Paper III: Typed Transfer, Nonlinear Packets, and Obstruction Theory

Search aliases: inputs operatorname

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• Notation table · visual_atlas.tex, line 36

  t,inputs(t) · A finite operation tree and its syntactically used input labels. · pagerefV20

  Exact mathematical passage

  t,\operatorname{inputs}(t)

Search aliases: t

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• Notation table · visual_atlas.tex, line 36

  t,inputs(t) · A finite operation tree and its syntactically used input labels. · pagerefV20

  Exact mathematical passage

  t,\operatorname{inputs}(t)

Search aliases: x

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• Notation table · visual_atlas.tex, line 35

  r(x,p) · A residual at packet p for the named candidate x. · pagerefV19

  Exact mathematical passage

  x

Search aliases: *

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 898

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  
  \Ucal(\Qcal,\Pcal)
  =
  S(\Qcal,\Pcal)
  \big(S(\Qcal,\Pcal)^*S(\Qcal,\Pcal)\big)^{-1/2}.
  

Search aliases: +

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 174

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    \tgt(e_r)=\src(e_{r+1})
    \qquad (1\le r<n).
  

Search aliases: =

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 67

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGr…

  Exact mathematical passage

  
    \operatorname{packetMatrix}(a,b;r)_{yx}
    =
    \begin{cases}
    r,&y=b\text{ and }x=a,\\
    0,&\text{otherwise}.
    \end{cases}
  

Search aliases: A

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 573

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  \mu:A^r\to A

Search aliases: a

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 66

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,

  Exact mathematical passage

  a,b\in V

Search aliases: a^2

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• Definition passage: Divided square · Read passage · 03_nonlinear_coordinates.tex, line 665

  [Divided square] A divided square on degree one elements is a map q satisfying q(x+y)=q(x)+q(y)+[x,y], q(ax)=a^2q(x), whenever the expressions are defined.

  Exact mathematical passage

  
  q(x+y)=q(x)+q(y)+[x,y],
  \qquad
  q(ax)=a^2q(x),
  

Search aliases: A^r

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 573

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  \mu:A^r\to A

Search aliases: a_0

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 266

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  p=(a_0,a_1,\ldots,a_n)

Search aliases: a_1

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 266

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  p=(a_0,a_1,\ldots,a_n)

Search aliases: A_g_1

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: A_g_1+⋯+g_r cdots

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: A_g_r

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: a_n

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 266

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  p=(a_0,a_1,\ldots,a_n)

Search aliases: a_n-1

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: Acal A mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 106

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}
  =
  \Acal_{\Qcal}F\Scal_{\Pcal}.
  

Search aliases: Acal_Qcal A_ Q Acal Qcal mathcal mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 106

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}
  =
  \Acal_{\Qcal}F\Scal_{\Pcal}.
  

Search aliases: Aut operatorname

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 13

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M)
  \qquad(a,b\in B)
  

Search aliases: Aut_R operatorname

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 13

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M)
  \qquad(a,b\in B)
  

Search aliases: B

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 13

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M)
  \qquad(a,b\in B)
  

Search aliases: b

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 66

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,

  Exact mathematical passage

  a,b\in V

Search aliases: binary operatorname

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: bullet

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1124

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  D:C^\bullet\to E^\bullet

Search aliases: C

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 119

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b
  \qquad(a,b\in B)
  

Search aliases: c

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 75

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  c:E\to R

Search aliases: C^bullet bullet

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1124

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  D:C^\bullet\to E^\bullet

Search aliases: C^n+1

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1125

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  \Cone(D)^n=E^n\oplus C^{n+1}
  

Search aliases: C_a

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 119

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b
  \qquad(a,b\in B)
  

Search aliases: C_a_0

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: C_b

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 119

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b
  \qquad(a,b\in B)
  

Search aliases: C_c

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• Definition passage: Cocycle defect · Read passage · 02_connections_obstructions.tex, line 176

  [Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect

  Exact mathematical passage

  
  \Delta_{c,b,a}
  =
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  -\Gamma_{c\leftarrow a}
  \in\Hom_R(C_a,C_c).
  

Search aliases: C_X

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• Definition passage: Nonlinear refinement defect · Read passage · 03_nonlinear_coordinates.tex, line 737

  [Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.

  Exact mathematical passage

  
  \Delta_F(x)
  =
  C_YF_f(x)-F_c(C_Xx).
  

Search aliases: C_Y

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• Definition passage: Nonlinear refinement defect · Read passage · 03_nonlinear_coordinates.tex, line 737

  [Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.

  Exact mathematical passage

  
  \Delta_F(x)
  =
  C_YF_f(x)-F_c(C_Xx).
  

Search aliases: cA A mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cA_b A_b cA mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: coker operatorname

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• Definition passage: Linear correction datum · Read passage · 03_nonlinear_coordinates.tex, line 912

  [Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.

  Exact mathematical passage

  
  \Ob_D(r)=[r]\in\coker D=V/\im D.
  

Search aliases: Cone operatorname

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1125

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  \Cone(D)^n=E^n\oplus C^{n+1}
  

Search aliases: constant operatorname

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: Coord operatorname

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cP P mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cP^a P^a cP mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cP^b P^b cP mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cS S mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cS_a S_a cS mathcal

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: cW W mathcal

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• Definition passage: Recursive word expansion · Read passage · 01_finite_coordinates.tex, line 213

  [Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion

  Exact mathematical passage

  
    \cW_0=I,
    \qquad
    \cW_{n+1}=T\cW_n.
  

Search aliases: cW_0 W_0 cW mathcal

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• Definition passage: Recursive word expansion · Read passage · 01_finite_coordinates.tex, line 213

  [Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion

  Exact mathematical passage

  
    \cW_0=I,
    \qquad
    \cW_{n+1}=T\cW_n.
  

Search aliases: cW_n W_n cW mathcal

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• Definition passage: Recursive word expansion · Read passage · 01_finite_coordinates.tex, line 213

  [Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion

  Exact mathematical passage

  
    \cW_0=I,
    \qquad
    \cW_{n+1}=T\cW_n.
  

Search aliases: cW_n+1 W_n+1 cW mathcal

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• Definition passage: Recursive word expansion · Read passage · 01_finite_coordinates.tex, line 213

  [Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion

  Exact mathematical passage

  
    \cW_0=I,
    \qquad
    \cW_{n+1}=T\cW_n.
  

Search aliases: D

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: d

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1129

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
  

Search aliases: D^r

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: d_C

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1129

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
  

Search aliases: d_Cone Cone operatorname

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1129

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
  

Search aliases: d_E

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1129

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
  

Search aliases: E

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 75

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  c:E\to R

Search aliases: e

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: E^bullet bullet

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1124

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  D:C^\bullet\to E^\bullet

Search aliases: E^n

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1125

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  \Cone(D)^n=E^n\oplus C^{n+1}
  

Search aliases: e_1

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 173

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  w=(e_1,\ldots,e_n)

Search aliases: e_n

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 173

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  w=(e_1,\ldots,e_n)

Search aliases: e_r

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 174

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    \tgt(e_r)=\src(e_{r+1})
    \qquad (1\le r<n).
  

Search aliases: e_r+1

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 174

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    \tgt(e_r)=\src(e_{r+1})
    \qquad (1\le r<n).
  

Search aliases: ev operatorname

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• Definition passage: Typed operation tree and evaluation · Read passage · 02_connections_obstructions.tex, line 848

  [Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.ColoredOperationTree and leannameColoredOperationTree.eval. This is stronger than the compiled uncolored unary--binary grammar.

  Exact mathematical passage

  
  \ev(\operatorname{node}(\sigma;t_1,\ldots,t_{r(\sigma)}))
  =
  \Phi_\sigma(\ev(t_1),\ldots,\ev(t_{r(\sigma)})).
  

Search aliases: F

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 105

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  F:X\to Y

Search aliases: f

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• Definition passage: Nonlinear refinement defect · Read passage · 03_nonlinear_coordinates.tex, line 737

  [Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.

  Exact mathematical passage

  
  \Delta_F(x)
  =
  C_YF_f(x)-F_c(C_Xx).
  

Search aliases: F^m

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• Definition passage: Depth obstruction · Read passage · 04_correction_transport.tex, line 247

  [Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.

  Exact mathematical passage

  F^mY/F^{m+1}Y

Search aliases: F^m+1

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• Definition passage: Depth obstruction · Read passage · 04_correction_transport.tex, line 247

  [Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.

  Exact mathematical passage

  F^mY/F^{m+1}Y

Search aliases: F_c

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• Definition passage: Nonlinear refinement defect · Read passage · 03_nonlinear_coordinates.tex, line 737

  [Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.

  Exact mathematical passage

  
  \Delta_F(x)
  =
  C_YF_f(x)-F_c(C_Xx).
  

Search aliases: F_f

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• Definition passage: Nonlinear refinement defect · Read passage · 03_nonlinear_coordinates.tex, line 737

  [Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.

  Exact mathematical passage

  
  \Delta_F(x)
  =
  C_YF_f(x)-F_c(C_Xx).
  

Search aliases: F_Qcal←Pcal F_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 106

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}
  =
  \Acal_{\Qcal}F\Scal_{\Pcal}.
  

Search aliases: Fin operatorname

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 831

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  \operatorname{in}_\sigma:\operatorname{Fin}(r(\sigma))\to V

Search aliases: G

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• Definition passage: Gauge change · Read passage · 02_connections_obstructions.tex, line 312

  [Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge

  Exact mathematical passage

  G_a\in\operatorname{Aut}_R(C_a)

Search aliases: g

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: g_1

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: G_a

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• Definition passage: Gauge change · Read passage · 02_connections_obstructions.tex, line 312

  [Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge

  Exact mathematical passage

  G_a\in\operatorname{Aut}_R(C_a)

Search aliases: G_a^-1

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• Definition passage: Gauge change · Read passage · 02_connections_obstructions.tex, line 314

  [Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge

  Exact mathematical passage

  
  \Gamma^G_{b\leftarrow a}
  =
  G_b\Gamma_{b\leftarrow a}G_a^{-1}.
  

Search aliases: G_b

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• Definition passage: Gauge change · Read passage · 02_connections_obstructions.tex, line 314

  [Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge

  Exact mathematical passage

  
  \Gamma^G_{b\leftarrow a}
  =
  G_b\Gamma_{b\leftarrow a}G_a^{-1}.
  

Search aliases: g_r

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: G_u^-1

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• Definition passage: Gauge change · Read passage · 04_correction_transport.tex, line 653

  [Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.

  Exact mathematical passage

  
  T'_{vu}=G_vT_{vu}G_u^{-1}.
  

Search aliases: G_v

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• Definition passage: Gauge change · Read passage · 04_correction_transport.tex, line 652

  [Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.

  Exact mathematical passage

  G_v\in\operatorname{GL}(X)

Search aliases: GL operatorname

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• Definition passage: Gauge change · Read passage · 04_correction_transport.tex, line 652

  [Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.

  Exact mathematical passage

  G_v\in\operatorname{GL}(X)

Search aliases: H

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• Definition passage: Triangle holonomy · Read passage · 02_connections_obstructions.tex, line 192

  [Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction

  Exact mathematical passage

  
  H_{c,b,a}
  =
  \Gamma_{a\leftarrow c}
  \Gamma_{c\leftarrow b}
  \Gamma_{b\leftarrow a}
  \in\operatorname{Aut}_R(C_a).
  

Search aliases: H_c,b,a

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• Definition passage: Triangle holonomy · Read passage · 02_connections_obstructions.tex, line 192

  [Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction

  Exact mathematical passage

  
  H_{c,b,a}
  =
  \Gamma_{a\leftarrow c}
  \Gamma_{c\leftarrow b}
  \Gamma_{b\leftarrow a}
  \in\operatorname{Aut}_R(C_a).
  

Search aliases: Hom operatorname

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• Definition passage: Cocycle defect · Read passage · 02_connections_obstructions.tex, line 176

  [Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect

  Exact mathematical passage

  
  \Delta_{c,b,a}
  =
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  -\Gamma_{c\leftarrow a}
  \in\Hom_R(C_a,C_c).
  

Search aliases: Hom_R Hom operatorname

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• Definition passage: Cocycle defect · Read passage · 02_connections_obstructions.tex, line 176

  [Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect

  Exact mathematical passage

  
  \Delta_{c,b,a}
  =
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  -\Gamma_{c\leftarrow a}
  \in\Hom_R(C_a,C_c).
  

Search aliases: I

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: i

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: id mathrm

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 18

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{a\leftarrow a}=\id_M.
  

Search aliases: id_C_a id mathrm

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: id_C_a_0 id mathrm

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: id_M id mathrm

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 18

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{a\leftarrow a}=\id_M.
  

Search aliases: im operatorname

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• Definition passage: Linear correction datum · Read passage · 03_nonlinear_coordinates.tex, line 912

  [Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.

  Exact mathematical passage

  
  \Ob_D(r)=[r]\in\coker D=V/\im D.
  

Search aliases: in operatorname

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 831

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  \operatorname{in}_\sigma:\operatorname{Fin}(r(\sigma))\to V

Search aliases: in_σ operatorname sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 831

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  \operatorname{in}_\sigma:\operatorname{Fin}(r(\sigma))\to V

Search aliases: input operatorname

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: IsSolution operatorname

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• Definition passage: Solution and obstruction packet for a candidate · Read passage · 02_connections_obstructions.tex, line 409

  [Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket

  Exact mathematical passage

  
  \operatorname{IsSolution}_r(x)
  \quad\Longleftrightarrow\quad
  \forall p,\ r(x,p)=0.
  

Search aliases: IsSolution_r operatorname

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• Definition passage: Solution and obstruction packet for a candidate · Read passage · 02_connections_obstructions.tex, line 409

  [Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket

  Exact mathematical passage

  
  \operatorname{IsSolution}_r(x)
  \quad\Longleftrightarrow\quad
  \forall p,\ r(x,p)=0.
  

Search aliases: J

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: J^X_b← a leftarrow

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: J_b← a leftarrow

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• Definition passage: Calibration and flatness · Read passage · 02_connections_obstructions.tex, line 141

  [Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Tra…

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}=J_{b\leftarrow a}
  \quad\text{for every }a,b.
  

Search aliases: K

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 770

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t)

  Exact mathematical passage

  K

Search aliases: k

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: L

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: L_e

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: L_e_1

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: L_e_n

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: L_w

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: L_∅ varnothing

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: Longleftrightarrow

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• Definition passage: Solution and obstruction packet for a candidate · Read passage · 02_connections_obstructions.tex, line 409

  [Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket

  Exact mathematical passage

  
  \operatorname{IsSolution}_r(x)
  \quad\Longleftrightarrow\quad
  \forall p,\ r(x,p)=0.
  

Search aliases: M

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 13

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M)
  \qquad(a,b\in B)
  

Search aliases: m

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 219

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  1\le r\le m

Search aliases: mid

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: N mathbb

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 829

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  r(\sigma)\in\N

Search aliases: n

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 173

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  w=(e_1,\ldots,e_n)

Search aliases: node operatorname

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• Definition passage: Typed operation tree and evaluation · Read passage · 02_connections_obstructions.tex, line 848

  [Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.ColoredOperationTree and leannameColoredOperationTree.eval. This is stronger than the compiled uncolored unary--binary grammar.

  Exact mathematical passage

  
  \ev(\operatorname{node}(\sigma;t_1,\ldots,t_{r(\sigma)}))
  =
  \Phi_\sigma(\ev(t_1),\ldots,\ev(t_{r(\sigma)})).
  

Search aliases: Ob operatorname

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Ob^μν_c,b,a Ob mu nu operatorname mu nu

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Ob_c,b,a Ob operatorname

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Ob_D Ob operatorname

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• Definition passage: Linear correction datum · Read passage · 03_nonlinear_coordinates.tex, line 912

  [Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.

  Exact mathematical passage

  
  \Ob_D(r)=[r]\in\coker D=V/\im D.
  

Search aliases: Ob_m Ob operatorname

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• Definition passage: Depth obstruction · Read passage · 04_correction_transport.tex, line 240

  [Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.

  Exact mathematical passage

  
  \Ob_m(x_m)
  =
  [\Phi(x_m)]
  \in
  \coker\overline L_m,
  

Search aliases: out operatorname

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 832

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  \operatorname{out}(\sigma)\in V

Search aliases: overline L overline

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• Definition passage: Depth obstruction · Read passage · 04_correction_transport.tex, line 240

  [Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.

  Exact mathematical passage

  
  \Ob_m(x_m)
  =
  [\Phi(x_m)]
  \in
  \coker\overline L_m,
  

Search aliases: overline L_m overline

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• Definition passage: Depth obstruction · Read passage · 04_correction_transport.tex, line 240

  [Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.

  Exact mathematical passage

  
  \Ob_m(x_m)
  =
  [\Phi(x_m)]
  \in
  \coker\overline L_m,
  

Search aliases: P

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 893

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  (P_\alpha)_{\alpha\in I}

Search aliases: p

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 266

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  p=(a_0,a_1,\ldots,a_n)

Search aliases: P_α alpha

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 893

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  (P_\alpha)_{\alpha\in I}

Search aliases: packetMatrix operatorname

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 67

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGr…

  Exact mathematical passage

  
    \operatorname{packetMatrix}(a,b;r)_{yx}
    =
    \begin{cases}
    r,&y=b\text{ and }x=a,\\
    0,&\text{otherwise}.
    \end{cases}
  

Search aliases: Pcal P mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 105

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  \Pcal

Search aliases: Q

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 112

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
  =
  \left(
  Q_\gamma F\left(\sum_\alpha x_\alpha\right)
  \right)_\gamma.
  

Search aliases: q

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: Q_α alpha

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 893

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  (Q_\alpha)_{\alpha\in I}

Search aliases: Q_γ gamma

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 112

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
  =
  \left(
  Q_\gamma F\left(\sum_\alpha x_\alpha\right)
  \right)_\gamma.
  

Search aliases: Qcal Q mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 105

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  \Qcal

Search aliases: R

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 66

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,

  Exact mathematical passage

  r\in R

Search aliases: r

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 66

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,

  Exact mathematical passage

  r\in R

Search aliases: S

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 894

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  
  S(\Qcal,\Pcal)=\sum_{\alpha\in I}Q_\alpha P_\alpha.
  

Search aliases: Scal S mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 106

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}
  =
  \Acal_{\Qcal}F\Scal_{\Pcal}.
  

Search aliases: Scal_Pcal S_ P Scal Pcal mathcal mathcal

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 106

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}
  =
  \Acal_{\Qcal}F\Scal_{\Pcal}.
  

Search aliases: src operatorname

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: T

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: T'_vu

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• Definition passage: Gauge change · Read passage · 04_correction_transport.tex, line 653

  [Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.

  Exact mathematical passage

  
  T'_{vu}=G_vT_{vu}G_u^{-1}.
  

Search aliases: t_1

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: t_2

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: t_r(σ) sigma

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• Definition passage: Typed operation tree and evaluation · Read passage · 02_connections_obstructions.tex, line 848

  [Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.ColoredOperationTree and leannameColoredOperationTree.eval. This is stronger than the compiled uncolored unary--binary grammar.

  Exact mathematical passage

  
  \ev(\operatorname{node}(\sigma;t_1,\ldots,t_{r(\sigma)}))
  =
  \Phi_\sigma(\ev(t_1),\ldots,\ev(t_{r(\sigma)})).
  

Search aliases: T_vu

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• Definition passage: Gauge change · Read passage · 04_correction_transport.tex, line 653

  [Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.

  Exact mathematical passage

  
  T'_{vu}=G_vT_{vu}G_u^{-1}.
  

Search aliases: tgt operatorname

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: U

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 770

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t)

  Exact mathematical passage

  U

Search aliases: u

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: Ucal U mathcal

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 898

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  
  \Ucal(\Qcal,\Pcal)
  =
  S(\Qcal,\Pcal)
  \big(S(\Qcal,\Pcal)^*S(\Qcal,\Pcal)\big)^{-1/2}.
  

Search aliases: unary operatorname

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 772

  [Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; K…

  Exact mathematical passage

  
  \begin{aligned}
  t ::={}&\operatorname{input}(i)
  \mid\operatorname{constant}(k)
  \mid\operatorname{unary}(u,t)\\
  &\mid\operatorname{binary}(b,t_1,t_2).
  \end{aligned}
  

Search aliases: V

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 66

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,

  Exact mathematical passage

  a,b\in V

Search aliases: v

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 834

  eldef:signature A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  X_v

Search aliases: w

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 173

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  w=(e_1,\ldots,e_n)

Search aliases: X

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: X_in_σ(q) operatorname sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: x_m

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• Definition passage: Depth obstruction · Read passage · 04_correction_transport.tex, line 239

  [Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.

  Exact mathematical passage

  x_m

Search aliases: X_out(σ) operatorname sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: X_v

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 834

  eldef:signature A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  X_v

Search aliases: x_α alpha

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 112

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
  =
  \left(
  Q_\gamma F\left(\sum_\alpha x_\alpha\right)
  \right)_\gamma.
  

Search aliases: X_α_1 alpha

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: X_α_r alpha

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: xrightarrow

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 119

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b
  \qquad(a,b\in B)
  

Search aliases: Y

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• Definition passage: Finite grammar for operation trees · Read passage · 02_connections_obstructions.tex, line 780

  · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval

  Exact mathematical passage

  Y

Search aliases: y

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 67

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGr…

  Exact mathematical passage

  
    \operatorname{packetMatrix}(a,b;r)_{yx}
    =
    \begin{cases}
    r,&y=b\text{ and }x=a,\\
    0,&\text{otherwise}.
    \end{cases}
  

Search aliases: Y_γ gamma

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 227

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  X_{\alpha_1}\times\cdots\times X_{\alpha_r}
  \longrightarrow
  Y_\gamma.
  

Search aliases: |

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: |_X_α_1×⋯× X_α_r alpha times cdots times alpha

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: × times

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: α alpha

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 112

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
  =
  \left(
  Q_\gamma F\left(\sum_\alpha x_\alpha\right)
  \right)_\gamma.
  

Search aliases: α_1 alpha

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: α_r alpha

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• Definition passage: Packet jet · Read passage · 03_nonlinear_coordinates.tex, line 220

  [Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.

  Exact mathematical passage

  
  J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
  =
  Q_\gamma D^rF(x)
  \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
  

Search aliases: Γ Gamma

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 13

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M)
  \qquad(a,b\in B)
  

Search aliases: γ gamma

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 112

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
  =
  \left(
  Q_\gamma F\left(\sum_\alpha x_\alpha\right)
  \right)_\gamma.
  

Search aliases: Γ^G_b← a Gamma leftarrow

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• Definition passage: Gauge change · Read passage · 02_connections_obstructions.tex, line 314

  [Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge

  Exact mathematical passage

  
  \Gamma^G_{b\leftarrow a}
  =
  G_b\Gamma_{b\leftarrow a}G_a^{-1}.
  

Search aliases: Γ_(a_0) Gamma

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: Γ_a_1← a_0 Gamma leftarrow

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: Γ_a_n← a_n-1 Gamma leftarrow

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: Γ_a← a Gamma leftarrow

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 18

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{a\leftarrow a}=\id_M.
  

Search aliases: Γ_a← b Gamma leftarrow

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 124

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{a\leftarrow a}=\id,
  \qquad
  \Gamma_{a\leftarrow b}=\Gamma_{b\leftarrow a}^{-1}.
  

Search aliases: Γ_a← c Gamma leftarrow

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• Definition passage: Triangle holonomy · Read passage · 02_connections_obstructions.tex, line 192

  [Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction

  Exact mathematical passage

  
  H_{c,b,a}
  =
  \Gamma_{a\leftarrow c}
  \Gamma_{c\leftarrow b}
  \Gamma_{b\leftarrow a}
  \in\operatorname{Aut}_R(C_a).
  

Search aliases: Γ_b← a Gamma leftarrow

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• Definition passage: Noncanonical connection normalized at each chart · Read passage · 02_connections_obstructions.tex, line 13

  [Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M)
  \qquad(a,b\in B)
  

Search aliases: Γ_b← a^-1 Gamma leftarrow

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 124

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{a\leftarrow a}=\id,
  \qquad
  \Gamma_{a\leftarrow b}=\Gamma_{b\leftarrow a}^{-1}.
  

Search aliases: Γ_c← a Gamma leftarrow

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• Definition passage: Calibration and flatness · Read passage · 02_connections_obstructions.tex, line 146

  [Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Tra…

  Exact mathematical passage

  
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  =
  \Gamma_{c\leftarrow a}
  \quad\text{for every }a,b,c.
  

Search aliases: Γ_c← b Gamma leftarrow

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• Definition passage: Calibration and flatness · Read passage · 02_connections_obstructions.tex, line 146

  [Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Tra…

  Exact mathematical passage

  
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  =
  \Gamma_{c\leftarrow a}
  \quad\text{for every }a,b,c.
  

Search aliases: Γ_p Gamma

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• Definition passage: Connection transport along a path · Read passage · 02_connections_obstructions.tex, line 267

  [Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport

  Exact mathematical passage

  
  \Gamma_p
  =
  \Gamma_{a_n\leftarrow a_{n-1}}\cdots
  \Gamma_{a_1\leftarrow a_0},
  \qquad
  \Gamma_{(a_0)}=\id_{C_{a_0}}.
  

Search aliases: Δ Delta

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• Definition passage: Cocycle defect · Read passage · 02_connections_obstructions.tex, line 176

  [Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect

  Exact mathematical passage

  
  \Delta_{c,b,a}
  =
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  -\Gamma_{c\leftarrow a}
  \in\Hom_R(C_a,C_c).
  

Search aliases: Δ_c,b,a Delta

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• Definition passage: Cocycle defect · Read passage · 02_connections_obstructions.tex, line 176

  [Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect

  Exact mathematical passage

  
  \Delta_{c,b,a}
  =
  \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}
  -\Gamma_{c\leftarrow a}
  \in\Hom_R(C_a,C_c).
  

Search aliases: Δ_F Delta

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• Definition passage: Nonlinear refinement defect · Read passage · 03_nonlinear_coordinates.tex, line 737

  [Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.

  Exact mathematical passage

  
  \Delta_F(x)
  =
  C_YF_f(x)-F_c(C_Xx).
  

Search aliases: μ mu

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: ν nu

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Π Pi

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Π^a_μ Pi mu

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Π^a_ν Pi nu

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• Definition passage: Holonomy obstruction and packet residuals · Read passage · 02_connections_obstructions.tex, line 472

  [Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…

  Exact mathematical passage

  
  \Ob_{c,b,a}=H_{c,b,a}-\id_{C_a},
  \qquad
  \Ob^{\mu\nu}_{c,b,a}
  =
  \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
  

Search aliases: Σ Sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 828

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  \Sigma

Search aliases: σ sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 829

  [Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  r(\sigma)\in\N

Search aliases: Φ Phi

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: Φ_σ Phi sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: … ldots

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 173

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  w=(e_1,\ldots,e_n)

Search aliases: ← leftarrow

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: → to

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 75

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  c:E\to R

Search aliases: ∀ forall

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• Definition passage: Solution and obstruction packet for a candidate · Read passage · 02_connections_obstructions.tex, line 409

  [Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket

  Exact mathematical passage

  
  \operatorname{IsSolution}_r(x)
  \quad\Longleftrightarrow\quad
  \forall p,\ r(x,p)=0.
  

Search aliases: ∅ varnothing

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: ∈ in

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 66

  [Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,

  Exact mathematical passage

  a,b\in V

Search aliases: ∏ prod

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: ∏_q∈Fin(r(σ)) prod in operatorname sigma

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• Definition passage: Finitary colored operation signature · Read passage · 02_connections_obstructions.tex, line 835

  lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.

  Exact mathematical passage

  
  \Phi_\sigma:
  \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)}
  \longrightarrow X_{\operatorname{out}(\sigma)}.
  

Search aliases: ∑ sum

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: ∑_e∈ E sum in

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• Definition passage: Packet graph and scalar edge system · Read passage · 01_finite_coordinates.tex, line 77

  0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp

  Exact mathematical passage

  
    L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)),
    \qquad
    T=\sum_{e\in E}L_e.
  

Search aliases: ∑_α sum alpha

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• Definition passage: Nonlinear packet coordinate map · Read passage · 03_nonlinear_coordinates.tex, line 112

  [Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.

  Exact mathematical passage

  
  F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
  =
  \left(
  Q_\gamma F\left(\sum_\alpha x_\alpha\right)
  \right)_\gamma.
  

Search aliases: ∑_α∈ I sum alpha in

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• Definition passage: Direct rotation of nearby packet systems · Read passage · 04_correction_transport.tex, line 894

  [Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.

  Exact mathematical passage

  
  S(\Qcal,\Pcal)=\sum_{\alpha\in I}Q_\alpha P_\alpha.
  

Search aliases: ∼ sim

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• Definition passage: Normalized reversible connection · Read passage · 02_connections_obstructions.tex, line 119

  [Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection

  Exact mathematical passage

  
  \Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b
  \qquad(a,b\in B)
  

Search aliases: ≠ ne

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• Definition passage: Solution and obstruction packet for a candidate · Read passage · 02_connections_obstructions.tex, line 415

  [Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket

  Exact mathematical passage

  r(x,p)\ne0

Search aliases: ≤ le

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 174

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    \tgt(e_r)=\src(e_{r+1})
    \qquad (1\le r<n).
  

Search aliases: ⊆ subseteq

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• Definition passage: Grade respecting operation · Read passage · 03_nonlinear_coordinates.tex, line 574

  [Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.

  Exact mathematical passage

  
  \mu(A_{g_1},\ldots,A_{g_r})
  \subseteq
  A_{g_1+\cdots+g_r}.
  

Search aliases: ⊕ oplus

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• Definition passage: Mapping cone · Read passage · 03_nonlinear_coordinates.tex, line 1125

  [Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).

  Exact mathematical passage

  
  \Cone(D)^n=E^n\oplus C^{n+1}
  

Search aliases: ⋯ cdots

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• Definition passage: Composable word and execution operator · Read passage · 01_finite_coordinates.tex, line 179

  [Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp

  Exact mathematical passage

  
    L_w=L_{e_n}\cdots L_{e_1},
    \qquad L_{\varnothing}=I.
  

Search aliases: ⟶ longrightarrow

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• Definition passage: Canonical exact coordinate transport · Read passage · 01_finite_coordinates.tex, line 472

  [Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv

  Exact mathematical passage

  
    J^X_{b\leftarrow a}
    =
    \cA_b\cS_a:
    \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
  

Search aliases: j

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• Introduction passage · Read passage · 01_finite_coordinates.tex, line 274

  Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.

  Exact mathematical passage

  \Walk_n(i,j)

Search aliases: Walk operatorname

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• Introduction passage · Read passage · 01_finite_coordinates.tex, line 274

  Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.

  Exact mathematical passage

  \Walk_n(i,j)

Search aliases: Walk_n Walk operatorname

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• Introduction passage · Read passage · 01_finite_coordinates.tex, line 274

  Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.

  Exact mathematical passage

  \Walk_n(i,j)

Search aliases: A mathcal

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• Use in source · visual_atlas.tex, line 28

  A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10

  Exact mathematical passage

  \mathcal A_a,\mathcal S_a

Search aliases: A^-1

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• Use in source · Read passage · 04_correction_transport.tex, line 540

  [Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.

  Exact mathematical passage

  
  DR
  =
  D_uF_P
  -
  D_vF_P A^{-1}D_uF_Q.
  

Search aliases: A^b_1

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• Use in source · Read passage · 04_correction_transport.tex, line 766

  [Abelian cohomology class] When the structure group A is abelian, gauge classes of transports are naturally H^1(Γ,A) ≅ Hom(H_1(Γ, Z),A). After a basis of H_1(Γ, Z) is chosen, this group is identified with A^b_1.

  Exact mathematical passage

  A^{b_1}

Search aliases: a_1,g_1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 583

  [Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).

  Exact mathematical passage

  
  P_k\mu(a_1,\ldots,a_r)
  =
  \sum_{g_1+\cdots+g_r=k}
  \mu(a_{1,g_1},\ldots,a_{r,g_r}).
  

Search aliases: A_a mathcal

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• Use in source · visual_atlas.tex, line 28

  A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10

  Exact mathematical passage

  \mathcal A_a,\mathcal S_a

Search aliases: A_g

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 567

  Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.

  Exact mathematical passage

  
  A=\bigoplus_{g\in G}A_g
  

Search aliases: A_i

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 597

  [Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.

  Exact mathematical passage

  
  A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j,
  \qquad
  A_iA_j\subseteq A_{i+j},
  

Search aliases: a_i

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 603

  [Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.

  Exact mathematical passage

  
  P_k(ab)=
  \sum_{i+j=k}a_ib_j.
  

Search aliases: A_i+j

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 597

  [Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.

  Exact mathematical passage

  
  A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j,
  \qquad
  A_iA_j\subseteq A_{i+j},
  

Search aliases: A_j

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 597

  [Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.

  Exact mathematical passage

  
  A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j,
  \qquad
  A_iA_j\subseteq A_{i+j},
  

Search aliases: a_j

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 582

  [Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).

  Exact mathematical passage

  a_j=\sum_{g\in G}a_{j,g}

Search aliases: a_j,g

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 582

  [Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).

  Exact mathematical passage

  a_j=\sum_{g\in G}a_{j,g}

Search aliases: A_k

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 592

  Expand every input into its homogeneous components. Each term lands in the component whose label is the sum of its input labels. Projection onto A_k retains exactly the displayed terms.

  Exact mathematical passage

  A_k

Search aliases: A_ki

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• Use in source · visual_atlas.tex, line 164

  vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.

  Exact mathematical passage

  C_{0j}B_{jk}A_{ki}

Search aliases: a_n+1

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• Use in source · Read passage · 02_connections_obstructions.tex, line 292

  Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.

  Exact mathematical passage

  a_{n+1}

Search aliases: a_r

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 583

  [Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).

  Exact mathematical passage

  
  P_k\mu(a_1,\ldots,a_r)
  =
  \sum_{g_1+\cdots+g_r=k}
  \mu(a_{1,g_1},\ldots,a_{r,g_r}).
  

Search aliases: a_r,g_r

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 583

  [Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).

  Exact mathematical passage

  
  P_k\mu(a_1,\ldots,a_r)
  =
  \sum_{g_1+\cdots+g_r=k}
  \mu(a_{1,g_1},\ldots,a_{r,g_r}).
  

Search aliases: A_vu

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• Use in source · Read passage · 04_correction_transport.tex, line 712

  [Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.

  Exact mathematical passage

  
  U_vA_{vu}=B_{vu}U_u
  

Search aliases: A_vu^-1

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• Use in source · Read passage · 04_correction_transport.tex, line 729

  Conversely, choose a spanning tree. Starting from U_r, propagate U_v along the unique tree path by the rule U_v=B_vuU_uA_vu^-1 on every tree edge. This is the only possible extension satisfying the tree edge equations. An edge outside the tree closes one fundamental loop. Its intertwining equation holds exactly when U_r intertwines the two corresponding fundamental holonomies. The second condition therefore supplies every remaining edge equation. Uniqueness follows from propagation along the tre…

  Exact mathematical passage

  
  U_v=B_{vu}U_uA_{vu}^{-1}
  

Search aliases: Acal_Pcal A_ P Acal Pcal mathcal mathcal

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 97

  Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.

  Exact mathematical passage

  
  \Acal_{\Pcal}x=(P_\alpha x)_\alpha,
  \qquad
  \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
  

Search aliases: Acal_Rcal A_ R Acal Rcal mathcal mathcal

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 146

  The identities S_ P A_ P=I_X and S_ Q A_ Q=I_Y give S_ QF_ Q← P A_ P = S_ Q A_ QF S_ P A_ P =F. For composition, G_ R← QF_ Q← P = A_ RG S_ Q A_ QF S_ P = A_ R(G∘ F) S_ P.

  Exact mathematical passage

  
  G_{\Rcal\leftarrow\Qcal}F_{\Qcal\leftarrow\Pcal}
  =
  \Acal_{\Rcal}G\Scal_{\Qcal}\Acal_{\Qcal}F\Scal_{\Pcal}
  =
  \Acal_{\Rcal}(G\circ F)\Scal_{\Pcal}.
  

Search aliases: Acal_Ucal A_ U Acal Ucal mathcal mathcal

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 170

  [Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.

  Exact mathematical passage

  
  J_{\Ucal\leftarrow\Vcal}
  =
  \Acal_{\Ucal}\Scal_{\Vcal}.
  

Search aliases: b^2

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• Use in source · visual_atlas.tex, line 497

  For a unary square, inserting a+b does not give a^2+b^2. The extra 2ab arises because the operation is nonlinear in its single argument. It can be analyzed by representing the square using a binary multiplication node with a repeated input, but that is an additional choice of syntax and semantics.

  Exact mathematical passage

  a^2+b^2

Search aliases: B_1

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• Use in source · Read passage · 02_connections_obstructions.tex, line 604

  The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.

  Exact mathematical passage

  B_1,\ldots,B_m\in R^{V\times V}

Search aliases: b_1

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• Use in source · Read passage · 04_correction_transport.tex, line 749

  be the chosen structure group and fix a root r. Restrict both edge transports and gauge values to G. Gauge equivalence classes of G valued transport systems are naturally identified with Hom(π_1(Γ,r),G)/G, where G acts by conjugation. After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of b_1=|E|-|V|+1 tuples in G.

  Exact mathematical passage

  
  b_1=|E|-|V|+1
  

Search aliases: b_j

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 603

  [Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.

  Exact mathematical passage

  
  P_k(ab)=
  \sum_{i+j=k}a_ib_j.
  

Search aliases: B_jk

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• Use in source · visual_atlas.tex, line 164

  vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.

  Exact mathematical passage

  C_{0j}B_{jk}A_{ki}

Search aliases: B_k

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 353

  di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.

  Exact mathematical passage

  
  \begin{aligned}
  &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x)
  [h_1,\ldots,h_n]\\
  &=
  \sum_{\pi=\{B_1,\ldots,B_k\}}
  \sum_{\gamma_1,\ldots,\gamma_k}
  J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx)
  \big[
  G_{\gamma_1,B_1}(x\mid h),\ldots,
  G_{\gamma_k,B_k}(x\mid h)
  \big].
  \end{aligned}
  

Search aliases: B_m

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• Use in source · Read passage · 02_connections_obstructions.tex, line 604

  The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.

  Exact mathematical passage

  B_1,\ldots,B_m\in R^{V\times V}

Search aliases: B_r

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• Use in source · Read passage · 02_connections_obstructions.tex, line 621

  rices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusleanthreeKomorebi.PaperIII.EdgeOperators.h…

  Exact mathematical passage

  B_r[q,p]

Search aliases: b_r

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 432

  Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.

  Exact mathematical passage

  
  a=\|s\|,
  \qquad
  b_r=\|B_r\|,
  \qquad
  \phi(t)=a+\sum_{r=2}^{d}b_rt^r.
  

Search aliases: B_vu

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• Use in source · Read passage · 04_correction_transport.tex, line 712

  [Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.

  Exact mathematical passage

  
  U_vA_{vu}=B_{vu}U_u
  

Search aliases: B_τ tau

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 468

  Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…

  Exact mathematical passage

  
  B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
  

Search aliases: B_τ_1 tau

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 468

  Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…

  Exact mathematical passage

  
  B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
  

Search aliases: B_τ_r tau

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 468

  Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…

  Exact mathematical passage

  
  B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
  

Search aliases: bbk 𝕜 Bbbk

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• Use in source · Read passage · 02_connections_obstructions.tex, line 665

  [A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …

  Exact mathematical passage

  C_0=C_1=C_2=\bbk

Search aliases: bbk^2 𝕜^2 bbk Bbbk

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 194

  [A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by the two diagonal restrictions would lose the product x_1x_2.

  Exact mathematical passage

  X=Y=\bbk^2

Search aliases: bbk^× 𝕜^× bbk times Bbbk times

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• Use in source · Read passage · 02_connections_obstructions.tex, line 666

  [A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …

  Exact mathematical passage

  u,v,w\in\bbk^\times

Search aliases: C mathbb

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 207

  Let X and Y be Banach spaces over the same field R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains containing the points and segments in question. For 1≤ r≤ m, the derivative D^rF(x):X^r⟶ Y is a continuous symmetric multilinear map.

  Exact mathematical passage

  \mathbb C

Search aliases: C^-1

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• Use in source · visual_atlas.tex, line 515

  vfig[.97]worked_holonomy_diagnosis platecaptionThe vector row starts with the source witness e_0 and follows it through A, then B, then C^-1. Returning to the base produces (1,1)^ T, whose displacement from the input is e_1.

  Exact mathematical passage

  C^{-1}

Search aliases: C^0

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• Use in source · Read passage · 04_correction_transport.tex, line 476

  [A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the class of x to the class of y. The total complex is acyclic. Thus nonzero fine obstruction classes can cancel through a later refinement differential.

  Exact mathematical passage

  C^0=\bbk x

Search aliases: C^1

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• Use in source · Read passage · 04_correction_transport.tex, line 476

  [A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the class of x to the class of y. The total complex is acyclic. Thus nonzero fine obstruction classes can cancel through a later refinement differential.

  Exact mathematical passage

  C^1=\bbk y

Search aliases: C^n

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 1150

  The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.

  Exact mathematical passage

  c\in C^n

Search aliases: C^p+q

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• Use in source · Read passage · 04_correction_transport.tex, line 435

  [Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.

  Exact mathematical passage

  
  E_0^{p,q}=\gr^pC^{p+q},
  \qquad
  E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
  

Search aliases: C^r

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• Use in source · Read passage · 04_correction_transport.tex, line 115

  [Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborho…

  Exact mathematical passage

  C^r

Search aliases: c^← leftarrow

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• Use in source · Read passage · 01_finite_coordinates.tex, line 249

  For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.

  Exact mathematical passage

  
    c^\leftarrow(w)=c(e_n)c(e_{n-1})\cdots c(e_1).
  

Search aliases: C_0

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• Use in source · Read passage · 02_connections_obstructions.tex, line 665

  [A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …

  Exact mathematical passage

  C_0=C_1=C_2=\bbk

Search aliases: C_0,m

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 860

  y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  
  \|C_{0,m}x_m-x_0\|
  \le
  \sum_{\ell=0}^{m-1}
  \|C_{0,\ell}\|
  \frac{\varepsilon_\ell}{1-q_\ell}.
  

Search aliases: C_0,ℓ ell

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 860

  y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  
  \|C_{0,m}x_m-x_0\|
  \le
  \sum_{\ell=0}^{m-1}
  \|C_{0,\ell}\|
  \frac{\varepsilon_\ell}{1-q_\ell}.
  

Search aliases: C_0j

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• Use in source · visual_atlas.tex, line 164

  vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.

  Exact mathematical passage

  C_{0j}B_{jk}A_{ki}

Search aliases: C_1

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• Use in source · Read passage · 02_connections_obstructions.tex, line 665

  [A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …

  Exact mathematical passage

  C_0=C_1=C_2=\bbk

Search aliases: C_2

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• Use in source · Read passage · 02_connections_obstructions.tex, line 665

  [A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …

  Exact mathematical passage

  C_0=C_1=C_2=\bbk

Search aliases: C_3

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• Use in source · Read passage · 04_correction_transport.tex, line 982

  Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.

  Exact mathematical passage

  
  S^*S=I+E_2(t,h),
  \qquad
  \|E_2(t,h)\|\le C_3h^2.
  

Search aliases: C_4

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• Use in source · Read passage · 04_correction_transport.tex, line 988

  P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.

  Exact mathematical passage

  
  (S^*S)^{-1/2}=I+E_3(t,h),
  \qquad
  \|E_3(t,h)\|\le C_4h^2.
  

Search aliases: C_a+1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 853

  [Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  
  C_{a,b}=C_aC_{a+1}\cdots C_{b-1}
  \quad\text{for }a<b,
  \qquad
  C_{a,a}=I.
  

Search aliases: C_a,a

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 853

  [Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  
  C_{a,b}=C_aC_{a+1}\cdots C_{b-1}
  \quad\text{for }a<b,
  \qquad
  C_{a,a}=I.
  

Search aliases: C_a,b

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 853

  [Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  
  C_{a,b}=C_aC_{a+1}\cdots C_{b-1}
  \quad\text{for }a<b,
  \qquad
  C_{a,a}=I.
  

Search aliases: C_b-1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 853

  [Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  
  C_{a,b}=C_aC_{a+1}\cdots C_{b-1}
  \quad\text{for }a<b,
  \qquad
  C_{a,a}=I.
  

Search aliases: C_n-1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 514

  Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.

  Exact mathematical passage

  C_{n-1}

Search aliases: C_Z

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 749

  [Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .

  Exact mathematical passage

  C_Z:Z_f\to Z_c

Search aliases: C_α^bullet alpha bullet

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• Use in source · Read passage · 04_correction_transport.tex, line 459

  [Direct packet complexes] Suppose C^bullet=⊕_α∈ IC_α^bullet is a finite direct sum of subcomplexes. Then H^n(C^bullet) ≅ ⊕_α∈ IH^n(C_α^bullet). A total cohomology class vanishes if and only if every packet class vanishes.

  Exact mathematical passage

  
  C^\bullet=\bigoplus_{\alpha\in I}C_\alpha^\bullet
  

Search aliases: C_ℓ ell

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 852

  [Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.

  Exact mathematical passage

  C_\ell:X_{\ell+1}\to X_\ell

Search aliases: cA^X_a A^X_a cA mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 713

  [Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…

  Exact mathematical passage

  \cA^X_a:X\xrightarrow{\sim}X_a

Search aliases: cA^X_b A^X_b cA mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 714

  [Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…

  Exact mathematical passage

  \cA^X_b:X\xrightarrow{\sim}X_b

Search aliases: cA^Y A^Y cA mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 752

  Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.

  Exact mathematical passage

  \cS^Y\circ\cA^Y

Search aliases: cA^Y_r A^Y_r cA mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 511

  If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.

  Exact mathematical passage

  
  \widehat F_{r\leftarrow a}
  =
  \cA^Y_rF\cS^X_a.
  

Search aliases: cA^Y_s A^Y_s cA mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 529

  Both sides reduce to A^Y_sF S^X_a after inserting the appropriate synthesis--analysis identity. The composition law is the same calculation with the middle ambient module inserted.

  Exact mathematical passage

  \cA^Y_sF\cS^X_a

Search aliases: cA^Z_t A^Z_t cA mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 737

  ^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthreeKomorebi.PaperIII.Transport…

  Exact mathematical passage

  \cA^Z_t:Z\xrightarrow{\sim}Z_t

Search aliases: cA_a A_a cA mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 450

  ordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).

  Exact mathematical passage

  
  \cA_a(x)=(P^a_ix)_i,
  \qquad
  \cS_a((x_i)_i)=\sum_ix_i.
  

Search aliases: cA_c A_c cA mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 498

  Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id; interchanging a and b gives the other composition, so the two transports are inverses.

  Exact mathematical passage

  
  J^X_{c\leftarrow b}J^X_{b\leftarrow a}
  =\cA_c\cS_b\cA_b\cS_a
  =\cA_c\cS_a
  =J^X_{c\leftarrow a}.
  

Search aliases: Candidate mathsf

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• Use in source · Read passage · 02_connections_obstructions.tex, line 397

  Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.

  Exact mathematical passage

  \mathsf{Candidate}

Search aliases: cos

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• Use in source · Read passage · 04_correction_transport.tex, line 1051

  [htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1

  Exact mathematical passage

  u(\theta)=(\cos\theta,\sin\theta)

Search aliases: cQ Q mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 510

  If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.

  Exact mathematical passage

  \cQ^r,\cQ^s

Search aliases: cQ^r Q^r cQ mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 510

  If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.

  Exact mathematical passage

  \cQ^r,\cQ^s

Search aliases: cQ^s Q^s cQ mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 510

  If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.

  Exact mathematical passage

  \cQ^r,\cQ^s

Search aliases: cS^X_a S^X_a cS mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 511

  If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.

  Exact mathematical passage

  
  \widehat F_{r\leftarrow a}
  =
  \cA^Y_rF\cS^X_a.
  

Search aliases: cS^X_b S^X_b cS mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 715

  [Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…

  Exact mathematical passage

  \cS^X_b

Search aliases: cS^Y S^Y cS mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 752

  Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.

  Exact mathematical passage

  \cS^Y\circ\cA^Y

Search aliases: cS^Y_r S^Y_r cS mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 718

  [Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…

  Exact mathematical passage

  \cS^Y_r

Search aliases: cS^Y_s S^Y_s cS mathcal

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• Use in source · Read passage · 02_connections_obstructions.tex, line 718

  beginproposition[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then …

  Exact mathematical passage

  \cS^Y_s

Search aliases: cS_b S_b cS mathcal

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• Use in source · Read passage · 01_finite_coordinates.tex, line 497

  Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id; interchanging a and b gives the other composition, so the two transports are inverses.

  Exact mathematical passage

  \cS_b\cA_b=\id_X

Search aliases: D mathfrak

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• Use in source · Read passage · 02_connections_obstructions.tex, line 889

  For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].

  Exact mathematical passage

  
  \mathfrak D^\Gamma_{b,a}(\Phi)
  =
  \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a
  -
  \widehat\Phi_b\circ
  \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
  

Search aliases: D'

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 953

  [Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).

  Exact mathematical passage

  
  \begin{array}{ccc}
  U & \xrightarrow{D} & V\\
  \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\
  U' & \xrightarrow{D'} & V'
  \end{array}
  

Search aliases: D^*

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 26

  A linear correction map D:U⟶ V places the obstruction to solving Du=-r in the canonical quotient coker D. This quotient has a universal property and behaves naturally under commuting squares. Mapping cones extend the same construction to cochain complexes. In a Hilbert space with closed range, the quotient is represented by the orthogonal projection onto Ker D^*. The Moore Penrose inverse then gives the least norm correction. Relevant background appears in [source reference].

  Exact mathematical passage

  \Ker D^*

Search aliases: D^-1

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• Use in source · Read passage · 04_correction_transport.tex, line 623

  C · D , where D is invertible. Eliminating the hidden variable replaces the retained derivative A by A-BD^-1C. The feedback term is exact. Keeping only A would ignore the path from retained variables into the hidden variable and back.

  Exact mathematical passage

  
  A-BD^{-1}C.
  

Search aliases: D^2

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 401

  [The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic interaction of F. The second records a genuine second order interaction created inside G before F is applied.

  Exact mathematical passage

  
  \begin{aligned}
  R_\delta D^2(F\circ G)(x)[h_1,h_2]
  ={}&
  \sum_{\gamma_1,\gamma_2}
  J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx)
  [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\
  &+
  \sum_\gamma
  J^1_{\delta\mid\gamma}F(Gx)
  [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]].
  \end{aligned}
  

Search aliases: D^m+1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 300

  · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.

  Exact mathematical passage

  
  R_{m+1}(x,h)
  =
  \frac1{m!}
  \int_0^1(1-t)^m
  D^{m+1}F(x+th)[h,\ldots,h]\,dt.
  

Search aliases: D^× times

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• Use in source · Read passage · 02_connections_obstructions.tex, line 61

  Flatness replaces the two-edge route in D^× by the direct route, which cancels its inverse. For H, flatness first collapses a→ b→ c to a→ c, then collapses a→ c→ a to the normalized transport a→ a.

  Exact mathematical passage

  D^\times

Search aliases: D^×_c,b,a times

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• Use in source · Read passage · 02_connections_obstructions.tex, line 33

  At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKomorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Kom…

  Exact mathematical passage

  
  D^\times_{c,b,a}
  =
  \bigl(\Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}\bigr)
  \Gamma_{c\leftarrow a}^{-1},
  \qquad
  H_{c,b,a}
  =
  \Gamma_{a\leftarrow c}
  \Gamma_{c\leftarrow b}
  \Gamma_{b\leftarrow a}.
  

Search aliases: D^Γ_b,a mathfrak Gamma

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• Use in source · Read passage · 02_connections_obstructions.tex, line 889

  For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].

  Exact mathematical passage

  
  \mathfrak D^\Gamma_{b,a}(\Phi)
  =
  \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a
  -
  \widehat\Phi_b\circ
  \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
  

Search aliases: D^† dagger

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• Use in source · Read passage · 04_correction_transport.tex, line 6

  [Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .

  Exact mathematical passage

  D^\dagger

Search aliases: D_0

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 1044

  [Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0

  Exact mathematical passage

  
  \begin{array}{ccccccccc}
  0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt]
  &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt]
  0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0.
  \end{array}
  

Search aliases: D_c

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 1044

  [Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0

  Exact mathematical passage

  
  \begin{array}{ccccccccc}
  0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt]
  &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt]
  0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0.
  \end{array}
  

Search aliases: D_f

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 1044

  [Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0

  Exact mathematical passage

  
  \begin{array}{ccccccccc}
  0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt]
  &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt]
  0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0.
  \end{array}
  

Search aliases: D_u

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• Use in source · Read passage · 04_correction_transport.tex, line 540

  [Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.

  Exact mathematical passage

  
  DR
  =
  D_uF_P
  -
  D_vF_P A^{-1}D_uF_Q.
  

Search aliases: D_v

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• Use in source · Read passage · 04_correction_transport.tex, line 508

  F_Q(u,v) . Assume F takes values in Y_P× Y_Q, is twice continuously Fr echet differentiable near (u_0,v_0), satisfies F_Q(u_0,v_0)=0, and A=D_vF_Q(u_0,v_0) is a Banach space isomorphism from V to Y_Q. The implicit function theorem gives a local map v=φ(u), with φ(u_0)=v_0, satisfying F_Q(u,φ(u))=0. Define the reduced equation R(u)=F_P(u,φ(u)).

  Exact mathematical passage

  
  A=D_vF_Q(u_0,v_0)
  

Search aliases: D_α alpha

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 982

  [Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.

  Exact mathematical passage

  
  D_\alpha:U_\alpha\longrightarrow V_\alpha
  

Search aliases: d_η eta

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• Use in source · Read passage · 04_correction_transport.tex, line 396

  Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.

  Exact mathematical passage

  
  \Omega(\eta+h)
  \equiv
  \Omega(\eta)+d_\eta h
  \pmod{F^{m+1}\Lcal^2},
  \qquad
  d_\eta=d+[\eta,\mathord\cdot].
  

Search aliases: det

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• Use in source · visual_atlas.tex, line 279

  The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1

  Exact mathematical passage

  
  \operatorname{tr}H=2,\qquad \det H=1,\qquad
  \chi_H(t)=(t-1)^2.
  

Search aliases: diag operatorname

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• Use in source · visual_atlas.tex, line 296

  vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSTraditional paper: Theorem 6.2 (p. 14) At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2

  Exact mathematical passage

  D=\operatorname{diag}(2,1)

Search aliases: dist operatorname

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• Use in source · Read passage · 04_correction_transport.tex, line 21

  agger be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .

  Exact mathematical passage

  
  \dist(r,\im D)=\|P_{\mathrm{ob}}r\|.
  

Search aliases: dot P dot

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• Use in source · Read passage · 04_correction_transport.tex, line 801

  Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).

  Exact mathematical passage

  
  K(t)=\sum_\alpha \dot P_\alpha(t)P_\alpha(t).
  

Search aliases: dot P_α dot alpha

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• Use in source · Read passage · 04_correction_transport.tex, line 801

  Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).

  Exact mathematical passage

  
  K(t)=\sum_\alpha \dot P_\alpha(t)P_\alpha(t).
  

Search aliases: dot P_β dot beta

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• Use in source · Read passage · 04_correction_transport.tex, line 810

  begingroup setlengthabovedisplayskip6pt plus 2pt minus 2pt setlengthbelowdisplayskip6pt plus 2pt minus 2pt [Complete packet commutator identity] For every packet label β, dot P_β(t)=[K(t),P_β(t)]. The generator is also skew adjoint: K(t)^*=-K(t).

  Exact mathematical passage

  
  \dot P_\beta(t)=[K(t),P_\beta(t)].
  

Search aliases: dot U dot

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• Use in source · Read passage · 04_correction_transport.tex, line 850

  [Canonical packet transport] Let U(t) solve dot U(t)=K(t)U(t), U(0)=I. Then U(t) is unitary and transports every packet exactly: P_α(t)U(t)=U(t)P_α(0) for all α.

  Exact mathematical passage

  
  \dot U(t)=K(t)U(t),
  \qquad
  U(0)=I.
  

Search aliases: dot V dot

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• Use in source · Read passage · 04_correction_transport.tex, line 863

  The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I; uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.

  Exact mathematical passage

  \dot V=-VK

Search aliases: dot W dot

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• Use in source · Read passage · 04_correction_transport.tex, line 863

  The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I; uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.

  Exact mathematical passage

  \dot W=KW-WK

Search aliases: downarrow

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 953

  [Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).

  Exact mathematical passage

  
  \begin{array}{ccc}
  U & \xrightarrow{D} & V\\
  \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\
  U' & \xrightarrow{D'} & V'
  \end{array}
  

Search aliases: e'

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 1150

  The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.

  Exact mathematical passage

  e'\in E^{n-1}

Search aliases: E^n-1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 1150

  The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.

  Exact mathematical passage

  e'\in E^{n-1}

Search aliases: e_0

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• Use in source · visual_atlas.tex, line 136

  vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSTraditional paper: Corollary 3.11 (p. 6) Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore exactly the span of the first two displayed columns.

  Exact mathematical passage

  e_0\in\QQ^5

Search aliases: E_01

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• Use in source · visual_atlas.tex, line 352

  vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside the diagonal encode two effects that mix packets.

  Exact mathematical passage

  
  H=I+2E_{01}+3E_{12},\qquad \Ob=H-I.
  

Search aliases: E_0^p,q

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• Use in source · Read passage · 04_correction_transport.tex, line 435

  [Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.

  Exact mathematical passage

  
  E_0^{p,q}=\gr^pC^{p+q},
  \qquad
  E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
  

Search aliases: E_10

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• Use in source · visual_atlas.tex, line 46

  vpage3Read the product in execution orderMODEL A / ONE EDGE IS ONE MATRIX ENTRYTraditional paper: Lemma 2.2 (p. 2), Theorem 2.3 (p. 2), Theorem 2.4 (p. 3) Work over ZZ with packets 0,1,2. Let e:0→1 have coefficient 2, and let f:1→2 have coefficient 3. The elementary operators are L_e=2E_10, L_f=3E_21. The location of the nonzero entry records the type of the transfer, not an arbitrary matrix convention.

  Exact mathematical passage

  
  L_e=2E_{10},\qquad L_f=3E_{21}.
  

Search aliases: E_12

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• Use in source · visual_atlas.tex, line 352

  vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside the diagonal encode two effects that mix packets.

  Exact mathematical passage

  
  H=I+2E_{01}+3E_{12},\qquad \Ob=H-I.
  

Search aliases: E_1^p,n-p

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• Use in source · Read passage · 04_correction_transport.tex, line 450

  [Fine obstruction survival] A class on the first page contributes to the total obstruction only if it is represented by a permanent cycle, so it is neither killed nor identified as a boundary on any later page. If the spectral sequence degenerates at the first page, then H^n(C^bullet)=0 exactly when E_1^p,n-p=0 for every p.

  Exact mathematical passage

  E_1^{p,n-p}=0

Search aliases: E_1^p,q

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• Use in source · Read passage · 04_correction_transport.tex, line 435

  [Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.

  Exact mathematical passage

  
  E_0^{p,q}=\gr^pC^{p+q},
  \qquad
  E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
  

Search aliases: E_2

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• Use in source · Read passage · 04_correction_transport.tex, line 982

  Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.

  Exact mathematical passage

  
  S^*S=I+E_2(t,h),
  \qquad
  \|E_2(t,h)\|\le C_3h^2.
  

Search aliases: e_2

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• Use in source · visual_atlas.tex, line 137

  vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSTraditional paper: Corollary 3.11 (p. 6) Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore exactly the span of the first two displayed columns.

  Exact mathematical passage

  
  Te_0=e_1+e_2,\qquad T^2e_0=0.
  

Search aliases: E_20

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• Use in source · visual_atlas.tex, line 55

  On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the entire product, even if both edge coefficients are nonzero.

  Exact mathematical passage

  
  L_fL_e=6E_{20},\qquad L_eL_f=0.
  

Search aliases: E_21

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• Use in source · visual_atlas.tex, line 46

  vpage3Read the product in execution orderMODEL A / ONE EDGE IS ONE MATRIX ENTRYTraditional paper: Lemma 2.2 (p. 2), Theorem 2.3 (p. 2), Theorem 2.4 (p. 3) Work over ZZ with packets 0,1,2. Let e:0→1 have coefficient 2, and let f:1→2 have coefficient 3. The elementary operators are L_e=2E_10, L_f=3E_21. The location of the nonzero entry records the type of the transfer, not an arbitrary matrix convention.

  Exact mathematical passage

  
  L_e=2E_{10},\qquad L_f=3E_{21}.
  

Search aliases: E_3

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• Use in source · Read passage · 04_correction_transport.tex, line 988

  P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.

  Exact mathematical passage

  
  (S^*S)^{-1/2}=I+E_3(t,h),
  \qquad
  \|E_3(t,h)\|\le C_4h^2.
  

Search aliases: e_3

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• Use in source · visual_atlas.tex, line 152

  headThe admissible operator grammar matters Allowing each individual edge operator is a different generation problem. For instance, applying the edge 0→1 and then 1→3 individually produces e_3. It does not follow that e_3 belongs to the orbit of the summed operator T. The paper's theorem on orbit support concerns the explicitly stated operator action.

  Exact mathematical passage

  e_3

Search aliases: e_4

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• Use in source · orbit_support.tex, line 63

  ,0.99500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.69000,0.65000) rectangle ++(0.69000,0.69000); node[] at (5.03500,0.99500) 0; node[] at (8.85000,3.80000) Reach(0)={0,1,2,3}; node[] at (8.85000,2.75000) Hull_T(e_0)={(a,b,b,0,0)}; node[] at (8.85000,1.65000) e_3: reachable but absent; node[] at (8.85000,0.60000) e_4: unreachable and absent;

  Exact mathematical passage

  e_4:\ \text{unreachable and absent}

Search aliases: e_i

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• Use in source · visual_atlas.tex, line 18

  vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLETraditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16) Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role is explicit.

  Exact mathematical passage

  E_{ji}e_i=e_j

Search aliases: e_j

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• Use in source · visual_atlas.tex, line 18

  vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLETraditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16) Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role is explicit.

  Exact mathematical passage

  E_{ji}e_i=e_j

Search aliases: E_ji

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• Use in source · visual_atlas.tex, line 18

  vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLETraditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16) Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role is explicit.

  Exact mathematical passage

  E_{ji}

Search aliases: e_n+1

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 815

  Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.

  Exact mathematical passage

  
  e_{n+1}\le\varepsilon_n+qe_n.
  

Search aliases: e_n-1

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• Use in source · Read passage · 01_finite_coordinates.tex, line 249

  For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.

  Exact mathematical passage

  
    c^\leftarrow(w)=c(e_n)c(e_{n-1})\cdots c(e_1).
  

Search aliases: E_rs

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 704

  [Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.

  Exact mathematical passage

  E_{rs}

Search aliases: E_rv

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 705

  [Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.

  Exact mathematical passage

  
  E_{rs}E_{uv}=\delta_{su}E_{rv},
  

Search aliases: E_tgt(e),src(e) tgt src operatorname operatorname

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• Use in source · visual_atlas.tex, line 24

  L_e=c(e)E_tgt(e),src(e) · One operator supported on a packet. · pagerefV3

  Exact mathematical passage

  L_e=c(e)E_{\tgt(e),\src(e)}

Search aliases: E_uv

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 705

  [Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.

  Exact mathematical passage

  
  E_{rs}E_{uv}=\delta_{su}E_{rv},
  

Search aliases: E_α alpha

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• Use in source · Read passage · 04_correction_transport.tex, line 966

  Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.

  Exact mathematical passage

  
  P_\alpha(t+h)
  =
  P_\alpha(t)+h\dot P_\alpha(t)+E_{\alpha}(t,h),
  \qquad
  \|E_{\alpha}(t,h)\|\le C_1h^2,
  

Search aliases: End operatorname

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• Use in source · Read passage · 03_nonlinear_coordinates.tex, line 80

  Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.

  Exact mathematical passage

  
  \Pcal=(P_\alpha)_{\alpha\in I}\subseteq\End(X)
  

Search aliases: exp

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• Use in source · Read passage · 04_correction_transport.tex, line 1062

  [Convergence of sampled transports] Assume K is Lipschitz on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T, define U_π = exp((t_N-t_N-1)K(t_N-1)) ⋯ exp((t_1-t_0)K(t_0)). As the mesh |π| tends to zero, U_π⟶ U(T) in operator norm. There is a constant C, depending only on T, the supremum norm of K, and its Lipschitz constant, such that U_π-U(T) ≤ C|π|.

  Exact mathematical passage

  
  U_\pi
  =
  \exp\big((t_N-t_{N-1})K(t_{N-1})\big)
  \cdots
  \exp\big((t_1-t_0)K(t_0)\big).
  

Search aliases: F^0

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• Use in source · Read passage · 04_correction_transport.tex, line 199

  Let X and Y be modules with decreasing filtrations X=F^0X⊇ F^1X⊇⋯, Y=F^0Y⊇ F^1Y⊇⋯. Assume both are separated and complete for their filtration topologies. Let Φ:X→ Y be continuous and filtration congruence preserving in the sense that h∈ F^mX Longrightarrow Φ(x+h)-Φ(x)∈ F^mY for every x∈ X and every m. This condition makes the statement Φ(x)∈ F^mY depend only on the class of x modulo F^mX.

  Exact mathematical passage

  
  X=F^0X\supseteq F^1X\supseteq\cdots,
  \qquad
  Y=F^0Y\supseteq F^1Y\supseteq\cdots.
  

Search aliases: F^1

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• Use in source · Read passage · 04_correction_transport.tex, line 199

  Let X and Y be modules with decreasing filtrations X=F^0X⊇ F^1X⊇⋯, Y=F^0Y⊇ F^1Y⊇⋯. Assume both are separated and complete for their filtration topologies. Let Φ:X→ Y be continuous and filtration congruence preserving in the sense that h∈ F^mX Longrightarrow Φ(x+h)-Φ(x)∈ F^mY for every x∈ X and every m. This condition makes the statement Φ(x)∈ F^mY depend only on the class of x modulo F^mX.

  Exact mathematical passage

  
  X=F^0X\supseteq F^1X\supseteq\cdots,
  \qquad
  Y=F^0Y\supseteq F^1Y\supseteq\cdots.
  

Search aliases: F^2

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• Use in source · Read passage · 04_correction_transport.tex, line 483

  [A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the class of x to the class of y. The total complex is acyclic. Thus nonzero fine obstruction classes can cancel through a later refinement differential.

  Exact mathematical passage

  
  F^0C^1=F^1C^1=\bbk y,
  \qquad
  F^2C^1=0.
  

Search aliases: F^2m

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• Use in source · Read passage · 04_correction_transport.tex, line 420

  Direct expansion gives Ω(η+h) = Ω(η)+dh+[η,h]+frac12[h,h]. The filtration rule places [h,h] in F^2m L^2, which is contained in F^m+1 L^2 because m≥1. The congruence follows. The last statement is [source reference] applied to the induced graded correction map.

  Exact mathematical passage

  F^{2m}\Lcal^2

Search aliases: F^i

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• Use in source · Read passage · 04_correction_transport.tex, line 386

  [Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.

  Exact mathematical passage

  
  dF^i\Lcal\subseteq F^i\Lcal,
  \qquad
  [F^i\Lcal,F^j\Lcal]\subseteq F^{i+j}\Lcal.
  

Search aliases: F^i+j