Notation by work
Large Signal Model
All notation · Glossary · Source coverage
957 symbol spellings in 2308 mathematical passages across 9 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 957 entries are available below.
*Introduction passage · 61 passages
Search aliases: *
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
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W_*
+Introduction passage · 246 passages
Search aliases: +
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta+\zeta^{-1}=6/5
=Introduction passage · 679 passages
Search aliases: =
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta=e^{i\theta}
AIntroduction passage · 107 passages
Search aliases: A
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
A,S
aIntroduction passage · 283 passages
Search aliases: a
Introduction passage · Read passage · FOUNDATIONS.md, line 3144
Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define
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P_t(\cdot\mid h,a)
BIntroduction passage · 195 passages
Search aliases: B
Introduction passage · Read passage · FOUNDATIONS.md, line 614
where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.
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B_0
B_0Introduction passage · 3 passages
Search aliases: B_0
Introduction passage · Read passage · FOUNDATIONS.md, line 614
where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.
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B_0
B_aIntroduction passage · 44 passages
Search aliases: B_a
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
Exact mathematical passage
F_a(u)=TB_a(u)T^{-1}
CIntroduction passage · 158 passages
Search aliases: C
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
C
\mathcal CIntroduction passage · 19 passages
Search aliases: C mathcal
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
\mathcal C=\im A
dIntroduction passage · 115 passages
Search aliases: d
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
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du/2
eIntroduction passage · 66 passages
Search aliases: e
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta=e^{i\theta}
e^{i\theta}Introduction passage · 1 passage
Search aliases: e^iθ theta
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta=e^{i\theta}
FIntroduction passage · 69 passages
Search aliases: F
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
Exact mathematical passage
F_a(u)=TB_a(u)T^{-1}
\mathcal FIntroduction passage · 11 passages
Search aliases: F mathcal
Introduction passage · Read passage · FOUNDATIONS.md, line 3040
Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put
Exact mathematical passage
\mathcal F_{t-1}
F_aIntroduction passage · 12 passages
Search aliases: F_a
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
Exact mathematical passage
F_a(u)=TB_a(u)T^{-1}
\mathcal F_{t-1}Introduction passage · 6 passages
Search aliases: F_t-1 mathcal
Introduction passage · Read passage · FOUNDATIONS.md, line 3040
Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put
Exact mathematical passage
\mathcal F_{t-1}
hIntroduction passage · 102 passages
Search aliases: h
Introduction passage · Read passage · FOUNDATIONS.md, line 3144
Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define
Exact mathematical passage
P_t(\cdot\mid h,a)
iIntroduction passage · 57 passages
Search aliases: i
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta=e^{i\theta}
\imIntroduction passage · 5 passages
Search aliases: im operatorname
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
\mathcal C=\im A
jIntroduction passage · 109 passages
Search aliases: j
Introduction passage · Read passage · FOUNDATIONS.md, line 3040
Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put
Exact mathematical passage
\phi_{t,j}
\midIntroduction passage · 32 passages
Search aliases: mid
Introduction passage · Read passage · FOUNDATIONS.md, line 3144
Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define
Exact mathematical passage
P_t(\cdot\mid h,a)
PIntroduction passage · 122 passages
Search aliases: P
Introduction passage · Read passage · FOUNDATIONS.md, line 3109
Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,
Exact mathematical passage
P_{t,j}
pIntroduction passage · 81 passages
Search aliases: p
Introduction passage · Read passage · FOUNDATIONS.md, line 3217
The subscript rm sp denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same T.
Exact mathematical passage
\rm sp
P_tIntroduction passage · 4 passages
Search aliases: P_t
Introduction passage · Read passage · FOUNDATIONS.md, line 3144
Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define
Exact mathematical passage
P_t(\cdot\mid h,a)
P_{t,j}Introduction passage · 2 passages
Search aliases: P_t,j
Introduction passage · Read passage · FOUNDATIONS.md, line 3109
Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,
Exact mathematical passage
P_{t,j}
QIntroduction passage · 55 passages
Search aliases: Q
Introduction passage · Read passage · FOUNDATIONS.md, line 3109
Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,
Exact mathematical passage
Q_{t,j}
Q_tIntroduction passage · 2 passages
Search aliases: Q_t
Introduction passage · Read passage · FOUNDATIONS.md, line 3144
Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define
Exact mathematical passage
Q_t(\cdot\mid h,a)
Q_{t,j}Introduction passage · 2 passages
Search aliases: Q_t,j
Introduction passage · Read passage · FOUNDATIONS.md, line 3109
Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,
Exact mathematical passage
Q_{t,j}
\rmIntroduction passage · 6 passages
Search aliases: rm
Introduction passage · Read passage · FOUNDATIONS.md, line 3217
The subscript rm sp denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same T.
Exact mathematical passage
\rm sp
SIntroduction passage · 49 passages
Search aliases: S
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
A,S
sIntroduction passage · 170 passages
Search aliases: s
Introduction passage · Read passage · FOUNDATIONS.md, line 3217
The subscript rm sp denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same T.
Exact mathematical passage
\rm sp
TIntroduction passage · 95 passages
Search aliases: T
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
Exact mathematical passage
F_a(u)=TB_a(u)T^{-1}
tIntroduction passage · 228 passages
Search aliases: t
Introduction passage · Read passage · FOUNDATIONS.md, line 3040
Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put
Exact mathematical passage
\mathcal F_{t-1}
T^{-1}Introduction passage · 13 passages
Search aliases: T^-1
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
Exact mathematical passage
F_a(u)=TB_a(u)T^{-1}
uIntroduction passage · 190 passages
Search aliases: u
Introduction passage · Read passage · FOUNDATIONS.md, line 3210
Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
Exact mathematical passage
F_a(u)=TB_a(u)T^{-1}
VIntroduction passage · 85 passages
Search aliases: V
Introduction passage · Read passage · FOUNDATIONS.md, line 614
where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.
Exact mathematical passage
V
WIntroduction passage · 66 passages
Search aliases: W
Introduction passage · Read passage · FOUNDATIONS.md, line 614
where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.
Exact mathematical passage
W
W_*Introduction passage · 11 passages
Search aliases: W_*
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
W_*
\mathbb ZIntroduction passage · 1 passage
Search aliases: Z mathbb
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
6/5\notin\mathbb Z
\cdotIntroduction passage · 14 passages
Search aliases: · cdot
Introduction passage · Read passage · FOUNDATIONS.md, line 3144
Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define
Exact mathematical passage
P_t(\cdot\mid h,a)
\gammaIntroduction passage · 42 passages
Search aliases: γ gamma
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
\gamma
\zetaIntroduction passage · 2 passages
Search aliases: ζ zeta
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta=e^{i\theta}
\zeta^{-1}Introduction passage · 1 passage
Search aliases: ζ^-1 zeta
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\zeta+\zeta^{-1}=6/5
\thetaIntroduction passage · 12 passages
Search aliases: θ theta
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
\theta
\lambdaIntroduction passage · 111 passages
Search aliases: λ lambda
Introduction passage · atlas.tex, line 373
bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introduced there.
Exact mathematical passage
\lambda
\piIntroduction passage · 33 passages
Search aliases: π pi
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
Exact mathematical passage
2\pi
\phiIntroduction passage · 18 passages
Search aliases: ϕ phi
Introduction passage · Read passage · FOUNDATIONS.md, line 3040
Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put
Exact mathematical passage
\phi_{t,j}
\phi_{t,j}Introduction passage · 3 passages
Search aliases: ϕ_t,j phi
Introduction passage · Read passage · FOUNDATIONS.md, line 3040
Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put
Exact mathematical passage
\phi_{t,j}
\daggerIntroduction passage · 8 passages
Search aliases: † dagger
Introduction passage · Read passage · FOUNDATIONS.md, line 614
where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.
Exact mathematical passage
\dagger
\notinIntroduction passage · 6 passages
Search aliases: ∉ notin
Introduction passage · Read passage · FOUNDATIONS.md, line 2058
Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite cl…
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6/5\notin\mathbb Z
\setminusIntroduction passage · 3 passages
Search aliases: ∖ setminus
Introduction passage · Read passage · FOUNDATIONS.md, line 614
where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.
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V\setminus W
\leqIntroduction passage · 187 passages
Search aliases: ≤ leq
Introduction passage · Read passage · FOUNDATIONS.md, line 3109
Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,
Exact mathematical passage
0\leq\phi_{t,j}\leq1
\mathbf1Use in source · 22 passages
Search aliases: 1 mathbf
Use in source · Read passage · FOUNDATIONS.md, line 276
C_π(a,i)= 1_i∈ B_a.
Exact mathematical passage
C_\pi(a,i)=\mathbf 1_{i\in B_a}.
\mathbf1^{\mathsf T}Use in source · 1 passage
Search aliases: 1^ T mathbf mathsf
Use in source · atlas.tex, line 272
A finite-generator witness. Let V have full row rank and normalized generator columns. Nonnegative lifts A_o with B_oV=VA_o and 1^ T∑_oA_o=1^ T certify the generated cone. The initial state must be a normalized nonnegative mixture of those columns. claimAll-horizon validity follows by induction from a finite exact witness. Statistical proximity to a generating process is a further obligation. boundaryA zero-probability branch does not define a normalized posterior by division. A finite positive …
Exact mathematical passage
\mathbf1^{\mathsf T}\sum_oA_o=\mathbf1^{\mathsf T}
\mathbf1^\topUse in source · 18 passages
Search aliases: 1^top mathbf top
Use in source · Read passage · FOUNDATIONS.md, line 1151
A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with
Exact mathematical passage
\mathbf1^\top p=1
\mathbf1_{i\in B_a}Use in source · 1 passage
Search aliases: 1_i∈ B_a mathbf in
Use in source · Read passage · FOUNDATIONS.md, line 276
C_π(a,i)= 1_i∈ B_a.
Exact mathematical passage
C_\pi(a,i)=\mathbf 1_{i\in B_a}.
\mathbf1_m^\topUse in source · 1 passage
Search aliases: 1_m^top mathbf top
Use in source · Read passage · FOUNDATIONS.md, line 1752
λ V=1_m^top.
Exact mathematical passage
\lambda V=\mathbf1_m^\top.
\mathbf1_{\{A_t=a\}}Use in source · 1 passage
Search aliases: 1_{A_t=a} mathbf
Use in source · Read passage · FOUNDATIONS.md, line 3085
μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).
Exact mathematical passage
\mu_t^a=\int\phi_t(y)P_t(dy\mid\mathcal H_{t-1},a),\qquad W_t^a=\frac{\mathbf1_{\{A_t=a\}}\phi_t(Y_t)}{\pi_t(a\mid\mathcal H_{t-1})}.
\mathcal AUse in source · 6 passages
Search aliases: A mathcal
Use in source · Read passage · FOUNDATIONS.md, line 51
Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.
Exact mathematical passage
\mathcal Ax=(w_aU_ax)_a,\qquad \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a, \qquad c=\sum_aw_a^2.
A^{\mathsf T}Use in source · 4 passages
Search aliases: A^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2850
A^ THA=H.
Exact mathematical passage
A^{\mathsf T}HA=H.
A^*Use in source · 10 passages
Search aliases: A^*
Use in source · Read passage · FOUNDATIONS.md, line 3326
Use the Frobenius norm on symmetric matrices and the induced product norm. Assume M_A^* is injective and M_A^*(H^*)=(0,1)=b. This fixes the scale of a one-dimensional invariant-form space.
Exact mathematical passage
\mathcal M_{A^*}
\mathcal A^*Use in source · 1 passage
Search aliases: A^* mathcal
Use in source · Read passage · FOUNDATIONS.md, line 57
Then S A=I and Ax ^2=c x ^2, as proved in [K1]. Here A S=c^-1 A A^* is the orthogonal projection onto the consistent coefficient subspace.
Exact mathematical passage
\mathcal A\mathcal S=c^{-1}\mathcal A\mathcal A^*
A_1Use in source · 1 passage
Search aliases: A_1
Use in source · Read passage · FOUNDATIONS.md, line 2455
Let A_1,…,A_n be a measurable partition of Y. Define the cell operators
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A_1,\ldots,A_n
a_1Use in source · 2 passages
Search aliases: a_1
Use in source · Read passage · FOUNDATIONS.md, line 1135
f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).
Exact mathematical passage
f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
A_aUse in source · 25 passages
Search aliases: A_a
Use in source · Read passage · FOUNDATIONS.md, line 27
A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.
Exact mathematical passage
A_a:X\longrightarrow C_a,\qquad S_a:C_a\longrightarrow X, \qquad S_aA_a=I_X.
A_{a,o}Use in source · 2 passages
Search aliases: A_a,o
Use in source · Read passage · FOUNDATIONS.md, line 1766
1^top∑_o A_a,o=1^top, λ B_wz_0=1^top A_wα_0.
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\mathbf1^\top\sum_o A_{a,o}=\mathbf1^\top, \qquad \lambda B_wz_0=\mathbf1^\top A_w\alpha_0.
A_a^\daggerUse in source · 1 passage
Search aliases: A_a^† dagger
Use in source · Read passage · FOUNDATIONS.md, line 986
For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis qualification matters.
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A_a^\dagger
A_bUse in source · 10 passages
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Use in source · Read passage · FOUNDATIONS.md, line 67
T_ba=A_bS_a: C_a⟶ C_b.
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T_{ba}=A_bS_a:\mathcal C_a\longrightarrow\mathcal C_b.
a_HUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 875
a_H=frac78-frac38 3^-H, Δ_H=frac38 3^-H, sup_t|z_t-widehat z_t|≤ frac3 3^-H1+3 3^-H.
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a_H=\frac78-\frac38\,3^{-H},\qquad \Delta_H=\frac38\,3^{-H},\qquad \boxed{\sup_t|z_t-\widehat z_t|\leq \frac{3\,3^{-H}}{1+3\,3^{-H}}.}
a_iUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 904
**Proposition 20. Packet-interaction truncation bound.** Put a_i= P_ix . Then
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a_i=\|P_ix\|
a_{i_j}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 906
F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.
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\boxed{ \|F(x)-F_{\mathcal I}(x)\| \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\notin\mathcal I_m} \beta_{m,\boldsymbol i}\prod_{j=1}^{m}a_{i_j}. }
A_jUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2457
B_a,j=∫_A_jB_a(y) μ(dy).
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B_{a,j}=\int_{A_j}B_a(y)\,\mu(dy).
a_jUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1352
Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),
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\Pr_\pi(w)=f(w)\prod_{j=1}^{|w|}\pi(a_j\mid h_{j-1}),
a_kUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1135
f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).
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f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
A_nUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2455
Let A_1,…,A_n be a measurable partition of Y. Define the cell operators
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A_1,\ldots,A_n
A_oUse in source · 3 passages
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Use in source · atlas.tex, line 272
A finite-generator witness. Let V have full row rank and normalized generator columns. Nonnegative lifts A_o with B_oV=VA_o and 1^ T∑_oA_o=1^ T certify the generated cone. The initial state must be a normalized nonnegative mixture of those columns. claimAll-horizon validity follows by induction from a finite exact witness. Statistical proximity to a generating process is a further obligation. boundaryA zero-probability branch does not define a normalized posterior by division. A finite positive …
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A_o
A_rUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 82
**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square
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F=S_rG_rA_r
A_sUse in source · 18 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1758
**Proposition 39. Finite positive-lift certificate.** This cone is invariant under B_s if and only if there is an entrywise nonnegative matrix A_s∈ R^m× m such that
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A_s\in\mathbb R^{m\times m}
A_s^{(j)}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1796
B_s^(j)V=VA_s^(j), A_s^(j)≥0.
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B_s^{(j)}V=VA_s^{(j)},\qquad A_s^{(j)}\geq0.
A_tUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3085
μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).
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\mu_t^a=\int\phi_t(y)P_t(dy\mid\mathcal H_{t-1},a),\qquad W_t^a=\frac{\mathbf1_{\{A_t=a\}}\phi_t(Y_t)}{\pi_t(a\mid\mathcal H_{t-1})}.
a_TUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2302
λ B_a_T(y_T)⋯ B_a_1(y_1)z_0
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\lambda B_{a_T}(y_T)\cdots B_{a_1}(y_1)z_0
A_wUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1766
1^top∑_o A_a,o=1^top, λ B_wz_0=1^top A_wα_0.
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\mathbf1^\top\sum_o A_{a,o}=\mathbf1^\top, \qquad \lambda B_wz_0=\mathbf1^\top A_w\alpha_0.
\mathcal BUse in source · 4 passages
Search aliases: B mathcal
Use in source · Read passage · FOUNDATIONS.md, line 282
B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.
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\mathcal B_{\ell_y}(p) =\frac{p\odot\ell_y}{\ell_y^\top p}.
bUse in source · 97 passages
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Use in source · Read passage · FOUNDATIONS.md, line 65
For a nonempty atlas family, define transport on realizable coefficients from atlas a to atlas b by
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b
B'_aUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2306
with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.
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B'_a(y)=B_a(y)/r(y)
B'_sUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2208
B'_s=TB_sT^-1, J'= JT^-1, λ'=λ T^-1.
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B'_s=TB_sT^{-1},\qquad \mathcal J'=\mathcal JT^{-1},\qquad \lambda'=\lambda T^{-1}.
b^*Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3352
For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has
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c^*=(I-A^*)^{-1}b^*
b^2Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 3075
Iterating this conditional inequality gives the same exponential bound for the martingale sum ∑_s≤ t(X_s,j-m_s,j). Optimizing the exponential Markov inequality gives the two-sided bound 2exp(-2b^2/t). At the displayed b_t,j this is α_j/[t(t+1)]. Sum over t, using ∑_t≥11/[t(t+1)]=1, and then over j. The resulting event holds at every time, so it also holds at a data-selected time. Scaling the variable proves the range version. square
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2\exp(-2b^2/t)
B^\PiUse in source · 1 passage
Search aliases: B^Π Pi
Use in source · Read passage · FOUNDATIONS.md, line 2600
The full-law comparison between B and B^Π permits any common measurable policy because both are continuous-output models on the same space. The exact equivalence to the finite bin model in Proposition 56 instead requires a bin-measurable policy. These are different conclusions with different observation information.
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B^\Pi
B_{\mathrm p,0}Use in source · 2 passages
Search aliases: B_ p,0 mathrm
Use in source · Read passage · FOUNDATIONS.md, line 2041
B_ p,0=z_0(1/2,1/2,0), B_ p,1=z_0(1/2,-1/2,0).
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B_{\mathrm p,0}=z_0(1/2,1/2,0),\qquad B_{\mathrm p,1}=z_0(1/2,-1/2,0).
B_{\mathrm p,1}Use in source · 2 passages
Search aliases: B_ p,1 mathrm
Use in source · Read passage · FOUNDATIONS.md, line 2041
B_ p,0=z_0(1/2,1/2,0), B_ p,1=z_0(1/2,-1/2,0).
Exact mathematical passage
B_{\mathrm p,0}=z_0(1/2,1/2,0),\qquad B_{\mathrm p,1}=z_0(1/2,-1/2,0).
B_{\mathrm r,0}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2035
B_ r,0=T, B_ r,1=0.
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B_{\mathrm r,0}=T,\qquad B_{\mathrm r,1}=0.
B_{\mathrm r,1}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2035
B_ r,0=T, B_ r,1=0.
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B_{\mathrm r,0}=T,\qquad B_{\mathrm r,1}=0.
b_0Use in source · 8 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1252
(G_s)_ij=f(h_jst_i), (b_0)_i=f(t_i).
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(G_s)_{ij}=f(h_jst_i),\qquad (b_0)_i=f(t_i).
B_{0,0}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1614
B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.
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\begin{aligned} B_{0,0}&=\begin{pmatrix}1&-1\\11/32&-1/4\end{pmatrix},& B_{0,1}&=\begin{pmatrix}0&1\\-3/32&3/4\end{pmatrix},\\[4pt] B_{1,0}&=\begin{pmatrix}5/48&5/6\\23/192&5/24\end{pmatrix},& B_{1,1}&=\begin{pmatrix}43/48&-5/6\\37/64&-5/8\end{pmatrix}. \end{aligned}
B_{0,1}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1614
B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.
Exact mathematical passage
\begin{aligned} B_{0,0}&=\begin{pmatrix}1&-1\\11/32&-1/4\end{pmatrix},& B_{0,1}&=\begin{pmatrix}0&1\\-3/32&3/4\end{pmatrix},\\[4pt] B_{1,0}&=\begin{pmatrix}5/48&5/6\\23/192&5/24\end{pmatrix},& B_{1,1}&=\begin{pmatrix}43/48&-5/6\\37/64&-5/8\end{pmatrix}. \end{aligned}
B_1Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2620
a(u)=frac1+9u^24, b(u)=frac3u2, c(u)=frac1-9u^24, [3pt] B_0(u)=a(u) · b(u) · 0 b(u) · a(u) · 0 0 · 0 · c(u), B_1(u)=B_0(u)T.
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\begin{gathered} a(u)=\frac{1+9u^2}{4},\qquad b(u)=\frac{3u}{2},\qquad c(u)=\frac{1-9u^2}{4},\\[3pt] B_0(u)=\begin{pmatrix}a(u)&b(u)&0\\b(u)&a(u)&0\\0&0&c(u)\end{pmatrix}, \qquad B_1(u)=B_0(u)T. \end{gathered}
B_{1,0}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1614
B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.
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\begin{aligned} B_{0,0}&=\begin{pmatrix}1&-1\\11/32&-1/4\end{pmatrix},& B_{0,1}&=\begin{pmatrix}0&1\\-3/32&3/4\end{pmatrix},\\[4pt] B_{1,0}&=\begin{pmatrix}5/48&5/6\\23/192&5/24\end{pmatrix},& B_{1,1}&=\begin{pmatrix}43/48&-5/6\\37/64&-5/8\end{pmatrix}. \end{aligned}
B_{1,1}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1614
B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.
Exact mathematical passage
\begin{aligned} B_{0,0}&=\begin{pmatrix}1&-1\\11/32&-1/4\end{pmatrix},& B_{0,1}&=\begin{pmatrix}0&1\\-3/32&3/4\end{pmatrix},\\[4pt] B_{1,0}&=\begin{pmatrix}5/48&5/6\\23/192&5/24\end{pmatrix},& B_{1,1}&=\begin{pmatrix}43/48&-5/6\\37/64&-5/8\end{pmatrix}. \end{aligned}
b_aUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2795
λ B_a(u)=(b_a,0(u),b_a(u)^ T), β_a(u)=b_a,0(u)+b_a(u)^ Tc.
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\lambda B_a(u)=(b_{a,0}(u),b_a(u)^{\mathsf T}),\qquad \beta_a(u)=b_{a,0}(u)+b_a(u)^{\mathsf T}c.
b_{a,0}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2795
λ B_a(u)=(b_a,0(u),b_a(u)^ T), β_a(u)=b_a,0(u)+b_a(u)^ Tc.
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\lambda B_a(u)=(b_{a,0}(u),b_a(u)^{\mathsf T}),\qquad \beta_a(u)=b_{a,0}(u)+b_a(u)^{\mathsf T}c.
B_{a,b}Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3790
B_a,b(u)=v(u) (1,au,bL_2(u)).
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\boxed{ B_{a,b}(u)=v(u)\,(1,au,bL_2(u)). }
B_{a,j}Use in source · 8 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2457
B_a,j=∫_A_jB_a(y) μ(dy).
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B_{a,j}=\int_{A_j}B_a(y)\,\mu(dy).
B_{a,o}Use in source · 9 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1188
Pr(omid z,a)=λ B_a,oz, z^+=fracB_a,ozλ B_a,oz.
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\Pr(o\mid z,a)=\lambda B_{a,o}z,\qquad z^+=\frac{B_{a,o}z}{\lambda B_{a,o}z}.
B_{a,t}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2563
sup_a,z∈ K∫ (B_a,t(y)-widetilde B_a,t(y))z _C μ(dy) ≤ϵ_t.
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\sup_{a,z\in K}\int\|(B_{a,t}(y)-\widetilde B_{a,t}(y))z\|_C\,\mu(dy) \leq\epsilon_t.
b_a^{\mathsf T}Use in source · 2 passages
Search aliases: b_a^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2822
For a nonzero z, write x=m(c+w) with w^ THw≤1. Its output mass is m(β_a+b_a^ Tw). Cauchy--Schwarz in the H inner product bounds this below by
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m(\beta_a+b_a^{\mathsf T}w)
B_a^{[n]}Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2913
B_a^[n](x)=∫_0^1D_n(x,v)B_a(v) dv =∑_k=0^n b_n,k(x)C_a,n,k,
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B_a^{[n]}(x)=\int_0^1D_n(x,v)B_a(v)\,dv =\sum_{k=0}^n b_{n,k}(x)C_{a,n,k},
B_a^\PiUse in source · 4 passages
Search aliases: B_a^Π Pi
Use in source · Read passage · FOUNDATIONS.md, line 2508
B_a^Π(y)=fracB_a,jμ(A_j) (y∈ A_j).
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B_a^\Pi(y)=\frac{B_{a,j}}{\mu(A_j)}\quad(y\in A_j).
B_{a_1}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2302
λ B_a_T(y_T)⋯ B_a_1(y_1)z_0
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\lambda B_{a_T}(y_T)\cdots B_{a_1}(y_1)z_0
B_{a_T}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2302
λ B_a_T(y_T)⋯ B_a_1(y_1)z_0
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\lambda B_{a_T}(y_T)\cdots B_{a_1}(y_1)z_0
B_bUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2838
F=B_b(u_*)B_a(u_*)^-1, λ F=λ.
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F=B_b(u_*)B_a(u_*)^{-1},\qquad \lambda F=\lambda.
\mathcal B_{\bar\ell}Use in source · 1 passage
Search aliases: B_barℓ mathcal bar ell
Use in source · Read passage · FOUNDATIONS.md, line 400
with one common c>0. For a fixed prior, put δ=r(p)-barℓ, bar b= B_barℓ(P) and Z_r=r(p)^top P. Then
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\bar b=\mathcal B_{\bar\ell}(P)
\mathcal B_{\bar\ell_y}Use in source · 1 passage
Search aliases: B_barℓ_y mathcal bar ell
Use in source · Read passage · FOUNDATIONS.md, line 289
C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).
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C_\pi K_u^\top=\bar K_u^\top C_\pi, \qquad C_\pi\mathcal B_{\ell_y}(p) =\mathcal B_{\bar\ell_y}(C_\pi p).
B_hUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1264
b(h)=(f(ht_i))_i=B_hb_0.
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b(h)=(f(ht_i))_i=B_hb_0.
B_mUse in source · 7 passages
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Use in source · Read passage · FOUNDATIONS.md, line 167
F(x)=b+∑_m=1^pB_m(x,…,x),
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F(x)=b+\sum_{m=1}^{p}B_m(x,\ldots,x),
b_nUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1053
∑_g∉ G_nπ(g)L_g(D)≤ b_n,
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\sum_{g\notin\mathcal G_n}\pi(g)L_g(D)\leq b_n,
b_{n,k}Use in source · 8 passages
Search aliases: b_n,k
Use in source · Read passage · FOUNDATIONS.md, line 2906
b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).
Exact mathematical passage
b_{n,k}(x)=\binom nk x^k(1-x)^{n-k},\qquad D_n(x,v)=(n+1)\sum_{k=0}^n b_{n,k}(x)b_{n,k}(v).
B_oUse in source · 4 passages
Search aliases: B_o
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
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5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
B_{o_1}Use in source · 1 passage
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Use in source · atlas.tex, line 267
y needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0. Summing over every branch repeatedly applies the mass identity and gives total probability one. A positive-probability posterior is B_oz/(λ B_oz).
Exact mathematical passage
\lambda B_{o_T}\cdots B_{o_1}z_0\geq0.
B_{o_T}Use in source · 1 passage
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Use in source · atlas.tex, line 267
y needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0. Summing over every branch repeatedly applies the mass identity and gives total probability one. A positive-probability posterior is B_oz/(λ B_oz).
Exact mathematical passage
\lambda B_{o_T}\cdots B_{o_1}z_0\geq0.
B_sUse in source · 54 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1180
Choose a full row rank matrix R with row space W_* and first row 1^top. There are unique matrices B_s with
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B_s
B_s'Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1663
z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.
Exact mathematical passage
z'=Tz,\qquad B_s'=TB_sT^{-1},\qquad \lambda'=\lambda T^{-1},\qquad C'=TC.
B_s^{(j)}Use in source · 2 passages
Search aliases: B_s^(j)
Use in source · Read passage · FOUNDATIONS.md, line 1792
Let U be the convex hull of finitely many **complete operator families** {B_s^(j)}_s. Every family must satisfy the exact per-action mass identities. The convex coefficient is shared across that family's symbols. A product of independent uncertainty sets can instead be handled by its joint vertices.
Exact mathematical passage
\{B_s^{(j)}\}_s
B_s^*Use in source · 2 passages
Search aliases: B_s^*
Use in source · Read passage · FOUNDATIONS.md, line 1844
B_s^*-widehat B_s ≤ϵ_s, widetilde B_s-widehat B_s ≤ϵ_s.
Exact mathematical passage
\|B_s^*-\widehat B_s\|\leq\epsilon_s, \qquad \|\widetilde B_s-\widehat B_s\|\leq\epsilon_s.
B_s^{-1}Use in source · 1 passage
Search aliases: B_s^-1
Use in source · Read passage · FOUNDATIONS.md, line 1733
C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.
Exact mathematical passage
C_{h+1}=C_h\cap\bigcap_s B_s^{-1}C_h, \qquad C_{h+1}\subseteq C_h.
B_{s_1}Use in source · 2 passages
Search aliases: B_s_1
Use in source · Read passage · FOUNDATIONS.md, line 1219
B_sv_h=v_hs, f(s_1⋯ s_k)=λ B_s_k⋯ B_s_1v_ϵ.
Exact mathematical passage
B_sv_h=v_{hs},\qquad f(s_1\cdots s_k)=\lambda B_{s_k}\cdots B_{s_1}v_\epsilon.
B_{s_1\cdots s_k}Use in source · 1 passage
Search aliases: B_s_1⋯ s_k cdots
Use in source · Read passage · FOUNDATIONS.md, line 1698
B_s_1⋯ s_k=B_s_k⋯ B_s_1, B_ϵ=I.
Exact mathematical passage
B_{s_1\cdots s_k}=B_{s_k}\cdots B_{s_1},\qquad B_\epsilon=I.
B_{s_k}Use in source · 2 passages
Search aliases: B_s_k
Use in source · Read passage · FOUNDATIONS.md, line 1219
B_sv_h=v_hs, f(s_1⋯ s_k)=λ B_s_k⋯ B_s_1v_ϵ.
Exact mathematical passage
B_sv_h=v_{hs},\qquad f(s_1\cdots s_k)=\lambda B_{s_k}\cdots B_{s_1}v_\epsilon.
B_{sv}Use in source · 1 passage
Search aliases: B_sv
Use in source · Read passage · FOUNDATIONS.md, line 1740
**Proof.** For a first symbol s followed by a word v, B_sv=B_vB_s. The inequalities for all words of length at most h+1 are therefore exactly the displayed recurrence. If z∈ C_∞, every test on B_sz is another test on z, so C_∞ is invariant. Conversely, invariance of a cone contained in C_0 makes every successive test nonnegative, hence places it in C_∞. Equality at one finite step establishes invariance immediately and therefore gives the intersection.
Exact mathematical passage
B_{sv}=B_vB_s
b_{t,j}Use in source · 4 passages
Search aliases: b_t,j
Use in source · Read passage · FOUNDATIONS.md, line 3059
|R_t,j-D_t,j|≤ b_t,j, b_t,j=sqrtfrac t2logfrac2t(t+1)α_j.
Exact mathematical passage
\boxed{ |R_{t,j}-D_{t,j}|\leq b_{t,j},\qquad b_{t,j}=\sqrt{\frac t2\log\frac{2t(t+1)}{\alpha_j}}. }
B_uUse in source · 6 passages
Search aliases: B_u
Use in source · Read passage · FOUNDATIONS.md, line 250
C_0=HR, RL_u=B_uR
Exact mathematical passage
C_0=HR,\qquad RL_u=B_uR
B_vUse in source · 1 passage
Search aliases: B_v
Use in source · Read passage · FOUNDATIONS.md, line 1740
**Proof.** For a first symbol s followed by a word v, B_sv=B_vB_s. The inequalities for all words of length at most h+1 are therefore exactly the displayed recurrence. If z∈ C_∞, every test on B_sz is another test on z, so C_∞ is invariant. Conversely, invariance of a cone contained in C_0 makes every successive test nonnegative, hence places it in C_∞. Equality at one finite step establishes invariance immediately and therefore gives the intersection.
Exact mathematical passage
B_{sv}=B_vB_s
B_wUse in source · 6 passages
Search aliases: B_w
Use in source · Read passage · FOUNDATIONS.md, line 1708
An initial state has λ z_0=1. The proposed probability of a word is λ B_wz_0. An invertible coordinate change must transform z_0, B_s, λ and the admissible cone together, as in Proposition 37.
Exact mathematical passage
\lambda B_wz_0
B_\epsilonUse in source · 1 passage
Search aliases: B_ϵ epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1698
B_s_1⋯ s_k=B_s_k⋯ B_s_1, B_ϵ=I.
Exact mathematical passage
B_{s_1\cdots s_k}=B_{s_k}\cdots B_{s_1},\qquad B_\epsilon=I.
\mathcal B_{\ell}Use in source · 1 passage
Search aliases: B_ℓ mathcal ell
Use in source · Read passage · FOUNDATIONS.md, line 402
C_π B_ℓ(p)-bar b = frac1Z_r(diagP-bar bP^top)δ.
Exact mathematical passage
\boxed{ C_\pi\mathcal B_{\ell}(p)-\bar b = \frac{1}{Z_r}\left(\operatorname{diag}P-\bar bP^\top\right)\delta. }
\mathcal B_{\ell_y}Use in source · 2 passages
Search aliases: B_ℓ_y mathcal ell
Use in source · Read passage · FOUNDATIONS.md, line 282
B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.
Exact mathematical passage
\mathcal B_{\ell_y}(p) =\frac{p\odot\ell_y}{\ell_y^\top p}.
\bar bUse in source · 2 passages
Search aliases: bar b bar
Use in source · Read passage · FOUNDATIONS.md, line 400
with one common c>0. For a fixed prior, put δ=r(p)-barℓ, bar b= B_barℓ(P) and Z_r=r(p)^top P. Then
Exact mathematical passage
\bar b=\mathcal B_{\bar\ell}(P)
\bar FUse in source · 6 passages
Search aliases: bar F bar
Use in source · Read passage · FOUNDATIONS.md, line 185
Let γ:X→ Z=γ(X) be a summary and let F_u:X→ X describe a deterministic transition for action u. The reduced description closes when there is bar F_u:Z→ Z such that
Exact mathematical passage
\bar F_u:Z\to Z
\bar F_uUse in source · 5 passages
Search aliases: bar F_u bar
Use in source · Read passage · FOUNDATIONS.md, line 185
Let γ:X→ Z=γ(X) be a summary and let F_u:X→ X describe a deterministic transition for action u. The reduced description closes when there is bar F_u:Z→ Z such that
Exact mathematical passage
\bar F_u:Z\to Z
\bar KUse in source · 8 passages
Search aliases: bar K bar
Use in source · Read passage · FOUNDATIONS.md, line 289
C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).
Exact mathematical passage
C_\pi K_u^\top=\bar K_u^\top C_\pi, \qquad C_\pi\mathcal B_{\ell_y}(p) =\mathcal B_{\bar\ell_y}(C_\pi p).
\bar K^{\mathsf T}Use in source · 2 passages
Search aliases: bar K^ T bar mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2482
∫ K(y)XK(y)^ T ν(dy) =bar KXbar K^ T +∫ (K(y)-bar K)X(K(y)-bar K)^ T ν(dy).
Exact mathematical passage
\boxed{ \int K(y)XK(y)^{\mathsf T}\,\nu(dy) =\bar KX\bar K^{\mathsf T} +\int (K(y)-\bar K)X(K(y)-\bar K)^{\mathsf T}\,\nu(dy). }
\bar K_uUse in source · 1 passage
Search aliases: bar K_u bar
Use in source · Read passage · FOUNDATIONS.md, line 306
**Proof.** At a stable partition, states in a common block have the same transition probability into every coarse block. Those probabilities define bar K_u and give the first commutation identity, by checking it on point-mass priors. Likelihood constancy on each block defines barℓ_y and gives the second identity by direct summation.
Exact mathematical passage
\bar K_u
\bar K_u^\topUse in source · 1 passage
Search aliases: bar K_u^top bar top
Use in source · Read passage · FOUNDATIONS.md, line 289
C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).
Exact mathematical passage
C_\pi K_u^\top=\bar K_u^\top C_\pi, \qquad C_\pi\mathcal B_{\ell_y}(p) =\mathcal B_{\bar\ell_y}(C_\pi p).
\bar\ellUse in source · 7 passages
Search aliases: barℓ bar ell
Use in source · Read passage · FOUNDATIONS.md, line 289
C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).
Exact mathematical passage
C_\pi K_u^\top=\bar K_u^\top C_\pi, \qquad C_\pi\mathcal B_{\ell_y}(p) =\mathcal B_{\bar\ell_y}(C_\pi p).
\bar\ell^\topUse in source · 1 passage
Search aliases: barℓ^top bar ell top
Use in source · Read passage · FOUNDATIONS.md, line 412
global factor gives commutation by summing the fine posterior. Conversely, compare priors supported on one state in each of two distinct coarse blocks. Equality of coarse posterior odds forces the ratio of those states' likelihoods to equal the ratio of the coarse likelihoods. Varying the representatives proves constancy within each block and the common factor across blocks. For the defect, use Z_r=barℓ^top P+P^topδ and subtract the two normalized vectors. square
Exact mathematical passage
Z_r=\bar\ell^\top P+P^\top\delta
\bar\ell_aUse in source · 1 passage
Search aliases: barℓ_a bar ell
Use in source · Read passage · FOUNDATIONS.md, line 396
ℓ_i=cbarℓ_a for every i∈ B_a
Exact mathematical passage
\ell_i=c\bar\ell_a\quad\text{for every }i\in B_a
\bar\ell_yUse in source · 2 passages
Search aliases: barℓ_y bar ell
Use in source · Read passage · FOUNDATIONS.md, line 289
C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).
Exact mathematical passage
C_\pi K_u^\top=\bar K_u^\top C_\pi, \qquad C_\pi\mathcal B_{\ell_y}(p) =\mathcal B_{\bar\ell_y}(C_\pi p).
\bottomruleUse in source · 1 passage
Search aliases: bottomrule
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
Exact mathematical passage
5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
\boundaryUse in source · 1 passage
Search aliases: boundary
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
Exact mathematical passage
5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
cUse in source · 56 passages
Search aliases: c
Use in source · Read passage · FOUNDATIONS.md, line 34
**Proposition 1. Exact realization and consistency.** The analysis map is injective, Π_a=A_aS_a is an idempotent with image C_a, and Π_ac=c exactly when c is a realizable coefficient vector. Moreover,
Exact mathematical passage
\Pi_ac=c
C'Use in source · 1 passage
Search aliases: C'
Use in source · Read passage · FOUNDATIONS.md, line 1663
z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.
Exact mathematical passage
z'=Tz,\qquad B_s'=TB_sT^{-1},\qquad \lambda'=\lambda T^{-1},\qquad C'=TC.
c^{\mathsf T}Use in source · 1 passage
Search aliases: c^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2418
**Proof.** A nonzero polynomial has strictly positive squared integral, so its coefficient vector c satisfies c^ TW_Dc>0. Thus W_D is invertible. Let T have rows t_i and S have columns h_j. Since H=TS is an invertible square core in an r-dimensional space, both T and S are invertible. Therefore G_a,jH^-1=TM_a,jT^-1. The relation M_a,j=∑_k(W_D)_jkC_a,k is exact integration of the polynomial expansion. Invert it and conjugate by T to obtain the displayed reconstruction. The error bound is the tria…
Exact mathematical passage
c^{\mathsf T}W_Dc>0
c^*Use in source · 3 passages
Search aliases: c^*
Use in source · Read passage · FOUNDATIONS.md, line 3352
For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has
Exact mathematical passage
c^*=(I-A^*)^{-1}b^*
c^{-1}Use in source · 1 passage
Search aliases: c^-1
Use in source · Read passage · FOUNDATIONS.md, line 57
Then S A=I and Ax ^2=c x ^2, as proved in [K1]. Here A S=c^-1 A A^* is the orthogonal projection onto the consistent coefficient subspace.
Exact mathematical passage
\mathcal A\mathcal S=c^{-1}\mathcal A\mathcal A^*
C_{\mathcal J}Use in source · 5 passages
Search aliases: C_ J mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2190
C_ J={z: J(z)succeq0}, K_ J={z∈ C_ J:λ z=1}.
Exact mathematical passage
C_{\mathcal J}=\{z:\mathcal J(z)\succeq0\}, \qquad K_{\mathcal J}=\{z\in C_{\mathcal J}:\lambda z=1\}.
C_{\mathrm L}Use in source · 8 passages
Search aliases: C_ L mathrm
Use in source · Read passage · FOUNDATIONS.md, line 2050
C_ L={(t,x,y):t≥sqrtx^2+y^2}
Exact mathematical passage
C_{\mathrm L}=\{(t,x,y):t\geq\sqrt{x^2+y^2}\}
C_0Use in source · 17 passages
Search aliases: C_0
Use in source · Read passage · FOUNDATIONS.md, line 241
Let the initial summary be C_0x. Regard its rows as linear observables. Starting from W_0=row(C_0), define
Exact mathematical passage
C_0x
c_1Use in source · 5 passages
Search aliases: c_1
Use in source · Read passage · FOUNDATIONS.md, line 3904
c_1=∫|u| dμ, c_2=∫|L_2(u)| dμ.
Exact mathematical passage
c_1=\int|u|\,d\mu, \qquad c_2=\int|L_2(u)|\,d\mu.
c_2Use in source · 6 passages
Search aliases: c_2
Use in source · Read passage · FOUNDATIONS.md, line 3904
c_1=∫|u| dμ, c_2=∫|L_2(u)| dμ.
Exact mathematical passage
c_1=\int|u|\,d\mu, \qquad c_2=\int|L_2(u)|\,d\mu.
c_3Use in source · 2 passages
Search aliases: c_3
Use in source · atlas.tex, line 78
subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A(x_1,x_2)=(x_1,x_2,x_1+x_2), S(c_1,c_2,c_3)=(c_1,c_2). Then Π(c_1,c_2,c_3)=(c_1,c_2,c_1+c_2). The shaded plane is the set of realizable coefficients. The red segment shows syn…
Exact mathematical passage
A(x_1,x_2)=(x_1,x_2,x_1+x_2),\qquad S(c_1,c_2,c_3)=(c_1,c_2).
C_aUse in source · 5 passages
Search aliases: C_a
Use in source · Read passage · FOUNDATIONS.md, line 25
Let X and C_a be finite-dimensional Hilbert spaces. A linear atlas consists of analysis and synthesis maps
Exact mathematical passage
C_a
\mathcal C_aUse in source · 15 passages
Search aliases: C_a mathcal
Use in source · Read passage · FOUNDATIONS.md, line 32
The coefficients representing actual signals form C_a=imA_a. This distinction matters for redundant representations, where C_a is a proper subspace of the ambient coefficient space.
Exact mathematical passage
\mathcal C_a=\operatorname{im}A_a
c_aUse in source · 1 passage
Search aliases: c_a
Use in source · Read passage · FOUNDATIONS.md, line 51
Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.
Exact mathematical passage
\mathcal Ax=(w_aU_ax)_a,\qquad \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a, \qquad c=\sum_aw_a^2.
C_{a,k}Use in source · 10 passages
Search aliases: C_a,k
Use in source · Read passage · FOUNDATIONS.md, line 2379
B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.
Exact mathematical passage
B_a(u)=\sum_{k=0}^D u^k C_{a,k}, \qquad M_{a,j}=\int_{-1}^1 u^j B_a(u)\,\frac{du}{2}.
C_{a,n,k}Use in source · 5 passages
Search aliases: C_a,n,k
Use in source · Read passage · FOUNDATIONS.md, line 2913
B_a^[n](x)=∫_0^1D_n(x,v)B_a(v) dv =∑_k=0^n b_n,k(x)C_a,n,k,
Exact mathematical passage
B_a^{[n]}(x)=\int_0^1D_n(x,v)B_a(v)\,dv =\sum_{k=0}^n b_{n,k}(x)C_{a,n,k},
\mathcal C_bUse in source · 1 passage
Search aliases: C_b mathcal
Use in source · Read passage · FOUNDATIONS.md, line 67
T_ba=A_bS_a: C_a⟶ C_b.
Exact mathematical passage
T_{ba}=A_bS_a:\mathcal C_a\longrightarrow\mathcal C_b.
C_{c,H}Use in source · 5 passages
Search aliases: C_c,H
Use in source · Read passage · FOUNDATIONS.md, line 2780
C_c,H={(m,x):m≥0, (x-mc)^ TH(x-mc)≤ m^2}.
Exact mathematical passage
C_{c,H}=\{(m,x):m\geq0,\ (x-mc)^{\mathsf T}H(x-mc)\leq m^2\}.
C_hUse in source · 5 passages
Search aliases: C_h
Use in source · Read passage · FOUNDATIONS.md, line 1724
C_h={z:λ B_wz≥0 for every |w|≤ h}, C_0={z:λ z≥0}.
Exact mathematical passage
C_h=\{z:\lambda B_wz\geq0\text{ for every }|w|\leq h\}, \qquad C_0=\{z:\lambda z\geq0\}.
c_hUse in source · 2 passages
Search aliases: c_h
Use in source · Read passage · FOUNDATIONS.md, line 1226
**Proof.** If ∑_hc_hv_h=0, then for every test t,
Exact mathematical passage
\sum_hc_hv_h=0
C_{h+1}Use in source · 2 passages
Search aliases: C_h+1
Use in source · Read passage · FOUNDATIONS.md, line 1733
C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.
Exact mathematical passage
C_{h+1}=C_h\cap\bigcap_s B_s^{-1}C_h, \qquad C_{h+1}\subseteq C_h.
c_tUse in source · 2 passages
Search aliases: c_t
Use in source · Read passage · FOUNDATIONS.md, line 491
Its coefficients c_t=Ux_t reconstruct as x_t=U^*c_t. The same dynamics in that atlas are ULU^*, not a separate physical law. Thus coordinate transport preserves the full dynamics.
Exact mathematical passage
c_t=Ux_t
C_\piUse in source · 3 passages
Search aliases: C_π pi
Use in source · Read passage · FOUNDATIONS.md, line 276
C_π(a,i)= 1_i∈ B_a.
Exact mathematical passage
C_\pi(a,i)=\mathbf 1_{i\in B_a}.
C_\inftyUse in source · 9 passages
Search aliases: C_∞ infty
Use in source · Read passage · FOUNDATIONS.md, line 1738
Their intersection C_∞ is the largest cone contained in C_0 and invariant under all B_s. If an exact equality C_h+1=C_h occurs, then C_h=C_∞ and the finite inequalities certify all future word probabilities. If the representation is observable, meaning that the rows λ B_w span E^*, then C_∞ is pointed and λ is strictly positive on its nonzero elements. If reachable vectors B_wz_0∈ C_∞ span E, the cone is also generating.
Exact mathematical passage
C_\infty
\operatorname{ch}Use in source · 1 passage
Search aliases: ch operatorname
Use in source · Read passage · FOUNDATIONS.md, line 129
V(v)=min(c(v),∑_u∈ch(v)V(u))
Exact mathematical passage
V(v)=\min\left(c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right)
\operatorname{conv}Use in source · 3 passages
Search aliases: conv operatorname
Use in source · Read passage · FOUNDATIONS.md, line 1756
Its columns generate C=V R_+^m and its normalized base is K=conv{v_1,…,v_m}. Finite generation makes the cone closed. Its normalized generators and full row rank make it pointed and generating, with a strictly positive mass functional.
Exact mathematical passage
K=\operatorname{conv}\{v_1,\ldots,v_m\}
\cosUse in source · 3 passages
Search aliases: cos
Use in source · Read passage · FOUNDATIONS.md, line 2093
α≤cos(π/n),
Exact mathematical passage
\alpha\leq\cos(\pi/n),
DUse in source · 64 passages
Search aliases: D
Use in source · Read passage · FOUNDATIONS.md, line 326
A=PLP, B=PLQ, C=QLP, D=QLQ.
Exact mathematical passage
A=PLP,\quad B=PLQ,\quad C=QLP,\quad D=QLQ.
\mathrm dUse in source · 1 passage
Search aliases: d mathrm
Use in source · Read passage · FOUNDATIONS.md, line 1332
P_ d=frac121 · -1 -1 · 1
Exact mathematical passage
P_{\mathrm d}=\frac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}
d^2Use in source · 2 passages
Search aliases: d^2
Use in source · Read passage · FOUNDATIONS.md, line 2320
For a real d^2× d^2 symmetric matrix F, define
Exact mathematical passage
d^2\times d^2
D^jUse in source · 1 passage
Search aliases: D^j
Use in source · atlas.tex, line 182
plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, retaining the last H history terms leaves a same-history tail at most Δ_H=tfrac38 3^-H.
Exact mathematical passage
y_{t+1}=Ay_t+\sum_{j=0}^{t-1}BD^jC\,y_{t-1-j}+BD^tw_0.
D^kUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 334
z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.
Exact mathematical passage
\boxed{ z_{t+1}=Az_t+ \sum_{k=0}^{t-1}BD^kCz_{t-1-k}+BD^tq_0. }
D^tUse in source · 6 passages
Search aliases: D^t
Use in source · Read passage · FOUNDATIONS.md, line 334
z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.
Exact mathematical passage
\boxed{ z_{t+1}=Az_t+ \sum_{k=0}^{t-1}BD^kCz_{t-1-k}+BD^tq_0. }
D^{t-1-s}Use in source · 1 passage
Search aliases: D^t-1-s
Use in source · Read passage · FOUNDATIONS.md, line 351
q_t=D^tq_0+∑_s=0^t-1D^t-1-sCz_s.
Exact mathematical passage
q_t=D^tq_0+\sum_{s=0}^{t-1}D^{t-1-s}Cz_s.
d_0Use in source · 1 passage
Search aliases: d_0
Use in source · Read passage · FOUNDATIONS.md, line 3232
(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.
Exact mathematical passage
(w_0+d_0)-\|w_{\rm sp}+d_{\rm sp}\|_2 \geq w_0-\|w_{\rm sp}\|_2-|d_0|-\|d_{\rm sp}\|_2.
D_2Use in source · 1 passage
Search aliases: D_2
Use in source · Read passage · FOUNDATIONS.md, line 709
D_2=Z_+Z_-^-1 =0 · 1 1/12 · 5/6.
Exact mathematical passage
\boxed{ D_2=Z_+Z_-^{-1} =\begin{pmatrix}0&1\\1/12&5/6\end{pmatrix}. }
d_aUse in source · 7 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3278
**Proposition 73. Interior repair with an explicit error budget.** Suppose a reference family F^* is valid on L_d, and the combined mass and spatial error of widehat F-F^* is at most d_a(u) on normalized inputs. Then the margin of widetilde F_a(u) is at least
Exact mathematical passage
d_a(u)
d_bUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3909
If |Δ a|≤ d_a and |Δ b|≤ d_b, the conditional observation laws on any common normalized state obey
Exact mathematical passage
|\Delta b|\leq d_b
d_{ba}Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 86
d_ba(x)=G_b(A_bx)-T_baG_a(A_ax).
Exact mathematical passage
d_{ba}(x)=G_b(A_bx)-T_{ba}G_a(A_ax).
D_hUse in source · 6 passages
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Use in source · Read passage · FOUNDATIONS.md, line 582
1. There is a linear map D_h:V_h→ V_h satisfying O_hL=D_hO_h. 2. ker O_h⊆ker(O_hL). 3. rankO_h+1=rankO_h.
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D_h:V_h\to V_h
D_nUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2906
b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).
Exact mathematical passage
b_{n,k}(x)=\binom nk x^k(1-x)^{n-k},\qquad D_n(x,v)=(n+1)\sum_{k=0}^n b_{n,k}(x)b_{n,k}(v).
d_{\rm sp}Use in source · 1 passage
Search aliases: d_rm sp rm
Use in source · Read passage · FOUNDATIONS.md, line 3232
(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.
Exact mathematical passage
(w_0+d_0)-\|w_{\rm sp}+d_{\rm sp}\|_2 \geq w_0-\|w_{\rm sp}\|_2-|d_0|-\|d_{\rm sp}\|_2.
D_tUse in source · 3 passages
Search aliases: D_t
Use in source · Read passage · FOUNDATIONS.md, line 1937
**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write
Exact mathematical passage
D_t=\sum_h\|u_h-\widetilde u_h\|_C
d_tUse in source · 6 passages
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Use in source · Read passage · FOUNDATIONS.md, line 830
An empty sum is zero. Let d_t be the exact right-hand side minus this truncated right-hand side evaluated on the **true** history.
Exact mathematical passage
d_t
D_{t+1}Use in source · 2 passages
Search aliases: D_t+1
Use in source · Read passage · FOUNDATIONS.md, line 1945
Sum the first term over observations and use Proposition 42. When λwidetilde u_h>0, normalize only to apply the definition of ϵ_t to the second term. Its sum is bounded by ϵ_tλwidetilde u_h. If the mass is zero, strict positivity implies widetilde u_h=0. Summing histories yields D_t+1≤ D_t+ϵ_t. Averaging over common policy probabilities gives the same inequality for randomized actions. Finally each word-probability difference is bounded by its state difference because |λ x|≤ x _C. Total variatio…
Exact mathematical passage
D_{t+1}\leq D_t+\epsilon_t
D_{t,j}Use in source · 4 passages
Search aliases: D_t,j
Use in source · Read passage · FOUNDATIONS.md, line 3052
R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).
Exact mathematical passage
R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
\detUse in source · 10 passages
Search aliases: det
Use in source · Read passage · FOUNDATIONS.md, line 707
Since det Z_-=1/4, these windows span the declared two-dimensional state. Exact arithmetic gives
Exact mathematical passage
\det Z_-=1/4
\diagUse in source · 3 passages
Search aliases: diag operatorname
Use in source · atlas.tex, line 164
plate6Compression can erase a return pathA two-step effect can survive when both compressed steps vanish.P7 Let P^2=P and Q=I-P. Inserting I=P+Q gives the exact defect PLMP-(PLP)(PMP)=PLQMP. diagram06_return_path Take P=diag(1,0) and L=M=0 · 1
Exact mathematical passage
P=\diag(1,0)
\operatorname{diag}Use in source · 9 passages
Search aliases: diag operatorname
Use in source · Read passage · FOUNDATIONS.md, line 402
C_π B_ℓ(p)-bar b = frac1Z_r(diagP-bar bP^top)δ.
Exact mathematical passage
\boxed{ C_\pi\mathcal B_{\ell}(p)-\bar b = \frac{1}{Z_r}\left(\operatorname{diag}P-\bar bP^\top\right)\delta. }
\dimUse in source · 2 passages
Search aliases: dim
Use in source · Read passage · FOUNDATIONS.md, line 248
**Proposition 8. Minimal linear predictive completion.** The sequence stabilizes after at most d-dim W_0 strict enlargements. Its limit W_* is the smallest row space containing W_0 and invariant under right multiplication by every L_u. If R is any full-row-rank matrix whose rows form a basis of W_*, then
Exact mathematical passage
d-\dim W_0
\operatorname{do}Use in source · 1 passage
Search aliases: do operatorname
Use in source · Read passage · FOUNDATIONS.md, line 1135
f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).
Exact mathematical passage
f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
\operatorname{dom}Use in source · 1 passage
Search aliases: dom operatorname
Use in source · Read passage · FOUNDATIONS.md, line 430
dom(g∘ f) ={x∈domf:f(x)∈domg}.
Exact mathematical passage
\operatorname{dom}(g\circ f) =\{x\in\operatorname{dom}f:f(x)\in\operatorname{dom}g\}.
\downarrowUse in source · 1 passage
Search aliases: downarrow
Use in source · Read passage · FOUNDATIONS.md, line 1065
Since the active sets exhaust the grammar and the evidence sum is finite, Z_nuparrow Z and τ_ndownarrow0. This is posterior approximation *inside the declared grammar*. It does not establish that the real signal-generating mechanism belongs to it.
Exact mathematical passage
\tau_n\downarrow0
EUse in source · 41 passages
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Use in source · Read passage · FOUNDATIONS.md, line 104
Within one orthogonal atlas, let E_i be pairwise orthogonal coordinate selectors with ∑_iE_i=I. Pull them back by a unitary map:
Exact mathematical passage
E_i
\mathbb EUse in source · 21 passages
Search aliases: E mathbb
Use in source · Read passage · FOUNDATIONS.md, line 367
**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then
Exact mathematical passage
\mathbb E\|Y\|^2<\infty
E^*Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1738
Their intersection C_∞ is the largest cone contained in C_0 and invariant under all B_s. If an exact equality C_h+1=C_h occurs, then C_h=C_∞ and the finite inequalities certify all future word probabilities. If the representation is observable, meaning that the rows λ B_w span E^*, then C_∞ is pointed and λ is strictly positive on its nonzero elements. If reachable vectors B_wz_0∈ C_∞ span E, the cone is also generating.
Exact mathematical passage
E^*
e^{7/10}Use in source · 1 passage
Search aliases: e^7/10
Use in source · Read passage · FOUNDATIONS.md, line 3862
**Proof.** The positive exponential series gives e^7/10>1+7/10+(7/10)^2/2+(7/10)^3/6+(7/10)^4/24>2. Hence log 2<7/10, so log V+2log(1/δ)≤(7/10)(k+2h). Divide the scalar regression bound by sqrt V. For a nonnegative rational x, the smallest integer j with (j/10^12)^2≥ x supplies a rational upper square-root enclosure. Integer square root and an exact final comparison compute it. Outward rounding only enlarges the interval. Intersecting with a true prior constraint preserves coverage. Apply the tw…
Exact mathematical passage
e^{7/10}>1+7/10+(7/10)^2/2+(7/10)^3/6+(7/10)^4/24>2
e^{h^2\sigma^2/2}Use in source · 1 passage
Search aliases: e^h^2σ^2/2 sigma
Use in source · Read passage · FOUNDATIONS.md, line 3660
Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2
Exact mathematical passage
Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t, \qquad \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}] \leq e^{h^2\sigma^2/2}
e^{h\xi_t}Use in source · 1 passage
Search aliases: e^hξ_t xi
Use in source · Read passage · FOUNDATIONS.md, line 3660
Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2
Exact mathematical passage
Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t, \qquad \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}] \leq e^{h^2\sigma^2/2}
E^\topUse in source · 1 passage
Search aliases: E^top top
Use in source · Read passage · FOUNDATIONS.md, line 657
**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. square
Exact mathematical passage
\|\widetilde Z_-^\top v\|\geq\|Z_-^\top v\|-\|E^\top v\|\geq s-\|E\|
e^{\theta(X-m)}Use in source · 1 passage
Search aliases: e^θ(X-m) theta
Use in source · Read passage · FOUNDATIONS.md, line 3071
E[e^θ(X-m)mid F]≤ e^θ^2/8.
Exact mathematical passage
\mathbb E[e^{\theta(X-m)}\mid\mathcal F]\leq e^{\theta^2/8}.
e^{\theta^2/8}Use in source · 1 passage
Search aliases: e^θ^2/8 theta
Use in source · Read passage · FOUNDATIONS.md, line 3071
E[e^θ(X-m)mid F]≤ e^θ^2/8.
Exact mathematical passage
\mathbb E[e^{\theta(X-m)}\mid\mathcal F]\leq e^{\theta^2/8}.
E_0Use in source · 5 passages
Search aliases: E_0
Use in source · Read passage · FOUNDATIONS.md, line 839
Let E_0 bound the initialization errors at indices -H,…,0. Then the recursively generated predictions satisfy
Exact mathematical passage
E_0
e_0Use in source · 15 passages
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Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
E_{00}Use in source · 2 passages
Search aliases: E_00
Use in source · Read passage · FOUNDATIONS.md, line 2254
J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,
Exact mathematical passage
\begin{aligned} \mathcal J(Tz)&=\frac15W\mathcal J(z)W^{\mathsf T},\\ \mathcal J(B_{\mathrm p,0}z)&=E_{00}\mathcal J(z)E_{00}^{\mathsf T},\\ \mathcal J(B_{\mathrm p,1}z)&=E_{01}\mathcal J(z)E_{01}^{\mathsf T}, \end{aligned}
E_{00}^{\mathsf T}Use in source · 2 passages
Search aliases: E_00^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2254
J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,
Exact mathematical passage
\begin{aligned} \mathcal J(Tz)&=\frac15W\mathcal J(z)W^{\mathsf T},\\ \mathcal J(B_{\mathrm p,0}z)&=E_{00}\mathcal J(z)E_{00}^{\mathsf T},\\ \mathcal J(B_{\mathrm p,1}z)&=E_{01}\mathcal J(z)E_{01}^{\mathsf T}, \end{aligned}
E_{01}Use in source · 2 passages
Search aliases: E_01
Use in source · Read passage · FOUNDATIONS.md, line 2254
J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,
Exact mathematical passage
\begin{aligned} \mathcal J(Tz)&=\frac15W\mathcal J(z)W^{\mathsf T},\\ \mathcal J(B_{\mathrm p,0}z)&=E_{00}\mathcal J(z)E_{00}^{\mathsf T},\\ \mathcal J(B_{\mathrm p,1}z)&=E_{01}\mathcal J(z)E_{01}^{\mathsf T}, \end{aligned}
E_{01}^{\mathsf T}Use in source · 2 passages
Search aliases: E_01^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2254
J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,
Exact mathematical passage
\begin{aligned} \mathcal J(Tz)&=\frac15W\mathcal J(z)W^{\mathsf T},\\ \mathcal J(B_{\mathrm p,0}z)&=E_{00}\mathcal J(z)E_{00}^{\mathsf T},\\ \mathcal J(B_{\mathrm p,1}z)&=E_{01}\mathcal J(z)E_{01}^{\mathsf T}, \end{aligned}
E_1Use in source · 2 passages
Search aliases: E_1
Use in source · Read passage · FOUNDATIONS.md, line 3487
δ=frac12000, E(u)=E_0+uE_1+u^2E_2,
Exact mathematical passage
\delta=\frac1{2000},\qquad E(u)=E_0+uE_1+u^2E_2,
e_1Use in source · 3 passages
Search aliases: e_1
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
Exact mathematical passage
Le_1=0
e_1^\topUse in source · 5 passages
Search aliases: e_1^top top
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
Exact mathematical passage
C=e_1^\top
E_2Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3487
δ=frac12000, E(u)=E_0+uE_1+u^2E_2,
Exact mathematical passage
\delta=\frac1{2000},\qquad E(u)=E_0+uE_1+u^2E_2,
e_AUse in source · 2 passages
Search aliases: e_A
Use in source · Read passage · FOUNDATIONS.md, line 3352
For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has
Exact mathematical passage
\|\widehat A-A^*\|_2\leq e_A
e_{a,0}Use in source · 3 passages
Search aliases: e_a,0
Use in source · Read passage · FOUNDATIONS.md, line 3221
|[Δ_a(u)(1,v)]_0| · ≤ e_a,0(u), [Δ_a(u)(1,v)]_rm sp _2 · ≤ e_a,1(u).
Exact mathematical passage
\begin{aligned} |[\Delta_a(u)(1,v)]_0|&\leq e_{a,0}(u),\\ \|[\Delta_a(u)(1,v)]_{\rm sp}\|_2&\leq e_{a,1}(u). \end{aligned}
e_{a,1}Use in source · 3 passages
Search aliases: e_a,1
Use in source · Read passage · FOUNDATIONS.md, line 3221
|[Δ_a(u)(1,v)]_0| · ≤ e_a,0(u), [Δ_a(u)(1,v)]_rm sp _2 · ≤ e_a,1(u).
Exact mathematical passage
\begin{aligned} |[\Delta_a(u)(1,v)]_0|&\leq e_{a,0}(u),\\ \|[\Delta_a(u)(1,v)]_{\rm sp}\|_2&\leq e_{a,1}(u). \end{aligned}
E_{a,k}Use in source · 1 passage
Search aliases: E_a,k
Use in source · Read passage · FOUNDATIONS.md, line 3424
Δ_a(u)=∑_k=0^Du^kE_a,k, |(E_a,k)_ij|≤ r_k.
Exact mathematical passage
\Delta_a(u)=\sum_{k=0}^{D}u^kE_{a,k},\qquad |(E_{a,k})_{ij}|\leq r_k.
e_bUse in source · 2 passages
Search aliases: e_b
Use in source · Read passage · FOUNDATIONS.md, line 3352
For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has
Exact mathematical passage
\|\widehat b-b^*\|_2\leq e_b
E_iUse in source · 3 passages
Search aliases: E_i
Use in source · Read passage · FOUNDATIONS.md, line 104
Within one orthogonal atlas, let E_i be pairwise orthogonal coordinate selectors with ∑_iE_i=I. Pull them back by a unitary map:
Exact mathematical passage
E_i
e_i^{\mathsf T}Use in source · 1 passage
Search aliases: e_i^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3749
|e_i^ T(widehat C_a,k-C_a,k)x| ≤ ∑_j |(W^-1)_kj| β_a,j,i x _V_a,t^-1.
Exact mathematical passage
\left|e_i^{\mathsf T}(\widehat C_{a,k}-C_{a,k})x\right| \leq \sum_j |(W^{-1})_{kj}|\, \beta_{a,j,i}\,\|x\|_{V_{a,t}^{-1}}.
E_{ij}Use in source · 1 passage
Search aliases: E_ij
Use in source · Read passage · FOUNDATIONS.md, line 2262
where E_ij is the corresponding 2×2 matrix unit. The other rotate branch is zero. Thus this exact model has a finite rational matrix certificate without attenuation or a polygonal replacement.
Exact mathematical passage
E_{ij}
e_jUse in source · 1 passage
Search aliases: e_j
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
Exact mathematical passage
Le_j=e_{j-1}
e_{j-1}Use in source · 1 passage
Search aliases: e_j-1
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
Exact mathematical passage
Le_j=e_{j-1}
e_MUse in source · 4 passages
Search aliases: e_M
Use in source · Read passage · FOUNDATIONS.md, line 3330
M_widehat A- M_A^* ≤ e_M, σ_min( M_widehat A)>e_M.
Exact mathematical passage
\|\mathcal M_{\widehat A}-\mathcal M_{A^*}\|\leq e_M, \qquad \sigma_{\min}(\mathcal M_{\widehat A})>e_M.
\mathbb E_{P,\pi}Use in source · 1 passage
Search aliases: E_P,π mathbb pi
Use in source · Read passage · FOUNDATIONS.md, line 3154
P_T-Q_T _TV ≤min(1,∑_t=0^T-1 E_P,π[d_t(H_t,A_t)]).
Exact mathematical passage
\boxed{ \|P_T-Q_T\|_{\mathrm{TV}} \leq\min\left(1,\sum_{t=0}^{T-1} \mathbb E_{P,\pi}[d_t(H_t,A_t)]\right). }
e_sUse in source · 1 passage
Search aliases: e_s
Use in source · Read passage · FOUNDATIONS.md, line 862
**Proof.** Subtract the two recurrences and bound every retained history error by the largest of the errors at indices t-H,…,t. This gives e_t+1≤ amax_t-H≤ s≤ te_s+Δ. The ball of radius E=max(E_0,Δ/(1-a)) is invariant because aE+Δ≤ E. Induction proves the bound. The estimate for Δ separates coefficient errors, the geometric kernel tail and the initialized forcing, using Proposition 10. square
Exact mathematical passage
e_{t+1}\leq a\max_{t-H\leq s\leq t}e_s+\Delta
e_TUse in source · 3 passages
Search aliases: e_T
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
e_tUse in source · 11 passages
Search aliases: e_t
Use in source · Read passage · FOUNDATIONS.md, line 218
e_t=PLQx_t, e_t ≤ PLQ Qx_t .
Exact mathematical passage
e_t=PLQx_t,\qquad \|e_t\|\leq\|PLQ\|\,\|Qx_t\|.
e_{t+1}Use in source · 4 passages
Search aliases: e_t+1
Use in source · Read passage · FOUNDATIONS.md, line 769
**Proof.** Add and subtract widehat f_t(γ_t(x_t)) to obtain e_t+1≤δ_t+λ_te_t. Repeated substitution yields the product-sum formula. Add and subtract H_t(γ_t(x_t)) for the output inequality. square
Exact mathematical passage
e_{t+1}\leq\delta_t+\lambda_te_t
e_{T+2}Use in source · 1 passage
Search aliases: e_T+2
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
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x_0=e_{T+2}
\mathbb E_\nuUse in source · 1 passage
Search aliases: E_ν mathbb nu
Use in source · Read passage · FOUNDATIONS.md, line 2492
**Proof.** Substitute K=bar K+(K-bar K) and expand. The two cross terms integrate to zero. The remaining integrand is a positive congruence when X is positive. For an affine matrix function the centered term is (u- E_ν u)K_1, giving the variance formula. Integrating the square of a centered uniform variable gives h^2/12. square
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(u-\mathbb E_\nu u)K_1
\expUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1525
**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the same bound. A union bound over the M words proves the simultaneous result. The spectral norm is bounded by the Frobenius norm, yielding the matrix an…
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g(t)=\log\mathbb E\exp(t(X-p))
fUse in source · 27 passages
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Use in source · Read passage · FOUNDATIONS.md, line 430
dom(g∘ f) ={x∈domf:f(x)∈domg}.
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\operatorname{dom}(g\circ f) =\{x\in\operatorname{dom}f:f(x)\in\operatorname{dom}g\}.
F^*Use in source · 6 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3278
**Proposition 73. Interior repair with an explicit error budget.** Suppose a reference family F^* is valid on L_d, and the combined mass and spatial error of widehat F-F^* is at most d_a(u) on normalized inputs. Then the margin of widetilde F_a(u) is at least
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F^*
F^2Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 960
∑_aw_a L-L_a _F^2
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\sum_aw_a\|L-L_a\|_F^2
F_{\mathcal I}Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 902
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
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F_{\mathcal I}
F_{(i,p),(j,q)}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2322
[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.
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[\Phi_F(X)]_{ij}=\sum_{p,q=1}^d F_{(i,p),(j,q)}X_{pq}.
f_{a,s}Use in source · 2 passages
Search aliases: f_a,s
Use in source · Read passage · FOUNDATIONS.md, line 3392
w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.
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\begin{aligned} w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\ f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\ h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2. \end{aligned}
F_bUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 75
T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.
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T_{aa}=I,\qquad T_{cb}T_{ba}=T_{ca},\qquad F_bT_{ba}=T_{ba}F_a.
F_sUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1410
widetilde H=H+E, widetilde G_s=G_s+F_s, E _2<s_H.
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\widetilde H=H+E,\qquad \widetilde G_s=G_s+F_s,\qquad \|E\|_2<s_H.
F_tUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 738
Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is
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x_{t+1}=F_t(x_t)
f_tUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 818
z_t+1=Az_t+∑_k=0^t-1K_kz_t-1-k+f_t, K_k=BD^kC, f_t=BD^tq_0.
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z_{t+1}=Az_t+\sum_{k=0}^{t-1}K_kz_{t-1-k}+f_t, \qquad K_k=BD^kC,\quad f_t=BD^tq_0.
F_uUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 185
Let γ:X→ Z=γ(X) be a summary and let F_u:X→ X describe a deterministic transition for action u. The reduced description closes when there is bar F_u:Z→ Z such that
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F_u:X\to X
F_\etaUse in source · 3 passages
Search aliases: F_η eta
Use in source · Read passage · FOUNDATIONS.md, line 3591
F_η=(1-η)widehat F+η N, 0≤η≤1.
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F_\eta=(1-\eta)\widehat F+\eta N, \qquad 0\leq\eta\leq1.
GUse in source · 37 passages
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Use in source · Read passage · FOUNDATIONS.md, line 80
Conversely, suppose maps G_a: C_a→ C_a satisfy G_bT_ba=T_baG_a for every pair. Then there is a unique map F:X→ X with G_a=A_aFS_a for every a.
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G_a:\mathcal C_a\to\mathcal C_a
\mathcal GUse in source · 12 passages
Search aliases: G mathcal
Use in source · Read passage · FOUNDATIONS.md, line 367
**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then
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\mathcal F\subseteq\mathcal G
gUse in source · 35 passages
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Use in source · Read passage · FOUNDATIONS.md, line 430
dom(g∘ f) ={x∈domf:f(x)∈domg}.
Exact mathematical passage
\operatorname{dom}(g\circ f) =\{x\in\operatorname{dom}f:f(x)\in\operatorname{dom}g\}.
g'Use in source · 1 passage
Search aliases: g'
Use in source · Read passage · FOUNDATIONS.md, line 1525
**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the same bound. A union bound over the M words proves the simultaneous result. The spectral norm is bounded by the Frobenius norm, yielding the matrix an…
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g(0)=g'(0)=0
g''Use in source · 1 passage
Search aliases: g''
Use in source · Read passage · FOUNDATIONS.md, line 1525
**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the same bound. A union bound over the M words proves the simultaneous result. The spectral norm is bounded by the Frobenius norm, yielding the matrix an…
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g''(t)
g_1Use in source · 1 passage
Search aliases: g_1
Use in source · Read passage · FOUNDATIONS.md, line 1069
The fixed prior need not be uniform. For an explicit enumeration g_1,g_2,…, π(g_k)=2^-k is a normalized example, with omitted prior mass 2^-n after n programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.
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g_1,g_2,\ldots
g_2Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1069
The fixed prior need not be uniform. For an explicit enumeration g_1,g_2,…, π(g_k)=2^-k is a normalized example, with omitted prior mass 2^-n after n programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.
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g_1,g_2,\ldots
G_aUse in source · 12 passages
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Use in source · Read passage · FOUNDATIONS.md, line 80
Conversely, suppose maps G_a: C_a→ C_a satisfy G_bT_ba=T_baG_a for every pair. Then there is a unique map F:X→ X with G_a=A_aFS_a for every a.
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G_a:\mathcal C_a\to\mathcal C_a
g_aUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3266
N_a(u)=g_a(u)e_0λ, g_a(u)≥ g_min>0, ∫ g_a(u) du/2=1.
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N_a(u)=g_a(u)e_0\lambda,\qquad g_a(u)\geq g_{\min}>0,\qquad \int g_a(u)\,du/2=1.
G_{a,0}Use in source · 2 passages
Search aliases: G_a,0
Use in source · Read passage · FOUNDATIONS.md, line 2341
Suppose each action has constant positive semidefinite matrices G_a,0,G_a,1, with the second term omitted when q=0, such that
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G_{a,0},G_{a,1}
G_{a,1}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2341
Suppose each action has constant positive semidefinite matrices G_a,0,G_a,1, with the second term omitted when q=0, such that
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G_{a,0},G_{a,1}
G_{a,j}Use in source · 6 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2394
H_ij=t_i h_j, (G_a,j)_iℓ=t_i M_a,jh_ℓ.
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H_{ij}=t_i h_j,\qquad (G_{a,j})_{i\ell}=t_i M_{a,j}h_\ell.
G_bUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 80
Conversely, suppose maps G_a: C_a→ C_a satisfy G_bT_ba=T_baG_a for every pair. Then there is a unique map F:X→ X with G_a=A_aFS_a for every a.
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G_bT_{ba}=T_{ba}G_a
g_iUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1804
g_i^topwidehat B_sv_j ≥ϵ_s g_i _2 v_j _2 for all s,i,j.
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\boxed{ g_i^\top\widehat B_sv_j \geq\epsilon_s\|g_i\|_2\|v_j\|_2 \quad\text{for all }s,i,j. }
g_i^\topUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1802
For a complete facet description C={z:Gz≥0}, let g_i^top be its rows. A sufficient norm-ball certificate around widehat B_s is
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g_i^\top
g_jUse in source · 8 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1077
**Proposition 24. Prior-preserving program refinement.** Suppose a parent program g is replaced by children g_j with weights α_j≥0, ∑_jα_j=1, and suppose its likelihood is the mixture
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g_j
g_kUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1069
The fixed prior need not be uniform. For an explicit enumeration g_1,g_2,…, π(g_k)=2^-k is a normalized example, with omitted prior mass 2^-n after n programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.
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\pi(g_k)=2^{-k}
g_{\min}Use in source · 2 passages
Search aliases: g_min min
Use in source · Read passage · FOUNDATIONS.md, line 3266
N_a(u)=g_a(u)e_0λ, g_a(u)≥ g_min>0, ∫ g_a(u) du/2=1.
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N_a(u)=g_a(u)e_0\lambda,\qquad g_a(u)\geq g_{\min}>0,\qquad \int g_a(u)\,du/2=1.
\mathcal G_nUse in source · 4 passages
Search aliases: G_n mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1040
Let finite active sets G_n increase to G. Their unnormalized evidence is Z_n=∑_g∈ G_nπ(g)L_g(D). Only stages with Z_n>0 are considered.
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\mathcal G_n
G_rUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 82
**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square
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F=S_rG_rA_r
G_sUse in source · 11 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1252
(G_s)_ij=f(h_jst_i), (b_0)_i=f(t_i).
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(G_s)_{ij}=f(h_jst_i),\qquad (b_0)_i=f(t_i).
g_\nuUse in source · 1 passage
Search aliases: g_ν nu
Use in source · Read passage · FOUNDATIONS.md, line 2366
Φ_u(X)=∑_ν w_ν g_ν(u)K_ν(u)XK_ν(u)^ T, g_ν(u)∈{1,1-u^2}.
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\Phi_u(X)=\sum_\nu w_\nu g_\nu(u)K_\nu(u)XK_\nu(u)^{\mathsf T}, \qquad g_\nu(u)\in\{1,1-u^2\}.
HUse in source · 92 passages
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Use in source · Read passage · FOUNDATIONS.md, line 250
C_0=HR, RL_u=B_uR
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C_0=HR,\qquad RL_u=B_uR
\mathcal HUse in source · 6 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3083
For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define
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\mathcal H_{t-1}
H^*Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3326
Use the Frobenius norm on symmetric matrices and the induced product norm. Assume M_A^* is injective and M_A^*(H^*)=(0,1)=b. This fixes the scale of a one-dimensional invariant-form space.
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\mathcal M_{A^*}(H^*)=(0,1)=b
H^{-1}Use in source · 9 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1258
B_s=G_sH^-1, λ=e_1^top, b_0=(f(t_i))_i
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\boxed{B_s=G_sH^{-1},\qquad \lambda=e_1^\top,\qquad b_0=(f(t_i))_i}
h^2Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2490
The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.
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h^2/12
H_0Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2856
**Proposition 63. Conditional operator-only ellipsoid discovery.** Suppose the symmetric solution space of A^ THA=H is one-dimensional and contains a positive definite matrix H_0. Let a rank-one fitted branch at a declared observation have a nonzero-mass image, normalized to v=(1,x_v). If x_v≠ c, define
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H_0
h_1Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1242
Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix
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h_1,\ldots,h_r
h_2Use in source · 1 passage
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Use in source · atlas.tex, line 240
plate10Behavior can supply the coordinatesFuture tests define a predictive state without naming hidden variables.P27--P28 For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space. c|ccc · h_1 · h_2 · ⋯ τ_1 · p(h_1τ_1) · p(h_2τ_1) · ⋯ τ_2 · p(h_1τ_2) · p(h_2τ_2) · ⋯ vdots · vdots · vdots · Assume the full behavior space has rank at most r. Choose an inver…
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\begin{array}{c|ccc} & h_1 & h_2 & \cdots\\\hline \tau_1 & p(h_1\tau_1)&p(h_2\tau_1)&\cdots\\ \tau_2 & p(h_1\tau_2)&p(h_2\tau_2)&\cdots\\ \vdots &\vdots&\vdots& \end{array}
h_{a,s}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3392
w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.
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\begin{aligned} w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\ f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\ h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2. \end{aligned}
H_{ij}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1244
H_ij=f(h_jt_i)
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H_{ij}=f(h_jt_i)
h_jUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1244
H_ij=f(h_jt_i)
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H_{ij}=f(h_jt_i)
h_{j-1}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1352
Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),
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\Pr_\pi(w)=f(w)\prod_{j=1}^{|w|}\pi(a_j\mid h_{j-1}),
h_rUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1242
Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix
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h_1,\ldots,h_r
H_sUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1786
There is a dual finite test. If a complete inequality description of the same cone is C={z:Gz≥0}, then GB_sv_j≥0 for every generator j is equivalent to invariance. One may also certify GB_s=H_sG with H_s≥0. The latter certificate is sufficient by direct multiplication and necessary by the polyhedral cone form of Farkas' lemma [R13]. The reference calculation uses the direct generator test, not an unreported numerical optimization certificate.
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GB_s=H_sG
H_tUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 760
Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then
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H_t
h_tUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 760
Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then
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h_t(x)
H_{t,h}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1211
mathscr H_t,h=f(ht).
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\mathscr H_{t,h}=f(ht).
\mathcal H_{t-1}Use in source · 6 passages
Search aliases: H_t-1 mathcal
Use in source · Read passage · FOUNDATIONS.md, line 3083
For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define
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\mathcal H_{t-1}
h_\ellUse in source · 1 passage
Search aliases: h_ℓ ell
Use in source · Read passage · FOUNDATIONS.md, line 2394
H_ij=t_i h_j, (G_a,j)_iℓ=t_i M_a,jh_ℓ.
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H_{ij}=t_i h_j,\qquad (G_{a,j})_{i\ell}=t_i M_{a,j}h_\ell.
\hlineUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3832
c|rrrrr u · -1 · -1/2 · 0 · 1/2 · 1 μ{u} · 7/90 · 16/45 · 2/15 · 16/45 · 7/90.
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\begin{array}{c|rrrrr} u&-1&-1/2&0&1/2&1\\\hline \mu\{u\}&7/90&16/45&2/15&16/45&7/90. \end{array}
IUse in source · 62 passages
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Use in source · Read passage · FOUNDATIONS.md, line 27
A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.
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A_a:X\longrightarrow C_a,\qquad S_a:C_a\longrightarrow X, \qquad S_aA_a=I_X.
\boldsymbol iUse in source · 4 passages
Search aliases: i boldsymbol
Use in source · Read passage · FOUNDATIONS.md, line 900
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
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\boldsymbol i=(i_1,\ldots,i_m)
\mathcal IUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 902
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
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\mathcal I_m
I_{\mathcal C_a}Use in source · 1 passage
Search aliases: I_ C_a mathcal
Use in source · atlas.tex, line 89
plate2One operation, several coordinatesA commuting square expresses coordinate independence.P2 Let C_a=im A_a and T_ba=A_bS_a. Coordinate realizations are F_a=A_aFS_a on C_a. Both paths through the square represent the same underlying operation. diagram02_transport T_cbT_ba=T_ca, T_aa=I_ C_a, F_bT_ba=T_baF_a. Conversely, a compatible family G_a on the realizable coefficient spaces determines one operation. With a supplied reference chart r, F=S_rG_rA_r, G_a=A_aFS_a. claimCompatibility is a desc…
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T_{cb}T_{ba}=T_{ca},\qquad T_{aa}=I_{\mathcal C_a},\qquad F_bT_{ba}=T_{ba}F_a.
i_1Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 173
Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).
Exact mathematical passage
Q_jF(x)=Q_jb+ \sum_{m=1}^{p}\ \sum_{i_1,\ldots,i_m} Q_jB_m(P_{i_1}x,\ldots,P_{i_m}x).
I_{d^2}Use in source · 1 passage
Search aliases: I_d^2
Use in source · Read passage · FOUNDATIONS.md, line 2337
Z_q(u)=I_d^2 uI_d^2 vdots u^qI_d^2.
Exact mathematical passage
Z_q(u)=\begin{pmatrix}I_{d^2}\\uI_{d^2}\\\vdots\\u^qI_{d^2}\end{pmatrix}.
i_jUse in source · 1 passage
Search aliases: i_j
Use in source · Read passage · FOUNDATIONS.md, line 906
F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.
Exact mathematical passage
\boxed{ \|F(x)-F_{\mathcal I}(x)\| \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\notin\mathcal I_m} \beta_{m,\boldsymbol i}\prod_{j=1}^{m}a_{i_j}. }
\mathcal I_mUse in source · 3 passages
Search aliases: I_m mathcal
Use in source · Read passage · FOUNDATIONS.md, line 902
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
Exact mathematical passage
\mathcal I_m
i_mUse in source · 4 passages
Search aliases: i_m
Use in source · Read passage · FOUNDATIONS.md, line 173
Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).
Exact mathematical passage
Q_jF(x)=Q_jb+ \sum_{m=1}^{p}\ \sum_{i_1,\ldots,i_m} Q_jB_m(P_{i_1}x,\ldots,P_{i_m}x).
I_rUse in source · 1 passage
Search aliases: I_r
Use in source · Read passage · FOUNDATIONS.md, line 658
**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. square
Exact mathematical passage
\widetilde Z_-\widetilde Z_-^\dagger=I_r
I_XUse in source · 4 passages
Search aliases: I_X
Use in source · Read passage · FOUNDATIONS.md, line 27
A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.
Exact mathematical passage
A_a:X\longrightarrow C_a,\qquad S_a:C_a\longrightarrow X, \qquad S_aA_a=I_X.
\iffUse in source · 1 passage
Search aliases: iff
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
Exact mathematical passage
5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
\operatorname{im}Use in source · 8 passages
Search aliases: im operatorname
Use in source · Read passage · FOUNDATIONS.md, line 32
The coefficients representing actual signals form C_a=imA_a. This distinction matters for redundant representations, where C_a is a proper subspace of the ambient coefficient space.
Exact mathematical passage
\mathcal C_a=\operatorname{im}A_a
\infUse in source · 1 passage
Search aliases: inf
Use in source · Read passage · FOUNDATIONS.md, line 1879
x _C=inf{λ u+λ v:x=u-v, u,v∈ C}.
Exact mathematical passage
\|x\|_C=\inf\{\lambda u+\lambda v:x=u-v,\ u,v\in C\}.
\mathrm{init}Use in source · 1 passage
Search aliases: init mathrm
Use in source · Read passage · FOUNDATIONS.md, line 304
where B ranges over the current partition. The process terminates after at most |Ω|-|π_init| strict refinement rounds. The final partition is the unique coarsest refinement of π_0 on which all declared predictions and Bayesian updates close for every prior.
Exact mathematical passage
|\Omega|-|\pi_{\mathrm{init}}|
JUse in source · 15 passages
Search aliases: J
Use in source · Read passage · FOUNDATIONS.md, line 125
This is a conservation law for information under complete refinement. Deleting a child, storing only its energy, or imposing a coefficient threshold changes the contract. For a retained subset J of an orthogonal decomposition, the lost signal is exactly ∑_i∉ JP_ix, and its squared norm is the sum of the omitted packet energies.
Exact mathematical passage
J
\mathcal JUse in source · 30 passages
Search aliases: J mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2183
J:E⟶ S^d, λ z=tr J(z).
Exact mathematical passage
\mathcal J:E\longrightarrow\mathbb S^d, \qquad \lambda z=\operatorname{tr}\mathcal J(z).
\mathcal J'Use in source · 1 passage
Search aliases: J' mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2208
B'_s=TB_sT^-1, J'= JT^-1, λ'=λ T^-1.
Exact mathematical passage
B'_s=TB_sT^{-1},\qquad \mathcal J'=\mathcal JT^{-1},\qquad \lambda'=\lambda T^{-1}.
J_tUse in source · 2 passages
Search aliases: J_t
Use in source · Read passage · FOUNDATIONS.md, line 1020
For adaptive prediction, an attempted lift J_t contributes its own initialization defect
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J_t
KUse in source · 75 passages
Search aliases: K
Use in source · Read passage · FOUNDATIONS.md, line 274
Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix
Exact mathematical passage
K_u(i,j)
\mathcal KUse in source · 3 passages
Search aliases: K mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1296
K={v∈ C:λ v=1}
Exact mathematical passage
\mathcal K=\{v\in\mathcal C:\lambda v=1\}
kUse in source · 75 passages
Search aliases: k
Use in source · Read passage · FOUNDATIONS.md, line 151
For another linear map M:Y→ Z and output projectors R_k,
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R_k
K^{\mathsf T}Use in source · 2 passages
Search aliases: K^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2216
Every term wKXK^ T is positive semidefinite when X is. The intertwining identity therefore proves invariance. Iteration gives invariance of every branch state, and the mass identity normalizes every finite observation tree. Substituting the transported quantities proves the final statement. square
Exact mathematical passage
wKXK^{\mathsf T}
K_{\mathcal J}Use in source · 4 passages
Search aliases: K_ J mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2190
C_ J={z: J(z)succeq0}, K_ J={z∈ C_ J:λ z=1}.
Exact mathematical passage
C_{\mathcal J}=\{z:\mathcal J(z)\succeq0\}, \qquad K_{\mathcal J}=\{z\in C_{\mathcal J}:\lambda z=1\}.
K_0Use in source · 4 passages
Search aliases: K_0
Use in source · Read passage · FOUNDATIONS.md, line 1625
K_0={(1,q)^top:3/8≤ q≤5/8}.
Exact mathematical passage
K_0=\{(1,q)^\top:3/8\leq q\leq5/8\}.
K_1Use in source · 6 passages
Search aliases: K_1
Use in source · Read passage · FOUNDATIONS.md, line 2490
The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.
Exact mathematical passage
K(u)=K_0+uK_1
K_1^{\mathsf T}Use in source · 3 passages
Search aliases: K_1^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2490
The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.
Exact mathematical passage
\operatorname{Var}_\nu(u)K_1XK_1^{\mathsf T}
K_jUse in source · 2 passages
Search aliases: K_j
Use in source · atlas.tex, line 186
plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, retaining the last H history terms leaves a same-history tail at most Δ_H=tfrac38 3^-H.
Exact mathematical passage
K_j=\tfrac14\,3^{-j}
K_kUse in source · 3 passages
Search aliases: K_k
Use in source · Read passage · FOUNDATIONS.md, line 818
z_t+1=Az_t+∑_k=0^t-1K_kz_t-1-k+f_t, K_k=BD^kC, f_t=BD^tq_0.
Exact mathematical passage
z_{t+1}=Az_t+\sum_{k=0}^{t-1}K_kz_{t-1-k}+f_t, \qquad K_k=BD^kC,\quad f_t=BD^tq_0.
K_{sj}Use in source · 1 passage
Search aliases: K_sj
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
Exact mathematical passage
\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
K_{sj}^{\mathsf T}Use in source · 1 passage
Search aliases: K_sj^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
Exact mathematical passage
\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
K_TUse in source · 2 passages
Search aliases: K_T
Use in source · Read passage · FOUNDATIONS.md, line 3206
K_T={T^-1(1,v): v _2≤1}.
Exact mathematical passage
K_T=\{T^{-1}(1,v):\|v\|_2\leq1\}.
K_uUse in source · 2 passages
Search aliases: K_u
Use in source · Read passage · FOUNDATIONS.md, line 274
Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix
Exact mathematical passage
K_u(i,j)
K_u^\topUse in source · 2 passages
Search aliases: K_u^top top
Use in source · Read passage · FOUNDATIONS.md, line 280
Distributions are column vectors. Prediction is p↦ K_u^top p, and a Bayesian update is
Exact mathematical passage
p\mapsto K_u^\top p
K_\nuUse in source · 3 passages
Search aliases: K_ν nu
Use in source · Read passage · FOUNDATIONS.md, line 2328
F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,
Exact mathematical passage
F=\sum_\nu w_\nu\operatorname{vec}(K_\nu) \operatorname{vec}(K_\nu)^{\mathsf T},\qquad w_\nu\geq0,
K_\nu^{\mathsf T}Use in source · 1 passage
Search aliases: K_ν^ T nu mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2333
then Φ_F(X)=∑_ν w_ν K_ν X K_ν^ T. Every positive semidefinite F has a real factorization of this kind. The matrix is the usual Choi/Gram representation, with an explicit index convention [R16].
Exact mathematical passage
\Phi_F(X)=\sum_\nu w_\nu K_\nu X K_\nu^{\mathsf T}
\kerUse in source · 9 passages
Search aliases: ker
Use in source · Read passage · FOUNDATIONS.md, line 264
Its annihilator is the largest L-invariant subspace inside ker C_0, the standard unobservable-subspace construction [R2]. With several actions, observables pulled back through action words are required, but the linear span still stabilizes in finite dimension.
Exact mathematical passage
\ker C_0
LUse in source · 130 passages
Search aliases: L
Use in source · Read passage · FOUNDATIONS.md, line 137
Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.
Exact mathematical passage
L:X\to Y
\mathcal LUse in source · 7 passages
Search aliases: L mathcal
Use in source · Read passage · FOUNDATIONS.md, line 3204
Let L_d={(m,v):m≥ v _2}, with mass row λ=(1,0,…,0). A mass preserving invertible matrix T transports a proposed predictive domain to this cone. Its normalized base in the original coordinates is
Exact mathematical passage
\mathcal L_d=\{(m,v):m\geq\|v\|_2\}
\mathrm LUse in source · 8 passages
Search aliases: L mathrm
Use in source · Read passage · FOUNDATIONS.md, line 2050
C_ L={(t,x,y):t≥sqrtx^2+y^2}
Exact mathematical passage
C_{\mathrm L}=\{(t,x,y):t\geq\sqrt{x^2+y^2}\}
lUse in source · 1 passage
Search aliases: l
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
Exact mathematical passage
5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
L'Use in source · 1 passage
Search aliases: L'
Use in source · Read passage · FOUNDATIONS.md, line 533
L'=0 · 1000 0 · 0, x=(0,1/1000)^top,
Exact mathematical passage
L'=\begin{pmatrix}0&1000\\0&0\end{pmatrix}, \qquad x=(0,1/1000)^\top,
L^{\mathsf T}Use in source · 2 passages
Search aliases: L^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3379
A rationalized invertible factor L specifies H=L^ TL exactly. Positive definiteness is then an algebraic fact even when the factor was suggested by floating-point calculations. It still does not establish operator invariance.
Exact mathematical passage
H=L^{\mathsf T}L
L^2Use in source · 3 passages
Search aliases: L^2
Use in source · Read passage · FOUNDATIONS.md, line 2983
An ordinary orthogonal truncation need not preserve that validity. The valid density p(v)=3v^2 has degree-one L^2 projection
Exact mathematical passage
L^2
L^dUse in source · 2 passages
Search aliases: L^d
Use in source · Read passage · FOUNDATIONS.md, line 588
,CL^h. All except the final block already occur in O_h. The kernel condition is therefore equivalent to CL^h being in the row space of O_h, which is exactly the rank equality. It also implies O_hLx∈ V_h because this vector is O_h applied to Lx. Uniqueness follows because every element of V_h has a representative. The history recurrence follows directly. Cayley-Hamilton expresses L^d as a linear combination of lower powers, so CL^d is in the row space of O_d. square
Exact mathematical passage
L^d
L^{d-1}Use in source · 1 passage
Search aliases: L^d-1
Use in source · Read passage · FOUNDATIONS.md, line 260
C_0, C_0L, …, C_0L^d-1.
Exact mathematical passage
C_0,\ C_0L,\ \ldots,\ C_0L^{d-1}.
L^hUse in source · 2 passages
Search aliases: L^h
Use in source · Read passage · FOUNDATIONS.md, line 588
**Proof.** The equivalence of the first two statements is the linear factorization criterion: define D_h(O_hx)=O_hLx and require independence from the representative x. The rows of O_hL are CL,…,CL^h. All except the final block already occur in O_h. The kernel condition is therefore equivalent to CL^h being in the row space of O_h, which is exactly the rank equality. It also implies O_hLx∈ V_h because this vector is O_h applied to Lx. Uniqueness follows because every element of V_h has a represe…
Exact mathematical passage
CL,\ldots,CL^h
L^{h-1}Use in source · 1 passage
Search aliases: L^h-1
Use in source · Read passage · FOUNDATIONS.md, line 566
O_h=C CL vdots CL^h-1, V_h=imO_h.
Exact mathematical passage
O_h=\begin{pmatrix}C\\CL\\\vdots\\CL^{h-1}\end{pmatrix}, \qquad V_h=\operatorname{im}O_h.
L^kUse in source · 3 passages
Search aliases: L^k
Use in source · Read passage · FOUNDATIONS.md, line 449
L_nx⟶ Lx, L_n^kx⟶ L^kx.
Exact mathematical passage
L_nx\longrightarrow Lx,\qquad L_n^kx\longrightarrow L^kx.
L^{k+1}Use in source · 1 passage
Search aliases: L^k+1
Use in source · Read passage · FOUNDATIONS.md, line 464
(L_n^k+1-L^k+1)x =L_n(L_n^k-L^k)x+(L_n-L)L^kx.
Exact mathematical passage
(L_n^{k+1}-L^{k+1})x =L_n(L_n^k-L^k)x+(L_n-L)L^kx.
L^tUse in source · 1 passage
Search aliases: L^t
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
Exact mathematical passage
CL^tx_0=0
L^{T+1}Use in source · 1 passage
Search aliases: L^T+1
Use in source · Read passage · FOUNDATIONS.md, line 668
**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square
Exact mathematical passage
CL^{T+1}x_0=1
L_*Use in source · 2 passages
Search aliases: L_*
Use in source · Read passage · FOUNDATIONS.md, line 964
over linear maps on X, where · _F is the Frobenius norm. For any target operator L_* and any chosen operator norm,
Exact mathematical passage
L_*
L_2Use in source · 15 passages
Search aliases: L_2
Use in source · Read passage · FOUNDATIONS.md, line 3768
v(u)=(1,u,u^2)^ T, L_2(u)=frac3u^2-12.
Exact mathematical passage
v(u)=(1,u,u^2)^{\mathsf T}, \qquad L_2(u)=\frac{3u^2-1}{2}.
L_3Use in source · 7 passages
Search aliases: L_3
Use in source · Read passage · FOUNDATIONS.md, line 2436
L_3(u)=frac5u^3-3u2, p_±(u)=1±ϵ L_3(u), 0<ϵ<1.
Exact mathematical passage
L_3(u)=\frac{5u^3-3u}{2},\qquad p_\pm(u)=1\pm\epsilon L_3(u),\qquad 0<\epsilon<1.
L_aUse in source · 5 passages
Search aliases: L_a
Use in source · Read passage · FOUNDATIONS.md, line 946
L_a=S_aG_aA_a.
Exact mathematical passage
L_a=S_aG_aA_a.
\mathcal L_dUse in source · 7 passages
Search aliases: L_d mathcal
Use in source · Read passage · FOUNDATIONS.md, line 3204
Let L_d={(m,v):m≥ v _2}, with mass row λ=(1,0,…,0). A mass preserving invertible matrix T transports a proposed predictive domain to this cone. Its normalized base in the original coordinates is
Exact mathematical passage
\mathcal L_d=\{(m,v):m\geq\|v\|_2\}
L_gUse in source · 8 passages
Search aliases: L_g
Use in source · Read passage · FOUNDATIONS.md, line 1034
Let G be a countable declared program set, with one fixed probability mass function π(g)>0 satisfying ∑_gπ(g)=1. Every program specifies a likelihood L_g(D) for the **same** observed data event or density relative to a common base measure. Assume
Exact mathematical passage
L_g(D)
L_{g_j}Use in source · 2 passages
Search aliases: L_g_j
Use in source · Read passage · FOUNDATIONS.md, line 1079
L_g(D)=∑_jα_jL_g_j(D).
Exact mathematical passage
L_g(D)=\sum_j\alpha_jL_{g_j}(D).
L_{ji}Use in source · 1 passage
Search aliases: L_ji
Use in source · Read passage · FOUNDATIONS.md, line 137
Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.
Exact mathematical passage
L_{ji}=Q_jLP_i
L_nUse in source · 6 passages
Search aliases: L_n
Use in source · Read passage · FOUNDATIONS.md, line 447
**Proposition 13. Strong fixed-horizon refinement limit.** Let X be a Hilbert space, L:X→ X bounded, and P_n orthogonal projections converging strongly to I. Put L_n=P_nLP_n. For every fixed x∈ X and every fixed nonnegative integer k,
Exact mathematical passage
L_n=P_nLP_n
L_n^kUse in source · 2 passages
Search aliases: L_n^k
Use in source · Read passage · FOUNDATIONS.md, line 449
L_nx⟶ Lx, L_n^kx⟶ L^kx.
Exact mathematical passage
L_nx\longrightarrow Lx,\qquad L_n^kx\longrightarrow L^kx.
L_n^{k+1}Use in source · 1 passage
Search aliases: L_n^k+1
Use in source · Read passage · FOUNDATIONS.md, line 464
(L_n^k+1-L^k+1)x =L_n(L_n^k-L^k)x+(L_n-L)L^kx.
Exact mathematical passage
(L_n^{k+1}-L^{k+1})x =L_n(L_n^k-L^k)x+(L_n-L)L^kx.
L_tUse in source · 1 passage
Search aliases: L_t
Use in source · Read passage · FOUNDATIONS.md, line 1485
e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.
Exact mathematical passage
e_{t+1}\leq\alpha_te_t+\beta_t,\qquad \alpha_t=\frac{(1+R\|\lambda\|_*)L_t}{\widehat\eta_t},\qquad \beta_t=\frac{(1+R\|\lambda\|_*)\epsilon_{B,t}R}{\widehat\eta_t}.
L_uUse in source · 11 passages
Search aliases: L_u
Use in source · Read passage · FOUNDATIONS.md, line 239
There is a constructive finite solution when the dynamics are a known finite family of linear maps L_u on a d-dimensional space. It identifies which additional observables are required, rather than merely requesting a larger state vector.
Exact mathematical passage
L_u
\lceilUse in source · 2 passages
Search aliases: lceil
Use in source · Read passage · FOUNDATIONS.md, line 3854
**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of
Exact mathematical passage
k=\lceil\log_2 V\rceil
\operatorname{Lip}Use in source · 1 passage
Search aliases: Lip operatorname
Use in source · Read passage · FOUNDATIONS.md, line 916
Lip(F_ I) ≤∑_m=1^p ∑_ i∈ I_m mβ_m, iR^m-1.
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\operatorname{Lip}(F_{\mathcal I}) \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\in\mathcal I_m} m\beta_{m,\boldsymbol i}R^{m-1}.
\logUse in source · 10 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1509
|widehat f(w)-f(w)|≤ ϵ_N, ϵ_N=sqrtfraclog(2M/δ)2N.
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|\widehat f(w)-f(w)|\leq \epsilon_N,\qquad \epsilon_N=\sqrt{\frac{\log(2M/\delta)}{2N}}.
\log_2Use in source · 2 passages
Search aliases: log_2 log
Use in source · Read passage · FOUNDATIONS.md, line 3854
**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of
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k=\lceil\log_2 V\rceil
\LongrightarrowUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 195
γ(x)=γ(x') Longrightarrow γ(F_u(x))=γ(F_u(x'))
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\gamma(x)=\gamma(x')\ \Longrightarrow\ \gamma(F_u(x))=\gamma(F_u(x'))
MUse in source · 57 passages
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Use in source · Read passage · FOUNDATIONS.md, line 151
For another linear map M:Y→ Z and output projectors R_k,
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M:Y\to Z
\mathcal MUse in source · 11 passages
Search aliases: M mathcal
Use in source · Read passage · FOUNDATIONS.md, line 3320
Let M_A map symmetric matrices into a Euclidean product space by
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\mathcal M_A
mUse in source · 70 passages
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Use in source · Read passage · FOUNDATIONS.md, line 167
F(x)=b+∑_m=1^pB_m(x,…,x),
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F(x)=b+\sum_{m=1}^{p}B_m(x,\ldots,x),
m^2Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2239
det J=fracm^2-x^2-y^24.
Exact mathematical passage
\det\mathcal J=\frac{m^2-x^2-y^2}{4}.
m_{\mathcal F}Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 367
**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then
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m_{\mathcal F}=\mathbb E[Y\mid\mathcal F]
m_{\mathcal G}Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 367
**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then
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m_{\mathcal G}=\mathbb E[Y\mid\mathcal G]
M_0Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 3699
M_t= (fracdet(γ I)det V_t)^1/2 exp(frac S_t _V_t^-1^22σ^2), M_0=1.
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M_t= \left(\frac{\det(\gamma I)}{\det V_t}\right)^{1/2} \exp\left(\frac{\|S_t\|_{V_t^{-1}}^2}{2\sigma^2}\right), \qquad M_0=1.
m_0Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1871
An empty maximum is zero. If every negative entry has magnitude at most d and every reference entry is at least m_0>0, the simpler sufficient choice is η=d/(m_0+d). These statements follow by solving each scalar inequality. Mass conservation is unchanged by convex mixing.
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m_0>0
M_aUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 986
For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis qualification matters.
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M_a
\mathcal M_AUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3320
Let M_A map symmetric matrices into a Euclidean product space by
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\mathcal M_A
m_aUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 986
For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis qualification matters.
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m_a>0
M_{a,j}Use in source · 9 passages
Search aliases: M_a,j
Use in source · Read passage · FOUNDATIONS.md, line 2379
B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.
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B_a(u)=\sum_{k=0}^D u^k C_{a,k}, \qquad M_{a,j}=\int_{-1}^1 u^j B_a(u)\,\frac{du}{2}.
M_{a,o}Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1153
1^top∑_oM_a,o=1^top.
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\mathbf1^\top\sum_oM_{a,o}=\mathbf1^\top.
\mathcal M_{A^*}Use in source · 6 passages
Search aliases: M_A^* mathcal
Use in source · Read passage · FOUNDATIONS.md, line 3326
Use the Frobenius norm on symmetric matrices and the induced product norm. Assume M_A^* is injective and M_A^*(H^*)=(0,1)=b. This fixes the scale of a one-dimensional invariant-form space.
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\mathcal M_{A^*}
M_bUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 987
For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis qualification matters.
Exact mathematical passage
\|T_{ba}\|\leq M_b/m_a
M_sUse in source · 14 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1151
A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with
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M_s\in\mathbb R^{n\times n}
m_sUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
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\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
m_{s,j}Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3052
R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).
Exact mathematical passage
R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
M_{s_1}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1159
Put M_w=M_s_k⋯ M_s_1. Then
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M_w=M_{s_k}\cdots M_{s_1}
M_{s_k}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1159
Put M_w=M_s_k⋯ M_s_1. Then
Exact mathematical passage
M_w=M_{s_k}\cdots M_{s_1}
M_tUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 3699
M_t= (fracdet(γ I)det V_t)^1/2 exp(frac S_t _V_t^-1^22σ^2), M_0=1.
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M_t= \left(\frac{\det(\gamma I)}{\det V_t}\right)^{1/2} \exp\left(\frac{\|S_t\|_{V_t^{-1}}^2}{2\sigma^2}\right), \qquad M_0=1.
m_{t,j}Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3042
X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.
Exact mathematical passage
X_{t,j}=\phi_{t,j}(Y_t),\quad m_{t,j}=\mathbb E[X_{t,j}\mid\mathcal F_{t-1}],\quad p_{t,j}=\text{the model's prediction for that test}.
M_wUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1159
Put M_w=M_s_k⋯ M_s_1. Then
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M_w=M_{s_k}\cdots M_{s_1}
\mathcal M_{\widehat A}Use in source · 6 passages
Search aliases: M_widehat A mathcal widehat
Use in source · Read passage · FOUNDATIONS.md, line 3328
**Proposition 74. A posteriori proposal error.** Given widehat H, let r= M_widehat A(widehat H)-b. Suppose
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r=\mathcal M_{\widehat A}(\widehat H)-b
\mathscrUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1211
mathscr H_t,h=f(ht).
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\mathscr H_{t,h}=f(ht).
\maxUse in source · 9 passages
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Use in source · Read passage · FOUNDATIONS.md, line 841
sup_t≥0 z_t-widehat z_t ≤max(E_0,fracΔ1-a).
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\boxed{ \sup_{t\geq0}\|z_t-\widehat z_t\| \leq\max\left(E_0,\frac{\Delta}{1-a}\right). }
\max_aUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 3016
fracLsqrt3(n+2)+ max_afrac1n+1∑_k=0^nη_a,k.
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\boxed{ \frac{L}{\sqrt{3(n+2)}}+ \max_a\frac1{n+1}\sum_{k=0}^n\eta_{a,k}. }
\max_{s,i,j:(A_s)_{ij}<0}Use in source · 1 passage
Search aliases: max_s,i,j:(A_s)_ij<0 max
Use in source · Read passage · FOUNDATIONS.md, line 1866
η≥max_s,i,j:(A_s)_ij<0 frac-(A_s)_ij(N_s)_ij-(A_s)_ij.
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\eta\geq\max_{s,i,j:(A_s)_{ij}<0} \frac{-(A_s)_{ij}}{(N_s)_{ij}-(A_s)_{ij}}.
\max_{t-H\leq s\leq t}Use in source · 1 passage
Search aliases: max_t-H≤ s≤ t max leq leq
Use in source · Read passage · FOUNDATIONS.md, line 862
**Proof.** Subtract the two recurrences and bound every retained history error by the largest of the errors at indices t-H,…,t. This gives e_t+1≤ amax_t-H≤ s≤ te_s+Δ. The ball of radius E=max(E_0,Δ/(1-a)) is invariant because aE+Δ≤ E. Induction proves the bound. The estimate for Δ separates coefficient errors, the geometric kernel tail and the initialized forcing, using Proposition 10. square
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e_{t+1}\leq a\max_{t-H\leq s\leq t}e_s+\Delta
\minUse in source · 36 passages
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Use in source · Read passage · FOUNDATIONS.md, line 129
V(v)=min(c(v),∑_u∈ch(v)V(u))
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V(v)=\min\left(c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right)
\min_{\|v\|_2\leq1}Use in source · 1 passage
Search aliases: min_ v _2≤1 min leq
Use in source · Read passage · FOUNDATIONS.md, line 3212
σ_a(u)=min_ v _2≤1 [(F_a(u)(1,v))_0- (F_a(u)(1,v))_rm sp _2].
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\sigma_a(u)=\min_{\|v\|_2\leq1} \left[(F_a(u)(1,v))_0-\|(F_a(u)(1,v))_{\rm sp}\|_2\right].
\min_{a,b\geq0}Use in source · 1 passage
Search aliases: min_a,b≥0 min geq
Use in source · Read passage · FOUNDATIONS.md, line 1912
x _C=min_a,b≥0 {1^top(a+b):V(a-b)=x}.
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\|x\|_C=\min_{a,b\geq0} \{\mathbf1^\top(a+b):V(a-b)=x\}.
NUse in source · 21 passages
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Use in source · Read passage · FOUNDATIONS.md, line 598
Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.
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Z_-,Z_+\in\mathbb R^{m\times N}
nUse in source · 101 passages
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Use in source · Read passage · FOUNDATIONS.md, line 243
W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.
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W_{n+1}=W_n+\sum_uW_nL_u, \qquad W_nL_u=\{wL_u:w\in W_n\}.
n^2Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2498
frac34n^2
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\boxed{\frac{3}{4n^2}}
N_aUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3266
N_a(u)=g_a(u)e_0λ, g_a(u)≥ g_min>0, ∫ g_a(u) du/2=1.
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N_a(u)=g_a(u)e_0\lambda,\qquad g_a(u)\geq g_{\min}>0,\qquad \int g_a(u)\,du/2=1.
N_sUse in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1858
When V is square and invertible, put A_s=V^-1widehat B_sV. Assume mass conservation has already been imposed. Let N_s be an entrywise positive reference instrument with the same per-action mass identities. For 0≤η≤1, the mixture
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N_s
OUse in source · 25 passages
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Use in source · Read passage · FOUNDATIONS.md, line 566
O_h=C CL vdots CL^h-1, V_h=imO_h.
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O_h=\begin{pmatrix}C\\CL\\\vdots\\CL^{h-1}\end{pmatrix}, \qquad V_h=\operatorname{im}O_h.
\mathcal OUse in source · 3 passages
Search aliases: O mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1131
Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.
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\mathcal O
oUse in source · 29 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1131
Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.
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s=(a,o)
O^{-1}Use in source · 1 passage
Search aliases: O^-1
Use in source · atlas.tex, line 132
For the running example, the rows of O=1 · 0 1/2 · 1, D=OLO^-1=0 · 1 1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) only after a one-step warm-up. Known-operator completion does not establish sensor…
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O=\begin{pmatrix}1&0\\1/2&1\end{pmatrix},\qquad D=OLO^{-1}=\begin{pmatrix}0&1\\1/12&5/6\end{pmatrix}
O_0Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1589
O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.
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O_0=\operatorname{diag}(1/4,3/4),\qquad O_1=\operatorname{diag}(3/4,1/4),\qquad M_{a,o}=O_oT_a.
O_1Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1589
O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.
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O_0=\operatorname{diag}(1/4,3/4),\qquad O_1=\operatorname{diag}(3/4,1/4),\qquad M_{a,o}=O_oT_a.
o_1Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1135
f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).
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f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
O_2Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 689
O_2=1 · 0 · 0 1/2 · 1 · 0
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O_2=\begin{pmatrix}1&0&0\\1/2&1&0\end{pmatrix}
O_3Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 693
has rank two and O_3 has the same rank. Proposition 14 gives an exact two-dimensional delay state. Choose x_0=(1,0,q)^top for any q. The first four observations are
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O_3
O_dUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 588
is therefore equivalent to CL^h being in the row space of O_h, which is exactly the rank equality. It also implies O_hLx∈ V_h because this vector is O_h applied to Lx. Uniqueness follows because every element of V_h has a representative. The history recurrence follows directly. Cayley-Hamilton expresses L^d as a linear combination of lower powers, so CL^d is in the row space of O_d. square
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O_d
O_hUse in source · 15 passages
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Use in source · Read passage · FOUNDATIONS.md, line 566
O_h=C CL vdots CL^h-1, V_h=imO_h.
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O_h=\begin{pmatrix}C\\CL\\\vdots\\CL^{h-1}\end{pmatrix}, \qquad V_h=\operatorname{im}O_h.
O_{h+1}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 584
1. There is a linear map D_h:V_h→ V_h satisfying O_hL=D_hO_h. 2. ker O_h⊆ker(O_hL). 3. rankO_h+1=rankO_h.
Exact mathematical passage
\operatorname{rank}O_{h+1}=\operatorname{rank}O_h
o_kUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1135
f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).
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f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
O_oUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1589
O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.
Exact mathematical passage
O_0=\operatorname{diag}(1/4,3/4),\qquad O_1=\operatorname{diag}(3/4,1/4),\qquad M_{a,o}=O_oT_a.
o_TUse in source · 2 passages
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Use in source · atlas.tex, line 266
g is geometricAn operator family needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0. Summing over every branch repeatedly applies the mass identity and gives total probability one. A positive-probability posterior is B_oz/(λ B_oz).
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o_1\cdots o_T
\odotUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 282
B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.
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\mathcal B_{\ell_y}(p) =\frac{p\odot\ell_y}{\ell_y^\top p}.
\overline LUse in source · 5 passages
Search aliases: overline L overline
Use in source · Read passage · FOUNDATIONS.md, line 952
overline L=∑_aw_aL_a, widetilde G_a=A_aoverline LS_a on C_a=imA_a.
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\overline L=\sum_aw_aL_a, \qquad\widetilde G_a=A_a\overline LS_a \quad\text{on }\mathcal C_a=\operatorname{im}A_a.
\mathrm pUse in source · 2 passages
Search aliases: p mathrm
Use in source · Read passage · FOUNDATIONS.md, line 2041
B_ p,0=z_0(1/2,1/2,0), B_ p,1=z_0(1/2,-1/2,0).
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B_{\mathrm p,0}=z_0(1/2,1/2,0),\qquad B_{\mathrm p,1}=z_0(1/2,-1/2,0).
p^+Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1161
f(w)=1^top M_wp_0, p^+=fracM_sp1^top M_sp
Exact mathematical passage
f(w)=\mathbf1^\top M_wp_0,\qquad p^+=\frac{M_sp}{\mathbf1^\top M_sp}
P^2Use in source · 1 passage
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Use in source · atlas.tex, line 159
plate6Compression can erase a return pathA two-step effect can survive when both compressed steps vanish.P7 Let P^2=P and Q=I-P. Inserting I=P+Q gives the exact defect PLMP-(PLP)(PMP)=PLQMP. diagram06_return_path Take P=diag(1,0) and L=M=0 · 1
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P^2=P
p^{[1]}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2991
p^[1](v)=v+frac12.
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p^{[1]}(v)=v+\frac12.
P^\topUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 402
C_π B_ℓ(p)-bar b = frac1Z_r(diagP-bar bP^top)δ.
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\boxed{ C_\pi\mathcal B_{\ell}(p)-\bar b = \frac{1}{Z_r}\left(\operatorname{diag}P-\bar bP^\top\right)\delta. }
P_{\mathrm d}Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1332
P_ d=frac121 · -1 -1 · 1
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P_{\mathrm d}=\frac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}
p_+Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2443
p_+μ-p_-μ _TV=frac13ϵ40.
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\boxed{\|p_+\mu-p_-\mu\|_{\mathrm{TV}}=\frac{13\epsilon}{40}.}
p_-Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2443
p_+μ-p_-μ _TV=frac13ϵ40.
Exact mathematical passage
\boxed{\|p_+\mu-p_-\mu\|_{\mathrm{TV}}=\frac{13\epsilon}{40}.}
p_0Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1161
f(w)=1^top M_wp_0, p^+=fracM_sp1^top M_sp
Exact mathematical passage
f(w)=\mathbf1^\top M_wp_0,\qquad p^+=\frac{M_sp}{\mathbf1^\top M_sp}
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · Read passage · FOUNDATIONS.md, line 1314
Let P_1,…,P_m be a complete family of linear packet maps on a predictive space, with ∑_iP_i=I. Algebraic reconstruction gives z=∑_iP_iz. It does not imply that P_iz is an unnormalized probability state.
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P_1,\ldots,P_m
p_1Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2681
p_1=frac1316, z_1=(1,12/13,0)^ T.
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p_1=\frac{13}{16},\qquad z_1=(1,12/13,0)^{\mathsf T}.
p_2Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2687
p_2=frac4131040, z_2=(1,-312/413,-240/413)^ T.
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p_2=\frac{413}{1040},\qquad z_2=(1,-312/413,-240/413)^{\mathsf T}.
P_aUse in source · 2 passages
Search aliases: P_a
Use in source · Read passage · FOUNDATIONS.md, line 386
P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).
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P_a=\sum_{i\in B_a}p_i,\qquad r_a(p)=\frac{\sum_{i\in B_a}p_i\ell_i}{P_a} \quad(P_a>0).
p_{a,b}Use in source · 4 passages
Search aliases: p_a,b
Use in source · Read passage · FOUNDATIONS.md, line 3800
p_a,b(umid x,y)=1+a x u+b yL_2(u)≥ρ.
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p_{a,b}(u\mid x,y)=1+a x u+b yL_2(u)\geq\rho.
P_iUse in source · 20 passages
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Use in source · Read passage · FOUNDATIONS.md, line 106
P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.
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P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i, \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
p_iUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 384
This section imports a specific existing Komorebi foundation rather than inventing another inference heuristic. Let p_i be a prior on a finite fine state set and let B_a be coarse blocks. Define
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p_i
P_{i,r}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 145
If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then
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P_i=\sum_rP_{i,r}
P_i^*Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 106
P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.
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P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i, \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
P_{i_1}Use in source · 3 passages
Search aliases: P_i_1
Use in source · Read passage · FOUNDATIONS.md, line 173
Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).
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Q_jF(x)=Q_jb+ \sum_{m=1}^{p}\ \sum_{i_1,\ldots,i_m} Q_jB_m(P_{i_1}x,\ldots,P_{i_m}x).
P_{i_m}Use in source · 3 passages
Search aliases: P_i_m
Use in source · Read passage · FOUNDATIONS.md, line 173
Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).
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Q_jF(x)=Q_jb+ \sum_{m=1}^{p}\ \sum_{i_1,\ldots,i_m} Q_jB_m(P_{i_1}x,\ldots,P_{i_m}x).
P_jUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 106
P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.
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P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i, \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
p_jUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2463
**Proposition 56. Exact bin coarsening and its posterior.** Every B_a,j preserves the probability cone, and λ∑_jB_a,j=λ. For a policy measurable with respect to bin histories, the finite model {B_a,j} reproduces exactly the binned observation law of the continuous model at every finite horizon. On an observed cell with p_j=λ B_a,jz>0,
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p_j=\lambda B_{a,j}z>0
P_mUse in source · 1 passage
Search aliases: P_m
Use in source · Read passage · FOUNDATIONS.md, line 1314
Let P_1,…,P_m be a complete family of linear packet maps on a predictive space, with ∑_iP_i=I. Algebraic reconstruction gives z=∑_iP_iz. It does not imply that P_iz is an unnormalized probability state.
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P_1,\ldots,P_m
P_nUse in source · 13 passages
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Use in source · Read passage · FOUNDATIONS.md, line 447
**Proposition 13. Strong fixed-horizon refinement limit.** Let X be a Hilbert space, L:X→ X bounded, and P_n orthogonal projections converging strongly to I. Put L_n=P_nLP_n. For every fixed x∈ X and every fixed nonnegative integer k,
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P_n
P_{s,j}Use in source · 2 passages
Search aliases: P_s,j
Use in source · Read passage · FOUNDATIONS.md, line 3117
frac1t∑_s=1^t P_s,j-Q_s,j _TV ≥frac(|R_t,j|-b_t,j)_+t.
Exact mathematical passage
\boxed{ \frac1t\sum_{s=1}^t\|P_{s,j}-Q_{s,j}\|_{\mathrm{TV}} \geq\frac{(|R_{t,j}|-b_{t,j})_+}{t}. }
p_{s,j}Use in source · 2 passages
Search aliases: p_s,j
Use in source · Read passage · FOUNDATIONS.md, line 3052
R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).
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R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
P_TUse in source · 9 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1925
**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy
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P_T,\widetilde P_T
p_{t,j}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3042
X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.
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X_{t,j}=\phi_{t,j}(Y_t),\quad m_{t,j}=\mathbb E[X_{t,j}\mid\mathcal F_{t-1}],\quad p_{t,j}=\text{the model's prediction for that test}.
P_T^{(\alpha)}Use in source · 1 passage
Search aliases: P_T^(α) alpha
Use in source · Read passage · FOUNDATIONS.md, line 2099
P_T-P_T^(α) _TV ≤min(1,T(1-α)/2).
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\boxed{\|P_T-P_T^{(\alpha)}\|_{\mathrm{TV}} \leq\min(1,T(1-\alpha)/2).}
P_vUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 117
∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.
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\sum_{v\in\Lambda}P_v=I,\qquad x=\sum_{v\in\Lambda}P_vx,\qquad \|x\|^2=\sum_{v\in\Lambda}\|P_vx\|^2.
p_\pmUse in source · 1 passage
Search aliases: p_± pm
Use in source · Read passage · FOUNDATIONS.md, line 2436
L_3(u)=frac5u^3-3u2, p_±(u)=1±ϵ L_3(u), 0<ϵ<1.
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L_3(u)=\frac{5u^3-3u}{2},\qquad p_\pm(u)=1\pm\epsilon L_3(u),\qquad 0<\epsilon<1.
\plateUse in source · 1 passage
Search aliases: plate
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
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5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
\PrUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1135
f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).
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f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
\Pr_\piUse in source · 1 passage
Search aliases: Pr_π Pr pi
Use in source · Read passage · FOUNDATIONS.md, line 1352
Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),
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\Pr_\pi(w)=f(w)\prod_{j=1}^{|w|}\pi(a_j\mid h_{j-1}),
\preceqUse in source · 1 passage
Search aliases: preceq
Use in source · Read passage · FOUNDATIONS.md, line 793
widehat B_u^top Mwidehat B_upreceqρ^2M for every declared action u.
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\widehat B_u^\top M\widehat B_u\preceq\rho^2M \quad\text{for every declared action }u.
qUse in source · 46 passages
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Use in source · Read passage · FOUNDATIONS.md, line 320
z_t=Px_t, q_t=Qx_t,
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z_t=Px_t,\qquad q_t=Qx_t,
q^{\mathsf T}Use in source · 1 passage
Search aliases: q^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3692
exp(q^ TS_t/σ -tfrac12q^ T(V_t-γ I)q)
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\exp\left(q^{\mathsf T}S_t/\sigma -\tfrac12q^{\mathsf T}(V_t-\gamma I)q\right)
q^+Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1633
p(1mid z,0)=q, q^+=frac-3/32+(3/4)qq =frac34-frac332q.
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p(1\mid z,0)=q,\qquad q^+=\frac{-3/32+(3/4)q}{q} =\frac34-\frac{3}{32q}.
Q^{-1}Use in source · 1 passage
Search aliases: Q^-1
Use in source · Read passage · FOUNDATIONS.md, line 2675
Proposition 54 reconstructs QB_a(u)Q^-1 from H and the six inserted-moment tables. The code that performs this reconstruction imports no generating matrices. The separate generator produces the tables and is used only in the verification suite. These are population values from an exact example, not learned sample estimates.
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QB_a(u)Q^{-1}
q_0Use in source · 11 passages
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Use in source · Read passage · FOUNDATIONS.md, line 334
z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.
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\boxed{ z_{t+1}=Az_t+ \sum_{k=0}^{t-1}BD^kCz_{t-1-k}+BD^tq_0. }
q_1Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1639
Starting at q_0=13/24, this gives probability 13/24 and q_1=15/26. If action one then produces observation zero, its probability is
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q_1=15/26
q_2Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1647
q_2=frac23/192+(5/24)(15/26)365/624=frac5991460.
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q_2=\frac{23/192+(5/24)(15/26)}{365/624}=\frac{599}{1460}.
Q_jUse in source · 10 passages
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Use in source · Read passage · FOUNDATIONS.md, line 137
Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.
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Q_j
Q_{j,s}Use in source · 2 passages
Search aliases: Q_j,s
Use in source · Read passage · FOUNDATIONS.md, line 145
If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then
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Q_j=\sum_sQ_{j,s}
Q_j^2Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 157
**Proof.** Insert the appropriate identity decomposition on the input, output or intermediate space. Use Q_j^2=Q_j in the composition formula. square
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Q_j^2=Q_j
Q_{s,j}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3117
frac1t∑_s=1^t P_s,j-Q_s,j _TV ≥frac(|R_t,j|-b_t,j)_+t.
Exact mathematical passage
\boxed{ \frac1t\sum_{s=1}^t\|P_{s,j}-Q_{s,j}\|_{\mathrm{TV}} \geq\frac{(|R_{t,j}|-b_{t,j})_+}{t}. }
Q_TUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 3154
P_T-Q_T _TV ≤min(1,∑_t=0^T-1 E_P,π[d_t(H_t,A_t)]).
Exact mathematical passage
\boxed{ \|P_T-Q_T\|_{\mathrm{TV}} \leq\min\left(1,\sum_{t=0}^{T-1} \mathbb E_{P,\pi}[d_t(H_t,A_t)]\right). }
q_tUse in source · 7 passages
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Use in source · Read passage · FOUNDATIONS.md, line 320
z_t=Px_t, q_t=Qx_t,
Exact mathematical passage
z_t=Px_t,\qquad q_t=Qx_t,
q_{t+1}Use in source · 2 passages
Search aliases: q_t+1
Use in source · Read passage · FOUNDATIONS.md, line 330
Thus z_t+1=Az_t+Bq_t and q_t+1=Cz_t+Dq_t.
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q_{t+1}=Cz_t+Dq_t
RUse in source · 64 passages
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Use in source · Read passage · FOUNDATIONS.md, line 151
For another linear map M:Y→ Z and output projectors R_k,
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R_k
\RUse in source · 4 passages
Search aliases: R mathbb
Use in source · atlas.tex, line 77
able coefficients form a subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A(x_1,x_2)=(x_1,x_2,x_1+x_2), S(c_1,c_2,c_3)=(c_1,c_2). Then Π(c_1,c_2,c_3)=(c_1,c_2,c_1+c_2). The shaded plane is the set of realizable coefficients. …
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X=\R^2
\mathbb RUse in source · 24 passages
Search aliases: R mathbb
Use in source · Read passage · FOUNDATIONS.md, line 558
Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:
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x_t\in\mathbb R^d
\mathrm rUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2035
B_ r,0=T, B_ r,1=0.
Exact mathematical passage
B_{\mathrm r,0}=T,\qquad B_{\mathrm r,1}=0.
rUse in source · 81 passages
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Use in source · Read passage · FOUNDATIONS.md, line 82
**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square
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r
\R^2Use in source · 1 passage
Search aliases: R^2 R mathbb
Use in source · atlas.tex, line 77
able coefficients form a subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A(x_1,x_2)=(x_1,x_2,x_1+x_2), S(c_1,c_2,c_3)=(c_1,c_2). Then Π(c_1,c_2,c_3)=(c_1,c_2,c_1+c_2). The shaded plane is the set of realizable coefficients. …
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X=\R^2
\R^{2\,*}Use in source · 1 passage
Search aliases: R^2 * R mathbb
Use in source · atlas.tex, line 136
1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) only after a one-step warm-up. Known-operator completion does not establish sensor access to arbitrary new observables. navInvariantCompletion.lean; Pr…
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W_*=\R^{2\,*}
\mathbb R^2_+Use in source · 1 passage
Search aliases: R^2_+ mathbb
Use in source · Read passage · FOUNDATIONS.md, line 1623
The coefficients include negatives. Nevertheless, every operator preserves the cone R R^2_+, because B_sR=RM_s. A sufficient normalized base is
Exact mathematical passage
R\mathbb R^2_+
\R^3Use in source · 1 passage
Search aliases: R^3 R mathbb
Use in source · atlas.tex, line 77
icients form a subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A(x_1,x_2)=(x_1,x_2,x_1+x_2), S(c_1,c_2,c_3)=(c_1,c_2). Then Π(c_1,c_2,c_3)=(c_1,c_2,c_1+c_2). The shaded plane is the set of realizable coefficients. The red se…
Exact mathematical passage
C=\R^3
\R^dUse in source · 1 passage
Search aliases: R^d R mathbb
Use in source · atlas.tex, line 261
plate11Probability meaning is geometricAn operator family needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0. Summing over every branch repeatedly applies the mass identity and gives total probability one. A positive-probability post…
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C\subseteq\R^d
\mathbb R^dUse in source · 4 passages
Search aliases: R^d mathbb
Use in source · Read passage · FOUNDATIONS.md, line 558
Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:
Exact mathematical passage
x_t\in\mathbb R^d
R^kUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2062
t± e_1^top R^k(x,y)^top≥0 for every k≥0.
Exact mathematical passage
t\pm e_1^\top R^k(x,y)^\top\geq0 \quad\text{for every }k\geq0.
R^{m-1}Use in source · 1 passage
Search aliases: R^m-1
Use in source · Read passage · FOUNDATIONS.md, line 916
Lip(F_ I) ≤∑_m=1^p ∑_ i∈ I_m mβ_m, iR^m-1.
Exact mathematical passage
\operatorname{Lip}(F_{\mathcal I}) \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\in\mathcal I_m} m\beta_{m,\boldsymbol i}R^{m-1}.
\mathbb R^{m\times m}Use in source · 1 passage
Search aliases: R^m× m mathbb times
Use in source · Read passage · FOUNDATIONS.md, line 1758
**Proposition 39. Finite positive-lift certificate.** This cone is invariant under B_s if and only if there is an entrywise nonnegative matrix A_s∈ R^m× m such that
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A_s\in\mathbb R^{m\times m}
\mathbb R^{m\times N}Use in source · 1 passage
Search aliases: R^m× N mathbb times
Use in source · Read passage · FOUNDATIONS.md, line 598
Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.
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Z_-,Z_+\in\mathbb R^{m\times N}
\mathbb R^{m\times n}Use in source · 1 passage
Search aliases: R^m× n mathbb times
Use in source · Read passage · FOUNDATIONS.md, line 1362
Let X∈ R^m× n have rank r≥1, let s=σ_r(X)>0, and observe Y=X+E with E _2≤ε<s/2. Let P project onto imX, and let widehat P project onto the leading r left singular vectors of Y.
Exact mathematical passage
X\in\mathbb R^{m\times n}
\mathbb R^nUse in source · 1 passage
Search aliases: R^n mathbb
Use in source · Read passage · FOUNDATIONS.md, line 1195
st, so the stable span is exactly the displayed word span. Invariance supplies the matrices B_s, and full row rank makes them unique. Apply R to the unnormalized and normalized hidden-state updates. If all test probabilities are linear functions of another summary, then each row 1^top M_w lies in its row space. Equality on all beliefs implies equality on their linear span, which is R^n. square
Exact mathematical passage
\mathbb R^n
\mathbb R^n_+Use in source · 3 passages
Search aliases: R^n_+ mathbb
Use in source · Read passage · FOUNDATIONS.md, line 1151
A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with
Exact mathematical passage
p\in\mathbb R^n_+
\mathbb R^{n\times n}Use in source · 1 passage
Search aliases: R^n× n mathbb times
Use in source · Read passage · FOUNDATIONS.md, line 1151
A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with
Exact mathematical passage
M_s\in\mathbb R^{n\times n}
\mathbb R^pUse in source · 1 passage
Search aliases: R^p mathbb
Use in source · Read passage · FOUNDATIONS.md, line 558
Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:
Exact mathematical passage
y_t\in\mathbb R^p
\mathbb R^rUse in source · 4 passages
Search aliases: R^r mathbb
Use in source · Read passage · FOUNDATIONS.md, line 656
**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. square
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v\in\mathbb R^r
\mathbb R^{r\times m}Use in source · 1 passage
Search aliases: R^r× m mathbb times
Use in source · Read passage · FOUNDATIONS.md, line 1750
Let V=[v_1 ⋯ v_m]∈ R^r× m have full row rank and satisfy
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V=[v_1\ \cdots\ v_m]\in\mathbb R^{r\times m}
\mathbb R^{r\times N}Use in source · 1 passage
Search aliases: R^r× N mathbb times
Use in source · Read passage · FOUNDATIONS.md, line 628
Work in a fixed r-dimensional coordinate system, where Z_-∈ R^r× N has full row rank. Let
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Z_-\in\mathbb R^{r\times N}
\mathbb R_+^mUse in source · 3 passages
Search aliases: R_+^m mathbb
Use in source · Read passage · FOUNDATIONS.md, line 1756
Its columns generate C=V R_+^m and its normalized base is K=conv{v_1,…,v_m}. Finite generation makes the cone closed. Its normalized generators and full row rank make it pointed and generating, with a strictly positive mass functional.
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C=V\mathbb R_+^m
R_aUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2800
**Proposition 62. Oriented quadratic probability certificate.** Suppose that for each action and every u∈[-1,1] there are τ_a(u)≥0 and R_a(u)succeq0 satisfying
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R_a(u)\succeq0
r_aUse in source · 2 passages
Search aliases: r_a
Use in source · Read passage · FOUNDATIONS.md, line 386
P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).
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P_a=\sum_{i\in B_a}p_i,\qquad r_a(p)=\frac{\sum_{i\in B_a}p_i\ell_i}{P_a} \quad(P_a>0).
R_kUse in source · 2 passages
Search aliases: R_k
Use in source · Read passage · FOUNDATIONS.md, line 151
For another linear map M:Y→ Z and output projectors R_k,
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R_k
r_kUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3424
Δ_a(u)=∑_k=0^Du^kE_a,k, |(E_a,k)_ij|≤ r_k.
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\Delta_a(u)=\sum_{k=0}^{D}u^kE_{a,k},\qquad |(E_{a,k})_{ij}|\leq r_k.
r_sUse in source · 1 passage
Search aliases: r_s
Use in source · Read passage · FOUNDATIONS.md, line 3385
d(t)=1+t^2, r_s(t)=fracsd(t)(1-t^2,2t).
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d(t)=1+t^2,\qquad r_s(t)=\frac{s}{d(t)}(1-t^2,2t).
r_tUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 740
r_t(x)=γ_t+1(F_t(x))-widehat f_t(γ_t(x)).
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r_t(x)=\gamma_{t+1}(F_t(x))-\widehat f_t(\gamma_t(x)).
R_{t,j}Use in source · 4 passages
Search aliases: R_t,j
Use in source · Read passage · FOUNDATIONS.md, line 3052
R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).
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R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
\operatorname{rank}Use in source · 3 passages
Search aliases: rank operatorname
Use in source · Read passage · FOUNDATIONS.md, line 584
1. There is a linear map D_h:V_h→ V_h satisfying O_hL=D_hO_h. 2. ker O_h⊆ker(O_hL). 3. rankO_h+1=rankO_h.
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\operatorname{rank}O_{h+1}=\operatorname{rank}O_h
\rceilUse in source · 2 passages
Search aliases: rceil
Use in source · Read passage · FOUNDATIONS.md, line 3854
**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of
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k=\lceil\log_2 V\rceil
\mathrm{repair}Use in source · 1 passage
Search aliases: repair mathrm
Use in source · Read passage · FOUNDATIONS.md, line 3637
ε_repair ≤frac3132frac24140000+frac116 =frac874711280000.
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\varepsilon_{\mathrm{repair}} \leq\frac{31}{32}\frac{241}{40000}+\frac1{16} =\frac{87471}{1280000}.
\operatorname{row}Use in source · 2 passages
Search aliases: row operatorname
Use in source · Read passage · FOUNDATIONS.md, line 241
Let the initial summary be C_0x. Regard its rows as linear observables. Starting from W_0=row(C_0), define
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W_0=\operatorname{row}(C_0)
\mathbb SUse in source · 3 passages
Search aliases: S mathbb
Use in source · Read passage · FOUNDATIONS.md, line 2181
Let E= R^r and let S^d be the real symmetric d× d matrices. Fix an injective linear map
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\mathbb S^d
\mathcal SUse in source · 3 passages
Search aliases: S mathcal
Use in source · Read passage · FOUNDATIONS.md, line 51
Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.
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\mathcal Ax=(w_aU_ax)_a,\qquad \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a, \qquad c=\sum_aw_a^2.
s^2Use in source · 2 passages
Search aliases: s^2
Use in source · Read passage · FOUNDATIONS.md, line 3806
p_a,b(umid s)=1+a s u+b s^2L_2(u).
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p_{a,b}(u\mid s)=1+a s u+b s^2L_2(u).
\mathbb S^dUse in source · 3 passages
Search aliases: S^d mathbb
Use in source · Read passage · FOUNDATIONS.md, line 2181
Let E= R^r and let S^d be the real symmetric d× d matrices. Fix an injective linear map
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\mathbb S^d
s_1Use in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1131
Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.
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w=s_1\cdots s_k
S_aUse in source · 26 passages
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Use in source · Read passage · FOUNDATIONS.md, line 27
A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.
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A_a:X\longrightarrow C_a,\qquad S_a:C_a\longrightarrow X, \qquad S_aA_a=I_X.
S_bUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 82
**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square
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S_bA_b=I
S_cUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2786
S_c=1 · 0 -c · I, Q=S_c^ T1 · 0 0 · -HS_c.
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S_c=\begin{pmatrix}1&0\\-c&I\end{pmatrix},\qquad Q=S_c^{\mathsf T}\begin{pmatrix}1&0\\0&-H\end{pmatrix}S_c.
S_c^{\mathsf T}Use in source · 1 passage
Search aliases: S_c^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2786
S_c=1 · 0 -c · I, Q=S_c^ T1 · 0 0 · -HS_c.
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S_c=\begin{pmatrix}1&0\\-c&I\end{pmatrix},\qquad Q=S_c^{\mathsf T}\begin{pmatrix}1&0\\0&-H\end{pmatrix}S_c.
s_HUse in source · 6 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1408
Let G_s=B_sH with s_H=σ_min(H)>0. Observe
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s_H=\sigma_{\min}(H)>0
s_kUse in source · 4 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1131
Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.
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w=s_1\cdots s_k
S_rUse in source · 2 passages
Search aliases: S_r
Use in source · Read passage · FOUNDATIONS.md, line 82
**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square
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F=S_rG_rA_r
S_tUse in source · 5 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3690
**Proof.** Put S_t=∑_s≤ tϕ_sξ_s. For every fixed q, the conditional moment bound makes
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S_t=\sum_{s\leq t}\phi_s\xi_s
\operatorname{span}Use in source · 3 passages
Search aliases: span operatorname
Use in source · Read passage · FOUNDATIONS.md, line 1168
**Proposition 26. Finite instrument closure.** Starting with W_0=span{1^top}, define
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W_0=\operatorname{span}\{\mathbf1^\top\}
\squareUse in source · 87 passages
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Use in source · Read passage · FOUNDATIONS.md, line 43
**Proof.** If A_ax=0, then x=S_aA_ax=0. Also Π_a^2=A_a(S_aA_a)S_a=Π_a. Its image is contained in C_a and it fixes every A_ax, so its image is exactly C_a. Both norm inequalities follow from the operator-norm inequality applied to S_a. square
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\square
\subsetneqUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 616
essity follows by applying B to every linear dependence among the input columns. Under the kernel condition, define B(Z_-a)=Z_+a. Two choices of a differ by a kernel vector and give the same answer. This defines and uniquely determines the map on W. It maps W into W exactly under the image condition. The pseudoinverse formula gives that map because a-Z_-^† Z_-a∈ker Z_-. If Wsubsetneq V, choose a nonzero linear functional φ on V vanishing on W and a nonzero v∈ V. The operators B and B+vφ agree on…
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W\subsetneq V
\succUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2188
Suppose some e∈ E satisfies J(e)succ0. Define
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\mathcal J(e)\succ0
\succeqUse in source · 9 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2190
C_ J={z: J(z)succeq0}, K_ J={z∈ C_ J:λ z=1}.
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C_{\mathcal J}=\{z:\mathcal J(z)\succeq0\}, \qquad K_{\mathcal J}=\{z\in C_{\mathcal J}:\lambda z=1\}.
\supUse in source · 14 passages
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Use in source · Read passage · FOUNDATIONS.md, line 834
a= widehat A +∑_k=0^H-1 widehat K_k <1, sup_t≥0 d_t ≤Δ.
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a=\|\widehat A\|+\sum_{k=0}^{H-1}\|\widehat K_k\|<1, \qquad\sup_{t\geq0}\|d_t\|\leq\Delta.
\sup_{\|v'\|_2\leq1}Use in source · 1 passage
Search aliases: sup_ v' _2≤1 sup leq
Use in source · Read passage · FOUNDATIONS.md, line 3247
sup_ v' _2≤1 Δ(1,v') _ L_d=ϵ
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\sup_{\|v'\|_2\leq1}\|\Delta(1,v')\|_{\mathcal L_d}=\epsilon
\sup_{(a,b)\in\Theta}Use in source · 1 passage
Search aliases: sup_(a,b)∈Θ sup in Theta
Use in source · atlas.tex, line 335
eta_t=V_t^-1∑_s≤ tϕ_sY_s. For 0<δ<1, with probability at least 1-δ, simultaneously for every finite t, widehatθ_t-θ_* _V_t ≤σsqrtlogfracdet V_tdet(η I)+2log(1/δ)+sqrtη S. diagram14_uncertainty In the structured moment family, a nonempty returned rectangle Θ is internally valid when its sufficient floor ρ_Θ=1-sup_(a,b)∈Θ(|a|+|b|) is nonnegative. Statistical coverage places the true parameter in Θ on a simultaneous event. The exact domain argument applies to every parameter in Θ. claimA data-selec…
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\rho_\Theta=1-\sup_{(a,b)\in\Theta}(|a|+|b|)
\sup_{a,z\in K}Use in source · 3 passages
Search aliases: sup_a,z∈ K sup in
Use in source · Read passage · FOUNDATIONS.md, line 2522
ϵ_Π:=sup_a,z∈ K∫ (B_a(y)-B_a^Π(y))z _C μ(dy) ≤ Lh.
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\epsilon_\Pi:=\sup_{a,z\in K}\int \|(B_a(y)-B_a^\Pi(y))z\|_C\,\mu(dy) \leq Lh.
\sup_sUse in source · 2 passages
Search aliases: sup_s sup
Use in source · Read passage · FOUNDATIONS.md, line 3944
Let P,Q be Markov kernels on a common state space. Suppose P(s,·)≥ρμ(·) for every s, where μ is one probability measure and 0<ρ≤1. Suppose also sup_s P(s,·)-Q(s,·) _TV≤Δ.
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\sup_s\|P(s,\cdot)-Q(s,\cdot)\|_{\mathrm{TV}}\leq\Delta
\sup_tUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 875
a_H=frac78-frac38 3^-H, Δ_H=frac38 3^-H, sup_t|z_t-widehat z_t|≤ frac3 3^-H1+3 3^-H.
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a_H=\frac78-\frac38\,3^{-H},\qquad \Delta_H=\frac38\,3^{-H},\qquad \boxed{\sup_t|z_t-\widehat z_t|\leq \frac{3\,3^{-H}}{1+3\,3^{-H}}.}
\sup_{t\geq0}Use in source · 3 passages
Search aliases: sup_t≥0 sup geq
Use in source · Read passage · FOUNDATIONS.md, line 834
a= widehat A +∑_k=0^H-1 widehat K_k <1, sup_t≥0 d_t ≤Δ.
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a=\|\widehat A\|+\sum_{k=0}^{H-1}\|\widehat K_k\|<1, \qquad\sup_{t\geq0}\|d_t\|\leq\Delta.
\sup_{z\in K}Use in source · 2 passages
Search aliases: sup_z∈ K sup in
Use in source · Read passage · FOUNDATIONS.md, line 1893
ϵ_a=sup_z∈ K∑_o (B_a,o-widetilde B_a,o)z _C.
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\epsilon_a=\sup_{z\in K}\sum_o \|(B_{a,o}-\widetilde B_{a,o})z\|_C.
\mathsf TUse in source · 57 passages
Search aliases: T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
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\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
T^2Use in source · 1 passage
Search aliases: T^2
Use in source · Read passage · FOUNDATIONS.md, line 2071
Finally the reachable columns z_0,Tz_0,T^2z_0 are independent. The mass row and probe rows before zero and one rotations span the dual space because the rotation has nonzero sine. Reachability and observability establish predictive dimension three. square
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z_0,Tz_0,T^2z_0
t^2Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1525
**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the same bound. A union bound over the M words proves the simultaneous result. The spectral norm is bounded by the Frobenius norm, yielding the matrix an…
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g(t)\leq t^2/8
T^kUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 2069
Any invariant cone with nonnegative mass lies inside this canonical cone, by Proposition 38. If it is closed, convex and contains z_0, it contains the orbit T^kz_0 and the closed convex hull of that orbit. On t=1, this hull is the full unit disk. Conical scaling gives all of C_ L. The unit disk has infinitely many extreme points, so it is not the convex hull of finitely many points.
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T^kz_0
T_*Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3455
**Proposition 77. Conditional search for strict rational curved certificates.** Consider finitely many rational polynomial operator families in three coordinates, each with exact mass, and finitely many rational normalized initial states. Suppose there exists a real mass preserving invertible T_* such that every transformed initial state lies strictly inside the Lorentz cone and every transformed operator family has a common positive margin on the observation interval. Then a rational mass prese…
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T_*
T_0Use in source · 1 passage
Search aliases: T_0
Use in source · Read passage · FOUNDATIONS.md, line 1581
T_0=3/4 · 1/4 1/4 · 3/4, T_1=1/4 · 2/3 3/4 · 1/3, p_0=2/3 1/3.
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T_0=\begin{pmatrix}3/4&1/4\\1/4&3/4\end{pmatrix},\qquad T_1=\begin{pmatrix}1/4&2/3\\3/4&1/3\end{pmatrix},\qquad p_0=\begin{pmatrix}2/3\\1/3\end{pmatrix}.
T_1Use in source · 1 passage
Search aliases: T_1
Use in source · Read passage · FOUNDATIONS.md, line 1581
T_0=3/4 · 1/4 1/4 · 3/4, T_1=1/4 · 2/3 3/4 · 1/3, p_0=2/3 1/3.
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T_0=\begin{pmatrix}3/4&1/4\\1/4&3/4\end{pmatrix},\qquad T_1=\begin{pmatrix}1/4&2/3\\3/4&1/3\end{pmatrix},\qquad p_0=\begin{pmatrix}2/3\\1/3\end{pmatrix}.
t_1Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1242
Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix
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t_1,\ldots,t_r
T_aUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1589
O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.
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O_0=\operatorname{diag}(1/4,3/4),\qquad O_1=\operatorname{diag}(3/4,1/4),\qquad M_{a,o}=O_oT_a.
T_{aa}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 75
T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.
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T_{aa}=I,\qquad T_{cb}T_{ba}=T_{ca},\qquad F_bT_{ba}=T_{ba}F_a.
T_{ba}Use in source · 11 passages
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Use in source · Read passage · FOUNDATIONS.md, line 67
T_ba=A_bS_a: C_a⟶ C_b.
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T_{ba}=A_bS_a:\mathcal C_a\longrightarrow\mathcal C_b.
T_{ca}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 75
T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.
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T_{aa}=I,\qquad T_{cb}T_{ba}=T_{ca},\qquad F_bT_{ba}=T_{ba}F_a.
T_{cb}Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 75
T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.
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T_{aa}=I,\qquad T_{cb}T_{ba}=T_{ca},\qquad F_bT_{ba}=T_{ba}F_a.
t_iUse in source · 7 passages
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Use in source · Read passage · FOUNDATIONS.md, line 1244
H_ij=f(h_jt_i)
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H_{ij}=f(h_jt_i)
t_rUse in source · 1 passage
Search aliases: t_r
Use in source · Read passage · FOUNDATIONS.md, line 1242
Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix
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t_1,\ldots,t_r
T_\alphaUse in source · 1 passage
Search aliases: T_α alpha
Use in source · Read passage · FOUNDATIONS.md, line 2085
T_α=diag(1,α R), 0≤α<1,
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T_\alpha=\operatorname{diag}(1,\alpha R),\qquad 0\leq\alpha<1,
\topUse in source · 50 passages
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Use in source · Read passage · FOUNDATIONS.md, line 280
Distributions are column vectors. Prediction is p↦ K_u^top p, and a Bayesian update is
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p\mapsto K_u^\top p
\operatorname{tr}Use in source · 4 passages
Search aliases: tr operatorname
Use in source · Read passage · FOUNDATIONS.md, line 2183
J:E⟶ S^d, λ z=tr J(z).
Exact mathematical passage
\mathcal J:E\longrightarrow\mathbb S^d, \qquad \lambda z=\operatorname{tr}\mathcal J(z).
\TVUse in source · 1 passage
Search aliases: TV operatorname
Use in source · atlas.tex, line 346
-P87 Start two processes from the same history or initial law and use the same action policy. If their conditional next-observation laws differ by at most Δ at every common history, telescoping gives the record bound min(1,TΔ) in total variation. diagram15_horizons For Markov kernels with a common probability measure μ and P(s,·)≥ρμ(·), 0<ρ≤1, and sup_s P(s,·)-Q(s,·) _TV≤Δ, their one-time law distances satisfy e_t≤min (1,(1-ρ)^te_0+ fracΔ[1-(1-ρ)^t]ρ). A length-k rolling window starting at time …
Exact mathematical passage
\sup_s\|P(s,\cdot)-Q(s,\cdot)\|_{\TV}\leq\Delta
\mathrm{TV}Use in source · 17 passages
Search aliases: TV mathrm
Use in source · Read passage · FOUNDATIONS.md, line 1044
P-P_n _TV=τ_n, τ_n=1-fracZ_nZ.
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\boxed{ \|P-P_n\|_{\mathrm{TV}}=\tau_n, \qquad \tau_n=1-\frac{Z_n}{Z}. }
UUse in source · 26 passages
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Use in source · Read passage · FOUNDATIONS.md, line 45
For a unitary atlas U_a, the synthesis is U_a^* and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.
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U_a
\mathcal UUse in source · 8 passages
Search aliases: U mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1792
Let U be the convex hull of finitely many **complete operator families** {B_s^(j)}_s. Every family must satisfy the exact per-action mass identities. The convex coefficient is shared across that family's symbols. A product of independent uncertainty sets can instead be handled by its joint vertices.
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\mathcal U
U^*Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 106
P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.
Exact mathematical passage
P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i, \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
U^2Use in source · 1 passage
Search aliases: U^2
Use in source · atlas.tex, line 318
x · ysucceq0, m-y≥0}. diagram13_moments Choose a reference probability measure on [-1,1] with moments E U= E U^3=0, E U^2=1/3 and E U^4=1/5. With L_2(u)=(3u^2-1)/2, define B_a,b(u)=v(u)(1,au,bL_2(u)). Its normalized-state density is p_a,b(umid x,y)=1+axu+byL_2(u). If |a|+|b|≤1-ρ for ρ∈[0,1], it is at least ρ, and integrated mass is exact. claimEvery positive-density posterior is the same boundary point v(u) throughout this parameter family. Uncertainty changes its probability without moving its …
Exact mathematical passage
\mathbb E U^2=1/3
u^2Use in source · 14 passages
Search aliases: u^2
Use in source · Read passage · FOUNDATIONS.md, line 181
For example, (u+v)^2=u^2+2uv+v^2. A grammar containing only independent packet squares misses 2uv. For nonpolynomial maps, one needs direct nonlinear semantics or a local expansion with a stated remainder bound. The word algebraic does not remove that obligation.
Exact mathematical passage
(u+v)^2=u^2+2uv+v^2
U^3Use in source · 1 passage
Search aliases: U^3
Use in source · atlas.tex, line 318
x · ysucceq0, m-y≥0}. diagram13_moments Choose a reference probability measure on [-1,1] with moments E U= E U^3=0, E U^2=1/3 and E U^4=1/5. With L_2(u)=(3u^2-1)/2, define B_a,b(u)=v(u)(1,au,bL_2(u)). Its normalized-state density is p_a,b(umid x,y)=1+axu+byL_2(u). If |a|+|b|≤1-ρ for ρ∈[0,1], it is at least ρ, and integrated mass is exact. claimEvery positive-density posterior is the same boundary point v(u) throughout this parameter family. Uncertainty changes its probability without moving its …
Exact mathematical passage
\mathbb E U=\mathbb E U^3=0
u^3Use in source · 1 passage
Search aliases: u^3
Use in source · Read passage · FOUNDATIONS.md, line 2436
L_3(u)=frac5u^3-3u2, p_±(u)=1±ϵ L_3(u), 0<ϵ<1.
Exact mathematical passage
L_3(u)=\frac{5u^3-3u}{2},\qquad p_\pm(u)=1\pm\epsilon L_3(u),\qquad 0<\epsilon<1.
U^4Use in source · 1 passage
Search aliases: U^4
Use in source · atlas.tex, line 318
x · ysucceq0, m-y≥0}. diagram13_moments Choose a reference probability measure on [-1,1] with moments E U= E U^3=0, E U^2=1/3 and E U^4=1/5. With L_2(u)=(3u^2-1)/2, define B_a,b(u)=v(u)(1,au,bL_2(u)). Its normalized-state density is p_a,b(umid x,y)=1+axu+byL_2(u). If |a|+|b|≤1-ρ for ρ∈[0,1], it is at least ρ, and integrated mass is exact. claimEvery positive-density posterior is the same boundary point v(u) throughout this parameter family. Uncertainty changes its probability without moving its …
Exact mathematical passage
\mathbb E U^4=1/5
u^jUse in source · 3 passages
Search aliases: u^j
Use in source · Read passage · FOUNDATIONS.md, line 2379
B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.
Exact mathematical passage
B_a(u)=\sum_{k=0}^D u^k C_{a,k}, \qquad M_{a,j}=\int_{-1}^1 u^j B_a(u)\,\frac{du}{2}.
u^{j+k}Use in source · 1 passage
Search aliases: u^j+k
Use in source · Read passage · FOUNDATIONS.md, line 2387
(W_D)_jk=∫_-1^1u^j+k fracdu2, 0≤ j,k≤ D.
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(W_D)_{jk}=\int_{-1}^1u^{j+k}\,\frac{du}{2}, \qquad 0\leq j,k\leq D.
u^kUse in source · 3 passages
Search aliases: u^k
Use in source · Read passage · FOUNDATIONS.md, line 2353
∫_-1^1u^k fracdu2 =0 · k odd, 1/(k+1) · k even.
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\int_{-1}^1u^k\,\frac{du}{2} =\begin{cases}0&k\text{ odd},\\1/(k+1)&k\text{ even}.\end{cases}
u^qUse in source · 1 passage
Search aliases: u^q
Use in source · Read passage · FOUNDATIONS.md, line 2337
Z_q(u)=I_d^2 uI_d^2 vdots u^qI_d^2.
Exact mathematical passage
Z_q(u)=\begin{pmatrix}I_{d^2}\\uI_{d^2}\\\vdots\\u^qI_{d^2}\end{pmatrix}.
U^\topUse in source · 1 passage
Search aliases: U^top top
Use in source · Read passage · FOUNDATIONS.md, line 1396
For equal-rank orthogonal projections, the latter norm equals P-widehat P _2. To see the equality, diagonalize U^topwidehat U by singular value decomposition. Each nontrivial paired principal plane contributes a sine to both norms, while common directions contribute zero. On the same planes the singular values of widehat U^top U are the corresponding cosines. This proves the last bound and the inverse formula. square
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U^\top\widehat U
u_*Use in source · 3 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2836
An operator identity can reveal a useful coordinate geometry without specifying that geometry in advance. Suppose two fitted operators evaluated at an anchor u_* give an invertible ratio
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u_*
u_1Use in source · 1 passage
Search aliases: u_1
Use in source · Read passage · FOUNDATIONS.md, line 2694
By contrast, the event u_1≥0 under action zero followed by u_2≥0 under action one has actual probability
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u_1\geq0
u_2Use in source · 1 passage
Search aliases: u_2
Use in source · Read passage · FOUNDATIONS.md, line 2694
By contrast, the event u_1≥0 under action zero followed by u_2≥0 under action one has actual probability
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u_2\geq0
U_aUse in source · 3 passages
Search aliases: U_a
Use in source · Read passage · FOUNDATIONS.md, line 45
For a unitary atlas U_a, the synthesis is U_a^* and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.
Exact mathematical passage
U_a
U_a^*Use in source · 2 passages
Search aliases: U_a^*
Use in source · Read passage · FOUNDATIONS.md, line 45
For a unitary atlas U_a, the synthesis is U_a^* and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.
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U_a^*
u_hUse in source · 3 passages
Search aliases: u_h
Use in source · Read passage · FOUNDATIONS.md, line 1937
**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write
Exact mathematical passage
u_h,\widetilde u_h\in C
\mathcal U_nUse in source · 3 passages
Search aliases: U_n mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2124
**Proposition 48. Sequential confidence accounting for certified proposals.** At stage n, let the proposal be measurable from previous data, and let a verification procedure return a set U_n for the true model in the declared identifiable coordinate system. Assume conditional coverage
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\mathcal U_n
U_tUse in source · 4 passages
Search aliases: U_t
Use in source · Read passage · FOUNDATIONS.md, line 3723
P(U_t∈ dumid F_t-1) =λ B_a(u)Z_t μ(du), Z_t+1=fracB_a(U_t)Z_tλ B_a(U_t)Z_t
Exact mathematical passage
P(U_t\in du\mid\mathcal F_{t-1}) =\lambda B_a(u)Z_t\,\mu(du), \qquad Z_{t+1}=\frac{B_a(U_t)Z_t}{\lambda B_a(U_t)Z_t}
U_{t-1}Use in source · 2 passages
Search aliases: U_t-1
Use in source · Read passage · FOUNDATIONS.md, line 3812
E[U_tmid U_t-1=s]=frac a3s, E[L_2(U_t)mid U_t-1=s]=frac b5s^2.
Exact mathematical passage
\boxed{ \mathbb E[U_t\mid U_{t-1}=s]=\frac a3s, \qquad \mathbb E[L_2(U_t)\mid U_{t-1}=s]=\frac b5s^2. }
U_{t-1}^2Use in source · 1 passage
Search aliases: U_t-1^2
Use in source · Read passage · FOUNDATIONS.md, line 3845
ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.
Exact mathematical passage
\begin{aligned} \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\ \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4. \end{aligned}
\uparrowUse in source · 1 passage
Search aliases: uparrow
Use in source · Read passage · FOUNDATIONS.md, line 1065
Since the active sets exhaust the grammar and the evidence sum is finite, Z_nuparrow Z and τ_ndownarrow0. This is posterior approximation *inside the declared grammar*. It does not establish that the real signal-generating mechanism belongs to it.
Exact mathematical passage
Z_n\uparrow Z
vUse in source · 107 passages
Search aliases: v
Use in source · Read passage · FOUNDATIONS.md, line 117
∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.
Exact mathematical passage
\sum_{v\in\Lambda}P_v=I,\qquad x=\sum_{v\in\Lambda}P_vx,\qquad \|x\|^2=\sum_{v\in\Lambda}\|P_vx\|^2.
V'Use in source · 3 passages
Search aliases: V'
Use in source · Read passage · FOUNDATIONS.md, line 2153
nfidence procedure supplies the stated event, not an inferred event of its own. Proposition 39 supplies internal validity for the fitted model. Proposition 40 extends the same domain to every member of the uncertainty set, hence to its true member on that event. Propositions 42 and 43 give the total-variation bound, and Proposition 44 gives the conditional bound. For a coordinate change T, use V'=TV, B_s'=TB_sT^-1 and λ'=λ T^-1. Then B_s'V'=V'A_s and λ'V'=1^top with the unchanged nonnegative wit…
Exact mathematical passage
V'=TV
v'Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 3247
sup_ v' _2≤1 Δ(1,v') _ L_d=ϵ
Exact mathematical passage
\sup_{\|v'\|_2\leq1}\|\Delta(1,v')\|_{\mathcal L_d}=\epsilon
v^{\mathsf T}Use in source · 1 passage
Search aliases: v^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3598
**Proposition 80. Convex repair bracketing.** The feasible repair weights form a closed interval [η_*,1]. If a cone-positive functional ℓ=(1,-v^ T), with v _2≤1, and a normalized input z give c=ℓwidehat F_a(u)z<κ, then every feasible weight satisfies
Exact mathematical passage
\ell=(1,-v^{\mathsf T})
V^{-1}Use in source · 3 passages
Search aliases: V^-1
Use in source · Read passage · FOUNDATIONS.md, line 1858
When V is square and invertible, put A_s=V^-1widehat B_sV. Assume mass conservation has already been imposed. Let N_s be an entrywise positive reference instrument with the same per-action mass identities. For 0≤η≤1, the mixture
Exact mathematical passage
A_s=V^{-1}\widehat B_sV
v^2Use in source · 3 passages
Search aliases: v^2
Use in source · Read passage · FOUNDATIONS.md, line 181
For example, (u+v)^2=u^2+2uv+v^2. A grammar containing only independent packet squares misses 2uv. For nonpolynomial maps, one needs direct nonlinear semantics or a local expansion with a stated remainder bound. The word algebraic does not remove that obligation.
Exact mathematical passage
(u+v)^2=u^2+2uv+v^2
v_1Use in source · 2 passages
Search aliases: v_1
Use in source · Read passage · FOUNDATIONS.md, line 1750
Let V=[v_1 ⋯ v_m]∈ R^r× m have full row rank and satisfy
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V=[v_1\ \cdots\ v_m]\in\mathbb R^{r\times m}
V_{a,t}Use in source · 1 passage
Search aliases: V_a,t
Use in source · Read passage · FOUNDATIONS.md, line 3747
Under the rowwise noise and parameter-bound assumptions of Proposition 81, simultaneous confidence ellipsoids for the rows of M_a,j give a confidence set for every C_a,k by this known linear map. If row i of widehat M_a,j-M_a,j has V_a,t-norm at most β_a,j,i, then for any direction x,
Exact mathematical passage
V_{a,t}
V_{a,t}^{-1}Use in source · 1 passage
Search aliases: V_a,t^-1
Use in source · Read passage · FOUNDATIONS.md, line 3749
|e_i^ T(widehat C_a,k-C_a,k)x| ≤ ∑_j |(W^-1)_kj| β_a,j,i x _V_a,t^-1.
Exact mathematical passage
\left|e_i^{\mathsf T}(\widehat C_{a,k}-C_{a,k})x\right| \leq \sum_j |(W^{-1})_{kj}|\, \beta_{a,j,i}\,\|x\|_{V_{a,t}^{-1}}.
V_hUse in source · 10 passages
Search aliases: V_h
Use in source · Read passage · FOUNDATIONS.md, line 566
O_h=C CL vdots CL^h-1, V_h=imO_h.
Exact mathematical passage
O_h=\begin{pmatrix}C\\CL\\\vdots\\CL^{h-1}\end{pmatrix}, \qquad V_h=\operatorname{im}O_h.
v_hUse in source · 8 passages
Search aliases: v_h
Use in source · Read passage · FOUNDATIONS.md, line 1205
v_h(t)=f(ht),
Exact mathematical passage
v_h(t)=f(ht),
v_{hs}Use in source · 2 passages
Search aliases: v_hs
Use in source · Read passage · FOUNDATIONS.md, line 1219
B_sv_h=v_hs, f(s_1⋯ s_k)=λ B_s_k⋯ B_s_1v_ϵ.
Exact mathematical passage
B_sv_h=v_{hs},\qquad f(s_1\cdots s_k)=\lambda B_{s_k}\cdots B_{s_1}v_\epsilon.
v_jUse in source · 6 passages
Search aliases: v_j
Use in source · Read passage · FOUNDATIONS.md, line 1774
**Proof.** Invariance places each B_sv_j in the conical hull of the columns of V. Choose its nonnegative coefficients as column j of A_s. Conversely, B_sVα=VA_sα∈ C for every α≥0. Multiplying the intertwining identity by λ and summing over observations gives
Exact mathematical passage
B_sv_j
v_mUse in source · 2 passages
Search aliases: v_m
Use in source · Read passage · FOUNDATIONS.md, line 1750
Let V=[v_1 ⋯ v_m]∈ R^r× m have full row rank and satisfy
Exact mathematical passage
V=[v_1\ \cdots\ v_m]\in\mathbb R^{r\times m}
V_tUse in source · 7 passages
Search aliases: V_t
Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
V_t^{-1}Use in source · 5 passages
Search aliases: V_t^-1
Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
v_\epsilonUse in source · 2 passages
Search aliases: v_ϵ epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1217
**Proposition 27. Minimal finite behavior realization.** If rankmathscr H=r<∞, the vector space V=span{v_h} has dimension r. There are well-defined linear maps B_s:V→ V, an initial vector v_ϵ, and a linear mass functional λ(v)=v(ϵ) satisfying
Exact mathematical passage
v_\epsilon
\operatorname{Var}Use in source · 4 passages
Search aliases: Var operatorname
Use in source · Read passage · FOUNDATIONS.md, line 2490
The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.
Exact mathematical passage
\operatorname{Var}_\nu(u)K_1XK_1^{\mathsf T}
\operatorname{Var}_\nuUse in source · 1 passage
Search aliases: Var_ν operatorname nu
Use in source · Read passage · FOUNDATIONS.md, line 2490
The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.
Exact mathematical passage
\operatorname{Var}_\nu(u)K_1XK_1^{\mathsf T}
\vdotsUse in source · 4 passages
Search aliases: vdots
Use in source · Read passage · FOUNDATIONS.md, line 566
O_h=C CL vdots CL^h-1, V_h=imO_h.
Exact mathematical passage
O_h=\begin{pmatrix}C\\CL\\\vdots\\CL^{h-1}\end{pmatrix}, \qquad V_h=\operatorname{im}O_h.
\operatorname{vec}Use in source · 1 passage
Search aliases: vec operatorname
Use in source · Read passage · FOUNDATIONS.md, line 2328
F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,
Exact mathematical passage
F=\sum_\nu w_\nu\operatorname{vec}(K_\nu) \operatorname{vec}(K_\nu)^{\mathsf T},\qquad w_\nu\geq0,
wUse in source · 56 passages
Search aliases: w
Use in source · Read passage · FOUNDATIONS.md, line 49
For unitary U_a and positive weights w_a, put
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w_a
W^{\mathsf T}Use in source · 5 passages
Search aliases: W^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2254
J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,
Exact mathematical passage
\begin{aligned} \mathcal J(Tz)&=\frac15W\mathcal J(z)W^{\mathsf T},\\ \mathcal J(B_{\mathrm p,0}z)&=E_{00}\mathcal J(z)E_{00}^{\mathsf T},\\ \mathcal J(B_{\mathrm p,1}z)&=E_{01}\mathcal J(z)E_{01}^{\mathsf T}, \end{aligned}
w^{\mathsf T}Use in source · 1 passage
Search aliases: w^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2822
For a nonzero z, write x=m(c+w) with w^ THw≤1. Its output mass is m(β_a+b_a^ Tw). Cauchy--Schwarz in the H inner product bounds this below by
Exact mathematical passage
w^{\mathsf T}Hw\leq1
W^{-1}Use in source · 2 passages
Search aliases: W^-1
Use in source · Read passage · FOUNDATIONS.md, line 3738
E[g_j(U_t)Z_t+1mid F_t-1] =M_a,jZ_t, C_a,k=∑_j(W^-1)_kjM_a,j.
Exact mathematical passage
\boxed{ \mathbb E[g_j(U_t)Z_{t+1}\mid\mathcal F_{t-1}] =M_{a,j}Z_t, \qquad C_{a,k}=\sum_j(W^{-1})_{kj}M_{a,j}. }
W_0Use in source · 9 passages
Search aliases: W_0
Use in source · Read passage · FOUNDATIONS.md, line 241
Let the initial summary be C_0x. Regard its rows as linear observables. Starting from W_0=row(C_0), define
Exact mathematical passage
W_0=\operatorname{row}(C_0)
w_0Use in source · 4 passages
Search aliases: w_0
Use in source · Read passage · FOUNDATIONS.md, line 3232
(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.
Exact mathematical passage
(w_0+d_0)-\|w_{\rm sp}+d_{\rm sp}\|_2 \geq w_0-\|w_{\rm sp}\|_2-|d_0|-\|d_{\rm sp}\|_2.
W_2Use in source · 1 passage
Search aliases: W_2
Use in source · Read passage · FOUNDATIONS.md, line 2668
W_2=1 · 0 · 1/3 0 · 1/3 · 0 1/3 · 0 · 1/5, W_2^-1=9/4 · 0 · -15/4 0 · 3 · 0 -15/4 · 0 · 45/4, det W_2=frac4135.
Exact mathematical passage
W_2=\begin{pmatrix}1&0&1/3\\0&1/3&0\\1/3&0&1/5\end{pmatrix}, \qquad W_2^{-1}=\begin{pmatrix}9/4&0&-15/4\\0&3&0\\-15/4&0&45/4\end{pmatrix}, \qquad\det W_2=\frac4{135}.
W_2^{-1}Use in source · 1 passage
Search aliases: W_2^-1
Use in source · Read passage · FOUNDATIONS.md, line 2668
W_2=1 · 0 · 1/3 0 · 1/3 · 0 1/3 · 0 · 1/5, W_2^-1=9/4 · 0 · -15/4 0 · 3 · 0 -15/4 · 0 · 45/4, det W_2=frac4135.
Exact mathematical passage
W_2=\begin{pmatrix}1&0&1/3\\0&1/3&0\\1/3&0&1/5\end{pmatrix}, \qquad W_2^{-1}=\begin{pmatrix}9/4&0&-15/4\\0&3&0\\-15/4&0&45/4\end{pmatrix}, \qquad\det W_2=\frac4{135}.
w_aUse in source · 6 passages
Search aliases: w_a
Use in source · Read passage · FOUNDATIONS.md, line 49
For unitary U_a and positive weights w_a, put
Exact mathematical passage
w_a
w_{a,s}Use in source · 1 passage
Search aliases: w_a,s
Use in source · Read passage · FOUNDATIONS.md, line 3392
w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.
Exact mathematical passage
\begin{aligned} w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\ f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\ h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2. \end{aligned}
w_{a,s,0}Use in source · 2 passages
Search aliases: w_a,s,0
Use in source · Read passage · FOUNDATIONS.md, line 3392
w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.
Exact mathematical passage
\begin{aligned} w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\ f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\ h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2. \end{aligned}
w_{a,s,1}Use in source · 2 passages
Search aliases: w_a,s,1
Use in source · Read passage · FOUNDATIONS.md, line 3392
w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.
Exact mathematical passage
\begin{aligned} w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\ f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\ h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2. \end{aligned}
w_{a,s,2}Use in source · 2 passages
Search aliases: w_a,s,2
Use in source · Read passage · FOUNDATIONS.md, line 3392
w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.
Exact mathematical passage
\begin{aligned} w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\ f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\ h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2. \end{aligned}
w_a^2Use in source · 1 passage
Search aliases: w_a^2
Use in source · Read passage · FOUNDATIONS.md, line 51
Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.
Exact mathematical passage
\mathcal Ax=(w_aU_ax)_a,\qquad \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a, \qquad c=\sum_aw_a^2.
W_DUse in source · 5 passages
Search aliases: W_D
Use in source · Read passage · FOUNDATIONS.md, line 2385
Define the moment Gram matrix W_D by
Exact mathematical passage
W_D
W_D^{-1}Use in source · 2 passages
Search aliases: W_D^-1
Use in source · Read passage · FOUNDATIONS.md, line 2403
widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.
Exact mathematical passage
\boxed{ \widehat M_{a,j}=G_{a,j}H^{-1}, \qquad \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}. }
w_iUse in source · 1 passage
Search aliases: w_i
Use in source · Read passage · FOUNDATIONS.md, line 1318
z=∑_i:w_i>0 w_i z_i, w_i=λ P_iz≥0, z_i=P_iz/w_i, ∑_iw_i=1.
Exact mathematical passage
z=\sum_{i:w_i>0} w_i z_i,\qquad w_i=\lambda P_iz\geq0,\qquad z_i=P_iz/w_i,\qquad\sum_iw_i=1.
W_jUse in source · 1 passage
Search aliases: W_j
Use in source · Read passage · FOUNDATIONS.md, line 1170
W_j+1=W_j+∑_s∈ A× OW_jM_s.
Exact mathematical passage
W_{j+1}=W_j+\sum_{s\in\mathcal A\times\mathcal O}W_jM_s.
W_{j+1}Use in source · 1 passage
Search aliases: W_j+1
Use in source · Read passage · FOUNDATIONS.md, line 1170
W_j+1=W_j+∑_s∈ A× OW_jM_s.
Exact mathematical passage
W_{j+1}=W_j+\sum_{s\in\mathcal A\times\mathcal O}W_jM_s.
W_{jk}Use in source · 2 passages
Search aliases: W_jk
Use in source · Read passage · FOUNDATIONS.md, line 3732
Let B_a(u)=∑_k=0^Dψ_k(u)C_a,k. Choose functions g_j so that the known matrix W_jk=∫ g_j(u)ψ_k(u) μ(du) is invertible.
Exact mathematical passage
W_{jk}=\int g_j(u)\psi_k(u)\,\mu(du)
W_nUse in source · 7 passages
Search aliases: W_n
Use in source · Read passage · FOUNDATIONS.md, line 243
W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.
Exact mathematical passage
W_{n+1}=W_n+\sum_uW_nL_u, \qquad W_nL_u=\{wL_u:w\in W_n\}.
W_{n+1}Use in source · 2 passages
Search aliases: W_n+1
Use in source · Read passage · FOUNDATIONS.md, line 243
W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.
Exact mathematical passage
W_{n+1}=W_n+\sum_uW_nL_u, \qquad W_nL_u=\{wL_u:w\in W_n\}.
w_{\rm sp}Use in source · 1 passage
Search aliases: w_rm sp rm
Use in source · Read passage · FOUNDATIONS.md, line 3232
(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.
Exact mathematical passage
(w_0+d_0)-\|w_{\rm sp}+d_{\rm sp}\|_2 \geq w_0-\|w_{\rm sp}\|_2-|d_0|-\|d_{\rm sp}\|_2.
W_s^aUse in source · 1 passage
Search aliases: W_s^a
Use in source · Read passage · FOUNDATIONS.md, line 3097
Consequently Proposition 66 gives a time-uniform certificate for ∑_s≤ tμ_s^a from ∑_s≤ tW_s^a, with its boundary multiplied by 1/π_min. The target is the average action-conditioned quantity along the encountered contexts, not a reset-population table or the state distribution under a different long-term policy.
Exact mathematical passage
\sum_{s\leq t}W_s^a
w_{sj}Use in source · 1 passage
Search aliases: w_sj
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
Exact mathematical passage
\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
w_tUse in source · 3 passages
Search aliases: w_t
Use in source · atlas.tex, line 181
plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, retaining the last H history terms leaves a same-history tail at most Δ_H=tfrac38 3^-H.
Exact mathematical passage
y_{t+1}=Ay_t+Bw_t
w_{t+1}Use in source · 1 passage
Search aliases: w_t+1
Use in source · atlas.tex, line 181
plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, retaining the last H history terms leaves a same-history tail at most Δ_H=tfrac38 3^-H.
Exact mathematical passage
w_{t+1}=Cy_t+Dw_t
W_t^aUse in source · 2 passages
Search aliases: W_t^a
Use in source · Read passage · FOUNDATIONS.md, line 3085
μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).
Exact mathematical passage
\mu_t^a=\int\phi_t(y)P_t(dy\mid\mathcal H_{t-1},a),\qquad W_t^a=\frac{\mathbf1_{\{A_t=a\}}\phi_t(Y_t)}{\pi_t(a\mid\mathcal H_{t-1})}.
w_\nuUse in source · 3 passages
Search aliases: w_ν nu
Use in source · Read passage · FOUNDATIONS.md, line 2328
F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,
Exact mathematical passage
F=\sum_\nu w_\nu\operatorname{vec}(K_\nu) \operatorname{vec}(K_\nu)^{\mathsf T},\qquad w_\nu\geq0,
\widehat AUse in source · 14 passages
Search aliases: widehat A widehat
Use in source · Read passage · FOUNDATIONS.md, line 825
widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.
Exact mathematical passage
\widehat z_{t+1}=\widehat A\widehat z_t+ \sum_{k=0}^{H-1}\widehat K_k\widehat z_{t-1-k}.
\widehat A^{\mathsf T}Use in source · 1 passage
Search aliases: widehat A^ T widehat mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3369
The least singular value of M_A^* is at least that of M_widehat A minus e_M. Apply this lower bound and the triangle inequality. To bound the map perturbation, insert and subtract widehat A^ THA^* and apply the Frobenius product inequality. The trace component is unchanged. Finally,
Exact mathematical passage
\widehat A^{\mathsf T}HA^*
\widehat BUse in source · 25 passages
Search aliases: widehat B widehat
Use in source · Read passage · FOUNDATIONS.md, line 640
widehat B=widetilde Z_+widetilde Z_-^†
Exact mathematical passage
\widehat B=\widetilde Z_+\widetilde Z_-^\dagger
\widehat bUse in source · 3 passages
Search aliases: widehat b widehat
Use in source · Read passage · FOUNDATIONS.md, line 3352
For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has
Exact mathematical passage
\widehat c=(I-\widehat A)^{-1}\widehat b
\widehat B_aUse in source · 1 passage
Search aliases: widehat B_a widehat
Use in source · Read passage · FOUNDATIONS.md, line 2740
**Proof.** All coefficient matrices are fixed, so the displayed constraints are linear. An invertible solution yields widehat B_a(u)=TB_a(u)T^-1 coefficient by coefficient. Substitution in the template intertwining identity proves the transported certificate. To prove uniqueness, let D be the difference of two solutions. It satisfies widehat C_a,kD=DC_a,k and widehatλ D=0. Therefore widehatλwidehat C_wD=0 for every coefficient word. Observability makes every column of D zero. An initial state ou…
Exact mathematical passage
\widehat B_a(u)=TB_a(u)T^{-1}
\widehat B_sUse in source · 15 passages
Search aliases: widehat B_s widehat
Use in source · Read passage · FOUNDATIONS.md, line 1416
**Proposition 33. Noisy core-block realization.** The estimate widehat B_s=widetilde G_swidetilde H^-1 satisfies
Exact mathematical passage
\widehat B_s=\widetilde G_s\widetilde H^{-1}
\widehat B_uUse in source · 5 passages
Search aliases: widehat B_u widehat
Use in source · Read passage · FOUNDATIONS.md, line 772
For a fixed linear representation R and an estimated transition widehat B_u, the defect is (RL_u-widehat B_uR)x. For a reference reduced operator B_u, it splits exactly as
Exact mathematical passage
\widehat B_u
\widehat B_u^\topUse in source · 1 passage
Search aliases: widehat B_u^top widehat top
Use in source · Read passage · FOUNDATIONS.md, line 793
widehat B_u^top Mwidehat B_upreceqρ^2M for every declared action u.
Exact mathematical passage
\widehat B_u^\top M\widehat B_u\preceq\rho^2M \quad\text{for every declared action }u.
\widehat CUse in source · 8 passages
Search aliases: widehat C widehat
Use in source · Read passage · FOUNDATIONS.md, line 2403
widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.
Exact mathematical passage
\boxed{ \widehat M_{a,j}=G_{a,j}H^{-1}, \qquad \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}. }
\widehat cUse in source · 3 passages
Search aliases: widehat c widehat
Use in source · Read passage · FOUNDATIONS.md, line 3352
For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has
Exact mathematical passage
\widehat c=(I-\widehat A)^{-1}\widehat b
\widehat C_{a,k}Use in source · 6 passages
Search aliases: widehat C_a,k widehat
Use in source · Read passage · FOUNDATIONS.md, line 2403
widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.
Exact mathematical passage
\boxed{ \widehat M_{a,j}=G_{a,j}H^{-1}, \qquad \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}. }
\widehat C_wUse in source · 2 passages
Search aliases: widehat C_w widehat
Use in source · Read passage · FOUNDATIONS.md, line 2738
are linear in T. Every invertible solution transports the template certificate to the fitted coordinates by widehat J= JT^-1. If the fitted coefficient family is observable, meaning the rows widehatλwidehat C_w span the dual space, the solution is unique. The fitted initial state must still be tested in the transported normalized cone.
Exact mathematical passage
\widehat\lambda\widehat C_w
\widehat FUse in source · 7 passages
Search aliases: widehat F widehat
Use in source · Read passage · FOUNDATIONS.md, line 3271
This reference instrument reports a state-independent density and resets to the center. It is a mathematical model modification, not a new observation. Given an estimate widehat F, define
Exact mathematical passage
\widehat F
\widehat fUse in source · 9 passages
Search aliases: widehat f widehat
Use in source · Read passage · FOUNDATIONS.md, line 738
Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is
Exact mathematical passage
\widehat f_t:Z_t\to Z_{t+1}
\widehat F_aUse in source · 2 passages
Search aliases: widehat F_a widehat
Use in source · Read passage · FOUNDATIONS.md, line 3273
widetilde F_a(u)=(1-η)widehat F_a(u)+η N_a(u), 0≤η≤1.
Exact mathematical passage
\widetilde F_a(u)=(1-\eta)\widehat F_a(u)+\eta N_a(u), \qquad 0\leq\eta\leq1.
\widehat f_tUse in source · 7 passages
Search aliases: widehat f_t widehat
Use in source · Read passage · FOUNDATIONS.md, line 738
Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is
Exact mathematical passage
\widehat f_t:Z_t\to Z_{t+1}
\widehat HUse in source · 4 passages
Search aliases: widehat H widehat
Use in source · Read passage · FOUNDATIONS.md, line 3328
**Proposition 74. A posteriori proposal error.** Given widehat H, let r= M_widehat A(widehat H)-b. Suppose
Exact mathematical passage
\widehat H
\widehat{\mathcal J}Use in source · 1 passage
Search aliases: widehat J widehat mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2738
are linear in T. Every invertible solution transports the template certificate to the fitted coordinates by widehat J= JT^-1. If the fitted coefficient family is observable, meaning the rows widehatλwidehat C_w span the dual space, the solution is unique. The fitted initial state must still be tested in the transported normalized cone.
Exact mathematical passage
\widehat{\mathcal J}=\mathcal JT^{-1}
\widehat KUse in source · 3 passages
Search aliases: widehat K widehat
Use in source · Read passage · FOUNDATIONS.md, line 825
widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.
Exact mathematical passage
\widehat z_{t+1}=\widehat A\widehat z_t+ \sum_{k=0}^{H-1}\widehat K_k\widehat z_{t-1-k}.
\widehat K_kUse in source · 3 passages
Search aliases: widehat K_k widehat
Use in source · Read passage · FOUNDATIONS.md, line 825
widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.
Exact mathematical passage
\widehat z_{t+1}=\widehat A\widehat z_t+ \sum_{k=0}^{H-1}\widehat K_k\widehat z_{t-1-k}.
\widehat LUse in source · 3 passages
Search aliases: widehat L widehat
Use in source · Read passage · FOUNDATIONS.md, line 471
If estimated operators widehat L_n are uniformly bounded and (widehat L_n-L_n)x→0 for every fixed x, the same argument gives fixed-horizon convergence for widehat L_n. These are explicit assumptions, not conclusions of the finite learning procedure.
Exact mathematical passage
\widehat L_n
\widehat L_nUse in source · 3 passages
Search aliases: widehat L_n widehat
Use in source · Read passage · FOUNDATIONS.md, line 471
If estimated operators widehat L_n are uniformly bounded and (widehat L_n-L_n)x→0 for every fixed x, the same argument gives fixed-horizon convergence for widehat L_n. These are explicit assumptions, not conclusions of the finite learning procedure.
Exact mathematical passage
\widehat L_n
\widehat MUse in source · 2 passages
Search aliases: widehat M widehat
Use in source · Read passage · FOUNDATIONS.md, line 2403
widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.
Exact mathematical passage
\boxed{ \widehat M_{a,j}=G_{a,j}H^{-1}, \qquad \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}. }
\widehat M_{a,j}Use in source · 2 passages
Search aliases: widehat M_a,j widehat
Use in source · Read passage · FOUNDATIONS.md, line 2403
widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.
Exact mathematical passage
\boxed{ \widehat M_{a,j}=G_{a,j}H^{-1}, \qquad \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}. }
\widehat PUse in source · 7 passages
Search aliases: widehat P widehat
Use in source · Read passage · FOUNDATIONS.md, line 1362
Let X∈ R^m× n have rank r≥1, let s=σ_r(X)>0, and observe Y=X+E with E _2≤ε<s/2. Let P project onto imX, and let widehat P project onto the leading r left singular vectors of Y.
Exact mathematical passage
\widehat P
\widehat pUse in source · 5 passages
Search aliases: widehat p widehat
Use in source · Read passage · FOUNDATIONS.md, line 1461
For a predicted state widehat z, form widehat u=widehat B_swidehat z, widehat p=λwidehat u, and define d=widehat u-u. Write e= widehat z-z , widehat B_s ≤ L, and widehat B_s-B_s ≤ϵ_B. Then
Exact mathematical passage
\widehat p=\lambda\widehat u
\widehat P_TUse in source · 1 passage
Search aliases: widehat P_T widehat
Use in source · Read passage · FOUNDATIONS.md, line 2146
P_T-widehat P_T _TV ≤min(1,frace_0+∑_t<Tϵ_t2).
Exact mathematical passage
\|P_T-\widehat P_T\|_{\mathrm{TV}} \leq\min\left(1,\frac{e_0+\sum_{t<T}\epsilon_t}{2}\right).
\widehat p_tUse in source · 1 passage
Search aliases: widehat p_t widehat
Use in source · Read passage · FOUNDATIONS.md, line 1483
Alternatively, if each realized fitted denominator is checked to satisfy widehat p_t≥widehatη_t>0, the pathwise recursion obeys
Exact mathematical passage
\widehat p_t\geq\widehat\eta_t>0
\widehat UUse in source · 6 passages
Search aliases: widehat U widehat
Use in source · Read passage · FOUNDATIONS.md, line 1372
The (r+1)st singular value is interpreted as zero when r is the maximum possible rank. If U and widehat U are orthonormal bases of the two ranges, then
Exact mathematical passage
\widehat U
\widehat uUse in source · 3 passages
Search aliases: widehat u widehat
Use in source · Read passage · FOUNDATIONS.md, line 1461
For a predicted state widehat z, form widehat u=widehat B_swidehat z, widehat p=λwidehat u, and define d=widehat u-u. Write e= widehat z-z , widehat B_s ≤ L, and widehat B_s-B_s ≤ϵ_B. Then
Exact mathematical passage
\widehat u=\widehat B_s\widehat z
\widehat U^\topUse in source · 4 passages
Search aliases: widehat U^top widehat top
Use in source · Read passage · FOUNDATIONS.md, line 1374
σ_min(widehat U^top U) ≥sqrt1-(2ε/s)^2.
Exact mathematical passage
\sigma_{\min}(\widehat U^\top U) \geq\sqrt{1-(2\varepsilon/s)^2}.
\widehat yUse in source · 1 passage
Search aliases: widehat y widehat
Use in source · atlas.tex, line 194
Here 0≤ y_t≤1 and the truncated recursion has gain a_H=frac12+∑_j=0^H-1K_j =frac78-frac38 3^-H<1. With the same initial state, recursive propagation therefore gives sup_t|y_t-widehat y_t|≤fracΔ_H1-a_H =frac33^H+3. boundaryP10's tail estimate uses the same resolved history. P19 supplies the separate stability argument for a predictor using its own history. For other initial states, retain the term BD^tw_0. navMemory.lean; MemoryBounds.lean; MemoryStability.lean.
Exact mathematical passage
\sup_t|y_t-\widehat y_t|\leq\frac{\Delta_H}{1-a_H} =\frac{3}{3^H+3}.
\widehat y_tUse in source · 1 passage
Search aliases: widehat y_t widehat
Use in source · atlas.tex, line 194
Here 0≤ y_t≤1 and the truncated recursion has gain a_H=frac12+∑_j=0^H-1K_j =frac78-frac38 3^-H<1. With the same initial state, recursive propagation therefore gives sup_t|y_t-widehat y_t|≤fracΔ_H1-a_H =frac33^H+3. boundaryP10's tail estimate uses the same resolved history. P19 supplies the separate stability argument for a predictor using its own history. For other initial states, retain the term BD^tw_0. navMemory.lean; MemoryBounds.lean; MemoryStability.lean.
Exact mathematical passage
\sup_t|y_t-\widehat y_t|\leq\frac{\Delta_H}{1-a_H} =\frac{3}{3^H+3}.
\widehat zUse in source · 16 passages
Search aliases: widehat z widehat
Use in source · Read passage · FOUNDATIONS.md, line 746
**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,
Exact mathematical passage
\widehat z_t
\widehat z^+Use in source · 6 passages
Search aliases: widehat z^+ widehat
Use in source · Read passage · FOUNDATIONS.md, line 1469
widehat z^+-z^+ =frac(I-z^+λ)dwidehat p,
Exact mathematical passage
\widehat z^+-z^+ =\frac{(I-z^+\lambda)d}{\widehat p},
\widehat z_tUse in source · 7 passages
Search aliases: widehat z_t widehat
Use in source · Read passage · FOUNDATIONS.md, line 746
**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,
Exact mathematical passage
\widehat z_t
\widehat z_{t+1}Use in source · 2 passages
Search aliases: widehat z_t+1 widehat
Use in source · Read passage · FOUNDATIONS.md, line 748
**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,
Exact mathematical passage
\widehat z_{t+1}=\widehat f_t(\widehat z_t)
\widehat z_{t-1-k}Use in source · 1 passage
Search aliases: widehat z_t-1-k widehat
Use in source · Read passage · FOUNDATIONS.md, line 825
widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.
Exact mathematical passage
\widehat z_{t+1}=\widehat A\widehat z_t+ \sum_{k=0}^{H-1}\widehat K_k\widehat z_{t-1-k}.
\widehat\etaUse in source · 2 passages
Search aliases: widehatη widehat eta
Use in source · Read passage · FOUNDATIONS.md, line 1483
Alternatively, if each realized fitted denominator is checked to satisfy widehat p_t≥widehatη_t>0, the pathwise recursion obeys
Exact mathematical passage
\widehat p_t\geq\widehat\eta_t>0
\widehat\eta_tUse in source · 2 passages
Search aliases: widehatη_t widehat eta
Use in source · Read passage · FOUNDATIONS.md, line 1483
Alternatively, if each realized fitted denominator is checked to satisfy widehat p_t≥widehatη_t>0, the pathwise recursion obeys
Exact mathematical passage
\widehat p_t\geq\widehat\eta_t>0
\widehat\thetaUse in source · 6 passages
Search aliases: widehatθ widehat theta
Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
\widehat\theta_tUse in source · 5 passages
Search aliases: widehatθ_t widehat theta
Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
\widehat\lambdaUse in source · 5 passages
Search aliases: widehatλ widehat lambda
Use in source · Read passage · FOUNDATIONS.md, line 2727
Suppose a structural library supplies a certified template with polynomial coefficients C_a,k, mass row λ and embedding J. Suppose observed population tables produce coefficients widehat C_a,k and mass row widehatλ of the same dimension.
Exact mathematical passage
\widehat\lambda
\widetilde AUse in source · 1 passage
Search aliases: widetilde A widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1860
widetilde A_s=(1-η)A_s+η N_s
Exact mathematical passage
\widetilde A_s=(1-\eta)A_s+\eta N_s
\widetilde A_sUse in source · 1 passage
Search aliases: widetilde A_s widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1860
widetilde A_s=(1-η)A_s+η N_s
Exact mathematical passage
\widetilde A_s=(1-\eta)A_s+\eta N_s
\widetilde BUse in source · 11 passages
Search aliases: widetilde B widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1829
Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints
Exact mathematical passage
\widetilde B_s
\widetilde bUse in source · 1 passage
Search aliases: widetilde b widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1517
widetilde H-H _2≤ rϵ_N, widetilde G_s-G_s _2≤ rϵ_N, widetilde b_0-b_0 _2≤sqrt rϵ_N.
Exact mathematical passage
\|\widetilde H-H\|_2\leq r\epsilon_N,\qquad \|\widetilde G_s-G_s\|_2\leq r\epsilon_N,\qquad \|\widetilde b_0-b_0\|_2\leq\sqrt r\epsilon_N.
\widetilde b_0Use in source · 1 passage
Search aliases: widetilde b_0 widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1517
widetilde H-H _2≤ rϵ_N, widetilde G_s-G_s _2≤ rϵ_N, widetilde b_0-b_0 _2≤sqrt rϵ_N.
Exact mathematical passage
\|\widetilde H-H\|_2\leq r\epsilon_N,\qquad \|\widetilde G_s-G_s\|_2\leq r\epsilon_N,\qquad \|\widetilde b_0-b_0\|_2\leq\sqrt r\epsilon_N.
\widetilde B_aUse in source · 2 passages
Search aliases: widetilde B_a widetilde
Use in source · Read passage · FOUNDATIONS.md, line 2561
**Proposition 59. Continuous-record total-variation bound.** Let B_a(y) and widetilde B_a(y) satisfy Proposition 52 on a common (C,λ,μ), and let the two models use the same measurable policy on every common observation history. Suppose their initial states satisfy z_0-widetilde z_0 _C≤ e_0 and their time-t instruments satisfy
Exact mathematical passage
\widetilde B_a(y)
\widetilde B_{a,o}Use in source · 3 passages
Search aliases: widetilde B_a,o widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1831
widetilde B_sV=VA_s, A_s≥0, λ∑_owidetilde B_a,o=λ, widetilde z_0=Vα_0, α_0≥0, 1^topα_0=1
Exact mathematical passage
\begin{gathered} \widetilde B_sV=VA_s,\quad A_s\geq0, \quad \lambda\sum_o\widetilde B_{a,o}=\lambda,\\ \widetilde z_0=V\alpha_0,\quad\alpha_0\geq0, \quad\mathbf1^\top\alpha_0=1 \end{gathered}
\widetilde B_{a,t}Use in source · 1 passage
Search aliases: widetilde B_a,t widetilde
Use in source · Read passage · FOUNDATIONS.md, line 2563
sup_a,z∈ K∫ (B_a,t(y)-widetilde B_a,t(y))z _C μ(dy) ≤ϵ_t.
Exact mathematical passage
\sup_{a,z\in K}\int\|(B_{a,t}(y)-\widetilde B_{a,t}(y))z\|_C\,\mu(dy) \leq\epsilon_t.
\widetilde B_a^{[n]}Use in source · 1 passage
Search aliases: widetilde B_a^[n] widetilde
Use in source · Read passage · FOUNDATIONS.md, line 3007
Then widetilde B_a^[n](x)=∑_k b_n,k(x)widetilde C_a,n,k is internally valid. If
Exact mathematical passage
\widetilde B_a^{[n]}(x)=\sum_k b_{n,k}(x)\widetilde C_{a,n,k}
\widetilde B_sUse in source · 5 passages
Search aliases: widetilde B_s widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1829
Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints
Exact mathematical passage
\widetilde B_s
\widetilde CUse in source · 6 passages
Search aliases: widetilde C widetilde
Use in source · Read passage · FOUNDATIONS.md, line 2413
widetilde C_a,k-widehat C_a,k ≤η∑_j|(W_D^-1)_kj|.
Exact mathematical passage
\|\widetilde C_{a,k}-\widehat C_{a,k}\| \leq\eta\sum_j|(W_D^{-1})_{kj}|.
\widetilde C_{a,k}Use in source · 1 passage
Search aliases: widetilde C_a,k widetilde
Use in source · Read passage · FOUNDATIONS.md, line 2413
widetilde C_a,k-widehat C_a,k ≤η∑_j|(W_D^-1)_kj|.
Exact mathematical passage
\|\widetilde C_{a,k}-\widehat C_{a,k}\| \leq\eta\sum_j|(W_D^{-1})_{kj}|.
\widetilde C_{a,n,k}Use in source · 5 passages
Search aliases: widetilde C_a,n,k widetilde
Use in source · Read passage · FOUNDATIONS.md, line 3001
**Proposition 65. Certified coefficient error and approximation bias.** Suppose every fitted coefficient widetilde C_a,n,k preserves the same cone C, and
Exact mathematical passage
\widetilde C_{a,n,k}
\widetilde FUse in source · 3 passages
Search aliases: widetilde F widetilde
Use in source · Read passage · FOUNDATIONS.md, line 3273
widetilde F_a(u)=(1-η)widehat F_a(u)+η N_a(u), 0≤η≤1.
Exact mathematical passage
\widetilde F_a(u)=(1-\eta)\widehat F_a(u)+\eta N_a(u), \qquad 0\leq\eta\leq1.
\widetilde F_aUse in source · 2 passages
Search aliases: widetilde F_a widetilde
Use in source · Read passage · FOUNDATIONS.md, line 3273
widetilde F_a(u)=(1-η)widehat F_a(u)+η N_a(u), 0≤η≤1.
Exact mathematical passage
\widetilde F_a(u)=(1-\eta)\widehat F_a(u)+\eta N_a(u), \qquad 0\leq\eta\leq1.
\widetilde GUse in source · 5 passages
Search aliases: widetilde G widetilde
Use in source · Read passage · FOUNDATIONS.md, line 952
overline L=∑_aw_aL_a, widetilde G_a=A_aoverline LS_a on C_a=imA_a.
Exact mathematical passage
\overline L=\sum_aw_aL_a, \qquad\widetilde G_a=A_a\overline LS_a \quad\text{on }\mathcal C_a=\operatorname{im}A_a.
\widetilde G_aUse in source · 2 passages
Search aliases: widetilde G_a widetilde
Use in source · Read passage · FOUNDATIONS.md, line 952
overline L=∑_aw_aL_a, widetilde G_a=A_aoverline LS_a on C_a=imA_a.
Exact mathematical passage
\overline L=\sum_aw_aL_a, \qquad\widetilde G_a=A_a\overline LS_a \quad\text{on }\mathcal C_a=\operatorname{im}A_a.
\widetilde G_sUse in source · 3 passages
Search aliases: widetilde G_s widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1410
widetilde H=H+E, widetilde G_s=G_s+F_s, E _2<s_H.
Exact mathematical passage
\widetilde H=H+E,\qquad \widetilde G_s=G_s+F_s,\qquad \|E\|_2<s_H.
\widetilde HUse in source · 7 passages
Search aliases: widetilde H widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1410
widetilde H=H+E, widetilde G_s=G_s+F_s, E _2<s_H.
Exact mathematical passage
\widetilde H=H+E,\qquad \widetilde G_s=G_s+F_s,\qquad \|E\|_2<s_H.
\widetilde H^{-1}Use in source · 2 passages
Search aliases: widetilde H^-1 widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1416
**Proposition 33. Noisy core-block realization.** The estimate widehat B_s=widetilde G_swidetilde H^-1 satisfies
Exact mathematical passage
\widehat B_s=\widetilde G_s\widetilde H^{-1}
\widetilde PUse in source · 5 passages
Search aliases: widetilde P widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1925
**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy
Exact mathematical passage
P_T,\widetilde P_T
\widetilde P_TUse in source · 5 passages
Search aliases: widetilde P_T widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1925
**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy
Exact mathematical passage
P_T,\widetilde P_T
\widetilde sUse in source · 4 passages
Search aliases: widetilde s widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1439
If widetilde s=σ_min(widetilde H) is known and E _2≤ e<widetilde s, the last denominator may be replaced by widetilde s-e.
Exact mathematical passage
\widetilde s=\sigma_{\min}(\widetilde H)
\widetilde uUse in source · 6 passages
Search aliases: widetilde u widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1937
**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write
Exact mathematical passage
u_h,\widetilde u_h\in C
\widetilde u_hUse in source · 6 passages
Search aliases: widetilde u_h widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1937
**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write
Exact mathematical passage
u_h,\widetilde u_h\in C
\widetilde vUse in source · 3 passages
Search aliases: widetilde v widetilde
Use in source · Read passage · FOUNDATIONS.md, line 2587
Taking the infimum proves integrated instrument contraction. At each continuous history let v_h,widetilde v_h be the unnormalized state densities, including the policy factors. Split the next difference as
Exact mathematical passage
v_h,\widetilde v_h
\widetilde v_hUse in source · 3 passages
Search aliases: widetilde v_h widetilde
Use in source · Read passage · FOUNDATIONS.md, line 2587
Taking the infimum proves integrated instrument contraction. At each continuous history let v_h,widetilde v_h be the unnormalized state densities, including the policy factors. Split the next difference as
Exact mathematical passage
v_h,\widetilde v_h
\widetilde ZUse in source · 6 passages
Search aliases: widetilde Z widetilde
Use in source · Read passage · FOUNDATIONS.md, line 630
Z_+=BZ_-+R, widetilde Z_-=Z_-+E, widetilde Z_+=Z_++F.
Exact mathematical passage
Z_+=BZ_-+R,\qquad \widetilde Z_-=Z_-+E,\qquad \widetilde Z_+=Z_++F.
\widetilde zUse in source · 5 passages
Search aliases: widetilde z widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1829
Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints
Exact mathematical passage
\widetilde z_0
\widetilde Z_+Use in source · 2 passages
Search aliases: widetilde Z_+ widetilde
Use in source · Read passage · FOUNDATIONS.md, line 630
Z_+=BZ_-+R, widetilde Z_-=Z_-+E, widetilde Z_+=Z_++F.
Exact mathematical passage
Z_+=BZ_-+R,\qquad \widetilde Z_-=Z_-+E,\qquad \widetilde Z_+=Z_++F.
\widetilde Z_-Use in source · 3 passages
Search aliases: widetilde Z_- widetilde
Use in source · Read passage · FOUNDATIONS.md, line 630
Z_+=BZ_-+R, widetilde Z_-=Z_-+E, widetilde Z_+=Z_++F.
Exact mathematical passage
Z_+=BZ_-+R,\qquad \widetilde Z_-=Z_-+E,\qquad \widetilde Z_+=Z_++F.
\widetilde Z_-^\topUse in source · 1 passage
Search aliases: widetilde Z_-^top widetilde top
Use in source · Read passage · FOUNDATIONS.md, line 657
**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. square
Exact mathematical passage
\|\widetilde Z_-^\top v\|\geq\|Z_-^\top v\|-\|E^\top v\|\geq s-\|E\|
\widetilde Z_-^\daggerUse in source · 3 passages
Search aliases: widetilde Z_-^† widetilde dagger
Use in source · Read passage · FOUNDATIONS.md, line 640
widehat B=widetilde Z_+widetilde Z_-^†
Exact mathematical passage
\widehat B=\widetilde Z_+\widetilde Z_-^\dagger
\widetilde z_0Use in source · 5 passages
Search aliases: widetilde z_0 widetilde
Use in source · Read passage · FOUNDATIONS.md, line 1829
Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints
Exact mathematical passage
\widetilde z_0
\widetilde\nuUse in source · 2 passages
Search aliases: widetildeν widetilde nu
Use in source · Read passage · FOUNDATIONS.md, line 3958
**Proof.** For ρ<1, write P(s,·)=ρμ+(1-ρ)R(s,·), where R is a Markov kernel. Applying a Markov kernel cannot increase total variation, by integration of bounded test functions. Thus ν P-widetildeν P _TV≤(1-ρ) ν-widetildeν _TV. Adding and subtracting widetildeν P yields e_t+1≤(1-ρ)e_t+Δ. Iterate this inequality. The case ρ=1 follows directly. Applying the finite-record telescoping argument from the initial time t gives the rolling-window statement.
Exact mathematical passage
\|\nu P-\widetilde\nu P\|_{\mathrm{TV}}\leq(1-\rho)\|\nu-\widetilde\nu\|_{\mathrm{TV}}
XUse in source · 67 passages
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Use in source · Read passage · FOUNDATIONS.md, line 11
An **observation** is an element x of a declared space X. Examples include a finite real signal, a complex field on a finite grid, or an array of categorical labels. Its type includes shape, scalar field or palette, and boundary convention. A zero-extended finite sequence and a periodic sequence are different objects even when their displayed samples agree.
Exact mathematical passage
X
xUse in source · 167 passages
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Use in source · Read passage · FOUNDATIONS.md, line 11
An **observation** is an element x of a declared space X. Examples include a finite real signal, a complex field on a finite grid, or an array of categorical labels. Its type includes shape, scalar field or palette, and boundary convention. A zero-extended finite sequence and a periodic sequence are different objects even when their displayed samples agree.
Exact mathematical passage
x
x'Use in source · 5 passages
Search aliases: x'
Use in source · Read passage · FOUNDATIONS.md, line 195
γ(x)=γ(x') Longrightarrow γ(F_u(x))=γ(F_u(x'))
Exact mathematical passage
\gamma(x)=\gamma(x')\ \Longrightarrow\ \gamma(F_u(x))=\gamma(F_u(x'))
X^*Use in source · 1 passage
Search aliases: X^*
Use in source · Read passage · FOUNDATIONS.md, line 266
This gives a precise role for obstruction-driven refinement: add observable directions created by the dynamics until the chosen prediction variables close. It does not guarantee that the completion is small. In some systems W_*=X^* and exact memoryless closure needs the full state.
Exact mathematical passage
W_*=X^*
x^2Use in source · 13 passages
Search aliases: x^2
Use in source · Read passage · FOUNDATIONS.md, line 2050
C_ L={(t,x,y):t≥sqrtx^2+y^2}
Exact mathematical passage
C_{\mathrm L}=\{(t,x,y):t\geq\sqrt{x^2+y^2}\}
x^kUse in source · 1 passage
Search aliases: x^k
Use in source · Read passage · FOUNDATIONS.md, line 2906
b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).
Exact mathematical passage
b_{n,k}(x)=\binom nk x^k(1-x)^{n-k},\qquad D_n(x,v)=(n+1)\sum_{k=0}^n b_{n,k}(x)b_{n,k}(v).
X^mUse in source · 1 passage
Search aliases: X^m
Use in source · Read passage · FOUNDATIONS.md, line 171
where each B_m:X^m→ Y is multilinear. Then
Exact mathematical passage
B_m:X^m\to Y
X^\daggerUse in source · 1 passage
Search aliases: X^† dagger
Use in source · Read passage · FOUNDATIONS.md, line 1389
(I-widehat P)X _2≤2ε, (I-widehat P)P _2 = (I-widehat P)XX^† _2≤2ε/s.
Exact mathematical passage
\|(I-\widehat P)X\|_2\leq2\varepsilon, \qquad \|(I-\widehat P)P\|_2 =\|(I-\widehat P)XX^\dagger\|_2\leq2\varepsilon/s.
x_0Use in source · 6 passages
Search aliases: x_0
Use in source · Read passage · FOUNDATIONS.md, line 479
x_t+1=Lx_t, L=tfrac12 · 1 [2pt]tfrac14 · tfrac13, x_0=1 0.
Exact mathematical passage
x_{t+1}=Lx_t,\qquad L=\begin{pmatrix}\tfrac12&1\\[2pt]\tfrac14&\tfrac13\end{pmatrix}, \qquad x_0=\begin{pmatrix}1\\0\end{pmatrix}.
x_1Use in source · 3 passages
Search aliases: x_1
Use in source · Read passage · FOUNDATIONS.md, line 1016
For example, γ(x_1,x_2)=x_1 and η(x_1,x_2)=(x_1,x_2). The states (0,1) and (0,-1) have the same coarse state, but any common point initialization has worst-case error at least one in Euclidean norm. Padding a vector with zeros is a choice, not information recovery.
Exact mathematical passage
\gamma(x_1,x_2)=x_1
x_2Use in source · 3 passages
Search aliases: x_2
Use in source · Read passage · FOUNDATIONS.md, line 1016
For example, γ(x_1,x_2)=x_1 and η(x_1,x_2)=(x_1,x_2). The states (0,1) and (0,-1) have the same coarse state, but any common point initialization has worst-case error at least one in Euclidean norm. Padding a vector with zeros is a choice, not information recovery.
Exact mathematical passage
\gamma(x_1,x_2)=x_1
X_{pq}Use in source · 1 passage
Search aliases: X_pq
Use in source · Read passage · FOUNDATIONS.md, line 2322
[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.
Exact mathematical passage
[\Phi_F(X)]_{ij}=\sum_{p,q=1}^d F_{(i,p),(j,q)}X_{pq}.
x_sUse in source · 3 passages
Search aliases: x_s
Use in source · Read passage · FOUNDATIONS.md, line 571
The coordinates of O_hx_s are future observations relative to time s. They are not available at time s merely because this matrix is written down. At time t≥ h-1, however, the same vector is the observed history
Exact mathematical passage
O_hx_s
X_{s,j}Use in source · 2 passages
Search aliases: X_s,j
Use in source · Read passage · FOUNDATIONS.md, line 3052
R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).
Exact mathematical passage
R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
x_tUse in source · 22 passages
Search aliases: x_t
Use in source · Read passage · FOUNDATIONS.md, line 210
Suppose X is a Hilbert space, P is an orthogonal projection, Q=I-P and x_t+1=Lx_t. The retained state is z_t=Px_t. Then
Exact mathematical passage
x_{t+1}=Lx_t
x_{t+1}Use in source · 4 passages
Search aliases: x_t+1
Use in source · Read passage · FOUNDATIONS.md, line 210
Suppose X is a Hilbert space, P is an orthogonal projection, Q=I-P and x_t+1=Lx_t. The retained state is z_t=Px_t. Then
Exact mathematical passage
x_{t+1}=Lx_t
X_{t,j}Use in source · 1 passage
Search aliases: X_t,j
Use in source · Read passage · FOUNDATIONS.md, line 3042
X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.
Exact mathematical passage
X_{t,j}=\phi_{t,j}(Y_t),\quad m_{t,j}=\mathbb E[X_{t,j}\mid\mathcal F_{t-1}],\quad p_{t,j}=\text{the model's prediction for that test}.
x_{t-h+1}Use in source · 1 passage
Search aliases: x_t-h+1
Use in source · Read passage · FOUNDATIONS.md, line 573
z_t=y_t-h+1 vdots y_t =O_hx_t-h+1.
Exact mathematical passage
z_t=\begin{pmatrix}y_{t-h+1}\\\vdots\\y_t\end{pmatrix} =O_hx_{t-h+1}.
x_vUse in source · 3 passages
Search aliases: x_v
Use in source · Read passage · FOUNDATIONS.md, line 2856
**Proposition 63. Conditional operator-only ellipsoid discovery.** Suppose the symmetric solution space of A^ THA=H is one-dimensional and contains a positive definite matrix H_0. Let a rank-one fitted branch at a declared observation have a nonzero-mass image, normalized to v=(1,x_v). If x_v≠ c, define
Exact mathematical passage
v=(1,x_v)
YUse in source · 26 passages
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Use in source · Read passage · FOUNDATIONS.md, line 137
Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.
Exact mathematical passage
L:X\to Y
\mathcal YUse in source · 1 passage
Search aliases: Y mathcal
Use in source · Read passage · FOUNDATIONS.md, line 2280
Let (Y, Y) be a standard Borel observation space and let μ be a fixed probability reference measure. Keep a finite action set. For each action a, let B_a(y):E→ E be measurable and integrable in operator norm. Let (C,λ) be a proper probability cone and mass functional as in Section 46. Assume that, outside one μ-null set for each action,
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(Y,\mathcal Y)
yUse in source · 76 passages
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Use in source · Read passage · FOUNDATIONS.md, line 274
Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix
Exact mathematical passage
\ell_y(i)>0
y^2Use in source · 9 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2050
C_ L={(t,x,y):t≥sqrtx^2+y^2}
Exact mathematical passage
C_{\mathrm L}=\{(t,x,y):t\geq\sqrt{x^2+y^2}\}
y_0Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 695
y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.
Exact mathematical passage
y_0=1,\quad y_1=1/2,\quad y_2=1/2,\quad y_3=11/24.
y_1Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 695
y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.
Exact mathematical passage
y_0=1,\quad y_1=1/2,\quad y_2=1/2,\quad y_3=11/24.
y_2Use in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 695
y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.
Exact mathematical passage
y_0=1,\quad y_1=1/2,\quad y_2=1/2,\quad y_3=11/24.
y_3Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 695
y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.
Exact mathematical passage
y_0=1,\quad y_1=1/2,\quad y_2=1/2,\quad y_3=11/24.
y_4Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 722
It predicts y_4=61/144, which was not used to identify the coefficients. Proposition 14 and the verified rank condition on the true observation maps establish why this recurrence continues for every initial state of this particular system. The four samples alone establish the fitted recurrence only within the declared closed two-state model class. Proposition 17 forbids confusing those two conclusions.
Exact mathematical passage
y_4=61/144
Y_{a,t}Use in source · 1 passage
Search aliases: Y_a,t
Use in source · Read passage · FOUNDATIONS.md, line 3845
ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.
Exact mathematical passage
\begin{aligned} \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\ \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4. \end{aligned}
Y_{b,t}Use in source · 1 passage
Search aliases: Y_b,t
Use in source · Read passage · FOUNDATIONS.md, line 3845
ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.
Exact mathematical passage
\begin{aligned} \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\ \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4. \end{aligned}
Y_sUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
y_sUse in source · 1 passage
Search aliases: y_s
Use in source · atlas.tex, line 136
1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) only after a one-step warm-up. Known-operator completion does not establish sensor access to arbitrary new observables. navInvariantCompletion.lean; Pr…
Exact mathematical passage
Ox_s=(y_s,y_{s+1})^{\mathsf T}
y_{s+1}Use in source · 1 passage
Search aliases: y_s+1
Use in source · atlas.tex, line 136
1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) only after a one-step warm-up. Known-operator completion does not establish sensor access to arbitrary new observables. navInvariantCompletion.lean; Pr…
Exact mathematical passage
Ox_s=(y_s,y_{s+1})^{\mathsf T}
Y_tUse in source · 5 passages
Search aliases: Y_t
Use in source · Read passage · FOUNDATIONS.md, line 3042
X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.
Exact mathematical passage
X_{t,j}=\phi_{t,j}(Y_t),\quad m_{t,j}=\mathbb E[X_{t,j}\mid\mathcal F_{t-1}],\quad p_{t,j}=\text{the model's prediction for that test}.
y_TUse in source · 1 passage
Search aliases: y_T
Use in source · Read passage · FOUNDATIONS.md, line 2302
λ B_a_T(y_T)⋯ B_a_1(y_1)z_0
Exact mathematical passage
\lambda B_{a_T}(y_T)\cdots B_{a_1}(y_1)z_0
y_tUse in source · 14 passages
Search aliases: y_t
Use in source · Read passage · FOUNDATIONS.md, line 558
Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:
Exact mathematical passage
y_t\in\mathbb R^p
y_{t+1}Use in source · 4 passages
Search aliases: y_t+1
Use in source · Read passage · FOUNDATIONS.md, line 718
y_t+1=frac56y_t+frac112y_t-1, t≥1.
Exact mathematical passage
\boxed{y_{t+1}=\frac56y_t+\frac1{12}y_{t-1},\qquad t\geq1.}
y_{t-1}Use in source · 3 passages
Search aliases: y_t-1
Use in source · Read passage · FOUNDATIONS.md, line 718
y_t+1=frac56y_t+frac112y_t-1, t≥1.
Exact mathematical passage
\boxed{y_{t+1}=\frac56y_t+\frac1{12}y_{t-1},\qquad t\geq1.}
y_{t-1-j}Use in source · 1 passage
Search aliases: y_t-1-j
Use in source · atlas.tex, line 182
plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, retaining the last H history terms leaves a same-history tail at most Δ_H=tfrac38 3^-H.
Exact mathematical passage
y_{t+1}=Ay_t+\sum_{j=0}^{t-1}BD^jC\,y_{t-1-j}+BD^tw_0.
y_{t-h+1}Use in source · 1 passage
Search aliases: y_t-h+1
Use in source · Read passage · FOUNDATIONS.md, line 573
z_t=y_t-h+1 vdots y_t =O_hx_t-h+1.
Exact mathematical passage
z_t=\begin{pmatrix}y_{t-h+1}\\\vdots\\y_t\end{pmatrix} =O_hx_{t-h+1}.
ZUse in source · 57 passages
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Use in source · Read passage · FOUNDATIONS.md, line 151
For another linear map M:Y→ Z and output projectors R_k,
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M:Y\to Z
zUse in source · 144 passages
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Use in source · Read passage · FOUNDATIONS.md, line 15
A **predictive state** is a summary z=γ(x), or more generally a summary of an observation history. It is sufficient only relative to a specified family of future quantities, actions and dynamics. There is no implication from exact storage of x to sufficiency of γ(x).
Exact mathematical passage
z=\gamma(x)
z'Use in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1663
z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.
Exact mathematical passage
z'=Tz,\qquad B_s'=TB_sT^{-1},\qquad \lambda'=\lambda T^{-1},\qquad C'=TC.
z^{\mathsf T}Use in source · 2 passages
Search aliases: z^ T mathsf
Use in source · Read passage · FOUNDATIONS.md, line 2791
Then C_c,H={z:m≥0, z^ TQz≥0}. The cone is proper and generating, and m is strictly positive on every nonzero cone element. These facts follow by an invertible linear transformation to the Lorentz cone. In particular, an output with zero mass in the cone is the zero vector.
Exact mathematical passage
C_{c,H}=\{z:m\geq0,\ z^{\mathsf T}Qz\geq0\}
z^+Use in source · 9 passages
Search aliases: z^+
Use in source · Read passage · FOUNDATIONS.md, line 1188
Pr(omid z,a)=λ B_a,oz, z^+=fracB_a,ozλ B_a,oz.
Exact mathematical passage
\Pr(o\mid z,a)=\lambda B_{a,o}z,\qquad z^+=\frac{B_{a,o}z}{\lambda B_{a,o}z}.
z^\topUse in source · 1 passage
Search aliases: z^top top
Use in source · Read passage · FOUNDATIONS.md, line 798
Then z _M=(z^top Mz)^1/2 gives widehat B_uz _M≤ρ z _M for all actions, directly from the displayed inequality. Proposition 18 therefore applies under arbitrary switching among them. This is the common quadratic Lyapunov construction from control theory [R8], used here as an explicit certificate rather than inferred from a list of eigenvalues.
Exact mathematical passage
\|z\|_M=(z^\top Mz)^{1/2}
Z_+Use in source · 14 passages
Search aliases: Z_+
Use in source · Read passage · FOUNDATIONS.md, line 598
Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.
Exact mathematical passage
Z_-,Z_+\in\mathbb R^{m\times N}
Z_-Use in source · 17 passages
Search aliases: Z_-
Use in source · Read passage · FOUNDATIONS.md, line 598
Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.
Exact mathematical passage
Z_-,Z_+\in\mathbb R^{m\times N}
Z_-^{-1}Use in source · 2 passages
Search aliases: Z_-^-1
Use in source · Read passage · FOUNDATIONS.md, line 709
D_2=Z_+Z_-^-1 =0 · 1 1/12 · 5/6.
Exact mathematical passage
\boxed{ D_2=Z_+Z_-^{-1} =\begin{pmatrix}0&1\\1/12&5/6\end{pmatrix}. }
Z_-^\topUse in source · 1 passage
Search aliases: Z_-^top top
Use in source · Read passage · FOUNDATIONS.md, line 657
**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. square
Exact mathematical passage
\|\widetilde Z_-^\top v\|\geq\|Z_-^\top v\|-\|E^\top v\|\geq s-\|E\|
Z_-^\daggerUse in source · 2 passages
Search aliases: Z_-^† dagger
Use in source · Read passage · FOUNDATIONS.md, line 610
B_0=Z_+Z_-^†,
Exact mathematical passage
B_0=Z_+Z_-^\dagger,
z_0Use in source · 30 passages
Search aliases: z_0
Use in source · Read passage · FOUNDATIONS.md, line 502
z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.
Exact mathematical passage
z_0=1,\qquad z_1=\tfrac12,\qquad z_2=\tfrac12, \qquad z_3=\tfrac{11}{24}.
z_1Use in source · 2 passages
Search aliases: z_1
Use in source · Read passage · FOUNDATIONS.md, line 502
z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.
Exact mathematical passage
z_0=1,\qquad z_1=\tfrac12,\qquad z_2=\tfrac12, \qquad z_3=\tfrac{11}{24}.
z_2Use in source · 4 passages
Search aliases: z_2
Use in source · Read passage · FOUNDATIONS.md, line 502
z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.
Exact mathematical passage
z_0=1,\qquad z_1=\tfrac12,\qquad z_2=\tfrac12, \qquad z_3=\tfrac{11}{24}.
z_3Use in source · 2 passages
Search aliases: z_3
Use in source · Read passage · FOUNDATIONS.md, line 502
z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.
Exact mathematical passage
z_0=1,\qquad z_1=\tfrac12,\qquad z_2=\tfrac12, \qquad z_3=\tfrac{11}{24}.
z_iUse in source · 1 passage
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Use in source · Read passage · FOUNDATIONS.md, line 1318
z=∑_i:w_i>0 w_i z_i, w_i=λ P_iz≥0, z_i=P_iz/w_i, ∑_iw_i=1.
Exact mathematical passage
z=\sum_{i:w_i>0} w_i z_i,\qquad w_i=\lambda P_iz\geq0,\qquad z_i=P_iz/w_i,\qquad\sum_iw_i=1.
Z_nUse in source · 9 passages
Search aliases: Z_n
Use in source · Read passage · FOUNDATIONS.md, line 1040
Let finite active sets G_n increase to G. Their unnormalized evidence is Z_n=∑_g∈ G_nπ(g)L_g(D). Only stages with Z_n>0 are considered.
Exact mathematical passage
Z_n=\sum_{g\in\mathcal G_n}\pi(g)L_g(D)
Z_qUse in source · 2 passages
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Use in source · Read passage · FOUNDATIONS.md, line 2337
Z_q(u)=I_d^2 uI_d^2 vdots u^qI_d^2.
Exact mathematical passage
Z_q(u)=\begin{pmatrix}I_{d^2}\\uI_{d^2}\\\vdots\\u^qI_{d^2}\end{pmatrix}.
Z_{q-1}Use in source · 1 passage
Search aliases: Z_q-1
Use in source · Read passage · FOUNDATIONS.md, line 2343
F_a(u) · =Z_q(u)^ TG_a,0Z_q(u) · +(1-u^2)Z_q-1(u)^ TG_a,1Z_q-1(u), J B_a(u) · =Φ_F_a(u) J.
Exact mathematical passage
\begin{aligned} F_a(u)&=Z_q(u)^{\mathsf T}G_{a,0}Z_q(u)\\ &\quad +(1-u^2)Z_{q-1}(u)^{\mathsf T}G_{a,1}Z_{q-1}(u),\\ \mathcal J B_a(u)&=\Phi_{F_a(u)}\mathcal J. \end{aligned}
Z_rUse in source · 3 passages
Search aliases: Z_r
Use in source · Read passage · FOUNDATIONS.md, line 400
with one common c>0. For a fixed prior, put δ=r(p)-barℓ, bar b= B_barℓ(P) and Z_r=r(p)^top P. Then
Exact mathematical passage
Z_r=r(p)^\top P
z_sUse in source · 3 passages
Search aliases: z_s
Use in source · Read passage · FOUNDATIONS.md, line 341
If D ≤ρ<1 and z_s ≤ Z over the relevant history, omitting terms with k≥ H changes the one-step right-hand side by at most
Exact mathematical passage
\|z_s\|\leq Z
Z_tUse in source · 8 passages
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Use in source · Read passage · FOUNDATIONS.md, line 738
Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is
Exact mathematical passage
\gamma_t:X\to Z_t
z_tUse in source · 17 passages
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Use in source · Read passage · FOUNDATIONS.md, line 210
Suppose X is a Hilbert space, P is an orthogonal projection, Q=I-P and x_t+1=Lx_t. The retained state is z_t=Px_t. Then
Exact mathematical passage
z_t=Px_t
Z_{t+1}Use in source · 4 passages
Search aliases: Z_t+1
Use in source · Read passage · FOUNDATIONS.md, line 738
Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is
Exact mathematical passage
\widehat f_t:Z_t\to Z_{t+1}
z_{t+1}Use in source · 8 passages
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Use in source · Read passage · FOUNDATIONS.md, line 212
z_t+1=PLPz_t+PLQx_t.
Exact mathematical passage
z_{t+1}=PLPz_t+PLQx_t.
z_{t-1-k}Use in source · 3 passages
Search aliases: z_t-1-k
Use in source · Read passage · FOUNDATIONS.md, line 334
z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.
Exact mathematical passage
\boxed{ z_{t+1}=Az_t+ \sum_{k=0}^{t-1}BD^kCz_{t-1-k}+BD^tq_0. }
|Use in source · 56 passages
Search aliases: |
Use in source · Read passage · FOUNDATIONS.md, line 304
where B ranges over the current partition. The process terminates after at most |Ω|-|π_init| strict refinement rounds. The final partition is the unique coarsest refinement of π_0 on which all declared predictions and Bayesian updates close for every prior.
Exact mathematical passage
|\Omega|-|\pi_{\mathrm{init}}|
|^kUse in source · 1 passage
Search aliases: |^k
Use in source · Read passage · FOUNDATIONS.md, line 3544
Using ∫_-1^1|u|^kdu/2=1/(k+1) gives the integrated upper budget 241/40000. The same center-reset reference is valid there. Both models have the same initial state.
Exact mathematical passage
\int_{-1}^1|u|^kdu/2=1/(k+1)
\pmUse in source · 7 passages
Search aliases: ± pm
Use in source · Read passage · FOUNDATIONS.md, line 2062
t± e_1^top R^k(x,y)^top≥0 for every k≥0.
Exact mathematical passage
t\pm e_1^\top R^k(x,y)^\top\geq0 \quad\text{for every }k\geq0.
\timesUse in source · 18 passages
Search aliases: × times
Use in source · Read passage · FOUNDATIONS.md, line 598
Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.
Exact mathematical passage
Z_-,Z_+\in\mathbb R^{m\times N}
\alphaUse in source · 34 passages
Search aliases: α alpha
Use in source · Read passage · FOUNDATIONS.md, line 1077
**Proposition 24. Prior-preserving program refinement.** Suppose a parent program g is replaced by children g_j with weights α_j≥0, ∑_jα_j=1, and suppose its likelihood is the mixture
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\alpha_j\geq0
\alpha_0Use in source · 10 passages
Search aliases: α_0 alpha
Use in source · Read passage · FOUNDATIONS.md, line 1764
If these identities hold for every symbol and z_0=Vα_0 for α_0≥0 with 1^topα_0=1, then
Exact mathematical passage
z_0=V\alpha_0
\alpha_jUse in source · 9 passages
Search aliases: α_j alpha
Use in source · Read passage · FOUNDATIONS.md, line 1077
**Proposition 24. Prior-preserving program refinement.** Suppose a parent program g is replaced by children g_j with weights α_j≥0, ∑_jα_j=1, and suppose its likelihood is the mixture
Exact mathematical passage
\alpha_j\geq0
\alpha_tUse in source · 1 passage
Search aliases: α_t alpha
Use in source · Read passage · FOUNDATIONS.md, line 1485
e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.
Exact mathematical passage
e_{t+1}\leq\alpha_te_t+\beta_t,\qquad \alpha_t=\frac{(1+R\|\lambda\|_*)L_t}{\widehat\eta_t},\qquad \beta_t=\frac{(1+R\|\lambda\|_*)\epsilon_{B,t}R}{\widehat\eta_t}.
\betaUse in source · 11 passages
Search aliases: β beta
Use in source · Read passage · FOUNDATIONS.md, line 900
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
Exact mathematical passage
\beta_{m,\boldsymbol i}
\beta_aUse in source · 5 passages
Search aliases: β_a beta
Use in source · Read passage · FOUNDATIONS.md, line 2795
λ B_a(u)=(b_a,0(u),b_a(u)^ T), β_a(u)=b_a,0(u)+b_a(u)^ Tc.
Exact mathematical passage
\lambda B_a(u)=(b_{a,0}(u),b_a(u)^{\mathsf T}),\qquad \beta_a(u)=b_{a,0}(u)+b_a(u)^{\mathsf T}c.
\beta_{a,j,i}Use in source · 2 passages
Search aliases: β_a,j,i beta
Use in source · Read passage · FOUNDATIONS.md, line 3747
Under the rowwise noise and parameter-bound assumptions of Proposition 81, simultaneous confidence ellipsoids for the rows of M_a,j give a confidence set for every C_a,k by this known linear map. If row i of widehat M_a,j-M_a,j has V_a,t-norm at most β_a,j,i, then for any direction x,
Exact mathematical passage
\beta_{a,j,i}
\beta_{m,\boldsymbol i}Use in source · 3 passages
Search aliases: β_m, i beta boldsymbol
Use in source · Read passage · FOUNDATIONS.md, line 900
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
Exact mathematical passage
\beta_{m,\boldsymbol i}
\beta_tUse in source · 1 passage
Search aliases: β_t beta
Use in source · Read passage · FOUNDATIONS.md, line 1485
e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.
Exact mathematical passage
e_{t+1}\leq\alpha_te_t+\beta_t,\qquad \alpha_t=\frac{(1+R\|\lambda\|_*)L_t}{\widehat\eta_t},\qquad \beta_t=\frac{(1+R\|\lambda\|_*)\epsilon_{B,t}R}{\widehat\eta_t}.
\gamma_tUse in source · 10 passages
Search aliases: γ_t gamma
Use in source · Read passage · FOUNDATIONS.md, line 738
Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is
Exact mathematical passage
\gamma_t:X\to Z_t
\gamma_{t+1}Use in source · 2 passages
Search aliases: γ_t+1 gamma
Use in source · Read passage · FOUNDATIONS.md, line 740
r_t(x)=γ_t+1(F_t(x))-widehat f_t(γ_t(x)).
Exact mathematical passage
r_t(x)=\gamma_{t+1}(F_t(x))-\widehat f_t(\gamma_t(x)).
\DeltaUse in source · 42 passages
Search aliases: Δ Delta
Use in source · Read passage · FOUNDATIONS.md, line 834
a= widehat A +∑_k=0^H-1 widehat K_k <1, sup_t≥0 d_t ≤Δ.
Exact mathematical passage
a=\|\widehat A\|+\sum_{k=0}^{H-1}\|\widehat K_k\|<1, \qquad\sup_{t\geq0}\|d_t\|\leq\Delta.
\deltaUse in source · 58 passages
Search aliases: δ delta
Use in source · Read passage · FOUNDATIONS.md, line 106
P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.
Exact mathematical passage
P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i, \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
\Delta_aUse in source · 5 passages
Search aliases: Δ_a Delta
Use in source · Read passage · FOUNDATIONS.md, line 3219
**Proposition 71. Robust oriented margin.** Suppose σ_a(u)≥κ_a(u) and an uncertain family has transported errors Δ_a(u) satisfying, for every normalized input,
Exact mathematical passage
\Delta_a(u)
\Delta_HUse in source · 3 passages
Search aliases: Δ_H Delta
Use in source · Read passage · FOUNDATIONS.md, line 875
a_H=frac78-frac38 3^-H, Δ_H=frac38 3^-H, sup_t|z_t-widehat z_t|≤ frac3 3^-H1+3 3^-H.
Exact mathematical passage
a_H=\frac78-\frac38\,3^{-H},\qquad \Delta_H=\frac38\,3^{-H},\qquad \boxed{\sup_t|z_t-\widehat z_t|\leq \frac{3\,3^{-H}}{1+3\,3^{-H}}.}
\delta_{ij}Use in source · 1 passage
Search aliases: δ_ij delta
Use in source · Read passage · FOUNDATIONS.md, line 106
P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.
Exact mathematical passage
P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i, \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
\delta_kUse in source · 2 passages
Search aliases: δ_k delta
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
\delta_nUse in source · 2 passages
Search aliases: δ_n delta
Use in source · Read passage · FOUNDATIONS.md, line 2126
Pr(true model∉ U_nmidprevious data) ≤δ_n, ∑_n≥1δ_n≤δ.
Exact mathematical passage
\Pr(\text{true model}\notin\mathcal U_n\mid\text{previous data}) \leq\delta_n, \qquad\sum_{n\geq1}\delta_n\leq\delta.
\delta_sUse in source · 7 passages
Search aliases: δ_s delta
Use in source · Read passage · FOUNDATIONS.md, line 1537
z=Rp, δ_s=(RM_s-B_sR)p.
Exact mathematical passage
z=Rp,\qquad \delta_s=(RM_s-B_sR)p.
\delta_tUse in source · 4 passages
Search aliases: δ_t delta
Use in source · Read passage · FOUNDATIONS.md, line 747
**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,
Exact mathematical passage
\|r_t(x_t)\|\leq\delta_t
\epsilonUse in source · 58 passages
Search aliases: ϵ epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1131
Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.
Exact mathematical passage
\epsilon
\varepsilonUse in source · 15 passages
Search aliases: ε varepsilon
Use in source · Read passage · FOUNDATIONS.md, line 924
If the declared polynomial itself approximates a nonpolynomial map with remainder at most ε_R on the ball, add ε_R to the defect bound. Without such a remainder estimate, the result is only a theorem for the polynomial, not for the unexpanded physical law.
Exact mathematical passage
\varepsilon_R
\epsilon^2Use in source · 1 passage
Search aliases: ϵ^2 epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1525
**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the same bound. A union bound over the M words proves the simultaneous result. The spectral norm is bounded by the Frobenius norm, yielding the matrix an…
Exact mathematical passage
\exp(-2N\epsilon^2)
\epsilon_aUse in source · 2 passages
Search aliases: ϵ_a epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1893
ϵ_a=sup_z∈ K∑_o (B_a,o-widetilde B_a,o)z _C.
Exact mathematical passage
\epsilon_a=\sup_{z\in K}\sum_o \|(B_{a,o}-\widetilde B_{a,o})z\|_C.
\epsilon_BUse in source · 2 passages
Search aliases: ϵ_B epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1461
For a predicted state widehat z, form widehat u=widehat B_swidehat z, widehat p=λwidehat u, and define d=widehat u-u. Write e= widehat z-z , widehat B_s ≤ L, and widehat B_s-B_s ≤ϵ_B. Then
Exact mathematical passage
\|\widehat B_s-B_s\|\leq\epsilon_B
\epsilon_{B,t}Use in source · 1 passage
Search aliases: ϵ_B,t epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1485
e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.
Exact mathematical passage
e_{t+1}\leq\alpha_te_t+\beta_t,\qquad \alpha_t=\frac{(1+R\|\lambda\|_*)L_t}{\widehat\eta_t},\qquad \beta_t=\frac{(1+R\|\lambda\|_*)\epsilon_{B,t}R}{\widehat\eta_t}.
\epsilon_NUse in source · 3 passages
Search aliases: ϵ_N epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1509
|widehat f(w)-f(w)|≤ ϵ_N, ϵ_N=sqrtfraclog(2M/δ)2N.
Exact mathematical passage
|\widehat f(w)-f(w)|\leq \epsilon_N,\qquad \epsilon_N=\sqrt{\frac{\log(2M/\delta)}{2N}}.
\varepsilon_RUse in source · 2 passages
Search aliases: ε_R varepsilon
Use in source · Read passage · FOUNDATIONS.md, line 924
If the declared polynomial itself approximates a nonpolynomial map with remainder at most ε_R on the ball, add ε_R to the defect bound. Without such a remainder estimate, the result is only a theorem for the polynomial, not for the unexpanded physical law.
Exact mathematical passage
\varepsilon_R
\varepsilon_{\mathrm{repair}}Use in source · 1 passage
Search aliases: ε_repair varepsilon mathrm
Use in source · Read passage · FOUNDATIONS.md, line 3637
ε_repair ≤frac3132frac24140000+frac116 =frac874711280000.
Exact mathematical passage
\varepsilon_{\mathrm{repair}} \leq\frac{31}{32}\frac{241}{40000}+\frac1{16} =\frac{87471}{1280000}.
\varepsilon_{\rm repaired}Use in source · 1 passage
Search aliases: ε_rm repaired varepsilon rm
Use in source · Read passage · FOUNDATIONS.md, line 3548
ε_rm repaired ≤frac78frac24140000+frac14 =frac81687320000.
Exact mathematical passage
\varepsilon_{\rm repaired} \leq\frac78\frac{241}{40000}+\frac14 =\frac{81687}{320000}.
\epsilon_sUse in source · 5 passages
Search aliases: ϵ_s epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1804
g_i^topwidehat B_sv_j ≥ϵ_s g_i _2 v_j _2 for all s,i,j.
Exact mathematical passage
\boxed{ g_i^\top\widehat B_sv_j \geq\epsilon_s\|g_i\|_2\|v_j\|_2 \quad\text{for all }s,i,j. }
\epsilon_tUse in source · 14 passages
Search aliases: ϵ_t epsilon
Use in source · Read passage · FOUNDATIONS.md, line 1925
**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy
Exact mathematical passage
\epsilon_t
\epsilon_\PiUse in source · 3 passages
Search aliases: ϵ_Π epsilon Pi
Use in source · Read passage · FOUNDATIONS.md, line 2522
ϵ_Π:=sup_a,z∈ K∫ (B_a(y)-B_a^Π(y))z _C μ(dy) ≤ Lh.
Exact mathematical passage
\epsilon_\Pi:=\sup_{a,z\in K}\int \|(B_a(y)-B_a^\Pi(y))z\|_C\,\mu(dy) \leq Lh.
\etaUse in source · 55 passages
Search aliases: η eta
Use in source · Read passage · FOUNDATIONS.md, line 761
Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then
Exact mathematical passage
\|h_t(x_t)-H_t(\gamma_t(x_t))\|\leq\eta_t
\eta_*Use in source · 3 passages
Search aliases: η_* eta
Use in source · Read passage · FOUNDATIONS.md, line 3598
**Proposition 80. Convex repair bracketing.** The feasible repair weights form a closed interval [η_*,1]. If a cone-positive functional ℓ=(1,-v^ T), with v _2≤1, and a normalized input z give c=ℓwidehat F_a(u)z<κ, then every feasible weight satisfies
Exact mathematical passage
[\eta_*,1]
\eta_{a,k}Use in source · 3 passages
Search aliases: η_a,k eta
Use in source · Read passage · FOUNDATIONS.md, line 3009
sup_z∈ K (C_a,n,k-widetilde C_a,n,k)z _C ≤η_a,k,
Exact mathematical passage
\sup_{z\in K}\|(C_{a,n,k}-\widetilde C_{a,n,k})z\|_C \leq\eta_{a,k},
\eta_tUse in source · 3 passages
Search aliases: η_t eta
Use in source · Read passage · FOUNDATIONS.md, line 761
Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then
Exact mathematical passage
\|h_t(x_t)-H_t(\gamma_t(x_t))\|\leq\eta_t
\eta_UUse in source · 1 passage
Search aliases: η_U eta
Use in source · Read passage · FOUNDATIONS.md, line 3604
A complete certificate at a weight η_U gives an upper bound on η_*. A negative Bernstein coefficient without a negative evaluated point does not give a lower bound.
Exact mathematical passage
\eta_U
\ThetaUse in source · 4 passages
Search aliases: Θ Theta
Use in source · atlas.tex, line 334
∑_s≤ tϕ_sϕ_s^ T and widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s. For 0<δ<1, with probability at least 1-δ, simultaneously for every finite t, widehatθ_t-θ_* _V_t ≤σsqrtlogfracdet V_tdet(η I)+2log(1/δ)+sqrtη S. diagram14_uncertainty In the structured moment family, a nonempty returned rectangle Θ is internally valid when its sufficient floor ρ_Θ=1-sup_(a,b)∈Θ(|a|+|b|) is nonnegative. Statistical coverage places the true parameter in Θ on a simultaneous event. The exact domain argument applies to every paramete…
Exact mathematical passage
\Theta
\theta^2Use in source · 1 passage
Search aliases: θ^2 theta
Use in source · Read passage · FOUNDATIONS.md, line 3071
E[e^θ(X-m)mid F]≤ e^θ^2/8.
Exact mathematical passage
\mathbb E[e^{\theta(X-m)}\mid\mathcal F]\leq e^{\theta^2/8}.
\theta_*Use in source · 5 passages
Search aliases: θ_* theta
Use in source · Read passage · FOUNDATIONS.md, line 3667
for every real h, with σ>0, and a declared bound θ_* _2≤ S. With ridge γ>0, set
Exact mathematical passage
\|\theta_*\|_2\leq S
\theta_*^{\mathsf T}Use in source · 2 passages
Search aliases: θ_*^ T theta mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3660
Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2
Exact mathematical passage
Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t, \qquad \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}] \leq e^{h^2\sigma^2/2}
\theta_jUse in source · 2 passages
Search aliases: θ_j theta
Use in source · Read passage · FOUNDATIONS.md, line 1814
**Proof.** For convex weights θ_j, take A_s=∑_jθ_jA_s^(j). The equality and nonnegativity survive convex combination, as does mass conservation. At each history, every selected family maps the current cone state back into the cone. Applying the mass identity at that node proves recursive normalization, regardless of how the family was selected. For the norm ball,
Exact mathematical passage
\theta_j
\kappaUse in source · 22 passages
Search aliases: κ kappa
Use in source · Read passage · FOUNDATIONS.md, line 760
Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then
Exact mathematical passage
\kappa_t
\kappa_aUse in source · 3 passages
Search aliases: κ_a kappa
Use in source · Read passage · FOUNDATIONS.md, line 3219
**Proposition 71. Robust oriented margin.** Suppose σ_a(u)≥κ_a(u) and an uncertain family has transported errors Δ_a(u) satisfying, for every normalized input,
Exact mathematical passage
\sigma_a(u)\geq\kappa_a(u)
\kappa_tUse in source · 2 passages
Search aliases: κ_t kappa
Use in source · Read passage · FOUNDATIONS.md, line 760
Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then
Exact mathematical passage
\kappa_t
\LambdaUse in source · 2 passages
Search aliases: Λ Lambda
Use in source · Read passage · FOUNDATIONS.md, line 115
**Proposition 3. Complete refinement preserves reconstruction.** Suppose a finite rooted tree has root projector I, and every internal projector is the sum of pairwise orthogonal child projectors. Every complete antichain Λ, meeting each root-to-leaf path exactly once, satisfies
Exact mathematical passage
\Lambda
\lambda'Use in source · 4 passages
Search aliases: λ' lambda
Use in source · Read passage · FOUNDATIONS.md, line 1663
z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.
Exact mathematical passage
z'=Tz,\qquad B_s'=TB_sT^{-1},\qquad \lambda'=\lambda T^{-1},\qquad C'=TC.
\lambda_jUse in source · 2 passages
Search aliases: λ_j lambda
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
\lambda_tUse in source · 4 passages
Search aliases: λ_t lambda
Use in source · Read passage · FOUNDATIONS.md, line 746
**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,
Exact mathematical passage
\lambda_t
\muUse in source · 39 passages
Search aliases: μ mu
Use in source · Read passage · FOUNDATIONS.md, line 2280
Let (Y, Y) be a standard Borel observation space and let μ be a fixed probability reference measure. Keep a finite action set. For each action a, let B_a(y):E→ E be measurable and integrable in operator norm. Let (C,λ) be a proper probability cone and mass functional as in Section 46. Assume that, outside one μ-null set for each action,
Exact mathematical passage
\mu
\mu'Use in source · 1 passage
Search aliases: μ' mu
Use in source · Read passage · FOUNDATIONS.md, line 2306
with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.
Exact mathematical passage
\mu'=r\mu
\mu^{\otimes T}Use in source · 1 passage
Search aliases: μ^⊗ T mu otimes
Use in source · Read passage · FOUNDATIONS.md, line 2306
with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.
Exact mathematical passage
\mu^{\otimes T}
\mu_s^aUse in source · 1 passage
Search aliases: μ_s^a mu
Use in source · Read passage · FOUNDATIONS.md, line 3097
Consequently Proposition 66 gives a time-uniform certificate for ∑_s≤ tμ_s^a from ∑_s≤ tW_s^a, with its boundary multiplied by 1/π_min. The target is the average action-conditioned quantity along the encountered contexts, not a reset-population table or the state distribution under a different long-term policy.
Exact mathematical passage
\sum_{s\leq t}\mu_s^a
\mu_t^aUse in source · 2 passages
Search aliases: μ_t^a mu
Use in source · Read passage · FOUNDATIONS.md, line 3085
μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).
Exact mathematical passage
\mu_t^a=\int\phi_t(y)P_t(dy\mid\mathcal H_{t-1},a),\qquad W_t^a=\frac{\mathbf1_{\{A_t=a\}}\phi_t(Y_t)}{\pi_t(a\mid\mathcal H_{t-1})}.
\nuUse in source · 12 passages
Search aliases: ν nu
Use in source · Read passage · FOUNDATIONS.md, line 2328
F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,
Exact mathematical passage
F=\sum_\nu w_\nu\operatorname{vec}(K_\nu) \operatorname{vec}(K_\nu)^{\mathsf T},\qquad w_\nu\geq0,
\nu_aUse in source · 2 passages
Search aliases: ν_a nu
Use in source · Read passage · FOUNDATIONS.md, line 3284
An additional uncertainty family with combined output error at most ν_a(u) is certified whenever
Exact mathematical passage
\nu_a(u)
\xiUse in source · 3 passages
Search aliases: ξ xi
Use in source · Read passage · FOUNDATIONS.md, line 3660
Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2
Exact mathematical passage
Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t, \qquad \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}] \leq e^{h^2\sigma^2/2}
\xi_sUse in source · 1 passage
Search aliases: ξ_s xi
Use in source · Read passage · FOUNDATIONS.md, line 3690
**Proof.** Put S_t=∑_s≤ tϕ_sξ_s. For every fixed q, the conditional moment bound makes
Exact mathematical passage
S_t=\sum_{s\leq t}\phi_s\xi_s
\xi_tUse in source · 2 passages
Search aliases: ξ_t xi
Use in source · Read passage · FOUNDATIONS.md, line 3660
Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2
Exact mathematical passage
Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t, \qquad \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}] \leq e^{h^2\sigma^2/2}
\PiUse in source · 14 passages
Search aliases: Π Pi
Use in source · Read passage · FOUNDATIONS.md, line 34
**Proposition 1. Exact realization and consistency.** The analysis map is injective, Π_a=A_aS_a is an idempotent with image C_a, and Π_ac=c exactly when c is a realizable coefficient vector. Moreover,
Exact mathematical passage
\Pi_a=A_aS_a
\Pi^2Use in source · 1 passage
Search aliases: Π^2 Pi
Use in source · atlas.tex, line 59
[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate…
Exact mathematical passage
5pt] Transport & $F_bT_{ba}=T_{ba}F_a$ & Do coordinate descriptions agree?\\[5pt] Closure & $\gamma F_u=\bar F_u\gamma$ & Does the summary determine its own future?\\[5pt] Probability & $B_oC\subseteq C$, $\lambda\sum_oB_o=\lambda$ & Are every branch and every finite record valid?\\\bottomrule \end{tabular} \vspace{10pt} \textbf{Reading route.} Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, lost paths, memory and observable identification. Plates 10--16 develop behavior spaces, positivity, curved geometry and statistical guarantees. Plate 17 locates the formal proofs. \textbf{Visual conventions.} Teal marks exact structure. Gold marks transported, refined or uncertain quantities. Red marks an obstruction or an omitted contribution. Every figure is drawn from a formula or an exact finite example. Curves of bounds are labeled as bounds. \boundary{The atlas selects core mechanisms from the 87 registered propositions. It introduces no new numbered theorem, fitted model, benchmark result or empirical coverage claim. Its diagrams supplement the proofs.} \plate{1}{Realizable coefficients form a subspace}{Exact storage includes a consistency condition.}{P1} For linear analysis and synthesis maps $A:X\to C$ and $S:C\to X$, the left-inverse law $SA=I_X$ gives injectivity of $A$. The coefficient projector is $\Pi=AS$, with \[ \Pi^2=\Pi,\qquad \im\Pi=\im A,\qquad c\text{ is realizable}\iff \Pi c=c.
\pi_0Use in source · 4 passages
Search aliases: π_0 pi
Use in source · Read passage · FOUNDATIONS.md, line 296
Fix a mandatory initial partition π_0 with at least two blocks. This preserves distinctions that the intended application requires and excludes the trivial one-state quotient.
Exact mathematical passage
\pi_0
\Pi_aUse in source · 4 passages
Search aliases: Π_a Pi
Use in source · Read passage · FOUNDATIONS.md, line 34
**Proposition 1. Exact realization and consistency.** The analysis map is injective, Π_a=A_aS_a is an idempotent with image C_a, and Π_ac=c exactly when c is a realizable coefficient vector. Moreover,
Exact mathematical passage
\Pi_a=A_aS_a
\Pi_a^2Use in source · 1 passage
Search aliases: Π_a^2 Pi
Use in source · Read passage · FOUNDATIONS.md, line 43
**Proof.** If A_ax=0, then x=S_aA_ax=0. Also Π_a^2=A_a(S_aA_a)S_a=Π_a. Its image is contained in C_a and it fixes every A_ax, so its image is exactly C_a. Both norm inequalities follow from the operator-norm inequality applied to S_a. square
Exact mathematical passage
\Pi_a^2=A_a(S_aA_a)S_a=\Pi_a
\pi_{\mathrm{init}}Use in source · 1 passage
Search aliases: π_init pi mathrm
Use in source · Read passage · FOUNDATIONS.md, line 304
where B ranges over the current partition. The process terminates after at most |Ω|-|π_init| strict refinement rounds. The final partition is the unique coarsest refinement of π_0 on which all declared predictions and Bayesian updates close for every prior.
Exact mathematical passage
|\Omega|-|\pi_{\mathrm{init}}|
\pi_{\min}Use in source · 4 passages
Search aliases: π_min pi min
Use in source · Read passage · FOUNDATIONS.md, line 3090
**Proposition 67. Covered adaptive-action evidence.** If the target action has π_t(amid H_t-1)≥π_min>0 at every time, then
Exact mathematical passage
\pi_t(a\mid\mathcal H_{t-1})\geq\pi_{\min}>0
\pi_tUse in source · 4 passages
Search aliases: π_t pi
Use in source · Read passage · FOUNDATIONS.md, line 3083
For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define
Exact mathematical passage
\pi_t
\rhoUse in source · 33 passages
Search aliases: ρ rho
Use in source · Read passage · FOUNDATIONS.md, line 341
If D ≤ρ<1 and z_s ≤ Z over the relevant history, omitting terms with k≥ H changes the one-step right-hand side by at most
Exact mathematical passage
\|D\|\leq\rho<1
\rho^2Use in source · 1 passage
Search aliases: ρ^2 rho
Use in source · Read passage · FOUNDATIONS.md, line 793
widehat B_u^top Mwidehat B_upreceqρ^2M for every declared action u.
Exact mathematical passage
\widehat B_u^\top M\widehat B_u\preceq\rho^2M \quad\text{for every declared action }u.
\rho^HUse in source · 2 passages
Search aliases: ρ^H rho
Use in source · Read passage · FOUNDATIONS.md, line 343
B C Zfracρ^H1-ρ.
Exact mathematical passage
\|B\|\,\|C\|Z\frac{\rho^H}{1-\rho}.
\rho^TUse in source · 1 passage
Search aliases: ρ^T rho
Use in source · Read passage · FOUNDATIONS.md, line 783
e_T≤ρ^Te_0+δfrac1-ρ^T1-ρ.
Exact mathematical passage
e_T\leq\rho^Te_0+\delta\frac{1-\rho^T}{1-\rho}.
\rho^tUse in source · 1 passage
Search aliases: ρ^t rho
Use in source · Read passage · FOUNDATIONS.md, line 347
Dropping the initial unresolved-state term adds at most B ρ^t q_0 .
Exact mathematical passage
\|B\|\rho^t\|q_0\|
\rho_\ThetaUse in source · 1 passage
Search aliases: ρ_Θ rho Theta
Use in source · atlas.tex, line 335
eta_t=V_t^-1∑_s≤ tϕ_sY_s. For 0<δ<1, with probability at least 1-δ, simultaneously for every finite t, widehatθ_t-θ_* _V_t ≤σsqrtlogfracdet V_tdet(η I)+2log(1/δ)+sqrtη S. diagram14_uncertainty In the structured moment family, a nonempty returned rectangle Θ is internally valid when its sufficient floor ρ_Θ=1-sup_(a,b)∈Θ(|a|+|b|) is nonnegative. Statistical coverage places the true parameter in Θ on a simultaneous event. The exact domain argument applies to every parameter in Θ. claimA data-selec…
Exact mathematical passage
\rho_\Theta=1-\sup_{(a,b)\in\Theta}(|a|+|b|)
\sigmaUse in source · 24 passages
Search aliases: σ sigma
Use in source · Read passage · FOUNDATIONS.md, line 638
**Proposition 16. Fixed-space identification bound.** Put s=σ_min(Z_-)>0. If E <s, then widetilde Z_- has full row rank, and
Exact mathematical passage
s=\sigma_{\min}(Z_-)>0
\sigma^2Use in source · 2 passages
Search aliases: σ^2 sigma
Use in source · Read passage · FOUNDATIONS.md, line 3660
Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2
Exact mathematical passage
Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t, \qquad \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}] \leq e^{h^2\sigma^2/2}
\sigma_aUse in source · 5 passages
Search aliases: σ_a sigma
Use in source · Read passage · FOUNDATIONS.md, line 3212
σ_a(u)=min_ v _2≤1 [(F_a(u)(1,v))_0- (F_a(u)(1,v))_rm sp _2].
Exact mathematical passage
\sigma_a(u)=\min_{\|v\|_2\leq1} \left[(F_a(u)(1,v))_0-\|(F_a(u)(1,v))_{\rm sp}\|_2\right].
\sigma_bUse in source · 1 passage
Search aliases: σ_b sigma
Use in source · Read passage · FOUNDATIONS.md, line 3845
ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.
Exact mathematical passage
\begin{aligned} \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\ \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4. \end{aligned}
\sigma_{\min}Use in source · 7 passages
Search aliases: σ_min sigma min
Use in source · Read passage · FOUNDATIONS.md, line 638
**Proposition 16. Fixed-space identification bound.** Put s=σ_min(Z_-)>0. If E <s, then widetilde Z_- has full row rank, and
Exact mathematical passage
s=\sigma_{\min}(Z_-)>0
\sigma_rUse in source · 2 passages
Search aliases: σ_r sigma
Use in source · Read passage · FOUNDATIONS.md, line 1362
Let X∈ R^m× n have rank r≥1, let s=σ_r(X)>0, and observe Y=X+E with E _2≤ε<s/2. Let P project onto imX, and let widehat P project onto the leading r left singular vectors of Y.
Exact mathematical passage
s=\sigma_r(X)>0
\sigma_{r+1}Use in source · 2 passages
Search aliases: σ_r+1 sigma
Use in source · Read passage · FOUNDATIONS.md, line 1366
σ_r(Y)≥ s-ε, σ_r+1(Y)≤ε, P-widehat P _2≤frac2εs<1.
Exact mathematical passage
\sigma_r(Y)\geq s-\varepsilon,\qquad \sigma_{r+1}(Y)\leq\varepsilon,\qquad \|P-\widehat P\|_2\leq\frac{2\varepsilon}{s}<1.
\tauUse in source · 13 passages
Search aliases: τ tau
Use in source · Read passage · FOUNDATIONS.md, line 1044
P-P_n _TV=τ_n, τ_n=1-fracZ_nZ.
Exact mathematical passage
\boxed{ \|P-P_n\|_{\mathrm{TV}}=\tau_n, \qquad \tau_n=1-\frac{Z_n}{Z}. }
\tau_1Use in source · 1 passage
Search aliases: τ_1 tau
Use in source · atlas.tex, line 240
plate10Behavior can supply the coordinatesFuture tests define a predictive state without naming hidden variables.P27--P28 For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space. c|ccc · h_1 · h_2 · ⋯ τ_1 · p(h_1τ_1) · p(h_2τ_1) · ⋯ τ_2 · p(h_1τ_2) · p(h_2τ_2) · ⋯ vdots · vdots · vdots · Assume the full behavior space has rank at most r. Choose an inver…
Exact mathematical passage
\begin{array}{c|ccc} & h_1 & h_2 & \cdots\\\hline \tau_1 & p(h_1\tau_1)&p(h_2\tau_1)&\cdots\\ \tau_2 & p(h_1\tau_2)&p(h_2\tau_2)&\cdots\\ \vdots &\vdots&\vdots& \end{array}
\tau_2Use in source · 1 passage
Search aliases: τ_2 tau
Use in source · atlas.tex, line 240
plate10Behavior can supply the coordinatesFuture tests define a predictive state without naming hidden variables.P27--P28 For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space. c|ccc · h_1 · h_2 · ⋯ τ_1 · p(h_1τ_1) · p(h_2τ_1) · ⋯ τ_2 · p(h_1τ_2) · p(h_2τ_2) · ⋯ vdots · vdots · vdots · Assume the full behavior space has rank at most r. Choose an inver…
Exact mathematical passage
\begin{array}{c|ccc} & h_1 & h_2 & \cdots\\\hline \tau_1 & p(h_1\tau_1)&p(h_2\tau_1)&\cdots\\ \tau_2 & p(h_1\tau_2)&p(h_2\tau_2)&\cdots\\ \vdots &\vdots&\vdots& \end{array}
\tau_aUse in source · 3 passages
Search aliases: τ_a tau
Use in source · Read passage · FOUNDATIONS.md, line 2800
**Proposition 62. Oriented quadratic probability certificate.** Suppose that for each action and every u∈[-1,1] there are τ_a(u)≥0 and R_a(u)succeq0 satisfying
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\tau_a(u)\geq0
\tau_nUse in source · 7 passages
Search aliases: τ_n tau
Use in source · Read passage · FOUNDATIONS.md, line 1044
P-P_n _TV=τ_n, τ_n=1-fracZ_nZ.
Exact mathematical passage
\boxed{ \|P-P_n\|_{\mathrm{TV}}=\tau_n, \qquad \tau_n=1-\frac{Z_n}{Z}. }
\PhiUse in source · 5 passages
Search aliases: Φ Phi
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
Exact mathematical passage
\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
\varphiUse in source · 2 passages
Search aliases: φ varphi
Use in source · Read passage · FOUNDATIONS.md, line 616
dence among the input columns. Under the kernel condition, define B(Z_-a)=Z_+a. Two choices of a differ by a kernel vector and give the same answer. This defines and uniquely determines the map on W. It maps W into W exactly under the image condition. The pseudoinverse formula gives that map because a-Z_-^† Z_-a∈ker Z_-. If Wsubsetneq V, choose a nonzero linear functional φ on V vanishing on W and a nonzero v∈ V. The operators B and B+vφ agree on every observed column and differ elsewhere. squar…
Exact mathematical passage
\varphi
\phi_{a,t}Use in source · 1 passage
Search aliases: ϕ_a,t phi
Use in source · Read passage · FOUNDATIONS.md, line 3845
ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.
Exact mathematical passage
\begin{aligned} \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\ \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4. \end{aligned}
\phi_{b,t}Use in source · 1 passage
Search aliases: ϕ_b,t phi
Use in source · Read passage · FOUNDATIONS.md, line 3845
ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.
Exact mathematical passage
\begin{aligned} \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\ \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4. \end{aligned}
\Phi_FUse in source · 2 passages
Search aliases: Φ_F Phi
Use in source · Read passage · FOUNDATIONS.md, line 2322
[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.
Exact mathematical passage
[\Phi_F(X)]_{ij}=\sum_{p,q=1}^d F_{(i,p),(j,q)}X_{pq}.
\Phi_{F_a(u)}Use in source · 1 passage
Search aliases: Φ_F_a(u) Phi
Use in source · Read passage · FOUNDATIONS.md, line 2343
F_a(u) · =Z_q(u)^ TG_a,0Z_q(u) · +(1-u^2)Z_q-1(u)^ TG_a,1Z_q-1(u), J B_a(u) · =Φ_F_a(u) J.
Exact mathematical passage
\begin{aligned} F_a(u)&=Z_q(u)^{\mathsf T}G_{a,0}Z_q(u)\\ &\quad +(1-u^2)Z_{q-1}(u)^{\mathsf T}G_{a,1}Z_{q-1}(u),\\ \mathcal J B_a(u)&=\Phi_{F_a(u)}\mathcal J. \end{aligned}
\Phi_sUse in source · 1 passage
Search aliases: Φ_s Phi
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
Exact mathematical passage
\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
\phi_sUse in source · 4 passages
Search aliases: ϕ_s phi
Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
\phi_s^{\mathsf T}Use in source · 2 passages
Search aliases: ϕ_s^ T phi mathsf
Use in source · Read passage · FOUNDATIONS.md, line 3669
V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.
Exact mathematical passage
V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T}, \qquad \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
\phi_tUse in source · 9 passages
Search aliases: ϕ_t phi
Use in source · Read passage · FOUNDATIONS.md, line 3083
For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define
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\phi_t\in[0,1]
\phi_t^2Use in source · 1 passage
Search aliases: ϕ_t^2 phi
Use in source · Read passage · FOUNDATIONS.md, line 3854
**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of
Exact mathematical passage
V=1+\sum\phi_t^2
\Phi_uUse in source · 1 passage
Search aliases: Φ_u Phi
Use in source · Read passage · FOUNDATIONS.md, line 2366
Φ_u(X)=∑_ν w_ν g_ν(u)K_ν(u)XK_ν(u)^ T, g_ν(u)∈{1,1-u^2}.
Exact mathematical passage
\Phi_u(X)=\sum_\nu w_\nu g_\nu(u)K_\nu(u)XK_\nu(u)^{\mathsf T}, \qquad g_\nu(u)\in\{1,1-u^2\}.
\psiUse in source · 2 passages
Search aliases: ψ psi
Use in source · Read passage · FOUNDATIONS.md, line 3732
Let B_a(u)=∑_k=0^Dψ_k(u)C_a,k. Choose functions g_j so that the known matrix W_jk=∫ g_j(u)ψ_k(u) μ(du) is invertible.
Exact mathematical passage
B_a(u)=\sum_{k=0}^{D}\psi_k(u)C_{a,k}
\psi_kUse in source · 2 passages
Search aliases: ψ_k psi
Use in source · Read passage · FOUNDATIONS.md, line 3732
Let B_a(u)=∑_k=0^Dψ_k(u)C_a,k. Choose functions g_j so that the known matrix W_jk=∫ g_j(u)ψ_k(u) μ(du) is invertible.
Exact mathematical passage
B_a(u)=\sum_{k=0}^{D}\psi_k(u)C_{a,k}
\OmegaUse in source · 2 passages
Search aliases: Ω Omega
Use in source · Read passage · FOUNDATIONS.md, line 274
Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix
Exact mathematical passage
\Omega
\ldotsUse in source · 21 passages
Search aliases: … ldots
Use in source · Read passage · FOUNDATIONS.md, line 167
F(x)=b+∑_m=1^pB_m(x,…,x),
Exact mathematical passage
F(x)=b+\sum_{m=1}^{p}B_m(x,\ldots,x),
\ellUse in source · 18 passages
Search aliases: ℓ ell
Use in source · Read passage · FOUNDATIONS.md, line 274
Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix
Exact mathematical passage
\ell_y(i)>0
\ell^1Use in source · 4 passages
Search aliases: ℓ^1 ell
Use in source · Read passage · FOUNDATIONS.md, line 1495
Rare evidence can create a large posterior change from a small absolute error. For example, take a true unnormalized vector (η,0) and a perturbed vector (η,η). Their ℓ^1 difference is η, while their normalized vectors differ by one in ℓ^1. Both arise as possible branch outputs of valid probability instruments when 0<η≤1/2, after assigning the remaining mass to another branch. There is no uniform posterior Lipschitz constant independent of a probability margin.
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\ell^1
\ell^2Use in source · 1 passage
Search aliases: ℓ^2 ell
Use in source · Read passage · FOUNDATIONS.md, line 473
This result is strong convergence, not operator-norm convergence and not a bound uniform in time. On ℓ^2, finite coordinate projections converge strongly to I while I-P_n =1 for every n. Unbounded operators require domain control. Adaptive data-dependent partitions and fitted coefficients require additional consistency arguments. These remain separate mathematical tasks.
Exact mathematical passage
\ell^2
\ell_iUse in source · 2 passages
Search aliases: ℓ_i ell
Use in source · Read passage · FOUNDATIONS.md, line 386
P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).
Exact mathematical passage
P_a=\sum_{i\in B_a}p_i,\qquad r_a(p)=\frac{\sum_{i\in B_a}p_i\ell_i}{P_a} \quad(P_a>0).
\ell_yUse in source · 4 passages
Search aliases: ℓ_y ell
Use in source · Read passage · FOUNDATIONS.md, line 274
Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix
Exact mathematical passage
\ell_y(i)>0
\ell_y^\topUse in source · 1 passage
Search aliases: ℓ_y^top ell top
Use in source · Read passage · FOUNDATIONS.md, line 282
B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.
Exact mathematical passage
\mathcal B_{\ell_y}(p) =\frac{p\odot\ell_y}{\ell_y^\top p}.
\toUse in source · 27 passages
Search aliases: → to
Use in source · Read passage · FOUNDATIONS.md, line 71
For an endomorphism F:X→ X, define its coordinate realization by F_a=A_aFS_a on C_a. In this paragraph F may be nonlinear. For a partial map, every formula is restricted to the transported admissible domain. Maps between different signal types use separate input and output atlases.
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F:X\to X
\mapstoUse in source · 4 passages
Search aliases: ↦ mapsto
Use in source · Read passage · FOUNDATIONS.md, line 280
Distributions are column vectors. Prediction is p↦ K_u^top p, and a Bayesian update is
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p\mapsto K_u^\top p
\inUse in source · 75 passages
Search aliases: ∈ in
Use in source · Read passage · FOUNDATIONS.md, line 117
∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.
Exact mathematical passage
\sum_{v\in\Lambda}P_v=I,\qquad x=\sum_{v\in\Lambda}P_vx,\qquad \|x\|^2=\sum_{v\in\Lambda}\|P_vx\|^2.
\prodUse in source · 4 passages
Search aliases: ∏ prod
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
\prod_{j=0}^{T-1}Use in source · 2 passages
Search aliases: ∏_j=0^T-1 prod
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
\prod_{j=1}^{m}Use in source · 1 passage
Search aliases: ∏_j=1^m prod
Use in source · Read passage · FOUNDATIONS.md, line 906
F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.
Exact mathematical passage
\boxed{ \|F(x)-F_{\mathcal I}(x)\| \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\notin\mathcal I_m} \beta_{m,\boldsymbol i}\prod_{j=1}^{m}a_{i_j}. }
\prod_{j=1}^{|w|}Use in source · 1 passage
Search aliases: ∏_j=1^|w| prod
Use in source · Read passage · FOUNDATIONS.md, line 1352
Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),
Exact mathematical passage
\Pr_\pi(w)=f(w)\prod_{j=1}^{|w|}\pi(a_j\mid h_{j-1}),
\prod_{j=k+1}^{T-1}Use in source · 2 passages
Search aliases: ∏_j=k+1^T-1 prod
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
\sumUse in source · 113 passages
Search aliases: ∑ sum
Use in source · Read passage · FOUNDATIONS.md, line 51
Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.
Exact mathematical passage
\mathcal Ax=(w_aU_ax)_a,\qquad \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a, \qquad c=\sum_aw_a^2.
\sum_{\boldsymbol i\in\mathcal I_m}Use in source · 1 passage
Search aliases: ∑_ i∈ I_m sum boldsymbol in mathcal
Use in source · Read passage · FOUNDATIONS.md, line 916
Lip(F_ I) ≤∑_m=1^p ∑_ i∈ I_m mβ_m, iR^m-1.
Exact mathematical passage
\operatorname{Lip}(F_{\mathcal I}) \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\in\mathcal I_m} m\beta_{m,\boldsymbol i}R^{m-1}.
\sum_{\boldsymbol i\notin\mathcal I_m}Use in source · 1 passage
Search aliases: ∑_ i∉ I_m sum boldsymbol notin mathcal
Use in source · Read passage · FOUNDATIONS.md, line 906
F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.
Exact mathematical passage
\boxed{ \|F(x)-F_{\mathcal I}(x)\| \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\notin\mathcal I_m} \beta_{m,\boldsymbol i}\prod_{j=1}^{m}a_{i_j}. }
\sum_aUse in source · 5 passages
Search aliases: ∑_a sum
Use in source · Read passage · FOUNDATIONS.md, line 51
Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.
Exact mathematical passage
\mathcal Ax=(w_aU_ax)_a,\qquad \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a, \qquad c=\sum_aw_a^2.
\sum_gUse in source · 1 passage
Search aliases: ∑_g sum
Use in source · Read passage · FOUNDATIONS.md, line 1034
Let G be a countable declared program set, with one fixed probability mass function π(g)>0 satisfying ∑_gπ(g)=1. Every program specifies a likelihood L_g(D) for the **same** observed data event or density relative to a common base measure. Assume
Exact mathematical passage
\sum_g\pi(g)=1
\sum_{g\in\mathcal G}Use in source · 1 passage
Search aliases: ∑_g∈ G sum in mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1036
0<Z(D)=∑_g∈ Gπ(g)L_g(D)<∞.
Exact mathematical passage
0<Z(D)=\sum_{g\in\mathcal G}\pi(g)L_g(D)<\infty.
\sum_{g\in\mathcal G_n}Use in source · 1 passage
Search aliases: ∑_g∈ G_n sum in mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1040
Let finite active sets G_n increase to G. Their unnormalized evidence is Z_n=∑_g∈ G_nπ(g)L_g(D). Only stages with Z_n>0 are considered.
Exact mathematical passage
Z_n=\sum_{g\in\mathcal G_n}\pi(g)L_g(D)
\sum_{g\notin\mathcal G_n}Use in source · 1 passage
Search aliases: ∑_g∉ G_n sum notin mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1053
∑_g∉ G_nπ(g)L_g(D)≤ b_n,
Exact mathematical passage
\sum_{g\notin\mathcal G_n}\pi(g)L_g(D)\leq b_n,
\sum_hUse in source · 3 passages
Search aliases: ∑_h sum
Use in source · Read passage · FOUNDATIONS.md, line 1226
**Proof.** If ∑_hc_hv_h=0, then for every test t,
Exact mathematical passage
\sum_hc_hv_h=0
\sum_iUse in source · 9 passages
Search aliases: ∑_i sum
Use in source · Read passage · FOUNDATIONS.md, line 104
Within one orthogonal atlas, let E_i be pairwise orthogonal coordinate selectors with ∑_iE_i=I. Pull them back by a unitary map:
Exact mathematical passage
\sum_iE_i=I
\sum_{i:w_i>0}Use in source · 1 passage
Search aliases: ∑_i:w_i>0 sum
Use in source · Read passage · FOUNDATIONS.md, line 1318
z=∑_i:w_i>0 w_i z_i, w_i=λ P_iz≥0, z_i=P_iz/w_i, ∑_iw_i=1.
Exact mathematical passage
z=\sum_{i:w_i>0} w_i z_i,\qquad w_i=\lambda P_iz\geq0,\qquad z_i=P_iz/w_i,\qquad\sum_iw_i=1.
\sum_{i_1,\ldots,i_m}Use in source · 2 passages
Search aliases: ∑_i_1,…,i_m sum ldots
Use in source · Read passage · FOUNDATIONS.md, line 173
Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).
Exact mathematical passage
Q_jF(x)=Q_jb+ \sum_{m=1}^{p}\ \sum_{i_1,\ldots,i_m} Q_jB_m(P_{i_1}x,\ldots,P_{i_m}x).
\sum_{i\in B_a}Use in source · 1 passage
Search aliases: ∑_i∈ B_a sum in
Use in source · Read passage · FOUNDATIONS.md, line 386
P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).
Exact mathematical passage
P_a=\sum_{i\in B_a}p_i,\qquad r_a(p)=\frac{\sum_{i\in B_a}p_i\ell_i}{P_a} \quad(P_a>0).
\sum_{i\notin J}Use in source · 1 passage
Search aliases: ∑_i∉ J sum notin
Use in source · Read passage · FOUNDATIONS.md, line 125
This is a conservation law for information under complete refinement. Deleting a child, storing only its energy, or imposing a coefficient threshold changes the contract. For a retained subset J of an orthogonal decomposition, the lost signal is exactly ∑_i∉ JP_ix, and its squared norm is the sum of the omitted packet energies.
Exact mathematical passage
\sum_{i\notin J}P_ix
\sum_jUse in source · 10 passages
Search aliases: ∑_j sum
Use in source · Read passage · FOUNDATIONS.md, line 153
R_kMLP_i=∑_j(R_kMQ_j)(Q_jLP_i).
Exact mathematical passage
R_kMLP_i=\sum_j(R_kMQ_j)(Q_jLP_i).
\sum_{j,i}Use in source · 2 passages
Search aliases: ∑_j,i sum
Use in source · Read passage · FOUNDATIONS.md, line 141
L=∑_j,iQ_jLP_i.
Exact mathematical passage
L=\sum_{j,i}Q_jLP_i.
\sum_{j=0}^DUse in source · 1 passage
Search aliases: ∑_j=0^D sum
Use in source · Read passage · FOUNDATIONS.md, line 2403
widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.
Exact mathematical passage
\boxed{ \widehat M_{a,j}=G_{a,j}H^{-1}, \qquad \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}. }
\sum_{j=0}^{H-1}Use in source · 1 passage
Search aliases: ∑_j=0^H-1 sum
Use in source · atlas.tex, line 189
Here 0≤ y_t≤1 and the truncated recursion has gain a_H=frac12+∑_j=0^H-1K_j =frac78-frac38 3^-H<1. With the same initial state, recursive propagation therefore gives sup_t|y_t-widehat y_t|≤fracΔ_H1-a_H =frac33^H+3. boundaryP10's tail estimate uses the same resolved history. P19 supplies the separate stability argument for a predictor using its own history. For other initial states, retain the term BD^tw_0. navMemory.lean; MemoryBounds.lean; MemoryStability.lean.
Exact mathematical passage
a_H=\frac12+\sum_{j=0}^{H-1}K_j =\frac78-\frac38\,3^{-H}<1.
\sum_{j=0}^{t-1}Use in source · 1 passage
Search aliases: ∑_j=0^t-1 sum
Use in source · atlas.tex, line 182
plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, retaining the last H history terms leaves a same-history tail at most Δ_H=tfrac38 3^-H.
Exact mathematical passage
y_{t+1}=Ay_t+\sum_{j=0}^{t-1}BD^jC\,y_{t-1-j}+BD^tw_0.
\sum_{j=1}^{m_s}Use in source · 1 passage
Search aliases: ∑_j=1^m_s sum
Use in source · Read passage · FOUNDATIONS.md, line 2199
J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.
Exact mathematical passage
\boxed{\mathcal J B_s=\Phi_s\mathcal J,} \qquad \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T}, \quad w_{sj}\geq0.
\sum_{j\in B}Use in source · 1 passage
Search aliases: ∑_j∈ B sum in
Use in source · Read passage · FOUNDATIONS.md, line 300
(∑_j∈ BK_u(i,j))_u,B,
Exact mathematical passage
\left(\sum_{j\in B}K_u(i,j)\right)_{u,B},
\sum_kUse in source · 5 passages
Search aliases: ∑_k sum
Use in source · Read passage · FOUNDATIONS.md, line 2418
**Proof.** A nonzero polynomial has strictly positive squared integral, so its coefficient vector c satisfies c^ TW_Dc>0. Thus W_D is invertible. Let T have rows t_i and S have columns h_j. Since H=TS is an invertible square core in an r-dimensional space, both T and S are invertible. Therefore G_a,jH^-1=TM_a,jT^-1. The relation M_a,j=∑_k(W_D)_jkC_a,k is exact integration of the polynomial expansion. Invert it and conjugate by T to obtain the displayed reconstruction. The error bound is the tria…
Exact mathematical passage
M_{a,j}=\sum_k(W_D)_{jk}C_{a,k}
\sum_{k=0}^DUse in source · 1 passage
Search aliases: ∑_k=0^D sum
Use in source · Read passage · FOUNDATIONS.md, line 2379
B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.
Exact mathematical passage
B_a(u)=\sum_{k=0}^D u^k C_{a,k}, \qquad M_{a,j}=\int_{-1}^1 u^j B_a(u)\,\frac{du}{2}.
\sum_{k=0}^{D}Use in source · 3 passages
Search aliases: ∑_k=0^D sum
Use in source · Read passage · FOUNDATIONS.md, line 3424
Δ_a(u)=∑_k=0^Du^kE_a,k, |(E_a,k)_ij|≤ r_k.
Exact mathematical passage
\Delta_a(u)=\sum_{k=0}^{D}u^kE_{a,k},\qquad |(E_{a,k})_{ij}|\leq r_k.
\sum_{k=0}^{H-1}Use in source · 3 passages
Search aliases: ∑_k=0^H-1 sum
Use in source · Read passage · FOUNDATIONS.md, line 825
widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.
Exact mathematical passage
\widehat z_{t+1}=\widehat A\widehat z_t+ \sum_{k=0}^{H-1}\widehat K_k\widehat z_{t-1-k}.
\sum_{k=0}^nUse in source · 4 passages
Search aliases: ∑_k=0^n sum
Use in source · Read passage · FOUNDATIONS.md, line 2906
b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).
Exact mathematical passage
b_{n,k}(x)=\binom nk x^k(1-x)^{n-k},\qquad D_n(x,v)=(n+1)\sum_{k=0}^n b_{n,k}(x)b_{n,k}(v).
\sum_{k=0}^{T-1}Use in source · 2 passages
Search aliases: ∑_k=0^T-1 sum
Use in source · Read passage · FOUNDATIONS.md, line 751
e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.
Exact mathematical passage
\boxed{ e_T\leq \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0 +\sum_{k=0}^{T-1} \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k. }
\sum_{k=0}^{t-1}Use in source · 3 passages
Search aliases: ∑_k=0^t-1 sum
Use in source · Read passage · FOUNDATIONS.md, line 334
z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.
Exact mathematical passage
\boxed{ z_{t+1}=Az_t+ \sum_{k=0}^{t-1}BD^kCz_{t-1-k}+BD^tq_0. }
\sum_{m=1}^{p}Use in source · 6 passages
Search aliases: ∑_m=1^p sum
Use in source · Read passage · FOUNDATIONS.md, line 167
F(x)=b+∑_m=1^pB_m(x,…,x),
Exact mathematical passage
F(x)=b+\sum_{m=1}^{p}B_m(x,\ldots,x),
\sum_{n\geq1}Use in source · 1 passage
Search aliases: ∑_n≥1 sum geq
Use in source · Read passage · FOUNDATIONS.md, line 2126
Pr(true model∉ U_nmidprevious data) ≤δ_n, ∑_n≥1δ_n≤δ.
Exact mathematical passage
\Pr(\text{true model}\notin\mathcal U_n\mid\text{previous data}) \leq\delta_n, \qquad\sum_{n\geq1}\delta_n\leq\delta.
\sum_oUse in source · 13 passages
Search aliases: ∑_o sum
Use in source · Read passage · FOUNDATIONS.md, line 1153
1^top∑_oM_a,o=1^top.
Exact mathematical passage
\mathbf1^\top\sum_oM_{a,o}=\mathbf1^\top.
\sum_{o\in\mathcal O}Use in source · 1 passage
Search aliases: ∑_o∈ O sum in mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1141
f(ϵ)=1, f(w)≥0, ∑_o∈ Of(w(a,o))=f(w) for every w,a.
Exact mathematical passage
f(\epsilon)=1,\qquad f(w)\geq0,\qquad \sum_{o\in\mathcal O}f(w(a,o))=f(w) \quad\text{for every }w,a.
\sum_{p,q=1}^dUse in source · 1 passage
Search aliases: ∑_p,q=1^d sum
Use in source · Read passage · FOUNDATIONS.md, line 2322
[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.
Exact mathematical passage
[\Phi_F(X)]_{ij}=\sum_{p,q=1}^d F_{(i,p),(j,q)}X_{pq}.
\sum_rUse in source · 1 passage
Search aliases: ∑_r sum
Use in source · Read passage · FOUNDATIONS.md, line 145
If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then
Exact mathematical passage
P_i=\sum_rP_{i,r}
\sum_sUse in source · 2 passages
Search aliases: ∑_s sum
Use in source · Read passage · FOUNDATIONS.md, line 145
If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then
Exact mathematical passage
Q_j=\sum_sQ_{j,s}
\sum_{s,r}Use in source · 1 passage
Search aliases: ∑_s,r sum
Use in source · Read passage · FOUNDATIONS.md, line 147
Q_jLP_i=∑_s,rQ_j,sLP_i,r.
Exact mathematical passage
Q_jLP_i=\sum_{s,r}Q_{j,s}LP_{i,r}.
\sum_{s=0}^{t-1}Use in source · 1 passage
Search aliases: ∑_s=0^t-1 sum
Use in source · Read passage · FOUNDATIONS.md, line 351
q_t=D^tq_0+∑_s=0^t-1D^t-1-sCz_s.
Exact mathematical passage
q_t=D^tq_0+\sum_{s=0}^{t-1}D^{t-1-s}Cz_s.
\sum_{s=1}^tUse in source · 2 passages
Search aliases: ∑_s=1^t sum
Use in source · Read passage · FOUNDATIONS.md, line 3052
R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).
Exact mathematical passage
R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
\sum_{s\in\mathcal A\times\mathcal O}Use in source · 1 passage
Search aliases: ∑_s∈ A× O sum in mathcal times mathcal
Use in source · Read passage · FOUNDATIONS.md, line 1170
W_j+1=W_j+∑_s∈ A× OW_jM_s.
Exact mathematical passage
W_{j+1}=W_j+\sum_{s\in\mathcal A\times\mathcal O}W_jM_s.
\sum_{s\leq t}Use in source · 7 passages
Search aliases: ∑_s≤ t sum leq
Use in source · Read passage · FOUNDATIONS.md, line 3075
Iterating this conditional inequality gives the same exponential bound for the martingale sum ∑_s≤ t(X_s,j-m_s,j). Optimizing the exponential Markov inequality gives the two-sided bound 2exp(-2b^2/t). At the displayed b_t,j this is α_j/[t(t+1)]. Sum over t, using ∑_t≥11/[t(t+1)]=1, and then over j. The resulting event holds at every time, so it also holds at a data-selected time. Scaling the variable proves the range version. square
Exact mathematical passage
\sum_{s\leq t}(X_{s,j}-m_{s,j})
\sum_{t<T}Use in source · 2 passages
Search aliases: ∑_t<T sum
Use in source · Read passage · FOUNDATIONS.md, line 2146
P_T-widehat P_T _TV ≤min(1,frace_0+∑_t<Tϵ_t2).
Exact mathematical passage
\|P_T-\widehat P_T\|_{\mathrm{TV}} \leq\min\left(1,\frac{e_0+\sum_{t<T}\epsilon_t}{2}\right).
\sum_{t=0}^{T-1}Use in source · 3 passages
Search aliases: ∑_t=0^T-1 sum
Use in source · Read passage · FOUNDATIONS.md, line 1927
P_T-widetilde P_T _TV ≤min(1,frac12 z_0-widetilde z_0 _C +frac12∑_t=0^T-1ϵ_t).
Exact mathematical passage
\boxed{ \|P_T-\widetilde P_T\|_{\mathrm{TV}} \leq\min\left(1,\frac12\|z_0-\widetilde z_0\|_C +\frac12\sum_{t=0}^{T-1}\epsilon_t\right). }
\sum_{t\geq1}Use in source · 1 passage
Search aliases: ∑_t≥1 sum geq
Use in source · Read passage · FOUNDATIONS.md, line 3075
Iterating this conditional inequality gives the same exponential bound for the martingale sum ∑_s≤ t(X_s,j-m_s,j). Optimizing the exponential Markov inequality gives the two-sided bound 2exp(-2b^2/t). At the displayed b_t,j this is α_j/[t(t+1)]. Sum over t, using ∑_t≥11/[t(t+1)]=1, and then over j. The resulting event holds at every time, so it also holds at a data-selected time. Scaling the variable proves the range version. square
Exact mathematical passage
\sum_{t\geq1}1/[t(t+1)]=1
\sum_uUse in source · 2 passages
Search aliases: ∑_u sum
Use in source · Read passage · FOUNDATIONS.md, line 243
W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.
Exact mathematical passage
W_{n+1}=W_n+\sum_uW_nL_u, \qquad W_nL_u=\{wL_u:w\in W_n\}.
\sum_{u\in\operatorname{ch}(v)}Use in source · 1 passage
Search aliases: ∑_u∈ch(v) sum in operatorname
Use in source · Read passage · FOUNDATIONS.md, line 129
V(v)=min(c(v),∑_u∈ch(v)V(u))
Exact mathematical passage
V(v)=\min\left(c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right)
\sum_{v\in\Lambda}Use in source · 1 passage
Search aliases: ∑_v∈Λ sum in Lambda
Use in source · Read passage · FOUNDATIONS.md, line 117
∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.
Exact mathematical passage
\sum_{v\in\Lambda}P_v=I,\qquad x=\sum_{v\in\Lambda}P_vx,\qquad \|x\|^2=\sum_{v\in\Lambda}\|P_vx\|^2.
\sum_\nuUse in source · 3 passages
Search aliases: ∑_ν sum nu
Use in source · Read passage · FOUNDATIONS.md, line 2328
F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,
Exact mathematical passage
F=\sum_\nu w_\nu\operatorname{vec}(K_\nu) \operatorname{vec}(K_\nu)^{\mathsf T},\qquad w_\nu\geq0,
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · Read passage · FOUNDATIONS.md, line 430
dom(g∘ f) ={x∈domf:f(x)∈domg}.
Exact mathematical passage
\operatorname{dom}(g\circ f) =\{x\in\operatorname{dom}f:f(x)\in\operatorname{dom}g\}.
\inftyUse in source · 12 passages
Search aliases: ∞ infty
Use in source · Read passage · FOUNDATIONS.md, line 367
**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then
Exact mathematical passage
\mathbb E\|Y\|^2<\infty
\capUse in source · 1 passage
Search aliases: ∩ cap
Use in source · Read passage · FOUNDATIONS.md, line 1733
C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.
Exact mathematical passage
C_{h+1}=C_h\cap\bigcap_s B_s^{-1}C_h, \qquad C_{h+1}\subseteq C_h.
\cupUse in source · 1 passage
Search aliases: ∪ cup
Use in source · Read passage · FOUNDATIONS.md, line 1883
The infimum is finite because the cone is generating. Compactness of K makes this a genuine norm. Its unit ball is conv(K∪(-K)). In particular u _C=λ u on C, and |λ x|≤ x _C. On the ordinary probability orthant this is the ℓ^1 norm. These are standard ordered-space constructions [R15], used here to avoid mixing Euclidean coefficient magnitude with probability mass.
Exact mathematical passage
\operatorname{conv}(K\cup(-K))
\intUse in source · 35 passages
Search aliases: ∫ int
Use in source · Read passage · FOUNDATIONS.md, line 2282
B_a(y)C⊆ C, λ∫_Y B_a(y) μ(dy)=λ.
Exact mathematical passage
B_a(y)C\subseteq C, \qquad \lambda\int_Y B_a(y)\,\mu(dy)=\lambda.
\int_{-1}^1Use in source · 4 passages
Search aliases: ∫_-1^1 int
Use in source · Read passage · FOUNDATIONS.md, line 2353
∫_-1^1u^k fracdu2 =0 · k odd, 1/(k+1) · k even.
Exact mathematical passage
\int_{-1}^1u^k\,\frac{du}{2} =\begin{cases}0&k\text{ odd},\\1/(k+1)&k\text{ even}.\end{cases}
\int_0^1Use in source · 4 passages
Search aliases: ∫_0^1 int
Use in source · Read passage · FOUNDATIONS.md, line 2913
B_a^[n](x)=∫_0^1D_n(x,v)B_a(v) dv =∑_k=0^n b_n,k(x)C_a,n,k,
Exact mathematical passage
B_a^{[n]}(x)=\int_0^1D_n(x,v)B_a(v)\,dv =\sum_{k=0}^n b_{n,k}(x)C_{a,n,k},
\int_AUse in source · 1 passage
Search aliases: ∫_A int
Use in source · Read passage · FOUNDATIONS.md, line 2292
P_a(Amid z)=∫_Aλ B_a(y)z μ(dy), z^+(y)=fracB_a(y)zλ B_a(y)z
Exact mathematical passage
\boxed{ P_a(A\mid z)=\int_A\lambda B_a(y)z\,\mu(dy), \qquad z^+(y)=\frac{B_a(y)z}{\lambda B_a(y)z} }
\int_{A_j}Use in source · 3 passages
Search aliases: ∫_A_j int
Use in source · Read passage · FOUNDATIONS.md, line 2457
B_a,j=∫_A_jB_a(y) μ(dy).
Exact mathematical passage
B_{a,j}=\int_{A_j}B_a(y)\,\mu(dy).
\int_YUse in source · 1 passage
Search aliases: ∫_Y int
Use in source · Read passage · FOUNDATIONS.md, line 2282
B_a(y)C⊆ C, λ∫_Y B_a(y) μ(dy)=λ.
Exact mathematical passage
B_a(y)C\subseteq C, \qquad \lambda\int_Y B_a(y)\,\mu(dy)=\lambda.
\simUse in source · 1 passage
Search aliases: ∼ sim
Use in source · Read passage · FOUNDATIONS.md, line 3697
a nonnegative supermartingale. Integrate against the fixed Gaussian law q∼ N(0,(γ I)^-1). Conditional Tonelli and completing the square give the nonnegative supermartingale
Exact mathematical passage
q\sim N(0,(\gamma I)^{-1})
\approxUse in source · 1 passage
Search aliases: ≈ approx
Use in source · Read passage · FOUNDATIONS.md, line 3966
sup_t≥0e_t≤frac243722320 ≈0.10919.
Exact mathematical passage
\boxed{ \sup_{t\geq0}e_t\leq\frac{2437}{22320} \approx0.10919. }
\neUse in source · 2 passages
Search aliases: ≠ ne
Use in source · Read passage · FOUNDATIONS.md, line 3253
**Proof.** If w≠0, put e=w/ w _2. If w=0, choose any spatial unit vector e. Define Δ=ϵ(0,e)λ. Its mass row vanishes and its value on every normalized input is (0,ϵ e), of base norm ϵ. If w≠0, the new spatial norm is m+ϵ. If w=0, the original output mass is zero and the new spatial norm is ϵ. Either image lies outside the cone. square
Exact mathematical passage
w\ne0
\neqUse in source · 2 passages
Search aliases: ≠ neq
Use in source · Read passage · FOUNDATIONS.md, line 1561
If M_s≠0 and both likelihoods and normalized updates agree on every strictly positive belief, then RM_s=B_sR. Agreement of normalized updates alone need not force this equality.
Exact mathematical passage
M_s\neq0
\geqUse in source · 85 passages
Search aliases: ≥ geq
Use in source · Read passage · FOUNDATIONS.md, line 36
A_ax ≥ frac x S_a , S_a(c+e)-S_ac ≤ S_a e .
Exact mathematical passage
\|A_ax\|\geq \frac{\|x\|}{\|S_a\|},\qquad \|S_a(c+e)-S_ac\|\leq\|S_a\|\,\|e\|.
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · Read passage · FOUNDATIONS.md, line 1710
A **probability-domain certificate** specifies a closed, pointed, generating cone C⊂ E, with λ strictly positive on C∖{0}, such that
Exact mathematical passage
C\subset E
\subseteqUse in source · 17 passages
Search aliases: ⊆ subseteq
Use in source · Read passage · FOUNDATIONS.md, line 367
**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then
Exact mathematical passage
\mathcal F\subseteq\mathcal G
\otimesUse in source · 2 passages
Search aliases: ⊗ otimes
Use in source · Read passage · FOUNDATIONS.md, line 2306
with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.
Exact mathematical passage
\mu^{\otimes T}
\bigcapUse in source · 1 passage
Search aliases: ⋂ bigcap
Use in source · Read passage · FOUNDATIONS.md, line 1733
C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.
Exact mathematical passage
C_{h+1}=C_h\cap\bigcap_s B_s^{-1}C_h, \qquad C_{h+1}\subseteq C_h.
\bigcap_sUse in source · 1 passage
Search aliases: ⋂_s bigcap
Use in source · Read passage · FOUNDATIONS.md, line 1733
C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.
Exact mathematical passage
C_{h+1}=C_h\cap\bigcap_s B_s^{-1}C_h, \qquad C_{h+1}\subseteq C_h.
\cdotsUse in source · 10 passages
Search aliases: ⋯ cdots
Use in source · Read passage · FOUNDATIONS.md, line 901
with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.
Exact mathematical passage
\operatorname{im}P_{i_1}\times\cdots\times\operatorname{im}P_{i_m}
\longrightarrowUse in source · 5 passages
Search aliases: ⟶ longrightarrow
Use in source · Read passage · FOUNDATIONS.md, line 27
A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.
Exact mathematical passage
A_a:X\longrightarrow C_a,\qquad S_a:C_a\longrightarrow X, \qquad S_aA_a=I_X.
\longmapstoUse in source · 2 passages
Search aliases: ⟼ longmapsto
Use in source · Read passage · FOUNDATIONS.md, line 1381
y⟼ U(widehat U^top U)^-1y,
Exact mathematical passage
y\longmapsto U(\widehat U^\top U)^{-1}y,
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- monographs/large-signal-model/Lean/COVERAGE.md · 0 mathematical passages
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PDF reading editions (20)
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- Realization_Calculus_Large_Signal_Model.pdf
- Realization_Calculus_Large_Signal_Model_Theoretical_Atlas.pdf
- 01_delay_prediction.pdf
- 02_memory_certificate.pdf
- 00_architecture.pdf
- 01_consistency.pdf
- 02_transport.pdf
- 03_fibers.pdf
- 04_refinement.pdf
- 05_packets.pdf
- 06_return_path.pdf
- 07_memory.pdf
- 08_defect.pdf
- 09_identification.pdf
- 10_domain.pdf
- 11_psd.pdf
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- 13_moments.pdf
- 14_uncertainty.pdf
- 15_horizons.pdf