Notation by work

Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates

All notation · Glossary · Source coverage

716 symbol spellings in 2117 mathematical passages across 32 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.

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

All 716 entries are available below.

=Notation table · 478 passages

Search aliases: =

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Notation table · Read passage · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
```
\mathfrak G=(Q,I)
```

ANotation table · 199 passages

Search aliases: A

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Notation table · Read passage · 01_orientation.tex, line 65
Hull_A(S) · Generated hull · The submodule AS, or its declared topological closure.
Exact mathematical passage
```
\Hull_A(S)
```

\ANotation table · 66 passages

Search aliases: A mathcal

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Notation table · Read passage · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
```
\A_X
```

\mathcal ANotation table · 14 passages

Search aliases: A mathcal

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Notation table · Read passage · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
```
\mathcal A_{\Pcal}
```

\mathcal A_{\Pcal}Notation table · 14 passages

Search aliases: A_Pcal A_ P mathcal Pcal mathcal mathcal

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Notation table · Read passage · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
```
\mathcal A_{\Pcal}
```

\A_XNotation table · 7 passages

Search aliases: A_X A mathcal

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Notation table · Read passage · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
```
\A_X
```

\bbkNotation table · 98 passages

Search aliases: bbk 𝕜 Bbbk

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Notation table · Read passage · 01_orientation.tex, line 62
Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.
Exact mathematical passage
```
\bbk Q/I
```

\EndNotation table · 24 passages

Search aliases: End operatorname

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Notation table · Read passage · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
```
\End_{\bbk}(|X|)
```

\End_{\bbk}Notation table · 12 passages

Search aliases: End_bbk End_𝕜 End bbk operatorname Bbbk

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Notation table · Read passage · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
```
\End_{\bbk}(|X|)
```

\mathfrak GNotation table · 14 passages

Search aliases: G mathfrak

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Notation table · Read passage · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
```
\mathfrak G=(Q,I)
```

\HullNotation table · 33 passages

Search aliases: Hull operatorname

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Notation table · Read passage · 01_orientation.tex, line 65
Hull_A(S) · Generated hull · The submodule AS, or its declared topological closure.
Exact mathematical passage
```
\Hull_A(S)
```

\Hull_ANotation table · 9 passages

Search aliases: Hull_A Hull operatorname

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Notation table · Read passage · 01_orientation.tex, line 65
Hull_A(S) · Generated hull · The submodule AS, or its declared topological closure.
Exact mathematical passage
```
\Hull_A(S)
```

INotation table · 112 passages

Search aliases: I

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Notation table · Read passage · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
```
\mathfrak G=(Q,I)
```

MNotation table · 158 passages

Search aliases: M

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Notation table · Read passage · 01_orientation.tex, line 64
S⊆ M · Seed family · The vectors from which the operator orbit is generated.
Exact mathematical passage
```
S\subseteq M
```

PNotation table · 185 passages

Search aliases: P

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Notation table · Read passage · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
```
\Pcal=(P_\alpha)
```

P_\alphaNotation table · 106 passages

Search aliases: P_α alpha

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Notation table · Read passage · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
```
\Pcal=(P_\alpha)
```

P_\betaNotation table · 33 passages

Search aliases: P_β beta

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Notation table · Read passage · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
```
P_\beta TP_\alpha:X_\alpha\to X_\beta
```

\PcalNotation table · 61 passages

Search aliases: Pcal P mathcal

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Notation table · Read passage · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
```
\Pcal=(P_\alpha)
```

\psrankNotation table · 9 passages

Search aliases: psrank operatorname

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Notation table · Read passage · 01_orientation.tex, line 73
psrank · Rank for seeds confined to individual packets · The minimal number of generating seeds each supported in a single packet.
Exact mathematical passage
```
\psrank
```

QNotation table · 62 passages

Search aliases: Q

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Notation table · Read passage · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
```
\mathfrak G=(Q,I)
```

SNotation table · 140 passages

Search aliases: S

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Notation table · Read passage · 01_orientation.tex, line 64
S⊆ M · Seed family · The vectors from which the operator orbit is generated.
Exact mathematical passage
```
S\subseteq M
```

\mathcal SNotation table · 16 passages

Search aliases: S mathcal

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Notation table · Read passage · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
```
\mathcal S_{\Pcal}
```

\mathcal S_{\Pcal}Notation table · 16 passages

Search aliases: S_Pcal S_ P mathcal Pcal mathcal mathcal

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Notation table · Read passage · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
```
\mathcal S_{\Pcal}
```

\srankNotation table · 43 passages

Search aliases: srank operatorname

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Notation table · Read passage · 01_orientation.tex, line 66
srank_A(M) · Seed rank · The smallest cardinality of a generating family. A finite value is the minimum number of seeds.
Exact mathematical passage
```
\srank_A(M)
```

\srank_ANotation table · 24 passages

Search aliases: srank_A srank operatorname

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Notation table · Read passage · 01_orientation.tex, line 66
srank_A(M) · Seed rank · The smallest cardinality of a generating family. A finite value is the minimum number of seeds.
Exact mathematical passage
```
\srank_A(M)
```

\SuppNotation table · 6 passages

Search aliases: Supp operatorname

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Notation table · Read passage · 01_orientation.tex, line 72
Supp_ P(x) · Packet support · The packet indices on which x has a nonzero coordinate.
Exact mathematical passage
```
\Supp_{\Pcal}(x)
```

\Supp_{\Pcal}Notation table · 6 passages

Search aliases: Supp_Pcal Supp_ P Supp Pcal operatorname mathcal

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Notation table · Read passage · 01_orientation.tex, line 72
Supp_ P(x) · Packet support · The packet indices on which x has a nonzero coordinate.
Exact mathematical passage
```
\Supp_{\Pcal}(x)
```

TNotation table · 123 passages

Search aliases: T

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Notation table · Read passage · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
```
T_{\beta\alpha}
```

T_{\beta\alpha}Notation table · 19 passages

Search aliases: T_βα beta alpha

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Notation table · Read passage · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
```
T_{\beta\alpha}
```

XNotation table · 283 passages

Search aliases: X

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Notation table · Read passage · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
```
\A_X
```

xNotation table · 168 passages

Search aliases: x

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Notation table · Read passage · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
```
x\mapsto(P_\alpha x)_\alpha
```

X_\alphaNotation table · 53 passages

Search aliases: X_α alpha

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Notation table · Read passage · 01_orientation.tex, line 68
X_α · Packet space · The range P_α X.
Exact mathematical passage
```
X_\alpha
```

x_\alphaNotation table · 8 passages

Search aliases: x_α alpha

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Notation table · Read passage · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
```
(x_\alpha)\mapsto\sum_\alpha x_\alpha
```

X_\betaNotation table · 13 passages

Search aliases: X_β beta

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Notation table · Read passage · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
```
P_\beta TP_\alpha:X_\alpha\to X_\beta
```

|Notation table · 25 passages

Search aliases: |

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Notation table · Read passage · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
```
\End_{\bbk}(|X|)
```

\alphaNotation table · 223 passages

Search aliases: α alpha

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Notation table · Read passage · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
```
\Pcal=(P_\alpha)
```

\betaNotation table · 89 passages

Search aliases: β beta

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Notation table · Read passage · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
```
T_{\beta\alpha}
```

\LambdaNotation table · 12 passages

Search aliases: Λ Lambda

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Notation table · Read passage · 01_orientation.tex, line 62
Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.
Exact mathematical passage
```
\Lambda_{\mathfrak G}
```

\Lambda_{\mathfrak G}Notation table · 10 passages

Search aliases: Λ_ G Lambda mathfrak

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Notation table · Read passage · 01_orientation.tex, line 62
Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.
Exact mathematical passage
```
\Lambda_{\mathfrak G}
```

\toNotation table · 53 passages

Search aliases: → to

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Notation table · Read passage · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
```
P_\beta TP_\alpha:X_\alpha\to X_\beta
```

\mapstoNotation table · 10 passages

Search aliases: ↦ mapsto

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Notation table · Read passage · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
```
x\mapsto(P_\alpha x)_\alpha
```

\sumNotation table · 83 passages

Search aliases: ∑ sum

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Notation table · Read passage · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
```
(x_\alpha)\mapsto\sum_\alpha x_\alpha
```

\sum_\alphaNotation table · 22 passages

Search aliases: ∑_α sum alpha

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Notation table · Read passage · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
```
(x_\alpha)\mapsto\sum_\alpha x_\alpha
```

\subseteqNotation table · 49 passages

Search aliases: ⊆ subseteq

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Notation table · Read passage · 01_orientation.tex, line 64
S⊆ M · Seed family · The vectors from which the operator orbit is generated.
Exact mathematical passage
```
S\subseteq M
```

aDefinition passage · 67 passages

Search aliases: a

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
X_a:X_{s(a)}\longrightarrow X_{t(a)}
\qquad(a\in Q_1)
```

A^{\oplus r}Definition passage · 9 passages

Search aliases: A^⊕ r oplus

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

a_1Definition passage · 6 passages

Search aliases: a_1

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\A_CDefinition passage · 5 passages

Search aliases: A_C A mathcal

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
P_C:=\sum_{\alpha\in C}P_\alpha,
\qquad
X_C:=P_CX,
\qquad
\A_C:=P_C\A P_C,
```

a_jDefinition passage · 6 passages

Search aliases: a_j

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

a_nDefinition passage · 8 passages

Search aliases: a_n

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

a_rDefinition passage · 1 passage

Search aliases: a_r

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\mathrm{abs}Definition passage · 2 passages

Search aliases: abs mathrm

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathbf{SeedSys}_{\mathrm{abs}}
```

\algDefinition passage · 2 passages

Search aliases: alg operatorname

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 919
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
\A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
```

\alg_{\bbk}Definition passage · 2 passages

Search aliases: alg_bbk alg_𝕜 alg bbk operatorname Bbbk

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 919
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
\A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
```

BDefinition passage · 44 passages

Search aliases: B

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Definition passage: Packet system surjective along edges · Read passage · 03_packets_and_graphs.tex, line 1058
[Packet system surjective along edges] The module resolved into packets is surjective along edges relative to Γ( G, P) when, for every graph edge α→β, there exists B∈ A such that the pure block P_β BP_α:X_α↠ X_β is surjective.
Exact mathematical passage
```
B\in\A
```

\BDefinition passage · 13 passages

Search aliases: B mathcal

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 50
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
(Y,\B,\Qcal)
```

\mathrm{boundary}Definition passage · 1 passage

Search aliases: boundary mathrm

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

CDefinition passage · 46 passages

Search aliases: C

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1019
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
C\subseteq I
```

\mathbf CDefinition passage · 3 passages

Search aliases: C mathbf

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathbf C_D
```

cDefinition passage · 7 passages

Search aliases: c

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Definition passage: Paper I export object · Read passage · 04_transport.tex, line 486
[Paper I export object] The canonical Paper I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, morphisms, invariant exact sequences, quotients, completions, realization functors and equational theorem transport ).
Exact mathematical passage
```
\boxed{
\mathsf{PSC}_I=
\left(
\begin{array}{c}
\text{typed grammars and realized operator algebras},\\
\text{seed systems, free presentations, and seed rank},\\
\text{module-top, semisimple, and single-operator formulas},\\
\text{finite and unconditional packet coordinates},\\
\text{transfer blocks and support algebras},\\
\text{reachability and packet-pure source generation},\\
\text{morphisms, invariant exact sequences, quotients, completions},\\
\text{realization functors and equational theorem transport}
\end{array}
\right).
}
```

\mathbf C_DDefinition passage · 2 passages

Search aliases: C_D mathbf

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathbf C_D
```

\clDefinition passage · 4 passages

Search aliases: cl

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Definition passage: Generated hull · Read passage · 02_foundations.tex, line 162
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
```
\Hull_{\A,\tau}(S):=\cl{\Hull_{\A}(S)}^{\,\tau}.
```

DDefinition passage · 49 passages

Search aliases: D

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathbf C_D
```

\mathrm{dependencies}Definition passage · 1 passage

Search aliases: dependencies mathrm

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

EDefinition passage · 19 passages

Search aliases: E

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 922
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
\alg_{\bbk}(E)
```

\mathsf{Ev}Definition passage · 1 passage

Search aliases: Ev mathsf

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 385
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathsf{Ev}_D:\mathbf{SeedSys}_{\mathrm{abs}}\longrightarrow\mathbf C_D
```

\mathsf{Ev}_DDefinition passage · 1 passage

Search aliases: Ev_D mathsf

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 385
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathsf{Ev}_D:\mathbf{SeedSys}_{\mathrm{abs}}\longrightarrow\mathbf C_D
```

\mathrm{exports}Definition passage · 1 passage

Search aliases: exports mathrm

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

FDefinition passage · 28 passages

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

fDefinition passage · 65 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 52
k G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f:X\to Y
```

f_{s(a)}Definition passage · 1 passage

Search aliases: f_s(a)

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 53
linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f_{t(a)}X_a=Y_af_{s(a)}
\qquad(a\in Q_1).
```

f_{t(a)}Definition passage · 1 passage

Search aliases: f_t(a)

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 53
linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f_{t(a)}X_a=Y_af_{s(a)}
\qquad(a\in Q_1).
```

f_vDefinition passage · 1 passage

Search aliases: f_v

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 52
l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f_v:X_v\to Y_v
```

GDefinition passage · 15 passages

Search aliases: G

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 927
l=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
P_\beta GP_\alpha\neq0
```

\mathcal GDefinition passage · 8 passages

Search aliases: G mathcal

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 918
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
\mathcal G\subseteq\A
```

\HomDefinition passage · 4 passages

Search aliases: Hom operatorname

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Definition passage: Transfer block · Read passage · 03_packets_and_graphs.tex, line 495
[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).
Exact mathematical passage
```
T_{\beta\alpha}:=P_\beta TP_\alpha
\in\Hom_{\bbk}(X_\alpha,X_\beta).
```

\Hom_{\bbk}Definition passage · 2 passages

Search aliases: Hom_bbk Hom_𝕜 Hom bbk operatorname Bbbk

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Definition passage: Transfer block · Read passage · 03_packets_and_graphs.tex, line 495
[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).
Exact mathematical passage
```
T_{\beta\alpha}:=P_\beta TP_\alpha
\in\Hom_{\bbk}(X_\alpha,X_\beta).
```

\Hull_{\A}Definition passage · 10 passages

Search aliases: Hull_A Hull_ A Hull A operatorname mathcal

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Definition passage: Generated hull · Read passage · 02_foundations.tex, line 158
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
```
\Hull_{\A}(S):=\sum_{s\in S}\A s.
```

\Hull_{\A,\tau}Definition passage · 1 passage

Search aliases: Hull_A,τ Hull_ A,τ Hull A tau operatorname mathcal tau

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Definition passage: Generated hull · Read passage · 02_foundations.tex, line 162
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
```
\Hull_{\A,\tau}(S):=\cl{\Hull_{\A}(S)}^{\,\tau}.
```

\Hull_{\B}Definition passage · 7 passages

Search aliases: Hull_B Hull_ B Hull B operatorname mathcal

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Definition passage: Morphism of seeded systems · Read passage · 04_transport.tex, line 80
[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.
Exact mathematical passage
```
\Phi(S)\subseteq\Hull_{\B}(S').
```

\mathrm{hypotheses}Definition passage · 1 passage

Search aliases: hypotheses mathrm

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

I_XDefinition passage · 6 passages

Search aliases: I_X

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Definition passage: Finite packet system · Read passage · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
```
P_\alpha^2=P_\alpha,
\qquad
P_\alpha P_\beta=0\quad(\alpha\neq\beta),
\qquad
\sum_{\alpha\in I}P_\alpha=I_X.
```

\mathrm{ID}Definition passage · 1 passage

Search aliases: ID mathrm

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

\imDefinition passage · 8 passages

Search aliases: im operatorname

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Definition passage: Realized operator algebra · Read passage · 02_foundations.tex, line 117
The realized action and faithful reduction. [Realized operator algebra] For a realization X, the realized operator algebra is A_X:=im (Λ_ G⟶End_𝕜(|X|)). It records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.
Exact mathematical passage
```
\A_X:=\im\!\left(\Lambda_{\mathfrak G}\longrightarrow\End_{\bbk}(|X|)\right).
```

JDefinition passage · 49 passages

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 50
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
J
```

jDefinition passage · 86 passages

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\mathfrak MDefinition passage · 14 passages

Search aliases: M mathfrak

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Definition passage: Support relation and support space · Read passage · 03_packets_and_graphs.tex, line 792
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
```
\mathfrak M_R
:=
\{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
```

M^rDefinition passage · 1 passage

Search aliases: M^r

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 217
gives a canonical map from a finite free module. The generated hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
S=(s_1,\ldots,s_r)\in M^r
```

\mathfrak M_RDefinition passage · 11 passages

Search aliases: M_R mathfrak

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Definition passage: Support relation and support space · Read passage · 03_packets_and_graphs.tex, line 792
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
```
\mathfrak M_R
:=
\{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
```

\minDefinition passage · 8 passages

Search aliases: min

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Definition passage: Seed rank · Read passage · 02_foundations.tex, line 373
[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.
Exact mathematical passage
```
\srank_A(M):=\min\{|S|:AS=M\}.
```

NDefinition passage · 60 passages

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Definition passage: Witness for seed rank · Read passage · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
```
\Phi:M\twoheadrightarrow N
```

\mathbb NDefinition passage · 3 passages

Search aliases: N mathbb

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

nDefinition passage · 103 passages

Search aliases: n

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 354
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
(X_n)_{n\geq1}
```

P_CDefinition passage · 6 passages

Search aliases: P_C

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
P_C:=\sum_{\alpha\in C}P_\alpha,
\qquad
X_C:=P_CX,
\qquad
\A_C:=P_C\A P_C,
```

P_\alpha^2Definition passage · 2 passages

Search aliases: P_α^2 alpha

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Definition passage: Finite packet system · Read passage · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
```
P_\alpha^2=P_\alpha,
\qquad
P_\alpha P_\beta=0\quad(\alpha\neq\beta),
\qquad
\sum_{\alpha\in I}P_\alpha=I_X.
```

P_{\sigma^{-1}(j)}Definition passage · 5 passages

Search aliases: P_σ^-1(j) sigma

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
Q_j\Phi
=
\Phi P_{\sigma^{-1}(j)},
\qquad
P_{\sigma^{-1}(j)}:=
\sum_{\sigma(\alpha)=j}P_\alpha.
```

\mathrm{proof}Definition passage · 1 passage

Search aliases: proof mathrm

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

\mathsf{PSC}Definition passage · 1 passage

Search aliases: PSC mathsf

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Definition passage: Paper I export object · Read passage · 04_transport.tex, line 486
[Paper I export object] The canonical Paper I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, morphisms, invariant exact sequences, quotients, completions, realization functors and equational theorem transport ).
Exact mathematical passage
```
\boxed{
\mathsf{PSC}_I=
\left(
\begin{array}{c}
\text{typed grammars and realized operator algebras},\\
\text{seed systems, free presentations, and seed rank},\\
\text{module-top, semisimple, and single-operator formulas},\\
\text{finite and unconditional packet coordinates},\\
\text{transfer blocks and support algebras},\\
\text{reachability and packet-pure source generation},\\
\text{morphisms, invariant exact sequences, quotients, completions},\\
\text{realization functors and equational theorem transport}
\end{array}
\right).
}
```

\mathsf{PSC}_IDefinition passage · 1 passage

Search aliases: PSC_I mathsf

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Definition passage: Paper I export object · Read passage · 04_transport.tex, line 486
[Paper I export object] The canonical Paper I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, morphisms, invariant exact sequences, quotients, completions, realization functors and equational theorem transport ).
Exact mathematical passage
```
\boxed{
\mathsf{PSC}_I=
\left(
\begin{array}{c}
\text{typed grammars and realized operator algebras},\\
\text{seed systems, free presentations, and seed rank},\\
\text{module-top, semisimple, and single-operator formulas},\\
\text{finite and unconditional packet coordinates},\\
\text{transfer blocks and support algebras},\\
\text{reachability and packet-pure source generation},\\
\text{morphisms, invariant exact sequences, quotients, completions},\\
\text{realization functors and equational theorem transport}
\end{array}
\right).
}
```

\psrank_{\A,\Pcal}Definition passage · 5 passages

Search aliases: psrank_A,Pcal psrank_ A, P psrank A Pcal operatorname mathcal mathcal

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1014
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
\psrank_{\A,\Pcal}(X)
:=
\min\{|S|:S\text{ is packet-pure and }\A S=X\}.
```

\psrank_{\A_C,\Pcal|_C}Definition passage · 1 passage

Search aliases: psrank_A_C,Pcal|_C psrank_ A_C, P|_C psrank A Pcal operatorname mathcal mathcal

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1028
f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
```

Q_0Definition passage · 5 passages

Search aliases: Q_0

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 39
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
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```
Q=(Q_0,Q_1,s,t)
```

Q_1Definition passage · 5 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 39
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
```
Q=(Q_0,Q_1,s,t)
```

Q_jDefinition passage · 14 passages

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
Q_j\Phi
=
\Phi P_{\sigma^{-1}(j)},
\qquad
P_{\sigma^{-1}(j)}:=
\sum_{\sigma(\alpha)=j}P_\alpha.
```

\QcalDefinition passage · 9 passages

Search aliases: Qcal Q mathcal

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 50
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
(Y,\B,\Qcal)
```

RDefinition passage · 53 passages

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Definition passage: Support relation and support space · Read passage · 03_packets_and_graphs.tex, line 791
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
```
R\subseteq I\times I
```

rDefinition passage · 57 passages

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Definition passage: Linear seed system · Read passage · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
```
S=(s_1,\ldots,s_r)
```

r_CDefinition passage · 6 passages

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1028
f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
```

\mathfrak SDefinition passage · 1 passage

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Definition passage: Linear seed system · Read passage · 02_foundations.tex, line 148
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
```
\mathfrak S=(X,\A,S,\Pcal,\tau),
```

sDefinition passage · 76 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 39
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
```
Q=(Q_0,Q_1,s,t)
```

S'Definition passage · 8 passages

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Definition passage: Morphism of seeded systems · Read passage · 04_transport.tex, line 76
[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.
Exact mathematical passage
```
(X,\A,S,\Pcal)\longrightarrow(Y,\B,S',\Qcal)
```

s_1Definition passage · 10 passages

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Definition passage: Linear seed system · Read passage · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
```
S=(s_1,\ldots,s_r)
```

s_jDefinition passage · 6 passages

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

s_rDefinition passage · 9 passages

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Definition passage: Linear seed system · Read passage · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
```
S=(s_1,\ldots,s_r)
```

\mathbf{SeedSys}Definition passage · 3 passages

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathbf{SeedSys}_{\mathrm{abs}}
```

\mathbf{SeedSys}_{\mathrm{abs}}Definition passage · 2 passages

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Definition passage: Realization functor and preservation profile · Read passage · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
```
\mathbf{SeedSys}_{\mathrm{abs}}
```

\srank_{\A,\tau}Definition passage · 1 passage

Search aliases: srank_A,τ srank_ A,τ srank A tau operatorname mathcal tau

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Definition passage: Seed rank · Read passage · 02_foundations.tex, line 377
[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.
Exact mathematical passage
```
\srank_{\A,\tau}(X):=\min\{|S|:\cl{\A S}^{\,\tau}=X\}.
```

\mathrm{statement}Definition passage · 1 passage

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

\mathrm{status}Definition passage · 1 passage

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Definition passage: Canonical theorem record · Read passage · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.
Exact mathematical passage
```
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
\mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
```

\SubsetDefinition passage · 2 passages

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

\supDefinition passage · 3 passages

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

\sup_{F\Subset\mathbb N}Definition passage · 2 passages

Search aliases: sup_FSubset N sup Subset mathbb

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

\sup_{|a_n|\leq1}Definition passage · 2 passages

Search aliases: sup_|a_n|≤1 sup leq

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

tDefinition passage · 31 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 39
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
```
Q=(Q_0,Q_1,s,t)
```

\triangleleftDefinition passage · 1 passage

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 43
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
```
I\triangleleft \bbk Q
```

uDefinition passage · 29 passages

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 355
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\bigoplus_{n\geq1}^{u}X_n
```

vDefinition passage · 40 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 47
r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a…
Exact mathematical passage
```
(X_v)_{v\in Q_0}
```

X_aDefinition passage · 3 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
X_a:X_{s(a)}\longrightarrow X_{t(a)}
\qquad(a\in Q_1)
```

X_CDefinition passage · 10 passages

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
P_C:=\sum_{\alpha\in C}P_\alpha,
\qquad
X_C:=P_CX,
\qquad
\A_C:=P_C\A P_C,
```

X_nDefinition passage · 7 passages

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 354
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
(X_n)_{n\geq1}
```

x_nDefinition passage · 8 passages

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 358
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
x=(x_n)
```

X_{s(a)}Definition passage · 1 passage

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
X_a:X_{s(a)}\longrightarrow X_{t(a)}
\qquad(a\in Q_1)
```

X_{t(a)}Definition passage · 1 passage

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
X_a:X_{s(a)}\longrightarrow X_{t(a)}
\qquad(a\in Q_1)
```

X_vDefinition passage · 5 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 47
r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a…
Exact mathematical passage
```
(X_v)_{v\in Q_0}
```

YDefinition passage · 51 passages

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 52
k G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f:X\to Y
```

Y_aDefinition passage · 1 passage

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 53
linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f_{t(a)}X_a=Y_af_{s(a)}
\qquad(a\in Q_1).
```

Y_vDefinition passage · 1 passage

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 52
l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
f_v:X_v\to Y_v
```

|_CDefinition passage · 1 passage

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1028
f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
```

\timesDefinition passage · 15 passages

Search aliases: × times

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Definition passage: Support relation and support space · Read passage · 03_packets_and_graphs.tex, line 791
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
```
R\subseteq I\times I
```

\GammaDefinition passage · 4 passages

Search aliases: Γ Gamma

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 922
efinition[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
\Gamma(\mathcal G,\Pcal)
```

\varepsilonDefinition passage · 20 passages

Search aliases: ε varepsilon

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\varepsilon_SDefinition passage · 9 passages

Search aliases: ε_S varepsilon

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\kappaDefinition passage · 4 passages

Search aliases: κ kappa

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Definition passage: Packet coordinate condition number · Read passage · 03_packets_and_graphs.tex, line 285
[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .
Exact mathematical passage
```
\kappa(\Pcal):=
\|\mathcal A_{\Pcal}\|\,
\|\mathcal S_{\Pcal}\|.
```

\sigmaDefinition passage · 15 passages

Search aliases: σ sigma

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 51
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
(\Phi,\varphi,\sigma)
```

\sigma^{-1}Definition passage · 5 passages

Search aliases: σ^-1 sigma

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
Q_j\Phi
=
\Phi P_{\sigma^{-1}(j)},
\qquad
P_{\sigma^{-1}(j)}:=
\sum_{\sigma(\alpha)=j}P_\alpha.
```

\tauDefinition passage · 10 passages

Search aliases: τ tau

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Definition passage: Linear seed system · Read passage · 02_foundations.tex, line 148
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
```
\mathfrak S=(X,\A,S,\Pcal,\tau),
```

\PhiDefinition passage · 57 passages

Search aliases: Φ Phi

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Definition passage: Witness for seed rank · Read passage · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
```
\Phi:M\twoheadrightarrow N
```

\varphiDefinition passage · 23 passages

Search aliases: φ varphi

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 51
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
(\Phi,\varphi,\sigma)
```

\ldotsDefinition passage · 17 passages

Search aliases: … ldots

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Definition passage: Linear seed system · Read passage · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
```
S=(s_1,\ldots,s_r)
```

\twoheadrightarrowDefinition passage · 12 passages

Search aliases: ↠ twoheadrightarrow

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Definition passage: Witness for seed rank · Read passage · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
```
\Phi:M\twoheadrightarrow N
```

\inDefinition passage · 185 passages

Search aliases: ∈ in

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 47
r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a…
Exact mathematical passage
```
(X_v)_{v\in Q_0}
```

\notinDefinition passage · 3 passages

Search aliases: ∉ notin

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Definition passage: Support relation and support space · Read passage · 03_packets_and_graphs.tex, line 792
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
```
\mathfrak M_R
:=
\{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
```

\sum_{j=1}^{r}Definition passage · 1 passage

Search aliases: ∑_j=1^r sum

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\sum_nDefinition passage · 5 passages

Search aliases: ∑_n sum

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 358
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\sum_nx_n
```

\sum_{n\in F}Definition passage · 6 passages

Search aliases: ∑_n∈ F sum in

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\|x\|_u
:=
\sup_{F\Subset\mathbb N}
\sup_{|a_n|\leq1}
\left\|\sum_{n\in F}a_nx_n\right\|.
```

\sum_{s\in S}Definition passage · 1 passage

Search aliases: ∑_s∈ S sum in

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Definition passage: Generated hull · Read passage · 02_foundations.tex, line 158
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
```
\Hull_{\A}(S):=\sum_{s\in S}\A s.
```

\sum_{\alpha\in C}Definition passage · 2 passages

Search aliases: ∑_α∈ C sum alpha in

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Definition passage: Packet pure seed rank · Read passage · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
```
P_C:=\sum_{\alpha\in C}P_\alpha,
\qquad
X_C:=P_CX,
\qquad
\A_C:=P_C\A P_C,
```

\sum_{\alpha\in I}Definition passage · 2 passages

Search aliases: ∑_α∈ I sum alpha in

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Definition passage: Finite packet system · Read passage · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
```
P_\alpha^2=P_\alpha,
\qquad
P_\alpha P_\beta=0\quad(\alpha\neq\beta),
\qquad
\sum_{\alpha\in I}P_\alpha=I_X.
```

\sum_{\sigma(\alpha)=j}Definition passage · 2 passages

Search aliases: ∑_σ(α)=j sum sigma alpha

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Definition passage: Packet compatible operator morphism · Read passage · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
```
Q_j\Phi
=
\Phi P_{\sigma^{-1}(j)},
\qquad
P_{\sigma^{-1}(j)}:=
\sum_{\sigma(\alpha)=j}P_\alpha.
```

\cupDefinition passage · 1 passage

Search aliases: ∪ cup

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Definition passage: Transfer graph of a family generating the algebra · Read passage · 03_packets_and_graphs.tex, line 919
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
```
\A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
```

\neqDefinition passage · 18 passages

Search aliases: ≠ neq

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Definition passage: Finite packet system · Read passage · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
```
P_\alpha^2=P_\alpha,
\qquad
P_\alpha P_\beta=0\quad(\alpha\neq\beta),
\qquad
\sum_{\alpha\in I}P_\alpha=I_X.
```

\leqDefinition passage · 29 passages

Search aliases: ≤ leq

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Definition passage: Packet support and packet pure vectors · Read passage · 03_packets_and_graphs.tex, line 36
[Packet support and packet pure vectors] For a finite packet system P=(P_α)_α∈ I and x∈ X, define Supp_ P(x) := {α∈ I:P_α x≠0}. The vector x is packet pure if |Supp_ P(x)|≤1. A finite family is packet pure if each of its members is packet pure, although different members may lie in different packets.
Exact mathematical passage
```
|\Supp_{\Pcal}(x)|\leq1
```

\geqDefinition passage · 16 passages

Search aliases: ≥ geq

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Definition passage: Witness for seed rank · Read passage · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
```
\srank(M)\geq r
```

\bigoplusDefinition passage · 26 passages

Search aliases: ⊕ bigoplus

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Definition passage: Realization on a single space · Read passage · 02_foundations.tex, line 63
[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.
Exact mathematical passage
```
|X|:=\bigoplus_{v\in Q_0}X_v.
```

\oplusDefinition passage · 37 passages

Search aliases: ⊕ oplus

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

\bigoplus_{n\geq1}^{u}Definition passage · 3 passages

Search aliases: ⊕_n≥1^u bigoplus geq

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Definition passage: Unconditional packet sum space · Read passage · 03_packets_and_graphs.tex, line 355
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
```
\bigoplus_{n\geq1}^{u}X_n
```

\bigoplus_{v\in Q_0}Definition passage · 1 passage

Search aliases: ⊕_v∈ Q_0 bigoplus in

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Definition passage: Realization on a single space · Read passage · 02_foundations.tex, line 63
[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.
Exact mathematical passage
```
|X|:=\bigoplus_{v\in Q_0}X_v.
```

\bigoplus_\alphaDefinition passage · 4 passages

Search aliases: ⊕_α bigoplus alpha

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Definition passage: Packet coordinate condition number · Read passage · 03_packets_and_graphs.tex, line 284
[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .
Exact mathematical passage
```
\bigoplus_\alpha X_\alpha
```

\bigoplus_{\alpha\in I}Definition passage · 3 passages

Search aliases: ⊕_α∈ I bigoplus alpha in

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Definition passage: Packet analysis and synthesis · Read passage · 03_packets_and_graphs.tex, line 72
[Packet analysis and synthesis] For a finite packet system, define A_ P:X⟶⊕_α∈ IX_α, x⟼(P_α x)_α, and S_ P:⊕_α∈ IX_α⟶ X, (x_α)_α⟼∑_α x_α.
Exact mathematical passage
```
\mathcal A_{\Pcal}:X\longrightarrow\bigoplus_{\alpha\in I}X_\alpha,
\qquad
x\longmapsto(P_\alpha x)_\alpha,
```

\longrightarrowDefinition passage · 19 passages

Search aliases: ⟶ longrightarrow

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Definition passage: Typed operator grammar · Read passage · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
```
X_a:X_{s(a)}\longrightarrow X_{t(a)}
\qquad(a\in Q_1)
```

\longmapstoDefinition passage · 5 passages

Search aliases: ⟼ longmapsto

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Definition passage: Seed evaluation map · Read passage · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
```
\varepsilon_S:A^{\oplus r}\longrightarrow M,
\qquad
(a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
```

+Introduction passage · 80 passages

Search aliases: +

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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
f=a_0+ya_1
```

a_0Introduction passage · 6 passages

Search aliases: a_0

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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
f=a_0+ya_1
```

a_iIntroduction passage · 6 passages

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Introduction passage · Read passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
Exact mathematical passage
```
m=\sum_{i\in F}a_im_i,
\qquad
n=\sum_{i\in F}b_in_i,
```

bIntroduction passage · 28 passages

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Introduction passage · Read passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
Exact mathematical passage
```
m=\sum_{i\in F}a_im_i,
\qquad
n=\sum_{i\in F}b_in_i,
```

b_0Introduction passage · 6 passages

Search aliases: b_0

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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
g=b_0+yb_1
```

b_1Introduction passage · 5 passages

Search aliases: b_1

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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
g=b_0+yb_1
```

b_iIntroduction passage · 2 passages

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Introduction passage · Read passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
Exact mathematical passage
```
m=\sum_{i\in F}a_im_i,
\qquad
n=\sum_{i\in F}b_in_i,
```

c_0Introduction passage · 2 passages

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Introduction passage · Read passage · 03_packets_and_graphs.tex, line 1116
[Unweighted source component count] Assume the module resolved into packets is packet propagating and X_α≠0 for every α∈ I. If c_0 denotes the number of source strongly connected components, then psrank_ A, P(X)=c_0. [source reference]
Exact mathematical passage
```
c_0
```

dIntroduction passage · 32 passages

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

D_iIntroduction passage · 5 passages

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

d_iIntroduction passage · 15 passages

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

D_i^{d_i}Introduction passage · 1 passage

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Introduction passage · Read passage · 02_foundations.tex, line 743
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
S_i=D_i^{d_i}
```

eIntroduction passage · 98 passages

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Introduction passage · Read passage · 02_foundations.tex, line 889
classicalinput[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
```
\bbk[t]/(p^e)
```

e_iIntroduction passage · 1 passage

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Introduction passage · visual_atlas.tex, line 249
vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills the other basis vectors.
Exact mathematical passage
```
e_i
```

E_{ij}Introduction passage · 1 passage

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Introduction passage · visual_atlas.tex, line 249
vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills the other basis vectors.
Exact mathematical passage
```
E_{ij}
```

e_jIntroduction passage · 6 passages

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Introduction passage · visual_atlas.tex, line 249
vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills the other basis vectors.
Exact mathematical passage
```
e_j
```

gIntroduction passage · 22 passages

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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
g=b_0+yb_1
```

iIntroduction passage · 76 passages

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Introduction passage · Read passage · 02_foundations.tex, line 543
The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)…
Exact mathematical passage
```
(m_i)_{i\in I}
```

I_0Introduction passage · 6 passages

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Introduction passage · Read passage · 03_packets_and_graphs.tex, line 997
For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.
Exact mathematical passage
```
s=\sum_{\alpha\in I_0}P_\alpha s
```

\IrrIntroduction passage · 2 passages

Search aliases: Irr operatorname

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Introduction passage · Read passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
```
X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
```

mIntroduction passage · 77 passages

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Introduction passage · Read passage · 02_foundations.tex, line 543
The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)…
Exact mathematical passage
```
(m_i)_{i\in I}
```

M_{d_i}Introduction passage · 5 passages

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

m_iIntroduction passage · 15 passages

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Introduction passage · Read passage · 02_foundations.tex, line 543
The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)…
Exact mathematical passage
```
(m_i)_{i\in I}
```

M_yIntroduction passage · 6 passages

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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
```

M_y^2Introduction passage · 2 passages

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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
```

\matIntroduction passage · 7 passages

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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
```

n_iIntroduction passage · 3 passages

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Introduction passage · Read passage · 02_foundations.tex, line 543
The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)…
Exact mathematical passage
```
(n_i)_{i\in I}
```

pIntroduction passage · 48 passages

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Introduction passage · Read passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
```
X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
```

p^eIntroduction passage · 2 passages

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Introduction passage · Read passage · 02_foundations.tex, line 889
classicalinput[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
```
\bbk[t]/(p^e)
```

qIntroduction passage · 25 passages

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

\QQIntroduction passage · 33 passages

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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
\QQ[x]
```

S_iIntroduction passage · 1 passage

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Introduction passage · Read passage · 02_foundations.tex, line 743
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
S_i=D_i^{d_i}
```

S_i^{\oplus d_i}Introduction passage · 1 passage

Search aliases: S_i^⊕ d_i oplus

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Introduction passage · Read passage · 02_foundations.tex, line 744
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
\boxed{A\cong\bigoplus_{i=1}^{q}S_i^{\oplus d_i}.}
```

x^2Introduction passage · 6 passages

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Introduction passage · visual_atlas.tex, line 434
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
(x^2+y)(x-y)=x+(x-x^2)y.
```

x^3Introduction passage · 17 passages

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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
```

X_pIntroduction passage · 9 passages

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Introduction passage · Read passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
```
X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
```

yIntroduction passage · 66 passages

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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
Exact mathematical passage
```
M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
```

\prodIntroduction passage · 10 passages

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

\prod_{i=1}^{q}Introduction passage · 4 passages

Search aliases: ∏_i=1^q prod

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Introduction passage · Read passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
A=\prod_{i=1}^{q}M_{d_i}(D_i)
```

\sum_{i\in F}Introduction passage · 2 passages

Search aliases: ∑_i∈ F sum in

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Introduction passage · Read passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
Exact mathematical passage
```
m=\sum_{i\in F}a_im_i,
\qquad
n=\sum_{i\in F}b_in_i,
```

\sum_{\alpha\in I_0}Introduction passage · 2 passages

Search aliases: ∑_α∈ I_0 sum alpha in

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Introduction passage · Read passage · 03_packets_and_graphs.tex, line 997
For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.
Exact mathematical passage
```
s=\sum_{\alpha\in I_0}P_\alpha s
```

\congIntroduction passage · 24 passages

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Introduction passage · Read passage · 02_foundations.tex, line 744
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
\boxed{A\cong\bigoplus_{i=1}^{q}S_i^{\oplus d_i}.}
```

\bigoplus_{i=1}^{q}Introduction passage · 3 passages

Search aliases: ⊕_i=1^q bigoplus

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Introduction passage · Read passage · 02_foundations.tex, line 744
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
```
\boxed{A\cong\bigoplus_{i=1}^{q}S_i^{\oplus d_i}.}
```

\bigoplus_{p\in\Irr(\bbk[t])}Introduction passage · 1 passage

Search aliases: ⊕_p∈Irr(bbk[t]) ⊕_p∈Irr(𝕜[t]) bigoplus in Irr bbk bigoplus in operatorname Bbbk

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Introduction passage · Read passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
```
X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
```

*Use in source · 21 passages

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Use in source · Read passage · 03_packets_and_graphs.tex, line 237
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
```
P_\alpha^*:X^*\to X^*,
\qquad
P_\alpha^*(\varphi)=\varphi\circ P_\alpha,
```

A^2Use in source · 1 passage

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Use in source · visual_atlas.tex, line 376
vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.
Exact mathematical passage
```
K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]
```

A^3Use in source · 1 passage

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Use in source · visual_atlas.tex, line 376
vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.
Exact mathematical passage
```
K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]
```

A^4Use in source · 1 passage

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Use in source · visual_atlas.tex, line 376
vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.
Exact mathematical passage
```
K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]
```

A^{n-1}Use in source · 1 passage

Search aliases: A^n-1

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Use in source · Read passage · 05_worked_models.tex, line 349
…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is zero when i>j, while the diagonal entry is one because there is a unique shorte…
Exact mathematical passage
```
K_n=[e_1,Ae_1,\ldots,A^{n-1}e_1].
```

\A^{\oplus r}Use in source · 1 passage

Search aliases: A^⊕ r A oplus mathcal oplus

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Use in source · Read passage · 02_foundations.tex, line 336
Topological completion of a presentation. [Completion of a dense free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and th…
Exact mathematical passage
```
E:=\A^{\oplus r}/N_S
```

A^{\oplus2}Use in source · 1 passage

Search aliases: A^⊕2 oplus

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Use in source · Read passage · 02_foundations.tex, line 701
[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.
Exact mathematical passage
```
M=A^{\oplus2}
```

A_0Use in source · 2 passages

Search aliases: A_0

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Use in source · Read passage · 05_worked_models.tex, line 88
The two packet spaces are A_0=𝕜[x], A_1=y𝕜[x]. Let P_0(a+yb)=a, P_1(a+yb)=yb. These idempotents are linear over 𝕜, annihilate one another, and satisfy P_0+P_1=I. They are not module maps over A, because multiplication by y exchanges the two packets.
Exact mathematical passage
```
A_0=\bbk[x],
\qquad
A_1=y\bbk[x].
```

A_1Use in source · 4 passages

Search aliases: A_1

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Use in source · Read passage · 05_worked_models.tex, line 88
The two packet spaces are A_0=𝕜[x], A_1=y𝕜[x]. Let P_0(a+yb)=a, P_1(a+yb)=yb. These idempotents are linear over 𝕜, annihilate one another, and satisfy P_0+P_1=I. They are not module maps over A, because multiplication by y exchanges the two packets.
Exact mathematical passage
```
A_0=\bbk[x],
\qquad
A_1=y\bbk[x].
```

A_2Use in source · 2 passages

Search aliases: A_2

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Use in source · visual_atlas.tex, line 337
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
```
A_2
```

a_2Use in source · 1 passage

Search aliases: a_2

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Use in source · quotient_heads.tex, line 35
00,0.4800) The quotient keeps only the two chain heads; node[font=scriptsize] at (0.4250,0.1900) e_0; node[font=scriptsize] at (1.2750,0.1900) e_1; node[font=scriptsize] at (2.1250,0.1900) e_2; node[font=scriptsize] at (2.9750,0.1900) f_0; node[font=scriptsize] at (3.8250,0.1900) f_1; node[font=scriptsize] at (4.6750,0.1900) f_2; node[] at (2.5500,-2.1800) q(a_0,a_1,a_2,b_0,b_1,b_2)=(a_0,b_0);
Exact mathematical passage
```
q(a_0,a_1,a_2,b_0,b_1,b_2)=(a_0,b_0)
```

A_3Use in source · 2 passages

Search aliases: A_3

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Use in source · visual_atlas.tex, line 337
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
```
A_3
```

A_{C_n}Use in source · 1 passage

Search aliases: A_C_n

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Use in source · Read passage · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
```
{
2=\min\left\{r\in\N:
\exists s_1,\ldots,s_r,\quad
\C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\}
\right\}.
}
```

A_GUse in source · 5 passages

Search aliases: A_G

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Use in source · Read passage · 05_worked_models.tex, line 274
Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.
Exact mathematical passage
```
A_G
```

A_iUse in source · 2 passages

Search aliases: A_i

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Use in source · visual_atlas.tex, line 513
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
```
x=\sum_i A_ix_i,\qquad y=\sum_j B_jy_j,
```

A_KUse in source · 3 passages

Search aliases: A_K

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Use in source · Read passage · 02_foundations.tex, line 595
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
```
A_K:=K\otimes_{\bbk}A,
\qquad
M_K:=K\otimes_{\bbk}M.
```

A_{P_n}Use in source · 1 passage

Search aliases: A_P_n

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Use in source · Read passage · 05_worked_models.tex, line 331
For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A…
Exact mathematical passage
```
{
1=\min\left\{r\in\N:
\exists s_1,\ldots,s_r,\quad
\bbk^n=\Span_{\bbk}\{p(A_{P_n})s_j:p\in\bbk[t],\ 1\le j\le r\}
\right\}.
}
```

A_xUse in source · 3 passages

Search aliases: A_x

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Use in source · Read passage · 02_foundations.tex, line 194
[Operators chosen for a target] For a fixed seed s and target x, the existence of a specially defined operator A_x satisfying A_xs=x does not establish a natural seed calculus unless A_x belongs to a grammar fixed independently of the target satisfying [source reference].
Exact mathematical passage
```
A_x
```

a_{\lambda}Use in source · 2 passages

Search aliases: a_λ lambda

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Use in source · Read passage · 05_worked_models.tex, line 282
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
```
a_{\lambda}\in\bbk
```

\AAUse in source · 7 passages

Search aliases: AA A mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\operatorname{Ann}Use in source · 8 passages

Search aliases: Ann operatorname

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Use in source · Read passage · 02_foundations.tex, line 286
[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]
Exact mathematical passage
```
\operatorname{Ann}_A(s):=\{a\in A:as=0\}
```

\operatorname{Ann}_AUse in source · 8 passages

Search aliases: Ann_A operatorname

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Use in source · Read passage · 02_foundations.tex, line 286
[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]
Exact mathematical passage
```
\operatorname{Ann}_A(s):=\{a\in A:as=0\}
```

\arraybackslashUse in source · 1 passage

Search aliases: arraybackslash

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\mathcal BUse in source · 5 passages

Search aliases: B mathcal

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Use in source · Read passage · 03_packets_and_graphs.tex, line 299
[Conditioned coordinate transport] For a finite Banach packet system with X≠0, κ( P)≥1. If T∈ B(X) and [T]_ P:= A_ PT S_ P, then [T]_ P ≤κ( P) T , T ≤κ( P) [T]_ P . [source reference]
Exact mathematical passage
```
T\in\mathcal B(X)
```

\mathrm bUse in source · 3 passages

Search aliases: b mathrm

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 835
[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]
Exact mathematical passage
```
\mathfrak M_R^{\mathrm b}:=\mathfrak M_R\cap\mathcal B(X).
```

b_2Use in source · 1 passage

Search aliases: b_2

Link to this entry ·

Use in source · quotient_heads.tex, line 35
00,0.4800) The quotient keeps only the two chain heads; node[font=scriptsize] at (0.4250,0.1900) e_0; node[font=scriptsize] at (1.2750,0.1900) e_1; node[font=scriptsize] at (2.1250,0.1900) e_2; node[font=scriptsize] at (2.9750,0.1900) f_0; node[font=scriptsize] at (3.8250,0.1900) f_1; node[font=scriptsize] at (4.6750,0.1900) f_2; node[] at (2.5500,-2.1800) q(a_0,a_1,a_2,b_0,b_1,b_2)=(a_0,b_0);
Exact mathematical passage
```
q(a_0,a_1,a_2,b_0,b_1,b_2)=(a_0,b_0)
```

B_jUse in source · 2 passages

Search aliases: B_j

Link to this entry ·

Use in source · visual_atlas.tex, line 513
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
```
x=\sum_i A_ix_i,\qquad y=\sum_j B_jy_j,
```

b_jUse in source · 1 passage

Search aliases: b_j

Link to this entry ·

Use in source · visual_atlas.tex, line 86
Define a surjective linear map q:X⟶ Q^2, q(∑ a_je_j+∑ b_jf_j)=(a_0,b_0).
Exact mathematical passage
```
q:X\longrightarrow\QQ^2,\qquad
q\Bigl(\sum a_je_j+\sum b_jf_j\Bigr)=(a_0,b_0).
```

B_{ji}Use in source · 3 passages

Search aliases: B_ji

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 549
[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. It does not claim that Lean extracts a routine that decides the edge list for every coefficient field.
Exact mathematical passage
```
B_{ji}=Q_jTP_i.
```

B_{\beta\alpha}Use in source · 3 passages

Search aliases: B_βα beta alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 692
Injectivity follows from exact reconstruction. Given arbitrary linear maps B_βα:X_α→ X_β, define T(∑_α x_α) := ∑_α,βB_βαx_α. The packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps. Thus Θ_ P is surjective. Compatibility with multiplication is [source reference].
Exact mathematical passage
```
B_{\beta\alpha}:X_\alpha\to X_\beta
```

b_{\lambda,j}Use in source · 2 passages

Search aliases: b_λ,j lambda

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 284
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
```
\{b_{\lambda,j}:\lambda\in\Lambda,\ 1\le j\le m_{\lambda}\}
```

\BBUse in source · 4 passages

Search aliases: BB B mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 312
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
```
\BB
```

\bbk^2Use in source · 6 passages

Search aliases: bbk^2 𝕜^2 bbk Bbbk

Link to this entry ·

Use in source · Read passage · 01_orientation.tex, line 87
[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.
Exact mathematical passage
```
M=(\bbk^2)^{\oplus3}
```

\bbk^3Use in source · 3 passages

Search aliases: bbk^3 𝕜^3 bbk Bbbk

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 460
[title=A quotient can strictly lower seed rank] Let A=𝕜 act by scalars on M=𝕜^3. Then srank_A(M)=3. Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is 2. Quotient monotonicity is therefore often strict.
Exact mathematical passage
```
M=\bbk^3
```

\bbk^{d_i}Use in source · 1 passage

Search aliases: bbk^d_i 𝕜^d_i bbk Bbbk

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Use in source · Read passage · 02_foundations.tex, line 812
[Algebraically closed multiplicity formula] Let 𝕜 be algebraically closed, let A be a unital algebra of finite dimension over 𝕜, put J=Jac(A), and let M be a finitely generated left module over A. Assume compatible decompositions indexed by an integer q≥0: A/J≅∏_i=1^qM_d_i(𝕜), M/JM≅⊕_i=1^q(𝕜^d_i)^⊕ m_i, where the d_i are positive integers and the m_i are nonnegative integers. The first isomorphism is an isomorphism of algebras over 𝕜. The second is an isomorphism of left modules over A/J, using …
Exact mathematical passage
```
A/J\cong\prod_{i=1}^{q}M_{d_i}(\bbk),
\qquad
M/JM\cong\bigoplus_{i=1}^{q}(\bbk^{d_i})^{\oplus m_i},
```

\bbk^nUse in source · 2 passages

Search aliases: bbk^n 𝕜^n bbk Bbbk

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Use in source · Read passage · 05_worked_models.tex, line 256
Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.
Exact mathematical passage
```
X=\bbk^n
```

\bbk^{V(G)}Use in source · 3 passages

Search aliases: bbk^V(G) 𝕜^V(G) bbk Bbbk

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Use in source · Read passage · 05_worked_models.tex, line 275
Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.
Exact mathematical passage
```
\bbk^{V(G)}
```

\bottomruleUse in source · 1 passage

Search aliases: bottomrule

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Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\boxtimesUse in source · 1 passage

Search aliases: boxtimes

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Use in source · Read passage · 03_packets_and_graphs.tex, line 191
[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
```
\Pcal\boxtimes\Qcal
:=
(P_\alpha\otimes Q_\gamma)_{(\alpha,\gamma)\in I\times J}
```

\CUse in source · 7 passages

Search aliases: C mathbb

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Use in source · Read passage · 05_worked_models.tex, line 195
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
\mathbb{F}\in\{\R,\C\}
```

\mathfrak CUse in source · 3 passages

Search aliases: C mathfrak

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1068
[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport forests and directed source rank calculations.
Exact mathematical passage
```
\mathfrak C_0
```

\C^nUse in source · 1 passage

Search aliases: C^n C mathbb

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Use in source · Read passage · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
```
{
2=\min\left\{r\in\N:
\exists s_1,\ldots,s_r,\quad
\C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\}
\right\}.
}
```

\mathfrak C_0Use in source · 3 passages

Search aliases: C_0 mathfrak

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1068
[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport forests and directed source rank calculations.
Exact mathematical passage
```
\mathfrak C_0
```

C_6Use in source · 1 passage

Search aliases: C_6

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Use in source · visual_atlas.tex, line 392
vpage16Closing a path changes the seed obstructionGRAPH MODEL / A CYCLE WITH SIX VERTICESMain text: Worked graph model. Proof of the minimum seed count for complex cycles For the adjacency matrix A of C_6, the distinct eigenvalues are 2,1,-1,-2, with multiplicities 1,2,2,1. A single vertex seed spans only four directions. Two adjacent vertex seeds span all six.
Exact mathematical passage
```
C_6
```

C_nUse in source · 1 passage

Search aliases: C_n

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Use in source · Read passage · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
```
{
2=\min\left\{r\in\N:
\exists s_1,\ldots,s_r,\quad
\C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\}
\right\}.
}
```

\cAUse in source · 3 passages

Search aliases: cA A mathcal

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Use in source · Read passage · 05_worked_models.tex, line 256
Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.
Exact mathematical passage
```
\cA=M_n(\bbk)=\End_{\bbk}(X)
```

\cA^{\oplus r}Use in source · 2 passages

Search aliases: cA^⊕ r A^⊕ r cA oplus mathcal oplus

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Use in source · Read passage · 05_worked_models.tex, line 438
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S),
Exact mathematical passage
```
\varepsilon_S:\cA^{\oplus r}\to X
```

\CCUse in source · 1 passage

Search aliases: CC C mathbb

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Use in source · visual_atlas.tex, line 402
0 · 1 · 1 · 0 · -1 · -1. Thus q: Q^6↠ Q^2 is a scalar quotient for the eigenvalue 1. Its extension over C gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.
Exact mathematical passage
```
\CC
```

\CcalUse in source · 3 passages

Search aliases: Ccal C mathcal

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Use in source · Read passage · 04_transport.tex, line 148
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ
Exact mathematical passage
```
(\Psi,\psi,\tau):(Y,\B,\Qcal)\to(Z,\Ccal,\Rcal)
```

\operatorname{char}Use in source · 1 passage

Search aliases: char operatorname

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Use in source · Read passage · 05_worked_models.tex, line 61
Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packe…
Exact mathematical passage
```
\operatorname{char}\bbk\ne2
```

\cosUse in source · 2 passages

Search aliases: cos

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Use in source · Read passage · 05_worked_models.tex, line 340
For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A…
Exact mathematical passage
```
\lambda_k=2\cos\frac{k\pi}{n+1},
\qquad 1\le k\le n,
```

\crefUse in source · 1 passage

Search aliases: cref

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Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\mathbf DUse in source · 2 passages

Search aliases: D mathbf

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 402
[Equational realization principle] Let F: C→ D be an additive functor between preadditive categories. Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by F. Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and F is linear for the declared enrichment. If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, …
Exact mathematical passage
```
F:\mathbf C\to\mathbf D
```

D^2Use in source · 3 passages

Search aliases: D^2

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Use in source · visual_atlas.tex, line 225
vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.
Exact mathematical passage
```
D^2e_1=e_4-e_4=0,\qquad D^2=0.
```

D^dUse in source · 2 passages

Search aliases: D^d

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 756
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
```
D^d
```

d_\alphaUse in source · 3 passages

Search aliases: d_α alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 738
[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]
Exact mathematical passage
```
d_\alpha=\dim X_\alpha
```

d_\betaUse in source · 2 passages

Search aliases: d_β beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 739
[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]
Exact mathematical passage
```
\dim\End_{\bbk}(X)
=
\sum_{\alpha,\beta}d_\alpha d_\beta
=
\left(\sum_\alpha d_\alpha\right)^2.
```

\detUse in source · 2 passages

Search aliases: det

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 353
There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is zero when i>j, while the diagonal entry is one because there is a unique shortest walk of length i-1 from vertex 1 to vertex i. Hence K_n is upper triangular with unit diagonal, so det K_n=1 (n≥1). Thus the endpoint is cyclic for every path size, not merely for the tested range.
Exact mathematical passage
```
\det K_n=1
\qquad(n\ge1).
```

\diagUse in source · 3 passages

Search aliases: diag operatorname

Link to this entry ·

Use in source · visual_atlas.tex, line 147
0 · 1 · 0. The typed relation is BA=I_U. In the opposite order, AB=diag(1,1,0)≠ I_V. The missing direction is Ker B= Q v_3, not an entirely missing vertex.
Exact mathematical passage
```
AB=\diag(1,1,0)\ne I_V.
```

\dimUse in source · 14 passages

Search aliases: dim

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Use in source · Read passage · 02_foundations.tex, line 712
[Local algebra formula] Let A be a local ring with Jacobson radical J and residue division ring D=A/J. For every finitely generated left module M over A, srank_A(M)=dim_D(M/JM), where the dimension is that of a left vector space over D. [source reference]
Exact mathematical passage
```
\boxed{
\srank_A(M)=\dim_D(M/JM),
}
```

\dim_{\bbk}Use in source · 1 passage

Search aliases: dim_bbk dim_𝕜 dim bbk dim Bbbk

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 974
[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]
Exact mathematical passage
```
\boxed{
\srank_{\bbk[T]}(X)
=
\max_{\lambda\in\operatorname{Spec}(T)}
\dim_{\bbk}\Ker(T-\lambda I).
}
```

\dim_{\bbk[t]/(p)}Use in source · 1 passage

Search aliases: dim_bbk[t]/(p) dim_𝕜[t]/(p) dim bbk dim Bbbk

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 896
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
```
\mu_p(T):=
\dim_{\bbk[t]/(p)}\bigl(X_p/p(T)X_p\bigr).
```

\dim_DUse in source · 1 passage

Search aliases: dim_D dim

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 712
[Local algebra formula] Let A be a local ring with Jacobson radical J and residue division ring D=A/J. For every finitely generated left module M over A, srank_A(M)=dim_D(M/JM), where the dimension is that of a left vector space over D. [source reference]
Exact mathematical passage
```
\boxed{
\srank_A(M)=\dim_D(M/JM),
}
```

\dim_KUse in source · 1 passage

Search aliases: dim_K dim

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
```
r_{\mathrm{spec}}(A)
=
\max_{\mu\in\Spec_K(A)}
\dim_K\ker(A-\mu I),
```

\dim_{K_p}Use in source · 1 passage

Search aliases: dim_K_p dim

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Use in source · Read passage · 02_foundations.tex, line 931
(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). Finally, every summand 𝕜[t]/(p^e) contributes one copy of K_p after quotienting by p, so the dimension equa…
Exact mathematical passage
```
\srank_{R_p}(X_p)
=
\dim_{K_p}X_p/p(T)X_p
=
\mu_p(T).
```

\DownarrowUse in source · 1 passage

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Use in source · Read passage · 05_worked_models.tex, line 627

Theorem spine at a glance. ConceptChain c free seed presentation [0.25em] Downarrow [-0.1em] minimal generator number [0.25em] Downarrow [-0.1em] reduction modulo the Jacobson radical [0.25em] Downarrow [-0.1em] semisimple or primary computation c packet idempotents [0.25em] Downarrow [-0.1em] exact analysis/synthesis [0.25em] Downarrow [-0.1em] transfer block algebra [0.25em] Downarrow [-0.1em] graphs, quotients, and realization

Exact mathematical passage


\begin{array}{c}
\text{free seed presentation}\\[0.25em]
\Downarrow\\[-0.1em]
\text{minimal generator number}\\[0.25em]
\Downarrow\\[-0.1em]
\text{reduction modulo the Jacobson radical}\\[0.25em]
\Downarrow\\[-0.1em]
\text{semisimple or primary computation}
\end{array}
\qquad\qquad
\begin{array}{c}
\text{packet idempotents}\\[0.25em]
\Downarrow\\[-0.1em]
\text{exact analysis/synthesis}\\[0.25em]
\Downarrow\\[-0.1em]
\text{transfer block algebra}\\[0.25em]
\Downarrow\\[-0.1em]
\text{graphs, quotients, and realization}
\end{array}

e_0Use in source · 14 passages

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Use in source · Read passage · 05_worked_models.tex, line 382
plies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(…
Exact mathematical passage
```
e_0
```

e_1Use in source · 35 passages

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Use in source · Read passage · 02_foundations.tex, line 328
[title=Why a seed annihilator need not be an ideal on both sides] Let A=M_2(𝕜) act on M=𝕜^2, and let s=e_1. Then Ann_A(s) consists of matrices whose first column is zero. It is a left ideal because left multiplication preserves a zero first column. It is not generally a right ideal. Thus the natural quotient for one seed is a quotient of the left regular module, not automatically a quotient algebra.
Exact mathematical passage
```
s=e_1
```

E_{11}Use in source · 1 passage

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Use in source · left_ideal.tex, line 32
t] (6.4000,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (6.8250,-1.2750) 0; filldraw[fill=black!5,draw=white,line width=.8pt] (7.2500,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (7.6750,-1.2750) 0; draw[black!60,line width=.5pt] (6.4,0) rectangle ++(1.7,-1.7); node[font=sffamilybfseriessmall] at (7.2500,0.4800) CB=E_11; node[] at (2.4800,-0.8000) ×; node[] at (5.6500,-0.8000) =; node[] at (0.8500,-2.1000) Ce_1=0; node[] at (7.250…
Exact mathematical passage
```
CB=E_{11}
```

E_{12}Use in source · 1 passage

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Use in source · left_ideal.tex, line 12
8pt] (0.0000,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (0.4250,-1.2750) 0; filldraw[fill=black!5,draw=white,line width=.8pt] (0.8500,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (1.2750,-1.2750) 0; draw[black!60,line width=.5pt] (0,0) rectangle ++(1.7,-1.7); node[font=sffamilybfseriessmall] at (0.8500,0.4800) C=E_12; filldraw[fill=black!5,draw=white,line width=.8pt] (3.2000,0.0000) rectangle ++(0.8500,-0.8500); node[text=black!…
Exact mathematical passage
```
C=E_{12}
```

e_2Use in source · 15 passages

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Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

E_{21}Use in source · 3 passages

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Use in source · visual_atlas.tex, line 255
The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
Exact mathematical passage
```
E_{32}E_{21}=E_{31}\ne0.
```

e_3Use in source · 7 passages

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Use in source · Read passage · 05_worked_models.tex, line 43
[Typed reachability certificate] In the preceding quiver realization, no seed supported at U can generate the full representation, whereas the two seeds e_1∈ U, e_3∈ V do generate it.
Exact mathematical passage
```
e_1\in U,
\qquad
e_3\in V
```

E_{31}Use in source · 1 passage

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Use in source · visual_atlas.tex, line 255
The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
Exact mathematical passage
```
E_{32}E_{21}=E_{31}\ne0.
```

E_{32}Use in source · 2 passages

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Use in source · visual_atlas.tex, line 255
The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
Exact mathematical passage
```
E_{32}E_{21}=E_{31}\ne0.
```

e_4Use in source · 6 passages

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Use in source · visual_atlas.tex, line 221
vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.
Exact mathematical passage
```
De_1=e_2+e_3,\qquad De_2=e_4,\qquad De_3=-e_4,\qquad De_4=0.
```

e_{j+1}Use in source · 3 passages

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Use in source · Read passage · 05_worked_models.tex, line 381
C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(…
Exact mathematical passage
```
Ae_j=e_{j-1}+e_{j+1}
```

e_{j-1}Use in source · 2 passages

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Use in source · Read passage · 05_worked_models.tex, line 381
C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(…
Exact mathematical passage
```
Ae_j=e_{j-1}+e_{j+1}
```

e_{jj}Use in source · 3 passages

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Use in source · Read passage · 02_foundations.tex, line 752
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
```
e_{jj}
```

e_pUse in source · 2 passages

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Use in source · Read passage · 02_foundations.tex, line 916
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
m_T(t)=\prod_{p}p(t)^{e_p}.
```

e_UUse in source · 1 passage

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Use in source · Read passage · 05_worked_models.tex, line 14
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1
Exact mathematical passage
```
BA=e_U,
```

e_vUse in source · 8 passages

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Use in source · Read passage · 02_foundations.tex, line 73
[Typed realization and module equivalence] The category of realizations of G is equivalent to the category of unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=ax. The two constructions are inverse up…
Exact mathematical passage
```
e_v
```

e_wUse in source · 2 passages

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Use in source · Read passage · 02_foundations.tex, line 93
n a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relations in I act as zero because M is a module over Λ_ G. Applying both constructions returns the original vertex spaces and …
Exact mathematical passage
```
a=e_wae_v
```

e_\lambdaUse in source · 7 passages

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Use in source · Read passage · 02_foundations.tex, line 352
X, continuity of a:X→ X supplies continuous seminorms controlling p(aj(e)), so the induced action on E is continuous and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X, choose a net (e_λ) in E with j(e_λ)→ x. The transported topology makes (e_λ) Cauchy, so it converges to some ξ∈widehat E, and continuity gives wideh…
Exact mathematical passage
```
(e_\lambda)
```

\mathbb FUse in source · 2 passages

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Use in source · Read passage · 05_worked_models.tex, line 196
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
X=\ell^2(\Z;\mathbb F)
```

F^mUse in source · 1 passage

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Use in source · visual_atlas.tex, line 135
headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.
Exact mathematical passage
```
F^m
```

F_0Use in source · 2 passages

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Use in source · Read passage · 03_packets_and_graphs.tex, line 396
Choose such a k≥ k_0. Since x^(k) is unconditionally summable, there is a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.
Exact mathematical passage
```
F_0
```

f_0Use in source · 11 passages

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Use in source · visual_atlas.tex, line 8
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
```
X=\QQ^6,\qquad N=J_3(0)\oplus J_3(0),\qquad
s=e_0+f_0.
```

f_1Use in source · 16 passages

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

f_2Use in source · 12 passages

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

f_jUse in source · 5 passages

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

f_{j+1}Use in source · 1 passage

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Use in source · linked_chains.tex, line 23
ode[text=black!65] at (9.0000,0.0000) 0; draw[dashed,black!50] (1.0000,0.4800) -- (1.0000,1.0200); node[] at (1.0000,-0.9200) ae_0+af_0; draw[dashed,black!50] (4.0000,0.4800) -- (4.0000,1.0200); node[] at (4.0000,-0.9200) be_1+bf_1; draw[dashed,black!50] (7.0000,0.4800) -- (7.0000,1.0200); node[] at (7.0000,-0.9200) ce_2+cf_2; node[font=sffamilysmall] at (4.0000,2.4500) N e_j=e_j+1, N f_j=f_j+1 (j=0,1);
Exact mathematical passage
```
N e_j=e_{j+1},\qquad N f_j=f_{j+1}\quad(j=0,1)
```

f_mUse in source · 1 passage

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Use in source · visual_atlas.tex, line 135
headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.
Exact mathematical passage
```
(f_1,\ldots,f_m)\mapsto\sum_j f_j(T)s_j
```

f_rUse in source · 2 passages

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

f_\alphaUse in source · 5 passages

Search aliases: f_α alpha

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Use in source · Read passage · 03_packets_and_graphs.tex, line 150
[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsA…
Exact mathematical passage
```
f_\alpha:X_\alpha\to Y
```

G_1Use in source · 1 passage

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Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

G_mUse in source · 1 passage

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Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

G_{m-1}Use in source · 1 passage

Search aliases: G_m-1

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

g_\alphaUse in source · 6 passages

Search aliases: g_α alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 154
] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsAssembly of packetwise operators, channels, corrections, …
Exact mathematical passage
```
g_\alpha:Y\to X_\alpha
```

g_\betaUse in source · 1 passage

Search aliases: g_β beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 177
For x=∑_α P_α x, any extension of the f_α must satisfy f(x)=∑_α f_α(P_α x), which proves both existence and uniqueness of the first formula. For the second, the sum g=∑_α g_α satisfies P_β g =∑_α P_β g_α =g_β because g_α(Y)⊆ X_α. Uniqueness follows from ∑_α P_α=I.
Exact mathematical passage
```
P_\beta g
=\sum_\alpha P_\beta g_\alpha
=g_\beta
```

HUse in source · 4 passages

Search aliases: H

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

hUse in source · 6 passages

Search aliases: h

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 862
lpha,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ∏_C∈overline IEnd_𝕜(X_C). [source reference] E…
Exact mathematical passage
```
h
```

\headUse in source · 1 passage

Search aliases: head

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\Hull_{(O_i)}Use in source · 1 passage

Search aliases: Hull_(O_i) Hull operatorname

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 533
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
W_{n+1}=W_n
\qquad\text{and}\qquad
W_n=\Hull_{(O_i)}(S),
```

\Hull_{\AA}Use in source · 2 passages

Search aliases: Hull_AA Hull_ A Hull AA operatorname mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\Hull_{\cA}Use in source · 1 passage

Search aliases: Hull_cA Hull_ A Hull cA operatorname mathcal

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
```
{
\begin{aligned}
&\text{generated hull: }\Hull_{\cA}(S),\\
&\text{seed complexity: }\srank_{\cA}(X),\\
&\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\
&\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\
&\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\
&\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\
&\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\
&\text{realization: }\Phi A=\varphi(A)\Phi.
\end{aligned}
}
```

\Hull_{\Ccal}Use in source · 2 passages

Search aliases: Hull_Ccal Hull_ C Hull Ccal operatorname mathcal

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 164
· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
```
\Phi(S)\subseteq\Hull_{\B}(S'),
\qquad
\Psi(S')\subseteq\Hull_{\Ccal}(S'').
```

\Hull_FUse in source · 1 passage

Search aliases: Hull_F Hull operatorname

Link to this entry ·

Use in source · visual_atlas.tex, line 118
IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
F/I_s\cong\Hull_F(s)
```

\Hull_{\QQ[D]}Use in source · 1 passage

Search aliases: Hull_QQ[D] Hull_ Q[D] Hull QQ operatorname mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 234
headWhat the seed generates With the fixed algebra Q[D] and seed e_1, Hull_ Q[D](e_1)=span{e_1,e_2+e_3}. It has dimension two, not four. Both the independent difference e_2-e_3 and the last coordinate e_4 are missing.
Exact mathematical passage
```
\Hull_{\QQ[D]}(e_1)=\Span\{e_1,e_2+e_3\}.
```

\Hull_{\varphi(\A)}Use in source · 3 passages

Search aliases: Hull_φ(A) Hull_φ( A) Hull varphi A operatorname varphi mathcal

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 90
[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j
Exact mathematical passage
```
\boxed{
\Phi(\Hull_{\A}(S))
=
\Hull_{\varphi(\A)}(\Phi(S))
\subseteq
\Hull_{\B}(\Phi(S)).
}
```

\Hull_{\varphi(\AA)}Use in source · 1 passage

Search aliases: Hull_φ(AA) Hull_φ( A) Hull varphi AA operatorname varphi mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 290
headWhen the map is not invertible For an intertwiner Φ A=φ(A)Φ, the hull identity uses the image algebra: Φ(Hull_ A(S))=Hull_φ( A)(Φ S). Replacing φ( A) with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not automatic equality.
Exact mathematical passage
```
\Phi\bigl(\Hull_{\AA}(S)\bigr)=\Hull_{\varphi(\AA)}(\Phi S).
```

I_{e_1}Use in source · 2 passages

Search aliases: I_e_1

Link to this entry ·

Use in source · visual_atlas.tex, line 125
headThe distinction is visible in two dimensions Use X= Q^2, seed e_1=(1,0)^ T, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column: I_e_1={mat0 · a 0 · b:a,b∈ Q}.
Exact mathematical passage
```
I_{e_1}=\left\{\mat{0&a\\0&b}:a,b\in\QQ\right\}.
```

I_sUse in source · 3 passages

Search aliases: I_s

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

I_UUse in source · 2 passages

Search aliases: I_U

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 17
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1
Exact mathematical passage
```
BA=I_U
```

I_VUse in source · 1 passage

Search aliases: I_V

Link to this entry ·

Use in source · visual_atlas.tex, line 147
0 · 1 · 0. The typed relation is BA=I_U. In the opposite order, AB=diag(1,1,0)≠ I_V. The missing direction is Ker B= Q v_3, not an entirely missing vertex.
Exact mathematical passage
```
AB=\diag(1,1,0)\ne I_V.
```

I_{X/K}Use in source · 1 passage

Search aliases: I_X/K

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 238
[Packet descent through an invariant quotient] Under the hypotheses of [source reference], the operators overline P_α(x+K)=P_α x+K are well defined pairwise annihilating idempotents with sum I_X/K, and the quotient map satisfies π P_α=overline P_απ. After zero packets are removed, they form the canonical quotient packet system. [source reference]
Exact mathematical passage
```
I_{X/K}
```

I_{X^*}Use in source · 1 passage

Search aliases: I_X^*

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Use in source · Read passage · 03_packets_and_graphs.tex, line 257
Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the direct sum decomposition.
Exact mathematical passage
```
\sum_\alpha P_\alpha^*=I_{X^*}
```

I_{X_\alpha}Use in source · 1 passage

Search aliases: I_X_α alpha

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Use in source · Read passage · 03_packets_and_graphs.tex, line 813
The identity has diagonal blocks I_X_α and no blocks outside the diagonal, so it belongs to M_R exactly when (α,α)∈ R for every α.
Exact mathematical passage
```
I_{X_\alpha}
```

I_YUse in source · 3 passages

Search aliases: I_Y

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Use in source · Read passage · 03_packets_and_graphs.tex, line 223
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
```
\sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma
=
\left(\sum_\alpha P_\alpha\right)
\otimes
\left(\sum_\gamma Q_\gamma\right)
=I_X\otimes I_Y.
```

I_{\oplus X_\alpha}Use in source · 1 passage

Search aliases: I_⊕ X_α oplus alpha

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Use in source · Read passage · 03_packets_and_graphs.tex, line 89
[Exact packet coordinate isomorphism] For every finite packet system, S_ P A_ P=I_X, A_ P S_ P=I_⊕ X_α. Hence X≅⊕_α∈ IX_α. [source reference] ExportsEvery exact packet reconstruction in the series.
Exact mathematical passage
```
\boxed{
\mathcal S_{\Pcal}\mathcal A_{\Pcal}=I_X,
\qquad
\mathcal A_{\Pcal}\mathcal S_{\Pcal}=I_{\oplus X_\alpha}.
}
```

\operatorname{im}Use in source · 2 passages

Search aliases: im operatorname

Link to this entry ·

Use in source · oblique_vectors.tex, line 21
,0.0000) -- (2.0000,2.0000); draw[->,line width=1.2pt] (0.0000,0.0000) -- (3.0000,2.0000); draw[dashed,black!55] (1.0000,0.0000) -- (3.0000,2.0000); draw[dashed,black!55] (2.0000,2.0000) -- (3.0000,2.0000); node[] at (1.0000,-0.5800) Px=(1,0); node[anchor=east] at (1.4700,2.2000) Qx=(2,2); node[anchor=west] at (3.0800,2.1700) x=(3,2); node[font=scriptsize] at (3.6000,0.2800) imP; node[anchor=west,font=scriptsize] at (2.9500,2.8500) imQ;
Exact mathematical passage
```
\operatorname{im}P
```

J_3Use in source · 3 passages

Search aliases: J_3

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Use in source · visual_atlas.tex, line 8
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
```
X=\QQ^6,\qquad N=J_3(0)\oplus J_3(0),\qquad
s=e_0+f_0.
```

\JacUse in source · 8 passages

Search aliases: Jac operatorname

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Use in source · Read passage · 01_orientation.tex, line 47
How many seeds are required? · srank_A(M) · For a finitely generated module, pass to M/Jac(A)M. When the quotient algebra is semisimple Artinian, compute its simple multiplicities.
Exact mathematical passage
```
M/\Jac(A)M
```

KUse in source · 62 passages

Search aliases: K

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Use in source · Read passage · 02_foundations.tex, line 594
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
```
K/\bbk
```

kUse in source · 31 passages

Search aliases: k

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Use in source · Read passage · 03_packets_and_graphs.tex, line 61
Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.
Exact mathematical passage
```
Q_j^2=Q_j,
\qquad
Q_jQ_k=0\quad(j\neq k).
```

k_0Use in source · 4 passages

Search aliases: k_0

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Use in source · Read passage · 03_packets_and_graphs.tex, line 383
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
```
k_0
```

K_nUse in source · 3 passages

Search aliases: K_n

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Use in source · Read passage · 05_worked_models.tex, line 349
…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is zero when i>j, while the diagonal entry is one because there is a unique shorte…
Exact mathematical passage
```
K_n=[e_1,Ae_1,\ldots,A^{n-1}e_1].
```

K_pUse in source · 3 passages

Search aliases: K_p

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Use in source · Read passage · 02_foundations.tex, line 930
T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). Finally, every summand 𝕜[t]/(p^e) contributes one copy of K_p after quotie…
Exact mathematical passage
```
K_p=\bbk[t]/(p)
```

K_\alphaUse in source · 5 passages

Search aliases: K_α alpha

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Use in source · Read passage · 04_transport.tex, line 188
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
```
K_\alpha:=P_\alpha K=K\cap X_\alpha
```

K_\betaUse in source · 2 passages

Search aliases: K_β beta

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Use in source · Read passage · 04_transport.tex, line 256
[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]
Exact mathematical passage
```
K_\beta
```

\KerUse in source · 9 passages

Search aliases: Ker operatorname

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Use in source · Read passage · 02_foundations.tex, line 229
[Finite free presentation] For the map in [source reference], imε_S=Hull_A(S), N_S:=Kerε_S≤ A^⊕ r, and the induced map is an isomorphism of left modules over A A^⊕ r/N_S≅Hull_A(S). Consequently, S generates M if and only if ε_S is surjective. [source reference] ExportsAll later free seed presentations.
Exact mathematical passage
```
\im\varepsilon_S=\Hull_A(S),
\qquad
N_S:=\Ker\varepsilon_S\leq A^{\oplus r},
```

\kerUse in source · 4 passages

Search aliases: ker

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Use in source · Read passage · 05_worked_models.tex, line 33
1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated representation is U⊕span{(1,0,0),(0,1,0)} subsetneq U⊕ V. Adding the seed (0,0,1)∈ V generates the full representation.
Exact mathematical passage
```
\ker B
```

\kkUse in source · 4 passages

Search aliases: kk 𝕜 Bbbk

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Use in source · visual_atlas.tex, line 113
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
F=\kk\langle z_1,\ldots,z_r\rangle
```

\kk^2Use in source · 1 passage

Search aliases: kk^2 𝕜^2 kk Bbbk

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Use in source · visual_atlas.tex, line 304
The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.
Exact mathematical passage
```
\srank_{\kk I}(\kk^2)=2,
\qquad
\srank_{\kk\times\kk}(\kk\oplus\kk)=1.
```

LUse in source · 20 passages

Search aliases: L

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Use in source · Read passage · 02_foundations.tex, line 304
One seed and the distinction between module and algebra relations. [Criterion for the annihilator to be an ideal on both sides] For s∈ M, the left ideal L=Ann_A(s) is an ideal on both sides if and only if L Hull_A(s)=0. If s is cyclic, these conditions are equivalent to Ann_A(s)=Ann_A(M). [source reference]
Exact mathematical passage
```
L=\operatorname{Ann}_A(s)
```

lUse in source · 2 passages

Search aliases: l

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Use in source · Read passage · 05_worked_models.tex, line 294
se there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with [2pt] 𝕜^V(G)=span_𝕜 {p(A_G)s_j:p∈𝕜[t], 1≤ j≤ r} }.
Exact mathematical passage
```
{
R=\min\left\{r\in\N:
\begin{array}{l}
\text{there exist }s_1,\ldots,s_r\text{ with}\\[2pt]
\bbk^{V(G)}=\Span_{\bbk}
  \{p(A_G)s_j:p\in\bbk[t],\ 1\le j\le r\}
\end{array}
\right\}.
}
```

\lceilUse in source · 8 passages

Search aliases: lceil

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Use in source · Read passage · 01_orientation.tex, line 88
[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.
Exact mathematical passage
```
\srank_A(M)=\left\lceil\frac{3}{2}\right\rceil=2.
```

\linewidthUse in source · 1 passage

Search aliases: linewidth

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Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\LongleftrightarrowUse in source · 4 passages

Search aliases: Longleftrightarrow

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Use in source · Read passage · 02_foundations.tex, line 387
[Cyclic characterization] For M≠0, srank_A(M)=1 Longleftrightarrow M contains a cyclic vector. [source reference]
Exact mathematical passage
```
\srank_A(M)=1
\quad\Longleftrightarrow\quad
M\text{ contains a cyclic vector}.
```

\LongrightarrowUse in source · 1 passage

Search aliases: Longrightarrow

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Use in source · normal_form.tex, line 26
1150) x^2y; filldraw[fill=black!18,draw=white] (3.9000000000000004,-0.8) rectangle ++(1.15,-.63); node[] at (4.4750,-1.1150) x^3y; filldraw[fill=black!18,draw=white] (5.2,-0.8) rectangle ++(1.15,-.63); node[] at (5.7750,-1.1150) x^4y; node[] at (6.9500,-1.1150) ⋯; node[] at (3.2000,0.5200) Independent coefficient slots: a(x) and b(x); node[] at (3.2000,-2.1800) y^2=x^3-x Longrightarrow y^3=(x^3-x)y;
Exact mathematical passage
```
y^2=x^3-x\quad\Longrightarrow\quad y^3=(x^3-x)y
```

M_2Use in source · 5 passages

Search aliases: M_2

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Use in source · Read passage · 01_orientation.tex, line 87
[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.
Exact mathematical passage
```
A=M_2(\bbk)
```

M_{2\times5}Use in source · 2 passages

Search aliases: M_2×5 times

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Use in source · visual_atlas.tex, line 325
vpage13How much row space can one seed supply?SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULAMain text: [source reference] Let M_2( Q) act on X=M_2×5( Q) by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a 2×5 matrix V.
Exact mathematical passage
```
X=M_{2\times5}(\QQ)
```

M_3Use in source · 1 passage

Search aliases: M_3

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Use in source · Read passage · 05_worked_models.tex, line 499
M_3( C) acting on seven copies, seed rank · 3
Exact mathematical passage
```
M_3(\C)
```

M_4Use in source · 1 passage

Search aliases: M_4

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Use in source · Read passage · 05_worked_models.tex, line 502
Full M_4( R) orbit dimensions for two sampled nonzero seeds · 4,4
Exact mathematical passage
```
M_4(\R)
```

M_6Use in source · 1 passage

Search aliases: M_6

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Use in source · grammar_budget.tex, line 27
filldraw[draw=black!35,fill=white] (1.34,-1.92) rectangle ++(.48,.40); filldraw[draw=black!35,fill=white] (2.0100000000000002,-1.92) rectangle ++(.48,.40); filldraw[draw=black!35,fill=white] (2.68,-1.92) rectangle ++(.48,.40); filldraw[draw=black!35,fill=white] (3.35,-1.92) rectangle ++(.48,.40); node[font=sffamilybfseries] at (4.5000,-1.7200) 1; node[anchor=east] at (-0.3000,-2.5800) M_6( Q); filldraw[draw=black!35,fill=black!75] (0.0,-2.7800000000000002) rectangle ++(.48,.40); filldraw[draw=bl…
Exact mathematical passage
```
M_6(\QQ)
```

M_aUse in source · 1 passage

Search aliases: M_a

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Use in source · Read passage · 02_foundations.tex, line 74
unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=ax. The two constructions are inverse up to canonical natural isomorphism. [source reference]
Exact mathematical passage
```
M_v=e_vM,
\qquad
M_a(x)=ax.
```

M_dUse in source · 7 passages

Search aliases: M_d

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Use in source · Read passage · 02_foundations.tex, line 752
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
```
M_d(D)
```

m_jUse in source · 2 passages

Search aliases: m_j

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Use in source · Read passage · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
```
m_j\in M
```

M_KUse in source · 6 passages

Search aliases: M_K

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Use in source · Read passage · 02_foundations.tex, line 595
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
```
A_K:=K\otimes_{\bbk}A,
\qquad
M_K:=K\otimes_{\bbk}M.
```

M_nUse in source · 2 passages

Search aliases: M_n

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Use in source · Read passage · 05_worked_models.tex, line 256
Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.
Exact mathematical passage
```
\cA=M_n(\bbk)=\End_{\bbk}(X)
```

m_nUse in source · 3 passages

Search aliases: m_n

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Use in source · Read passage · 03_packets_and_graphs.tex, line 346
For x∈ X, choose m_n∈ M with m_n→ x. Continuity gives P_α m_n→ P_α x, so P_α X⊆clP_α M. The reverse inclusion follows because P_α X is closed and contains P_α M. The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.
Exact mathematical passage
```
m_n\in M
```

\mathfrak M_R^{\mathrm b}Use in source · 3 passages

Search aliases: M_R^ b mathfrak mathrm

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Use in source · Read passage · 03_packets_and_graphs.tex, line 835
[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]
Exact mathematical passage
```
\mathfrak M_R^{\mathrm b}:=\mathfrak M_R\cap\mathcal B(X).
```

\mathfrak M_{R_\Gamma}Use in source · 1 passage

Search aliases: M_R_Γ mathfrak Gamma

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Use in source · Read passage · 03_packets_and_graphs.tex, line 942
[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]
Exact mathematical passage
```
\A\subseteq\mathfrak M_{R_\Gamma},
\qquad
R_\Gamma:=\{(\beta,\alpha):\beta\in\Reach(\alpha)\}.
```

m_TUse in source · 2 passages

Search aliases: m_T

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Use in source · Read passage · 02_foundations.tex, line 916
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
m_T(t)=\prod_{p}p(t)^{e_p}.
```

M_vUse in source · 1 passage

Search aliases: M_v

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Use in source · Read passage · 02_foundations.tex, line 74
unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=ax. The two constructions are inverse up to canonical natural isomorphism. [source reference]
Exact mathematical passage
```
M_v=e_vM,
\qquad
M_a(x)=ax.
```

M_xUse in source · 7 passages

Search aliases: M_x

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Use in source · Read passage · 05_worked_models.tex, line 81
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
```
M_x,M_y
```

M_x^3Use in source · 1 passage

Search aliases: M_x^3

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Use in source · Read passage · 05_worked_models.tex, line 126
The relation is reconstructed as an operator identity: M_y^2=M_x^3-M_x.
Exact mathematical passage
```
M_y^2=M_x^3-M_x.
```

M_{x^3-x}Use in source · 2 passages

Search aliases: M_x^3-x

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Use in source · Read passage · 05_worked_models.tex, line 107
Identify the packet coefficients by A_0⊕ A_1⟶𝕜[x]⊕𝕜[x], (a,yb)⟼(a,b). In these ordered coefficient coordinates, with M_x and M_x^3-x denoting multiplication on 𝕜[x], M_x= M_x · 0
Exact mathematical passage
```
M_{x^3-x}
```

M_x^iUse in source · 1 passage

Search aliases: M_x^i

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Use in source · Read passage · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
```
M_x^iM_y^js=x^iy^j,
```

M_y^jUse in source · 1 passage

Search aliases: M_y^j

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Use in source · Read passage · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
```
M_x^iM_y^js=x^iy^j,
```

m_\lambdaUse in source · 3 passages

Search aliases: m_λ lambda

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Use in source · Read passage · 02_foundations.tex, line 452
For the topological statement, let S generate M densely. Given n=Φ(m), choose a net m_λ∈ AS converging to m. Continuity gives Φ(m_λ)⟶ n, and every Φ(m_λ) lies in BΦ(S). Thus BΦ(S) is dense in N.
Exact mathematical passage
```
m_\lambda\in AS
```

m_{\lambda}Use in source · 6 passages

Search aliases: m_λ lambda

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Use in source · Read passage · 05_worked_models.tex, line 283
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
```
m_{\lambda}\in\N
```

m_{\lambda_*}Use in source · 1 passage

Search aliases: m_λ_* lambda

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Use in source · Read passage · 05_worked_models.tex, line 292
r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
```
m_{\lambda_*}=R
```

\maxUse in source · 13 passages

Search aliases: max

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Use in source · Read passage · 02_foundations.tex, line 12
vector space over a complete valued field, together with a unital algebra of continuous linear operators. Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement. Conventions for empty families. For finite families of nonnegative integers we use max∅=0. The height of an empty finite partially ordered set is 0, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for …
Exact mathematical passage
```
\max\varnothing=0
```

\max_{1\leq i\leq q}Use in source · 2 passages

Search aliases: max_1≤ i≤ q max leq leq

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Use in source · Read passage · 02_foundations.tex, line 770
[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A/Jac(A) computes srank_A(M). [source reference] ExportsEvery calculation of seed rank for semisimple algebras of finite dimension in the series.
Exact mathematical passage
```
\boxed{
\srank_A(M)
=
\max_{1\leq i\leq q}
\left\lceil\frac{m_i}{d_i}\right\rceil.
}
```

\max_pUse in source · 3 passages

Search aliases: max_p max

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Use in source · Read passage · 02_foundations.tex, line 901
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
```
\boxed{
\srank_{\bbk[T]}(X)=\max_p\mu_p(T).
}
```

\max_{\lambda\in\operatorname{Spec}(T)}Use in source · 1 passage

Search aliases: max_λ∈Spec(T) max lambda in operatorname

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Use in source · Read passage · 02_foundations.tex, line 974
[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]
Exact mathematical passage
```
\boxed{
\srank_{\bbk[T]}(X)
=
\max_{\lambda\in\operatorname{Spec}(T)}
\dim_{\bbk}\Ker(T-\lambda I).
}
```

\max_{\mu\in\Spec_K(A)}Use in source · 1 passage

Search aliases: max_μ∈Spec_K(A) max mu in Spec max mu in operatorname

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Use in source · Read passage · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
```
r_{\mathrm{spec}}(A)
=
\max_{\mu\in\Spec_K(A)}
\dim_K\ker(A-\mu I),
```

\midUse in source · 2 passages

Search aliases: mid

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

\midruleUse in source · 1 passage

Search aliases: midrule

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Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\NUse in source · 7 passages

Search aliases: N mathbb

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Use in source · Read passage · 05_worked_models.tex, line 213
[Shift seed] For n∈ N, let G(n) mean that some finite seed set with n elements has dense hull under the declared family {S_m:m∈ Z}. Then 1=min{n∈ N:G(n)}. Equivalently, one seed suffices and no empty seed family generates densely.
Exact mathematical passage
```
n\in\N
```

\mathfrak NUse in source · 8 passages

Search aliases: N mathfrak

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Use in source · Read passage · 03_packets_and_graphs.tex, line 859
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
```
\mathfrak N_R\subseteq\mathfrak M_R
```

N^2Use in source · 1 passage

Search aliases: N^2

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Use in source · visual_atlas.tex, line 60
vpage3Two chains can stay linkedMODEL A / AN OPERATOR IN SIX DIMENSIONSMain text: [source reference]. Matrices over Q Use the ordered basis (e_0,e_1,e_2,f_0,f_1,f_2). Define Ne_0=e_1, Ne_1=e_2, Ne_2=0, Nf_0=f_1, Nf_1=f_2, Nf_2=0. Starting with s=e_0+f_0, the two chains advance together: s=e_0+f_0, Ns=e_1+f_1, N^2s=e_2+f_2, N^3s=0.
Exact mathematical passage
```
s=e_0+f_0,\qquad Ns=e_1+f_1,\qquad N^2s=e_2+f_2,\qquad N^3s=0.
```

N^3Use in source · 2 passages

Search aliases: N^3

Link to this entry ·

Use in source · visual_atlas.tex, line 60
vpage3Two chains can stay linkedMODEL A / AN OPERATOR IN SIX DIMENSIONSMain text: [source reference]. Matrices over Q Use the ordered basis (e_0,e_1,e_2,f_0,f_1,f_2). Define Ne_0=e_1, Ne_1=e_2, Ne_2=0, Nf_0=f_1, Nf_1=f_2, Nf_2=0. Starting with s=e_0+f_0, the two chains advance together: s=e_0+f_0, Ns=e_1+f_1, N^2s=e_2+f_2, N^3s=0.
Exact mathematical passage
```
s=e_0+f_0,\qquad Ns=e_1+f_1,\qquad N^2s=e_2+f_2,\qquad N^3s=0.
```

n_jUse in source · 7 passages

Search aliases: n_j

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Use in source · Read passage · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
```
(n_j)_{j\in J}
```

\mathfrak N_RUse in source · 7 passages

Search aliases: N_R mathfrak

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Use in source · Read passage · 03_packets_and_graphs.tex, line 859
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
```
\mathfrak N_R\subseteq\mathfrak M_R
```

\mathfrak N_R^{\,h+1}Use in source · 1 passage

Search aliases: N_R^ h+1 mathfrak

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Use in source · Read passage · 03_packets_and_graphs.tex, line 863
rtial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ∏_C∈overline IEnd_𝕜(X_C). [source reference] ExportsFiltered transfer algebras and triangular packet inver…
Exact mathematical passage
```
\mathfrak N_R^{\,h+1}=0.
```

N_SUse in source · 5 passages

Search aliases: N_S

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Use in source · Read passage · 02_foundations.tex, line 229
[Finite free presentation] For the map in [source reference], imε_S=Hull_A(S), N_S:=Kerε_S≤ A^⊕ r, and the induced map is an isomorphism of left modules over A A^⊕ r/N_S≅Hull_A(S). Consequently, S generates M if and only if ε_S is surjective. [source reference] ExportsAll later free seed presentations.
Exact mathematical passage
```
\im\varepsilon_S=\Hull_A(S),
\qquad
N_S:=\Ker\varepsilon_S\leq A^{\oplus r},
```

\normUse in source · 2 passages

Search aliases: norm

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Use in source · visual_atlas.tex, line 185
headWhat the algebra does not preserve For the displayed x, normx^2=13, normPx^2+normQx^2=1+8=9. There is no Parseval identity here. The cross term is nonzero because the packet directions are not perpendicular. Likewise, the simple formula for reversing adjoints of packet blocks requires selfadjoint projectors. It is not justified for this oblique pair.
Exact mathematical passage
```
\norm{x}^2=13,\qquad \norm{Px}^2+\norm{Qx}^2=1+8=9.
```

OUse in source · 7 passages

Search aliases: O

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Use in source · Read passage · 05_worked_models.tex, line 525
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
(O_i)_{i\in I}
```

oUse in source · 1 passage

Search aliases: o

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

O_iUse in source · 6 passages

Search aliases: O_i

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 525
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
(O_i)_{i\in I}
```

\overline IUse in source · 4 passages

Search aliases: overline I overline

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Use in source · Read passage · 03_packets_and_graphs.tex, line 855
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
```
\overline I=I/{\sim}
```

\overline PUse in source · 9 passages

Search aliases: overline P overline

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Use in source · Read passage · 04_transport.tex, line 192
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
```
\overline P_\alpha(x+K):=P_\alpha x+K.
```

\overline P_\alphaUse in source · 9 passages

Search aliases: overline P_α overline alpha

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Use in source · Read passage · 04_transport.tex, line 192
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
```
\overline P_\alpha(x+K):=P_\alpha x+K.
```

\overline P_\betaUse in source · 1 passage

Search aliases: overline P_β overline beta

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Use in source · Read passage · 04_transport.tex, line 275
If x∈ K_α, then Tx∈ K, so T_βαx=P_β Tx∈ K∩ X_β=K_β. On the quotient, overline T_βαπ = overline P_βoverline Toverline P_απ = π P_β TP_α.
Exact mathematical passage
```
\overline T_{\beta\alpha}\pi
=
\overline P_\beta\overline T\overline P_\alpha\pi
=
\pi P_\beta TP_\alpha.
```

\overline TUse in source · 3 passages

Search aliases: overline T overline

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Use in source · Read passage · 04_transport.tex, line 256
[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]
Exact mathematical passage
```
\overline T
```

\overline T_{\beta\alpha}Use in source · 2 passages

Search aliases: overline T_βα overline beta alpha

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Use in source · Read passage · 04_transport.tex, line 257
[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]
Exact mathematical passage
```
\boxed{
\overline T_{\beta\alpha}\,\pi
=
\pi T_{\beta\alpha},
}
```

P^2Use in source · 1 passage

Search aliases: P^2

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Use in source · visual_atlas.tex, line 172
0 · 1. These maps satisfy P^2=P, Q^2=Q, PQ=QP=0 and P+Q=I. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.
Exact mathematical passage
```
P^2=P
```

p^{e_p}Use in source · 1 passage

Search aliases: p^e_p

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Use in source · Read passage · 02_foundations.tex, line 920
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
\bbk[T]\cong\bbk[t]/(m_T)
\cong\prod_pR_p,
\qquad
R_p:=\bbk[t]/(p^{e_p}),
```

P_0Use in source · 4 passages

Search aliases: P_0

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Use in source · Read passage · 05_worked_models.tex, line 94
The two packet spaces are A_0=𝕜[x], A_1=y𝕜[x]. Let P_0(a+yb)=a, P_1(a+yb)=yb. These idempotents are linear over 𝕜, annihilate one another, and satisfy P_0+P_1=I. They are not module maps over A, because multiplication by y exchanges the two packets.
Exact mathematical passage
```
P_0(a+yb)=a,
\qquad
P_1(a+yb)=yb.
```

p_0Use in source · 2 passages

Search aliases: p_0

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Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
```
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
```

P_1Use in source · 9 passages

Search aliases: P_1

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Use in source · Read passage · 01_orientation.tex, line 96
[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
```
P_1=\begin{pmatrix}1&-1\\0&0\end{pmatrix},
\qquad
P_2=\begin{pmatrix}0&1\\0&1\end{pmatrix}.
```

P_2Use in source · 10 passages

Search aliases: P_2

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Use in source · Read passage · 01_orientation.tex, line 96
[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
```
P_1=\begin{pmatrix}1&-1\\0&0\end{pmatrix},
\qquad
P_2=\begin{pmatrix}0&1\\0&1\end{pmatrix}.
```

p_2Use in source · 3 passages

Search aliases: p_2

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Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
```
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
```

P_3Use in source · 1 passage

Search aliases: P_3

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Use in source · visual_atlas.tex, line 199
headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.
Exact mathematical passage
```
P_1+P_2+P_3=I
```

P_eUse in source · 8 passages

Search aliases: P_e

Link to this entry ·

Use in source · visual_atlas.tex, line 12
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
```
P_e,P_f
```

P_fUse in source · 8 passages

Search aliases: P_f

Link to this entry ·

Use in source · visual_atlas.tex, line 12
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
```
P_e,P_f
```

P_iUse in source · 7 passages

Search aliases: P_i

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Use in source · Read passage · 01_orientation.tex, line 101
0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
```
P_i^2=P_i
```

P_i^2Use in source · 1 passage

Search aliases: P_i^2

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Use in source · Read passage · 01_orientation.tex, line 101
0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
```
P_i^2=P_i
```

P_jUse in source · 4 passages

Search aliases: P_j

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Use in source · visual_atlas.tex, line 193
vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.
Exact mathematical passage
```
T_{ij}=P_iTP_j
```

P_mUse in source · 3 passages

Search aliases: P_m

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Use in source · Read passage · 03_packets_and_graphs.tex, line 410
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
```
P_nP_m=\delta_{nm}P_n
```

P_nUse in source · 11 passages

Search aliases: P_n

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Use in source · Read passage · 03_packets_and_graphs.tex, line 409
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
```
(P_n)_{n\geq1}\subseteq\mathcal B(X)
```

P_{N,j}Use in source · 2 passages

Search aliases: P_N,j

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Use in source · Read passage · 05_worked_models.tex, line 227
For N≥1, define residue projectors (P_N,jx)[n] = x[n], · n≡ jpmod N, 0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.
Exact mathematical passage
```
(P_{N,j}x)[n]
=
\begin{cases}
x[n],&n\equiv j\pmod N,\\0,&\text{otherwise.}\end{cases}
```

P_{N,j+1}Use in source · 1 passage

Search aliases: P_N,j+1

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Use in source · Read passage · 05_worked_models.tex, line 234
0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.
Exact mathematical passage
```
S_1P_{N,j}=P_{N,j+1}S_1,
```

P_UUse in source · 2 passages

Search aliases: P_U

Link to this entry ·

Use in source · visual_atlas.tex, line 159
headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The paper proves that no seed supported at U is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.
Exact mathematical passage
```
P_U,P_V
```

P_VUse in source · 2 passages

Search aliases: P_V

Link to this entry ·

Use in source · visual_atlas.tex, line 159
headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The paper proves that no seed supported at U is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.
Exact mathematical passage
```
P_U,P_V
```

P_{\alpha}Use in source · 1 passage

Search aliases: P_α alpha

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
```
{
\begin{aligned}
&\text{generated hull: }\Hull_{\cA}(S),\\
&\text{seed complexity: }\srank_{\cA}(X),\\
&\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\
&\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\
&\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\
&\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\
&\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\
&\text{realization: }\Phi A=\varphi(A)\Phi.
\end{aligned}
}
```

P_\alpha^*Use in source · 6 passages

Search aliases: P_α^* alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 237
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
```
P_\alpha^*:X^*\to X^*,
\qquad
P_\alpha^*(\varphi)=\varphi\circ P_\alpha,
```

P_{\beta}Use in source · 1 passage

Search aliases: P_β beta

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
```
{
\begin{aligned}
&\text{generated hull: }\Hull_{\cA}(S),\\
&\text{seed complexity: }\srank_{\cA}(X),\\
&\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\
&\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\
&\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\
&\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\
&\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\
&\text{realization: }\Phi A=\varphi(A)\Phi.
\end{aligned}
}
```

P_\beta^*Use in source · 1 passage

Search aliases: P_β^* beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 253
Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the direct sum decomposition.
Exact mathematical passage
```
P_\alpha^*P_\beta^*=(P_\beta P_\alpha)^*
=\delta_{\alpha\beta}P_\alpha^*.
```

P_\gammaUse in source · 3 passages

Search aliases: P_γ gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 546
Insert I=∑_β P_β between S and T: P_γ STP_α = ∑_β P_γ SP_β TP_α.
Exact mathematical passage
```
P_\gamma STP_\alpha
=
\sum_\beta
P_\gamma SP_\beta TP_\alpha.
```

P_{\gamma_1}Use in source · 1 passage

Search aliases: P_γ_1 gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

P_{\gamma_{m-1}}Use in source · 1 passage

Search aliases: P_γ_m-1 gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

P_{\sigma^{-1}(i)}Use in source · 1 passage

Search aliases: P_σ^-1(i) sigma

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 122
For the block formula, use intertwining and packet compatibility twice: Q_jφ(T)Q_iΦ · =Q_jφ(T)Φ P_σ^-1(i) · =Q_jΦ TP_σ^-1(i) · =Φ P_σ^-1(j)TP_σ^-1(i) · =∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and Hull_ A(S)=X, the hull equality gives Y=Hull_φ( A)(Φ(S)). If Φ(S)⊆Hull_ B(S'), then Hull_ B(Φ(S))⊆Hull_ B(S'), so S' generates Y.
Exact mathematical passage
```
Q_j\varphi(T)Q_i\Phi
&=Q_j\varphi(T)\Phi P_{\sigma^{-1}(i)}\\
&=Q_j\Phi TP_{\sigma^{-1}(i)}\\
&=\Phi P_{\sigma^{-1}(j)}TP_{\sigma^{-1}(i)}\\
&=\sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}}
\Phi P_\beta TP_\alpha.
```

\mathrm{packets}Use in source · 1 passage

Search aliases: packets mathrm

Link to this entry ·

Use in source · visual_atlas.tex, line 481
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1 would preserve the phase graph but would lose the operator's action on the infinite sequences.
Exact mathematical passage
```
S_{\mathrm{packets}}=\mat{0&0&L\\I&0&0\\0&I&0}.
```

\pagerefVUse in source · 1 passage

Search aliases: pagerefV

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\mathbf{PktMod}Use in source · 1 passage

Search aliases: PktMod mathbf

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 140
[Category of seeded packet systems] Packet compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions. Thus packet resolved operator modules form a category PktMod and seeded packet systems form a category SeedSys. DependsnolinkurlPSC-I-D-PACKET-MORPHISM, nolinkurlPSC-I-D-SEED-MORPHISM, and nolinkurlPSC-I-T-NATURALITY.
Exact mathematical passage
```
\mathbf{PktMod}
```

\pmodUse in source · 3 passages

Search aliases: pmod

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 227
For N≥1, define residue projectors (P_N,jx)[n] = x[n], · n≡ jpmod N, 0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.
Exact mathematical passage
```
(P_{N,j}x)[n]
=
\begin{cases}
x[n],&n\equiv j\pmod N,\\0,&\text{otherwise.}\end{cases}
```

\psrank_{A,\Pcal}Use in source · 1 passage

Search aliases: psrank_A,Pcal psrank_A, P psrank Pcal operatorname mathcal

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 1037
[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]
Exact mathematical passage
```
\boxed{
\srank_A(X)
\leq
\psrank_{A,\Pcal}(X)
\leq
\sum_{\alpha\in I}
\srank_{P_\alpha A P_\alpha}(X_\alpha).
}
```

Q^2Use in source · 1 passage

Search aliases: Q^2

Link to this entry ·

Use in source · visual_atlas.tex, line 172
0 · 1. These maps satisfy P^2=P, Q^2=Q, PQ=QP=0 and P+Q=I. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.
Exact mathematical passage
```
Q^2=Q
```

Q_2Use in source · 2 passages

Search aliases: Q_2

Link to this entry ·

Use in source · visual_atlas.tex, line 281
Indeed, Q_1T^UQ_2=Q_2T^UQ_1=0, Q_iT^UQ_j=U(P_iTP_j)U^-1. A change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.
Exact mathematical passage
```
Q_1T^UQ_2=Q_2T^UQ_1=0,
\qquad
Q_iT^UQ_j=U(P_iTP_j)U^{-1}.
```

Q_iUse in source · 4 passages

Search aliases: Q_i

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 100
[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in addition, the morphism is seeded, then S' generates Y. [source reference] ExportsThe canonical naturality theorem for every later realization.
Exact mathematical passage
```
\boxed{
Q_j\varphi(T)Q_i\Phi
=
\sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}}
\Phi P_\beta TP_\alpha.
}
```

Q_j^2Use in source · 1 passage

Search aliases: Q_j^2

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 61
Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.
Exact mathematical passage
```
Q_j^2=Q_j,
\qquad
Q_jQ_k=0\quad(j\neq k).
```

Q_kUse in source · 1 passage

Search aliases: Q_k

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 61
Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.
Exact mathematical passage
```
Q_j^2=Q_j,
\qquad
Q_jQ_k=0\quad(j\neq k).
```

Q_\alphaUse in source · 5 passages

Search aliases: Q_α alpha

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 10
[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family G is transported to U G U^-1, then its transfer graph and reachability relation are also preserved. [source reference]
Exact mathematical passage
```
Q_\alpha:=UP_\alpha U^{-1}.
```

Q_\betaUse in source · 3 passages

Search aliases: Q_β beta

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 14
[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family G is transported to U G U^-1, then its transfer graph and reachability relation are also preserved. [source reference]
Exact mathematical passage
```
\boxed{
Q_\beta\widetilde TQ_\alpha
=
U(P_\beta TP_\alpha)U^{-1}.
}
```

Q_\gammaUse in source · 13 passages

Search aliases: Q_γ gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 190
[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
```
\Qcal=(Q_\gamma)_{\gamma\in J}
```

Q_\gamma^2Use in source · 1 passage

Search aliases: Q_γ^2 gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 217
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
```
(P_\alpha\otimes Q_\gamma)^2
=P_\alpha^2\otimes Q_\gamma^2
=P_\alpha\otimes Q_\gamma.
```

Q_\deltaUse in source · 5 passages

Search aliases: Q_δ delta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 222
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
```
Q_\gamma Q_\delta=0
```

Q_{\tau^{-1}(k)}Use in source · 1 passage

Search aliases: Q_τ^-1(k) tau

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
```
R_k\Psi\Phi
&=\Psi Q_{\tau^{-1}(k)}\Phi\\
&=\Psi\Phi
\sum_{\tau(\sigma(\alpha))=k}P_\alpha.
```

\QQ^2Use in source · 7 passages

Search aliases: QQ^2 Q^2 QQ mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 86
Define a surjective linear map q:X⟶ Q^2, q(∑ a_je_j+∑ b_jf_j)=(a_0,b_0).
Exact mathematical passage
```
q:X\longrightarrow\QQ^2,\qquad
q\Bigl(\sum a_je_j+\sum b_jf_j\Bigr)=(a_0,b_0).
```

\QQ^3Use in source · 2 passages

Search aliases: QQ^3 Q^3 QQ mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 139
vpage6A reached type can still hide a directionTYPED MODEL / U= Q^2 AND V= Q^3Main text: [source reference] The paper's typed model has a rotation R:U→ U, an inclusion A:U→ V, and a retraction B:V→ U: R=mat0 · -1
Exact mathematical passage
```
V=\QQ^3
```

\QQ^4Use in source · 1 passage

Search aliases: QQ^4 Q^4 QQ mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 220
vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.
Exact mathematical passage
```
\QQ^4
```

\QQ^6Use in source · 4 passages

Search aliases: QQ^6 Q^6 QQ mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 8
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
```
X=\QQ^6,\qquad N=J_3(0)\oplus J_3(0),\qquad
s=e_0+f_0.
```

\RUse in source · 2 passages

Search aliases: R mathbb

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 195
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
\mathbb{F}\in\{\R,\C\}
```

\mathbb RUse in source · 1 passage

Search aliases: R mathbb

Link to this entry ·

Use in source · Read passage · 01_orientation.tex, line 101
0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
```
\mathbb R
```

R_kUse in source · 1 passage

Search aliases: R_k

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
```
R_k\Psi\Phi
&=\Psi Q_{\tau^{-1}(k)}\Phi\\
&=\Psi\Phi
\sum_{\tau(\sigma(\alpha))=k}P_\alpha.
```

R_pUse in source · 5 passages

Search aliases: R_p

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 920
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
\bbk[T]\cong\bbk[t]/(m_T)
\cong\prod_pR_p,
\qquad
R_p:=\bbk[t]/(p^{e_p}),
```

r_{\mathrm{spec}}Use in source · 2 passages

Search aliases: r_spec mathrm

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
```
r_{\mathrm{spec}}(A)
=
\max_{\mu\in\Spec_K(A)}
\dim_K\ker(A-\mu I),
```

R_\GammaUse in source · 1 passage

Search aliases: R_Γ Gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 942
[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]
Exact mathematical passage
```
\A\subseteq\mathfrak M_{R_\Gamma},
\qquad
R_\Gamma:=\{(\beta,\alpha):\beta\in\Reach(\alpha)\}.
```

\raggedrightUse in source · 1 passage

Search aliases: raggedright

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\RcalUse in source · 1 passage

Search aliases: Rcal R mathcal

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 148
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ
Exact mathematical passage
```
(\Psi,\psi,\tau):(Y,\B,\Qcal)\to(Z,\Ccal,\Rcal)
```

\rceilUse in source · 8 passages

Search aliases: rceil

Link to this entry ·

Use in source · Read passage · 01_orientation.tex, line 88
[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.
Exact mathematical passage
```
\srank_A(M)=\left\lceil\frac{3}{2}\right\rceil=2.
```

\ReachUse in source · 3 passages

Search aliases: Reach operatorname

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 937
[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]
Exact mathematical passage
```
P_\beta AP_\alpha=0
\qquad\text{whenever }\beta\notin\Reach(\alpha).
```

\RRUse in source · 4 passages

Search aliases: RR R mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 168
vpage7Exact does not mean orthogonalPACKET COORDINATES / AN OBLIQUE SPLITTINGMain text: [source reference] In R^2, take P=mat1 · -1
Exact mathematical passage
```
\RR^2
```

\RR^2Use in source · 1 passage

Search aliases: RR^2 R^2 RR mathbb

Link to this entry ·

Use in source · visual_atlas.tex, line 168
vpage7Exact does not mean orthogonalPACKET COORDINATES / AN OBLIQUE SPLITTINGMain text: [source reference] In R^2, take P=mat1 · -1
Exact mathematical passage
```
\RR^2
```

S''Use in source · 2 passages

Search aliases: S''

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 164
· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
```
\Phi(S)\subseteq\Hull_{\B}(S'),
\qquad
\Psi(S')\subseteq\Hull_{\Ccal}(S'').
```

S^{\oplus m}Use in source · 2 passages

Search aliases: S^⊕ m oplus

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 847
[Multiplicity obstruction] For the diagonal action of M_d(D) on S^⊕ m, srank_M_d(D)(S^⊕ m) = lceilfracmdrceil. [source reference]
Exact mathematical passage
```
S^{\oplus m}
```

S_0Use in source · 1 passage

Search aliases: S_0

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
S_0=I
```

S_1Use in source · 3 passages

Search aliases: S_1

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Use in source · Read passage · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
S_1
```

S_CUse in source · 2 passages

Search aliases: S_C

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 1093
For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in A S. Choose an incoming condensation edge from such a component to D, represented by a graph edge α→β with α already generated and β∈ D. Surjectivity …
Exact mathematical passage
```
S_C
```

s_iUse in source · 4 passages

Search aliases: s_i

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 446
Let S generate M. For every n∈ N, choose m∈ M with Φ(m)=n and write m=∑_ia_is_i. Then n=∑_iφ(a_i)Φ(s_i), so Φ(S) generates N.
Exact mathematical passage
```
m=\sum_ia_is_i
```

S_i^{\oplus m_i}Use in source · 1 passage

Search aliases: S_i^⊕ m_i oplus

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 766
[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A/Jac(A) computes srank_A(M). [source reference] ExportsEvery calculation of seed rank for semisimple algebras of finite dimension in the series.
Exact mathematical passage
```
M\cong\bigoplus_{i=1}^{q}S_i^{\oplus m_i}
```

S_i^{\oplus rd_i}Use in source · 1 passage

Search aliases: S_i^⊕ rd_i oplus

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 786
By [source reference], M is generated by r elements exactly when it is a quotient of A^⊕ r. By [source reference], A^⊕ r≅⊕_iS_i^⊕ rd_i. Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to M exists if and only if m_i≤ rd_i for every i. The smallest such r is the displayed maximum. For a nonsemisimple algebra of finite dimension, [source reference] reduces the calculation to the semisimple module M/Jac(A)M over A/Jac(A).
Exact mathematical passage
```
A^{\oplus r}\cong\bigoplus_iS_i^{\oplus rd_i}.
```

S_mUse in source · 7 passages

Search aliases: S_m

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 200
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
(S_mx)[n]=x[n-m]
\qquad(m,n\in\Z).
```

s_mUse in source · 1 passage

Search aliases: s_m

Link to this entry ·

Use in source · visual_atlas.tex, line 135
headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.
Exact mathematical passage
```
(s_1,\ldots,s_m)
```

S_{m+r}Use in source · 1 passage

Search aliases: S_m+r

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
S_{m+r}=S_mS_r
```

S_{\mathrm{packets}}Use in source · 1 passage

Search aliases: S_packets mathrm

Link to this entry ·

Use in source · visual_atlas.tex, line 481
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1 would preserve the phase graph but would lose the operator's action on the infinite sequences.
Exact mathematical passage
```
S_{\mathrm{packets}}=\mat{0&0&L\\I&0&0\\0&I&0}.
```

S_rUse in source · 1 passage

Search aliases: S_r

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
S_{m+r}=S_mS_r
```

S_\alphaUse in source · 3 passages

Search aliases: S_α alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 1053
Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates X. Its cardinality is at most the displayed sum.
Exact mathematical passage
```
S_\alpha\subseteq X_\alpha
```

S_{\beta\alpha}Use in source · 2 passages

Search aliases: S_βα beta alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 601
[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.
Exact mathematical passage
```
\boxed{
(S\otimes T)_{(\beta,\delta),(\alpha,\gamma)}
=
S_{\beta\alpha}\otimes T_{\delta\gamma}.
}
```

S_{\gamma\beta}Use in source · 2 passages

Search aliases: S_γβ gamma beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 532
[Packet matrix multiplication] For S,T∈End_𝕜(X), (ST)_γα = ∑_β∈ IS_γβT_βα. [source reference] ExportsAll packet path expansions.
Exact mathematical passage
```
\boxed{
(ST)_{\gamma\alpha}
=
\sum_{\beta\in I}S_{\gamma\beta}T_{\beta\alpha}.
}
```

S_{\gamma\delta}Use in source · 3 passages

Search aliases: S_γδ gamma delta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 626
[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
```
T_{\beta\alpha}=P_\beta TP_\alpha,
\qquad
S_{\gamma\delta}=Q_\gamma SQ_\delta,
\qquad
U_{\gamma\alpha}=Q_\gamma UP_\alpha.
```

\sinUse in source · 1 passage

Search aliases: sin

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 345
quently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is ze…
Exact mathematical passage
```
\sin\frac{k\pi}{n+1}\ne0.
```

\smallUse in source · 1 passage

Search aliases: small

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\SpanUse in source · 8 passages

Search aliases: Span span operatorname

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 34
2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated representation is U⊕span{(1,0,0),(0,1,0)} subsetneq U⊕ V. Adding the seed (0,0,1)∈ V generates the full representation.
Exact mathematical passage
```
U\oplus\Span\{(1,0,0),(0,1,0)\}
\subsetneq U\oplus V.
```

\Span_{\bbk}Use in source · 3 passages

Search aliases: Span_bbk span_𝕜 Span bbk operatorname Bbbk

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 294
se there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with [2pt] 𝕜^V(G)=span_𝕜 {p(A_G)s_j:p∈𝕜[t], 1≤ j≤ r} }.
Exact mathematical passage
```
{
R=\min\left\{r\in\N:
\begin{array}{l}
\text{there exist }s_1,\ldots,s_r\text{ with}\\[2pt]
\bbk^{V(G)}=\Span_{\bbk}
  \{p(A_G)s_j:p\in\bbk[t],\ 1\le j\le r\}
\end{array}
\right\}.
}
```

\Span_{\C}Use in source · 1 passage

Search aliases: Span_C span_ C Span C operatorname mathbb

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
```
{
2=\min\left\{r\in\N:
\exists s_1,\ldots,s_r,\quad
\C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\}
\right\}.
}
```

\SpecUse in source · 1 passage

Search aliases: Spec operatorname

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
```
r_{\mathrm{spec}}(A)
=
\max_{\mu\in\Spec_K(A)}
\dim_K\ker(A-\mu I),
```

\mathrm{spec}Use in source · 2 passages

Search aliases: spec mathrm

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
```
r_{\mathrm{spec}}(A)
=
\max_{\mu\in\Spec_K(A)}
\dim_K\ker(A-\mu I),
```

\operatorname{Spec}Use in source · 1 passage

Search aliases: Spec operatorname

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 974
[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]
Exact mathematical passage
```
\boxed{
\srank_{\bbk[T]}(X)
=
\max_{\lambda\in\operatorname{Spec}(T)}
\dim_{\bbk}\Ker(T-\lambda I).
}
```

\Spec_KUse in source · 1 passage

Search aliases: Spec_K Spec operatorname

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
```
r_{\mathrm{spec}}(A)
=
\max_{\mu\in\Spec_K(A)}
\dim_K\ker(A-\mu I),
```

\srank_{\A}Use in source · 1 passage

Search aliases: srank_A srank_ A srank A operatorname mathcal

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 1143
] The source component formulas concern packet pure seeds. They do not give the unrestricted seed rank. For example, let A=𝕜×𝕜, X=𝕜⊕𝕜 with the coordinate action and the two coordinate packet projectors. The transfer graph has two source components and psrank_ A, P(X)=2, but the mixed vector (1,1) is cyclic, so srank_ A(X)=1.
Exact mathematical passage
```
\srank_{\A}(X)=1.
```

\srank_{A,\tau_M}Use in source · 1 passage

Search aliases: srank_A,τ_M srank tau operatorname tau

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
```
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
```

\srank_{A/J}Use in source · 2 passages

Search aliases: srank_A/J srank operatorname

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 678
radical] Let A be a unital ring, let J=Jac(A), and let M be a finitely generated left module over A. A family S⊆ M generates M if and only if its image generates M/JM as a module over A/J. When A/J is semisimple Artinian, this quotient is the usual semisimple top. No semisimplicity of A/J is required for the general equality. Consequently, srank_A(M)=srank_A/J(M/JM). [source reference] ExportsGenerator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modu…
Exact mathematical passage
```
\boxed{
\srank_A(M)=\srank_{A/J}(M/JM).
}
```

\srank_{A_K}Use in source · 1 passage

Search aliases: srank_A_K srank operatorname

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 601
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
```
\boxed{
\srank_{A_K}(M_K)\leq\srank_A(M).
}
```

\srank_{\AA}Use in source · 2 passages

Search aliases: srank_AA srank_ A srank AA operatorname mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 313
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
```
\srank_{\AA\times\BB}(X\oplus Y)
=\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
```

\srank_{\AA\times\BB}Use in source · 1 passage

Search aliases: srank_AA×BB srank_ A× B srank AA times BB operatorname mathcal times mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 313
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
```
\srank_{\AA\times\BB}(X\oplus Y)
=\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
```

\srank_{\AA\otimes\BB}Use in source · 1 passage

Search aliases: srank_AA⊗BB srank_ A⊗ B srank AA otimes BB operatorname mathcal otimes mathcal

Link to this entry ·

Use in source · visual_atlas.tex, line 528
headThe resulting upper bound Since simple tensors span the algebraic tensor product, the complete rectangular family proves srank_ A⊗ B(X⊗ Y) ≤srank_ A(X) srank_ B(Y). This is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. Nor does the algebraic argument itself establish a theorem for completed topological tensor products.
Exact mathematical passage
```
\boxed{\srank_{\AA\otimes\BB}(X\otimes Y)
\le\srank_{\AA}(X)\,\srank_{\BB}(Y).}
```

\srank_{A\times B}Use in source · 1 passage

Search aliases: srank_A× B srank times operatorname times

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Use in source · Read passage · 02_foundations.tex, line 531
[Direct sum formula for independent actions] Let M be a left module over A and N a left module over B. Under the action of the product algebra (a,b)(m,n)=(am,bn), one has srank_A× B(M⊕ N) = max{srank_A(M),srank_B(N)}. [source reference]
Exact mathematical passage
```
\boxed{
\srank_{A\times B}(M\oplus N)
=
\max\{\srank_A(M),\srank_B(N)\}.
}
```

\srank_{A\otimes B}Use in source · 1 passage

Search aliases: srank_A⊗ B srank otimes operatorname otimes

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Use in source · Read passage · 02_foundations.tex, line 560
Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]
Exact mathematical passage
```
\boxed{
\srank_{A\otimes B}(M\otimes_{\bbk}N)
\leq
\srank_A(M)\srank_B(N).
}
```

\srank_BUse in source · 4 passages

Search aliases: srank_B srank operatorname

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Use in source · Read passage · 02_foundations.tex, line 403
[Monotonicity under enlargement of the acting algebra] Let B⊆ A⊆End_𝕜(X) be unital operator algebras with the same identity. Then srank_A(X)≤srank_B(X). The same statement holds for topological seed rank when both algebras act continuously. [source reference]
Exact mathematical passage
```
\boxed{\srank_A(X)\leq\srank_B(X).}
```

\srank_{B,\tau_N}Use in source · 1 passage

Search aliases: srank_B,τ_N srank tau operatorname tau

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Use in source · Read passage · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
```
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
```

\srank_{\BB}Use in source · 2 passages

Search aliases: srank_BB srank_ B srank BB operatorname mathcal

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Use in source · visual_atlas.tex, line 313
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
```
\srank_{\AA\times\BB}(X\oplus Y)
=\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
```

\srank_{\bbk[T]}Use in source · 5 passages

Search aliases: srank_bbk[T] srank_𝕜[T] srank bbk operatorname Bbbk

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Use in source · Read passage · 02_foundations.tex, line 901
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
```
\boxed{
\srank_{\bbk[T]}(X)=\max_p\mu_p(T).
}
```

\srank_{\cA}Use in source · 1 passage

Search aliases: srank_cA srank_ A srank cA operatorname mathcal

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Use in source · Read passage · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
```
{
\begin{aligned}
&\text{generated hull: }\Hull_{\cA}(S),\\
&\text{seed complexity: }\srank_{\cA}(X),\\
&\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\
&\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\
&\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\
&\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\
&\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\
&\text{realization: }\Phi A=\varphi(A)\Phi.
\end{aligned}
}
```

\srank_{\kk I}Use in source · 1 passage

Search aliases: srank_kk I srank_𝕜 I srank kk operatorname Bbbk

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Use in source · visual_atlas.tex, line 304
The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.
Exact mathematical passage
```
\srank_{\kk I}(\kk^2)=2,
\qquad
\srank_{\kk\times\kk}(\kk\oplus\kk)=1.
```

\srank_{\kk\times\kk}Use in source · 1 passage

Search aliases: srank_kk×kk srank_𝕜×𝕜 srank kk times kk operatorname Bbbk times Bbbk

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Use in source · visual_atlas.tex, line 304
The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.
Exact mathematical passage
```
\srank_{\kk I}(\kk^2)=2,
\qquad
\srank_{\kk\times\kk}(\kk\oplus\kk)=1.
```

\srank_{M_2(\QQ)}Use in source · 1 passage

Search aliases: srank_M_2(QQ) srank_M_2( Q) srank QQ operatorname mathbb

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Use in source · visual_atlas.tex, line 342
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
```
\boxed{\srank_{M_2(\QQ)}M_{2\times5}(\QQ)=\lceil5/2\rceil=3.}
```

\srank_{M_d(D)}Use in source · 2 passages

Search aliases: srank_M_d(D) srank operatorname

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Use in source · Read passage · 02_foundations.tex, line 833
[Full matrix algebra] For the natural module S=D^d over M_d(D), srank_M_d(D)(S)=1. More strongly, every nonzero vector of S is cyclic. [source reference]
Exact mathematical passage
```
\srank_{M_d(D)}(S)=1.
```

\srank_{P_\alpha A P_\alpha}Use in source · 1 passage

Search aliases: srank_P_α A P_α srank alpha alpha operatorname alpha alpha

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1037
[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]
Exact mathematical passage
```
\boxed{
\srank_A(X)
\leq
\psrank_{A,\Pcal}(X)
\leq
\sum_{\alpha\in I}
\srank_{P_\alpha A P_\alpha}(X_\alpha).
}
```

\srank_{\QQ[N]}Use in source · 1 passage

Search aliases: srank_QQ[N] srank_ Q[N] srank QQ operatorname mathbb

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Use in source · visual_atlas.tex, line 102
headLower and upper bounds meet If r seeds generated X, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, r≥2. Conversely, the two seeds e_0,f_0 generate their respective chains, so srank_ Q[N](X)=2. This proof uses the displayed operator and quotient. It does not ask a computer to search every possible seed, and it does not need an existence theorem for Jordan form.
Exact mathematical passage
```
\boxed{\srank_{\QQ[N]}(X)=2.}
```

\srank_{R_p}Use in source · 2 passages

Search aliases: srank_R_p srank operatorname

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Use in source · Read passage · 02_foundations.tex, line 927
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
\srank_{\bbk[T]}(X)=\max_p\srank_{R_p}(X_p).
```

\subsetneqUse in source · 1 passage

Search aliases: subsetneq

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Use in source · Read passage · 05_worked_models.tex, line 34
2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated representation is U⊕span{(1,0,0),(0,1,0)} subsetneq U⊕ V. Adding the seed (0,0,1)∈ V generates the full representation.
Exact mathematical passage
```
U\oplus\Span\{(1,0,0),(0,1,0)\}
\subsetneq U\oplus V.
```

\sup_{F,a}Use in source · 1 passage

Search aliases: sup_F,a sup

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Use in source · Read passage · 03_packets_and_graphs.tex, line 443
Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Completeness of the coordinate space is [source reference].
Exact mathematical passage
```
\|\mathcal A_{\Pcal}x\|_u
=
\sup_{F,a}
\left\|\sum_{n\in F}a_nP_nx\right\|
\leq K\|x\|.
```

\mathsf TUse in source · 1 passage

Search aliases: T mathsf

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Use in source · visual_atlas.tex, line 124
headThe distinction is visible in two dimensions Use X= Q^2, seed e_1=(1,0)^ T, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column: I_e_1={mat0 · a
Exact mathematical passage
```
e_1=(1,0)^\mathsf T
```

T^*Use in source · 2 passages

Search aliases: T^*

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 582
[Adjoint reversal] If X is Hilbert, every P_α is orthogonal, and T∈ B(X), then (T_βα)^*=(T^*)_αβ. [source reference]
Exact mathematical passage
```
\boxed{(T_{\beta\alpha})^*=(T^*)_{\alpha\beta}.}
```

T^2Use in source · 3 passages

Search aliases: T^2

Link to this entry ·

Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
=1\cdot2+2\cdot1+0=4.
```

t^2Use in source · 1 passage

Search aliases: t^2

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Use in source · Read passage · 02_foundations.tex, line 263
[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.
Exact mathematical passage
```
s,ts,t^2s
```

t^3Use in source · 5 passages

Search aliases: t^3

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Use in source · Read passage · 02_foundations.tex, line 257
[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.
Exact mathematical passage
```
M=\bbk[t]/(t^3)
```

t^4Use in source · 1 passage

Search aliases: t^4

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Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
```
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
```

t^5Use in source · 1 passage

Search aliases: t^5

Link to this entry ·

Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
```
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
```

T^mUse in source · 1 passage

Search aliases: T^m

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

T^UUse in source · 2 passages

Search aliases: T^U

Link to this entry ·

Use in source · visual_atlas.tex, line 275
vpage11Move the packets with the coordinatesCONJUGACY / STRUCTURE VERSUS RAW SPARSITYMain text: [source reference] Start with T=diag(1,2) and the two coordinate projections. Apply the invertible shear U. The transformed operator is T^U=UTU^-1, and the transformed projectors are Q_i=UP_iU^-1.
Exact mathematical passage
```
T^U=UTU^{-1}
```

T_1Use in source · 1 passage

Search aliases: T_1

Link to this entry ·

Use in source · visual_atlas.tex, line 114
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
\Phi_s:F\to X,\qquad f\mapsto f(T_1,\ldots,T_r)s.
```

T_{11}Use in source · 1 passage

Search aliases: T_11

Link to this entry ·

Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
=1\cdot2+2\cdot1+0=4.
```

T_{12}Use in source · 2 passages

Search aliases: T_12

Link to this entry ·

Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
=1\cdot2+2\cdot1+0=4.
```

T_{13}Use in source · 1 passage

Search aliases: T_13

Link to this entry ·

Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
=1\cdot2+2\cdot1+0=4.
```

T_{22}Use in source · 1 passage

Search aliases: T_22

Link to this entry ·

Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
=1\cdot2+2\cdot1+0=4.
```

T_{23}Use in source · 1 passage

Search aliases: T_23

Link to this entry ·

Use in source · visual_atlas.tex, line 212
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{13}=T_{12}T_{23}=2\cdot3=6.
```

T_{32}Use in source · 1 passage

Search aliases: T_32

Link to this entry ·

Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
```
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
=1\cdot2+2\cdot1+0=4.
```

T_{ij}Use in source · 1 passage

Search aliases: T_ij

Link to this entry ·

Use in source · visual_atlas.tex, line 193
vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.
Exact mathematical passage
```
T_{ij}=P_iTP_j
```

T_rUse in source · 1 passage

Search aliases: T_r

Link to this entry ·

Use in source · visual_atlas.tex, line 114
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
\Phi_s:F\to X,\qquad f\mapsto f(T_1,\ldots,T_r)s.
```

T_{\beta_1\beta_0}Use in source · 1 passage

Search aliases: T_β_1β_0 beta beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

T_{\beta_m\beta_{m-1}}Use in source · 1 passage

Search aliases: T_β_mβ_m-1 beta beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

T_{\delta\gamma}Use in source · 2 passages

Search aliases: T_δγ delta gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 601
[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.
Exact mathematical passage
```
\boxed{
(S\otimes T)_{(\beta,\delta),(\alpha,\gamma)}
=
S_{\beta\alpha}\otimes T_{\delta\gamma}.
}
```

\takeawayUse in source · 1 passage

Search aliases: takeaway

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\topruleUse in source · 1 passage

Search aliases: toprule

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

UUse in source · 57 passages

Search aliases: U

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Use in source · Read passage · 03_packets_and_graphs.tex, line 625
[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
```
U:X\to Y
```

U^{-1}Use in source · 13 passages

Search aliases: U^-1

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Use in source · Read passage · 04_transport.tex, line 10
[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family G is transported to U G U^-1, then its transfer graph and reachability relation are also preserved. [source reference]
Exact mathematical passage
```
Q_\alpha:=UP_\alpha U^{-1}.
```

u_1Use in source · 7 passages

Search aliases: u_1

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Use in source · Read passage · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
```
R=
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
A(u_1,u_2)=(u_1,u_2,0),
\qquad
B(v_1,v_2,v_3)=(v_1,v_2).
```

u_2Use in source · 4 passages

Search aliases: u_2

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Use in source · Read passage · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
```
R=
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
A(u_1,u_2)=(u_1,u_2,0),
\qquad
B(v_1,v_2,v_3)=(v_1,v_2).
```

U_{\gamma\alpha}Use in source · 1 passage

Search aliases: U_γα gamma alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 626
[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
```
T_{\beta\alpha}=P_\beta TP_\alpha,
\qquad
S_{\gamma\delta}=Q_\gamma SQ_\delta,
\qquad
U_{\gamma\alpha}=Q_\gamma UP_\alpha.
```

U_{\gamma\beta}Use in source · 2 passages

Search aliases: U_γβ gamma beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 638
et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
```
\boxed{
\sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha}
=
\sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}.
}
```

U_{\delta\alpha}Use in source · 2 passages

Search aliases: U_δα delta alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 638
et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
```
\boxed{
\sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha}
=
\sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}.
}
```

VUse in source · 32 passages

Search aliases: V

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Use in source · Read passage · 05_worked_models.tex, line 5
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1
Exact mathematical passage
```
U,V
```

V_1Use in source · 2 passages

Search aliases: V_1

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Use in source · visual_atlas.tex, line 338
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
```
A_1V_1+A_2V_2+A_3V_3=W.
```

v_1Use in source · 5 passages

Search aliases: v_1

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Use in source · Read passage · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
```
R=
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
A(u_1,u_2)=(u_1,u_2,0),
\qquad
B(v_1,v_2,v_3)=(v_1,v_2).
```

V_2Use in source · 2 passages

Search aliases: V_2

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Use in source · visual_atlas.tex, line 338
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
```
A_1V_1+A_2V_2+A_3V_3=W.
```

v_2Use in source · 5 passages

Search aliases: v_2

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Use in source · Read passage · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
```
R=
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
A(u_1,u_2)=(u_1,u_2,0),
\qquad
B(v_1,v_2,v_3)=(v_1,v_2).
```

V_3Use in source · 2 passages

Search aliases: V_3

Link to this entry ·

Use in source · visual_atlas.tex, line 338
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
```
A_1V_1+A_2V_2+A_3V_3=W.
```

v_3Use in source · 10 passages

Search aliases: v_3

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Use in source · Read passage · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
```
R=
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
A(u_1,u_2)=(u_1,u_2,0),
\qquad
B(v_1,v_2,v_3)=(v_1,v_2).
```

v_jUse in source · 1 passage

Search aliases: v_j

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Use in source · Read passage · 02_foundations.tex, line 842
The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.
Exact mathematical passage
```
v_j\neq0
```

v_j^{-1}Use in source · 1 passage

Search aliases: v_j^-1

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Use in source · Read passage · 02_foundations.tex, line 842
The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.
Exact mathematical passage
```
w_iv_j^{-1}
```

\vpageUse in source · 1 passage

Search aliases: vpage

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

WUse in source · 11 passages

Search aliases: W

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Use in source · Read passage · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
W_0=\Span_{\bbk}S,
\qquad
W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
```

wUse in source · 9 passages

Search aliases: w

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Use in source · Read passage · 02_foundations.tex, line 93
ambda_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relations in I act as zero because M is a module over Λ_ G. Applying both constructions returns the original vert…
Exact mathematical passage
```
a:v\to w
```

W_0Use in source · 2 passages

Search aliases: W_0

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Use in source · Read passage · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
W_0=\Span_{\bbk}S,
\qquad
W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
```

w_iUse in source · 2 passages

Search aliases: w_i

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Use in source · Read passage · 02_foundations.tex, line 842
The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.
Exact mathematical passage
```
w=(w_i)
```

W_nUse in source · 4 passages

Search aliases: W_n

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Use in source · Read passage · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
W_0=\Span_{\bbk}S,
\qquad
W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
```

W_{n+1}Use in source · 3 passages

Search aliases: W_n+1

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Use in source · Read passage · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
W_0=\Span_{\bbk}S,
\qquad
W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
```

\widehat EUse in source · 5 passages

Search aliases: widehat E widehat

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Use in source · Read passage · 02_foundations.tex, line 343
_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and therefore extends to widehat E. If Hull_ A(S) is dense in X, then j extends uniquely to a topological isomorphism of modules over A widehat E≅ X. [source reference]
Exact mathematical passage
```
\widehat E
```

\widehat jUse in source · 5 passages

Search aliases: widehat j widehat

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Use in source · Read passage · 02_foundations.tex, line 352
that subspace. For a∈ A and a continuous seminorm p on X, continuity of a:X→ X supplies continuous seminorms controlling p(aj(e)), so the induced action on E is continuous and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X, choose a net (e_λ) in E with j(e_λ)→ x. The transported topology makes (e_λ) Cauchy, so it co…
Exact mathematical passage
```
\widehat j:\widehat E\to X
```

\widetilde pUse in source · 2 passages

Search aliases: widetilde p widetilde

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Use in source · Read passage · 02_foundations.tex, line 340
free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and therefore extends to widehat E. If Hull_ A(S) is dense in X, then j…
Exact mathematical passage
```
\widetilde p(e):=p(j(e))
```

\widetilde TUse in source · 3 passages

Search aliases: widetilde T widetilde

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Use in source · Read passage · 04_transport.tex, line 13
[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family G is transported to U G U^-1, then its transfer graph and reachability relation are also preserved. [source reference]
Exact mathematical passage
```
\widetilde T=UTU^{-1}
```

\widetilde\PhiUse in source · 4 passages

Search aliases: widetildeΦ widetilde Phi

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Use in source · Read passage · 04_transport.tex, line 336
[Quotient topology and completed realization] Let Φ:X↠ Y be a continuous linear quotient map between Hausdorff topological vector spaces, and put K=KerΦ. Then the induced map widetildeΦ:X/K⟶ Y is a topological isomorphism. If D⊆ X is a linear subspace whose image Φ(D) is dense in a complete space Y, then the completion of D/(D∩ K) in the topology transported from Y is canonically isomorphic to Y. [source reference]
Exact mathematical passage
```
\widetilde\Phi:X/K\longrightarrow Y
```

x^{(3)}Use in source · 1 passage

Search aliases: x^(3)

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Use in source · visual_atlas.tex, line 451
vfig[.93]shift_tail atlasCaptionA finite window of the actual sequence x[n]=2^-|n|. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation x^(3), not a periodic wraparound.
Exact mathematical passage
```
x^{(3)}
```

x^{(k)}Use in source · 6 passages

Search aliases: x^(k)

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Use in source · Read passage · 03_packets_and_graphs.tex, line 377
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
```
x^{(k)}=(x_n^{(k)})
```

x^{(N)}Use in source · 2 passages

Search aliases: x^(N)

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Use in source · visual_atlas.tex, line 453
This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.
Exact mathematical passage
```
x^{(N)}
```

x^{(\ell)}Use in source · 2 passages

Search aliases: x^(ℓ) ell

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Use in source · Read passage · 03_packets_and_graphs.tex, line 378
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
```
\|x_n^{(k)}-x_n^{(\ell)}\|
\leq
\|x^{(k)}-x^{(\ell)}\|_u,
```

X^*Use in source · 5 passages

Search aliases: X^*

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Use in source · Read passage · 03_packets_and_graphs.tex, line 237
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
```
P_\alpha^*:X^*\to X^*,
\qquad
P_\alpha^*(\varphi)=\varphi\circ P_\alpha,
```

x^4Use in source · 2 passages

Search aliases: x^4

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Use in source · normal_form.tex, line 12
(1.3,0.0) rectangle ++(1.15,-.63); node[] at (1.8750,-0.3150) x; filldraw[fill=black!8,draw=white] (2.6,0.0) rectangle ++(1.15,-.63); node[] at (3.1750,-0.3150) x^2; filldraw[fill=black!8,draw=white] (3.9000000000000004,0.0) rectangle ++(1.15,-.63); node[] at (4.4750,-0.3150) x^3; filldraw[fill=black!8,draw=white] (5.2,0.0) rectangle ++(1.15,-.63); node[] at (5.7750,-0.3150) x^4; node[] at (6.9500,-0.3150) ⋯; filldraw[fill=black!18,draw=white] (0.0,-0.8) rectangle ++(1.15,-.63); node[] at (0.575…
Exact mathematical passage
```
x^4
```

x^iUse in source · 1 passage

Search aliases: x^i

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Use in source · Read passage · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
```
M_x^iM_y^js=x^iy^j,
```

x_0Use in source · 1 passage

Search aliases: x_0

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Use in source · visual_atlas.tex, line 477
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L
Exact mathematical passage
```
z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
```

X_1Use in source · 9 passages

Search aliases: X_1

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Use in source · Read passage · 01_orientation.tex, line 105
[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.
Exact mathematical passage
```
X=X_1\oplus X_2\oplus X_3
```

x_1Use in source · 2 passages

Search aliases: x_1

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1150
[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be 1.
Exact mathematical passage
```
x_1+x_2
```

X_2Use in source · 6 passages

Search aliases: X_2

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Use in source · Read passage · 01_orientation.tex, line 105
[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.
Exact mathematical passage
```
X=X_1\oplus X_2\oplus X_3
```

x_2Use in source · 2 passages

Search aliases: x_2

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1150
[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be 1.
Exact mathematical passage
```
x_1+x_2
```

X_3Use in source · 1 passage

Search aliases: X_3

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Use in source · Read passage · 01_orientation.tex, line 105
[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.
Exact mathematical passage
```
X=X_1\oplus X_2\oplus X_3
```

X_DUse in source · 1 passage

Search aliases: X_D

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1097
a:X_α↠ X_β. Because X_α⊆ A S and A S is invariant under A, surjectivity implies X_β⊆ A S. For any other vertex γ∈ D, choose a directed path inside the strongly connected component from β to γ. Successive surjective edge blocks propagate the full packet space along the path, so X_γ⊆ A S. Hence X_D⊆ A S. Induction over the condensation graph gives X= A S, proving the upper bound.
Exact mathematical passage
```
X_D\subseteq\A S
```

x_iUse in source · 4 passages

Search aliases: x_i

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Use in source · visual_atlas.tex, line 512
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
```
(x_i)
```

x_jUse in source · 3 passages

Search aliases: x_j

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Use in source · Read passage · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
```
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j,
             \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
```

x_n^{(k)}Use in source · 5 passages

Search aliases: x_n^(k)

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Use in source · Read passage · 03_packets_and_graphs.tex, line 377
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
```
x^{(k)}=(x_n^{(k)})
```

x_n^{(\ell)}Use in source · 1 passage

Search aliases: x_n^(ℓ) ell

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Use in source · Read passage · 03_packets_and_graphs.tex, line 378
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
```
\|x_n^{(k)}-x_n^{(\ell)}\|
\leq
\|x^{(k)}-x^{(\ell)}\|_u,
```

X_\alpha^*Use in source · 4 passages

Search aliases: X_α^* alpha

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Use in source · Read passage · 03_packets_and_graphs.tex, line 242
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
```
X_\alpha^*
```

x_\betaUse in source · 1 passage

Search aliases: x_β beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 114
For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.
Exact mathematical passage
```
\mathcal A_{\Pcal}\mathcal S_{\Pcal}(x_\alpha)_\alpha
=
\left(P_\beta\sum_\alpha x_\alpha\right)_\beta
=(x_\beta)_\beta.
```

X_\beta^*Use in source · 1 passage

Search aliases: X_β^* beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 829
Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero forbidden (γ,α) block. This proves necessity. Combining reflexivity and transitivity gives (iii).
Exact mathematical passage
```
g\in X_\beta^*
```

X_\gammaUse in source · 3 passages

Search aliases: X_γ gamma

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Use in source · Read passage · 03_packets_and_graphs.tex, line 829
Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero forbidden (γ,α) block. This proves necessity. Combining reflexivity and transitivity gives (iii).
Exact mathematical passage
```
w\in X_\gamma
```

X_\lambdaUse in source · 1 passage

Search aliases: X_λ lambda

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Use in source · Read passage · 02_foundations.tex, line 987
The irreducibles are p=t-λ. On the primary component for λ, the quotient by (T-λ I)X_λ has dimension equal to the number of Jordan blocks, which is also dimKer(T-λ I).
Exact mathematical passage
```
(T-\lambda I)X_\lambda
```

x_\lambdaUse in source · 3 passages

Search aliases: x_λ lambda

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Use in source · Read passage · 04_transport.tex, line 323
Given y=Φ(x), choose a net x_λ∈Hull_ A(S) converging to x. Then Φ(x_λ)→ y, and naturality places every Φ(x_λ) in Hull_ B(Φ(S)). Hence the latter hull is dense. For packets, one inclusion follows from compatibility. Conversely, if y=Q_jy, choose x with Φ(x)=y. Then y=Q_jΦ(x)=Φ P_σ^-1(j)x.
Exact mathematical passage
```
x_\lambda\in\Hull_{\A}(S)
```

\xrightarrowUse in source · 2 passages

Search aliases: xrightarrow

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Use in source · Read passage · 02_foundations.tex, line 106
[title=A grammar with two vertices] Take two vertices and one arrow a:1→2. A realization is simply a pair of spaces and a linear map X_1xrightarrow X_a X_2. After collapse, |X|=X_1⊕ X_2, the vertex idempotents are the two coordinate projectors, and a is a single block outside the diagonal. This elementary case already contains the viewpoint of packets and transfer blocks used later.
Exact mathematical passage
```
X_1\xrightarrow{\,X_a\,}X_2.
```

y^2Use in source · 8 passages

Search aliases: y^2

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Use in source · Read passage · 05_worked_models.tex, line 62
Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packe…
Exact mathematical passage
```
A=\bbk[x,y]/(y^2-x^3+x).
```

y^3Use in source · 1 passage

Search aliases: y^3

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Use in source · normal_form.tex, line 26
1150) x^2y; filldraw[fill=black!18,draw=white] (3.9000000000000004,-0.8) rectangle ++(1.15,-.63); node[] at (4.4750,-1.1150) x^3y; filldraw[fill=black!18,draw=white] (5.2,-0.8) rectangle ++(1.15,-.63); node[] at (5.7750,-1.1150) x^4y; node[] at (6.9500,-1.1150) ⋯; node[] at (3.2000,0.5200) Independent coefficient slots: a(x) and b(x); node[] at (3.2000,-2.1800) y^2=x^3-x Longrightarrow y^3=(x^3-x)y;
Exact mathematical passage
```
y^2=x^3-x\quad\Longrightarrow\quad y^3=(x^3-x)y
```

y^jUse in source · 1 passage

Search aliases: y^j

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Use in source · Read passage · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
```
M_x^iM_y^js=x^iy^j,
```

y^mUse in source · 1 passage

Search aliases: y^m

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Use in source · Read passage · 05_worked_models.tex, line 70
alize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring. Every class has a unique normal form f(x,y)=a(x)+y b(x). Indeed, y^2-(x^3-x) is monic of degree two in y, so division in 𝕜[x][y] gives a unique remainder of degree less than two in y. Concretely, every power y^m with m≥2 can be reduced using y^2=x^3-x.
Exact mathematical passage
```
y^m
```

y_jUse in source · 4 passages

Search aliases: y_j

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Use in source · visual_atlas.tex, line 512
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
```
(y_j)
```

y_mUse in source · 1 passage

Search aliases: y_m

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Use in source · Read passage · 03_packets_and_graphs.tex, line 460
For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Completeness of the coordinate space is [source reference].
Exact mathematical passage
```
P_m\mathcal S_{\Pcal}y
=
\sum_nP_my_n
=y_m.
```

y_nUse in source · 3 passages

Search aliases: y_n

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Use in source · Read passage · 03_packets_and_graphs.tex, line 450
Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Completeness of the coordinate space is [source reference].
Exact mathematical passage
```
y=(y_n)\in\bigoplus^uP_nX
```

Y_\gammaUse in source · 3 passages

Search aliases: Y_γ gamma

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Use in source · Read passage · 03_packets_and_graphs.tex, line 197
ct packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
```
(P_\alpha\otimes Q_\gamma)(X\otimes Y)
=X_\alpha\otimes Y_\gamma.
```

ZUse in source · 1 passage

Search aliases: Z

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Use in source · Read passage · 04_transport.tex, line 148
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ
Exact mathematical passage
```
(\Psi,\psi,\tau):(Y,\B,\Qcal)\to(Z,\Ccal,\Rcal)
```

\ZUse in source · 7 passages

Search aliases: Z mathbb

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Use in source · Read passage · 05_worked_models.tex, line 196
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
X=\ell^2(\Z;\mathbb F)
```

zUse in source · 4 passages

Search aliases: z

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Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

z_0Use in source · 1 passage

Search aliases: z_0

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Use in source · visual_atlas.tex, line 477
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L
Exact mathematical passage
```
z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
```

z_1Use in source · 2 passages

Search aliases: z_1

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Use in source · visual_atlas.tex, line 113
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
F=\kk\langle z_1,\ldots,z_r\rangle
```

z_2Use in source · 1 passage

Search aliases: z_2

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Use in source · visual_atlas.tex, line 477
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L
Exact mathematical passage
```
z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
```

z_rUse in source · 1 passage

Search aliases: z_r

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Use in source · visual_atlas.tex, line 113
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
F=\kk\langle z_1,\ldots,z_r\rangle
```

\ZZUse in source · 3 passages

Search aliases: ZZ Z mathbb

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Use in source · visual_atlas.tex, line 444
vpage18Density is not finite generation of each vectorINFINITE MODEL / THE BILATERAL SHIFTMain text: [source reference] On ℓ^2( Z), declare every integer shift (S_mx)[n]=x[n-m]. The seed δ_0 generates δ_m=S_mδ_0. Finite linear combinations give precisely the sequences with finite support. Their norm closure is all of ℓ^2( Z).
Exact mathematical passage
```
\ell^2(\ZZ)
```

|_KUse in source · 1 passage

Search aliases: |_K

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Use in source · Read passage · 04_transport.tex, line 197
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
```
P_\alpha|_K
```

|_{X_\alpha}Use in source · 1 passage

Search aliases: |_X_α alpha

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Use in source · Read passage · 03_packets_and_graphs.tex, line 150
[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsA…
Exact mathematical passage
```
f|_{X_\alpha}=f_\alpha
```

\cdotUse in source · 8 passages

Search aliases: · cdot

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Use in source · Read passage · 01_orientation.tex, line 83
[title=R1: one operator with several primary blocks] Let T act on a finite dimensional vector space. Regard X as a 𝕜[t]-module by t· x=Tx. If T has three Jordan blocks for the same eigenvalue, then three seeds are necessary and sufficient for that primary component. The largest number of blocks over any irreducible polynomial is the seed rank.
Exact mathematical passage
```
t\cdot x=Tx
```

\beta_0Use in source · 1 passage

Search aliases: β_0 beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

\beta_1Use in source · 1 passage

Search aliases: β_1 beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

\beta_mUse in source · 1 passage

Search aliases: β_m beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

\beta_{m-1}Use in source · 1 passage

Search aliases: β_m-1 beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

\gammaUse in source · 37 passages

Search aliases: γ gamma

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Use in source · Read passage · 03_packets_and_graphs.tex, line 190
[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
```
\Qcal=(Q_\gamma)_{\gamma\in J}
```

\gamma_1Use in source · 2 passages

Search aliases: γ_1 gamma

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Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

\gamma_{m-1}Use in source · 2 passages

Search aliases: γ_m-1 gamma

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Use in source · Read passage · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
P_\beta G_mP_{\gamma_{m-1}}
G_{m-1}\cdots
P_{\gamma_1}G_1P_\alpha.
```

\DeltaUse in source · 4 passages

Search aliases: Δ Delta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 757
[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invariance and packet leakage decompositions.
Exact mathematical passage
```
\Delta_{\Pcal}(T):=\sum_{\alpha}P_\alpha TP_\alpha.
```

\deltaUse in source · 18 passages

Search aliases: δ delta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 113
For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.
Exact mathematical passage
```
P_\beta x_\alpha=\delta_{\alpha\beta}x_\alpha
```

\delta_0Use in source · 6 passages

Search aliases: δ_0 delta

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Use in source · Read passage · 05_worked_models.tex, line 205
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
\delta_0
```

\delta_mUse in source · 2 passages

Search aliases: δ_m delta

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 206
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
S_m\delta_0=\delta_m
```

\delta_{nm}Use in source · 1 passage

Search aliases: δ_nm delta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 410
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
```
P_nP_m=\delta_{nm}P_n
```

\Delta_{\Pcal}Use in source · 4 passages

Search aliases: Δ_Pcal Δ_ P Delta Pcal Delta mathcal

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Use in source · Read passage · 03_packets_and_graphs.tex, line 757
[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invariance and packet leakage decompositions.
Exact mathematical passage
```
\Delta_{\Pcal}(T):=\sum_{\alpha}P_\alpha TP_\alpha.
```

\Delta_{\Pcal}^2Use in source · 1 passage

Search aliases: Δ_Pcal^2 Δ_ P^2 Delta Pcal Delta mathcal

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Use in source · Read passage · 03_packets_and_graphs.tex, line 772
Pairwise annihilation gives Δ_ P^2=Δ_ P and identifies its range with diagonal block families. If T commutes with all P_α, then for β≠α, P_β TP_α=P_β P_α T=0. Thus (i) implies (ii). Exact reconstruction shows (ii) implies (iii). If T=∑_α P_α TP_α, then TP_γ=P_γ TP_γ=P_γ T, so (iii) implies (i).
Exact mathematical passage
```
\Delta_{\Pcal}^2=\Delta_{\Pcal}
```

\delta_{\alpha\beta}Use in source · 3 passages

Search aliases: δ_αβ delta alpha beta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 113
For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.
Exact mathematical passage
```
P_\beta x_\alpha=\delta_{\alpha\beta}x_\alpha
```

\varepsilon^2Use in source · 1 passage

Search aliases: ε^2 varepsilon

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Use in source · Read passage · 02_foundations.tex, line 705
[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.
Exact mathematical passage
```
\varepsilon^2M
```

\varepsilon^3Use in source · 1 passage

Search aliases: ε^3 varepsilon

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Use in source · Read passage · 02_foundations.tex, line 701
[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.
Exact mathematical passage
```
A=\bbk[\varepsilon]/(\varepsilon^3)
```

\varepsilon_sUse in source · 1 passage

Search aliases: ε_s varepsilon

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Use in source · Read passage · 02_foundations.tex, line 258
[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.
Exact mathematical passage
```
\varepsilon_s:A\to M,
\qquad
f(t)\mapsto f(t)[1]
```

\zetaUse in source · 5 passages

Search aliases: ζ zeta

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Use in source · Read passage · 05_worked_models.tex, line 383
ery n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are ind…
Exact mathematical passage
```
\zeta
```

\zeta^{-1}Use in source · 3 passages

Search aliases: ζ^-1 zeta

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Use in source · Read passage · 05_worked_models.tex, line 388
[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar quotient of dimension two, and [source reference] gives the m…
Exact mathematical passage
```
q(Ax)=(\zeta+\zeta^{-1})q(x)
```

\zeta^{-j}Use in source · 1 passage

Search aliases: ζ^-j zeta

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Use in source · Read passage · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
```
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j,
             \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
```

\zeta^jUse in source · 1 passage

Search aliases: ζ^j zeta

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Use in source · Read passage · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
```
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j,
             \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
```

\ThetaUse in source · 2 passages

Search aliases: Θ Theta

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Use in source · Read passage · 03_packets_and_graphs.tex, line 678
[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.
Exact mathematical passage
```
\Theta_{\Pcal}:\End_{\bbk}(X)
\longrightarrow
\prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta),
\qquad
T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
```

\Theta_{\Pcal}Use in source · 2 passages

Search aliases: Θ_Pcal Θ_ P Theta Pcal Theta mathcal

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 678
[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.
Exact mathematical passage
```
\Theta_{\Pcal}:\End_{\bbk}(X)
\longrightarrow
\prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta),
\qquad
T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
```

\iotaUse in source · 3 passages

Search aliases: ι iota

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Use in source · Read passage · 02_foundations.tex, line 478
[Short exact sequence bounds] Let 0⟶ LxrightarrowιMxrightarrowπN⟶0 be a short exact sequence of left modules over A. Then srank_A(N)≤srank_A(M) ≤srank_A(L)+srank_A(N). [source reference]
Exact mathematical passage
```
0\longrightarrow L\xrightarrow{\iota}M\xrightarrow{\pi}N\longrightarrow0
```

\lambdaUse in source · 38 passages

Search aliases: λ lambda

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Use in source · Read passage · 02_foundations.tex, line 136
The action map ρ:Λ_ G↠ A_X is surjective by definition and satisfies λ s=ρ(λ)s. Hence every grammar orbit vector lies in A_XS. The reverse inclusion holds because every element of A_X is ρ(λ) for some λ.
Exact mathematical passage
```
\lambda s=\rho(\lambda)s
```

\lambda_*Use in source · 3 passages

Search aliases: λ_* lambda

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Use in source · Read passage · 05_worked_models.tex, line 292
r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
```
m_{\lambda_*}=R
```

\lambda_kUse in source · 2 passages

Search aliases: λ_k lambda

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Use in source · Read passage · 05_worked_models.tex, line 340
For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A…
Exact mathematical passage
```
\lambda_k=2\cos\frac{k\pi}{n+1},
\qquad 1\le k\le n,
```

\muUse in source · 7 passages

Search aliases: μ mu

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Use in source · Read passage · 02_foundations.tex, line 896
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
```
\mu_p(T):=
\dim_{\bbk[t]/(p)}\bigl(X_p/p(T)X_p\bigr).
```

\mu_pUse in source · 6 passages

Search aliases: μ_p mu

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Use in source · Read passage · 02_foundations.tex, line 896
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
```
\mu_p(T):=
\dim_{\bbk[t]/(p)}\bigl(X_p/p(T)X_p\bigr).
```

\xiUse in source · 5 passages

Search aliases: ξ xi

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Use in source · Read passage · 02_foundations.tex, line 352
us and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X, choose a net (e_λ) in E with j(e_λ)→ x. The transported topology makes (e_λ) Cauchy, so it converges to some ξ∈widehat E, and continuity gives widehat j(ξ)=x. Thus widehat j is surjective. If widehat j(ξ)=0, represent ξ by a Cauchy net (e_λ). Then j(e_λ)→0, so e…
Exact mathematical passage
```
\xi\in\widehat E
```

\piUse in source · 10 passages

Search aliases: π pi

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Use in source · Read passage · 02_foundations.tex, line 478
[Short exact sequence bounds] Let 0⟶ LxrightarrowιMxrightarrowπN⟶0 be a short exact sequence of left modules over A. Then srank_A(N)≤srank_A(M) ≤srank_A(L)+srank_A(N). [source reference]
Exact mathematical passage
```
0\longrightarrow L\xrightarrow{\iota}M\xrightarrow{\pi}N\longrightarrow0
```

\rhoUse in source · 6 passages

Search aliases: ρ rho

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Use in source · Read passage · 02_foundations.tex, line 136
The action map ρ:Λ_ G↠ A_X is surjective by definition and satisfies λ s=ρ(λ)s. Hence every grammar orbit vector lies in A_XS. The reverse inclusion holds because every element of A_X is ρ(λ) for some λ.
Exact mathematical passage
```
\rho:\Lambda_{\mathfrak G}\twoheadrightarrow\A_X
```

\tau^{-1}Use in source · 1 passage

Search aliases: τ^-1 tau

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
```
R_k\Psi\Phi
&=\Psi Q_{\tau^{-1}(k)}\Phi\\
&=\Psi\Phi
\sum_{\tau(\sigma(\alpha))=k}P_\alpha.
```

\tau_MUse in source · 1 passage

Search aliases: τ_M tau

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
```
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
```

\tau_NUse in source · 1 passage

Search aliases: τ_N tau

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Use in source · Read passage · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
```
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
```

\Phi_sUse in source · 2 passages

Search aliases: Φ_s Phi

Link to this entry ·

Use in source · visual_atlas.tex, line 114
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
```
\Phi_s:F\to X,\qquad f\mapsto f(T_1,\ldots,T_r)s.
```

\chiUse in source · 1 passage

Search aliases: χ chi

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Use in source · Read passage · 02_foundations.tex, line 641
[Scalar quotient witness] Let M be a module over A with coefficient field 𝕜. Suppose M has a quotient N of dimension r on which A acts through a character χ:A⟶𝕜. Then srank_A(M)≥ r. [source reference]
Exact mathematical passage
```
\chi:A\longrightarrow\bbk.
```

\PsiUse in source · 5 passages

Search aliases: Ψ Psi

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Use in source · Read passage · 04_transport.tex, line 148
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ
Exact mathematical passage
```
(\Psi,\psi,\tau):(Y,\B,\Qcal)\to(Z,\Ccal,\Rcal)
```

\psiUse in source · 2 passages

Search aliases: ψ psi

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Use in source · Read passage · 04_transport.tex, line 148
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ
Exact mathematical passage
```
(\Psi,\psi,\tau):(Y,\B,\Qcal)\to(Z,\Ccal,\Rcal)
```

\ellUse in source · 14 passages

Search aliases: ℓ ell

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Use in source · Read passage · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
```
(\ell_i)_{i\in I}
```

\ell^2Use in source · 9 passages

Search aliases: ℓ^2 ell

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Use in source · Read passage · 03_packets_and_graphs.tex, line 330
The quantity κ( P) measures how strongly the chosen packet coordinates can amplify perturbations. An orthogonal Hilbert decomposition has the best possible behavior in the natural ℓ^2 sum norm. An oblique packet system can reconstruct every vector exactly and still have a large condition number.
Exact mathematical passage
```
\ell^2
```

\ell_iUse in source · 2 passages

Search aliases: ℓ_i ell

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Use in source · Read passage · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
```
(\ell_i)_{i\in I}
```

\leftrightarrowUse in source · 2 passages

Search aliases: ↔ leftrightarrow

Link to this entry ·

Use in source · Read passage · 01_orientation.tex, line 48
How are vectors and operators resolved? · Complete idempotents (P_α) · X≅⊕_α P_α X and T↔(P_β TP_α).
Exact mathematical passage
```
T\leftrightarrow(P_\beta TP_\alpha)
```

\RightarrowUse in source · 1 passage

Search aliases: ⇒ Rightarrow

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Use in source · Read passage · 05_worked_models.tex, line 65
Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packe…
Exact mathematical passage
```
2\ne0\Rightarrow64\ne0
```

\LeftrightarrowUse in source · 1 passage

Search aliases: ⇔ Leftrightarrow

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Use in source · Read passage · 02_foundations.tex, line 1006
ii) holds exactly when every μ_p(T) belonging to a nonzero primary component equals one, which is (iii). Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv). In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)⇔(v). Scalar extension preserves the equality of minimal and characteristic polynomials. Over an algebraic …
Exact mathematical passage
```
\Leftrightarrow
```

\existsUse in source · 3 passages

Search aliases: ∃ exists

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Use in source · Read passage · 03_packets_and_graphs.tex, line 706
[Propagation of packet support] Let T∈End(X) and x∈ X. Then Supp_ P(Tx) ⊆ {β:∃α∈Supp_ P(x) with T_βα≠0}. Equality holds whenever, for every target β in the set on the right, the active contributions (T_βαP_α x: α∈Supp_ P(x), T_βα≠0) have nonzero sum. In particular, equality holds if at most one active source packet feeds each such β and that source coordinate does not lie in the kernel of the corresponding block. [source reference] ExportsGraph reachability, leakage bounds, and support pruning.
Exact mathematical passage
```
\boxed{
\Supp_{\Pcal}(Tx)
\subseteq
\left\{\beta:\exists\alpha\in\Supp_{\Pcal}(x)
\text{ with }T_{\beta\alpha}\neq0\right\}.
}
```

\varnothingUse in source · 2 passages

Search aliases: ∅ varnothing

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 12
vector space over a complete valued field, together with a unital algebra of continuous linear operators. Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement. Conventions for empty families. For finite families of nonnegative integers we use max∅=0. The height of an empty finite partially ordered set is 0, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for …
Exact mathematical passage
```
\max\varnothing=0
```

\prod_CUse in source · 1 passage

Search aliases: ∏_C prod

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Use in source · Read passage · 03_packets_and_graphs.tex, line 884
Let P_C=∑_α∈ CP_α. Define ρ: M_R⟶∏_CEnd(X_C), ρ(T)=(P_CTP_C)_C. If an intermediate class D contributes to P_CSTP_C, then the preorder contains transfers C→ D and D→ C, forcing C=D in the quotient poset. Hence P_CSTP_C=(P_CSP_C)(P_CTP_C), so ρ is an algebra homomorphism. Every collection of endomorphisms of the X_C defines an allowed operator that is block diagonal with respect to the classes, so ρ is surjective. Its kernel is exactly N_R. The first isomorphism theorem gives the quotient formula.
Exact mathematical passage
```
\rho:\mathfrak M_R\longrightarrow\prod_C\End(X_C),
\qquad
\rho(T)=(P_CTP_C)_C.
```

\prod_{C\in\overline I}Use in source · 1 passage

Search aliases: ∏_C∈overline I prod in overline

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 867
k M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ∏_C∈overline IEnd_𝕜(X_C). [source reference] ExportsFiltered transfer algebras and triangular packet inverses in Paper II.
Exact mathematical passage
```
\boxed{
\mathfrak M_R/\mathfrak N_R
\cong
\prod_{C\in\overline I}\End_{\bbk}(X_C).
}
```

\prod_pUse in source · 1 passage

Search aliases: ∏_p prod

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 920
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
\bbk[T]\cong\bbk[t]/(m_T)
\cong\prod_pR_p,
\qquad
R_p:=\bbk[t]/(p^{e_p}),
```

\prod_{p}Use in source · 1 passage

Search aliases: ∏_p prod

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 916
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
```
m_T(t)=\prod_{p}p(t)^{e_p}.
```

\prod_\alphaUse in source · 1 passage

Search aliases: ∏_α prod alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 160
enumerate item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsAssembly of packetwise operators, channels, corrections, and local transport data.
Exact mathematical passage
```
\Hom(X,Y)\cong\prod_\alpha\Hom(X_\alpha,Y),
\qquad
\Hom(Y,X)\cong\prod_\alpha\Hom(Y,X_\alpha),
```

\prod_{\beta,\alpha\in I}Use in source · 1 passage

Search aliases: ∏_β,α∈ I prod beta alpha in

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 678
[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.
Exact mathematical passage
```
\Theta_{\Pcal}:\End_{\bbk}(X)
\longrightarrow
\prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta),
\qquad
T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
```

\sum_{C\in\mathfrak C_0}Use in source · 2 passages

Search aliases: ∑_C∈ C_0 sum in mathfrak

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 1069
[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport forests and directed source rank calculations.
Exact mathematical passage
```
\boxed{
\psrank_{\A,\Pcal}(X)
=
\sum_{C\in\mathfrak C_0}r_C.
}
```

\sum_iUse in source · 5 passages

Search aliases: ∑_i sum

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 446
Let S generate M. For every n∈ N, choose m∈ M with Φ(m)=n and write m=∑_ia_is_i. Then n=∑_iφ(a_i)Φ(s_i), so Φ(S) generates N.
Exact mathematical passage
```
m=\sum_ia_is_i
```

\sum_{i,j=1}^3Use in source · 1 passage

Search aliases: ∑_i,j=1^3 sum

Link to this entry ·

Use in source · visual_atlas.tex, line 200
headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.
Exact mathematical passage
```
T=\sum_{i,j=1}^3 P_iTP_j.
```

\sum_{i=1}^rUse in source · 2 passages

Search aliases: ∑_i=1^r sum

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 614
Let S={s_1,…,s_r} generate M. Every pure tensor λ⊗ m∈ M_K has an expression m=∑_i=1^r a_is_i, so λ⊗ m = ∑_i=1^r(λ⊗ a_i)(1⊗ s_i). Pure tensors span M_K, hence 1⊗ S generates M_K over A_K. Taking the minimum over S proves the inequality.
Exact mathematical passage
```
m=\sum_{i=1}^r a_is_i,
```

\sum_{i\in I}Use in source · 2 passages

Search aliases: ∑_i∈ I sum in

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
```
W_0=\Span_{\bbk}S,
\qquad
W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
```

\sum_jUse in source · 4 passages

Search aliases: ∑_j sum

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 245
The image consists exactly of finite sums ∑_ja_js_j, which is Hull_A(S). Because ε_S is linear over A, its kernel is a left submodule. The first isomorphism theorem gives A^⊕ r/N_S≅imε_S=Hull_A(S). The final equivalence is the equality imε_S=M.
Exact mathematical passage
```
\sum_ja_js_j
```

\sum_{j=0}^{n-1}Use in source · 1 passage

Search aliases: ∑_j=0^n-1 sum

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
```
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j,
             \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
```

\sum_{j\in F}Use in source · 2 passages

Search aliases: ∑_j∈ F sum in

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 494
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
```
\pi(m)=\sum_{j\in F}a_jn_j.
```

\sum_{j\in J}Use in source · 1 passage

Search aliases: ∑_j∈ J sum in

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 553
[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. It does not claim that Lean extracts a routine that decides the edge list for every coefficient field.
Exact mathematical passage
```
\sum_{j\in J}\sum_{i\in I}B_{ji}=T,
```

\sum_{n>N}Use in source · 1 passage

Search aliases: ∑_n>N sum

Link to this entry ·

Use in source · visual_atlas.tex, line 454
This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.
Exact mathematical passage
```
\norm{x-x^{(N)}}_2^2
=2\sum_{n>N}4^{-n}
=\frac{2}{3}\,4^{-N}\longrightarrow0.
```

\sum_{n\geq1}Use in source · 1 passage

Search aliases: ∑_n≥1 sum geq

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 414
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
```
x=\sum_{n\geq1}P_nx
```

\sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}}Use in source · 2 passages

Search aliases: ∑_substackσ(β)=j σ(α)=i sum substack sigma beta sigma alpha

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 100
[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in addition, the morphism is seeded, then S' generates Y. [source reference] ExportsThe canonical naturality theorem for every later realization.
Exact mathematical passage
```
\boxed{
Q_j\varphi(T)Q_i\Phi
=
\sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}}
\Phi P_\beta TP_\alpha.
}
```

\sum_vUse in source · 1 passage

Search aliases: ∑_v sum

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 90
, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relations in I act as zero because M is a module o…
Exact mathematical passage
```
M=1M=\sum_ve_vM.
```

\sum_{v\in Q_0}Use in source · 1 passage

Search aliases: ∑_v∈ Q_0 sum in

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 85
Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relatio…
Exact mathematical passage
```
1=\sum_{v\in Q_0}e_v
```

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

Search aliases: ∑_α sum alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 757
[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invariance and packet leakage decompositions.
Exact mathematical passage
```
\Delta_{\Pcal}(T):=\sum_{\alpha}P_\alpha TP_\alpha.
```

\sum_{\alpha,\beta}Use in source · 3 passages

Search aliases: ∑_α,β sum alpha beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 514
Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.
Exact mathematical passage
```
T=
\left(\sum_\beta P_\beta\right)
T
\left(\sum_\alpha P_\alpha\right)
=
\sum_{\alpha,\beta}P_\beta TP_\alpha.
```

\sum_{\alpha,\beta\in I}Use in source · 1 passage

Search aliases: ∑_α,β∈ I sum alpha beta in

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 505
[Exact operator reconstruction] Every T∈End_𝕜(X) satisfies T=∑_α,β∈ IT_βα. [source reference]
Exact mathematical passage
```
\boxed{T=\sum_{\alpha,\beta\in I}T_{\beta\alpha}.}
```

\sum_{\alpha,\gamma}Use in source · 1 passage

Search aliases: ∑_α,γ sum alpha gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 223
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
```
\sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma
=
\left(\sum_\alpha P_\alpha\right)
\otimes
\left(\sum_\gamma Q_\gamma\right)
=I_X\otimes I_Y.
```

\sum_{\alpha\in\Supp_{\Pcal}(x)}Use in source · 1 passage

Search aliases: ∑_α∈Supp_Pcal(x) ∑_α∈Supp_ P(x) sum alpha in Supp Pcal sum alpha in operatorname mathcal

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 727
For every target packet β, P_β Tx = ∑_α∈Supp_ P(x)T_βαP_α x. If every block in this sum is zero, then the coordinate with index β vanishes. This proves the inclusion. Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right. The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.
Exact mathematical passage
```
P_\beta Tx
=
\sum_{\alpha\in\Supp_{\Pcal}(x)}T_{\beta\alpha}P_\alpha x.
```

\sum_\betaUse in source · 5 passages

Search aliases: ∑_β sum beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 514
Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.
Exact mathematical passage
```
T=
\left(\sum_\beta P_\beta\right)
T
\left(\sum_\alpha P_\alpha\right)
=
\sum_{\alpha,\beta}P_\beta TP_\alpha.
```

\sum_{\beta_1,\ldots,\beta_{m-1}}Use in source · 1 passage

Search aliases: ∑_β_1,…,β_m-1 sum beta ldots beta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
```
(T^m)_{\beta_m\beta_0}
=
\sum_{\beta_1,\ldots,\beta_{m-1}}
T_{\beta_m\beta_{m-1}}
\cdots
T_{\beta_1\beta_0}.
```

\sum_{\beta\in I}Use in source · 2 passages

Search aliases: ∑_β∈ I sum beta in

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 532
[Packet matrix multiplication] For S,T∈End_𝕜(X), (ST)_γα = ∑_β∈ IS_γβT_βα. [source reference] ExportsAll packet path expansions.
Exact mathematical passage
```
\boxed{
(ST)_{\gamma\alpha}
=
\sum_{\beta\in I}S_{\gamma\beta}T_{\beta\alpha}.
}
```

\sum_\gammaUse in source · 1 passage

Search aliases: ∑_γ sum gamma

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 223
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
```
\sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma
=
\left(\sum_\alpha P_\alpha\right)
\otimes
\left(\sum_\gamma Q_\gamma\right)
=I_X\otimes I_Y.
```

\sum_{\gamma\in I}Use in source · 1 passage

Search aliases: ∑_γ∈ I sum gamma in

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 953
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
```
I=\sum_{\gamma\in I}P_\gamma
```

\sum_\deltaUse in source · 2 passages

Search aliases: ∑_δ sum delta

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 659
Apply Q_γ on the left and P_α on the right. Inserting I_X=∑_β P_β gives Q_γ UTP_α = ∑_β Q_γ UP_β TP_α = ∑_β U_γβT_βα. Inserting I_Y=∑_δ Q_δ gives Q_γ SUP_α = ∑_δ Q_γ SQ_δ UP_α = ∑_δ S_γδU_δα. Thus UT=SU implies every block equation. Conversely, if every block equation holds, then all blocks of UT-SU vanish. Exact operator reconstruction yields UT-SU=0.
Exact mathematical passage
```
I_Y=\sum_\delta Q_\delta
```

\sum_{\delta\in J}Use in source · 1 passage

Search aliases: ∑_δ∈ J sum delta in

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 638
et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
```
\boxed{
\sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha}
=
\sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}.
}
```

\sum_{\tau(\sigma(\alpha))=k}Use in source · 1 passage

Search aliases: ∑_τ(σ(α))=k sum tau sigma alpha

Link to this entry ·

Use in source · Read passage · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
```
R_k\Psi\Phi
&=\Psi Q_{\tau^{-1}(k)}\Phi\\
&=\Psi\Phi
\sum_{\tau(\sigma(\alpha))=k}P_\alpha.
```

\setminusUse in source · 1 passage

Search aliases: ∖ setminus

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 396
Choose such a k≥ k_0. Since x^(k) is unconditionally summable, there is a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.
Exact mathematical passage
```
F\subseteq\mathbb N\setminus F_0
```

\circUse in source · 4 passages

Search aliases: ∘ circ

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 237
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
```
P_\alpha^*:X^*\to X^*,
\qquad
P_\alpha^*(\varphi)=\varphi\circ P_\alpha,
```

\inftyUse in source · 2 passages

Search aliases: ∞ infty

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 388
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
```
\ell\to\infty
```

\capUse in source · 9 passages

Search aliases: ∩ cap

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 835
[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]
Exact mathematical passage
```
\mathfrak M_R^{\mathrm b}:=\mathfrak M_R\cap\mathcal B(X).
```

\simUse in source · 2 passages

Search aliases: ∼ sim

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 850
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
```
\alpha\sim\beta
\quad\Longleftrightarrow\quad
(\beta,\alpha)\in R\text{ and }(\alpha,\beta)\in R.
```

\neUse in source · 12 passages

Search aliases: ≠ ne

Link to this entry ·

Use in source · Read passage · 02_foundations.tex, line 992
[Criterion for a cyclic matrix] For X≠0 of finite dimension, the following are equivalent: [label=(roman*)] item T has a cyclic vector. item srank_𝕜[T](X)=1. item Every nonzero primary component for p is cyclic. item The rational canonical form has one invariant factor. item The minimal polynomial equals the characteristic polynomial. item After scalar extension to an algebraic closure, every eigenvalue has geometric multiplicity one. [source reference]
Exact mathematical passage
```
X\ne0
```

\equivUse in source · 3 passages

Search aliases: ≡ equiv

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 227
For N≥1, define residue projectors (P_N,jx)[n] = x[n], · n≡ jpmod N, 0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.
Exact mathematical passage
```
(P_{N,j}x)[n]
=
\begin{cases}
x[n],&n\equiv j\pmod N,\\0,&\text{otherwise.}\end{cases}
```

\leUse in source · 14 passages

Search aliases: ≤ le

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Use in source · Read passage · 05_worked_models.tex, line 284
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
```
\{b_{\lambda,j}:\lambda\in\Lambda,\ 1\le j\le m_{\lambda}\}
```

\geUse in source · 8 passages

Search aliases: ≥ ge

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Use in source · Read passage · 05_worked_models.tex, line 70
ctive smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring. Every class has a unique normal form f(x,y)=a(x)+y b(x). Indeed, y^2-(x^3-x) is monic of degree two in y, so division in 𝕜[x][y] gives a unique remainder of degree less than two in y. Concretely, every power y^m with m≥2 can be reduced using y^2=x^3-x.
Exact mathematical passage
```
m\ge2
```

\subsetUse in source · 1 passage

Search aliases: ⊂ subset

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Use in source · Read passage · 05_worked_models.tex, line 55
This example shows why a typed grammar is more than a notational convenience. The obstruction lies in the subspace ker B⊂ V, which is unreachable from seeds supported at U, and is visible before any coordinate calculation.
Exact mathematical passage
```
\ker B\subset V
```

\bigoplus^uUse in source · 2 passages

Search aliases: ⊕^u bigoplus

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Use in source · Read passage · 03_packets_and_graphs.tex, line 404
s a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.
Exact mathematical passage
```
x\in\bigoplus^uX_n
```

\bigoplus_{(\alpha,\gamma)\in I\times J}Use in source · 1 passage

Search aliases: ⊕_(α,γ)∈ I× J bigoplus alpha gamma in times

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 202
lpha∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
```
\boxed{
X\otimes Y
\cong
\bigoplus_{(\alpha,\gamma)\in I\times J}
X_\alpha\otimes Y_\gamma.
}
```

\bigoplus_iUse in source · 1 passage

Search aliases: ⊕_i bigoplus

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Use in source · Read passage · 02_foundations.tex, line 786
By [source reference], M is generated by r elements exactly when it is a quotient of A^⊕ r. By [source reference], A^⊕ r≅⊕_iS_i^⊕ rd_i. Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to M exists if and only if m_i≤ rd_i for every i. The smallest such r is the displayed maximum. For a nonsemisimple algebra of finite dimension, [source reference] reduces the calculation to the semisimple module M/Jac(A)M over A/Jac(A).
Exact mathematical passage
```
A^{\oplus r}\cong\bigoplus_iS_i^{\oplus rd_i}.
```

\bigoplus_{j=1}^{d}Use in source · 1 passage

Search aliases: ⊕_j=1^d bigoplus

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Use in source · Read passage · 02_foundations.tex, line 753
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
```
M_d(D)=\bigoplus_{j=1}^{d}M_d(D)e_{jj}.
```

\bigoplus_{j=1}^{r}Use in source · 1 passage

Search aliases: ⊕_j=1^r bigoplus

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

\bigoplus_vUse in source · 1 passage

Search aliases: ⊕_v bigoplus

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Use in source · Read passage · 02_foundations.tex, line 86
Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relatio…
Exact mathematical passage
```
|X|=\bigoplus_v e_v|X|=\bigoplus_vX_v.
```

\bigoplus_{\alpha\in C}Use in source · 1 passage

Search aliases: ⊕_α∈ C bigoplus alpha in

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Use in source · Read passage · 03_packets_and_graphs.tex, line 856
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
```
X_C:=\bigoplus_{\alpha\in C}X_\alpha.
```

\bigoplus_{\alpha\in I_0}Use in source · 1 passage

Search aliases: ⊕_α∈ I_0 bigoplus alpha in

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Use in source · Read passage · 03_packets_and_graphs.tex, line 980
Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]
Exact mathematical passage
```
S\subseteq\bigoplus_{\alpha\in I_0}X_\alpha.
```

\bigoplus_{\beta\in\Reach(I_0)}Use in source · 1 passage

Search aliases: ⊕_β∈Reach(I_0) bigoplus beta in Reach bigoplus beta in operatorname

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 984
Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]
Exact mathematical passage
```
\boxed{
\Hull_{\A}(S)
\subseteq
\bigoplus_{\beta\in\Reach(I_0)}X_\beta.
}
```

\bigoplus_{\sigma(\alpha)=j}Use in source · 1 passage

Search aliases: ⊕_σ(α)=j bigoplus sigma alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 52
[Finite packet aggregation] Let σ:I↠ J be a surjection and define Q_j:=∑_σ(α)=jP_α. Then (Q_j)_j∈ J is a finite packet system and Q_jX=⊕_σ(α)=jX_α. [source reference]
Exact mathematical passage
```
Q_jX=\bigoplus_{\sigma(\alpha)=j}X_\alpha.
```

\otimesUse in source · 27 passages

Search aliases: ⊗ otimes

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Use in source · Read passage · 02_foundations.tex, line 559
Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]
Exact mathematical passage
```
A\otimes_{\bbk}B
```

\otimes_{\bbk}Use in source · 4 passages

Search aliases: ⊗_bbk ⊗_𝕜 otimes bbk otimes Bbbk

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Use in source · Read passage · 02_foundations.tex, line 559
Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]
Exact mathematical passage
```
A\otimes_{\bbk}B
```

\bigcupUse in source · 2 passages

Search aliases: ⋃ bigcup

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1053
Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates X. Its cardinality is at most the displayed sum.
Exact mathematical passage
```
S=\bigcup_\alpha S_\alpha
```

\bigcup_CUse in source · 1 passage

Search aliases: ⋃_C bigcup

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Use in source · Read passage · 03_packets_and_graphs.tex, line 1093
For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in A S. Choose an incoming condensation edge from such a component to D, represented by a graph edge α→β with α already generated and β∈ D. Surjectivity …
Exact mathematical passage
```
S=\bigcup_CS_C
```

\bigcup_\alphaUse in source · 1 passage

Search aliases: ⋃_α bigcup alpha

Link to this entry ·

Use in source · Read passage · 03_packets_and_graphs.tex, line 1053
Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates X. Its cardinality is at most the displayed sum.
Exact mathematical passage
```
S=\bigcup_\alpha S_\alpha
```

\cdotsUse in source · 7 passages

Search aliases: ⋯ cdots

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Use in source · Read passage · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
```
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
\qquad
f_1\mid f_2\mid\cdots\mid f_r,
```

\langleUse in source · 4 passages

Search aliases: ⟨ langle

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\rangleUse in source · 4 passages

Search aliases: ⟩ rangle

Link to this entry ·

Use in source · visual_atlas.tex, line 19

[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…

Exact mathematical passage

4pt]
\textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
\textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
\end{tabularx}

A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.

\takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}

\vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.

{\small
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
\textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
$X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
$S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
$\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
$\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
$P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
$X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
$T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
$F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
$U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
$\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
$\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
\end{tabularx}}

\head{Read products from right to left}
In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.

\head{Where to begin}
For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.

\vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
\[
Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
\qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.

\mathbb{F}Use in source · 1 passage

Search aliases: 𝔽 mathbb

Link to this entry ·

Use in source · Read passage · 05_worked_models.tex, line 195
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
```
\mathbb{F}\in\{\R,\C\}
```

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