · continuum.tex, line 53
[Two tails and a finite core] Let V and W be linear isometries on a normed space. For any a and any approximation b, normVa-Wa≤ 2norma-b+normVb-Wb. Write the residual as V(a-b)+(Vb-Wb)+W(b-a). The triangle inequality bounds its norm by the sum of the three norms. Each outer term has norm norma-b because its operator is an isometry.
Exact mathematical passage
```
\norm{Va-Wa}\le 2\norm{a-b}+\norm{Vb-Wb}.
```

\geUse in source · 20 passages

Search aliases: ≥ ge

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Use in source · continuum.tex, line 72
[Compact-time strong convergence] For every a∈KK, every T≥0, and every ε>0, there exists δ>0 such that 0<h<δ, |t|≤ T Longrightarrow normU_h(t)a-U(t)a<ε. Choose a finite S such that norma-P_Sa<ε/4. The finite-packet theorem gives a sufficiently small positive δ for which the core residual is less than ε/2 throughout the time interval. The two-tail inequality then makes the full residual less than ε. The choice of S precedes the choice of h and works for every time in the same interval.
Exact mathematical passage
```
T\ge0
```

\subseteqUse in source · 3 passages

Search aliases: ⊆ subseteq

Link to this entry ·

Use in source · continuum.tex, line 154
The adjoint domain and self-adjointness. Modewise Hermiticity gives symmetry on the maximal domain. If a,b belong to that domain, the two series in innerwidehat H_mab =∑_ninnerH_na_nb_n =∑_ninnera_nH_nb_n =innerawidehat H_mb converge absolutely by Cauchy--Schwarz. This establishes the inclusion widehat H_m⊆widehat H_m^*, including the corresponding domain inclusion.
Exact mathematical passage
```
\widehat H_m\subseteq\widehat H_m^*
```

\bigoplusUse in source · 7 passages

Search aliases: ⊕ bigoplus

Link to this entry ·

Use in source · finite.tex, line 305
The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an additional bound on the detail weights of the state.
Exact mathematical passage
```
 \mathcal H_{p,h}=\bigoplus_{r=0}^h\Ran D_r^{(h)},\qquad
 \norm f^2=\sum_{r=0}^h\norm{D_r^{(h)}f}^2.
 \label{a-finite-parseval}
```

\oplusUse in source · 4 passages

Search aliases: ⊕ oplus

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Use in source · finite.tex, line 520
QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.
Exact mathematical passage
```
\Ran P\oplus\Ran Q
```

\bigoplus_{\mathbf r}Use in source · 1 passage

Search aliases: ⊕_ r bigoplus mathbf

Link to this entry ·

Use in source · limit.tex, line 599
[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber
Exact mathematical passage
```
\HH_{\mathcal P}^{\infty}=\bigoplus_{\mathbf r}\KK_{\mathbf r}
```

\bigoplus_aUse in source · 1 passage

Search aliases: ⊕_a bigoplus

Link to this entry ·

Use in source · extensions.tex, line 79
Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_…
Exact mathematical passage
```
V:\HH\to\KK=\bigoplus_a\KK_a
```

\bigoplus_{j=0}^{M-1}Use in source · 1 passage

Search aliases: ⊕_j=0^M-1 bigoplus

Link to this entry ·

Use in source · spectral.tex, line 112
e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.
Exact mathematical passage
```
\bigoplus_{j=0}^{M-1}\mathsf H_p(\theta_j)
```

\bigoplus_{m=0}^{\infty}Use in source · 1 passage

Search aliases: ⊕_m=0^∞ bigoplus infty

Link to this entry ·

Use in source · limit.tex, line 164
Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.
Exact mathematical passage
```
 \HH_p=\bigoplus_{m=0}^{\infty}\operatorname{Ran}\mathsf F_m,
 \qquad \norm{f}_2^2=\sum_{m=0}^{\infty}\norm{\mathsf F_mf}_2^2.
 \label{a-limit-parseval}
```

\bigoplus_{n\in S}Use in source · 1 passage

Search aliases: ⊕_n∈ S bigoplus in

Link to this entry ·

Use in source · finite_core.tex, line 50
K_S=⊕_n∈ S C^4, a _ K_S^2=∑_n∈ S a_n _2^2.
Exact mathematical passage
```
\mathcal K_S=\bigoplus_{n\in S}\mathbb C^4,
\qquad
\|a\|_{\mathcal K_S}^2=\sum_{n\in S}\|a_n\|_2^2.
```

\bigoplus_{r=0}^hUse in source · 1 passage

Search aliases: ⊕_r=0^h bigoplus

Link to this entry ·

Use in source · finite.tex, line 305
The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an additional bound on the detail weights of the state.
Exact mathematical passage
```
 \mathcal H_{p,h}=\bigoplus_{r=0}^h\Ran D_r^{(h)},\qquad
 \norm f^2=\sum_{r=0}^h\norm{D_r^{(h)}f}^2.
 \label{a-finite-parseval}
```

\bigoplus_vUse in source · 1 passage

Search aliases: ⊕_v bigoplus

Link to this entry ·

Use in source · limit.tex, line 838
for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary bec…
Exact mathematical passage
```
 C=\bigoplus_v(G_v\otimes R_v).
```

\bigotimesUse in source · 6 passages

Search aliases: ⊗ bigotimes

Link to this entry ·

Use in source · finite.tex, line 126
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 \mathcal U_{\mathrm{CRT}}:
 \ell^2\bigl((\Z/M\Z)^\times\bigr)
 \longrightarrow
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^{h_2}\Z)
 \otimes\bigotimes_{p\in\mathcal P_{\mathrm o}}
 \left(\C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^{h_p}\Z)\right).
 \label{a-finite-crt}
```

\bigotimes_pUse in source · 2 passages

Search aliases: ⊗_p bigotimes

Link to this entry ·

Use in source · limit.tex, line 543
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections an…
Exact mathematical passage
```
I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathcal U_{p,h_p}
```

\bigotimes_{p\in\mathcal P}Use in source · 2 passages

Search aliases: ⊗_p∈ P bigotimes in mathcal

Link to this entry ·

Use in source · limit.tex, line 536
Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections an…
Exact mathematical passage
```
 \HH_{\mathcal P}^{\infty}
 =\KK_{\mathrm{tor}}\otimes
 \bigotimes_{p\in\mathcal P}L^2(\Z_p,\mu_p).
 \label{a-limit-multiprime-space}
```

\bigotimes_{p\in\mathcal P_{\mathrm o}}Use in source · 2 passages

Search aliases: ⊗_p∈ P_ o bigotimes in mathcal mathrm

Link to this entry ·

Use in source · finite.tex, line 126
Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are pre…
Exact mathematical passage
```
 \mathcal U_{\mathrm{CRT}}:
 \ell^2\bigl((\Z/M\Z)^\times\bigr)
 \longrightarrow
 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^{h_2}\Z)
 \otimes\bigotimes_{p\in\mathcal P_{\mathrm o}}
 \left(\C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^{h_p}\Z)\right).
 \label{a-finite-crt}
```

\perpUse in source · 5 passages

Search aliases: ⊥ perp

Link to this entry ·

Use in source · limit.tex, line 353
Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.
Exact mathematical passage
```
\mathcal V_M^\perp
```

\bigcapUse in source · 1 passage

Search aliases: ⋂ bigcap

Link to this entry ·

Use in source · spectral.tex, line 1112
E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say that each lane acts on every vector.
Exact mathematical passage
```
 \sum_jf_j(H_\infty)\psi=e^{-itH_\infty}\psi,
 \qquad \psi\in\bigcap_j\Dom(f_j(H_\infty)).
 \label{a-spectral-unbounded-reconstruction}
```

\bigcap_jUse in source · 1 passage

Search aliases: ⋂_j bigcap

Link to this entry ·

Use in source · spectral.tex, line 1112
E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say that each lane acts on every vector.
Exact mathematical passage
```
 \sum_jf_j(H_\infty)\psi=e^{-itH_\infty}\psi,
 \qquad \psi\in\bigcap_j\Dom(f_j(H_\infty)).
 \label{a-spectral-unbounded-reconstruction}
```

\bigcupUse in source · 2 passages

Search aliases: ⋃ bigcup

Link to this entry ·

Use in source · limit.tex, line 118
sets on which ν is zero contains the whole space, is closed under complements, and is closed under countable disjoint unions. The elementary monotone-class extension from the cylinder algebra therefore gives ν(A)=0 for every Borel set. Applying this to the positive and negative level sets of the real and imaginary parts of f yields f=0 almost everywhere. The orthogonal complement of ⋃_m V_m is thus zero.
Exact mathematical passage
```
\bigcup_m\mathcal V_m
```

\bigcup_KUse in source · 1 passage

Search aliases: ⋃_K bigcup

Link to this entry ·

Use in source · constraints.tex, line 188
for products On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.
Exact mathematical passage
```
\bigcup_K\mathcal T_K
```

\bigcup_mUse in source · 1 passage

Search aliases: ⋃_m bigcup

Link to this entry ·

Use in source · limit.tex, line 118
sets on which ν is zero contains the whole space, is closed under complements, and is closed under countable disjoint unions. The elementary monotone-class extension from the cylinder algebra therefore gives ν(A)=0 for every Borel set. Applying this to the positive and negative level sets of the real and imaginary parts of f yields f=0 almost everywhere. The orthogonal complement of ⋃_m V_m is thus zero.
Exact mathematical passage
```
\bigcup_m\mathcal V_m
```

\starUse in source · 6 passages

Search aliases: ⋆ star

Link to this entry ·

Use in source · constraints.tex, line 334
Variation of φ gives the unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol_e. The s…
Exact mathematical passage
```
\alpha\wedge\star\beta=\langle\alpha,\beta\rangle\operatorname{vol}_e
```

\star^2Use in source · 1 passage

Search aliases: ⋆^2 star

Link to this entry ·

Use in source · constraints.tex, line 411
Reality is built into e and ω. For real Lorentz bivectors, the internal dual obeys ⋆^2=-id and the projectors P_±=tfrac12(id∓ i⋆) satisfy overlineP_+X=P_-X. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical starting point. It does not prove a missing finite reality-preserving step for the earlier isolated self-dual coordinates.
Exact mathematical passage
```
\star^2=-\mathrm{id}
```

\cdotsUse in source · 9 passages

Search aliases: ⋯ cdots

Link to this entry ·

Use in source · finite.tex, line 411
D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.
Exact mathematical passage
```
 V_{p,h+w-1}\cdots V_{p,h}|a\rangle
 =p^{-w/2}\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}|a,\mathbf d\rangle,
 \label{a-finite-nested-refinement}
```

\rangle_hUse in source · 5 passages

Search aliases: ⟩_h rangle

Link to this entry ·

Use in source · core.tex, line 682
dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic gradient is 2π_ij-δ_ijπ_rm tr. Lemma~[source reference] makes the potential gradient G(h). The lapse gradient follows by two spatial integrations by parts. Integrating the shift term once gives -⟨δ_in_j+δ_jn_i,π_ij⟩_h in the Lagrangian, fixing the sign in the first equation.
Exact mathematical passage
```
-\langle\delta_in_j+\delta_jn_i,\pi_{ij}\rangle_h
```

\rangle_{L^2}Use in source · 4 passages

Search aliases: ⟩_L^2 rangle

Link to this entry ·

Use in source · finite_core.tex, line 35
⟨ f,g⟩_L^2 =∫_ T^3⟨ f(x),g(x)⟩ dx
Exact mathematical passage
```
\langle f,g\rangle_{L^2}
=\int_{\mathbb T^3}\langle f(x),g(x)\rangle\,dx
```

\longmapstoUse in source · 1 passage

Search aliases: ⟼ longmapsto

Link to this entry ·

Use in source · finite.tex, line 32
[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ, (s,a) · ⟼(-1)^s5^apmod2^h+2, · (s,a) · ∈Z/2Z×Z/2^hZ are group isomorphisms onto U_p,h and U_2,h respectively.
Exact mathematical passage
```
 (t,a)&\longmapsto\zeta_{p,h}^{\,t}(1+p)^a
       \pmod{p^{h+1}},
 & (t,a)&\in\Z/(p-1)\Z\times\Z/p^h\Z,
 \label{a-finite-odd-coordinate}\\
 (s,a)&\longmapsto(-1)^s5^a\pmod{2^{h+2}},
 & (s,a)&\in\Z/2\Z\times\Z/2^h\Z
 \label{a-finite-dyadic-coordinate}
```

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monographs/physics/source/monographs/Physics_Gravity_Monograph.tex · 0 mathematical passages
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monographs/physics/source/monographs/adic/finite.tex · 374 mathematical passages
monographs/physics/source/monographs/adic/limit.tex · 402 mathematical passages
monographs/physics/source/monographs/adic/orientation.tex · 10 mathematical passages
monographs/physics/source/monographs/adic/primary_context.tex · 5 mathematical passages
monographs/physics/source/monographs/adic/references.tex · 0 mathematical passages
monographs/physics/source/monographs/adic/spectral.tex · 564 mathematical passages
monographs/physics/source/monographs/chapters/boundary.tex · 1 mathematical passage
monographs/physics/source/monographs/chapters/continuum.tex · 159 mathematical passages
monographs/physics/source/monographs/chapters/finite_core.tex · 429 mathematical passages
monographs/physics/source/monographs/chapters/orientation.tex · 20 mathematical passages
monographs/physics/source/monographs/foundations/conclusion.tex · 0 mathematical passages
monographs/physics/source/monographs/foundations/core.tex · 246 mathematical passages
monographs/physics/source/monographs/foundations/extensions.tex · 323 mathematical passages
monographs/physics/source/monographs/foundations/orientation.tex · 24 mathematical passages
monographs/physics/source/monographs/foundations/references.tex · 0 mathematical passages
monographs/physics/source/monographs/foundations/refinement.tex · 45 mathematical passages
monographs/physics/source/monographs/gauge/conclusion.tex · 1 mathematical passage
monographs/physics/source/monographs/gauge/core.tex · 264 mathematical passages
monographs/physics/source/monographs/gauge/orientation.tex · 22 mathematical passages
monographs/physics/source/monographs/gauge/references.tex · 0 mathematical passages
monographs/physics/source/monographs/gauge/refinement.tex · 261 mathematical passages
monographs/physics/source/monographs/gravity/conclusion.tex · 0 mathematical passages
monographs/physics/source/monographs/gravity/constraints.tex · 358 mathematical passages
monographs/physics/source/monographs/gravity/core.tex · 328 mathematical passages
monographs/physics/source/monographs/gravity/orientation.tex · 5 mathematical passages
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monographs/physics/source/monographs/gravity/strong_fields.tex · 334 mathematical passages
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