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Seeded Two-Axis Calculus

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934 symbol spellings in 2490 mathematical passages across 4 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 934 entries are available below.

+Definition passage · 129 passages

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 366
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_{n+1}\ \subseteq\ \sim_n,
\qquad \sim_0=S\times S.
```

=Definition passage · 441 passages

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 124
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
\Tcat_r:=F(r).
```

ADefinition passage · 178 passages

Search aliases: A

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Definition passage: One-step zoo operator · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 448
[One-step zoo operator] For A⊆ U, define Φ(A):= A ∪ mathscr T[A] ∪ mathscr C[A] ∪ mathscr L[A] ∪ mathscr D[A], where each term denotes all defined outputs of the corresponding constructors applied to inputs from A.
Exact mathematical passage
```
A\subseteq U
```

aDefinition passage · 138 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 905
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
T_1,\ldots,T_a
```

A_rDefinition passage · 13 passages

Search aliases: A_r

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 532
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
A_r:\Tcat_r\to\Tcat_r
```

A_sDefinition passage · 8 passages

Search aliases: A_s

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 536
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{s,r}:
A_rC_{s\to r}
\Longrightarrow
C_{s\to r}A_s
\qquad(r\le s),
```

\mathrm{add}Definition passage · 20 passages

Search aliases: add mathrm

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Definition passage: Additive two-axis system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3635
[Additive two-axis system] An additive two-axis system over a small resolution category R is a functor F: R^op⟶ Cat^add. Thus every fiber F(r) is additive and every coarsening functor C_s→ r is additive.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow \mathbf{Cat}^{\mathrm{add}}.
```

BDefinition passage · 96 passages

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Definition passage: Interaction residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1074
[Interaction residual] The interaction residual of this specified square is Θ=Cofib(Cofib(A→ B)⟶ Cofib(C→ D)). The specified square, including its coherent homotopy, is part of the input.
Exact mathematical passage
```
 \Theta=\Cofib\bigl(\Cofib(A\to B)\longrightarrow
                         \Cofib(C\to D)\bigr).
```

bDefinition passage · 105 passages

Search aliases: b

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Definition passage: Resolution branching number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 226
[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.
Exact mathematical passage
```
b_{r\to s}(X):=|\Lift_{r\to s}(X)|.
```

b_{r\to s}Definition passage · 3 passages

Search aliases: b_r→ s to

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Definition passage: Resolution branching number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 226
[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.
Exact mathematical passage
```
b_{r\to s}(X):=|\Lift_{r\to s}(X)|.
```

CDefinition passage · 148 passages

Search aliases: C

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 128
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
C_{s\to r}:=F(r\le s):\Tcat_s\longrightarrow\Tcat_r.
```

C_1Definition passage · 3 passages

Search aliases: C_1

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 905
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
C_1,\ldots,C_b
```

C_bDefinition passage · 2 passages

Search aliases: C_b

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 905
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
C_1,\ldots,C_b
```

C_jDefinition passage · 8 passages

Search aliases: C_j

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 906
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
(i,j)\in E(W)
\quad\Longleftrightarrow\quad
T_i\text{ occurs to the left of }C_j\text{ in }W.
```

C_{s\to r}Definition passage · 33 passages

Search aliases: C_s→ r to

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 128
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
C_{s\to r}:=F(r\le s):\Tcat_s\longrightarrow\Tcat_r.
```

C_{t\to r}Definition passage · 8 passages

Search aliases: C_t→ r to

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Definition passage: Indistinguishability at resolution r · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 158
[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.
Exact mathematical passage
```
C_{t\to r}(X)\cong C_{t\to r}(Y)
```

C_{t\to s}Definition passage · 6 passages

Search aliases: C_t→ s to

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 548
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{t,r}
=
\bigl(C_{s\to r}\eta^A_{t,s}\bigr)
\circ
\bigl(\eta^A_{s,r}C_{t\to s}\bigr).
```

\CatDefinition passage · 4 passages

Search aliases: Cat mathbf

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 120
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow\Cat.
```

\mathbf{Cat}Definition passage · 8 passages

Search aliases: Cat mathbf

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Definition passage: Additive two-axis system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3635
[Additive two-axis system] An additive two-axis system over a small resolution category R is a functor F: R^op⟶ Cat^add. Thus every fiber F(r) is additive and every coarsening functor C_s→ r is additive.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow \mathbf{Cat}^{\mathrm{add}}.
```

\mathbf{Cat}^{\mathrm{add}}Definition passage · 6 passages

Search aliases: Cat^add mathbf mathrm

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Definition passage: Additive two-axis system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3635
[Additive two-axis system] An additive two-axis system over a small resolution category R is a functor F: R^op⟶ Cat^add. Thus every fiber F(r) is additive and every coarsening functor C_s→ r is additive.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow \mathbf{Cat}^{\mathrm{add}}.
```

\CofibDefinition passage · 30 passages

Search aliases: Cofib operatorname

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Definition passage: Stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 599
[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.
Exact mathematical passage
```
\Delta^A_{s,r}(X)
:=
\Cofib\bigl(\eta^A_{s,r,X}\bigr)
\in\Tcat_r.
```

DDefinition passage · 80 passages

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

\mathbb DDefinition passage · 5 passages

Search aliases: D mathbb

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Definition passage: Seeded two-axis double computad · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3602
class of vertical generating 1-cells, interpreted as resolution generators; item designated generating 2-cells, interpreted as exchange or compatibility cells. Generator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of S in a strict double category D is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the…
Exact mathematical passage
```
\mathbb D
```

D_iDefinition passage · 15 passages

Search aliases: D_i

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 744
th] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
D_i:=\Cofib(f_i),
\qquad
D_{\mathrm{tot}}:=\Cofib(X_0\to X_m).
```

D_{\mathrm{tot}}Definition passage · 10 passages

Search aliases: D_tot mathrm

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 744
th] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
D_i:=\Cofib(f_i),
\qquad
D_{\mathrm{tot}}:=\Cofib(X_0\to X_m).
```

EDefinition passage · 51 passages

Search aliases: E

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 711
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
TC(X)\cong CT'(X)\oplus E_{T,C}(X).
```

\mathcal EDefinition passage · 9 passages

Search aliases: E mathcal

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1022
[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤…
Exact mathematical passage
```
\mathcal E_j
```

\mathcal E_jDefinition passage · 3 passages

Search aliases: E_j mathcal

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1022
[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤…
Exact mathematical passage
```
\mathcal E_j
```

\mathcal E_{j-1}Definition passage · 1 passage

Search aliases: E_j-1 mathcal

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1023
hain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤ b). The …
Exact mathematical passage
```
 C_j:\mathcal E_j\to\mathcal E_{j-1},\qquad
 T_{i,j}:\mathcal E_j\to\mathcal E_j,
```

E_{T,C}Definition passage · 3 passages

Search aliases: E_T,C

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 711
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
TC(X)\cong CT'(X)\oplus E_{T,C}(X).
```

FDefinition passage · 53 passages

Search aliases: F

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 120
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow\Cat.
```

fDefinition passage · 57 passages

Search aliases: f

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

f_1Definition passage · 3 passages

Search aliases: f_1

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

f_2Definition passage · 1 passage

Search aliases: f_2

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

f_iDefinition passage · 17 passages

Search aliases: f_i

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 743
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
f_i
```

f_mDefinition passage · 3 passages

Search aliases: f_m

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

\mathcal GDefinition passage · 6 passages

Search aliases: G mathcal

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Definition passage: Operation orbit of the seeds · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3877
[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact refinement functor has been separately specified. A multivalued lift relation is not silently treated as an exact functor.
Exact mathematical passage
```
\mathcal G_r
```

gDefinition passage · 83 passages

Search aliases: g

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Definition passage: Growth function · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 519
[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.
Exact mathematical passage
```
g_{\Seeds_0}(n):=|\Psi^n(\varnothing)|.
```

\mathcal G_rDefinition passage · 3 passages

Search aliases: G_r mathcal

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Definition passage: Operation orbit of the seeds · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3877
[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact refinement functor has been separately specified. A multivalued lift relation is not silently treated as an exact functor.
Exact mathematical passage
```
\mathcal G_r
```

g_{\Seeds_0}Definition passage · 1 passage

Search aliases: g_Seeds_0 g_Σ_0 Seeds Sigma

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Definition passage: Growth function · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 519
[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.
Exact mathematical passage
```
g_{\Seeds_0}(n):=|\Psi^n(\varnothing)|.
```

HDefinition passage · 76 passages

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Definition passage: Resolution branching number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 230
[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.
Exact mathematical passage
```
H_{r\to s}(X):=\log b_{r\to s}(X)
```

hDefinition passage · 77 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 912
ymbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist, (h,j) · prec(i,j) · · when h>i and both events exist. These relations are acyclic: along a generating relation the t…
Exact mathematical passage
```
(i,k)&\prec(i,j) &&\text{when }k<j\text{ and both events exist},\\
(h,j)&\prec(i,j) &&\text{when }h>i\text{ and both events exist}.
```

H_{r\to s}Definition passage · 1 passage

Search aliases: H_r→ s to

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Definition passage: Resolution branching number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 230
[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.
Exact mathematical passage
```
H_{r\to s}(X):=\log b_{r\to s}(X)
```

IDefinition passage · 116 passages

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Definition passage: Coherent exchange-event diagram · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 962
[Coherent exchange-event diagram] For a finite event poset P, a coherent exchange-event diagram is a functor X:NJ(P)⟶ S into a stable ∞-category. Here N denotes the nerve. Its value X_I is the semantic state after the ideal I of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the compatibility data.
Exact mathematical passage
```
X_I
```

iDefinition passage · 139 passages

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 743
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
f_i
```

\idDefinition passage · 15 passages

Search aliases: id operatorname

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 544
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{r,r}=\id
```

\mathrm{inv}Definition passage · 9 passages

Search aliases: inv mathrm

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Definition passage: Constructor-invariant thick saturation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1498
[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.
Exact mathematical passage
```
q^{\mathrm{inv}}:\mathcal Z\longrightarrow
\mathcal O^{\mathrm{inv}}_{\mathscr M}
:=\mathcal Z/\mathcal J_{\mathscr M}.
```

JDefinition passage · 66 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 916
(h,j) · prec(i,j) · · when h>i and both events exist. These relations are acyclic: along a generating relation the transformation index decreases or the coarsening index increases, with at least one strict change. Let J(P_W) be the poset of order ideals, ordered by inclusion.
Exact mathematical passage
```
J(P_W)
```

\mathcal JDefinition passage · 23 passages

Search aliases: J mathcal

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Definition passage: Constructor-invariant thick saturation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1492
[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.
Exact mathematical passage
```
\mathcal J_{\mathscr M}
:=\operatorname{Thick}\{W(K):W\in\mathscr M,
\ K\in\mathcal K_{\mathscr M}\}.
```

jDefinition passage · 77 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 906
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
(i,j)\in E(W)
\quad\Longleftrightarrow\quad
T_i\text{ occurs to the left of }C_j\text{ in }W.
```

\mathcal J_{\mathscr M}Definition passage · 10 passages

Search aliases: J_mathscr M mathcal mathscr

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Definition passage: Constructor-invariant thick saturation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1492
[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.
Exact mathematical passage
```
\mathcal J_{\mathscr M}
:=\operatorname{Thick}\{W(K):W\in\mathscr M,
\ K\in\mathcal K_{\mathscr M}\}.
```

KDefinition passage · 41 passages

Search aliases: K

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Definition passage: Constructor-invariant thick saturation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1492
[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.
Exact mathematical passage
```
\mathcal J_{\mathscr M}
:=\operatorname{Thick}\{W(K):W\in\mathscr M,
\ K\in\mathcal K_{\mathscr M}\}.
```

\mathcal KDefinition passage · 17 passages

Search aliases: K mathcal

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1255
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal K_{\mathscr M}
:=
\{X\in\mathcal Z:\exists W\in\mathscr M,\;W(X)\simeq0\}.
```

kDefinition passage · 65 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 912
ymbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist, (h,j) · prec(i,j) · · when h>i and both events exist. These relations are acyclic: along a generating relation the t…
Exact mathematical passage
```
(i,k)&\prec(i,j) &&\text{when }k<j\text{ and both events exist},\\
(h,j)&\prec(i,j) &&\text{when }h>i\text{ and both events exist}.
```

\mathcal K_{\mathscr M}Definition passage · 15 passages

Search aliases: K_mathscr M mathcal mathscr

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1255
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal K_{\mathscr M}
:=
\{X\in\mathcal Z:\exists W\in\mathscr M,\;W(X)\simeq0\}.
```

LDefinition passage · 41 passages

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

L_jDefinition passage · 1 passage

Search aliases: L_j

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1022
[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤…
Exact mathematical passage
```
L_j:\mathcal E_j\to\mathcal S_j
```

\LiftDefinition passage · 5 passages

Search aliases: Lift operatorname

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Definition passage: Lift set · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 205
[Lift set] For r≤ s and X∈ T_r, define the set of isomorphism classes of lifts Lift_r→ s(X) := { Y∈Ob( T_s):C_s→ r(Y)≅ X }/≅.
Exact mathematical passage
```
\Lift_{r\to s}(X)
:=
\left\{
Y\in\Ob(\Tcat_s):C_{s\to r}(Y)\cong X
\right\}/\cong.
```

\Lift_{r\to s}Definition passage · 5 passages

Search aliases: Lift_r→ s Lift to operatorname to

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Definition passage: Lift set · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 205
[Lift set] For r≤ s and X∈ T_r, define the set of isomorphism classes of lifts Lift_r→ s(X) := { Y∈Ob( T_s):C_s→ r(Y)≅ X }/≅.
Exact mathematical passage
```
\Lift_{r\to s}(X)
:=
\left\{
Y\in\Ob(\Tcat_s):C_{s\to r}(Y)\cong X
\right\}/\cong.
```

\logDefinition passage · 2 passages

Search aliases: log

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Definition passage: Resolution branching number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 230
[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.
Exact mathematical passage
```
H_{r\to s}(X):=\log b_{r\to s}(X)
```

\LongleftrightarrowDefinition passage · 8 passages

Search aliases: Longleftrightarrow

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 906
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
(i,j)\in E(W)
\quad\Longleftrightarrow\quad
T_i\text{ occurs to the left of }C_j\text{ in }W.
```

\LongrightarrowDefinition passage · 7 passages

Search aliases: Longrightarrow

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 536
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{s,r}:
A_rC_{s\to r}
\Longrightarrow
C_{s\to r}A_s
\qquad(r\le s),
```

MDefinition passage · 191 passages

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1255
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal K_{\mathscr M}
:=
\{X\in\mathcal Z:\exists W\in\mathscr M,\;W(X)\simeq0\}.
```

mDefinition passage · 131 passages

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

\mathscrDefinition passage · 82 passages

Search aliases: mathscr

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

NDefinition passage · 58 passages

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Definition passage: Coherent exchange-event diagram · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 959
[Coherent exchange-event diagram] For a finite event poset P, a coherent exchange-event diagram is a functor X:NJ(P)⟶ S into a stable ∞-category. Here N denotes the nerve. Its value X_I is the semantic state after the ideal I of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the compatibility data.
Exact mathematical passage
```
X:NJ(P)\longrightarrow\mathcal S
```

\mathbb NDefinition passage · 5 passages

Search aliases: N mathbb

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 365
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
n\in\mathbb N=\{0,1,\ldots\}
```

\mathcal NDefinition passage · 19 passages

Search aliases: N mathcal

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1261
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal N_{\mathscr M}
:=
\operatorname{Thick}(\mathcal K_{\mathscr M})
```

nDefinition passage · 340 passages

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 365
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_n
```

\mathcal N_{\mathscr M}Definition passage · 14 passages

Search aliases: N_mathscr M mathcal mathscr

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1261
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal N_{\mathscr M}
:=
\operatorname{Thick}(\mathcal K_{\mathscr M})
```

\mathrm{nf}Definition passage · 3 passages

Search aliases: nf mathrm

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

\mathcal ODefinition passage · 13 passages

Search aliases: O mathcal

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Definition passage: Stable obstruction category and obstruction class · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1273
[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).
Exact mathematical passage
```
\mathcal O_{\mathscr M}
:=
\mathcal Z/\mathcal N_{\mathscr M},
```

oDefinition passage · 16 passages

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 120
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow\Cat.
```

\mathcal O^{\mathrm{inv}}_{\mathscr M}Definition passage · 4 passages

Search aliases: O^inv_mathscr M mathcal mathrm mathscr

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Definition passage: Constructor-invariant thick saturation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1498
[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.
Exact mathematical passage
```
q^{\mathrm{inv}}:\mathcal Z\longrightarrow
\mathcal O^{\mathrm{inv}}_{\mathscr M}
:=\mathcal Z/\mathcal J_{\mathscr M}.
```

\mathcal O_{\mathscr M}Definition passage · 7 passages

Search aliases: O_mathscr M mathcal mathscr

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Definition passage: Stable obstruction category and obstruction class · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1273
[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).
Exact mathematical passage
```
\mathcal O_{\mathscr M}
:=
\mathcal Z/\mathcal N_{\mathscr M},
```

\ObDefinition passage · 3 passages

Search aliases: Ob operatorname

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Definition passage: Resolved object · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 145
[Resolved object] A resolved object is a pair (r,X) with X∈Ob( T_r).
Exact mathematical passage
```
X\in\Ob(\Tcat_r)
```

\ObsDefinition passage · 5 passages

Search aliases: Obs operatorname

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Definition passage: Stable obstruction category and obstruction class · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1283
[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).
Exact mathematical passage
```
\Obs_{\mathscr M}(X):=q(X).
```

\Obs_{\mathscr M}Definition passage · 3 passages

Search aliases: Obs_mathscr M Obs mathscr operatorname mathscr

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Definition passage: Stable obstruction category and obstruction class · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1283
[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).
Exact mathematical passage
```
\Obs_{\mathscr M}(X):=q(X).
```

PDefinition passage · 96 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 911
set Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
P_W
```

pDefinition passage · 561 passages

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 120
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow\Cat.
```

P_WDefinition passage · 11 passages

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 911
set Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
P_W
```

\precDefinition passage · 1 passage

Search aliases: prec

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 912
ymbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist, (h,j) · prec(i,j) · · when h>i and both events exist. These relations are acyclic: along a generating relation the t…
Exact mathematical passage
```
(i,k)&\prec(i,j) &&\text{when }k<j\text{ and both events exist},\\
(h,j)&\prec(i,j) &&\text{when }h>i\text{ and both events exist}.
```

QDefinition passage · 39 passages

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Definition passage: Directed annihilator property · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1346
[Directed annihilator property] The constructor system mathscr M has the directed annihilator property if for every U,V∈mathscr M there exist P,Q,W∈mathscr M and natural equivalences W≃ P∘ U≃ Q∘ V. Thus any two constructor words admit a common postcomposition. With multiplication written as composition, P∘ U=Q∘ V is a common left multiple. The displayed equation, rather than a left/right name, fixes the convention.
Exact mathematical passage
```
P,Q,W\in\mathscr M
```

qDefinition passage · 65 passages

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Definition passage: Stable obstruction category and obstruction class · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1279
[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).
Exact mathematical passage
```
q:\mathcal Z\longrightarrow\mathcal O_{\mathscr M}.
```

q^{\mathrm{inv}}Definition passage · 7 passages

Search aliases: q^inv mathrm

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Definition passage: Constructor-invariant thick saturation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1498
[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.
Exact mathematical passage
```
q^{\mathrm{inv}}:\mathcal Z\longrightarrow
\mathcal O^{\mathrm{inv}}_{\mathscr M}
:=\mathcal Z/\mathcal J_{\mathscr M}.
```

RDefinition passage · 148 passages

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Definition passage: Resolution category · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 113
[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.
Exact mathematical passage
```
(R,\le)
```

rDefinition passage · 214 passages

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Definition passage: Resolution category · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 113
[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.
Exact mathematical passage
```
r\to s
```

r_0Definition passage · 3 passages

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1022
[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤…
Exact mathematical passage
```
r_0\le\cdots\le r_b
```

r_bDefinition passage · 1 passage

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1022
[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤…
Exact mathematical passage
```
r_0\le\cdots\le r_b
```

\RcatDefinition passage · 20 passages

Search aliases: Rcat R mathbf

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Definition passage: Resolution category · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 113
[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.
Exact mathematical passage
```
\Rcat
```

\Rcat^{op}Definition passage · 8 passages

Search aliases: Rcat^op R^op Rcat mathbf

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 120
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow\Cat.
```

SDefinition passage · 61 passages

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 365
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
S
```

\mathbb SDefinition passage · 8 passages

Search aliases: S mathbb

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Definition passage: Seeded two-axis double computad · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3595
[Seeded two-axis double computad] A seeded two-axis double computad S consists of a small double computad together with: item a distinguished set Σ⊆ S_0 of seed objects; item a designated class of horizontal generating 1-cells, interpreted as transformation generators; item a designated class of vertical generating 1-cells, interpreted as resolution generators; item designated generating 2-cells, interpreted as exchange or compatibility cells. Generator boundaries may be well-typed words, but no…
Exact mathematical passage
```
\mathbb S
```

\mathcal SDefinition passage · 17 passages

Search aliases: S mathcal

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 739
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
\mathcal S
```

\mathfrak SDefinition passage · 1 passage

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

sDefinition passage · 148 passages

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Definition passage: Resolution category · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 113
[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.
Exact mathematical passage
```
r\to s
```

\mathbb S_0Definition passage · 1 passage

Search aliases: S_0 mathbb

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Definition passage: Seeded two-axis double computad · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3597
[Seeded two-axis double computad] A seeded two-axis double computad S consists of a small double computad together with: item a distinguished set Σ⊆ S_0 of seed objects; item a designated class of horizontal generating 1-cells, interpreted as transformation generators; item a designated class of vertical generating 1-cells, interpreted as resolution generators; item designated generating 2-cells, interpreted as exchange or compatibility cells. Generator boundaries may be well-typed words, but no…
Exact mathematical passage
```
\Seeds\subseteq \mathbb S_0
```

\mathcal S_jDefinition passage · 1 passage

Search aliases: S_j mathcal

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1022
[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤…
Exact mathematical passage
```
L_j:\mathcal E_j\to\mathcal S_j
```

\SeedsDefinition passage · 20 passages

Search aliases: Seeds Σ Sigma

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

\Seeds_0Definition passage · 13 passages

Search aliases: Seeds_0 Σ_0 Seeds Sigma

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

\Seeds_rDefinition passage · 2 passages

Search aliases: Seeds_r Σ_r Seeds Sigma

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3784
[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\Seeds_r\subseteq\Ob(F(r))
```

\operatorname{St}Definition passage · 16 passages

Search aliases: St operatorname

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Definition passage: Universal stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3732
[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).
Exact mathematical passage
```
X\in\operatorname{St}^{\mathrm{add}}(F(s))
```

\operatorname{St}^{\mathrm{add}}Definition passage · 13 passages

Search aliases: St^add operatorname mathrm

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Definition passage: Universal stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3732
[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).
Exact mathematical passage
```
X\in\operatorname{St}^{\mathrm{add}}(F(s))
```

\operatorname{St}^{\mathrm{add}}_{\Rcat}Definition passage · 4 passages

Search aliases: St^add_Rcat St^add_ R operatorname mathrm Rcat operatorname mathrm mathbf

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3784
[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\operatorname{St}^{\mathrm{add}}_{\Rcat}F
```

\supDefinition passage · 1 passage

Search aliases: sup

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 371
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\nu(X,Y):=\sup\{n\in\mathbb N:X\sim_nY\}\in\mathbb N\cup\{\infty\}.
```

TDefinition passage · 61 passages

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 432
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
\mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
```

tDefinition passage · 56 passages

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Definition passage: Indistinguishability at resolution r · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 158
[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.
Exact mathematical passage
```
C_{t\to r}(X)\cong C_{t\to r}(Y)
```

T'Definition passage · 7 passages

Search aliases: T'

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 710
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
\eta:TC\Rightarrow CT'
```

T_1Definition passage · 2 passages

Search aliases: T_1

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 905
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
T_1,\ldots,T_a
```

T_aDefinition passage · 1 passage

Search aliases: T_a

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 905
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
T_1,\ldots,T_a
```

T_iDefinition passage · 6 passages

Search aliases: T_i

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Definition passage: Exchange event poset · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 906
[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · · when k<j and both events exist,
Exact mathematical passage
```
(i,j)\in E(W)
\quad\Longleftrightarrow\quad
T_i\text{ occurs to the left of }C_j\text{ in }W.
```

T_{i,j}Definition passage · 3 passages

Search aliases: T_i,j

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1023
hain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤ b). The …
Exact mathematical passage
```
 C_j:\mathcal E_j\to\mathcal E_{j-1},\qquad
 T_{i,j}:\mathcal E_j\to\mathcal E_j,
```

T_{i,j-1}Definition passage · 1 passage

Search aliases: T_i,j-1

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1028
or L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤ b). The components of χ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are…
Exact mathematical passage
```
 \chi_{i,j}:T_{i,j-1}C_j\Longrightarrow C_jT_{i,j}
 \qquad(1\le j\le b).
```

\TcatDefinition passage · 25 passages

Search aliases: Tcat T mathbf

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 124
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
\Tcat_r:=F(r).
```

\Tcat_rDefinition passage · 15 passages

Search aliases: Tcat_r T_r Tcat mathbf

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 124
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
\Tcat_r:=F(r).
```

\Tcat_sDefinition passage · 7 passages

Search aliases: Tcat_s T_s Tcat mathbf

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 128
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
C_{s\to r}:=F(r\le s):\Tcat_s\longrightarrow\Tcat_r.
```

\mathrm{thick}Definition passage · 3 passages

Search aliases: thick mathrm

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Definition passage: Operation orbit of the seeds · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3878
[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact refinement functor has been separately specified. A multivalued lift relation is not silently treated as an exact functor.
Exact mathematical passage
```
\mathcal Z_r^{\mathrm{thick}}:=\operatorname{Thick}(\mathcal G_r).
```

\operatorname{Thick}Definition passage · 12 passages

Search aliases: Thick operatorname

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1261
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal N_{\mathscr M}
:=
\operatorname{Thick}(\mathcal K_{\mathscr M})
```

\mathrm{tot}Definition passage · 10 passages

Search aliases: tot mathrm

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 744
th] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
D_i:=\Cofib(f_i),
\qquad
D_{\mathrm{tot}}:=\Cofib(X_0\to X_m).
```

UDefinition passage · 71 passages

Search aliases: U

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Definition passage: Seed signature · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 431
[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refinement classes; item mathscr D is a collection of partial residual constructors.
Exact mathematical passage
```
U
```

VDefinition passage · 56 passages

Search aliases: V

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Definition passage: Directed annihilator property · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1346
[Directed annihilator property] The constructor system mathscr M has the directed annihilator property if for every U,V∈mathscr M there exist P,Q,W∈mathscr M and natural equivalences W≃ P∘ U≃ Q∘ V. Thus any two constructor words admit a common postcomposition. With multiplication written as composition, P∘ U=Q∘ V is a common left multiple. The displayed equation, rather than a left/right name, fixes the convention.
Exact mathematical passage
```
U,V\in\mathscr M
```

WDefinition passage · 44 passages

Search aliases: W

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Definition passage: Inversion number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 678
[Inversion number] For a word W, let ι(W) be the number of ordered pairs (T,C) in which a transformation symbol occurs to the left of a resolution symbol when the chosen normal form requires all resolution symbols first.
Exact mathematical passage
```
W
```

W_{\mathrm{nf}}Definition passage · 3 passages

Search aliases: W_nf mathrm

Link to this entry ·

Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

\widehat ADefinition passage · 13 passages

Search aliases: widehat A widehat

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3791
[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\widehat A_r(\mathcal Z_r)\subseteq\mathcal Z_r
```

\widehat A_rDefinition passage · 4 passages

Search aliases: widehat A_r widehat

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3791
[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\widehat A_r(\mathcal Z_r)\subseteq\mathcal Z_r
```

\widehat CDefinition passage · 4 passages

Search aliases: widehat C widehat

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3792
phstable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\widehat C_{s\to r}(\mathcal Z_s)\subseteq\mathcal Z_r
```

\widehat C_{s\to r}Definition passage · 4 passages

Search aliases: widehat C_s→ r widehat to

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3792
phstable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\widehat C_{s\to r}(\mathcal Z_s)\subseteq\mathcal Z_r
```

\widehat\etaDefinition passage · 4 passages

Search aliases: widehatη widehat eta

Link to this entry ·

Definition passage: Universal stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3733
[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).
Exact mathematical passage
```
\Delta^A_{s,r}(X)
:=
\Cofib\!\left(
\widehat\eta^A_{s,r,X}
\right)
\in
\operatorname{St}^{\mathrm{add}}(F(r)).
```

\widehat\eta^A_{s,r,X}Definition passage · 3 passages

Search aliases: widehatη^A_s,r,X widehat eta

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Definition passage: Universal stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3733
[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).
Exact mathematical passage
```
\Delta^A_{s,r}(X)
:=
\Cofib\!\left(
\widehat\eta^A_{s,r,X}
\right)
\in
\operatorname{St}^{\mathrm{add}}(F(r)).
```

XDefinition passage · 185 passages

Search aliases: X

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Definition passage: Resolved object · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 145
[Resolved object] A resolved object is a pair (r,X) with X∈Ob( T_r).
Exact mathematical passage
```
(r,X)
```

X_0Definition passage · 9 passages

Search aliases: X_0

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

X_1Definition passage · 4 passages

Search aliases: X_1

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

X_IDefinition passage · 7 passages

Search aliases: X_I

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Definition passage: Coherent exchange-event diagram · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 962
[Coherent exchange-event diagram] For a finite event poset P, a coherent exchange-event diagram is a functor X:NJ(P)⟶ S into a stable ∞-category. Here N denotes the nerve. Its value X_I is the semantic state after the ideal I of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the compatibility data.
Exact mathematical passage
```
X_I
```

X_mDefinition passage · 5 passages

Search aliases: X_m

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

\xrightarrowDefinition passage · 29 passages

Search aliases: xrightarrow

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

YDefinition passage · 61 passages

Search aliases: Y

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Definition passage: Indistinguishability at resolution r · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 154
[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.
Exact mathematical passage
```
X\sim_r Y
```

\mathcal ZDefinition passage · 42 passages

Search aliases: Z mathcal

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1255
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal K_{\mathscr M}
:=
\{X\in\mathcal Z:\exists W\in\mathscr M,\;W(X)\simeq0\}.
```

\mathcal Z_rDefinition passage · 8 passages

Search aliases: Z_r mathcal

Link to this entry ·

Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3785
[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\mathcal Z_r\subseteq\operatorname{St}^{\mathrm{add}}(F(r))
```

\mathcal Z_r^{\mathrm{thick}}Definition passage · 2 passages

Search aliases: Z_r^thick mathcal mathrm

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Definition passage: Operation orbit of the seeds · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3878
[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact refinement functor has been separately specified. A multivalued lift relation is not silently treated as an exact functor.
Exact mathematical passage
```
\mathcal Z_r^{\mathrm{thick}}:=\operatorname{Thick}(\mathcal G_r).
```

\mathcal Z_sDefinition passage · 2 passages

Search aliases: Z_s mathcal

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Definition passage: Stable seeded zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3792
phstable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.
Exact mathematical passage
```
\widehat C_{s\to r}(\mathcal Z_s)\subseteq\mathcal Z_r
```

\ZooDefinition passage · 8 passages

Search aliases: Zoo Z mathcal

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Definition passage: Generated zoo · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 485
[Generated zoo] The least fixed point in [source reference] is denoted Z(Σ_0).
Exact mathematical passage
```
\Zoo(\Seeds_0).
```

|Definition passage · 25 passages

Search aliases: |

Link to this entry ·

Definition passage: Resolution branching number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 226
[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.
Exact mathematical passage
```
b_{r\to s}(X):=|\Lift_{r\to s}(X)|.
```

\timesDefinition passage · 20 passages

Search aliases: × times

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 366
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_{n+1}\ \subseteq\ \sim_n,
\qquad \sim_0=S\times S.
```

\DeltaDefinition passage · 38 passages

Search aliases: Δ Delta

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Definition passage: Stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 599
[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.
Exact mathematical passage
```
\Delta^A_{s,r}(X)
:=
\Cofib\bigl(\eta^A_{s,r,X}\bigr)
\in\Tcat_r.
```

\deltaDefinition passage · 25 passages

Search aliases: δ delta

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Definition passage: First distinguishing scale · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 416
[First distinguishing scale] When finite, define δ(X,Y):=ν(X,Y)+1. This is the first resolution at which X and Y become distinguishable.
Exact mathematical passage
```
\delta(X,Y):=\nu(X,Y)+1.
```

\Delta^A_{s,r}Definition passage · 18 passages

Search aliases: Δ^A_s,r Delta

Link to this entry ·

Definition passage: Stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 599
[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.
Exact mathematical passage
```
\Delta^A_{s,r}(X)
:=
\Cofib\bigl(\eta^A_{s,r,X}\bigr)
\in\Tcat_r.
```

\Delta_{T,C}Definition passage · 2 passages

Search aliases: Δ_T,C Delta

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 715
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
\Cofib(\eta_X)\simeq E_{T,C}(X)[1],
\qquad E_{T,C}(X)\simeq\Delta_{T,C}(X)[-1].
```

\etaDefinition passage · 26 passages

Search aliases: η eta

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 536
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{s,r}:
A_rC_{s\to r}
\Longrightarrow
C_{s\to r}A_s
\qquad(r\le s),
```

\eta^A_{r,r}Definition passage · 2 passages

Search aliases: η^A_r,r eta

Link to this entry ·

Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 544
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{r,r}=\id
```

\eta^A_{s,r}Definition passage · 6 passages

Search aliases: η^A_s,r eta

Link to this entry ·

Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 536
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{s,r}:
A_rC_{s\to r}
\Longrightarrow
C_{s\to r}A_s
\qquad(r\le s),
```

\eta^A_{s,r,X}Definition passage · 5 passages

Search aliases: η^A_s,r,X eta

Link to this entry ·

Definition passage: Stable mixed residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 599
[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.
Exact mathematical passage
```
\Delta^A_{s,r}(X)
:=
\Cofib\bigl(\eta^A_{s,r,X}\bigr)
\in\Tcat_r.
```

\eta^A_{t,r}Definition passage · 1 passage

Search aliases: η^A_t,r eta

Link to this entry ·

Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 548
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{t,r}
=
\bigl(C_{s\to r}\eta^A_{t,s}\bigr)
\circ
\bigl(\eta^A_{s,r}C_{t\to s}\bigr).
```

\eta^A_{t,s}Definition passage · 1 passage

Search aliases: η^A_t,s eta

Link to this entry ·

Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 548
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{t,r}
=
\bigl(C_{s\to r}\eta^A_{t,s}\bigr)
\circ
\bigl(\eta^A_{s,r}C_{t\to s}\bigr).
```

\eta_XDefinition passage · 2 passages

Search aliases: η_X eta

Link to this entry ·

Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 715
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
\Cofib(\eta_X)\simeq E_{T,C}(X)[1],
\qquad E_{T,C}(X)\simeq\Delta_{T,C}(X)[-1].
```

\ThetaDefinition passage · 7 passages

Search aliases: Θ Theta

Link to this entry ·

Definition passage: Interaction residual · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1074
[Interaction residual] The interaction residual of this specified square is Θ=Cofib(Cofib(A→ B)⟶ Cofib(C→ D)). The specified square, including its coherent homotopy, is part of the input.
Exact mathematical passage
```
 \Theta=\Cofib\bigl(\Cofib(A\to B)\longrightarrow
                         \Cofib(C\to D)\bigr).
```

\iotaDefinition passage · 4 passages

Search aliases: ι iota

Link to this entry ·

Definition passage: Inversion number · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 678
[Inversion number] For a word W, let ι(W) be the number of ordered pairs (T,C) in which a transformation symbol occurs to the left of a resolution symbol when the chosen normal form requires all resolution symbols first.
Exact mathematical passage
```
\iota(W)
```

\nuDefinition passage · 7 passages

Search aliases: ν nu

Link to this entry ·

Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 371
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\nu(X,Y):=\sup\{n\in\mathbb N:X\sim_nY\}\in\mathbb N\cup\{\infty\}.
```

\PhiDefinition passage · 13 passages

Search aliases: Φ Phi

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Definition passage: One-step zoo operator · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 449
[One-step zoo operator] For A⊆ U, define Φ(A):= A ∪ mathscr T[A] ∪ mathscr C[A] ∪ mathscr L[A] ∪ mathscr D[A], where each term denotes all defined outputs of the corresponding constructors applied to inputs from A.
Exact mathematical passage
```
\Phi(A):=
A
\cup \mathscr T[A]
\cup \mathscr C[A]
\cup \mathscr L[A]
\cup \mathscr D[A],
```

\chiDefinition passage · 8 passages

Search aliases: χ chi

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1028
or L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤ b). The components of χ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are…
Exact mathematical passage
```
 \chi_{i,j}:T_{i,j-1}C_j\Longrightarrow C_jT_{i,j}
 \qquad(1\le j\le b).
```

\chi_{i,j}Definition passage · 1 passage

Search aliases: χ_i,j chi

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Definition passage: Strict chain-natural realization · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1028
or L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_jT_i,j (1≤ j≤ b). The components of χ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are…
Exact mathematical passage
```
 \chi_{i,j}:T_{i,j-1}C_j\Longrightarrow C_jT_{i,j}
 \qquad(1\le j\le b).
```

\PsiDefinition passage · 6 passages

Search aliases: Ψ Psi

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Definition passage: Growth function · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 519
[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.
Exact mathematical passage
```
g_{\Seeds_0}(n):=|\Psi^n(\varnothing)|.
```

\Psi^nDefinition passage · 2 passages

Search aliases: Ψ^n Psi

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Definition passage: Growth function · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 519
[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.
Exact mathematical passage
```
g_{\Seeds_0}(n):=|\Psi^n(\varnothing)|.
```

\ldotsDefinition passage · 29 passages

Search aliases: … ldots

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 365
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
n\in\mathbb N=\{0,1,\ldots\}
```

\toDefinition passage · 146 passages

Search aliases: → to

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Definition passage: Resolution category · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 113
[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.
Exact mathematical passage
```
r\to s
```

\RightarrowDefinition passage · 10 passages

Search aliases: ⇒ Rightarrow

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 710
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
\eta:TC\Rightarrow CT'
```

\existsDefinition passage · 3 passages

Search aliases: ∃ exists

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Definition passage: Operationally killable subcategory · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1255
[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.
Exact mathematical passage
```
\mathcal K_{\mathscr M}
:=
\{X\in\mathcal Z:\exists W\in\mathscr M,\;W(X)\simeq0\}.
```

\varnothingDefinition passage · 15 passages

Search aliases: ∅ varnothing

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Definition passage: Growth function · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 519
[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.
Exact mathematical passage
```
g_{\Seeds_0}(n):=|\Psi^n(\varnothing)|.
```

\inDefinition passage · 156 passages

Search aliases: ∈ in

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 123
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
r\in R
```

\circDefinition passage · 14 passages

Search aliases: ∘ circ

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Definition passage: Lax resolution-compatible transformation · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 548
[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s). If every η^A_s,r is a natural isomorphism, A is called invertibly resolution-compatible. This is distinct from exactness of a functor between stable…
Exact mathematical passage
```
\eta^A_{t,r}
=
\bigl(C_{s\to r}\eta^A_{t,s}\bigr)
\circ
\bigl(\eta^A_{s,r}C_{t\to s}\bigr).
```

\inftyDefinition passage · 22 passages

Search aliases: ∞ infty

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 371
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\nu(X,Y):=\sup\{n\in\mathbb N:X\sim_nY\}\in\mathbb N\cup\{\infty\}.
```

\cupDefinition passage · 14 passages

Search aliases: ∪ cup

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 371
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\nu(X,Y):=\sup\{n\in\mathbb N:X\sim_nY\}\in\mathbb N\cup\{\infty\}.
```

\simDefinition passage · 21 passages

Search aliases: ∼ sim

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Definition passage: Indistinguishability at resolution r · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 154
[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.
Exact mathematical passage
```
X\sim_r Y
```

\sim_0Definition passage · 1 passage

Search aliases: ∼_0 sim

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 366
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_{n+1}\ \subseteq\ \sim_n,
\qquad \sim_0=S\times S.
```

\sim_nDefinition passage · 8 passages

Search aliases: ∼_n sim

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 365
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_n
```

\sim_{n+1}Definition passage · 1 passage

Search aliases: ∼_n+1 sim

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 366
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_{n+1}\ \subseteq\ \sim_n,
\qquad \sim_0=S\times S.
```

\sim_rDefinition passage · 6 passages

Search aliases: ∼_r sim

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Definition passage: Indistinguishability at resolution r · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 154
[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.
Exact mathematical passage
```
X\sim_r Y
```

\simeqDefinition passage · 98 passages

Search aliases: ≃ simeq

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 715
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
\Cofib(\eta_X)\simeq E_{T,C}(X)[1],
\qquad E_{T,C}(X)\simeq\Delta_{T,C}(X)[-1].
```

\congDefinition passage · 34 passages

Search aliases: ≅ cong

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Definition passage: Indistinguishability at resolution r · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 158
[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.
Exact mathematical passage
```
C_{t\to r}(X)\cong C_{t\to r}(Y)
```

\leDefinition passage · 32 passages

Search aliases: ≤ le

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Definition passage: Resolution category · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 113
[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.
Exact mathematical passage
```
(R,\le)
```

\subseteqDefinition passage · 28 passages

Search aliases: ⊆ subseteq

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Definition passage: Separation depth · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 366
[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.
Exact mathematical passage
```
\sim_{n+1}\ \subseteq\ \sim_n,
\qquad \sim_0=S\times S.
```

\oplusDefinition passage · 29 passages

Search aliases: ⊕ oplus

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Definition passage: Split residual exchange · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 711
[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].
Exact mathematical passage
```
TC(X)\cong CT'(X)\oplus E_{T,C}(X).
```

\cdotsDefinition passage · 11 passages

Search aliases: ⋯ cdots

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Definition passage: Normalization path · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 740
[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i), D_tot:=Cofib(X_0→ X_m).
Exact mathematical passage
```
X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
```

\longrightarrowDefinition passage · 59 passages

Search aliases: ⟶ longrightarrow

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Definition passage: Indexed transformation system · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 120
[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.
Exact mathematical passage
```
F:\Rcat^{op}\longrightarrow\Cat.
```

\longmapstoDefinition passage · 4 passages

Search aliases: ⟼ longmapsto

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Definition passage: Residual promotion · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 642
[Residual promotion] If X is a seed-generated object and Δ^A_s,r(X) is nonzero, the operation X⟼Δ^A_s,r(X) is called residual promotion when the residual is added to the active seed set.
Exact mathematical passage
```
X\longmapsto\Delta^A_{s,r}(X)
```

\mathcal AIntroduction passage · 18 passages

Search aliases: A mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1107
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 \mathcal A_e=\{I\in J(P):D_e\subseteq I,\ e\notin I\}.
```

\mathcal A_eIntroduction passage · 3 passages

Search aliases: A_e mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1107
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 \mathcal A_e=\{I\in J(P):D_e\subseteq I,\ e\notin I\}.
```

A_gIntroduction passage · 13 passages

Search aliases: A_g

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2728
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
 E(g,h)=A_gA_hA_{gh}^{-1}-I,
 \qquad \alpha_g(X)=A_gXA_g^{-1}.
```

A_g^{-1}Introduction passage · 1 passage

Search aliases: A_g^-1

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2728
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
 E(g,h)=A_gA_hA_{gh}^{-1}-I,
 \qquad \alpha_g(X)=A_gXA_g^{-1}.
```

A_{gh}^{-1}Introduction passage · 3 passages

Search aliases: A_gh^-1

Link to this entry ·

Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2728
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
 E(g,h)=A_gA_hA_{gh}^{-1}-I,
 \qquad \alpha_g(X)=A_gXA_g^{-1}.
```

A_hIntroduction passage · 3 passages

Search aliases: A_h

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2728
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
 E(g,h)=A_gA_hA_{gh}^{-1}-I,
 \qquad \alpha_g(X)=A_gXA_g^{-1}.
```

B_nIntroduction passage · 18 passages

Search aliases: B_n

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2001
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \operatorname{Perf}(B_n)/\mathcal J_n
 \simeq \prod_{p\in\mathcal P\setminus S}\operatorname{Perf}(\mathbb Z/p^n)
 \simeq\operatorname{Perf}(\mathbb Z/D_0^n).
```

\mathcal CIntroduction passage · 15 passages

Search aliases: C mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3649
For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].
Exact mathematical passage
```
\operatorname{Fun}^{\mathrm{ex}}
\bigl(\operatorname{St}^{\mathrm{add}}(\mathcal A),\mathcal C\bigr)
\simeq
\operatorname{Fun}^{\oplus}(\mathcal A,U\mathcal C)
```

cIntroduction passage · 54 passages

Search aliases: c

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
d^1(c)(g,u)=g c(u)-c(gu)+c(g)
```

C^0_{m,n}Introduction passage · 1 passage

Search aliases: C^0_m,n

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2007
cular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is nonzero exactly when S≠ P. When S= P, use the zero category and the zero ring Z/1. At a prime outside S, each designated interaction has…
Exact mathematical passage
```
 q_nC_{m,n}\simeq C^0_{m,n}q_m
```

C_mIntroduction passage · 3 passages

Search aliases: C_m

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2410
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
 M_m^{C_m}=0,\qquad
 H^1(C_m,M_m)\cong O/mO.
```

C_{m,n}Introduction passage · 8 passages

Search aliases: C_m,n

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2007
cular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is nonzero exactly when S≠ P. When S= P, use the zero category and the zero ring Z/1. At a prime outside S, each designated interaction has…
Exact mathematical passage
```
 q_nC_{m,n}\simeq C^0_{m,n}q_m
```

\mathbf{Cat}^{\mathrm{st}}Introduction passage · 5 passages

Search aliases: Cat^st mathbf mathrm

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3631
The residual layer becomes canonical after passing to stable categories. Let Cat^add denote the ∞-category of small additive ∞-categories and additive functors, and let Cat^st denote small stable ∞-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive ∞-category, not merely its homotopy category.
Exact mathematical passage
```
\mathbf{Cat}^{\mathrm{st}}
```

\operatorname{coker}Introduction passage · 4 passages

Search aliases: coker operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
\operatorname{coker}d_G
```

dIntroduction passage · 121 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
d^1(c)(g,u)=g c(u)-c(gu)+c(g)
```

d^1Introduction passage · 4 passages

Search aliases: d^1

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
d^1(c)(g,u)=g c(u)-c(gu)+c(g)
```

D_0^nIntroduction passage · 1 passage

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2001
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \operatorname{Perf}(B_n)/\mathcal J_n
 \simeq \prod_{p\in\mathcal P\setminus S}\operatorname{Perf}(\mathbb Z/p^n)
 \simeq\operatorname{Perf}(\mathbb Z/D_0^n).
```

D_eIntroduction passage · 10 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1102
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 D_e=\{p\in P:p<e\},\qquad
 \downarrow e=D_e\cup\{e\}.
```

d_GIntroduction passage · 4 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
\operatorname{coker}d_G
```

\downarrowIntroduction passage · 10 passages

Search aliases: downarrow

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1102
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 D_e=\{p\in P:p<e\},\qquad
 \downarrow e=D_e\cup\{e\}.
```

eIntroduction passage · 117 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1101
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
e\in P
```

E_eIntroduction passage · 3 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1111
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 E_e(I)=\Cofib(X_I\to X_{I\cup\{e\}}).
```

\mathrm{ex}Introduction passage · 2 passages

Search aliases: ex mathrm

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3649
For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].
Exact mathematical passage
```
\operatorname{Fun}^{\mathrm{ex}}
\bigl(\operatorname{St}^{\mathrm{add}}(\mathcal A),\mathcal C\bigr)
\simeq
\operatorname{Fun}^{\oplus}(\mathcal A,U\mathcal C)
```

\operatorname{Exc}Introduction passage · 2 passages

Search aliases: Exc operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1162
[Principal-ideal reconstruction] Let P be finite and S stable. Write Exc_0(J(P), S) for the full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an excisive event diagram X its reduced residual diagram satisfies…
Exact mathematical passage
```
\operatorname{Exc}_0(J(P),\mathcal S)
```

\operatorname{Exc}_0Introduction passage · 2 passages

Search aliases: Exc_0 operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1162
[Principal-ideal reconstruction] Let P be finite and S stable. Write Exc_0(J(P), S) for the full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an excisive event diagram X its reduced residual diagram satisfies…
Exact mathematical passage
```
\operatorname{Exc}_0(J(P),\mathcal S)
```

\operatorname{Fun}Introduction passage · 6 passages

Search aliases: Fun operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1167
elthm:principal-reconstruction Let P be finite and S stable. Write Exc_0(J(P), S) for the full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an excisive event diagram X its reduced residual diagram satisfies F…
Exact mathematical passage
```
 \operatorname{Exc}_0(J(P),\mathcal S)
 \simeq\operatorname{Fun}(NP,\mathcal S).
```

\operatorname{Fun}^{\mathrm{ex}}Introduction passage · 2 passages

Search aliases: Fun^ex operatorname mathrm

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3649
For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].
Exact mathematical passage
```
\operatorname{Fun}^{\mathrm{ex}}
\bigl(\operatorname{St}^{\mathrm{add}}(\mathcal A),\mathcal C\bigr)
\simeq
\operatorname{Fun}^{\oplus}(\mathcal A,U\mathcal C)
```

\operatorname{Fun}^{\oplus}Introduction passage · 2 passages

Search aliases: Fun^⊕ operatorname oplus

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3649
For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].
Exact mathematical passage
```
\operatorname{Fun}^{\mathrm{ex}}
\bigl(\operatorname{St}^{\mathrm{add}}(\mathcal A),\mathcal C\bigr)
\simeq
\operatorname{Fun}^{\oplus}(\mathcal A,U\mathcal C)
```

GIntroduction passage · 106 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
\operatorname{coker}d_G
```

\operatorname{GL}Introduction passage · 31 passages

Search aliases: GL operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2727
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
A:G\to\operatorname{GL}_d(\mathbb Z_p)
```

\operatorname{GL}_dIntroduction passage · 26 passages

Search aliases: GL_d operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2727
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
A:G\to\operatorname{GL}_d(\mathbb Z_p)
```

H^1Introduction passage · 18 passages

Search aliases: H^1

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
H^1
```

\operatorname{Hom}Introduction passage · 6 passages

Search aliases: Hom operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2860
[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication table and finite matrix data. The finite-horizon theorem gives exactly h_G(s)=2e+1, so every element of mathscr L_s lifts and every integral action reduce…
Exact mathematical passage
```
 \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^h))
 \longrightarrow
 \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^s)).
```

\mathcal J_nIntroduction passage · 4 passages

Search aliases: J_n mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

\mathcal J_PIntroduction passage · 4 passages

Search aliases: J_P mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1739
[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
 \boxed{\mathcal J_P=
 \operatorname{Thick}\{\mathbb Z/p:p\in S_P\}.}
```

\mathcal J_{\square}Introduction passage · 2 passages

Search aliases: J_square mathcal square

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1589
Let Z be a small stable category, let mathscr M be a coherently acting monoid of exact endofunctors, and let mathscr X be a small family of finite coherent event diagrams in Z. Denote by mathscr H the family of interaction residuals of all their elementary independent-event squares. Put J_square =Thick{T(H):T∈mathscr M, H∈mathscr H}. This notation does not assert that H was operationally killable before localization.
Exact mathematical passage
```
 \mathcal J_{\square}
 =\operatorname{Thick}\{T(H):T\in\mathscr M,\ H\in\mathscr H\}.
```

\mathbf LIntroduction passage · 24 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1744
ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
 q_P(M)\simeq0
 \quad\Longleftrightarrow\quad
 M\otimes^{\mathbf L}_{\mathbb Z}\mathbb Z[S_P^{-1}]\simeq0.
```

L_sIntroduction passage · 3 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2859
[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication table and finite matrix data. The finite-horizon theorem gives exactly h_G(s)=2e+1, so every element of mathscr L_s lifts and every integral action reduce…
Exact mathematical passage
```
\mathscr L_s
```

\longleftrightarrowIntroduction passage · 1 passage

Search aliases: longleftrightarrow

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2866
[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication table and finite matrix data. The finite-horizon theorem gives exactly h_G(s)=2e+1, so every element of mathscr L_s lifts and every integral action reduce…
Exact mathematical passage
```
 \boxed{
 \frac{\operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z_p))}
 {\operatorname{GL}_d(\mathbb Z_p)}
 \ \longleftrightarrow\
 \frac{\mathscr L_s}{\operatorname{GL}_d(\mathbb Z/p^s)}.}
```

M_mIntroduction passage · 5 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2410
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
 M_m^{C_m}=0,\qquad
 H^1(C_m,M_m)\cong O/mO.
```

M_m^{C_m}Introduction passage · 1 passage

Search aliases: M_m^C_m

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2410
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
 M_m^{C_m}=0,\qquad
 H^1(C_m,M_m)\cong O/mO.
```

\minIntroduction passage · 6 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2415
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
 (M_m/p^nM_m)^{C_m}\cong O/p^{\min(a,n)}O.
```

N_mIntroduction passage · 6 passages

Search aliases: N_m

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2421
C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, the element t-1 is invertible modulo N_m(t) because N_m(1)=m≠0. Hence its kernel over O is zero. The norm operator is multiplication by N_m…
Exact mathematical passage
```
O/(N_m(1))=O/mO
```

OIntroduction passage · 35 passages

Search aliases: O

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2251
Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.
Exact mathematical passage
```
O=\mathbb Z_p
```

O_nIntroduction passage · 6 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2251
Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.
Exact mathematical passage
```
O_n=O/p^nO
```

\mathcal PIntroduction passage · 21 passages

Search aliases: P mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

p^aIntroduction passage · 15 passages

Search aliases: p^a

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
p^a c=d_G(m)
```

p^hIntroduction passage · 14 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2860
[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication table and finite matrix data. The finite-horizon theorem gives exactly h_G(s)=2e+1, so every element of mathscr L_s lifts and every integral action reduce…
Exact mathematical passage
```
 \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^h))
 \longrightarrow
 \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^s)).
```

p^{\min(a,n)}Introduction passage · 1 passage

Search aliases: p^min(a,n) min

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2415
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
 (M_m/p^nM_m)^{C_m}\cong O/p^{\min(a,n)}O.
```

p^nIntroduction passage · 90 passages

Search aliases: p^n

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

p^sIntroduction passage · 8 passages

Search aliases: p^s

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2860
[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication table and finite matrix data. The finite-horizon theorem gives exactly h_G(s)=2e+1, so every element of mathscr L_s lifts and every integral action reduce…
Exact mathematical passage
```
 \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^h))
 \longrightarrow
 \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^s)).
```

p^{s-n}Introduction passage · 1 passage

Search aliases: p^s-n

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2418
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
p^{s-n}
```

P_dIntroduction passage · 2 passages

Search aliases: P_d

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3210
There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.
Exact mathematical passage
```
\mathscr P_d(n,U)
```

\operatorname{Perf}Introduction passage · 19 passages

Search aliases: Perf operatorname

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1738
[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
\operatorname{Perf}(\mathbb Z)
```

\mathbb QIntroduction passage · 17 passages

Search aliases: Q mathbb

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2421
)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, the element t-1 is invertible modulo N_m(t) because N_m(1)=m≠0. Hence its kernel over O is zero. The norm operator is multiplication by N_m(t) and therefor…
Exact mathematical passage
```
\mathbb Q_p
```

\mathcal QIntroduction passage · 3 passages

Search aliases: Q mathcal

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3210
There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.
Exact mathematical passage
```
\mathcal Q_\Gamma
```

q_mIntroduction passage · 1 passage

Search aliases: q_m

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2007
cular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is nonzero exactly when S≠ P. When S= P, use the zero category and the zero ring Z/1. At a prime outside S, each designated interaction has…
Exact mathematical passage
```
 q_nC_{m,n}\simeq C^0_{m,n}q_m
```

q_nIntroduction passage · 2 passages

Search aliases: q_n

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2006
cal P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is nonzero exactly when S≠ P. When S= P, use the zero category and the zero ring Z/1. At a prime outside S, each designa…
Exact mathematical passage
```
q_n
```

\mathbb Q_pIntroduction passage · 14 passages

Search aliases: Q_p mathbb

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2421
)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, the element t-1 is invertible modulo N_m(t) because N_m(1)=m≠0. Hence its kernel over O is zero. The norm operator is multiplication by N_m(t) and therefor…
Exact mathematical passage
```
\mathbb Q_p
```

q_PIntroduction passage · 4 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1743
[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
q_P
```

\mathcal Q_\GammaIntroduction passage · 2 passages

Search aliases: Q_Γ mathcal Gamma

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3210
There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.
Exact mathematical passage
```
\mathcal Q_\Gamma
```

R_n^dIntroduction passage · 1 passage

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3210
There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.
Exact mathematical passage
```
R_n^d
```

\rightleftarrowsIntroduction passage · 1 passage

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3642
For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].
Exact mathematical passage
```
\operatorname{St}^{\mathrm{add}}:
\mathbf{Cat}^{\mathrm{add}}
\rightleftarrows
\mathbf{Cat}^{\mathrm{st}}:U,
```

S_PIntroduction passage · 6 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1739
[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
 \boxed{\mathcal J_P=
 \operatorname{Thick}\{\mathbb Z/p:p\in S_P\}.}
```

S_P^{-1}Introduction passage · 3 passages

Search aliases: S_P^-1

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1744
ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
 q_P(M)\simeq0
 \quad\Longleftrightarrow\quad
 M\otimes^{\mathbf L}_{\mathbb Z}\mathbb Z[S_P^{-1}]\simeq0.
```

\squareIntroduction passage · 9 passages

Search aliases: square

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1589
Let Z be a small stable category, let mathscr M be a coherently acting monoid of exact endofunctors, and let mathscr X be a small family of finite coherent event diagrams in Z. Denote by mathscr H the family of interaction residuals of all their elementary independent-event squares. Put J_square =Thick{T(H):T∈mathscr M, H∈mathscr H}. This notation does not assert that H was operationally killable before localization.
Exact mathematical passage
```
 \mathcal J_{\square}
 =\operatorname{Thick}\{T(H):T\in\mathscr M,\ H\in\mathscr H\}.
```

\mathrm{st}Introduction passage · 5 passages

Search aliases: st mathrm

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3631
The residual layer becomes canonical after passing to stable categories. Let Cat^add denote the ∞-category of small additive ∞-categories and additive functors, and let Cat^st denote small stable ∞-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive ∞-category, not merely its homotopy category.
Exact mathematical passage
```
\mathbf{Cat}^{\mathrm{st}}
```

\stackrelIntroduction passage · 1 passage

Search aliases: stackrel

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4030
Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.
Exact mathematical passage
```
\Zoo(X_\infty)
\stackrel{?}{\simeq}
\varprojlim_r\Zoo(X_r).
```

uIntroduction passage · 31 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2364
Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and the sharp-bound assertion.
Exact mathematical passage
```
d^1(c)(g,u)=g c(u)-c(gu)+c(g)
```

U'Introduction passage · 9 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3210
There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.
Exact mathematical passage
```
U\to U'
```

vIntroduction passage · 46 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2414
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
a=v_p(m)
```

v_pIntroduction passage · 10 passages

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2414
[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, …
Exact mathematical passage
```
a=v_p(m)
```

\varprojlimIntroduction passage · 10 passages

Search aliases: varprojlim

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4026
Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.
Exact mathematical passage
```
X_\infty\simeq\varprojlim_r X_r.
```

\varprojlim_rIntroduction passage · 2 passages

Search aliases: varprojlim_r varprojlim

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4026
Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.
Exact mathematical passage
```
X_\infty\simeq\varprojlim_r X_r.
```

X_{I\cup\{e\}}Introduction passage · 3 passages

Search aliases: X_I∪{e} cup

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1111
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 E_e(I)=\Cofib(X_I\to X_{I\cup\{e\}}).
```

X_MIntroduction passage · 4 passages

Search aliases: X_M

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1749
Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
q_PX_M
```

X_rIntroduction passage · 2 passages

Search aliases: X_r

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4026
Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.
Exact mathematical passage
```
X_\infty\simeq\varprojlim_r X_r.
```

X_\inftyIntroduction passage · 2 passages

Search aliases: X_∞ infty

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4026
Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.
Exact mathematical passage
```
X_\infty\simeq\varprojlim_r X_r.
```

\mathbb ZIntroduction passage · 223 passages

Search aliases: Z mathbb

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1738
[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
\operatorname{Perf}(\mathbb Z)
```

\mathbb Z_pIntroduction passage · 75 passages

Search aliases: Z_p mathbb

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2251
Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.
Exact mathematical passage
```
O=\mathbb Z_p
```

\alphaIntroduction passage · 27 passages

Search aliases: α alpha

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2728
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
 E(g,h)=A_gA_hA_{gh}^{-1}-I,
 \qquad \alpha_g(X)=A_gXA_g^{-1}.
```

\alpha_gIntroduction passage · 7 passages

Search aliases: α_g alpha

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2728
Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p^mM_d( Z_p) for every pair. The operators α_g are not assumed to give a group action. This distinction is essential before the repair has succeeded.
Exact mathematical passage
```
 E(g,h)=A_gA_hA_{gh}^{-1}-I,
 \qquad \alpha_g(X)=A_gXA_g^{-1}.
```

\GammaIntroduction passage · 34 passages

Search aliases: Γ Gamma

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3210
There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.
Exact mathematical passage
```
\mathcal Q_\Gamma
```

\Theta_{e,f}Introduction passage · 4 passages

Search aliases: Θ_e,f Theta

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1114
m X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
\Theta_{e,f}(I)
```

\sigmaIntroduction passage · 12 passages

Search aliases: σ sigma

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2421
mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, the element t-1 is invertible modulo N_m(t) because N_m(1)=m≠0. Hence its kernel over O is ze…
Exact mathematical passage
```
\sigma-1
```

\notinIntroduction passage · 2 passages

Search aliases: ∉ notin

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1107
Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their elementary square.
Exact mathematical passage
```
 \mathcal A_e=\{I\in J(P):D_e\subseteq I,\ e\notin I\}.
```

\prodIntroduction passage · 17 passages

Search aliases: ∏ prod

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

\prod_{p\in\mathcal P\setminus S}Introduction passage · 6 passages

Search aliases: ∏_p∈ P∖ S prod in mathcal setminus

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

\prod_{p\in S}Introduction passage · 4 passages

Search aliases: ∏_p∈ S prod in

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

\setminusIntroduction passage · 14 passages

Search aliases: ∖ setminus

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1995
[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is…
Exact mathematical passage
```
 \boxed{\mathcal J_n=
 \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
 \times\prod_{p\in\mathcal P\setminus S}0.}
```

\neIntroduction passage · 10 passages

Search aliases: ≠ ne

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2010
athcal P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is nonzero exactly when S≠ P. When S= P, use the zero category and the zero ring Z/1. At a prime outside S, each designated interaction has a unit label and is acyclic. At a prime in S, at least one pair h…
Exact mathematical passage
```
S\ne\mathcal P
```

\geqIntroduction passage · 40 passages

Search aliases: ≥ geq

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2251
Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.
Exact mathematical passage
```
n\geq1
```

\otimesIntroduction passage · 32 passages

Search aliases: ⊗ otimes

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1744
ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
 q_P(M)\simeq0
 \quad\Longleftrightarrow\quad
 M\otimes^{\mathbf L}_{\mathbb Z}\mathbb Z[S_P^{-1}]\simeq0.
```

\otimes^{\mathbf L}_{\mathbb Z}Introduction passage · 7 passages

Search aliases: ⊗^ L_ Z otimes mathbf mathbb

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Introduction passage · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1744
ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduced residual diagram is reconstructed from principal ideals.
Exact mathematical passage
```
 q_P(M)\simeq0
 \quad\Longleftrightarrow\quad
 M\otimes^{\mathbf L}_{\mathbb Z}\mathbb Z[S_P^{-1}]\simeq0.
```

*Use in source · 20 passages

Search aliases: *

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1188
Now let F be reduced and excisive. We prove the counit j_!j^*F→ F is an equivalence on an ideal I by induction on |I|. The empty ideal is the zero object on both sides. Choose a maximal element e∈ I. Put I'=I∖{e}. Then I=I'∪downarrow e, I'∩downarrow e=D_e. The ideal-excision lemma expresses F(I) as the pushout of F(I') and F(downarrow e) over F(D_e). The same holds for j_!j^*F. Induction handles I' and D_e, and the terminal-object observation handles downarrow e. Therefore the counit is an equiv…
Exact mathematical passage
```
j_!j^*F\to F
```

\operatorname*Use in source · 1 passage

Search aliases: * operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3547
The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.
Exact mathematical passage
```
(T^\ast,R^\ast)
\in
\operatorname*{arg\,min}_{T,R}
\mathcal C\bigl(\Omega(T,R;P)\bigr),
```

A'Use in source · 1 passage

Search aliases: A'

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2740
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
A'
```

A'_gUse in source · 3 passages

Search aliases: A'_g

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2736
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
 b(g)=\frac1N\sum_{t\in G}E(g,t),\qquad
 A'_g=(I-b(g))A_g.
```

A'_{gh}Use in source · 1 passage

Search aliases: A'_gh

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2741
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
 \boxed{A'_gA'_h(A'_{gh})^{-1}-I\in
 p^{2m-2e}M_d(\mathbb Z_p).}
```

A'_hUse in source · 1 passage

Search aliases: A'_h

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2741
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
 \boxed{A'_gA'_h(A'_{gh})^{-1}-I\in
 p^{2m-2e}M_d(\mathbb Z_p).}
```

a'_{ij}Use in source · 1 passage

Search aliases: a'_ij

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3254
be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1({U_i}, A) is independent of the chosen local sections; item Ω(P;{U_i})=0 if and only if the local sections can be modified by admissible local repair…
Exact mathematical passage
```
    a'_{ij}=a_{ij}+b_j-b_i;
    
```

A^2Use in source · 1 passage

Search aliases: A^2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2537
3 · 1, G=C_3. Then A^3=I over R, I+A+A^2=3I, and M^G=3M. The invariant cochain complex is quasi-isomorphic to ( F_3)^2 in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has ( F_3)^2 in every degree. Ordinary tensor of the original nonnegative complex would incorrectly reproduce only the right-hand answer.
Exact mathematical passage
```
I+A+A^2=3I
```

a^2Use in source · 1 passage

Search aliases: a^2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2826
Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is -1, which is 7pmod8, not the selected witness 3pmod8. This is exactly the distinction between preserving the requested seed and preserving every explo…
Exact mathematical passage
```
a^2=1
```

A^3Use in source · 2 passages

Search aliases: A^3

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2427
For p=m=3, in the basis 1,t, A=0 · -1 1 · -1, A^3=I, det(A-I)=3. The fixed vectors modulo 3^n are exactly {b 3^n-1binom1-1:b∈{0,1,2}}. The three vectors at level n+1 all reduce to zero at level n. Thus a nonzero residual appears at every level, while its inverse system has zero limit. This is not persistent nonexistence of the lattice; it is the failure of a particular finite-level vector to be the reduction of an invariant integral vector.
Exact mathematical passage
```
 A=\begin{pmatrix}0&-1\\1&-1\end{pmatrix},\qquad
 A^3=I,\qquad \det(A-I)=3.
```

A^aUse in source · 1 passage

Search aliases: A^a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2980
[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_…
Exact mathematical passage
```
A_a=A^a
```

A^bUse in source · 1 passage

Search aliases: A^b

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2676
Any alternative matrix lift is uniquely of the form (I+p^nb(g))A_g, for a normalized cochain b. Expanding its multiplicativity residual and discarding square-zero terms gives c_A^b=c_A+δ b. Thus the class is unchanged and the new matrices satisfy every group relation exactly if and only if δ b=-c_A. Such a cochain exists precisely when [c_A]=0. Invertibility is automatic because I+p^nb(g) has inverse I-p^nb(g) at this level.
Exact mathematical passage
```
 c_{A^b}=c_A+\delta b.
```

A^\bulletUse in source · 2 passages

Search aliases: A^bullet bullet

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3318
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
A^\bullet,B^\bullet:\Delta\to\Cat_\infty
```

a^dUse in source · 1 passage

Search aliases: a^d

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1925
_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step …
Exact mathematical passage
```
 N\otimes^{\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\qquad
 K_R(u,v)=C_R(u)\otimes^{\mathbf L}_R C_R(v).
```

A^iUse in source · 1 passage

Search aliases: A^i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2509
zero on H^0. Its functor-intrinsic cofiber satisfies H^iCofib(β)≅ k^d for every i∈ Z. Both composites defining β are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect. The binomial expansion gives (I+pJ)^p=I modulo p^2. Also A^i=I+ipJ modulo p^2, so N=pI+pfracp(p-1)2J=pIpmodp^2, using that p is odd. The standard cyclic free resolution of the trivial R[G]-module gives the cochain complex MxrightarrowpJMxright…
Exact mathematical passage
```
A^i=I+ipJ
```

A^mUse in source · 1 passage

Search aliases: A^m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3029
[When an integral matrix has continuous procyclic powers] Let A∈GL_d( Z_p). The map m↦ A^m on Z extends uniquely to a continuous homomorphism Z_p→GL_d( Z_p) if and only if the reduction of A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion s…
Exact mathematical passage
```
m\mapsto A^m
```

A^nUse in source · 1 passage

Search aliases: A^n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3319
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
\lim_{[n]\in\Delta} A^n\simeq\lim_{[n]\in\Delta} B^n.
```

A^pUse in source · 2 passages

Search aliases: A^p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2492
[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,
Exact mathematical passage
```
A^p=I
```

A^{p-1}Use in source · 1 passage

Search aliases: A^p-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2492
[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,
Exact mathematical passage
```
N=I+A+\cdots+A^{p-1}=pI
```

A^{p^2}Use in source · 1 passage

Search aliases: A^p^2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2980
[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_…
Exact mathematical passage
```
A^{p^2}=I
```

A^{p^t}Use in source · 1 passage

Search aliases: A^p^t

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3032
a continuous homomorphism Z_p→GL_d( Z_p) if and only if the reduction of A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows…
Exact mathematical passage
```
A^{p^t}\in I+pM_d(\mathbb Z_p)
```

A^{p^{t+j}}Use in source · 1 passage

Search aliases: A^p^t+j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3033
p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponents congruent modulo p^t+j give powers congruent modulo p^j+1, also for negative integer exponents by invertibility. Completeness extends the map conti…
Exact mathematical passage
```
 A^{p^{t+j}}\in I+p^{j+1}M_d(\mathbb Z_p).
```

A^qUse in source · 1 passage

Search aliases: A^q

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2884
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
A^q=I
```

A^rUse in source · 2 passages

Search aliases: A^r

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2201
bda⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_…
Exact mathematical passage
```
V=A^r
```

A^xUse in source · 1 passage

Search aliases: A^x

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2995
delta u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_p^2 put b(x)=lfloorfracxprfloor Jpmod p (0≤ x<p^2). Writing each x as its low digit plus p times its high digit shows δ b=-Inf(c): the high-digit carry is the negative of the low-digit carry modulo p. The repaired map is exactly x↦ A^x. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.
Exact mathematical passage
```
x\mapsto A^x
```

A_*Use in source · 1 passage

Search aliases: A_*

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3136
[Complete rank-two profinite lifting test] Let p be any prime and let ρ_n: Z_p^r→GL_2(R_n) be continuous, with finitely many topological generators as above. item If a=n, all the A_i are scalar and ρ_n has a continuous integral lift. item If a<n, choose any generator A_* with scalar depth a. Take λ_i to be an integer representative of (A_i)_11 and form B_i modulo p^n-a. Then ρ_n lifts over Z_p Longleftrightarrow [B_*,B_i]=0 in M_2(R_n-a) (1≤ i≤ r). The test is finite matrix arithmetic on the exi…
Exact mathematical passage
```
A_*
```

A_0Use in source · 5 passages

Search aliases: A_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 842
mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B, these choices are respectively the identity and P…
Exact mathematical passage
```
A_0=P_2,\quad A_1=A_2=\id,
\qquad
B_0=B_1=P_1,\quad B_2=\id.
```

a_0Use in source · 1 passage

Search aliases: a_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2822
Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is -1, which is 7pmod8, not the selected witness 3pmod8. This is exactly the distinction between preserving the requested seed and preserving every explo…
Exact mathematical passage
```
a_0=3
```

A_1Use in source · 8 passages

Search aliases: A_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 842
mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B, these choices are respectively the identity and P…
Exact mathematical passage
```
A_0=P_2,\quad A_1=A_2=\id,
\qquad
B_0=B_1=P_1,\quad B_2=\id.
```

a_1Use in source · 12 passages

Search aliases: a_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1679
[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other cohomology group vanishes. In particular the interaction vanishes exactly when g=1.
Exact mathematical passage
```
a_1,\ldots,a_m
```

A_{11}Use in source · 1 passage

Search aliases: A_11

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3130
For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).
Exact mathematical passage
```
A_{12},A_{21},A_{11}-A_{22}
```

A_{12}Use in source · 1 passage

Search aliases: A_12

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3130
For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).
Exact mathematical passage
```
A_{12},A_{21},A_{11}-A_{22}
```

A_2Use in source · 4 passages

Search aliases: A_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 842
mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B, these choices are respectively the identity and P…
Exact mathematical passage
```
A_0=P_2,\quad A_1=A_2=\id,
\qquad
B_0=B_1=P_1,\quad B_2=\id.
```

A_{21}Use in source · 1 passage

Search aliases: A_21

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3130
For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).
Exact mathematical passage
```
A_{12},A_{21},A_{11}-A_{22}
```

A_{22}Use in source · 1 passage

Search aliases: A_22

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3130
For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).
Exact mathematical passage
```
A_{12},A_{21},A_{11}-A_{22}
```

A_aUse in source · 1 passage

Search aliases: A_a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2980
[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_…
Exact mathematical passage
```
A_a=A^a
```

A_g^bUse in source · 1 passage

Search aliases: A_g^b

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2659
heorem[Complete one-step relation obstruction] The residual c_A is a 2-cocycle. Its cohomology class o_n(ρ_n)=[c_A]∈ H^2(G,V) is independent of the chosen matrix lifts. Moreover, o_n(ρ_n)=0 Longleftrightarrow ρ_n lifts to GL_d(R_n+1). More precisely, all lifts with the fixed reduction have the form A_g^b=(I+p^n b(g))A_g, where representatives of b(g) may be chosen arbitrarily, and the necessary and sufficient equation is δ b=-c_A. Associativity of the threefold product gives the exact identity (…
Exact mathematical passage
```
 A_g^b=(I+p^n b(g))A_g,
```

A_iUse in source · 10 passages

Search aliases: A_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1628
[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S,
Exact mathematical passage
```
f_i:A_i\to B_i
```

a_iUse in source · 9 passages

Search aliases: a_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1649
For a commutative ring R and a∈ R, set C_R(a)=[RxrightarrowaR], with the source in cohomological degree -1 and the target in degree 0. The tensor product ⊗_i C_R(a_i) is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see [source reference]. We use them with an explicit sign convention and direct proofs below.
Exact mathematical passage
```
\bigotimes_i C_R(a_i)
```

A_i'Use in source · 1 passage

Search aliases: A_i'

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3118
alent: ρ_n lifts over Z_p, ρ_n extends over R_n+a, Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B_i' are defined modulo p^n and reduce to B_i modulo p^n-a. The equality p^2a[B_*',B_i']=0 modulo p^n+a implies [B_*',B_i'…
Exact mathematical passage
```
A_i'
```

a_{ij}Use in source · 8 passages

Search aliases: a_ij

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3239
[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a…
Exact mathematical passage
```
a_{ij}\in\mathcal A(U_{ij})
```

a_{ik}Use in source · 2 passages

Search aliases: a_ik

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3271
On a triple overlap, s_k=s_j+a_jk=s_i+a_ij+a_jk, while also s_k=s_i+a_ik. Freeness of the torsor action gives a_ij+a_jk=a_ik, which is the vCech cocycle identity.
Exact mathematical passage
```
s_k=s_i+a_{ik}
```

a_jUse in source · 1 passage

Search aliases: a_j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2823
Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is -1, which is 7pmod8, not the selected witness 3pmod8. This is exactly the distinction between preserving the requested seed and preserving every explo…
Exact mathematical passage
```
 a_{j+1}=\frac{3-a_j^2}{2}\,a_j.
```

a_{j+1}Use in source · 1 passage

Search aliases: a_j+1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2823
Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is -1, which is 7pmod8, not the selected witness 3pmod8. This is exactly the distinction between preserving the requested seed and preserving every explo…
Exact mathematical passage
```
 a_{j+1}=\frac{3-a_j^2}{2}\,a_j.
```

a_j^2Use in source · 1 passage

Search aliases: a_j^2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2823
Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is -1, which is 7pmod8, not the selected witness 3pmod8. This is exactly the distinction between preserving the requested seed and preserving every explo…
Exact mathematical passage
```
 a_{j+1}=\frac{3-a_j^2}{2}\,a_j.
```

a_{jk}Use in source · 2 passages

Search aliases: a_jk

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3268
On a triple overlap, s_k=s_j+a_jk=s_i+a_ij+a_jk, while also s_k=s_i+a_ik. Freeness of the torsor action gives a_ij+a_jk=a_ik, which is the vCech cocycle identity.
Exact mathematical passage
```
s_k=s_j+a_{jk}=s_i+a_{ij}+a_{jk},
```

A_{L,m}Use in source · 1 passage

Search aliases: A_L,m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1904
For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For scalar unit templates take L= Z, so that the transformation functors are identities but the coarsening functors …
Exact mathematical passage
```
 \theta_{L;m,n}:A_{L,n}C_{m,n}\simeq C_{m,n}A_{L,m}.
```

A_{L,n}Use in source · 4 passages

Search aliases: A_L,n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1900
For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For scalar unit templates take L= Z, so that the transformation functors are identities but the coarsening functors …
Exact mathematical passage
```
 A_{L,n}(M)=(L\otimes^{\mathbf L}_{\mathbb Z}B_n)\otimes^{\mathbf L}_{B_n}M.
```

A_mUse in source · 1 passage

Search aliases: A_m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1642
For m=1 this is the definition. Taking cofibers in the last coordinate replaces its factor A_m→ B_m by Cofib(f_m) because tensoring by the other factors is exact. Repeat for the remaining coordinates. Interchanging finite colimits shows that this iterated construction is the cofiber of the punctured-cube colimit map. The universal properties give naturality and coherent interchange. In a complex model, these interchange maps incorporate the tensor differential's graded signs.
Exact mathematical passage
```
A_m\to B_m
```

a_mUse in source · 6 passages

Search aliases: a_m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1679
[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other cohomology group vanishes. In particular the interaction vanishes exactly when g=1.
Exact mathematical passage
```
a_1,\ldots,a_m
```

A_NUse in source · 2 passages

Search aliases: A_N

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 500
For a constructor output from ⋃_n A_n, its finitely many inputs all belong to a common A_N. The output therefore already belongs to Φ(A_N). This proves preservation of increasing unions. The union on the right is a fixed point because Ψ preserves that union. It contains Σ_0 and is contained in every fixed point by induction on n, hence is least.
Exact mathematical passage
```
A_N
```

A_nUse in source · 1 passage

Search aliases: A_n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 500
For a constructor output from ⋃_n A_n, its finitely many inputs all belong to a common A_N. The output therefore already belongs to Φ(A_N). This proves preservation of increasing unions. The union on the right is a fixed point because Ψ preserves that union. It contains Σ_0 and is contained in every fixed point by induction on n, hence is least.
Exact mathematical passage
```
\bigcup_n A_n
```

A_RUse in source · 2 passages

Search aliases: A_R

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2457
A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by notation. A fixed ambient universe is understood. A small stable subdiagram can instead be generated by the stated inputs and their images, without m…
Exact mathematical passage
```
 \mathcal K_R=D(R[G])\times D(R),\qquad
 A_R(M,N)=(0,\Gamma_R(M)),
```

a_rUse in source · 2 passages

Search aliases: a_r

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2288
Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.
Exact mathematical passage
```
p^{a_1},\ldots,p^{a_r}
```

A_SUse in source · 1 passage

Search aliases: A_S

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2461
A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by notation. A fixed ambient universe is understood. A small stable subdiagram can instead be generated by the stated inputs and their images, without m…
Exact mathematical passage
```
C_{R,S}A_R\to A_SC_{R,S}
```

a_tUse in source · 4 passages

Search aliases: a_t

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1944
[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose …
Exact mathematical passage
```
a_1,\ldots,a_t
```

\operatorname{Ad}Use in source · 2 passages

Search aliases: Ad operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3001
[Continuous reconstruction from unobstructed stages] If H^2_cts(Γ,Adbarρ)=0, every given ρ_n with that residual action extends to a continuous representation ρ:Γ⟶GL_d( Z_p). Without the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If Γ is topologically finitely generated, existence of some extension of the fixed ρ_n to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.…
Exact mathematical passage
```
H^2_{\mathrm{cts}}(\Gamma,\operatorname{Ad}\bar\rho)=0
```

\amalgUse in source · 2 passages

Search aliases: amalg

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1083
[Interaction is the total cofiber] There are natural equivalences Θ≃Cofib(Bamalg_A C⟶ D) ≃Cofib(Cofib(A→ C)⟶ Cofib(B→ D)). Consequently Θ≃0 if and only if the square is a homotopy pushout, equivalently a homotopy pullback. An exact functor preserves these equivalences.
Exact mathematical passage
```
 \Theta\simeq\Cofib\bigl(B\amalg_A C\longrightarrow D\bigr)
 \simeq\Cofib\bigl(\Cofib(A\to C)\longrightarrow
                         \Cofib(B\to D)\bigr).
```

\amalg_AUse in source · 2 passages

Search aliases: amalg_A amalg

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1083
[Interaction is the total cofiber] There are natural equivalences Θ≃Cofib(Bamalg_A C⟶ D) ≃Cofib(Cofib(A→ C)⟶ Cofib(B→ D)). Consequently Θ≃0 if and only if the square is a homotopy pushout, equivalently a homotopy pullback. An exact functor preserves these equivalences.
Exact mathematical passage
```
 \Theta\simeq\Cofib\bigl(B\amalg_A C\longrightarrow D\bigr)
 \simeq\Cofib\bigl(\Cofib(A\to C)\longrightarrow
                         \Cofib(B\to D)\bigr).
```

\mathrm{arith}Use in source · 2 passages

Search aliases: arith mathrm

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3351
Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.
Exact mathematical passage
```
\mathbb K_{\mathrm{arith}}
\qquad\text{and}\qquad
\mathbb K_{\mathrm{aut}}
```

\arrowUse in source · 5 passages

Search aliases: arrow

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 559
The square is [column sep=large,row sep=large] A_rC_s→ r(X) arrow[r,"η^A_s,r,X"] · C_s→ rA_s(X). It compares transforming after coarsening with coarsening after transforming.
Exact mathematical passage
```
\begin{tikzcd}[column sep=large,row sep=large]
A_rC_{s\to r}(X) \arrow[r,"\eta^A_{s,r,X}"] & C_{s\to r}A_s(X).
\end{tikzcd}
```

\mathrm{aut}Use in source · 2 passages

Search aliases: aut mathrm

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3351
Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.
Exact mathematical passage
```
\mathbb K_{\mathrm{arith}}
\qquad\text{and}\qquad
\mathbb K_{\mathrm{aut}}
```

\mathcal BUse in source · 7 passages

Search aliases: B mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1382
[Idempotent projection model] Let Z= A× B for stable categories A, B, and let L(a,b)=(a,0). With constructor monoid generated by the idempotent exact functor L, an object is operationally killable exactly when it lies in 0× B. The obstruction quotient is equivalent to A, and Obs(a,b)≃ a. Thus the quotient literally retains the component that cannot be annihilated by the admissible constructor.
Exact mathematical passage
```
\mathcal Z=\mathcal A\times\mathcal B
```

b^{-1}Use in source · 2 passages

Search aliases: b^-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2202
s of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r⊆ E_p⊆ b^-1 Z_p^r for all p. The intersection defining Λ therefore satisfies b Z^r⊆Λ⊆ b^-1 Z^r. The right inclusion uses the same integrality-of-denominators …
Exact mathematical passage
```
 b\mathbb Z_p^r\subseteq E_p\subseteq b^{-1}\mathbb Z_p^r
 \quad\text{for all }p.
```

b^2Use in source · 4 passages

Search aliases: b^2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2213
Here is also an explicit finite construction proving the local identifications. Put F_p=bE_p, so b^2 Z_p^r⊆ F_p⊆ Z_p^r. The Chinese remainder isomorphism identifies Z^r/b^2 Z^r≅ ∏_p∈ P Z_p^r/b^2 Z_p^r. Let H⊆ Z^r be the inverse image of the product of the subgroups F_p/b^2 Z_p^r. Directly from the definitions, H=bΛ. The quotient Z^r/H is the product of the corresponding finite p-primary quotients. Tensoring with Z_p retains exactly the p component. Thus H⊗ Z_p is the inverse image of F_p/b^2 Z_p…
Exact mathematical passage
```
 b^2\mathbb Z_p^r\subseteq F_p\subseteq\mathbb Z_p^r.
```

B^\bulletUse in source · 2 passages

Search aliases: B^bullet bullet

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3318
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
A^\bullet,B^\bullet:\Delta\to\Cat_\infty
```

B^{d-1}Use in source · 1 passage

Search aliases: B^d-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3041
[Cyclic centralizers over a coefficient ring] Let R be a commutative ring, B∈ M_d(R), and suppose a vector v makes Q=(v,Bv,…,B^d-1v)∈GL_d(R). Then every matrix commuting with B is uniquely f(B) for a polynomial f∈ R[t] of degree less than d. The coefficients of f are the entries of Q^-1Mv. In particular, the centralizer is commutative.
Exact mathematical passage
```
 Q=(v,Bv,\ldots,B^{d-1}v)\in\operatorname{GL}_d(R).
```

b^eUse in source · 1 passage

Search aliases: b^e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1925
_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step …
Exact mathematical passage
```
 N\otimes^{\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\qquad
 K_R(u,v)=C_R(u)\otimes^{\mathbf L}_R C_R(v).
```

B^jUse in source · 1 passage

Search aliases: B^j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3049
Over R= Z/p^s, it is enough that bar B have a cyclic vector over F_p. Every nonscalar 2×2 matrix over a field has such a vector. There is a unique polynomial of degree less than d satisfying f(B)v=Mv, since the columns of Q are a basis. If MB=BM, then M-f(B) annihilates every B^jv, so it is zero. Conversely every polynomial in B commutes with B. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo p makes det Q a unit modulo p^s.
Exact mathematical passage
```
B^jv
```

b^{l-m}Use in source · 1 passage

Search aliases: b^l-m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1932
independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step cell has scalar b^m-nb^l-m=b^l-n. The associativity coherence of derived tensor identifies its underlying θ with θ_L;l,n. Identity intervals have scalar one. Tensoring transformation templates gives their us…
Exact mathematical passage
```
b^{m-n}b^{l-m}=b^{l-n}
```

b^{l-n}Use in source · 1 passage

Search aliases: b^l-n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1932
independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step cell has scalar b^m-nb^l-m=b^l-n. The associativity coherence of derived tensor identifies its underlying θ with θ_L;l,n. Identity intervals have scalar one. Tensoring transformation templates gives their us…
Exact mathematical passage
```
b^{m-n}b^{l-m}=b^{l-n}
```

b^{m-n}Use in source · 4 passages

Search aliases: b^m-n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1908
ut A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For scalar unit templates take L= Z, so that the transformation functors are identities but the coarsening functors are not.
Exact mathematical passage
```
 \eta_{L,b;m,n}=b^{m-n}\theta_{L;m,n}.
```

B^nUse in source · 1 passage

Search aliases: B^n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3319
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
\lim_{[n]\in\Delta} A^n\simeq\lim_{[n]\in\Delta} B^n.
```

B^pUse in source · 1 passage

Search aliases: B^p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3032
A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponents congruent modulo p^t+j g…
Exact mathematical passage
```
B^p\in I+p^{s+1}M_d(\mathbb Z_p)
```

B_*Use in source · 6 passages

Search aliases: B_*

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3067
d integer scalars λ_i such that A_i≡λ_i Ipmodp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modulo p. Define the finite obstruction record Ω_a,*(ρ_n)= ([B_*,B_i])_i=1^r∈ M_d(R_n-a)^r, [B,C]=BC-CB.
Exact mathematical passage
```
B_*
```

B_*'Use in source · 2 passages

Search aliases: B_*'

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3119
thbb Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B_i' are defined modulo p^n and reduce to B_i modulo p^n-a. The equality p^2a[B_*',B_i']=0 modulo p^n+a implies [B_*',B_i']=0 modulo p^n-a, so Ω_a,*=0. The normalized criterion supplies an integral lift. This is a bound obtained from…
Exact mathematical passage
```
p^{2a}[B_*',B_i']=0
```

B_0Use in source · 1 passage

Search aliases: B_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 842
mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B, these choices are respectively the identity and P…
Exact mathematical passage
```
A_0=P_2,\quad A_1=A_2=\id,
\qquad
B_0=B_1=P_1,\quad B_2=\id.
```

b_0Use in source · 2 passages

Search aliases: b_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2692
For a fixed multiplication table and residual action, deciding existence is a finite linear problem over k. A solution b is a positive certificate. A linear functional λ on normalized 2-cochains with λ∘δ=0, λ(c_A)≠0 is a negative certificate. If b_0 is one solution, every other solution is b_0+z with δ z=0. Conjugating the matrices by I+p^nX replaces b(g) by b(g)+X-g· X. These are exactly the negatives of the usual 0-coboundaries and yield the same quotient subgroup. Conjugation fixes the action…
Exact mathematical passage
```
b_0
```

B_1Use in source · 6 passages

Search aliases: B_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 842
mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B, these choices are respectively the identity and P…
Exact mathematical passage
```
A_0=P_2,\quad A_1=A_2=\id,
\qquad
B_0=B_1=P_1,\quad B_2=\id.
```

B_2Use in source · 4 passages

Search aliases: B_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 842
mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B, these choices are respectively the identity and P…
Exact mathematical passage
```
A_0=P_2,\quad A_1=A_2=\id,
\qquad
B_0=B_1=P_1,\quad B_2=\id.
```

B_iUse in source · 10 passages

Search aliases: B_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1628
[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S,
Exact mathematical passage
```
f_i:A_i\to B_i
```

b_iUse in source · 10 passages

Search aliases: b_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1812
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,
Exact mathematical passage
```
f_i:\mathbb Z^{a_i}\to\mathbb Z^{b_i}
```

B_i'Use in source · 3 passages

Search aliases: B_i'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3118
Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B_i' are defined modulo p^n and reduce to B_i modulo p^n-a. The equality p^2a[B_*',B_i']=0 modulo p^n+a implies [B_*',B_i']=0 modulo p^n-a, so Ω_a,*=0. The normalized criter…
Exact mathematical passage
```
B_i'
```

B_jUse in source · 1 passage

Search aliases: B_j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3102
[What normalization reveals] The input relations only force p^2a[B_i,B_j]≡0pmodp^n. When a>0, this can conceal a nonzero commutator in R_n-a. The obstruction is therefore not just the original zero commutator re-labelled. It measures the additional commutativity required after removing a common scalar precision. Different permitted scalar choices change B_i by scalar matrices and leave its commutators unchanged. Changing representatives changes them only by zero in R_n-a. No preferred normalized…
Exact mathematical passage
```
 p^{2a}[B_i,B_j]\equiv0\pmod{p^n}.
```

b_jUse in source · 2 passages

Search aliases: b_j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3254
be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1({U_i}, A) is independent of the chosen local sections; item Ω(P;{U_i})=0 if and only if the local sections can be modified by admissible local repair…
Exact mathematical passage
```
    a'_{ij}=a_{ij}+b_j-b_i;
    
```

B_kUse in source · 1 passage

Search aliases: B_k

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1920
orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C…
Exact mathematical passage
```
 C_{n,k}\bigl(N\otimes^{\mathbf L}_{B_n}C_{B_n}(a)\bigr)
 \simeq C_{n,k}(N)\otimes^{\mathbf L}_{B_k}C_{B_k}(a).
```

B_mUse in source · 4 passages

Search aliases: B_m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1642
For m=1 this is the definition. Taking cofibers in the last coordinate replaces its factor A_m→ B_m by Cofib(f_m) because tensoring by the other factors is exact. Repeat for the remaining coordinates. Interchanging finite colimits shows that this iterated construction is the cofiber of the punctured-cube colimit map. The universal properties give naturality and coherent interchange. In a complex model, these interchange maps incorporate the tensor differential's graded signs.
Exact mathematical passage
```
A_m\to B_m
```

B_{p,n}Use in source · 2 passages

Search aliases: B_p,n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1866
Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bo…
Exact mathematical passage
```
B_{p,n}=\mathbb Z/p^n\mathbb Z
```

b_{r_0\to r_1}Use in source · 1 passage

Search aliases: b_r_0→ r_1 to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3488
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
B_X=(b_{r_0\to r_1}(X_0),\ldots,b_{r_{n-1}\to r_n}(X_{n-1})).
```

b_{r_{n-1}\to r_n}Use in source · 1 passage

Search aliases: b_r_n-1→ r_n to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3488
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
B_X=(b_{r_0\to r_1}(X_0),\ldots,b_{r_{n-1}\to r_n}(X_{n-1})).
```

B_XUse in source · 1 passage

Search aliases: B_X

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3488
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
B_X=(b_{r_0\to r_1}(X_0),\ldots,b_{r_{n-1}\to r_n}(X_{n-1})).
```

\bar AUse in source · 4 passages

Search aliases: bar A bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2885
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
 c=\frac{\widetilde A^q-I}{p^n}\pmod p,
 \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
```

\bar A^iUse in source · 1 passage

Search aliases: bar A^i bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2885
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
 c=\frac{\widetilde A^q-I}{p^n}\pmod p,
 \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
```

\bar A^{q-1-i}Use in source · 1 passage

Search aliases: bar A^q-1-i bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2885
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
 c=\frac{\widetilde A^q-I}{p^n}\pmod p,
 \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
```

\bar BUse in source · 1 passage

Search aliases: bar B bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3046
Over R= Z/p^s, it is enough that bar B have a cyclic vector over F_p. Every nonscalar 2×2 matrix over a field has such a vector. There is a unique polynomial of degree less than d satisfying f(B)v=Mv, since the columns of Q are a basis. If MB=BM, then M-f(B) annihilates every B^jv, so it is zero. Conversely every polynomial in B commutes with B. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo p makes det Q a unit modulo p^s.
Exact mathematical passage
```
\bar B
```

\bar EUse in source · 4 passages

Search aliases: bar E bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1294
item every operationally killable object maps to zero; item q(X)≃0 if and only if X∈ N_mathscr M; item for every exact functor E: Z→ D into a stable ∞-category such that E(K)≃0 for every K∈ K_mathscr M, there is an exact factorization, unique up to contractible choice together with its comparison to E, Zxrightarrowq O_mathscr Mxrightarrowbar E D. Thus O_mathscr M is universal among exact targets in which all operationally killable residuals vanish.
Exact mathematical passage
```
    \mathcal Z\xrightarrow{q}\mathcal O_{\mathscr M}\xrightarrow{\bar E}\mathcal D.
    
```

\bar mUse in source · 3 passages

Search aliases: bar m bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2342
[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^1(G,M) is finite and killed by |G|. In particular it is killed by p^e, where e=v_p(|G|), and the delay e is a valid uniform lifting bound.
Exact mathematical passage
```
\bar m
```

\bar TUse in source · 2 passages

Search aliases: bar T bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1519
Now q^invT annihilates J_mathscr M, so the quotient universal property gives bar T with bar T q^inv≃ q^invT. The full faithfulness of precomposition in the functor-category universal property descends any supplied unit, composition, and higher coherence data. It does not create action data absent from the original system.
Exact mathematical passage
```
\bar T
```

\bar UUse in source · 2 passages

Search aliases: bar U bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1484
en its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently N_mathscr M= C× C×0, Z/ N_mathscr M≃ C. Nevertheless U(0,A,0)=(0,0,A) for every A∈ C. The input lies in N_mathscr M and the output generally does not. If U descended to an exact bar U on the quotient, then qU(0,A,0)≃bar Uq(0,A,0)≃0, a contradiction for Anot≃0.
Exact mathematical passage
```
\bar U
```

\bar vUse in source · 3 passages

Search aliases: bar v bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2268
[Intrinsic kernel exchange and its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural comparison α_n is Q[p^n], with these specified transition maps. Its cone, viewing both modules in degree zero in D(O_n), …
Exact mathematical passage
```
\bar v\in K_n
```

\bar\LambdaUse in source · 6 passages

Search aliases: barΛ bar Lambda

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1403
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\bar\Lambda:\mathcal O_{\mathscr M_A}\longrightarrow\mathcal O_{\mathscr M_B}.
```

\bar\rhoUse in source · 4 passages

Search aliases: barρ bar rho

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2627
Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^nR_n+1 with k. It is not inversion of p in R_n+1. The square-zero assertion uses 2n≥ n+1.
Exact mathematical passage
```
 g\cdot X=\bar\rho(g)X\bar\rho(g)^{-1},
 \qquad \bar\rho=\rho_n\bmod p.
```

\bigwedgeUse in source · 1 passage

Search aliases: bigwedge

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1700
The Euclidean algorithm produces a unimodular matrix U∈GL_m( Z) with (a_1,…,a_m)U=(g,0,…,0). The Koszul complex is the exterior algebra on a free module in degree -1, with differential the derivation determined by this row vector. Hence the maps bigwedge^jU give an isomorphism of complexes from K(g,0,…,0) to K(a_1,…,a_m).
Exact mathematical passage
```
\bigwedge^jU
```

\bigwedge^jUse in source · 1 passage

Search aliases: bigwedge^j bigwedge

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1700
The Euclidean algorithm produces a unimodular matrix U∈GL_m( Z) with (a_1,…,a_m)U=(g,0,…,0). The Koszul complex is the exterior algebra on a free module in degree -1, with differential the derivation determined by this row vector. Hence the maps bigwedge^jU give an isomorphism of complexes from K(g,0,…,0) to K(a_1,…,a_m).
Exact mathematical passage
```
\bigwedge^jU
```

\bmodUse in source · 2 passages

Search aliases: bmod

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2627
Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^nR_n+1 with k. It is not inversion of p in R_n+1. The square-zero assertion uses 2n≥ n+1.
Exact mathematical passage
```
 g\cdot X=\bar\rho(g)X\bar\rho(g)^{-1},
 \qquad \bar\rho=\rho_n\bmod p.
```

\bulletUse in source · 3 passages

Search aliases: bullet

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3318
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
A^\bullet,B^\bullet:\Delta\to\Cat_\infty
```

C'Use in source · 1 passage

Search aliases: C'

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3359
A strong correspondence should include: Λ(TX) · ≃ T'Λ(X), Λ(CX) · ≃ C'Λ(X), Λ(Δ_T,CX) · ≃Δ_T',C'Λ(X), barΛ(Obs X) · ≃Obs'Λ(X).
Exact mathematical passage
```
\Lambda(TX)&\simeq T'\Lambda(X),\\
\Lambda(CX)&\simeq C'\Lambda(X),\\
\Lambda(\Delta_{T,C}X)&\simeq\Delta_{T',C'}\Lambda(X),\\
\bar\Lambda(\Obs X)&\simeq\Obs'\Lambda(X).
```

\mathcal C^3Use in source · 2 passages

Search aliases: C^3 mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1311
[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so both X and Y are operationally killable. However X⊕ Y=(A,A,0), and both U and V send this object to (0,0,A), which is fixed by U and…
Exact mathematical passage
```
\mathcal Z=\mathcal C^3.
```

C_{\mathbb Z}Use in source · 16 passages

Search aliases: C_ Z mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1661
For an enabled event e, E_e(I)≃ M⊗^ L_ ZC_ Z(n_e). The comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)≃ M⊗^ L_ ZC_ Z(N), N=∏_e∈ Pn_e. This is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vert…
Exact mathematical passage
```
 E_e(I)\simeq M\otimes^{\mathbf L}_{\mathbb Z}C_{\mathbb Z}(n_e).
```

C_{\mathbb Z_p}Use in source · 1 passage

Search aliases: C_ Z_p mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2063
put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-ax. Reduction of p^m-ax gives p^n-ap^m-nx. Thus for each target …
Exact mathematical passage
```
 C_{\mathbb Z_p}(p^a),\qquad
 H^{-1}=0,\quad H^0=\mathbb Z/p^a.
```

C_{1\to0}Use in source · 2 passages

Search aliases: C_1→0 to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 850
On the seed x=(k[0],k^2[0]), the word W=A_0C_1→0B_1C_2→1 evaluates to zero. Exchanging its left inversion first yields P_1x. Exchanging its right inversion first yields P_2x. The second exchange in either path yields x. Both paths then perform the same identity exchange to reach the normal word C_1→0C_2→1A_2B_2. Thus the complete paths have residual filtrations 0→ P_1x→ xxrightarrowidx, 0→ P_2x→ xxrightarrowidx. Their first nonzero stages are not equivalent. The transformations, contexts, and co…
Exact mathematical passage
```
W=A_0C_{1\to0}B_1C_{2\to1}
```

C_2Use in source · 5 passages

Search aliases: C_2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 668
Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformation at the new resolution.
Exact mathematical passage
```
W=T_1C_1T_2T_3C_2T_4.
```

C_{2\to1}Use in source · 2 passages

Search aliases: C_2→1 to

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 850
On the seed x=(k[0],k^2[0]), the word W=A_0C_1→0B_1C_2→1 evaluates to zero. Exchanging its left inversion first yields P_1x. Exchanging its right inversion first yields P_2x. The second exchange in either path yields x. Both paths then perform the same identity exchange to reach the normal word C_1→0C_2→1A_2B_2. Thus the complete paths have residual filtrations 0→ P_1x→ xxrightarrowidx, 0→ P_2x→ xxrightarrowidx. Their first nonzero stages are not equivalent. The transformations, contexts, and co…
Exact mathematical passage
```
W=A_0C_{1\to0}B_1C_{2\to1}
```

C_3Use in source · 1 passage

Search aliases: C_3

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2533
[The two-dimensional Z/9 instance] Take R= Z/9, A=1 · 3 3 · 1, G=C_3. Then A^3=I over R, I+A+A^2=3I, and M^G=3M. The invariant cochain complex is quasi-isomorphic to ( F_3)^2 in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has ( F_3)^2 in every degree. Ordinary tensor of the original nonnegative complex would incorrectly reproduc…
Exact mathematical passage
```
 R=\mathbb Z/9,\qquad
 A=\begin{pmatrix}1&3\\3&1\end{pmatrix},\qquad G=C_3.
```

c_AUse in source · 18 passages

Search aliases: c_A

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2632
Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^nR_n+1 with k. It is not inversion of p in R_n+1. The square-zero assertion uses 2n≥ n+1.
Exact mathematical passage
```
 c_A(g,h)=\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\pmod p\quad\in V.
```

c_{A^b}Use in source · 1 passage

Search aliases: c_A^b

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2676
Any alternative matrix lift is uniquely of the form (I+p^nb(g))A_g, for a normalized cochain b. Expanding its multiplicativity residual and discarding square-zero terms gives c_A^b=c_A+δ b. Thus the class is unchanged and the new matrices satisfy every group relation exactly if and only if δ b=-c_A. Such a cochain exists precisely when [c_A]=0. Invertibility is automatic because I+p^nb(g) has inverse I-p^nb(g) at this level.
Exact mathematical passage
```
 c_{A^b}=c_A+\delta b.
```

C_{B_k}Use in source · 1 passage

Search aliases: C_B_k

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1920
orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C…
Exact mathematical passage
```
 C_{n,k}\bigl(N\otimes^{\mathbf L}_{B_n}C_{B_n}(a)\bigr)
 \simeq C_{n,k}(N)\otimes^{\mathbf L}_{B_k}C_{B_k}(a).
```

C_{B_n}Use in source · 2 passages

Search aliases: C_B_n

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1915
[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u…
Exact mathematical passage
```
 \Cofib(\eta_{L,b;m,n,M})\simeq
 A_{L,n}C_{m,n}(M)\otimes^{\mathbf L}_{B_n}C_{B_n}(b^{m-n}).
```

c_{\bar m}Use in source · 1 passage

Search aliases: c_bar m bar

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2343
[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^1(G,M) is finite and killed by |G|. In particular it is killed by p^e, where e=v_p(|G|), and the delay e is a valid uniform lifting bound.
Exact mathematical passage
```
 c_{\bar m}(g)=\frac{gm-m}{p^n}.
```

C_kUse in source · 2 passages

Search aliases: C_k

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 934
Before event (i,j) can occur, every initially intervening C_k with k<j must already have crossed T_i, and every initially intervening T_h with h>i must already have crossed C_j. Thus all predecessors of a performed event have already occurred. The performed events form an ideal.
Exact mathematical passage
```
C_k
```

C_{l,m}Use in source · 1 passage

Search aliases: C_l,m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1874
mathbb Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise re…
Exact mathematical passage
```
C_{l,n}\simeq C_{m,n}C_{l,m}
```

C_{l,n}Use in source · 1 passage

Search aliases: C_l,n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1874
mathbb Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise re…
Exact mathematical passage
```
C_{l,n}\simeq C_{m,n}C_{l,m}
```

C_{n,k}Use in source · 1 passage

Search aliases: C_n,k

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1920
orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C…
Exact mathematical passage
```
 C_{n,k}\bigl(N\otimes^{\mathbf L}_{B_n}C_{B_n}(a)\bigr)
 \simeq C_{n,k}(N)\otimes^{\mathbf L}_{B_k}C_{B_k}(a).
```

C_{n\to m}Use in source · 2 passages

Search aliases: C_n→ m to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3406
Let the resolution level be n≥1 and take X_n= Z/p^n Z. For m≤ n, coarsening is reduction C_n→ m: Z/p^n Z→ Z/p^m Z. For any polynomial f∈ Z[x], the induced transformation satisfies C_n→ m(f(x)) = f(C_n→ m(x)). Thus the transformation-resolution square commutes exactly. This is a set-level equality, not a cofiber calculation. Regarding the sets as discrete categories gives an identity comparison, while a stable residual requires a separately specified stable realization. This is a simple instance …
Exact mathematical passage
```
C_{n\to m}:\mathbb Z/p^n\mathbb Z\to\mathbb Z/p^m\mathbb Z.
```

C_pUse in source · 13 passages

Search aliases: C_p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2482
The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1
Exact mathematical passage
```
 R=\mathbb Z/p^2,\qquad k=\mathbb F_p,\qquad G=C_p,
```

C_{p^2}Use in source · 1 passage

Search aliases: C_p^2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2989
ion of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_p^2 put b(x)=lfloorfracxprfloor Jpmod p (0≤ x<p^2). Writing each x as its low digit plus p times its high digit shows δ b=-Inf(c): the high-digit carry is the negative of the lo…
Exact mathematical passage
```
C_{p^2}
```

C_qUse in source · 1 passage

Search aliases: C_q

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2884
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
G=C_q
```

C_RUse in source · 15 passages

Search aliases: C_R

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1646
For a commutative ring R and a∈ R, set C_R(a)=[RxrightarrowaR], with the source in cohomological degree -1 and the target in degree 0. The tensor product ⊗_i C_R(a_i) is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see [source reference]. We use them with an explicit sign convention and direct proofs below.
Exact mathematical passage
```
 C_R(a)=\bigl[R\xrightarrow{a}R\bigr],
```

C_{R,S}Use in source · 2 passages

Search aliases: C_R,S

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2461
A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by notation. A fixed ambient universe is understood. A small stable subdiagram can instead be generated by the stated inputs and their images, without m…
Exact mathematical passage
```
C_{R,S}
```

C_{r_{i+1}\to r_i}Use in source · 1 passage

Search aliases: C_r_i+1→ r_i to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3487
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
C_{r_{i+1}\to r_i}X_{i+1}\cong X_i
```

C_{R_n}Use in source · 1 passage

Search aliases: C_R_n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2058
[Transient negative cohomology in the resolution direction] Fix p and an integer a≥1. For n≥ a put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and id…
Exact mathematical passage
```
 H^0C_{R_n}(p^a)\cong\mathbb Z/p^a,
 \qquad H^{-1}C_{R_n}(p^a)\cong\mathbb Z/p^a.
```

C_{r\to r}Use in source · 2 passages

Search aliases: C_r→ r to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 135
Strictness gives C_r→ r=id_ T_r and, for r≤ s≤ t, C_t→ r=C_s→ r∘ C_t→ s. A pseudofunctor version is more natural in some applications. The strict version isolates the main structure without coherence bookkeeping.
Exact mathematical passage
```
C_{r\to r}=\id_{\Tcat_r}
```

C_{r\to u}Use in source · 1 passage

Search aliases: C_r→ u to

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 357
This is the formal version of the two-axis statement mixed movement = within-resolution transformation + resolution movement. Categorically, π: K→ R is the Grothendieck fibration associated with F. Indeed, a map (u,Z)→(s,Y) above u≤ r≤ s is exactly a map Z→ C_s→ uY=C_r→ uC_s→ rY. This uniquely factors through κ_r,s,Y as a morphism to (r,C_s→ rY) above u≤ r, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object Y. It is not a procedur…
Exact mathematical passage
```
Z\to C_{s\to u}Y=C_{r\to u}C_{s\to r}Y
```

C_{s\to u}Use in source · 1 passage

Search aliases: C_s→ u to

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 357
This is the formal version of the two-axis statement mixed movement = within-resolution transformation + resolution movement. Categorically, π: K→ R is the Grothendieck fibration associated with F. Indeed, a map (u,Z)→(s,Y) above u≤ r≤ s is exactly a map Z→ C_s→ uY=C_r→ uC_s→ rY. This uniquely factors through κ_r,s,Y as a morphism to (r,C_s→ rY) above u≤ r, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object Y. It is not a procedur…
Exact mathematical passage
```
Z\to C_{s\to u}Y=C_{r\to u}C_{s\to r}Y
```

C_{\widehat B}Use in source · 1 passage

Search aliases: C_widehat B widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2133
The generic-fiber criterion also gives a positive case. For M=C_widehat B(u) with a nonzero integer u, its selected generic fibers vanish, because u becomes invertible in each Q_p. Thus this comparison is an equivalence on that torsion residual even though it is not an equivalence on the unit. Commutation depends on the object, not only on the existence of coherent finite-level squares. The distinction is consistent with the standard derived-completion fact that a complex on which a defining ele…
Exact mathematical passage
```
M=C_{\widehat B}(u)
```

\Cat_\inftyUse in source · 1 passage

Search aliases: Cat_∞ Cat infty mathbf infty

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3318
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
A^\bullet,B^\bullet:\Delta\to\Cat_\infty
```

\mathsf{Cert}Use in source · 1 passage

Search aliases: Cert mathsf

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3542
The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.
Exact mathematical passage
```
\mathsf S=
(\Seeds,\mathscr T,\mathscr R,\Delta,\Omega,\mathsf{Cert}).
```

\checkUse in source · 1 passage

Search aliases: check

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3258
). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1({U_i}, A) is independent of the chosen local sections; item Ω(P;{U_i})=0 if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of P over U.
Exact mathematical passage
```
    \Omega(P;\{U_i\})=[a]\in\check H^1(\{U_i\},\mathcal A)
    
```

\colimUse in source · 2 passages

Search aliases: colim *colim operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1172
e full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an excisive event diagram X its reduced residual diagram satisfies F_X(I)≃ *colim_e∈ I F_X(downarrow e).
Exact mathematical passage
```
 (j_!G)(I)=\colim_{e\in I}G(e),
 \qquad (j_!G)(\varnothing)=0.
```

\operatorname{colim}Use in source · 1 passage

Search aliases: colim operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2969
and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open nor…
Exact mathematical passage
```
 H^j_{\mathrm{cts}}(\Gamma,V)
 \cong\underset{U'\subseteq U}{\operatorname{colim}}
 H^j(\Gamma/U',V).
```

\colim_{e\in I}Use in source · 2 passages

Search aliases: colim_e∈ I *colim_e∈ I colim in operatorname in

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1172
e full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an excisive event diagram X its reduced residual diagram satisfies F_X(I)≃ *colim_e∈ I F_X(downarrow e).
Exact mathematical passage
```
 (j_!G)(I)=\colim_{e\in I}G(e),
 \qquad (j_!G)(\varnothing)=0.
```

\operatorname{const}Use in source · 2 passages

Search aliases: const operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1122
[Criterion for context-independent event residuals] The following conditions are equivalent. item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e), listed in the order in which that history performs the events. The…
Exact mathematical passage
```
 \operatorname{const}_{\mathcal A_e}E_e(D_e)\longrightarrow E_e
```

\operatorname{const}_{\mathcal A_e}Use in source · 1 passage

Search aliases: const_ A_e operatorname mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1122
[Criterion for context-independent event residuals] The following conditions are equivalent. item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e), listed in the order in which that history performs the events. The…
Exact mathematical passage
```
 \operatorname{const}_{\mathcal A_e}E_e(D_e)\longrightarrow E_e
```

\mathrm{cts}Use in source · 8 passages

Search aliases: cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2947
[Complete continuous one-step obstruction] The class o_n^cts(ρ_n)=[c_A]∈ H^2_cts(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, re…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=[c_A]\in H^2_{\mathrm{cts}}(\Gamma,V)
```

\mathcal DUse in source · 7 passages

Search aliases: D mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1293
[Universal stable obstruction quotient] The quotient q is exact and satisfies: item every operationally killable object maps to zero; item q(X)≃0 if and only if X∈ N_mathscr M; item for every exact functor E: Z→ D into a stable ∞-category such that E(K)≃0 for every K∈ K_mathscr M, there is an exact factorization, unique up to contractible choice together with its comparison to E, Zxrightarrowq O_mathscr Mxrightarrowbar E D. Thus O_mathscr M is universal among exact targets in which all operation…
Exact mathematical passage
```
E:\mathcal Z\to\mathcal D
```

d^2Use in source · 4 passages

Search aliases: d^2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2692
tive certificate. If b_0 is one solution, every other solution is b_0+z with δ z=0. Conjugating the matrices by I+p^nX replaces b(g) by b(g)+X-g· X. These are exactly the negatives of the usual 0-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when X is invariant. The cochain spaces have dimensions (|G|-1)d^2 and (|G|-1)^2d^2. Gaussian elimination decides whether -c_A is in the image of δ. If it is not, elementary linear algebra supplies a functional ann…
Exact mathematical passage
```
(|G|-1)d^2
```

d^iUse in source · 1 passage

Search aliases: d^i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2043
Rvarprojlim_n M_n. Finite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels. Every matrix entry is a compatible residue sequence and therefore determines a unique element of widehat B. The equation d^i+1d^i=0 holds because it holds modulo every D^n and the inverse-limit ring is separated. The same argument proves the equations for a…
Exact mathematical passage
```
d^{i+1}d^i=0
```

d^{i+1}Use in source · 1 passage

Search aliases: d^i+1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2043
Rvarprojlim_n M_n. Finite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels. Every matrix entry is a compatible residue sequence and therefore determines a unique element of widehat B. The equation d^i+1d^i=0 holds because it holds modulo every D^n and the inverse-limit ring is separated. The same argument proves the equations for a…
Exact mathematical passage
```
d^{i+1}d^i=0
```

D^mUse in source · 1 passage

Search aliases: D^m

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1886
Finally D^n is not zero modulo D^m, but is zero modulo D^n. Since Hom_D(B_m)(B_m,B_m)=B_m, this is genuine loss of morphism information, not just a change of representatives.
Exact mathematical passage
```
D^m
```

D^NUse in source · 4 passages

Search aliases: D^N

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2165
For surjectivity take (y_p)∈ G. Choose N large enough that D^Ny_p∈ Z_p for every p. For each p, choose the residue of D^Ny_p modulo p^N. The ordinary Chinese remainder theorem provides z∈ Z with all these residues. Set a=z/D^N∈ A. Then a-y_p∈ Z_p for every p, so (a,(a-y_p)_p) maps to (y_p)_p. This proves exactness.
Exact mathematical passage
```
D^Ny_p\in\mathbb Z_p
```

D^nUse in source · 5 passages

Search aliases: D^n

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1863
Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bo…
Exact mathematical passage
```
 D=\prod_{p\in\mathcal P}p,\qquad B_n=\mathbb Z/D^n\mathbb Z\quad(n\geq1).
```

\mathbb D_{\mathbb S}Use in source · 3 passages

Search aliases: D_ S mathbb mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3608
[Free seeded two-axis double category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying computadic data. If the target carries distinguished subsets for the same labels, restriction gives the analogous bijection with label-preservation impo…
Exact mathematical passage
```
\mathbb D_{\mathbb S}
```

D_0Use in source · 1 passage

Search aliases: D_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1986
Specify finitely many positive scalar labels and a collection I of allowed independent pairs. Include their unit-seed interaction diagrams as the designated tests. This hypothesis ensures that the minimal kernel below is not being inferred from templates that accidentally annihilate an interaction already. Put S={p∈ P:some (a,b)∈ I satisfies pmid a and pmid b}, D_0=∏_p∈ P∖ Sp. Let J_n be the tensor-invariant thick closure of those interactions in Perf(B_n). Positive lengths in [source reference]…
Exact mathematical passage
```
 S=\{p\in\mathcal P:\text{some }(a,b)\in\mathcal I
                    \text{ satisfies }p\mid a\text{ and }p\mid b\},
 \qquad D_0=\prod_{p\in\mathcal P\setminus S}p.
```

d_eUse in source · 2 passages

Search aliases: d_e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1128
nenumerate item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e), listed in the order in which that history performs the events. Their labels and equivalence classes are independent of context, but their ordering a…
Exact mathematical passage
```
 d_e:=\Cofib(X_{D_e}\to X_{\downarrow e}),
```

D_fUse in source · 1 passage

Search aliases: D_f

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1753
First, every pair of incomparable events occurs in an elementary square. Take the ideal D_e∪ D_f. It contains neither e nor f, contains their predecessors, and enables both. The square residual at the unit is K(n_e,n_f).
Exact mathematical passage
```
D_e\cup D_f
```

d_{i,1}Use in source · 1 passage

Search aliases: d_i,1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1812
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,
Exact mathematical passage
```
d_{i,1},\ldots,d_{i,r_i}
```

d_{i,k}Use in source · 3 passages

Search aliases: d_i,k

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1813
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has …
Exact mathematical passage
```
 \epsilon_i=\begin{cases}
 1&a_i+b_i-2r_i>0,\\
 0&a_i+b_i-2r_i=0,
 \end{cases}
 \qquad
 P_i=\{p:p\mid d_{i,k}\text{ for some }k\}.
```

d_{i,r_i}Use in source · 1 passage

Search aliases: d_i,r_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1812
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,
Exact mathematical passage
```
d_{i,1},\ldots,d_{i,r_i}
```

d_{j,l}Use in source · 1 passage

Search aliases: d_j,l

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1850
If no allowed pair has two free summands, every generating tensor is torsion. A free summand tensored with a torsion cone preserves that torsion's prime support. Two torsion summands have tensor product K(d_i,k,d_j,l), whose thick support is the prime intersection by [source reference]. Taking finite direct sums gives exactly the listed sets S_ij. The argument of [source reference] identifies the generated thick subcategory with finite S-primary cohomology and identifies its kernel with localiza…
Exact mathematical passage
```
K(d_{i,k},d_{j,l})
```

d_RUse in source · 4 passages

Search aliases: d_R

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 379
[Resolution ultrametric] Define d_R(X,Y):= 0, · X=Y, 2^-ν(X,Y), · X≠ Y. Then d_R is an ultrametric: d_R(X,Z)≤max{d_R(X,Y),d_R(Y,Z)}.
Exact mathematical passage
```
d_R(X,Y):=
\begin{cases}
0,&X=Y,\\
2^{-\nu(X,Y)},&X\ne Y.
\end{cases}
```

\dashvUse in source · 1 passage

Search aliases: dashv

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3685
Any adjunction Ldashv U induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation α:L∘ F⇒ G has pointwise adjoint mates α_r^flat:F(r)⟶ U(G(r)). Naturality of the unit and counit implies that the family {α_r^flat}_r is natural in r. The mate construction is invertible and respects higher coherence. Applying this to L=St^add gives the result.
Exact mathematical passage
```
L\dashv U
```

\operatorname{DblCat}Use in source · 1 passage

Search aliases: DblCat operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3613
category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying computadic data. If the target carries distinguished subsets for the same labels, restriction gives the analogous bijection with label-preservation imposed on both sides.
Exact mathematical passage
```
\operatorname{DblCat}(\mathbb D_{\mathbb S},\mathbb D)
\cong
\operatorname{DblCptd}(\mathbb S,U\mathbb D),
```

\operatorname{DblCptd}Use in source · 1 passage

Search aliases: DblCptd operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3613
category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying computadic data. If the target carries distinguished subsets for the same labels, restriction gives the analogous bijection with label-preservation imposed on both sides.
Exact mathematical passage
```
\operatorname{DblCat}(\mathbb D_{\mathbb S},\mathbb D)
\cong
\operatorname{DblCptd}(\mathbb S,U\mathbb D),
```

\degUse in source · 1 passage

Search aliases: deg

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3080
[Normalized cyclic lifting criterion] Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(wide…
Exact mathematical passage
```
\deg f_i<d
```

\detUse in source · 3 passages

Search aliases: det

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1714
re calculation is visible: K(a,b)=[ Zxrightarrowbinom-ba Z^2xrightarrow(a b) Z ], degrees -2,-1,0. The product of the two differentials is zero. The kernel of (a b) is generated by (-b/g,a/g), and the incoming image is g times that generator. Both H^-1 and H^0 are Z/g. Equivalently, if au+bv=g, then U=u · -b/g v · a/g, det U=1, (a b)U=(g 0).
Exact mathematical passage
```
 U=\begin{pmatrix}u&-b/g\\v&a/g\end{pmatrix},
 \qquad\det U=1,
 \qquad(a\ b)U=(g\ 0).
```

\operatorname{diag}Use in source · 1 passage

Search aliases: diag operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3200
3 · 2pmod9, B_2=2I+4B_1pmod9. The vector e_2 is cyclic for B_1 modulo 3. The displayed integer representatives of A_1,A_2 do not commute exactly: their additive commutator is diag(-81,81). The criterion nevertheless passes. Retain B_1 integrally and replace B_2 by the exact polynomial 2I+4B_1. This constructs widehat A_1=1 · 3
Exact mathematical passage
```
\operatorname{diag}(-81,81)
```

\dimUse in source · 1 passage

Search aliases: dim

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2893
nameGL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutative product (widetilde A+p^nB)^q. Terms with two or more factor…
Exact mathematical passage
```
p^{\dim_k\ker L_{\bar A}}
```

\dim_kUse in source · 1 passage

Search aliases: dim_k dim

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2893
nameGL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutative product (widetilde A+p^nB)^q. Terms with two or more factor…
Exact mathematical passage
```
p^{\dim_k\ker L_{\bar A}}
```

E'Use in source · 2 passages

Search aliases: E'

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1070
C arrow[r,"u'"'] · D. Write E=Cofib(u) and E'=Cofib(u'). The square induces E→ E'.
Exact mathematical passage
```
E'=\Cofib(u')
```

e_+Use in source · 1 passage

Search aliases: e_+

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2715
M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is invertible in R, the central orthogonal idempotents e_+=(1+τ)/2, e_-=(1-τ)/2 split R[G] into a product of rings. The trivial module R is in the plus factor and M in the minus factor. Every derived Hom between the two factors is zero, as is seen by r…
Exact mathematical passage
```
 e_+=(1+\tau)/2,\qquad e_-=(1-\tau)/2
```

e_-Use in source · 1 passage

Search aliases: e_-

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2715
M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is invertible in R, the central orthogonal idempotents e_+=(1+τ)/2, e_-=(1-τ)/2 split R[G] into a product of rings. The trivial module R is in the plus factor and M in the minus factor. Every derived Hom between the two factors is zero, as is seen by r…
Exact mathematical passage
```
 e_+=(1+\tau)/2,\qquad e_-=(1-\tau)/2
```

\mathcal E_0Use in source · 2 passages

Search aliases: E_0 mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1035
Coarsening symbols occur in their fixed order C_1,…,C_b. A transformation symbol with exactly j coarsening symbols to its left is interpreted as T_i,j. Hence every reachable word represents a functor from E_b to E_0. Moving T_i across C_j reindexes its level rather than creating a new untyped symbol. No rule in this subsection contracts coarsenings or introduces a new exchange relation.
Exact mathematical passage
```
\mathcal E_0
```

E_1Use in source · 1 passage

Search aliases: E_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 732
At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.
Exact mathematical passage
```
X_0\cong X_m\oplus E_m(s)\oplus\cdots\oplus E_1(s)
```

e_1Use in source · 2 passages

Search aliases: e_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3051
Finally, if a 2×2 matrix has no cyclic vector, it takes each vector into its own span. On a basis e_1,e_2 it is diagonal, and applying this condition to e_1+e_2 makes the two diagonal entries equal. Thus it is scalar. This argument includes the field F_2.
Exact mathematical passage
```
e_1,e_2
```

E_{12}Use in source · 3 passages

Search aliases: E_12

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3164
For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde B-widetilde Bwidetilde A ≡ p^2n-2 1 · 0
Exact mathematical passage
```
 A=I+p^{n-1}E_{12},\qquad B=I+p^{n-1}E_{21}.
```

e_2Use in source · 3 passages

Search aliases: e_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3051
Finally, if a 2×2 matrix has no cyclic vector, it takes each vector into its own span. On a basis e_1,e_2 it is diagonal, and applying this condition to e_1+e_2 makes the two diagonal entries equal. Thus it is scalar. This argument includes the field F_2.
Exact mathematical passage
```
e_1,e_2
```

E_{21}Use in source · 3 passages

Search aliases: E_21

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3164
For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde B-widetilde Bwidetilde A ≡ p^2n-2 1 · 0
Exact mathematical passage
```
 A=I+p^{n-1}E_{12},\qquad B=I+p^{n-1}E_{21}.
```

\mathcal E_AUse in source · 2 passages

Search aliases: E_A mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3467
Fix X∈ T_s. First record the exact commutation profile E_A(X;s)={r≤ s:Δ^A_s,r(X)≃0}. No interval or monotonicity property is assumed. For R= N and finite s, a well-defined consecutive coarsening depth is κ_A(X;s):=max{k∈{0,…,s}:Δ^A_s,r(X)≃0 for all s-k≤ r≤ s}. The set is nonempty because Δ^A_s,s(X)≃0. A larger value means that more consecutive coarsenings preserve this exchange. The preceding version's supremum over exact target levels was uninformative on a fixed finite tower because the identi…
Exact mathematical passage
```
\mathcal E_A(X;s)=\{r\le s:\Delta^A_{s,r}(X)\simeq0\}.
```

\mathcal E_bUse in source · 3 passages

Search aliases: E_b mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1035
Coarsening symbols occur in their fixed order C_1,…,C_b. A transformation symbol with exactly j coarsening symbols to its left is interpreted as T_i,j. Hence every reachable word represents a functor from E_b to E_0. Moving T_i across C_j reindexes its level rather than creating a new untyped symbol. No rule in this subsection contracts coarsenings or introduces a new exchange relation.
Exact mathematical passage
```
\mathcal E_b
```

E_iUse in source · 4 passages

Search aliases: E_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 723
[Finite split residual expansion] Fix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then W(s)≅ W_nf(s)⊕⊕_i=1^m E_i(s), where E_i(s) is the surrounding context applied to the source-side remainder of step i. In the stable case, E_i(s)≃ D_i[-1] for the step cofibers D_i defined below.
Exact mathematical passage
```
W(s)\cong W_{\mathrm{nf}}(s)\oplus\bigoplus_{i=1}^m E_i(s),
```

E_mUse in source · 1 passage

Search aliases: E_m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 732
At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.
Exact mathematical passage
```
X_0\cong X_m\oplus E_m(s)\oplus\cdots\oplus E_1(s)
```

E_pUse in source · 4 passages

Search aliases: E_p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2201
bb Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r…
Exact mathematical passage
```
E_p=\theta_p^{-1}(L_p)
```

\operatorname{End}Use in source · 4 passages

Search aliases: End operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2234
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
f_1,\ldots,f_s\in\operatorname{End}_A(V)
```

\operatorname{End}_{\mathbb Z_p}Use in source · 1 passage

Search aliases: End_ Z_p operatorname mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2234
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
g_{p,1},\ldots,g_{p,s}\in\operatorname{End}_{\mathbb Z_p}(L_p)
```

\operatorname{End}_AUse in source · 1 passage

Search aliases: End_A operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2234
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
f_1,\ldots,f_s\in\operatorname{End}_A(V)
```

\operatorname{End}_kUse in source · 2 passages

Search aliases: End_k operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2626
Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^nR_n+1 with k. It is not inversion of p in R_n+1. The square-zero assertion uses 2n≥ n+1.
Exact mathematical passage
```
V=\operatorname{End}_k(k^d)
```

\mathbb FUse in source · 34 passages

Search aliases: F mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1880
. In particular, for m>n, the nonzero endomorphism D^n:B_m→ B_m becomes zero under C_m,n. Exactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over R= Z/p^m is zero if its derived reduction to F_p is zero.
Exact mathematical passage
```
\mathbb F_p
```

\mathcal FUse in source · 12 passages

Search aliases: F mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 969
[Ideal-indexed residual diagram] A coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its terminal object is the total residual Cofib(X_∅→ X_P), independent of the linear extension used to reach it; item ex…
Exact mathematical passage
```
\mathcal F_X:NJ(P)\longrightarrow\mathcal S,
\qquad
\mathcal F_X(I)=\Cofib(X_{\varnothing}\longrightarrow X_I),
```

f_*Use in source · 1 passage

Search aliases: f_*

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3080
[Normalized cyclic lifting criterion] Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(wide…
Exact mathematical passage
```
f_*(t)=t
```

F_0Use in source · 4 passages

Search aliases: F_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 753
[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals D_i.
Exact mathematical passage
```
0=F_0\longrightarrow F_1\longrightarrow\cdots\longrightarrow F_m=D_{\mathrm{tot}}
```

F_1Use in source · 1 passage

Search aliases: F_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 753
[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals D_i.
Exact mathematical passage
```
0=F_0\longrightarrow F_1\longrightarrow\cdots\longrightarrow F_m=D_{\mathrm{tot}}
```

\mathbb F_2Use in source · 4 passages

Search aliases: F_2 mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3051
Finally, if a 2×2 matrix has no cyclic vector, it takes each vector into its own span. On a basis e_1,e_2 it is diagonal, and applying this condition to e_1+e_2 makes the two diagonal entries equal. Thus it is scalar. This argument includes the field F_2.
Exact mathematical passage
```
\mathbb F_2
```

\mathbb F_2^2Use in source · 1 passage

Search aliases: F_2^2 mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4403
icient pairs in the eight listed rings Z/q with q∈{2,3,4,5,8,9,16,25}. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instances, all 30 pairs of subspaces of F_2^2 and F_3^2, 2,430 operator-preservation instances, 6,100 rank-one rational lattice membership tests, and 972 pri…
Exact mathematical passage
```
\mathbb F_2^2
```

\mathbb F_3Use in source · 5 passages

Search aliases: F_3 mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2537
3 · 1, G=C_3. Then A^3=I over R, I+A+A^2=3I, and M^G=3M. The invariant cochain complex is quasi-isomorphic to ( F_3)^2 in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has ( F_3)^2 in every degree. Ordinary tensor of the original nonnegative complex would incorrectly reproduce only the right-hand answer.
Exact mathematical passage
```
(\mathbb F_3)^2
```

\mathbb F_3^2Use in source · 1 passage

Search aliases: F_3^2 mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4403
eight listed rings Z/q with q∈{2,3,4,5,8,9,16,25}. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instances, all 30 pairs of subspaces of F_2^2 and F_3^2, 2,430 operator-preservation instances, 6,100 rank-one rational lattice membership tests, and 972 principal-part reconstr…
Exact mathematical passage
```
\mathbb F_3^2
```

\mathbb F_5Use in source · 2 passages

Search aliases: F_5 mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4357
Cohomology dimensions over F_2, F_3, F_5, F_7 · 29,856
Exact mathematical passage
```
\mathbb F_2,\mathbb F_3,\mathbb F_5,\mathbb F_7
```

\mathbb F_7Use in source · 2 passages

Search aliases: F_7 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4357
Cohomology dimensions over F_2, F_3, F_5, F_7 · 29,856
Exact mathematical passage
```
\mathbb F_2,\mathbb F_3,\mathbb F_5,\mathbb F_7
```

F_iUse in source · 6 passages

Search aliases: F_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 757
[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals D_i.
Exact mathematical passage
```
F_{i-1}\longrightarrow F_i\longrightarrow D_i.
```

F_{i-1}Use in source · 5 passages

Search aliases: F_i-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 757
[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals D_i.
Exact mathematical passage
```
F_{i-1}\longrightarrow F_i\longrightarrow D_i.
```

f_jUse in source · 1 passage

Search aliases: f_j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1821
0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0,
Exact mathematical passage
```
\Cofib(f_i)\otimes^{\mathbf L}\Cofib(f_j)
```

F_mUse in source · 4 passages

Search aliases: F_m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 753
[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals D_i.
Exact mathematical passage
```
0=F_0\longrightarrow F_1\longrightarrow\cdots\longrightarrow F_m=D_{\mathrm{tot}}
```

F_pUse in source · 5 passages

Search aliases: F_p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2212
Here is also an explicit finite construction proving the local identifications. Put F_p=bE_p, so b^2 Z_p^r⊆ F_p⊆ Z_p^r. The Chinese remainder isomorphism identifies Z^r/b^2 Z^r≅ ∏_p∈ P Z_p^r/b^2 Z_p^r. Let H⊆ Z^r be the inverse image of the product of the subgroups F_p/b^2 Z_p^r. Directly from the definitions, H=bΛ. The quotient Z^r/H is the product of the corresponding finite p-primary quotients. Tensoring with Z_p retains exactly the p component. Thus H⊗ Z_p is the inverse image of F_p/b^2 Z_p…
Exact mathematical passage
```
F_p=bE_p
```

\mathbb F_pUse in source · 21 passages

Search aliases: F_p mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1880
. In particular, for m>n, the nonzero endomorphism D^n:B_m→ B_m becomes zero under C_m,n. Exactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over R= Z/p^m is zero if its derived reduction to F_p is zero.
Exact mathematical passage
```
\mathbb F_p
```

\mathbb F_p^\timesUse in source · 1 passage

Search aliases: F_p^× mathbb times

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3146
ting input. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in [source reference] produces exact commuting integral generator matrices. If all generators are scalar, lift their scalar entries integrally. Their residues modulo p have p-power order and therefore equal 1, since F_p^× has order prime to p. The scalar lifts are units and define continuous powers.
Exact mathematical passage
```
\mathbb F_p^\times
```

F_r^{(p)}Use in source · 3 passages

Search aliases: F_r^(p)

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3014
[Unrestricted coefficient lifting for a free pro-p source] Let F_r^(p) be the free pro-p group on finitely many generators. Every continuous F_r^(p)⟶GL_d(R_n) has a continuous lift to GL_d( Z_p). One may choose arbitrary integral lifts of its generator matrices. Construct F_r^(p) as the inverse limit of the finite p-group quotients of the abstract free group on these generators. A map from the generators to a pro-p group induces compatible maps to each of its finite p-group quotients, and hence …
Exact mathematical passage
```
F_r^{(p)}
```

f_sUse in source · 1 passage

Search aliases: f_s

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2234
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
f_1,\ldots,f_s\in\operatorname{End}_A(V)
```

f_vUse in source · 3 passages

Search aliases: f_v

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 324
[Two-axis factorization] Every morphism f:(r,X)⟶(s,Y) in K factors uniquely as (r,X) xrightarrowf_v (r,C_s→ rY) xrightarrowκ_r,s,Y (s,Y), where f_v lies entirely in the fiber T_r.
Exact mathematical passage
```
(r,X)
\xrightarrow{f_v}
(r,C_{s\to r}Y)
\xrightarrow{\kappa_{r,s,Y}}
(s,Y),
```

\mathcal F_XUse in source · 11 passages

Search aliases: F_X mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 969
[Ideal-indexed residual diagram] A coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its terminal object is the total residual Cofib(X_∅→ X_P), independent of the linear extension used to reach it; item ex…
Exact mathematical passage
```
\mathcal F_X:NJ(P)\longrightarrow\mathcal S,
\qquad
\mathcal F_X(I)=\Cofib(X_{\varnothing}\longrightarrow X_I),
```

\mathcal F_{X_M}Use in source · 1 passage

Search aliases: F_X_M mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1671
comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)≃ M⊗^ L_ ZC_ Z(N), N=∏_e∈ Pn_e. This is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vertices, whereas a cube's total cofiber measures their hig…
Exact mathematical passage
```
 \mathcal F_{X_M}(P)\simeq
 M\otimes^{\mathbf L}_{\mathbb Z}C_{\mathbb Z}(N),
 \qquad N=\prod_{e\in P}n_e.
```

\operatorname{Fib}Use in source · 1 passage

Search aliases: Fib operatorname

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2109
For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(widehat B); the field summands need not be perfect over widehat B. In the p-primary factor of B_n, inverting T gives zero if p∈ S and does nothing otherwise…
Exact mathematical passage
```
 \boxed{\operatorname{Fib}(\chi_M)\simeq
 \bigoplus_{p\in S}M\otimes^{\mathbf L}_{\widehat B}\mathbb Q_p.}
```

\flatUse in source · 2 passages

Search aliases: flat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3690
Any adjunction Ldashv U induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation α:L∘ F⇒ G has pointwise adjoint mates α_r^flat:F(r)⟶ U(G(r)). Naturality of the unit and counit implies that the family {α_r^flat}_r is natural in r. The mate construction is invertible and respects higher coherence. Applying this to L=St^add gives the result.
Exact mathematical passage
```
\alpha_r^{\flat}:F(r)\longrightarrow U(G(r)).
```

g_{p,1}Use in source · 1 passage

Search aliases: g_p,1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2234
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
g_{p,1},\ldots,g_{p,s}\in\operatorname{End}_{\mathbb Z_p}(L_p)
```

g_{p,i}Use in source · 2 passages

Search aliases: g_p,i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2235
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
 \theta_p(f_i\otimes1)=(g_{p,i}\otimes1)\theta_p.
```

g_{p,s}Use in source · 1 passage

Search aliases: g_p,s

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2234
[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If the operators and their inverses are supplied compatibly, they glue as automorphisms. For v∈Λ, the displayed identity puts θ_p(f_i v)=g_p,iθ_p(v) in …
Exact mathematical passage
```
g_{p,1},\ldots,g_{p,s}\in\operatorname{End}_{\mathbb Z_p}(L_p)
```

g_RUse in source · 1 passage

Search aliases: g_R

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3496
Instead of a single growth function, separate growth by generator type: g_T(n), g_R(n), g_Δ(n). This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.
Exact mathematical passage
```
g_T(n),\qquad g_R(n),\qquad g_\Delta(n).
```

\mathcal G_sUse in source · 3 passages

Search aliases: G_s mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3894
[Orbit-generator test for an exchange obstruction] Fix an exchange η^A_s,r between exact functors, and an exact quotient q_r of the target fiber. If q_rΔ^A_s,r(g)≃0 for every g∈ G_s, then q_rΔ^A_s,r(X)≃0 for every X∈ Z_s^thick. The same assertion holds for q_r=id, in which case the exchange is an equivalence on the entire generated thick zoo.
Exact mathematical passage
```
q_r\Delta^A_{s,r}(g)\simeq0
\quad\text{for every }g\in\mathcal G_s,
```

g_TUse in source · 1 passage

Search aliases: g_T

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3496
Instead of a single growth function, separate growth by generator type: g_T(n), g_R(n), g_Δ(n). This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.
Exact mathematical passage
```
g_T(n),\qquad g_R(n),\qquad g_\Delta(n).
```

g_\DeltaUse in source · 1 passage

Search aliases: g_Δ Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3496
Instead of a single growth function, separate growth by generator type: g_T(n), g_R(n), g_Δ(n). This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.
Exact mathematical passage
```
g_T(n),\qquad g_R(n),\qquad g_\Delta(n).
```

\gcdUse in source · 4 passages

Search aliases: gcd

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1679
[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other cohomology group vanishes. In particular the interaction vanishes exactly when g=1.
Exact mathematical passage
```
g=\gcd(a_1,\ldots,a_m)
```

\operatorname{GL}_2Use in source · 4 passages

Search aliases: GL_2 operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3133
[Complete rank-two profinite lifting test] Let p be any prime and let ρ_n: Z_p^r→GL_2(R_n) be continuous, with finitely many topological generators as above. item If a=n, all the A_i are scalar and ρ_n has a continuous integral lift. item If a<n, choose any generator A_* with scalar depth a. Take λ_i to be an integer representative of (A_i)_11 and form B_i modulo p^n-a. Then ρ_n lifts over Z_p Longleftrightarrow [B_*,B_i]=0 in M_2(R_n-a) (1≤ i≤ r). The test is finite matrix arithmetic on the exi…
Exact mathematical passage
```
\rho_n:\mathbb Z_p^r\to\operatorname{GL}_2(R_n)
```

\operatorname{GL}_mUse in source · 1 passage

Search aliases: GL_m operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1696
The Euclidean algorithm produces a unimodular matrix U∈GL_m( Z) with (a_1,…,a_m)U=(g,0,…,0). The Koszul complex is the exterior algebra on a free module in degree -1, with differential the derivation determined by this row vector. Hence the maps bigwedge^jU give an isomorphism of complexes from K(g,0,…,0) to K(a_1,…,a_m).
Exact mathematical passage
```
U\in\operatorname{GL}_m(\mathbb Z)
```

\mathbf{Gpd}Use in source · 1 passage

Search aliases: Gpd mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3214
[The source--coefficient diagram has mixed variance] Reduction of coefficients and inflation of source define a functor mathscr P_d: N_≥1^op× Q_Γ ⟶Gpd. Each source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at (m,U') is a compatible refinement of ρ_n at (n,U) precisely when its reduction to level n equals, or is supplied with an isomorphism to, the inflation of ρ_n. There is no total source-coarsening functor on arbitrary representations obtained s…
Exact mathematical passage
```
 \mathscr P_d:\mathbb N_{\geq1}^{op}\times\mathcal Q_\Gamma
 \longrightarrow\mathbf{Gpd}.
```

H^{-1}Use in source · 6 passages

Search aliases: H^-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1713
For two events the entire calculation is visible: K(a,b)=[ Zxrightarrowbinom-ba Z^2xrightarrow(a b) Z ], degrees -2,-1,0. The product of the two differentials is zero. The kernel of (a b) is generated by (-b/g,a/g), and the incoming image is g times that generator. Both H^-1 and H^0 are Z/g. Equivalently, if au+bv=g, then U=u · -b/g
Exact mathematical passage
```
H^{-1}
```

H^{-j}Use in source · 2 passages

Search aliases: H^-j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1687
Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other cohomology group vanishes. In particular the interaction vanishes exactly when g=1.
Exact mathematical passage
```
 H^{-j}K(a_1,\ldots,a_m)\cong
 (\mathbb Z/g)^{\oplus\binom{m-1}{j}}
 \quad(0\le j\le m-1),
```

H^0Use in source · 14 passages

Search aliases: H^0

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1702
Each zero entry contributes C_ Z(0)≅ Z⊕ Z[1] with zero differential. Distributing these m-1 factors gives the stated shifted copies, with the standard shift isomorphisms. Since g>0, the map g: Z→ Z is injective. Thus C_ Z(g) has only H^0= Z/g, proving the cohomology formula and the vanishing criterion.
Exact mathematical passage
```
H^0=\mathbb Z/g
```

H^0_{\mathrm{cts}}Use in source · 1 passage

Search aliases: H^0_cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2954
lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, respectively. The associativity and square-zero computations in [source reference] hold pointwise without a finiteness assumption on the source. All the resulting cochains are co…
Exact mathematical passage
```
H^0_{\mathrm{cts}}
```

H^1_{\mathrm{cts}}Use in source · 1 passage

Search aliases: H^1_cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2954
n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, respectively. The associativity and square-zero computations in [source reference] hold pointwise without a finiteness assumption on the source. All the result…
Exact mathematical passage
```
H^1_{\mathrm{cts}}
```

H^2Use in source · 10 passages

Search aliases: H^2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2564
This is a concrete relation obstruction for a prescribed deformation problem. It is not asserted to classify arbitrary deformation obstructions or to identify the entire unbounded exchange cone with an H^2 obstruction group. In contrast, the norm-quotient actions in [source reference] genuinely exist over Z_p at all precisions, even though some of their finite-level invariant vectors do not lift. These are different failures.
Exact mathematical passage
```
H^2
```

H^2_{\mathrm{cts}}Use in source · 3 passages

Search aliases: H^2_cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2947
[Complete continuous one-step obstruction] The class o_n^cts(ρ_n)=[c_A]∈ H^2_cts(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, re…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=[c_A]\in H^2_{\mathrm{cts}}(\Gamma,V)
```

H^iUse in source · 4 passages

Search aliases: H^i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2497
[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0, 0 · i>0, H^i(V)≅0 · i<0, k^d · i≥0. The natural comparison β:U→ V is zero on H^0. Its functor-intrinsic cofiber satisfies H^iCofib(β)≅ k^d for every i∈ Z. Both composites defining β are exact. The nonzero residual is therefore a genuine defect between fully derived exact fun…
Exact mathematical passage
```
 H^i(U)\cong\begin{cases}k^d&i\leq0,\\0&i>0,\end{cases}
 \qquad
 H^i(V)\cong\begin{cases}0&i<0,\\k^d&i\geq0.\end{cases}
```

H^{i+1}Use in source · 1 passage

Search aliases: H^i+1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2526
The evaluation map on degree-zero invariants is the inclusion pM↪ M. On tensoring with k its degree-zero map is zero. Evaluation on H^0(V)=(k^d)^G=k^d is an isomorphism. Naturality of evaluation defining β therefore proves H^0(β)=0. The long exact cohomology sequence of its cofiber now gives k^d in every degree: the groups in negative degrees come from H^i+1(U), those in positive degrees from H^i(V), and the two middle groups use the zero degree-zero comparison.
Exact mathematical passage
```
H^{i+1}(U)
```

H^jUse in source · 1 passage

Search aliases: H^j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2969
and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open nor…
Exact mathematical passage
```
 H^j_{\mathrm{cts}}(\Gamma,V)
 \cong\underset{U'\subseteq U}{\operatorname{colim}}
 H^j(\Gamma/U',V).
```

H^j_{\mathrm{cts}}Use in source · 1 passage

Search aliases: H^j_cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2969
and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open nor…
Exact mathematical passage
```
 H^j_{\mathrm{cts}}(\Gamma,V)
 \cong\underset{U'\subseteq U}{\operatorname{colim}}
 H^j(\Gamma/U',V).
```

h_GUse in source · 6 passages

Search aliases: h_G

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2799
[Finite horizon for fixed finite-group representations] Let G be a finite group, d,n≥1, and e=v_p(|G|). Set h_G(n)=max{n+e, 2e+1}. For every ρ_n:G→GL_d( Z/p^n), the following are equivalent: item There is a representation ρ:G→GL_d( Z_p) with reduction ρ_n. item There is a representation ρ_h:G→GL_d( Z/p^h_G(n)) with reduction ρ_n. The implication from (2) to (1) is constructive to arbitrary requested precision by [source reference]. It preserves the original level n, not necessarily every digit o…
Exact mathematical passage
```
 \boxed{h_G(n)=\max\{n+e,\,2e+1\}.}
```

\operatorname{Hom}_{D(B_m)}Use in source · 1 passage

Search aliases: Hom_D(B_m) operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1886
Finally D^n is not zero modulo D^m, but is zero modulo D^n. Since Hom_D(B_m)(B_m,B_m)=B_m, this is genuine loss of morphism information, not just a change of representatives.
Exact mathematical passage
```
\operatorname{Hom}_{D(B_m)}(B_m,B_m)=B_m
```

\operatorname{Hom}_OUse in source · 1 passage

Search aliases: Hom_O operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2373
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)
Exact mathematical passage
```
 V=\operatorname{Hom}_O(L,M),\qquad
 (g\cdot f)=\rho_M(g)f\rho_L(g)^{-1}.
```

\operatorname{Hom}_{O[G]}Use in source · 1 passage

Search aliases: Hom_O[G] operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2378
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM) ⟶ · H^1(G,V)[p^n]⟶0 is exact and natural. A modular equivariant map lifts to an integral equivariant map if and only if its boundary class vanishes. If p^h annihilates the p-primary cohomology as in [source reference], this is equivalent to lifting the same map equivariantly to precision n+h.
Exact mathematical passage
```
 \begin{split}
 0\longrightarrow&\operatorname{Hom}_{O[G]}(L,M)/p^n
 \longrightarrow\operatorname{Hom}_{O_n[G]}(L/p^nL,M/p^nM)\\
 \longrightarrow&H^1(G,V)[p^n]\longrightarrow0
 \end{split}
```

\operatorname{Hom}_{O_n[G]}Use in source · 1 passage

Search aliases: Hom_O_n[G] operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2378
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM) ⟶ · H^1(G,V)[p^n]⟶0 is exact and natural. A modular equivariant map lifts to an integral equivariant map if and only if its boundary class vanishes. If p^h annihilates the p-primary cohomology as in [source reference], this is equivalent to lifting the same map equivariantly to precision n+h.
Exact mathematical passage
```
 \begin{split}
 0\longrightarrow&\operatorname{Hom}_{O[G]}(L,M)/p^n
 \longrightarrow\operatorname{Hom}_{O_n[G]}(L/p^nL,M/p^nM)\\
 \longrightarrow&H^1(G,V)[p^n]\longrightarrow0
 \end{split}
```

\operatorname{Hom}_{R[G]}Use in source · 1 passage

Search aliases: Hom_R[G] operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2442
For a commutative ring R, a finite group G, and M∈ D(R[G]), define Γ_R(M)= RHom_R[G](R,M)∈ D(R). This is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or colimit.
Exact mathematical passage
```
 \Gamma_R(M)=\mathbf R\operatorname{Hom}_{R[G]}(R,M)\in D(R).
```

\mathcal IUse in source · 7 passages

Search aliases: I mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1791
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
\mathcal I
```

I'Use in source · 4 passages

Search aliases: I'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1188
Now let F be reduced and excisive. We prove the counit j_!j^*F→ F is an equivalence on an ideal I by induction on |I|. The empty ideal is the zero object on both sides. Choose a maximal element e∈ I. Put I'=I∖{e}. Then I=I'∪downarrow e, I'∩downarrow e=D_e. The ideal-excision lemma expresses F(I) as the pushout of F(I') and F(downarrow e) over F(D_e). The same holds for j_!j^*F. Induction handles I' and D_e, and the terminal-object observation handles downarrow e. Therefore the counit is an equiv…
Exact mathematical passage
```
I'=I\setminus\{e\}
```

\id_{C_{s\to r}Y}Use in source · 2 passages

Search aliases: id_C_s→ rY id to operatorname to

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 314
For r≤ s and Y∈ T_s, define the canonical morphism κ_r,s,Y: (r,C_s→ rY)⟶(s,Y) represented by the identity map id_C_s→ rY.
Exact mathematical passage
```
\id_{C_{s\to r}Y}.
```

\id_kUse in source · 1 passage

Search aliases: id_k id operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 872
The normal-form endpoint alone does not determine a residual. For example, the maps id_k and 0:k→ k have the same source and target, but cofibers 0 and k⊕ k[1]. Comparison maps therefore belong to the data.
Exact mathematical passage
```
\id_k
```

\id_MUse in source · 1 passage

Search aliases: id_M id operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1654
Fix a finite poset P and assign a positive integer n_e to every event e∈ P. For M∈Perf( Z) define X_M(I)=M, X_M(I⊆ J)=(∏_e∈ J∖ In_e)id_M. Empty products are 1. For nested ideals these products multiply exactly, so X_M is a strict chain-level diagram followed by the stable realization. When P=P_W, take identity transformation and coarsening functors and let the exchange cell for event e be multiplication by n_e. This realizes the diagram inside the typed two-axis grammar of [source reference]. Th…
Exact mathematical passage
```
 X_M(I)=M,\qquad
 X_M(I\subseteq J)=\left(\prod_{e\in J\setminus I}n_e\right)\id_M.
```

\id_{\Tcat_r}Use in source · 1 passage

Search aliases: id_Tcat_r id_ T_r id Tcat operatorname mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 135
Strictness gives C_r→ r=id_ T_r and, for r≤ s≤ t, C_t→ r=C_s→ r∘ C_t→ s. A pseudofunctor version is more natural in some applications. The strict version isolates the main structure without coherence bookkeeping.
Exact mathematical passage
```
C_{r\to r}=\id_{\Tcat_r}
```

\id_XUse in source · 1 passage

Search aliases: id_X id operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 281
Identity at (r,X) is id_X:X→ C_r→ r(X)=X. For associativity, let (r,X)xrightarrowα(s,Y) xrightarrowβ(t,Z) xrightarrowγ(u,W). One composite gives C_t→ r(γ)∘ C_s→ r(β)∘α. The other gives the same expression by functoriality and C_t→ r=C_s→ rC_t→ s. Thus composition is associative.
Exact mathematical passage
```
\id_X:X\to C_{r\to r}(X)=X
```

\iffUse in source · 2 passages

Search aliases: iff

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1359
[When operational killability is already thick] If mathscr M consists of exact endofunctors and has the directed annihilator property, then K_mathscr M is a thick stable subcategory. Hence N_mathscr M= K_mathscr M and Obs_mathscr M(X)≃0 iff X is operationally killable.
Exact mathematical passage
```
\boxed{
\Obs_{\mathscr M}(X)\simeq0
\iff
X\text{ is operationally killable}.}
```

\operatorname{im}Use in source · 2 passages

Search aliases: im operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2302
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 \boxed{\operatorname{im}(K_{n+h}\longrightarrow K_n)
       =\operatorname{im}(K/p^nK\longrightarrow K_n).}
```

\operatorname{Inf}Use in source · 2 passages

Search aliases: Inf operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2962
[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets o…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=0
 \quad\Longleftrightarrow\quad
 \exists\, U'\triangleleft_{\mathrm{open}}\Gamma,\ U'\subseteq U:
 \operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}[c_A]=0.
```

\operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}Use in source · 1 passage

Search aliases: Inf_Γ/U^Γ/U' operatorname Gamma Gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2962
[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets o…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=0
 \quad\Longleftrightarrow\quad
 \exists\, U'\triangleleft_{\mathrm{open}}\Gamma,\ U'\subseteq U:
 \operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}[c_A]=0.
```

j^*Use in source · 2 passages

Search aliases: j^*

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1188
Now let F be reduced and excisive. We prove the counit j_!j^*F→ F is an equivalence on an ideal I by induction on |I|. The empty ideal is the zero object on both sides. Choose a maximal element e∈ I. Put I'=I∖{e}. Then I=I'∪downarrow e, I'∩downarrow e=D_e. The ideal-excision lemma expresses F(I) as the pushout of F(I') and F(downarrow e) over F(D_e). The same holds for j_!j^*F. Induction handles I' and D_e, and the terminal-object observation handles downarrow e. Therefore the counit is an equiv…
Exact mathematical passage
```
j_!j^*F\to F
```

J^jUse in source · 1 passage

Search aliases: J^j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2905
[A profinite lift need not preserve the finite quotient] Let p be odd and J∈ M_d( Z_p) be invertible modulo p. The action C_p→GL_d( Z/p^2) given by I+pJ has no integral lift as an action of C_p. Its pullback along Z_p↠ C_p, however, has a continuous integral lift ρ(x)=(I+pJ)^x =∑_j≥0binomxjp^jJ^j, x∈ Z_p. For each j, the polynomial binomxj is continuous on Z_p and takes values there: approximate x by nonnegative integers, where the binomial coefficient is integral, and use closedness of Z_p in Q…
Exact mathematical passage
```
 \rho(x)=(I+pJ)^x
 =\sum_{j\geq0}\binom{x}{j}p^jJ^j,
 \qquad x\in\mathbb Z_p.
```

j_{\mathcal B}Use in source · 1 passage

Search aliases: j_ B mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3714
The universal property above is an equivalence of functor ∞-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend η:f⇒ g between additive functors A→ B, compose it with j_ B: B→St^add( B) and lift the resulting morphism through Fun^ex(St^add( A),St^add( B)) ≃Fun^⊕( A,St^add( B)). Full faithfulness transports the supplied composition coherences as well. Accordingly A_r, C_s→ r, and their coherently supplied exchange cells extend, essentially…
Exact mathematical passage
```
j_{\mathcal B}:\mathcal B\to\operatorname{St}^{\mathrm{add}}(\mathcal B)
```

j_!Use in source · 6 passages

Search aliases: j_!

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1172
e full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an excisive event diagram X its reduced residual diagram satisfies F_X(I)≃ *colim_e∈ I F_X(downarrow e).
Exact mathematical passage
```
 (j_!G)(I)=\colim_{e\in I}G(e),
 \qquad (j_!G)(\varnothing)=0.
```

\KUse in source · 10 passages

Search aliases: K mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 244
[Total Komorebi category] Given F: R^op→Cat, define the category K:=∫_ RF as follows.
Exact mathematical passage
```
\K:=\int_{\Rcat}F
```

\mathbb KUse in source · 1 passage

Search aliases: K mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3351
Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.
Exact mathematical passage
```
\mathbb K_{\mathrm{arith}}
\qquad\text{and}\qquad
\mathbb K_{\mathrm{aut}}
```

k^2Use in source · 2 passages

Search aliases: k^2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 814
Work in the stable category of perfect complexes over a field k. Put M=k[0] and N=k^2[0], and consider the square [column sep=large,row sep=large] 0 arrow[r] arrow[d] · M arrow[d,hook]
Exact mathematical passage
```
N=k^2[0]
```

k^dUse in source · 11 passages

Search aliases: k^d

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2493
[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,
Exact mathematical passage
```
 \Gamma_R(M)\simeq pM[0]\cong k^d[0].
```

\mathbb K_{\mathrm{arith}}Use in source · 1 passage

Search aliases: K_arith mathbb mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3351
Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.
Exact mathematical passage
```
\mathbb K_{\mathrm{arith}}
\qquad\text{and}\qquad
\mathbb K_{\mathrm{aut}}
```

\mathbb K_{\mathrm{aut}}Use in source · 1 passage

Search aliases: K_aut mathbb mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3351
Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.
Exact mathematical passage
```
\mathbb K_{\mathrm{arith}}
\qquad\text{and}\qquad
\mathbb K_{\mathrm{aut}}
```

K_{B_n}Use in source · 1 passage

Search aliases: K_B_n

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1925
_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step …
Exact mathematical passage
```
 N\otimes^{\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\qquad
 K_R(u,v)=C_R(u)\otimes^{\mathbf L}_R C_R(v).
```

K_mUse in source · 1 passage

Search aliases: K_m

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2272
its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural comparison α_n is Q[p^n], with these specified transition maps. Its cone, viewing both modules in degree zero in D(O_n), is canonically equivalent to th…
Exact mathematical passage
```
r_{m,n}:K_m\to K_n
```

\mathcal K_{\mathscr M_A}Use in source · 1 passage

Search aliases: K_mathscr M_A mathcal mathscr

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1399
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\Lambda(\mathcal K_{\mathscr M_A})\subseteq\mathcal N_{\mathscr M_B},
```

K_nUse in source · 10 passages

Search aliases: K_n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2256
For a map d:V→ W of lattices, put K=ker d, Q=cokerd, K_n=ker(d⊗_O O_n). The module K is saturated in V: if p^n v∈ K, then p^n d(v)=0 in the torsion-free module W, so v∈ K. It is therefore a finite free direct summand as an O-module, although a splitting is not canonical. Reduction supplies a natural injection α_n:K/p^nK→ K_n.
Exact mathematical passage
```
 K=\ker d,\qquad Q=\operatorname{coker}d,\qquad
 K_n=\ker(d\otimes_O O_n).
```

K_{n+h}Use in source · 2 passages

Search aliases: K_n+h

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2302
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 \boxed{\operatorname{im}(K_{n+h}\longrightarrow K_n)
       =\operatorname{im}(K/p^nK\longrightarrow K_n).}
```

K_RUse in source · 3 passages

Search aliases: K_R

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1925
_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step …
Exact mathematical passage
```
 N\otimes^{\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\qquad
 K_R(u,v)=C_R(u)\otimes^{\mathbf L}_R C_R(v).
```

\mathcal K_RUse in source · 1 passage

Search aliases: K_R mathcal

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2457
A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by notation. A fixed ambient universe is understood. A small stable subdiagram can instead be generated by the stated inputs and their images, without m…
Exact mathematical passage
```
 \mathcal K_R=D(R[G])\times D(R),\qquad
 A_R(M,N)=(0,\Gamma_R(M)),
```

\kerUse in source · 6 passages

Search aliases: ker

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 633
In an abelian fiber one may instead record Δ^-_s,r(X)=kerη^A_s,r,X, Δ^+_s,r(X)=cokerη^A_s,r,X. These vanish simultaneously exactly when the comparison is an isomorphism.
Exact mathematical passage
```
\Delta^-_{s,r}(X)=\ker\eta^A_{s,r,X},
\qquad
\Delta^+_{s,r}(X)=\operatorname{coker}\eta^A_{s,r,X}.
```

lUse in source · 7 passages

Search aliases: l

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 559
The square is [column sep=large,row sep=large] A_rC_s→ r(X) arrow[r,"η^A_s,r,X"] · C_s→ rA_s(X). It compares transforming after coarsening with coarsening after transforming.
Exact mathematical passage
```
\begin{tikzcd}[column sep=large,row sep=large]
A_rC_{s\to r}(X) \arrow[r,"\eta^A_{s,r,X}"] & C_{s\to r}A_s(X).
\end{tikzcd}
```

L_0Use in source · 1 passage

Search aliases: L_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1050
Evaluate this ordinary functor on s, take its nerve, and compose with L_0. An ordinary functor gives all simplices of its nerve, so the output is a coherent diagram, not merely a diagram in a homotopy category. The prescribed vertices and edges are unchanged by this construction.
Exact mathematical passage
```
L_0
```

L_{\bar A}Use in source · 4 passages

Search aliases: L_bar A bar

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2885
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
 c=\frac{\widetilde A^q-I}{p^n}\pmod p,
 \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
```

L_IUse in source · 1 passage

Search aliases: L_I

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2899
For A=I+pJ modulo p^2 and q=p, the residual is c=J while L_I(B)=pB=0 in k. This recovers the one-step obstruction without exhaustive testing. At other inputs the Jacobian may have nonzero rank, so some nonzero initial relation residuals are removable rather than persistent.
Exact mathematical passage
```
L_I(B)=pB=0
```

L_pUse in source · 8 passages

Search aliases: L_p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2184
[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence f…
Exact mathematical passage
```
L_p
```

\leadstoUse in source · 3 passages

Search aliases: leadsto

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 879
c diamond criterion] Let a rewrite graph have a height h in N that decreases on every edge, and suppose every vertex x has a unique reachable normal vertex n(x). Assign an object X_x in a stable ∞-category to each vertex and a morphism to each edge. Assume every elementary peak xxrightarrowey, xxrightarrowfz has joining paths a:yleadsto u and b:zleadsto u whose semantic composites satisfy [a] [e]=[b] [f] in the homotopy category. Then every normalization path from x to n(x) induces the same homo…
Exact mathematical passage
```
a:y\leadsto u
```

\lfloorUse in source · 2 passages

Search aliases: lfloor

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2981
[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_…
Exact mathematical passage
```
 c(a,b)=\left\lfloor\frac{a+b}{p}\right\rfloor J\pmod p.
```

\limUse in source · 1 passage

Search aliases: lim

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3319
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
\lim_{[n]\in\Delta} A^n\simeq\lim_{[n]\in\Delta} B^n.
```

\lim_{[n]\in\Delta}Use in source · 1 passage

Search aliases: lim_[n]∈Δ lim in Delta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3319
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
\lim_{[n]\in\Delta} A^n\simeq\lim_{[n]\in\Delta} B^n.
```

M^GUse in source · 3 passages

Search aliases: M^G

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2331
Let G be a finite group acting O-linearly on a lattice M. The natural comparison is α_G,n:M^G/p^nM^G⟶(M/p^nM)^G. The first path takes invariant vectors and then reduces. The second reduces the representation and then takes invariant vectors. Its direction is fixed by reduction of an invariant vector; no arbitrary scalar enters this map.
Exact mathematical passage
```
 \alpha_{G,n}:M^G/p^nM^G\longrightarrow(M/p^nM)^G.
```

m_0Use in source · 1 passage

Search aliases: m_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2788
[ ρ:G⟶GL_d( Z_p) with ρ(g)≡ A_gpmodp^m-e. After j iterations the certified defect depth is m_j=2e+2^j(m-2e), and the next change has depth at least m_j-e. In particular, the sequence converges uniformly on G and all prescribed multiplication relations hold exactly in the limit. Set m_0=m and m_j+1=2m_j-2e. Induction gives the displayed formula, so m_j>2e at every stage and m_j→∞. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinit…
Exact mathematical passage
```
m_0=m
```

M_2Use in source · 4 passages

Search aliases: M_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3125
For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).
Exact mathematical passage
```
A\in M_2(R_n)
```

M_AUse in source · 6 passages

Search aliases: M_A

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1394
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
(\mathcal Z_A,\mathscr M_A)
```

M_BUse in source · 5 passages

Search aliases: M_B

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1394
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
(\mathcal Z_B,\mathscr M_B)
```

M_dUse in source · 23 passages

Search aliases: M_d

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2485
The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1
Exact mathematical passage
```
J\in M_d(\mathbb Z)
```

m_jUse in source · 6 passages

Search aliases: m_j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2782
[Convergent finite-group action repair] If the initial defect depth satisfies m>2e, the preceding averaging operation iterates to a representation ρ:G⟶GL_d( Z_p) with ρ(g)≡ A_gpmodp^m-e. After j iterations the certified defect depth is m_j=2e+2^j(m-2e), and the next change has depth at least m_j-e. In particular, the sequence converges uniformly on G and all prescribed multiplication relations hold exactly in the limit. Set m_0=m and m_j+1=2m_j-2e. Induction gives the displayed formula, so m_j>2…
Exact mathematical passage
```
 m_j=2e+2^j(m-2e),
```

m_{j+1}Use in source · 1 passage

Search aliases: m_j+1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2788
ngrightarrowGL_d( Z_p) with ρ(g)≡ A_gpmodp^m-e. After j iterations the certified defect depth is m_j=2e+2^j(m-2e), and the next change has depth at least m_j-e. In particular, the sequence converges uniformly on G and all prescribed multiplication relations hold exactly in the limit. Set m_0=m and m_j+1=2m_j-2e. Induction gives the displayed formula, so m_j>2e at every stage and m_j→∞. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to i…
Exact mathematical passage
```
m_{j+1}=2m_j-2e
```

M_nUse in source · 2 passages

Search aliases: M_n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2034
[Uniform based-complex tower reconstruction] Suppose M_n are based finite free cochain complexes over B_n, concentrated in a common finite interval with fixed ranks in each degree. Suppose their differential matrices reduce exactly from level m to level n for m≥ n. Then there is a unique based complex widehat M with those ranks over widehat B whose matrices reduce to the supplied ones.
Exact mathematical passage
```
M_n
```

M_n^iUse in source · 1 passage

Search aliases: M_n^i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2046
Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map 1-shift:∏_n M_n^i⟶∏_n M_n^i is surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the homotopy fiber is represented by that kernel complex. This is the degreewise Mittag--Leffler case of derived inverse limits [source reference].
Exact mathematical passage
```
 1-\mathrm{shift}:\prod_n M_n^i\longrightarrow\prod_n M_n^i
```

\operatorname{Map}Use in source · 1 passage

Search aliases: Map operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3675
hbfCat^st), F⟼ St^add∘ F. It is left adjoint to pointwise forgetting U_ R: Fun( R^op,Cat^st) ⟶ Fun( R^op,Cat^add). Equivalently, for every additive two-axis system F and stable two-axis system G there is a natural equivalence Map (St^add_ RF,G) ≃ Map (F,U_ RG).
Exact mathematical passage
```
\operatorname{Map}
\bigl(\operatorname{St}^{\mathrm{add}}_{\Rcat}F,G\bigr)
\simeq
\operatorname{Map}
\bigl(F,U_{\Rcat}G\bigr).
```

\maxUse in source · 7 passages

Search aliases: max

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 387
2^-ν(X,Y), · X≠ Y. Then d_R is an ultrametric: d_R(X,Z)≤max{d_R(X,Y),d_R(Y,Z)}.
Exact mathematical passage
```
d_R(X,Z)\le\max\{d_R(X,Y),d_R(Y,Z)\}.
```

\midUse in source · 5 passages

Search aliases: mid

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1731
For the fixed labelled poset define the finite prime set S_P=⋃_eparallel f {p prime:pmidgcd(n_e,n_f)}, where eparallel f means incomparable. Let J_P be the invariant thick closure of the square interaction residuals for the unit seed Z.
Exact mathematical passage
```
 S_P=\bigcup_{e\parallel f}
 \{p\text{ prime}:p\mid\gcd(n_e,n_f)\},
```

\min_iUse in source · 1 passage

Search aliases: min_i min

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3130
For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).
Exact mathematical passage
```
a=\min_i a(A_i)
```

n_1Use in source · 2 passages

Search aliases: n_1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1791
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
n_1,\ldots,n_t
```

n_eUse in source · 9 passages

Search aliases: n_e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1653
Fix a finite poset P and assign a positive integer n_e to every event e∈ P. For M∈Perf( Z) define X_M(I)=M, X_M(I⊆ J)=(∏_e∈ J∖ In_e)id_M. Empty products are 1. For nested ideals these products multiply exactly, so X_M is a strict chain-level diagram followed by the stable realization. When P=P_W, take identity transformation and coarsening functors and let the exchange cell for event e be multiplication by n_e. This realizes the diagram inside the typed two-axis grammar of [source reference]. Th…
Exact mathematical passage
```
n_e
```

n_fUse in source · 5 passages

Search aliases: n_f

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1664
For an enabled event e, E_e(I)≃ M⊗^ L_ ZC_ Z(n_e). The comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)≃ M⊗^ L_ ZC_ Z(N), N=∏_e∈ Pn_e. This is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vert…
Exact mathematical passage
```
n_f
```

n_iUse in source · 2 passages

Search aliases: n_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1792
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
 S_{\mathcal I}=\bigcup_{(i,j)\in\mathcal I}
 \{p:p\mid\gcd(n_i,n_j)\}.
```

n_jUse in source · 2 passages

Search aliases: n_j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1792
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
 S_{\mathcal I}=\bigcup_{(i,j)\in\mathcal I}
 \{p:p\mid\gcd(n_i,n_j)\}.
```

n_mUse in source · 1 passage

Search aliases: n_m

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4340
quare's commutativity is not its excision. The second cofiber vanishes precisely for a homotopy pushout, not for every commuting square. item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the Z/p^2 chain is an explicit counterexample. item The total normalization residual C(N) and a cube's total cofiber K(n_1,…,n_m) are different objects. The (6,10,15) example refutes an inference from highest interaction vanishing t…
Exact mathematical passage
```
K(n_1,\ldots,n_m)
```

\mathcal N_{\mathscr M_A}Use in source · 3 passages

Search aliases: N_mathscr M_A mathcal mathscr

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1406
mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\mathcal N_{\mathscr M_A}
```

\mathcal N_{\mathscr M_B}Use in source · 3 passages

Search aliases: N_mathscr M_B mathcal mathscr

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1399
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\Lambda(\mathcal K_{\mathscr M_A})\subseteq\mathcal N_{\mathscr M_B},
```

n_tUse in source · 1 passage

Search aliases: n_t

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1791
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
n_1,\ldots,n_t
```

\mathbb N_{\geq1}^{op}Use in source · 1 passage

Search aliases: N_≥1^op mathbb geq

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3214
[The source--coefficient diagram has mixed variance] Reduction of coefficients and inflation of source define a functor mathscr P_d: N_≥1^op× Q_Γ ⟶Gpd. Each source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at (m,U') is a compatible refinement of ρ_n at (n,U) precisely when its reduction to level n equals, or is supplied with an isomorphism to, the inflation of ρ_n. There is no total source-coarsening functor on arbitrary representations obtained s…
Exact mathematical passage
```
 \mathscr P_d:\mathbb N_{\geq1}^{op}\times\mathcal Q_\Gamma
 \longrightarrow\mathbf{Gpd}.
```

\nleqUse in source · 1 passage

Search aliases: nleq

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 259
For r≤ s, a morphism (r,X)⟶(s,Y) is a morphism α:X⟶ C_s→ r(Y) in T_r. If rnleq s, there is no such morphism in this directed version.
Exact mathematical passage
```
r\nleq s
```

\nmidUse in source · 1 passage

Search aliases: nmid

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2817
For pnmid|G|, one has e=0 and h_G(n)=n. Every finite-level representation in this fixed finite-group problem then lifts. When e>0, the test is still finite, but its number of branches may be enormous. No polynomial-time or uniformly practical algorithm is claimed.
Exact mathematical passage
```
p\nmid|G|
```

\notUse in source · 10 passages

Search aliases: not

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1330
ex:killable-not-thick Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so both X and Y are operationally killable. However X⊕ Y=(A,A,0), and both U and V send this object to (0,0,A), which is fixed by U and V. Hence no construc…
Exact mathematical passage
```
0\not\simeq A\in\mathcal C
```

\mathcal O_{\mathscr M_A}Use in source · 1 passage

Search aliases: O_mathscr M_A mathcal mathscr

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1403
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\bar\Lambda:\mathcal O_{\mathscr M_A}\longrightarrow\mathcal O_{\mathscr M_B}.
```

\mathcal O_{\mathscr M_B}Use in source · 1 passage

Search aliases: O_mathscr M_B mathcal mathscr

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1403
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\bar\Lambda:\mathcal O_{\mathscr M_A}\longrightarrow\mathcal O_{\mathscr M_B}.
```

o_nUse in source · 3 passages

Search aliases: o_n

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2649
[Complete one-step relation obstruction] The residual c_A is a 2-cocycle. Its cohomology class o_n(ρ_n)=[c_A]∈ H^2(G,V) is independent of the chosen matrix lifts. Moreover, o_n(ρ_n)=0 Longleftrightarrow ρ_n lifts to GL_d(R_n+1). More precisely, all lifts with the fixed reduction have the form A_g^b=(I+p^n b(g))A_g, where representatives of b(g) may be chosen arbitrarily, and the necessary and sufficient equation is δ b=-c_A. Associativity of the threefold product gives the exact identity (I+p^nc…
Exact mathematical passage
```
 o_n(\rho_n)=[c_A]\in H^2(G,V)
```

o_n^{\mathrm{cts}}Use in source · 2 passages

Search aliases: o_n^cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2947
[Complete continuous one-step obstruction] The class o_n^cts(ρ_n)=[c_A]∈ H^2_cts(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, re…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=[c_A]\in H^2_{\mathrm{cts}}(\Gamma,V)
```

\Obs'Use in source · 1 passage

Search aliases: Obs' Obs operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3359
A strong correspondence should include: Λ(TX) · ≃ T'Λ(X), Λ(CX) · ≃ C'Λ(X), Λ(Δ_T,CX) · ≃Δ_T',C'Λ(X), barΛ(Obs X) · ≃Obs'Λ(X).
Exact mathematical passage
```
\Lambda(TX)&\simeq T'\Lambda(X),\\
\Lambda(CX)&\simeq C'\Lambda(X),\\
\Lambda(\Delta_{T,C}X)&\simeq\Delta_{T',C'}\Lambda(X),\\
\bar\Lambda(\Obs X)&\simeq\Obs'\Lambda(X).
```

\mathrm{open}Use in source · 1 passage

Search aliases: open mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2962
[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets o…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=0
 \quad\Longleftrightarrow\quad
 \exists\, U'\triangleleft_{\mathrm{open}}\Gamma,\ U'\subseteq U:
 \operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}[c_A]=0.
```

p^2Use in source · 32 passages

Search aliases: p^2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1209
For an antichain, each principal ideal is a singleton, D_e=∅, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take P to be a two-element chain and a reduced diagram in Perf( Z) 0⟶ Z/pxrightarrow p Z/p^2 for a prime p, where the second map sends the class of x to the class of px. Both successive cofibers are Z/p in degree zero, but the terminal object is not Z/p⊕ Z/p: the latter is annihilated by p and the former is not. There are no independent-ev…
Exact mathematical passage
```
 0\longrightarrow\mathbb Z/p\xrightarrow{\,p\,}\mathbb Z/p^2
```

p^{2a}Use in source · 7 passages

Search aliases: p^2a

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2769
] Consequently E(g,h)+E(gh,t)-α_g E(h,t)-E(g,ht) ∈ p^2mM_d( Z_p). Sum over t and divide by N. The permutation t↦ ht identifies the last average with b(g), giving E(g,h)-b(g)-α_g b(h)+b(gh) ∈ p^2m-eM_d( Z_p). Put a=m-e>0. The repaired multiplicativity product is exactly (I-b(g))(I-α_g b(h))(I+E(g,h))(I-b(gh))^-1. Modulo p^2a its difference from I is the linear expression just displayed: products of two corrections vanish at that precision, and (I-b)^-1≡ I+bpmodp^2a. That expression is divisible b…
Exact mathematical passage
```
p^{2a}
```

p^{2m}Use in source · 1 passage

Search aliases: p^2m

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2756
Associativity gives an identity without assuming an exact adjoint action: (I+E(g,h))(I+E(gh,t)) = (I+α_g E(h,t))(I+E(g,ht)). Consequently E(g,h)+E(gh,t)-α_g E(h,t)-E(g,ht) ∈ p^2mM_d( Z_p). Sum over t and divide by N. The permutation t↦ ht identifies the last average with b(g), giving E(g,h)-b(g)-α_g b(h)+b(gh) ∈ p^2m-eM_d( Z_p). Put a=m-e>0. The repaired multiplicativity product is exactly (I-b(g))(I-α_g b(h))(I+E(g,h))(I-b(gh))^-1. Modulo p^2a its difference from I is the linear expression just…
Exact mathematical passage
```
 E(g,h)+E(gh,t)-\alpha_g E(h,t)-E(g,ht)
 \in p^{2m}M_d(\mathbb Z_p).
```

p^{2m-2e}Use in source · 1 passage

Search aliases: p^2m-2e

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2741
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
 \boxed{A'_gA'_h(A'_{gh})^{-1}-I\in
 p^{2m-2e}M_d(\mathbb Z_p).}
```

p^{2m-e}Use in source · 2 passages

Search aliases: p^2m-e

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2761
Associativity gives an identity without assuming an exact adjoint action: (I+E(g,h))(I+E(gh,t)) = (I+α_g E(h,t))(I+E(g,ht)). Consequently E(g,h)+E(gh,t)-α_g E(h,t)-E(g,ht) ∈ p^2mM_d( Z_p). Sum over t and divide by N. The permutation t↦ ht identifies the last average with b(g), giving E(g,h)-b(g)-α_g b(h)+b(gh) ∈ p^2m-eM_d( Z_p). Put a=m-e>0. The repaired multiplicativity product is exactly (I-b(g))(I-α_g b(h))(I+E(g,h))(I-b(gh))^-1. Modulo p^2a its difference from I is the linear expression just…
Exact mathematical passage
```
 E(g,h)-b(g)-\alpha_g b(h)+b(gh)
 \in p^{2m-e}M_d(\mathbb Z_p).
```

p^{2n-1}Use in source · 1 passage

Search aliases: p^2n-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3169
For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde B-widetilde Bwidetilde A ≡ p^2n-2 1 · 0 0 · -1 not≡0pmodp^2n-1. Indeed every correction term has valuation at least 2n-1. This proves nonextension at …
Exact mathematical passage
```
 \widetilde A\widetilde B-\widetilde B\widetilde A
 \equiv p^{2n-2}
 \begin{pmatrix}1&0\\0&-1\end{pmatrix}
 \not\equiv0\pmod{p^{2n-1}}.
```

p^{2n-2}Use in source · 2 passages

Search aliases: p^2n-2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3167
For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde B-widetilde Bwidetilde A ≡ p^2n-2 1 · 0
Exact mathematical passage
```
p^{2n-2}
```

p^3Use in source · 12 passages

Search aliases: p^3

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2552
[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite free Z_p-representation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ…
Exact mathematical passage
```
(\mathbb Z/p^3)^d
```

p^{a_1}Use in source · 1 passage

Search aliases: p^a_1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2288
Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.
Exact mathematical passage
```
p^{a_1},\ldots,p^{a_r}
```

p^{a_i}Use in source · 1 passage

Search aliases: p^a_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2315
The residual system (Q[p^n],p^m-n) is pro-zero, and Kxrightarrow∼varprojlim_n K_n. Here pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish. In Smith coordinates a nonzero diagonal entry imposes p^a_ix_i=0 modulo p^n, giving min(a_i,n) units of O-length. Each of the v-r zero columns contributes a free O_n coordinate. This proves the formulas.
Exact mathematical passage
```
p^{a_i}x_i=0
```

p^{a_r}Use in source · 1 passage

Search aliases: p^a_r

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2288
Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.
Exact mathematical passage
```
p^{a_1},\ldots,p^{a_r}
```

p^cUse in source · 1 passage

Search aliases: p^c

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2319
If c<h, choose an element of exact order p^h in Q_tors and take n≥ h. Surjectivity of δ_n+c supplies a preimage. Its reduction to level n has nonzero boundary after multiplication by p^c, so it is not the reduction of an exact solution. This proves sharpness. Multiplication by p^h also makes every transition with gap at least h zero on residuals.
Exact mathematical passage
```
p^c
```

p^{\dim_k\ker L_{\bar A}}Use in source · 1 passage

Search aliases: p^dim_kker L_bar A dim ker bar

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2893
nameGL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutative product (widetilde A+p^nB)^q. Terms with two or more factor…
Exact mathematical passage
```
p^{\dim_k\ker L_{\bar A}}
```

p^eUse in source · 2 passages

Search aliases: p^e

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2346
M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^1(G,M) is finite and killed by |G|. In particular it is killed by p^e, where e=v_p(|G|), and the delay e is a valid uniform lifting bound.
Exact mathematical passage
```
p^e
```

p^{h-e}Use in source · 1 passage

Search aliases: p^h-e

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2810
, not necessarily every digit of the chosen level-h witness. Reduction proves (1) implies (2). Conversely, lift every matrix of ρ_h arbitrarily to an invertible integral matrix, keeping the identity equal to I. Its multiplicativity defects are divisible by p^h. Since h>2e, convergent repair applies and returns an action agreeing with these matrices modulo p^h-e. Because h-e≥ n, it returns precisely the specified ρ_n on reduction.
Exact mathematical passage
```
p^{h-e}
```

p^{h_G(n)}Use in source · 1 passage

Search aliases: p^h_G(n)

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2805
group representations] Let G be a finite group, d,n≥1, and e=v_p(|G|). Set h_G(n)=max{n+e, 2e+1}. For every ρ_n:G→GL_d( Z/p^n), the following are equivalent: item There is a representation ρ:G→GL_d( Z_p) with reduction ρ_n. item There is a representation ρ_h:G→GL_d( Z/p^h_G(n)) with reduction ρ_n. The implication from (2) to (1) is constructive to arbitrary requested precision by [source reference]. It preserves the original level n, not necessarily every digit of the chosen level-h witness. Red…
Exact mathematical passage
```
\rho_h:G\to\operatorname{GL}_d(\mathbb Z/p^{h_G(n)})
```

p^jUse in source · 4 passages

Search aliases: p^j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1757
For pmid g, tensor C_ Z(g) by C_ Z(p). The gcd theorem identifies the result with C_ Z(p)⊕ C_ Z(p)[1]. Hence Z/p≃ C_ Z(p) belongs to the thick closure. Conversely, the Chinese remainder decomposition of Z/g and the short exact sequences building Z/p^j from copies of Z/p put C_ Z(g)≃ Z/g in the thick subcategory generated by its prime divisors. This proves the boxed equality.
Exact mathematical passage
```
\mathbb Z/p^j
```

p^{j+1}Use in source · 2 passages

Search aliases: p^j+1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3033
p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponents congruent modulo p^t+j give powers congruent modulo p^j+1, also for negative integer exponents by invertibility. Completeness extends the map conti…
Exact mathematical passage
```
 A^{p^{t+j}}\in I+p^{j+1}M_d(\mathbb Z_p).
```

p^mUse in source · 7 passages

Search aliases: p^m

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1880
vative on perfect complexes. They need not be faithful. In particular, for m>n, the nonzero endomorphism D^n:B_m→ B_m becomes zero under C_m,n. Exactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over R= Z/p^m is zero if its derived reduction to F_p is zero.
Exact mathematical passage
```
R=\mathbb Z/p^m
```

p^{m-2e}Use in source · 1 passage

Search aliases: p^m-2e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2879
This is a classification of isomorphism classes, not an equivalence with a finite groupoid. Integral automorphism groups and noninvertible morphisms need not be determined by finite reduction. Likewise the averaging construction is not asserted to be a canonical framed lift. Two outputs agreeing modulo p^m-e are conjugate by a matrix congruent to I modulo p^m-2e when m>2e, but need not be literally the same matrices.
Exact mathematical passage
```
p^{m-2e}
```

p^{m-a}Use in source · 1 passage

Search aliases: p^m-a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2069
. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-ax. Reduction of p^m-ax gives p^n-ap^m-nx. Thus for each target level n, every source level m≥ n+a maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem rec…
Exact mathematical passage
```
p^{m-a}x
```

p^{m-e}Use in source · 5 passages

Search aliases: p^m-e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2740
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
b(g)\in p^{m-e}M_d(\mathbb Z_p)
```

p^{m-n}Use in source · 6 passages

Search aliases: p^m-n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2062
[Transient negative cohomology in the resolution direction] Fix p and an integer a≥1. For n≥ a put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and id…
Exact mathematical passage
```
p^{m-n}
```

p^{\min(a_i,n)}Use in source · 1 passage

Search aliases: p^min(a_i,n) min

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2296
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
 \qquad
 \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
```

p^NUse in source · 1 passage

Search aliases: p^N

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2165
For surjectivity take (y_p)∈ G. Choose N large enough that D^Ny_p∈ Z_p for every p. For each p, choose the residue of D^Ny_p modulo p^N. The ordinary Chinese remainder theorem provides z∈ Z with all these residues. Set a=z/D^N∈ A. Then a-y_p∈ Z_p for every p, so (a,(a-y_p)_p) maps to (y_p)_p. This proves exactness.
Exact mathematical passage
```
p^N
```

p^{n+1}Use in source · 3 passages

Search aliases: p^n+1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 87
t directions: transformation: change of representation and resolution: change of distinguishability. The distinction is operational rather than rhetorical. A Fourier transform, a duality, or a change of coordinates may alter representation while preserving the level of information being resolved. Coarsening a filtration, passing from modulus p^n+1 to modulus p^n, truncating a spectral expansion, or forgetting fine ramification data changes the level at which distinctions are visible.
Exact mathematical passage
```
p^{n+1}
```

p^{n+a}Use in source · 2 passages

Search aliases: p^n+a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3118
o_n lifts over Z_p, ρ_n extends over R_n+a, Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B_i' are defined modulo p^n and reduce to B_i modulo p^n-a. The equality p^2a[B_*',B_i']=0 modulo p^n+a implies [B_*',B_i']=0 mod…
Exact mathematical passage
```
p^{n+a}
```

p^{n-1}Use in source · 1 passage

Search aliases: p^n-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3164
For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde B-widetilde Bwidetilde A ≡ p^2n-2 1 · 0
Exact mathematical passage
```
 A=I+p^{n-1}E_{12},\qquad B=I+p^{n-1}E_{21}.
```

p^{n-a}Use in source · 8 passages

Search aliases: p^n-a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2069
m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-ax. Reduction of p^m-ax gives p^n-ap^m-nx. Thus for each target level n, every source level m≥ n+a maps to zero. This is exactly the pro-zero assertion…
Exact mathematical passage
```
p^{n-a}R_n
```

p^{n-h}Use in source · 2 passages

Search aliases: p^n-h

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1963
Each zero entry contributes R⊕ R[1]. This proves the complex decomposition. Multiplication by p^h on R has cokernel Q_h and kernel p^n-hR, the latter isomorphic to Q_h by multiplication by p^n-h. Hence C_R(p^h) has Q_h in degrees -1 and 0. Counting the shifted copies and using Pascal's identity gives the formula. The unit case is contractible, and otherwise H^0=Q_h≠0.
Exact mathematical passage
```
p^{n-h}R
```

p^{s+1}Use in source · 1 passage

Search aliases: p^s+1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3032
A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponents congruent modulo p^t+j g…
Exact mathematical passage
```
B^p\in I+p^{s+1}M_d(\mathbb Z_p)
```

p^{s-e}Use in source · 2 passages

Search aliases: p^s-e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2837
[Close integral actions are conjugate] Let ρ,σ:G→GL_d( Z_p) be representations satisfying σ(g)≡ρ(g)pmodp^s (g∈ G), s>e. Then T=frac1|G|∑_g∈ Gσ(g)ρ(g)^-1 ∈ I+p^s-eM_d( Z_p) is invertible and satisfies σ(g)T=Tρ(g), σ(g)=Tρ(g)T^-1. Every summand is congruent to I modulo p^s. Its average is therefore I modulo p^s-e, so it is integral and invertible. For h∈ G, σ(h)Tρ(h)^-1 =frac1|G|∑_gσ(hg)ρ(hg)^-1=T, by reindexing the finite sum. This is the required intertwining identity.
Exact mathematical passage
```
 T=\frac1{|G|}\sum_{g\in G}\sigma(g)\rho(g)^{-1}
 \in I+p^{s-e}M_d(\mathbb Z_p)
```

p^tUse in source · 2 passages

Search aliases: p^t

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3032
a continuous homomorphism Z_p→GL_d( Z_p) if and only if the reduction of A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows…
Exact mathematical passage
```
A^{p^t}\in I+pM_d(\mathbb Z_p)
```

p^{t+j}Use in source · 2 passages

Search aliases: p^t+j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3033
p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponents congruent modulo p^t+j give powers congruent modulo p^j+1, also for negative integer exponents by invertibility. Completeness extends the map conti…
Exact mathematical passage
```
 A^{p^{t+j}}\in I+p^{j+1}M_d(\mathbb Z_p).
```

P_1Use in source · 5 passages

Search aliases: P_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 838
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
P_1(x,y)=(x,0),\qquad P_2(x,y)=(0,y).
```

P_2Use in source · 5 passages

Search aliases: P_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 838
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
P_1(x,y)=(x,0),\qquad P_2(x,y)=(0,y).
```

P_iUse in source · 2 passages

Search aliases: P_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1813
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has …
Exact mathematical passage
```
 \epsilon_i=\begin{cases}
 1&a_i+b_i-2r_i>0,\\
 0&a_i+b_i-2r_i=0,
 \end{cases}
 \qquad
 P_i=\{p:p\mid d_{i,k}\text{ for some }k\}.
```

P_jUse in source · 1 passage

Search aliases: P_j

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1829
gory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0, P_j · ϵ_i=1, ϵ_j=0, P_i · ϵ_i=0, ϵ_j=1. Localization at S is an exact nonzero detector with that kernel. It makes the prescribed tensor-cube squares excisive. A supplied Smith form is a finite certificate: …
Exact mathematical passage
```
 S_{ij}=\begin{cases}
 P_i\cap P_j&\epsilon_i=\epsilon_j=0,\\
 P_j&\epsilon_i=1,\ \epsilon_j=0,\\
 P_i&\epsilon_i=0,\ \epsilon_j=1.
 \end{cases}
```

P_rUse in source · 2 passages

Search aliases: P_r

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4056
Resolution spectrum. For a fixed X∈ T_s and a specified detection predicate P_r at each level r≤ s, define: Spec_R(P;X) = {r≤ s:P_r(C_s→ rX)}. This could behave like a support or spectrum for distinguishability.
Exact mathematical passage
```
P_r
```

\parallelUse in source · 2 passages

Search aliases: parallel

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1731
For the fixed labelled poset define the finite prime set S_P=⋃_eparallel f {p prime:pmidgcd(n_e,n_f)}, where eparallel f means incomparable. Let J_P be the invariant thick closure of the square interaction residuals for the unit seed Z.
Exact mathematical passage
```
 S_P=\bigcup_{e\parallel f}
 \{p\text{ prime}:p\mid\gcd(n_e,n_f)\},
```

\pmodUse in source · 26 passages

Search aliases: pmod

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2486
The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1
Exact mathematical passage
```
 A=I+pJ\pmod{p^2}.
```

Q^{-1}Use in source · 1 passage

Search aliases: Q^-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3044
[Cyclic centralizers over a coefficient ring] Let R be a commutative ring, B∈ M_d(R), and suppose a vector v makes Q=(v,Bv,…,B^d-1v)∈GL_d(R). Then every matrix commuting with B is uniquely f(B) for a polynomial f∈ R[t] of degree less than d. The coefficients of f are the entries of Q^-1Mv. In particular, the centralizer is commutative.
Exact mathematical passage
```
Q^{-1}Mv
```

Q_0Use in source · 1 passage

Search aliases: Q_0

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1948
[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose …
Exact mathematical passage
```
Q_0=0
```

\mathbb Q_2Use in source · 2 passages

Search aliases: Q_2 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2130
:repair-before-completion For P={p} and two independent labels divisible by p, every finite excisive quotient is zero, whereas (varprojlim_n Z/p^n)[1/p]= Q_p≠0. For a nonzero finite-level quotient, take D=30 and labels 6,10. Only the prime 2 is shared, so the finite repair is B_n[1/2]≅ Z/15^n. The two completed outputs are Q_2× Z_3× Z_5 and Z_3× Z_5, with fiber Q_2. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite…
Exact mathematical passage
```
\mathbb Q_2\times\mathbb Z_3\times\mathbb Z_5
```

q_BUse in source · 1 passage

Search aliases: q_B

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1414
The hypothesis and exactness imply Λ( N_mathscr M_A)⊆ N_mathscr M_B, because exact functors preserve the operations used to generate a thick subcategory. Therefore q_BΛ kills N_mathscr M_A, so the universal property in [source reference] gives barΛ. If Λ is an equivalence identifying the two thick kernels, a quasi-inverse descends in the same way and yields a quasi-inverse to barΛ.
Exact mathematical passage
```
q_B\Lambda
```

Q_hUse in source · 6 passages

Search aliases: Q_h

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1945
[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose …
Exact mathematical passage
```
 h=\min\{n,v_p(a_1),\ldots,v_p(a_t)\},\qquad Q_h=R/(p^h).
```

Q_h^{\oplus\binom tj}Use in source · 1 passage

Search aliases: Q_h^⊕binom tj oplus binom

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1954
rime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose an…
Exact mathematical passage
```
 \boxed{H^{-j}K_R(a_1,\ldots,a_t)\cong Q_h^{\oplus\binom tj}
 \quad(0\leq j\leq t),}
```

Q_nUse in source · 1 passage

Search aliases: Q_n

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1948
[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose …
Exact mathematical passage
```
Q_n=R
```

\mathbb Q_p^rUse in source · 1 passage

Search aliases: Q_p^r mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2201
side V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r⊆ E_p⊆ b^-1 Z_p^r for all p. The intersect…
Exact mathematical passage
```
\mathbb Q_p^r
```

q_rUse in source · 5 passages

Search aliases: q_r

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3893
[Orbit-generator test for an exchange obstruction] Fix an exchange η^A_s,r between exact functors, and an exact quotient q_r of the target fiber. If q_rΔ^A_s,r(g)≃0 for every g∈ G_s, then q_rΔ^A_s,r(X)≃0 for every X∈ Z_s^thick. The same assertion holds for q_r=id, in which case the exchange is an equivalence on the entire generated thick zoo.
Exact mathematical passage
```
q_r
```

\mathcal Q_{\square}Use in source · 1 passage

Search aliases: Q_square mathcal square

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1597
[Universal constructor-compatible excisive quotient] The stable Verdier quotient q_square: Z⟶ Q_square := Z/ J_square has the following properties. item The constructors descend, with their supplied coherent action data. item Every diagram q_squareX, for X∈mathscr X, has context-independent event residuals and satisfies principal-ideal reconstruction. item Every exact equivariant functor E: Z→ D for which all the diagrams EX are excisive factors through q_square, with the enhanced universal uniq…
Exact mathematical passage
```
 q_{\square}:\mathcal Z\longrightarrow\mathcal Q_{\square}
 :=\mathcal Z/\mathcal J_{\square}
```

q_{\square}Use in source · 8 passages

Search aliases: q_square square

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1597
[Universal constructor-compatible excisive quotient] The stable Verdier quotient q_square: Z⟶ Q_square := Z/ J_square has the following properties. item The constructors descend, with their supplied coherent action data. item Every diagram q_squareX, for X∈mathscr X, has context-independent event residuals and satisfies principal-ideal reconstruction. item Every exact equivariant functor E: Z→ D for which all the diagrams EX are excisive factors through q_square, with the enhanced universal uniq…
Exact mathematical passage
```
 q_{\square}:\mathcal Z\longrightarrow\mathcal Q_{\square}
 :=\mathcal Z/\mathcal J_{\square}
```

Q_{\mathrm{tors}}Use in source · 3 passages

Search aliases: Q_tors mathrm

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2292
Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.
Exact mathematical passage
```
p^h Q_{\mathrm{tors}}=0
```

\mathbb RUse in source · 1 passage

Search aliases: R mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3423
Let a fine state be x=(x_1,x_2)∈ R^2 and let coarse resolution record only the average C(x)=fracx_1+x_22. Let the fine transformation be A_1(x_1,x_2)=(ax_1,bx_2) and the coarse transformation be multiplication by c: A_0(y)=cy. Then C(A_1x)-A_0(Cx) = frac(a-c)x_1+(b-c)x_22. Choosing c=fraca+b2 gives C(A_1x)-A_0(Cx) = fraca-b4(x_1-x_2). The defect depends only on the antisymmetric fine mode x_1-x_2, precisely the information erased by averaging. This is a concrete model of a mixed residual exposin…
Exact mathematical passage
```
x=(x_1,x_2)\in\mathbb R^2
```

\mathbf RUse in source · 1 passage

Search aliases: R mathbf

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2442
For a commutative ring R, a finite group G, and M∈ D(R[G]), define Γ_R(M)= RHom_R[G](R,M)∈ D(R). This is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or colimit.
Exact mathematical passage
```
 \Gamma_R(M)=\mathbf R\operatorname{Hom}_{R[G]}(R,M)\in D(R).
```

\mathbb R^2Use in source · 1 passage

Search aliases: R^2 mathbb

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3423
Let a fine state be x=(x_1,x_2)∈ R^2 and let coarse resolution record only the average C(x)=fracx_1+x_22. Let the fine transformation be A_1(x_1,x_2)=(ax_1,bx_2) and the coarse transformation be multiplication by c: A_0(y)=cy. Then C(A_1x)-A_0(Cx) = frac(a-c)x_1+(b-c)x_22. Choosing c=fraca+b2 gives C(A_1x)-A_0(Cx) = fraca-b4(x_1-x_2). The defect depends only on the antisymmetric fine mode x_1-x_2, precisely the information erased by averaging. This is a concrete model of a mixed residual exposin…
Exact mathematical passage
```
x=(x_1,x_2)\in\mathbb R^2
```

R^dUse in source · 2 passages

Search aliases: R^d

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2485
The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1
Exact mathematical passage
```
M=R^d
```

R^rUse in source · 1 passage

Search aliases: R^r

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1882
A finite projective module over R is free. Indeed, lift a basis modulo (p) to obtain a surjection R^r→ P. It is surjective because its cokernel Q satisfies Q=pQ, whence Q=p^mQ=0. The surjection splits, and its kernel has zero reduction modulo p, so the same argument kills the kernel. A perfect complex therefore has a bounded finite free representative.
Exact mathematical passage
```
R^r\to P
```

R^\astUse in source · 1 passage

Search aliases: R^∗ ast

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3547
The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.
Exact mathematical passage
```
(T^\ast,R^\ast)
\in
\operatorname*{arg\,min}_{T,R}
\mathcal C\bigl(\Omega(T,R;P)\bigr),
```

r_1Use in source · 2 passages

Search aliases: r_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3487
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
r_0<r_1<\cdots<r_n
```

R_{2n-1}Use in source · 1 passage

Search aliases: R_2n-1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3153
[A sharp uniform rank-two horizon] For every p,n,r as above, ρ_n lifts continuously over Z_p Longleftrightarrow ρ_n extends continuously to R_2n-1. For n≥2 and r≥2, 2n-1 is optimal as a uniform coefficient level. For n=1, all continuous rank-two representations of Z_p^r over F_p lift. For a nonscalar tuple, a≤ n-1, so an extension to 2n-1 restricts to one at n+a. Apply [source reference]. Scalar tuples lift directly. This also proves the assertion at n=1.
Exact mathematical passage
```
 \boxed{\rho_n\text{ lifts continuously over }\mathbb Z_p
 \quad\Longleftrightarrow\quad
 \rho_n\text{ extends continuously to }R_{2n-1}.}
```

r_iUse in source · 6 passages

Search aliases: r_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1812
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,
Exact mathematical passage
```
r_i
```

r_{i+1}Use in source · 1 passage

Search aliases: r_i+1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3487
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
C_{r_{i+1}\to r_i}X_{i+1}\cong X_i
```

r_{m,n}Use in source · 2 passages

Search aliases: r_m,n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2272
its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural comparison α_n is Q[p^n], with these specified transition maps. Its cone, viewing both modules in degree zero in D(O_n), is canonically equivalent to th…
Exact mathematical passage
```
r_{m,n}:K_m\to K_n
```

R_nUse in source · 13 passages

Search aliases: R_n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2057
[Transient negative cohomology in the resolution direction] Fix p and an integer a≥1. For n≥ a put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and id…
Exact mathematical passage
```
R_n=\mathbb Z/p^n
```

r_nUse in source · 2 passages

Search aliases: r_n

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3487
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
r_0<r_1<\cdots<r_n
```

R_{n+1}Use in source · 9 passages

Search aliases: R_n+1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2631
Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^nR_n+1 with k. It is not inversion of p in R_n+1. The square-zero assertion uses 2n≥ n+1.
Exact mathematical passage
```
A_g\in\operatorname{GL}_d(R_{n+1})
```

R_{n+a}Use in source · 1 passage

Search aliases: R_n+a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3110
[A source-fixed finite horizon in the cyclic stratum] Under the hypotheses of [source reference], the following are equivalent: ρ_n lifts over Z_p, ρ_n extends over R_n+a, Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B…
Exact mathematical passage
```
 \rho_n\text{ lifts over }\mathbb Z_p,
 \qquad \rho_n\text{ extends over }R_{n+a},
 \qquad \Omega_{a,*}(\rho_n)=0.
```

r_{n-1}Use in source · 1 passage

Search aliases: r_n-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3488
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
B_X=(b_{r_0\to r_1}(X_0),\ldots,b_{r_{n-1}\to r_n}(X_{n-1})).
```

R_{n-a}Use in source · 6 passages

Search aliases: R_n-a

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3063
Choose 0≤ a<n and integer scalars λ_i such that A_i≡λ_i Ipmodp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modulo p. Define the finite obstruction record Ω_a,*(ρ_n)= ([B_*,B_i])_i=1^r∈ M_d(R_n-a)^r, [B,C]=BC-CB.
Exact mathematical passage
```
 B_i=\frac{A_i-\lambda_i I}{p^a}
 \quad\in M_d(R_{n-a}).
```

\operatorname{rank}Use in source · 1 passage

Search aliases: rank operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2295
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
v=\operatorname{rank}_O V
```

\operatorname{rank}_OUse in source · 1 passage

Search aliases: rank_O operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2295
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
v=\operatorname{rank}_O V
```

\rfloorUse in source · 2 passages

Search aliases: rfloor

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2981
[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_…
Exact mathematical passage
```
 c(a,b)=\left\lfloor\frac{a+b}{p}\right\rfloor J\pmod p.
```

\mathsf SUse in source · 1 passage

Search aliases: S mathsf

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3542
The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.
Exact mathematical passage
```
\mathsf S=
(\Seeds,\mathscr T,\mathscr R,\Delta,\Omega,\mathsf{Cert}).
```

S_{\mathcal I}Use in source · 3 passages

Search aliases: S_ I mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1792
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
 S_{\mathcal I}=\bigcup_{(i,j)\in\mathcal I}
 \{p:p\mid\gcd(n_i,n_j)\}.
```

\mathcal S_0Use in source · 1 passage

Search aliases: S_0 mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1039
[Strict semantic realization of the event diagram] For a typed pure word W in a strict chain-natural realization and a seed s∈ E_b, evaluation gives a functor X_W:NJ(P_W)⟶ S_0. Its vertices evaluate the reachable words and its cover arrows evaluate the whiskered exchange maps. Thus the residual diagram in [source reference] is constructed, rather than assumed, for this class of realizations.
Exact mathematical passage
```
 X_W:NJ(P_W)\longrightarrow\mathcal S_0.
```

S_3Use in source · 1 passage

Search aliases: S_3

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4493
Finite computation contract. All five exact regression scripts passed, including reruns of the four inherited suites. The new script uses integer and modular arithmetic only in mathematical checks. It examines eight specified cochain fixtures, including the nonabelian group S_3, with 593 cocycle triples and 56 changes of section. Positive solutions and negative left-nullspace witnesses are checked directly against their equations.
Exact mathematical passage
```
S_3
```

s_iUse in source · 10 passages

Search aliases: s_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3238
[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a…
Exact mathematical passage
```
s_i\in P(U_i)
```

s_i'Use in source · 4 passages

Search aliases: s_i'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3250
e fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1({U_i}, A) is independent of the chosen local sections; item Ω(P;{U_i})=0 if and o…
Exact mathematical passage
```
    s_i'=s_i+b_i,
    
```

S_{ij}Use in source · 2 passages

Search aliases: S_ij

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1829
gory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0, P_j · ϵ_i=1, ϵ_j=0, P_i · ϵ_i=0, ϵ_j=1. Localization at S is an exact nonzero detector with that kernel. It makes the prescribed tensor-cube squares excisive. A supplied Smith form is a finite certificate: …
Exact mathematical passage
```
 S_{ij}=\begin{cases}
 P_i\cap P_j&\epsilon_i=\epsilon_j=0,\\
 P_j&\epsilon_i=1,\ \epsilon_j=0,\\
 P_i&\epsilon_i=0,\ \epsilon_j=1.
 \end{cases}
```

s_jUse in source · 3 passages

Search aliases: s_j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3243
[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a…
Exact mathematical passage
```
s_j=s_i+a_{ij}.
```

s_j'Use in source · 1 passage

Search aliases: s_j'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3278
If s_i'=s_i+b_i, then s_j'=s_j+b_j=s_i+a_ij+b_j =s_i'+(a_ij+b_j-b_i), so the residual changes by the coboundary δ b. Hence its cohomology class is independent of the chosen local sections.
Exact mathematical passage
```
s_j'=s_j+b_j=s_i+a_{ij}+b_j
=s_i'+(a_{ij}+b_j-b_i),
```

s_kUse in source · 2 passages

Search aliases: s_k

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3268
On a triple overlap, s_k=s_j+a_jk=s_i+a_ij+a_jk, while also s_k=s_i+a_ik. Freeness of the torsor action gives a_ij+a_jk=a_ik, which is the vCech cocycle identity.
Exact mathematical passage
```
s_k=s_j+a_{jk}=s_i+a_{ij}+a_{jk},
```

\mathcal S_mUse in source · 1 passage

Search aliases: S_m mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1914
[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u…
Exact mathematical passage
```
M\in\mathcal S_m
```

\mathcal S_nUse in source · 1 passage

Search aliases: S_n mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1870
Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bo…
Exact mathematical passage
```
\mathcal S_n=\operatorname{Perf}(B_n)
```

\Seeds_1Use in source · 1 passage

Search aliases: Seeds_1 Σ_1 Seeds Sigma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 504
This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).
Exact mathematical passage
```
\Seeds_0
\subseteq
\Seeds_1
\subseteq
\Seeds_2
\subseteq\cdots,
\qquad
\Seeds_{n+1}=\Psi(\Seeds_n).
```

\Seeds_2Use in source · 1 passage

Search aliases: Seeds_2 Σ_2 Seeds Sigma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 504
This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).
Exact mathematical passage
```
\Seeds_0
\subseteq
\Seeds_1
\subseteq
\Seeds_2
\subseteq\cdots,
\qquad
\Seeds_{n+1}=\Psi(\Seeds_n).
```

\Seeds_nUse in source · 1 passage

Search aliases: Seeds_n Σ_n Seeds Sigma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 504
This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).
Exact mathematical passage
```
\Seeds_0
\subseteq
\Seeds_1
\subseteq
\Seeds_2
\subseteq\cdots,
\qquad
\Seeds_{n+1}=\Psi(\Seeds_n).
```

\Seeds_{n+1}Use in source · 1 passage

Search aliases: Seeds_n+1 Σ_n+1 Seeds Sigma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 504
This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).
Exact mathematical passage
```
\Seeds_0
\subseteq
\Seeds_1
\subseteq
\Seeds_2
\subseteq\cdots,
\qquad
\Seeds_{n+1}=\Psi(\Seeds_n).
```

\mathrm{shift}Use in source · 1 passage

Search aliases: shift mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2046
Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map 1-shift:∏_n M_n^i⟶∏_n M_n^i is surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the homotopy fiber is represented by that kernel complex. This is the degreewise Mittag--Leffler case of derived inverse limits [source reference].
Exact mathematical passage
```
 1-\mathrm{shift}:\prod_n M_n^i\longrightarrow\prod_n M_n^i
```

\operatorname{Spec}Use in source · 1 passage

Search aliases: Spec operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4057
Resolution spectrum. For a fixed X∈ T_s and a specified detection predicate P_r at each level r≤ s, define: Spec_R(P;X) = {r≤ s:P_r(C_s→ rX)}. This could behave like a support or spectrum for distinguishability.
Exact mathematical passage
```
\operatorname{Spec}_R(P;X)
=
\{r\le s:P_r(C_{s\to r}X)\}.
```

\operatorname{Spec}_RUse in source · 1 passage

Search aliases: Spec_R operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4057
Resolution spectrum. For a fixed X∈ T_s and a specified detection predicate P_r at each level r≤ s, define: Spec_R(P;X) = {r≤ s:P_r(C_s→ rX)}. This could behave like a support or spectrum for distinguishability.
Exact mathematical passage
```
\operatorname{Spec}_R(P;X)
=
\{r\le s:P_r(C_{s\to r}X)\}.
```

T^{-1}Use in source · 1 passage

Search aliases: T^-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2842
[Close integral actions are conjugate] Let ρ,σ:G→GL_d( Z_p) be representations satisfying σ(g)≡ρ(g)pmodp^s (g∈ G), s>e. Then T=frac1|G|∑_g∈ Gσ(g)ρ(g)^-1 ∈ I+p^s-eM_d( Z_p) is invertible and satisfies σ(g)T=Tρ(g), σ(g)=Tρ(g)T^-1. Every summand is congruent to I modulo p^s. Its average is therefore I modulo p^s-e, so it is integral and invertible. For h∈ G, σ(h)Tρ(h)^-1 =frac1|G|∑_gσ(hg)ρ(hg)^-1=T, by reindexing the finite sum. This is the required intertwining identity.
Exact mathematical passage
```
 \sigma(g)T=T\rho(g),\qquad
 \sigma(g)=T\rho(g)T^{-1}.
```

t^mUse in source · 1 passage

Search aliases: t^m

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2406
Let m≥2, and put N_m(t)=1+t+⋯+t^m-1, M_m=O[t]/(N_m(t)). This is free of rank m-1. Let a generator σ of C_m act by multiplication by t. Since (t-1)N_m(t)=t^m-1, this is a genuine cyclic-group action.
Exact mathematical passage
```
(t-1)N_m(t)=t^m-1
```

t^{m-1}Use in source · 1 passage

Search aliases: t^m-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2402
Let m≥2, and put N_m(t)=1+t+⋯+t^m-1, M_m=O[t]/(N_m(t)). This is free of rank m-1. Let a generator σ of C_m act by multiplication by t. Since (t-1)N_m(t)=t^m-1, this is a genuine cyclic-group action.
Exact mathematical passage
```
 N_m(t)=1+t+\cdots+t^{m-1},\qquad
 M_m=O[t]/(N_m(t)).
```

T^\astUse in source · 1 passage

Search aliases: T^∗ ast

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3547
The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.
Exact mathematical passage
```
(T^\ast,R^\ast)
\in
\operatorname*{arg\,min}_{T,R}
\mathcal C\bigl(\Omega(T,R;P)\bigr),
```

T_2Use in source · 1 passage

Search aliases: T_2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 668
Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformation at the new resolution.
Exact mathematical passage
```
W=T_1C_1T_2T_3C_2T_4.
```

T_3Use in source · 1 passage

Search aliases: T_3

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 668
Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformation at the new resolution.
Exact mathematical passage
```
W=T_1C_1T_2T_3C_2T_4.
```

T_4Use in source · 1 passage

Search aliases: T_4

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 668
Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformation at the new resolution.
Exact mathematical passage
```
W=T_1C_1T_2T_3C_2T_4.
```

T_hUse in source · 2 passages

Search aliases: T_h

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 934
Before event (i,j) can occur, every initially intervening C_k with k<j must already have crossed T_i, and every initially intervening T_h with h>i must already have crossed C_j. Thus all predecessors of a performed event have already occurred. The performed events form an ideal.
Exact mathematical passage
```
T_h
```

\Tcat_0Use in source · 1 passage

Search aliases: Tcat_0 T_0 Tcat mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 834
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
\Tcat_0=\Tcat_1=\Tcat_2=\operatorname{Perf}(k)\times\operatorname{Perf}(k),
```

\Tcat_1Use in source · 1 passage

Search aliases: Tcat_1 T_1 Tcat mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 834
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
\Tcat_0=\Tcat_1=\Tcat_2=\operatorname{Perf}(k)\times\operatorname{Perf}(k),
```

\Tcat_2Use in source · 1 passage

Search aliases: Tcat_2 T_2 Tcat mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 834
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
\Tcat_0=\Tcat_1=\Tcat_2=\operatorname{Perf}(k)\times\operatorname{Perf}(k),
```

\Tcat_{r_i}Use in source · 1 passage

Search aliases: Tcat_r_i T_r_i Tcat mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3487
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
X_i\in\Tcat_{r_i}
```

\Tcat_tUse in source · 3 passages

Search aliases: Tcat_t T_t Tcat mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 150
Let X,Y∈ T_t and let r≤ t.
Exact mathematical passage
```
X,Y\in\Tcat_t
```

\operatorname{tcofib}Use in source · 1 passage

Search aliases: tcofib operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1634
A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib(f_i). Equivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural in the arrows, with the coherent symmetries of the monoidal structure, rather than literal equality of chain matrices for different coordinate orders.
Exact mathematical passage
```
 \operatorname{tcofib}(Y)\simeq
 \bigotimes_{i=1}^m\Cofib(f_i).
```

\operatorname{Tor}Use in source · 1 passage

Search aliases: Tor operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1892
stitutes] Over R= Z/p^2, the exact sequence 0→(p)→ R→ F_p→0 ceases to be left exact after ordinary tensor with F_p: the first map becomes zero. The module F_p has the infinite periodic free resolution with every differential multiplication by p. Tensoring that resolution with F_p gives zero differentials, hence Tor_i^R( F_p, F_p)= F_p for every i≥0.
Exact mathematical passage
```
\operatorname{Tor}_i^R(\mathbb F_p,\mathbb F_p)=\mathbb F_p
```

\operatorname{Tor}_i^RUse in source · 1 passage

Search aliases: Tor_i^R operatorname

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1892
stitutes] Over R= Z/p^2, the exact sequence 0→(p)→ R→ F_p→0 ceases to be left exact after ordinary tensor with F_p: the first map becomes zero. The module F_p has the infinite periodic free resolution with every differential multiplication by p. Tensoring that resolution with F_p gives zero differentials, hence Tor_i^R( F_p, F_p)= F_p for every i≥0.
Exact mathematical passage
```
\operatorname{Tor}_i^R(\mathbb F_p,\mathbb F_p)=\mathbb F_p
```

\mathrm{tors}Use in source · 3 passages

Search aliases: tors mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2292
Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.
Exact mathematical passage
```
p^h Q_{\mathrm{tors}}=0
```

\triangleleftUse in source · 1 passage

Search aliases: triangleleft

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2962
[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets o…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=0
 \quad\Longleftrightarrow\quad
 \exists\, U'\triangleleft_{\mathrm{open}}\Gamma,\ U'\subseteq U:
 \operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}[c_A]=0.
```

\triangleleft_{\mathrm{open}}Use in source · 1 passage

Search aliases: triangleleft_open triangleleft mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2962
[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets o…
Exact mathematical passage
```
 o_n^{\mathrm{cts}}(\rho_n)=0
 \quad\Longleftrightarrow\quad
 \exists\, U'\triangleleft_{\mathrm{open}}\Gamma,\ U'\subseteq U:
 \operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}[c_A]=0.
```

u'Use in source · 2 passages

Search aliases: u'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1064
Let a specified coherent square in a stable ∞-category S be A arrow[r,"u"] arrow[d,"v"'] · B arrow[d,"v'"] C arrow[r,"u'"'] · D. Write E=Cofib(u) and E'=Cofib(u'). The square induces E→ E'.
Exact mathematical passage
```
\begin{tikzcd}
 A \arrow[r,"u"] \arrow[d,"v"'] & B \arrow[d,"v'"]\\
 C \arrow[r,"u'"'] & D.
\end{tikzcd}
```

U^2Use in source · 1 passage

Search aliases: U^2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1321
[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so both X and Y are operationally killable. However X⊕ Y=(A,A,0), and both U and V send this object to (0,0,A), which is fixed by U and…
Exact mathematical passage
```
U^2\simeq U,
\qquad
V^2\simeq V,
\qquad
U\circ V\simeq V,
\qquad
V\circ U\simeq U,
```

U_iUse in source · 6 passages

Search aliases: U_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3238
[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a…
Exact mathematical passage
```
\{U_i\to U\}
```

u_iUse in source · 1 passage

Search aliases: u_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3148
Otherwise every A_i is scalar modulo p^a, while B_* is nonscalar modulo p by the definition of a. It is cyclic by [source reference]. Thus every hypothesis of [source reference] holds. In rank two the polynomials have the form u_i+v_it and can be recovered from a two-column cyclic basis by solving one invertible 2×2 linear system modulo p^n-a. The equivalence with liftability also shows independence of the distinguished generator and of basis choices, although the selected lift itself need not b…
Exact mathematical passage
```
u_i+v_it
```

U_{ij}Use in source · 2 passages

Search aliases: U_ij

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3239
[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a…
Exact mathematical passage
```
a_{ij}\in\mathcal A(U_{ij})
```

U_{\Rcat}Use in source · 2 passages

Search aliases: U_Rcat U_ R Rcat mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3668
intwise-stable Let R be small. Postcomposition with St^add defines a functor St^add_ R: Fun( R^op,Cat^add) ⟶ Fun( R^op,Cat^st), F⟼ St^add∘ F. It is left adjoint to pointwise forgetting U_ R: Fun( R^op,Cat^st) ⟶ Fun( R^op,Cat^add). Equivalently, for every additive two-axis system F and stable two-axis system G there is a natural equivalence Map (St^add_ RF,G) ≃ Map (F,U_ RG).
Exact mathematical passage
```
U_{\Rcat}:
\operatorname{Fun}(\Rcat^{op},\mathbf{Cat}^{\mathrm{st}})
\longrightarrow
\operatorname{Fun}(\Rcat^{op},\mathbf{Cat}^{\mathrm{add}}).
```

\undersetUse in source · 1 passage

Search aliases: underset

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2969
and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open nor…
Exact mathematical passage
```
 H^j_{\mathrm{cts}}(\Gamma,V)
 \cong\underset{U'\subseteq U}{\operatorname{colim}}
 H^j(\Gamma/U',V).
```

V'Use in source · 1 passage

Search aliases: V'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2225
Any other lattice with the specified generic and local identifications has analogous bounds by powers of D. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals Λ. A compatible A-linear map V→ V' sends every θ_p-integral element to a θ'_p-integral element and hence restricts to Λ→Λ'. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in V. Thes…
Exact mathematical passage
```
V\to V'
```

v'Use in source · 1 passage

Search aliases: v'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1064
Let a specified coherent square in a stable ∞-category S be A arrow[r,"u"] arrow[d,"v"'] · B arrow[d,"v'"] C arrow[r,"u'"'] · D. Write E=Cofib(u) and E'=Cofib(u'). The square induces E→ E'.
Exact mathematical passage
```
\begin{tikzcd}
 A \arrow[r,"u"] \arrow[d,"v"'] & B \arrow[d,"v'"]\\
 C \arrow[r,"u'"'] & D.
\end{tikzcd}
```

V^2Use in source · 1 passage

Search aliases: V^2

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1321
[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so both X and Y are operationally killable. However X⊕ Y=(A,A,0), and both U and V send this object to (0,0,A), which is fixed by U and…
Exact mathematical passage
```
U^2\simeq U,
\qquad
V^2\simeq V,
\qquad
U\circ V\simeq V,
\qquad
V\circ U\simeq U,
```

v_iUse in source · 1 passage

Search aliases: v_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3148
Otherwise every A_i is scalar modulo p^a, while B_* is nonscalar modulo p by the definition of a. It is cyclic by [source reference]. Thus every hypothesis of [source reference] holds. In rank two the polynomials have the form u_i+v_it and can be recovered from a two-column cyclic basis by solving one invertible 2×2 linear system modulo p^n-a. The equivalence with liftability also shows independence of the distinguished generator and of basis choices, although the selected lift itself need not b…
Exact mathematical passage
```
u_i+v_it
```

\varprojlim_nUse in source · 8 passages

Search aliases: varprojlim_n varprojlim

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2028
Write widehat B=varprojlim_n B_n≅∏_p∈ P Z_p. The isomorphism follows by taking inverse limits of the compatible Chinese remainder decompositions. Here Z_p is the ring of compatible residue sequences modulo p^n, and Q_p= Z_p[1/p].
Exact mathematical passage
```
 \widehat B=\varprojlim_n B_n\cong\prod_{p\in\mathcal P}\mathbb Z_p.
```

wUse in source · 5 passages

Search aliases: w

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 559
The square is [column sep=large,row sep=large] A_rC_s→ r(X) arrow[r,"η^A_s,r,X"] · C_s→ rA_s(X). It compares transforming after coarsening with coarsening after transforming.
Exact mathematical passage
```
\begin{tikzcd}[column sep=large,row sep=large]
A_rC_{s\to r}(X) \arrow[r,"\eta^A_{s,r,X}"] & C_{s\to r}A_s(X).
\end{tikzcd}
```

W'Use in source · 1 passage

Search aliases: W'

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1526
The last universal property follows directly from thick closure and Verdier localization. If E is equivariant and kills K, then E(WK)≃ W'E(K)≃ W'(0)≃0. Thus it kills all the generators and factors through the quotient.
Exact mathematical passage
```
E(WK)\simeq W'E(K)\simeq W'(0)\simeq0
```

\widehat A_*Use in source · 1 passage

Search aliases: widehat A_* widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3091
at B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preserves the prescribed reduction modulo p^n and does not pass to an obstruction quotient. Suppose first that widehat A_i are any commuting integral lifts. Then widehat B_i=(widehat A_i-λ_iI)/p^a are integral. The identities 0=[widehat A_*,widehat A_i] =p^2a[widehat B_*,widehat B_i] imply [widehat B_*,widehat B_i]=0, since matrices over Z_p have no p-torsion. Reducing modulo p^n-a proves necessit…
Exact mathematical passage
```
 0=[\widehat A_*,\widehat A_i]
   =p^{2a}[\widehat B_*,\widehat B_i]
```

\widehat A_1Use in source · 1 passage

Search aliases: widehat A_1 widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3201
3 · 2pmod9, B_2=2I+4B_1pmod9. The vector e_2 is cyclic for B_1 modulo 3. The displayed integer representatives of A_1,A_2 do not commute exactly: their additive commutator is diag(-81,81). The criterion nevertheless passes. Retain B_1 integrally and replace B_2 by the exact polynomial 2I+4B_1. This constructs widehat A_1=1 · 3 9 · 1, widehat A_2=7 · 12 36 · 7. They commute exactly, remain units over Z_3, and have the original reduction modulo 27. Their continuous powers give the integral represe…
Exact mathematical passage
```
 \widehat A_1=\begin{pmatrix}1&3\\9&1\end{pmatrix},\qquad
 \widehat A_2=\begin{pmatrix}7&12\\36&7\end{pmatrix}.
```

\widehat A_2Use in source · 1 passage

Search aliases: widehat A_2 widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3201
3 · 2pmod9, B_2=2I+4B_1pmod9. The vector e_2 is cyclic for B_1 modulo 3. The displayed integer representatives of A_1,A_2 do not commute exactly: their additive commutator is diag(-81,81). The criterion nevertheless passes. Retain B_1 integrally and replace B_2 by the exact polynomial 2I+4B_1. This constructs widehat A_1=1 · 3 9 · 1, widehat A_2=7 · 12 36 · 7. They commute exactly, remain units over Z_3, and have the original reduction modulo 27. Their continuous powers give the integral represe…
Exact mathematical passage
```
 \widehat A_1=\begin{pmatrix}1&3\\9&1\end{pmatrix},\qquad
 \widehat A_2=\begin{pmatrix}7&12\\36&7\end{pmatrix}.
```

\widehat A_iUse in source · 5 passages

Search aliases: widehat A_i widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3081
ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preser…
Exact mathematical passage
```
 \widehat A_i=\lambda_i I+p^a\widehat f_i(\widehat B),
 \qquad
 \rho(x_1,\ldots,x_r)=\prod_i\widehat A_i^{x_i}.
```

\widehat A_i^{x_i}Use in source · 1 passage

Search aliases: widehat A_i^x_i widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3081
ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preser…
Exact mathematical passage
```
 \widehat A_i=\lambda_i I+p^a\widehat f_i(\widehat B),
 \qquad
 \rho(x_1,\ldots,x_r)=\prod_i\widehat A_i^{x_i}.
```

\widehat A_sUse in source · 3 passages

Search aliases: widehat A_s widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3720
cal A),St^add( B)) ≃Fun^⊕( A,St^add( B)). Full faithfulness transports the supplied composition coherences as well. Accordingly A_r, C_s→ r, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations widehat A_r, widehat C_s→ r, widehatη^A_s,r: widehat A_rwidehat C_s→ r Longrightarrow widehat C_s→ rwidehat A_s.
Exact mathematical passage
```
\widehat A_r,
\qquad
\widehat C_{s\to r},
\qquad
\widehat\eta^A_{s,r}:
\widehat A_r\widehat C_{s\to r}
\Longrightarrow
\widehat C_{s\to r}\widehat A_s.
```

\widehat BUse in source · 23 passages

Search aliases: widehat B widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2028
Write widehat B=varprojlim_n B_n≅∏_p∈ P Z_p. The isomorphism follows by taking inverse limits of the compatible Chinese remainder decompositions. Here Z_p is the ring of compatible residue sequences modulo p^n, and Q_p= Z_p[1/p].
Exact mathematical passage
```
 \widehat B=\varprojlim_n B_n\cong\prod_{p\in\mathcal P}\mathbb Z_p.
```

\widehat B_*Use in source · 2 passages

Search aliases: widehat B_* widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3091
at B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preserves the prescribed reduction modulo p^n and does not pass to an obstruction quotient. Suppose first that widehat A_i are any commuting integral lifts. Then widehat B_i=(widehat A_i-λ_iI)/p^a are integral. The identities 0=[widehat A_*,widehat A_i] =p^2a[widehat B_*,widehat B_i] imply [widehat B_*,widehat B_i]=0, since matrices over Z_p have no p-torsion. Reducing modulo p^n-a proves necessit…
Exact mathematical passage
```
 0=[\widehat A_*,\widehat A_i]
   =p^{2a}[\widehat B_*,\widehat B_i]
```

\widehat B_iUse in source · 3 passages

Search aliases: widehat B_i widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3090
widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preserves the prescribed reduction modulo p^n and does not pass to an obstruction quotient. Suppose first that widehat A_i are any commuting integral lifts. Then widehat B_i=(widehat A_i-λ_iI)/p^a are integral. The identities 0=[widehat A_*,widehat A_i] =p^2a[widehat B_*,widehat B_i] imply [widehat B_*,widehat B_i]=0, since matrices over Z_…
Exact mathematical passage
```
\widehat B_i=(\widehat A_i-\lambda_iI)/p^a
```

\widehat fUse in source · 2 passages

Search aliases: widehat f widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3080
ic-lift Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwideha…
Exact mathematical passage
```
\widehat f_i
```

\widehat f_iUse in source · 2 passages

Search aliases: widehat f_i widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3080
ic-lift Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwideha…
Exact mathematical passage
```
\widehat f_i
```

\widehat MUse in source · 2 passages

Search aliases: widehat M widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2034
[Uniform based-complex tower reconstruction] Suppose M_n are based finite free cochain complexes over B_n, concentrated in a common finite interval with fixed ranks in each degree. Suppose their differential matrices reduce exactly from level m to level n for m≥ n. Then there is a unique based complex widehat M with those ranks over widehat B whose matrices reduce to the supplied ones.
Exact mathematical passage
```
\widehat M
```

\widehat{(-)}Use in source · 1 passage

Search aliases: widehat(-) widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4042
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{(-)}_T,
\qquad
\widehat{(-)}_R.
```

\widehat{(-)}_RUse in source · 1 passage

Search aliases: widehat(-)_R widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4042
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{(-)}_T,
\qquad
\widehat{(-)}_R.
```

\widehat{(-)}_TUse in source · 1 passage

Search aliases: widehat(-)_T widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4042
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{(-)}_T,
\qquad
\widehat{(-)}_R.
```

\widehat{\widehat X_R}Use in source · 1 passage

Search aliases: widehatwidehat X_R widehat widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4048
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{\widehat X_T}_R
\simeq
\widehat{\widehat X_R}_T.
```

\widehat{\widehat X_R}_TUse in source · 1 passage

Search aliases: widehatwidehat X_R_T widehat widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4048
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{\widehat X_T}_R
\simeq
\widehat{\widehat X_R}_T.
```

\widehat{\widehat X_T}Use in source · 1 passage

Search aliases: widehatwidehat X_T widehat widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4048
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{\widehat X_T}_R
\simeq
\widehat{\widehat X_R}_T.
```

\widehat{\widehat X_T}_RUse in source · 1 passage

Search aliases: widehatwidehat X_T_R widehat widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4048
Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in [source reference]: even compatible finite-level repairs can fail to commute with complet…
Exact mathematical passage
```
\widehat{\widehat X_T}_R
\simeq
\widehat{\widehat X_R}_T.
```

\widehat\eta^A_{s,r}Use in source · 1 passage

Search aliases: widehatη^A_s,r widehat eta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3720
cal A),St^add( B)) ≃Fun^⊕( A,St^add( B)). Full faithfulness transports the supplied composition coherences as well. Accordingly A_r, C_s→ r, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations widehat A_r, widehat C_s→ r, widehatη^A_s,r: widehat A_rwidehat C_s→ r Longrightarrow widehat C_s→ rwidehat A_s.
Exact mathematical passage
```
\widehat A_r,
\qquad
\widehat C_{s\to r},
\qquad
\widehat\eta^A_{s,r}:
\widehat A_r\widehat C_{s\to r}
\Longrightarrow
\widehat C_{s\to r}\widehat A_s.
```

\widetilde AUse in source · 8 passages

Search aliases: widetilde A widetilde

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2552
[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite free Z_p-representation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ…
Exact mathematical passage
```
\widetilde A
```

\widetilde A^pUse in source · 1 passage

Search aliases: widetilde A^p widetilde

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2553
[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite free Z_p-representation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ…
Exact mathematical passage
```
 \boxed{\widetilde A^p-I\equiv p^2J\not\equiv0\pmod{p^3}.}
```

\widetilde A^qUse in source · 1 passage

Search aliases: widetilde A^q widetilde

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2885
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
 c=\frac{\widetilde A^q-I}{p^n}\pmod p,
 \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
```

\widetilde BUse in source · 2 passages

Search aliases: widetilde B widetilde

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3168
For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde B-widetilde Bwidetilde A ≡ p^2n-2 1 · 0
Exact mathematical passage
```
\widetilde B=B+p^nQ
```

\widetilde fUse in source · 3 passages

Search aliases: widetilde f widetilde

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2390
ies existence of an integral one whose reduction at precision n is the specified map. The coarser bound v_p(|G|) is always valid. Finite freeness identifies V/p^nV with the full module of linear maps between the reduced lattices. Its G-invariants are exactly the equivariant maps. Apply [source reference]. Explicitly, for a lift widetilde f the boundary cocycle is g⟼fracρ_M(g)widetilde fρ_L(g)^-1-widetilde fp^n. If this is g· b-b, then widetilde f-p^n b is an integral equivariant lift. In the equ…
Exact mathematical passage
```
\widetilde f
```

xUse in source · 52 passages

Search aliases: x

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 838
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
P_1(x,y)=(x,0),\qquad P_2(x,y)=(0,y).
```

X^jUse in source · 1 passage

Search aliases: X^j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2559
hoice of the lift. There is also no finite free Z_p-representation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ+p^2B and use the binomial identity for I+X. The linear term is pX=p^2J modulo p^3. For 2≤ j<p, the coefficient binom pj is divisible by p and X^j by p^j, so those terms vanish modulo p^3. The last term X^p also vanishes because p≥3. Since J is invertible modulo p, the surviving term is nonzero. Hence t…
Exact mathematical passage
```
X^j
```

x^kUse in source · 1 passage

Search aliases: x^k

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1972
ve ring R and nilpotent x∈ R, the thick subcategory generated by C_R(x) is all of Perf(R). Therefore, over R= Z/p^n, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero. Choose N with x^N=0. For composable scalar multiplications, the octahedral cofiber relation builds C_R(x^k) from C_R(x^k-1) and C_R(x). Induction puts C_R(x^N) in Thick(C_R(x)). But C_R(x^N)=C_R(0)≅ R⊕ R[1]. Retract closure therefore includes R, and R thickly genera…
Exact mathematical passage
```
C_R(x^k)
```

x^{k-1}Use in source · 1 passage

Search aliases: x^k-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1972
nilpotent x∈ R, the thick subcategory generated by C_R(x) is all of Perf(R). Therefore, over R= Z/p^n, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero. Choose N with x^N=0. For composable scalar multiplications, the octahedral cofiber relation builds C_R(x^k) from C_R(x^k-1) and C_R(x). Induction puts C_R(x^N) in Thick(C_R(x)). But C_R(x^N)=C_R(0)≅ R⊕ R[1]. Retract closure therefore includes R, and R thickly generates Perf(R). T…
Exact mathematical passage
```
C_R(x^{k-1})
```

x^NUse in source · 3 passages

Search aliases: x^N

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1972
t interaction forces excisive collapse] For any commutative ring R and nilpotent x∈ R, the thick subcategory generated by C_R(x) is all of Perf(R). Therefore, over R= Z/p^n, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero. Choose N with x^N=0. For composable scalar multiplications, the octahedral cofiber relation builds C_R(x^k) from C_R(x^k-1) and C_R(x). Induction puts C_R(x^N) in Thick(C_R(x)). But C_R(x^N)=C_R(0)≅ R⊕ R[1]. R…
Exact mathematical passage
```
x^N=0
```

X^pUse in source · 1 passage

Search aliases: X^p

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2559
esentation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ+p^2B and use the binomial identity for I+X. The linear term is pX=p^2J modulo p^3. For 2≤ j<p, the coefficient binom pj is divisible by p and X^j by p^j, so those terms vanish modulo p^3. The last term X^p also vanishes because p≥3. Since J is invertible modulo p, the surviving term is nonzero. Hence the relation σ^p=1 cannot hold at the next precision.
Exact mathematical passage
```
X^p
```

x_1Use in source · 7 passages

Search aliases: x_1

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3081
ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preser…
Exact mathematical passage
```
 \widehat A_i=\lambda_i I+p^a\widehat f_i(\widehat B),
 \qquad
 \rho(x_1,\ldots,x_r)=\prod_i\widehat A_i^{x_i}.
```

X_2Use in source · 2 passages

Search aliases: X_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1315
[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so both X and Y are operationally killable. However X⊕ Y=(A,A,0), and both U and V send this object to (0,0,A), which is fixed by U and…
Exact mathematical passage
```
U(X_1,X_2,X_3)=(0,0,X_2\oplus X_3),
\qquad
V(X_1,X_2,X_3)=(0,0,X_1\oplus X_3).
```

x_2Use in source · 6 passages

Search aliases: x_2

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3423
Let a fine state be x=(x_1,x_2)∈ R^2 and let coarse resolution record only the average C(x)=fracx_1+x_22. Let the fine transformation be A_1(x_1,x_2)=(ax_1,bx_2) and the coarse transformation be multiplication by c: A_0(y)=cy. Then C(A_1x)-A_0(Cx) = frac(a-c)x_1+(b-c)x_22. Choosing c=fraca+b2 gives C(A_1x)-A_0(Cx) = fraca-b4(x_1-x_2). The defect depends only on the antisymmetric fine mode x_1-x_2, precisely the information erased by averaging. This is a concrete model of a mixed residual exposin…
Exact mathematical passage
```
x=(x_1,x_2)\in\mathbb R^2
```

X_3Use in source · 1 passage

Search aliases: X_3

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1315
[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so both X and Y are operationally killable. However X⊕ Y=(A,A,0), and both U and V send this object to (0,0,A), which is fixed by U and…
Exact mathematical passage
```
U(X_1,X_2,X_3)=(0,0,X_2\oplus X_3),
\qquad
V(X_1,X_2,X_3)=(0,0,X_1\oplus X_3).
```

X_{D_e}Use in source · 1 passage

Search aliases: X_D_e

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1128
nenumerate item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e), listed in the order in which that history performs the events. Their labels and equivalence classes are independent of context, but their ordering a…
Exact mathematical passage
```
 d_e:=\Cofib(X_{D_e}\to X_{\downarrow e}),
```

X_{\downarrow e}Use in source · 1 passage

Search aliases: X_downarrow e downarrow

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1128
nenumerate item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e), listed in the order in which that history performs the events. Their labels and equivalence classes are independent of context, but their ordering a…
Exact mathematical passage
```
 d_e:=\Cofib(X_{D_e}\to X_{\downarrow e}),
```

X_iUse in source · 11 passages

Search aliases: X_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 731
At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.
Exact mathematical passage
```
X_{i-1}\cong X_i\oplus E_i(s)
```

x_iUse in source · 2 passages

Search aliases: x_i

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2315
The residual system (Q[p^n],p^m-n) is pro-zero, and Kxrightarrow∼varprojlim_n K_n. Here pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish. In Smith coordinates a nonzero diagonal entry imposes p^a_ix_i=0 modulo p^n, giving min(a_i,n) units of O-length. Each of the v-r zero columns contributes a free O_n coordinate. This proves the formulas.
Exact mathematical passage
```
p^{a_i}x_i=0
```

X_{i+1}Use in source · 1 passage

Search aliases: X_i+1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3487
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
C_{r_{i+1}\to r_i}X_{i+1}\cong X_i
```

X_{i-1}Use in source · 3 passages

Search aliases: X_i-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 731
At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.
Exact mathematical passage
```
X_{i-1}\cong X_i\oplus E_i(s)
```

X_{I\cap J}Use in source · 1 passage

Search aliases: X_I∩ J cap

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1146
[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.
Exact mathematical passage
```
\begin{tikzcd}
 X_{I\cap J}\arrow[r]\arrow[d] &X_I\arrow[d]\\
 X_J\arrow[r]&X_{I\cup J}
\end{tikzcd}
```

X_{I\cup J}Use in source · 1 passage

Search aliases: X_I∪ J cup

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1146
[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.
Exact mathematical passage
```
\begin{tikzcd}
 X_{I\cap J}\arrow[r]\arrow[d] &X_I\arrow[d]\\
 X_J\arrow[r]&X_{I\cup J}
\end{tikzcd}
```

X_JUse in source · 1 passage

Search aliases: X_J

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1146
[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.
Exact mathematical passage
```
\begin{tikzcd}
 X_{I\cap J}\arrow[r]\arrow[d] &X_I\arrow[d]\\
 X_J\arrow[r]&X_{I\cup J}
\end{tikzcd}
```

X_jUse in source · 2 passages

Search aliases: X_j

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3180
[The first hidden commutator] Suppose n=2 and all generator matrices are A_i=I+pX_i modulo p^2. They always commute at that coefficient level. For rank two, [source reference] gives ρ_2 lifts over Z_p Longleftrightarrow [X_i,X_j]=0 in M_2( F_p) for all i,j. If some X_i is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For X_1=E_12 and X_2=E_21 it fails. Each generator separately has a co…
Exact mathematical passage
```
 \rho_2\text{ lifts over }\mathbb Z_p
 \quad\Longleftrightarrow\quad
 [X_i,X_j]=0\text{ in }M_2(\mathbb F_p)\text{ for all }i,j.
```

X_nUse in source · 1 passage

Search aliases: X_n

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3402
Let the resolution level be n≥1 and take X_n= Z/p^n Z. For m≤ n, coarsening is reduction C_n→ m: Z/p^n Z→ Z/p^m Z. For any polynomial f∈ Z[x], the induced transformation satisfies C_n→ m(f(x)) = f(C_n→ m(x)). Thus the transformation-resolution square commutes exactly. This is a set-level equality, not a cofiber calculation. Regarding the sets as discrete categories gives an identity comparison, while a stable residual requires a separately specified stable realization. This is a simple instance …
Exact mathematical passage
```
X_n=\mathbb Z/p^n\mathbb Z.
```

X_{n-1}Use in source · 1 passage

Search aliases: X_n-1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3488
For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.
Exact mathematical passage
```
B_X=(b_{r_0\to r_1}(X_0),\ldots,b_{r_{n-1}\to r_n}(X_{n-1})).
```

X_PUse in source · 1 passage

Search aliases: X_P

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 983
ginenumerate item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its terminal object is the total residual Cofib(X_∅→ X_P), independent of the linear extension used to reach it; item exact functors preserve the entire diagram, naturally in the specified coherent input.
Exact mathematical passage
```
\Cofib(X_{\varnothing}\to X_P)
```

x_rUse in source · 1 passage

Search aliases: x_r

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3081
ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preser…
Exact mathematical passage
```
 \widehat A_i=\lambda_i I+p^a\widehat f_i(\widehat B),
 \qquad
 \rho(x_1,\ldots,x_r)=\prod_i\widehat A_i^{x_i}.
```

X_WUse in source · 1 passage

Search aliases: X_W

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1039
[Strict semantic realization of the event diagram] For a typed pure word W in a strict chain-natural realization and a seed s∈ E_b, evaluation gives a functor X_W:NJ(P_W)⟶ S_0. Its vertices evaluate the reachable words and its cover arrows evaluate the whiskered exchange maps. Thus the residual diagram in [source reference] is constructed, rather than assumed, for this class of realizations.
Exact mathematical passage
```
 X_W:NJ(P_W)\longrightarrow\mathcal S_0.
```

X_xUse in source · 1 passage

Search aliases: X_x

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 875
[Semantic diamond criterion] Let a rewrite graph have a height h in N that decreases on every edge, and suppose every vertex x has a unique reachable normal vertex n(x). Assign an object X_x in a stable ∞-category to each vertex and a morphism to each edge. Assume every elementary peak xxrightarrowey, xxrightarrowfz has joining paths a:yleadsto u and b:zleadsto u whose semantic composites satisfy [a] [e]=[b] [f] in the homotopy category. Then every normalization path from x to n(x) induces the s…
Exact mathematical passage
```
X_x
```

X_{\varnothing}Use in source · 6 passages

Search aliases: X_∅ varnothing

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 969
[Ideal-indexed residual diagram] A coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its terminal object is the total residual Cofib(X_∅→ X_P), independent of the linear extension used to reach it; item ex…
Exact mathematical passage
```
\mathcal F_X:NJ(P)\longrightarrow\mathcal S,
\qquad
\mathcal F_X(I)=\Cofib(X_{\varnothing}\longrightarrow X_I),
```

yUse in source · 13 passages

Search aliases: y

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 838
[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id. For A, the comparison across 1→0 is the inclusion P_2⇒id, and across 2→1 it is the identity. For B…
Exact mathematical passage
```
P_1(x,y)=(x,0),\qquad P_2(x,y)=(0,y).
```

y_pUse in source · 6 passages

Search aliases: y_p

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2165
For surjectivity take (y_p)∈ G. Choose N large enough that D^Ny_p∈ Z_p for every p. For each p, choose the residue of D^Ny_p modulo p^N. The ordinary Chinese remainder theorem provides z∈ Z with all these residues. Set a=z/D^N∈ A. Then a-y_p∈ Z_p for every p, so (a,(a-y_p)_p) maps to (y_p)_p. This proves exactness.
Exact mathematical passage
```
(y_p)\in G
```

ZUse in source · 14 passages

Search aliases: Z

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 263
Given (r,X)xrightarrowα(s,Y) and (s,Y)xrightarrowβ(t,Z), define β∘_ Kα := C_s→ r(β)∘α: X→ C_t→ r(Z).
Exact mathematical passage
```
(r,X)\xrightarrow{\alpha}(s,Y)
\quad\text{and}\quad
(s,Y)\xrightarrow{\beta}(t,Z),
```

zUse in source · 7 passages

Search aliases: z

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 876
[Semantic diamond criterion] Let a rewrite graph have a height h in N that decreases on every edge, and suppose every vertex x has a unique reachable normal vertex n(x). Assign an object X_x in a stable ∞-category to each vertex and a morphism to each edge. Assume every elementary peak xxrightarrowey, xxrightarrowfz has joining paths a:yleadsto u and b:zleadsto u whose semantic composites satisfy [a] [e]=[b] [f] in the homotopy category. Then every normalization path from x to n(x) induces the s…
Exact mathematical passage
```
x\xrightarrow{e}y,\qquad x\xrightarrow{f}z
```

Z^1Use in source · 1 passage

Search aliases: Z^1

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2683
[Repair ambiguity, conjugacy, and finite certificates] If the one-step lift set is nonempty, its framed lifts form a torsor for Z^1(G,V). Conjugation by matrices I+p^nX, X∈ V, quotients this torsor by coboundaries. Hence the strict-conjugacy classes form a torsor for H^1(G,V), and the stabilizer of a lift under this conjugation is H^0(G,V) as an additive group.
Exact mathematical passage
```
Z^1(G,V)
```

Z^1_{\mathrm{cts}}Use in source · 1 passage

Search aliases: Z^1_cts mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2954
(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, respectively. The associativity and square-zero computations in [source reference]…
Exact mathematical passage
```
Z^1_{\mathrm{cts}}(\Gamma,V)
```

\mathbb Z^2Use in source · 1 passage

Search aliases: Z^2 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1706
For two events the entire calculation is visible: K(a,b)=[ Zxrightarrowbinom-ba Z^2xrightarrow(a b) Z ], degrees -2,-1,0. The product of the two differentials is zero. The kernel of (a b) is generated by (-b/g,a/g), and the incoming image is g times that generator. Both H^-1 and H^0 are Z/g. Equivalently, if au+bv=g, then U=u · -b/g
Exact mathematical passage
```
 K(a,b)=\left[
 \mathbb Z\xrightarrow{\binom{-b}{a}}
 \mathbb Z^2\xrightarrow{(a\ \ b)}\mathbb Z
 \right],
 \qquad\text{degrees }-2,-1,0.
```

\mathbb Z^{a_i}Use in source · 1 passage

Search aliases: Z^a_i mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1812
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,
Exact mathematical passage
```
f_i:\mathbb Z^{a_i}\to\mathbb Z^{b_i}
```

\mathbb Z^{a_i-r_i}Use in source · 1 passage

Search aliases: Z^a_i-r_i mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1843
Changing bases in the source and target yields a chain isomorphism Cofib(f_i)≃ Z^b_i-r_i⊕ Z^a_i-r_i[1] ⊕⊕_k=1^r_iC_ Z(d_i,k). Unit diagonal entries contribute contractible complexes. Thus ϵ_i=1 means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the unit. A thick subcategory containing it is all of Perf( Z).
Exact mathematical passage
```
 \Cofib(f_i)\simeq
 \mathbb Z^{b_i-r_i}\oplus\mathbb Z^{a_i-r_i}[1]
 \oplus\bigoplus_{k=1}^{r_i}C_{\mathbb Z}(d_{i,k}).
```

\mathbb Z^{b_i}Use in source · 1 passage

Search aliases: Z^b_i mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1812
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,
Exact mathematical passage
```
f_i:\mathbb Z^{a_i}\to\mathbb Z^{b_i}
```

\mathbb Z^{b_i-r_i}Use in source · 1 passage

Search aliases: Z^b_i-r_i mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1843
Changing bases in the source and target yields a chain isomorphism Cofib(f_i)≃ Z^b_i-r_i⊕ Z^a_i-r_i[1] ⊕⊕_k=1^r_iC_ Z(d_i,k). Unit diagonal entries contribute contractible complexes. Thus ϵ_i=1 means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the unit. A thick subcategory containing it is all of Perf( Z).
Exact mathematical passage
```
 \Cofib(f_i)\simeq
 \mathbb Z^{b_i-r_i}\oplus\mathbb Z^{a_i-r_i}[1]
 \oplus\bigoplus_{k=1}^{r_i}C_{\mathbb Z}(d_{i,k}).
```

\mathbb Z^rUse in source · 4 passages

Search aliases: Z^r mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2207
ic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r⊆ E_p⊆ b^-1 Z_p^r for all p. The intersection defining Λ therefore satisfies b Z^r⊆Λ⊆ b^-1 Z^r. The right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently Λ is a finite-rank free abelian lattice of rank r, and localizing the …
Exact mathematical passage
```
 b\mathbb Z^r\subseteq\Lambda\subseteq b^{-1}\mathbb Z^r.
```

\mathbb Z_2Use in source · 1 passage

Search aliases: Z_2 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2826
Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is -1, which is 7pmod8, not the selected witness 3pmod8. This is exactly the distinction between preserving the requested seed and preserving every explo…
Exact mathematical passage
```
\mathbb Z_2
```

\mathbb Z_3Use in source · 3 passages

Search aliases: Z_3 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2130
:repair-before-completion For P={p} and two independent labels divisible by p, every finite excisive quotient is zero, whereas (varprojlim_n Z/p^n)[1/p]= Q_p≠0. For a nonzero finite-level quotient, take D=30 and labels 6,10. Only the prime 2 is shared, so the finite repair is B_n[1/2]≅ Z/15^n. The two completed outputs are Q_2× Z_3× Z_5 and Z_3× Z_5, with fiber Q_2. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite…
Exact mathematical passage
```
\mathbb Q_2\times\mathbb Z_3\times\mathbb Z_5
```

\mathbb Z_5Use in source · 2 passages

Search aliases: Z_5 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2130
:repair-before-completion For P={p} and two independent labels divisible by p, every finite excisive quotient is zero, whereas (varprojlim_n Z/p^n)[1/p]= Q_p≠0. For a nonzero finite-level quotient, take D=30 and labels 6,10. Only the prime 2 is shared, so the finite repair is B_n[1/2]≅ Z/15^n. The two completed outputs are Q_2× Z_3× Z_5 and Z_3× Z_5, with fiber Q_2. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite…
Exact mathematical passage
```
\mathbb Q_2\times\mathbb Z_3\times\mathbb Z_5
```

\mathcal Z_AUse in source · 2 passages

Search aliases: Z_A mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1394
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
(\mathcal Z_A,\mathscr M_A)
```

\mathcal Z_BUse in source · 2 passages

Search aliases: Z_B mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1394
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
(\mathcal Z_B,\mathscr M_B)
```

Z_iUse in source · 1 passage

Search aliases: Z_i

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1629
[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S, A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib(f_i). Equivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural…
Exact mathematical passage
```
 Y(S)=\bigotimes_{i=1}^m Z_i(S),\qquad
 Z_i(S)=\begin{cases}B_i&i\in S,\\A_i&i\notin S.\end{cases}
```

\mathbb Z_p^2Use in source · 1 passage

Search aliases: Z_p^2 mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3185
[X_i,X_j]=0 in M_2( F_p) for all i,j. If some X_i is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For X_1=E_12 and X_2=E_21 it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of Z_p^2. On the free pro-p source the same pair does lift, by [source reference]. The surviving obstruction is therefore an actual source…
Exact mathematical passage
```
\mathbb Z_p^2
```

\mathbb Z_p^rUse in source · 18 passages

Search aliases: Z_p^r mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 70
For continuous representations of Z_p^r, removing a common scalar precision exposes normalized commutators. When one normalized generator is cyclic modulo p, their vanishing is necessary and sufficient for an integral lift, constructed by finite polynomial certificates. This condition covers every rank-two input. It yields a complete finite-matrix decision and a sharp uniform horizon 2n-1 for lifting a representation modulo p^n. Explicit positive and negative examples distinguish genuine abelian…
Exact mathematical passage
```
\mathbb Z_p^r
```

\mathbb Z_p^\timesUse in source · 1 passage

Search aliases: Z_p^× mathbb times

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2748
define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
u\in\mathbb Z_p^\times
```

\mathcal Z_s^{\mathrm{thick}}Use in source · 1 passage

Search aliases: Z_s^thick mathcal mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3899
[Orbit-generator test for an exchange obstruction] Fix an exchange η^A_s,r between exact functors, and an exact quotient q_r of the target fiber. If q_rΔ^A_s,r(g)≃0 for every g∈ G_s, then q_rΔ^A_s,r(X)≃0 for every X∈ Z_s^thick. The same assertion holds for q_r=id, in which case the exchange is an equivalence on the entire generated thick zoo.
Exact mathematical passage
```
q_r\Delta^A_{s,r}(X)\simeq0
\quad\text{for every }X\in\mathcal Z_s^{\mathrm{thick}}.
```

|_IUse in source · 1 passage

Search aliases: |_I

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1186
For arbitrary ideals I,J, every chain in I∪ J lies wholly in I or wholly in J. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on G|_I∪ J are precisely compatible cocones on G|_I and G|_J. The resulting colimit is the pushout of the two colimits over the colimit on I∩ J. Thus j_!G is reduced and excisive.
Exact mathematical passage
```
G|_I
```

|_{I\cup J}Use in source · 1 passage

Search aliases: |_I∪ J cup

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1186
For arbitrary ideals I,J, every chain in I∪ J lies wholly in I or wholly in J. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on G|_I∪ J are precisely compatible cocones on G|_I and G|_J. The resulting colimit is the pushout of the two colimits over the colimit on I∩ J. Thus j_!G is reduced and excisive.
Exact mathematical passage
```
G|_{I\cup J}
```

|_JUse in source · 1 passage

Search aliases: |_J

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1186
For arbitrary ideals I,J, every chain in I∪ J lies wholly in I or wholly in J. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on G|_I∪ J are precisely compatible cocones on G|_I and G|_J. The resulting colimit is the pushout of the two colimits over the colimit on I∩ J. Thus j_!G is reduced and excisive.
Exact mathematical passage
```
G|_J
```

|_{U_{ij}}Use in source · 1 passage

Search aliases: |_U_ij

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3308
Given a cover {U_i→ U}, local objects X_i∈mathscr K(U_i) carry descent isomorphisms ϕ_ij:X_i|_U_ijxrightarrow∼X_j|_U_ij satisfying the cocycle condition on triple overlaps.
Exact mathematical passage
```
\phi_{ij}:X_i|_{U_{ij}}\xrightarrow{\sim}X_j|_{U_{ij}}
```

\cdotUse in source · 8 passages

Search aliases: · cdot

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2373
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)
Exact mathematical passage
```
 V=\operatorname{Hom}_O(L,M),\qquad
 (g\cdot f)=\rho_M(g)f\rho_L(g)^{-1}.
```

\alpha_{G,n}Use in source · 1 passage

Search aliases: α_G,n alpha

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2331
Let G be a finite group acting O-linearly on a lattice M. The natural comparison is α_G,n:M^G/p^nM^G⟶(M/p^nM)^G. The first path takes invariant vectors and then reduces. The second reduces the representation and then takes invariant vectors. Its direction is fixed by reduction of an invariant vector; no arbitrary scalar enters this map.
Exact mathematical passage
```
 \alpha_{G,n}:M^G/p^nM^G\longrightarrow(M/p^nM)^G.
```

\alpha_nUse in source · 4 passages

Search aliases: α_n alpha

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2260
For a map d:V→ W of lattices, put K=ker d, Q=cokerd, K_n=ker(d⊗_O O_n). The module K is saturated in V: if p^n v∈ K, then p^n d(v)=0 in the torsion-free module W, so v∈ K. It is therefore a finite free direct summand as an O-module, although a splitting is not canonical. Reduction supplies a natural injection α_n:K/p^nK→ K_n.
Exact mathematical passage
```
\alpha_n:K/p^nK\to K_n
```

\alpha_r^{\flat}Use in source · 2 passages

Search aliases: α_r^flat alpha flat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3690
Any adjunction Ldashv U induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation α:L∘ F⇒ G has pointwise adjoint mates α_r^flat:F(r)⟶ U(G(r)). Naturality of the unit and counit implies that the family {α_r^flat}_r is natural in r. The mate construction is invertible and respects higher coherence. Applying this to L=St^add gives the result.
Exact mathematical passage
```
\alpha_r^{\flat}:F(r)\longrightarrow U(G(r)).
```

\betaUse in source · 24 passages

Search aliases: β beta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 263
Given (r,X)xrightarrowα(s,Y) and (s,Y)xrightarrowβ(t,Z), define β∘_ Kα := C_s→ r(β)∘α: X→ C_t→ r(Z).
Exact mathematical passage
```
(r,X)\xrightarrow{\alpha}(s,Y)
\quad\text{and}\quad
(s,Y)\xrightarrow{\beta}(t,Z),
```

\beta_{O,R,N}Use in source · 2 passages

Search aliases: β_O,R,N beta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2583
For O= Z_p, R= Z/p^m, and S= Z/p^n with m≥ n≥1, this applies at every pair of levels. It applies in particular to reductions of a supplied integral G-lattice. The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite Γ_O(N)⊗_O^ LS · xrightarrowβ_O,R,N⊗_R S Γ_R(N⊗_O^ LR)⊗_R^ LS · xrightarrowβ_R,S,M Γ_S(N⊗_O^ LS) is β_O,S,N. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same …
Exact mathematical passage
```
 \begin{split}
 \Gamma_O(N)\otimes_O^{\mathbf L}S
 &\xrightarrow{\beta_{O,R,N}\otimes_R S}
 \Gamma_R(N\otimes_O^{\mathbf L}R)\otimes_R^{\mathbf L}S\\
 &\xrightarrow{\beta_{R,S,M}}
 \Gamma_S(N\otimes_O^{\mathbf L}S)
 \end{split}
```

\beta_{O,S,N}Use in source · 2 passages

Search aliases: β_O,S,N beta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2592
· xrightarrowβ_R,S,M Γ_S(N⊗_O^ LS) is β_O,S,N. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same evaluation map from homotopy invariants to N. The coefficient extension adjunctions compose by derived tensor associativity. Their mates therefore agree. No arbitrary homotopy-category cone is chosen in this argument.
Exact mathematical passage
```
\beta_{O,S,N}
```

\beta_{R,k,M}Use in source · 1 passage

Search aliases: β_R,k,M beta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2706
[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is inverti…
Exact mathematical passage
```
 \Gamma_R(M)\simeq0,
 \qquad \Gamma_k(M\otimes_R^{\mathbf L}k)\simeq0,
 \qquad \Cofib(\beta_{R,k,M})\simeq0.
```

\beta_{R,S,-}Use in source · 1 passage

Search aliases: β_R,S,- beta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2597
The cofiber of β_R,S,- is an exact functor. Its vanishing class is therefore thick and contains every coefficient-lifted input, as well as the thick closure of those inputs. The example in [source reference] lies outside that class. This is a functor-intrinsic necessary lifting obstruction, not merely a failed scalar equality.
Exact mathematical passage
```
\beta_{R,S,-}
```

\beta_{R,S,M}Use in source · 6 passages

Search aliases: β_R,S,M beta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2448
For a ring map R→ S there is a canonical comparison β_R,S,M:Γ_R(M)⊗_R^ LS ⟶Γ_S(M⊗_R^ LS). It is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to M. Its construction is natural in M and uses the actual coefficient map. No exchange weight is selected.
Exact mathematical passage
```
 \boxed{\beta_{R,S,M}:\Gamma_R(M)\otimes_R^{\mathbf L}S
 \longrightarrow\Gamma_S(M\otimes_R^{\mathbf L}S).}
```

\gammaUse in source · 2 passages

Search aliases: γ gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 282
Identity at (r,X) is id_X:X→ C_r→ r(X)=X. For associativity, let (r,X)xrightarrowα(s,Y) xrightarrowβ(t,Z) xrightarrowγ(u,W). One composite gives C_t→ r(γ)∘ C_s→ r(β)∘α. The other gives the same expression by functoriality and C_t→ r=C_s→ rC_t→ s. Thus composition is associative.
Exact mathematical passage
```
(r,X)\xrightarrow{\alpha}(s,Y)
\xrightarrow{\beta}(t,Z)
\xrightarrow{\gamma}(u,W).
```

\Gamma^jUse in source · 1 passage

Search aliases: Γ^j Gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2976
U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with U. The map then factors th…
Exact mathematical passage
```
\Gamma^j\to V
```

\Gamma_kUse in source · 2 passages

Search aliases: Γ_k Gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2496
[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,
Exact mathematical passage
```
V=\Gamma_k(M\otimes_R^{\mathbf L}k)
```

\Gamma_OUse in source · 1 passage

Search aliases: Γ_O Gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2583
For O= Z_p, R= Z/p^m, and S= Z/p^n with m≥ n≥1, this applies at every pair of levels. It applies in particular to reductions of a supplied integral G-lattice. The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite Γ_O(N)⊗_O^ LS · xrightarrowβ_O,R,N⊗_R S Γ_R(N⊗_O^ LR)⊗_R^ LS · xrightarrowβ_R,S,M Γ_S(N⊗_O^ LS) is β_O,S,N. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same …
Exact mathematical passage
```
 \begin{split}
 \Gamma_O(N)\otimes_O^{\mathbf L}S
 &\xrightarrow{\beta_{O,R,N}\otimes_R S}
 \Gamma_R(N\otimes_O^{\mathbf L}R)\otimes_R^{\mathbf L}S\\
 &\xrightarrow{\beta_{R,S,M}}
 \Gamma_S(N\otimes_O^{\mathbf L}S)
 \end{split}
```

\Gamma_RUse in source · 11 passages

Search aliases: Γ_R Gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2442
For a commutative ring R, a finite group G, and M∈ D(R[G]), define Γ_R(M)= RHom_R[G](R,M)∈ D(R). This is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or colimit.
Exact mathematical passage
```
 \Gamma_R(M)=\mathbf R\operatorname{Hom}_{R[G]}(R,M)\in D(R).
```

\Gamma_SUse in source · 3 passages

Search aliases: Γ_S Gamma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2448
For a ring map R→ S there is a canonical comparison β_R,S,M:Γ_R(M)⊗_R^ LS ⟶Γ_S(M⊗_R^ LS). It is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to M. Its construction is natural in M and uses the actual coefficient map. No exchange weight is selected.
Exact mathematical passage
```
 \boxed{\beta_{R,S,M}:\Gamma_R(M)\otimes_R^{\mathbf L}S
 \longrightarrow\Gamma_S(M\otimes_R^{\mathbf L}S).}
```

\Delta^+_{s,r}Use in source · 1 passage

Search aliases: Δ^+_s,r Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 633
In an abelian fiber one may instead record Δ^-_s,r(X)=kerη^A_s,r,X, Δ^+_s,r(X)=cokerη^A_s,r,X. These vanish simultaneously exactly when the comparison is an isomorphism.
Exact mathematical passage
```
\Delta^-_{s,r}(X)=\ker\eta^A_{s,r,X},
\qquad
\Delta^+_{s,r}(X)=\operatorname{coker}\eta^A_{s,r,X}.
```

\Delta^-_{s,r}Use in source · 1 passage

Search aliases: Δ^-_s,r Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 633
In an abelian fiber one may instead record Δ^-_s,r(X)=kerη^A_s,r,X, Δ^+_s,r(X)=cokerη^A_s,r,X. These vanish simultaneously exactly when the comparison is an isomorphism.
Exact mathematical passage
```
\Delta^-_{s,r}(X)=\ker\eta^A_{s,r,X},
\qquad
\Delta^+_{s,r}(X)=\operatorname{coker}\eta^A_{s,r,X}.
```

\Delta^A_{s,s}Use in source · 1 passage

Search aliases: Δ^A_s,s Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3475
Fix X∈ T_s. First record the exact commutation profile E_A(X;s)={r≤ s:Δ^A_s,r(X)≃0}. No interval or monotonicity property is assumed. For R= N and finite s, a well-defined consecutive coarsening depth is κ_A(X;s):=max{k∈{0,…,s}:Δ^A_s,r(X)≃0 for all s-k≤ r≤ s}. The set is nonempty because Δ^A_s,s(X)≃0. A larger value means that more consecutive coarsenings preserve this exchange. The preceding version's supremum over exact target levels was uninformative on a fixed finite tower because the identi…
Exact mathematical passage
```
\Delta^A_{s,s}(X)\simeq0
```

\Delta^B_{s,r}Use in source · 1 passage

Search aliases: Δ^B_s,r Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3758
mathrmadd_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).
Exact mathematical passage
```
\Phi_r\bigl(\Delta^A_{s,r}(X)\bigr)
\simeq
\Delta^B_{s,r}(\Phi_sX).
```

\Delta_1Use in source · 1 passage

Search aliases: Δ_1 Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4066
Residual genealogy. Residual promotion creates a directed graph s→Δ_1→Δ_2→⋯. Classify whether genealogies terminate, become periodic, stabilize, or grow indefinitely. A terminating genealogy would correspond to finite obstruction resolution. Periodic genealogy may encode a persistent but structured defect.
Exact mathematical passage
```
s\to\Delta_1\to\Delta_2\to\cdots.
```

\Delta_2Use in source · 1 passage

Search aliases: Δ_2 Delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4066
Residual genealogy. Residual promotion creates a directed graph s→Δ_1→Δ_2→⋯. Classify whether genealogies terminate, become periodic, stabilize, or grow indefinitely. A terminating genealogy would correspond to finite obstruction resolution. Periodic genealogy may encode a persistent but structured defect.
Exact mathematical passage
```
s\to\Delta_1\to\Delta_2\to\cdots.
```

\delta_mUse in source · 1 passage

Search aliases: δ_m delta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2273
ockstein There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural comparison α_n is Q[p^n], with these specified transition maps. Its cone, viewing both modules in degree zero in D(O_n), is canonically equivalent to that co…
Exact mathematical passage
```
 \boxed{\delta_n r_{m,n}=p^{m-n}\delta_m.}
```

\delta_nUse in source · 3 passages

Search aliases: δ_n delta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2264
[Intrinsic kernel exchange and its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural comparison α_n is Q[p^n], with these specified transition maps. Its cone, viewing both modules in degree zero in D(O_n), …
Exact mathematical passage
```
 \boxed{0\longrightarrow K/p^nK\xrightarrow{\alpha_n}K_n
 \xrightarrow{\delta_n}Q[p^n]\longrightarrow0.}
```

\delta_{n+c}Use in source · 1 passage

Search aliases: δ_n+c delta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2319
If c<h, choose an element of exact order p^h in Q_tors and take n≥ h. Surjectivity of δ_n+c supplies a preimage. Its reduction to level n has nonzero boundary after multiplication by p^c, so it is not the reduction of an exact solution. This proves sharpness. Multiplication by p^h also makes every transition with gap at least h zero on residuals.
Exact mathematical passage
```
\delta_{n+c}
```

\delta_{n+h}Use in source · 1 passage

Search aliases: δ_n+h delta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2317
By [source reference], the image of an element of K_n+h has boundary p^hδ_n+h=0, so lies in the image of K/p^nK. The reverse inclusion follows by reducing any exact lift at level n+h.
Exact mathematical passage
```
p^h\delta_{n+h}=0
```

\Delta_{T',C'}Use in source · 1 passage

Search aliases: Δ_T',C' Delta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3359
A strong correspondence should include: Λ(TX) · ≃ T'Λ(X), Λ(CX) · ≃ C'Λ(X), Λ(Δ_T,CX) · ≃Δ_T',C'Λ(X), barΛ(Obs X) · ≃Obs'Λ(X).
Exact mathematical passage
```
\Lambda(TX)&\simeq T'\Lambda(X),\\
\Lambda(CX)&\simeq C'\Lambda(X),\\
\Lambda(\Delta_{T,C}X)&\simeq\Delta_{T',C'}\Lambda(X),\\
\bar\Lambda(\Obs X)&\simeq\Obs'\Lambda(X).
```

\Delta_\etaUse in source · 4 passages

Search aliases: Δ_η Delta eta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3862
[Residual formation is exact in its input] Let P,Q: C→ D be exact functors between small stable ∞-categories, and let η:P⇒ Q be a natural transformation. Then Δ_η(X):=Cofib(η_X) is an exact functor C→ D. If q: D→ O is exact, the subcategory {X:qΔ_η(X)≃0} is thick.
Exact mathematical passage
```
\Delta_\eta(X):=\Cofib(\eta_X)
```

\epsilonUse in source · 4 passages

Search aliases: ϵ epsilon

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1813
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has …
Exact mathematical passage
```
 \epsilon_i=\begin{cases}
 1&a_i+b_i-2r_i>0,\\
 0&a_i+b_i-2r_i=0,
 \end{cases}
 \qquad
 P_i=\{p:p\mid d_{i,k}\text{ for some }k\}.
```

\epsilon_iUse in source · 4 passages

Search aliases: ϵ_i epsilon

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1813
[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has …
Exact mathematical passage
```
 \epsilon_i=\begin{cases}
 1&a_i+b_i-2r_i>0,\\
 0&a_i+b_i-2r_i=0,
 \end{cases}
 \qquad
 P_i=\{p:p\mid d_{i,k}\text{ for some }k\}.
```

\epsilon_jUse in source · 2 passages

Search aliases: ϵ_j epsilon

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1823
0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0,
Exact mathematical passage
```
\epsilon_i=\epsilon_j=1
```

\eta^AUse in source · 1 passage

Search aliases: η^A eta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 590
A morphism (r,X)→(s,Y) in K is represented by α:X→ C_s→ rY. Define widehat A(α) to be A_rX xrightarrowA_r(α) A_rC_s→ rY xrightarrowη^A_s,r,Y C_s→ rA_sY. Identity preservation follows from η^A_r,r=id. Composition preservation is exactly the coherence condition in [source reference], together with naturality of η^A.
Exact mathematical passage
```
\eta^A
```

\eta^A_{s,r,Y}Use in source · 1 passage

Search aliases: η^A_s,r,Y eta

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Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 583
A morphism (r,X)→(s,Y) in K is represented by α:X→ C_s→ rY. Define widehat A(α) to be A_rX xrightarrowA_r(α) A_rC_s→ rY xrightarrowη^A_s,r,Y C_s→ rA_sY. Identity preservation follows from η^A_r,r=id. Composition preservation is exactly the coherence condition in [source reference], together with naturality of η^A.
Exact mathematical passage
```
A_rX
\xrightarrow{A_r(\alpha)}
A_rC_{s\to r}Y
\xrightarrow{\eta^A_{s,r,Y}}
C_{s\to r}A_sY.
```

\eta^BUse in source · 1 passage

Search aliases: η^B eta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3757
ratornameSt^add_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).
Exact mathematical passage
```
\eta^B
```

\eta_{L,b;m,n}Use in source · 2 passages

Search aliases: η_L,b;m,n eta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1908
ut A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For scalar unit templates take L= Z, so that the transformation functors are identities but the coarsening functors are not.
Exact mathematical passage
```
 \eta_{L,b;m,n}=b^{m-n}\theta_{L;m,n}.
```

\eta_{L,b;m,n,M}Use in source · 1 passage

Search aliases: η_L,b;m,n,M eta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1915
[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u…
Exact mathematical passage
```
 \Cofib(\eta_{L,b;m,n,M})\simeq
 A_{L,n}C_{m,n}(M)\otimes^{\mathbf L}_{B_n}C_{B_n}(b^{m-n}).
```

\thetaUse in source · 17 passages

Search aliases: θ theta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 222
0, |Lift_r→ s(X)|=1, |Lift_r→ s(X)|>1. They correspond respectively to obstruction to refinement, uniqueness of the fine object's isomorphism class, and hidden fine-scale multiplicity. The middle case does not imply a contractible space of refinements. It forgets the identification C_s→ rY≅ X and all automorphisms. When these matter, use the groupoid of pairs (Y,θ) with θ:C_s→ rYxrightarrow∼X, and isomorphisms commuting with θ. In the enhanced setting this is the homotopy fiber of the functor on…
Exact mathematical passage
```
(Y,\theta)
```

\theta'_pUse in source · 1 passage

Search aliases: θ'_p theta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2225
Any other lattice with the specified generic and local identifications has analogous bounds by powers of D. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals Λ. A compatible A-linear map V→ V' sends every θ_p-integral element to a θ'_p-integral element and hence restricts to Λ→Λ'. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in V. Thes…
Exact mathematical passage
```
\theta'_p
```

\theta_{L;l,n}Use in source · 1 passage

Search aliases: θ_L;l,n theta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1932
B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step cell has scalar b^m-nb^l-m=b^l-n. The associativity coherence of derived tensor identifies its underlying θ with θ_L;l,n. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred…
Exact mathematical passage
```
\theta_{L;l,n}
```

\theta_{L;m,n}Use in source · 2 passages

Search aliases: θ_L;m,n theta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1904
For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For scalar unit templates take L= Z, so that the transformation functors are identities but the coarsening functors …
Exact mathematical passage
```
 \theta_{L;m,n}:A_{L,n}C_{m,n}\simeq C_{m,n}A_{L,m}.
```

\theta_pUse in source · 7 passages

Search aliases: θ_p theta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2185
[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence f…
Exact mathematical passage
```
 \theta_p:V\otimes_A\mathbb Q_p\xrightarrow{\sim}
 L_p\otimes_{\mathbb Z_p}\mathbb Q_p.
```

\theta_p^{-1}Use in source · 1 passage

Search aliases: θ_p^-1 theta

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2201
bb Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r…
Exact mathematical passage
```
E_p=\theta_p^{-1}(L_p)
```

\iota_{\mathbb S}Use in source · 2 passages

Search aliases: ι_ S iota mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3609
[Free seeded two-axis double category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying computadic data. If the target carries distinguished subsets for the same labels, restriction gives the analogous bijection with label-preservation impo…
Exact mathematical passage
```
\iota_{\mathbb S}:\mathbb S\longrightarrow U\mathbb D_{\mathbb S}
```

\kappaUse in source · 5 passages

Search aliases: κ kappa

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 309
For r≤ s and Y∈ T_s, define the canonical morphism κ_r,s,Y: (r,C_s→ rY)⟶(s,Y) represented by the identity map id_C_s→ rY.
Exact mathematical passage
```
\kappa_{r,s,Y}:
(r,C_{s\to r}Y)\longrightarrow(s,Y)
```

\kappa_AUse in source · 1 passage

Search aliases: κ_A kappa

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3471
Fix X∈ T_s. First record the exact commutation profile E_A(X;s)={r≤ s:Δ^A_s,r(X)≃0}. No interval or monotonicity property is assumed. For R= N and finite s, a well-defined consecutive coarsening depth is κ_A(X;s):=max{k∈{0,…,s}:Δ^A_s,r(X)≃0 for all s-k≤ r≤ s}. The set is nonempty because Δ^A_s,s(X)≃0. A larger value means that more consecutive coarsenings preserve this exchange. The preceding version's supremum over exact target levels was uninformative on a fixed finite tower because the identi…
Exact mathematical passage
```
\kappa_A(X;s):=\max\{k\in\{0,\ldots,s\}:\Delta^A_{s,r}(X)\simeq0
\text{ for all }s-k\le r\le s\}.
```

\kappa_{r,s,Y}Use in source · 4 passages

Search aliases: κ_r,s,Y kappa

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 309
For r≤ s and Y∈ T_s, define the canonical morphism κ_r,s,Y: (r,C_s→ rY)⟶(s,Y) represented by the identity map id_C_s→ rY.
Exact mathematical passage
```
\kappa_{r,s,Y}:
(r,C_{s\to r}Y)\longrightarrow(s,Y)
```

\LambdaUse in source · 27 passages

Search aliases: Λ Lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1395
[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved and reflected.
Exact mathematical passage
```
\Lambda:\mathcal Z_A\longrightarrow\mathcal Z_B
```

\lambdaUse in source · 10 passages

Search aliases: λ lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2685
For a fixed multiplication table and residual action, deciding existence is a finite linear problem over k. A solution b is a positive certificate. A linear functional λ on normalized 2-cochains with λ∘δ=0, λ(c_A)≠0 is a negative certificate. If b_0 is one solution, every other solution is b_0+z with δ z=0. Conjugating the matrices by I+p^nX replaces b(g) by b(g)+X-g· X. These are exactly the negatives of the usual 0-coboundaries and yield the same quotient subgroup. Conjugation fixes the action…
Exact mathematical passage
```
\lambda
```

\Lambda'Use in source · 1 passage

Search aliases: Λ' Lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2225
ther lattice with the specified generic and local identifications has analogous bounds by powers of D. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals Λ. A compatible A-linear map V→ V' sends every θ_p-integral element to a θ'_p-integral element and hence restricts to Λ→Λ'. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in V. These con…
Exact mathematical passage
```
\Lambda\to\Lambda'
```

\Lambda^\bulletUse in source · 2 passages

Search aliases: Λ^bullet Lambda bullet

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3318
[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
Exact mathematical passage
```
\Lambda^\bullet:A^\bullet\to B^\bullet
```

\Lambda_*Use in source · 2 passages

Search aliases: Λ_* Lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3392
Interpretation. This reframes a possible route to local-global reciprocity as: identify the globalization obstruction and prove that local correspondence transports it. If Λ_*(Ω_arith)=Ω_aut, and Λ_* preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative exact functor, suffices for this extra detection condition. A general exact functor only preserves vanishing. Effective reconstruction then gives the s…
Exact mathematical passage
```
\Lambda_*(\Omega_{\mathrm{arith}})=\Omega_{\mathrm{aut}},
```

\lambda_1Use in source · 1 passage

Search aliases: λ_1 lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3194
9 · 7pmod27. They commute modulo 27 and both are the identity modulo 3. Choose a=1, λ_1=λ_2=1. Then B_1=0 · 1
Exact mathematical passage
```
\lambda_1=\lambda_2=1
```

\lambda_2Use in source · 1 passage

Search aliases: λ_2 lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3194
9 · 7pmod27. They commute modulo 27 and both are the identity modulo 3. Choose a=1, λ_1=λ_2=1. Then B_1=0 · 1
Exact mathematical passage
```
\lambda_1=\lambda_2=1
```

\lambda_iUse in source · 7 passages

Search aliases: λ_i lambda

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3058
Choose 0≤ a<n and integer scalars λ_i such that A_i≡λ_i Ipmodp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modulo p. Define the finite obstruction record Ω_a,*(ρ_n)= ([B_*,B_i])_i=1^r∈ M_d(R_n-a)^r, [B,C]=BC-CB.
Exact mathematical passage
```
\lambda_i
```

\muUse in source · 1 passage

Search aliases: μ mu

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 4246

Resolution system: · F: R^op→Cat. [0.4em] Fiber: · T_r=F(r). [0.4em] Coarsening: · C_s→ r: T_s→ T_r. [0.4em] Total category: · K=∫_ RF. [0.4em] Resolution equivalence: · X∼_rYiff C_t→ rX≅ C_t→ rY. [0.4em] Lift set: · Lift_r→ s(X)={Y:C_s→ rY≅ X}/≅. [0.4em] Exchange map: · η^A_s,r:A_rC_s→ r⇒ C_s→ rA_s. [0.4em] Stable residual: · Δ^A_s,r(X)=Cofib(η^A_s,r,X). [0.4em] Stable obstruction quotient: · O_mathscr M= Z/Thick( K_mathscr M). [0.4em] Invariant obstruction quotient: · O^inv_mathscr M= Z/ J_mat…

Exact mathematical passage


\text{Resolution system:}\quad
&F:\Rcat^{op}\to\Cat.\\[0.4em]
\text{Fiber:}\quad
&\Tcat_r=F(r).\\[0.4em]
\text{Coarsening:}\quad
&C_{s\to r}:\Tcat_s\to\Tcat_r.\\[0.4em]
\text{Total category:}\quad
&\K=\int_{\Rcat}F.\\[0.4em]
\text{Resolution equivalence:}\quad
&X\sim_rY\iff C_{t\to r}X\cong C_{t\to r}Y.\\[0.4em]
\text{Lift set:}\quad
&\Lift_{r\to s}(X)=\{Y:C_{s\to r}Y\cong X\}/\cong.\\[0.4em]
\text{Exchange map:}\quad
&\eta^A_{s,r}:A_rC_{s\to r}\Rightarrow C_{s\to r}A_s.\\[0.4em]
\text{Stable residual:}\quad
&\Delta^A_{s,r}(X)=\Cofib(\eta^A_{s,r,X}).\\[0.4em]
\text{Stable obstruction quotient:}\quad
&\mathcal O_{\mathscr M}=\mathcal Z/\operatorname{Thick}(\mathcal K_{\mathscr M}).\\[0.4em]
\text{Invariant obstruction quotient:}\quad
&\mathcal O^{\mathrm{inv}}_{\mathscr M}=\mathcal Z/\mathcal J_{\mathscr M}.\\[0.4em]
\text{Event residual diagram:}\quad
&\mathcal F_X(I)=\Cofib(X_{\varnothing}\to X_I).\\[0.4em]
\text{Obstruction class:}\quad
&\Obs_{\mathscr M}(X)=q(X).\\[0.4em]
\text{Seed zoo:}\quad
&\Zoo(\Seeds_0)=\mu A.\bigl(\Seeds_0\cup\Phi(A)\bigr).\\[0.4em]
\text{Resolution ultrametric:}\quad
&d_R(X,Y)=2^{-\nu(X,Y)},\qquad 2^{-\infty}=0.\\[0.4em]
\text{Reconstruction target:}\quad
&\Omega_G=0\iff\text{the specified descent datum is effective}.

\piUse in source · 3 passages

Search aliases: π pi

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 299
Define π: K→ R, π(r,X)=r. The fiber over r is canonically equivalent to T_r.
Exact mathematical passage
```
\pi:\K\to\Rcat,
\qquad
\pi(r,X)=r.
```

\rhoUse in source · 52 passages

Search aliases: ρ rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2373
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)
Exact mathematical passage
```
 V=\operatorname{Hom}_O(L,M),\qquad
 (g\cdot f)=\rho_M(g)f\rho_L(g)^{-1}.
```

\rho_2Use in source · 1 passage

Search aliases: ρ_2 rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3180
[The first hidden commutator] Suppose n=2 and all generator matrices are A_i=I+pX_i modulo p^2. They always commute at that coefficient level. For rank two, [source reference] gives ρ_2 lifts over Z_p Longleftrightarrow [X_i,X_j]=0 in M_2( F_p) for all i,j. If some X_i is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For X_1=E_12 and X_2=E_21 it fails. Each generator separately has a co…
Exact mathematical passage
```
 \rho_2\text{ lifts over }\mathbb Z_p
 \quad\Longleftrightarrow\quad
 [X_i,X_j]=0\text{ in }M_2(\mathbb F_p)\text{ for all }i,j.
```

\rho_hUse in source · 2 passages

Search aliases: ρ_h rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2805
group representations] Let G be a finite group, d,n≥1, and e=v_p(|G|). Set h_G(n)=max{n+e, 2e+1}. For every ρ_n:G→GL_d( Z/p^n), the following are equivalent: item There is a representation ρ:G→GL_d( Z_p) with reduction ρ_n. item There is a representation ρ_h:G→GL_d( Z/p^h_G(n)) with reduction ρ_n. The implication from (2) to (1) is constructive to arbitrary requested precision by [source reference]. It preserves the original level n, not necessarily every digit of the chosen level-h witness. Red…
Exact mathematical passage
```
\rho_h:G\to\operatorname{GL}_d(\mathbb Z/p^{h_G(n)})
```

\rho_LUse in source · 2 passages

Search aliases: ρ_L rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2373
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)
Exact mathematical passage
```
 V=\operatorname{Hom}_O(L,M),\qquad
 (g\cdot f)=\rho_M(g)f\rho_L(g)^{-1}.
```

\rho_MUse in source · 2 passages

Search aliases: ρ_M rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2373
[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)
Exact mathematical passage
```
 V=\operatorname{Hom}_O(L,M),\qquad
 (g\cdot f)=\rho_M(g)f\rho_L(g)^{-1}.
```

\rho_mUse in source · 1 passage

Search aliases: ρ_m rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3008
f Γ is topologically finitely generated, existence of some extension of the fixed ρ_n to every coefficient level is enough to obtain a compatible sequence and hence an integral lift. The residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, [source reference] permits recursive choice of compatible lifts ρ_m for all m≥ n. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because thei…
Exact mathematical passage
```
\rho_m
```

\rho_nUse in source · 32 passages

Search aliases: ρ_n rho

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2623
Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^nR_n+1 with k. It is not inversion of p in R_n+1. The square-zero assertion uses 2n≥ n+1.
Exact mathematical passage
```
 \rho_n:G\longrightarrow\operatorname{GL}_d(R_n),\qquad n\geq1.
```

\sigma^pUse in source · 1 passage

Search aliases: σ^p sigma

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2559
has the form I+pJ+p^2B modulo p^3. Put X=pJ+p^2B and use the binomial identity for I+X. The linear term is pX=p^2J modulo p^3. For 2≤ j<p, the coefficient binom pj is divisible by p and X^j by p^j, so those terms vanish modulo p^3. The last term X^p also vanishes because p≥3. Since J is invertible modulo p, the surviving term is nonzero. Hence the relation σ^p=1 cannot hold at the next precision.
Exact mathematical passage
```
\sigma^p=1
```

\tauUse in source · 2 passages

Search aliases: τ tau

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2701
[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is inverti…
Exact mathematical passage
```
 G=C_p\times C_2=\langle\sigma,\tau\rangle,
 \qquad \sigma\mapsto I+pJ,\quad \tau\mapsto-I.
```

\phiUse in source · 1 passage

Search aliases: ϕ phi

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3308
Given a cover {U_i→ U}, local objects X_i∈mathscr K(U_i) carry descent isomorphisms ϕ_ij:X_i|_U_ijxrightarrow∼X_j|_U_ij satisfying the cocycle condition on triple overlaps.
Exact mathematical passage
```
\phi_{ij}:X_i|_{U_{ij}}\xrightarrow{\sim}X_j|_{U_{ij}}
```

\phi_{ij}Use in source · 1 passage

Search aliases: ϕ_ij phi

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3308
Given a cover {U_i→ U}, local objects X_i∈mathscr K(U_i) carry descent isomorphisms ϕ_ij:X_i|_U_ijxrightarrow∼X_j|_U_ij satisfying the cocycle condition on triple overlaps.
Exact mathematical passage
```
\phi_{ij}:X_i|_{U_{ij}}\xrightarrow{\sim}X_j|_{U_{ij}}
```

\Phi_rUse in source · 4 passages

Search aliases: Φ_r Phi

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3750
niversal residual preservation] Let G: R^op→Cat^st be a stable two-axis system and let Φ:St^add_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).
Exact mathematical passage
```
\Phi_r\bigl(\Delta^A_{s,r}(X)\bigr)
\simeq
\Cofib\!\left(
\Phi_r(\widehat\eta^A_{s,r,X})
\right).
```

\Phi_sUse in source · 1 passage

Search aliases: Φ_s Phi

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3758
mathrmadd_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).
Exact mathematical passage
```
\Phi_r\bigl(\Delta^A_{s,r}(X)\bigr)
\simeq
\Delta^B_{s,r}(\Phi_sX).
```

\chi_MUse in source · 3 passages

Search aliases: χ_M chi

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2103
For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(widehat B); the field summands need not be perfect over widehat B. In the p-primary factor of B_n, inverting T gives zero if p∈ S and does nothing otherwise…
Exact mathematical passage
```
 \chi_M:M\otimes^{\mathbf L}_{\widehat B}\widehat B[1/T]
 \longrightarrow R\varprojlim_n
 \bigl(M\otimes^{\mathbf L}_{\widehat B}B_n[1/T]\bigr)
```

\OmegaUse in source · 15 passages

Search aliases: Ω Omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1236
A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.
Exact mathematical passage
```
\boxed{
\Omega=(\Omega_R,\Omega_T,\Omega_{TR},\Omega_G)
}
```

\omegaUse in source · 2 passages

Search aliases: ω omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 491
[Finitary iteration] Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage A. Then Φ preserves increasing unions of ω-chains. Then Z(Σ_0) = ⋃_n<ωΨ^n(∅).
Exact mathematical passage
```
\omega
```

\Omega_{a,*}Use in source · 4 passages

Search aliases: Ω_a,* Omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3068
modp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modulo p. Define the finite obstruction record Ω_a,*(ρ_n)= ([B_*,B_i])_i=1^r∈ M_d(R_n-a)^r, [B,C]=BC-CB.
Exact mathematical passage
```
 \boxed{\Omega_{a,*}(\rho_n)=
 \bigl([B_*,B_i]\bigr)_{i=1}^r\in M_d(R_{n-a})^r,}
 \qquad [B,C]=BC-CB.
```

\Omega_{\mathrm{arith}}Use in source · 1 passage

Search aliases: Ω_arith Omega mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3392
Interpretation. This reframes a possible route to local-global reciprocity as: identify the globalization obstruction and prove that local correspondence transports it. If Λ_*(Ω_arith)=Ω_aut, and Λ_* preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative exact functor, suffices for this extra detection condition. A general exact functor only preserves vanishing. Effective reconstruction then gives the s…
Exact mathematical passage
```
\Lambda_*(\Omega_{\mathrm{arith}})=\Omega_{\mathrm{aut}},
```

\Omega_{\mathrm{aut}}Use in source · 1 passage

Search aliases: Ω_aut Omega mathrm

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3392
Interpretation. This reframes a possible route to local-global reciprocity as: identify the globalization obstruction and prove that local correspondence transports it. If Λ_*(Ω_arith)=Ω_aut, and Λ_* preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative exact functor, suffices for this extra detection condition. A general exact functor only preserves vanishing. Effective reconstruction then gives the s…
Exact mathematical passage
```
\Lambda_*(\Omega_{\mathrm{arith}})=\Omega_{\mathrm{aut}},
```

\Omega_GUse in source · 3 passages

Search aliases: Ω_G Omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1236
A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.
Exact mathematical passage
```
\boxed{
\Omega=(\Omega_R,\Omega_T,\Omega_{TR},\Omega_G)
}
```

\Omega_RUse in source · 2 passages

Search aliases: Ω_R Omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1236
A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.
Exact mathematical passage
```
\boxed{
\Omega=(\Omega_R,\Omega_T,\Omega_{TR},\Omega_G)
}
```

\Omega_TUse in source · 2 passages

Search aliases: Ω_T Omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1236
A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.
Exact mathematical passage
```
\boxed{
\Omega=(\Omega_R,\Omega_T,\Omega_{TR},\Omega_G)
}
```

\Omega_{TR}Use in source · 2 passages

Search aliases: Ω_TR Omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1236
A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.
Exact mathematical passage
```
\boxed{
\Omega=(\Omega_R,\Omega_T,\Omega_{TR},\Omega_G)
}
```

\ellUse in source · 1 passage

Search aliases: ℓ ell

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2296
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
 \qquad
 \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
```

\ell_OUse in source · 1 passage

Search aliases: ℓ_O ell

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2296
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
 \qquad
 \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
```

\twoheadrightarrowUse in source · 1 passage

Search aliases: ↠ twoheadrightarrow

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2904
[A profinite lift need not preserve the finite quotient] Let p be odd and J∈ M_d( Z_p) be invertible modulo p. The action C_p→GL_d( Z/p^2) given by I+pJ has no integral lift as an action of C_p. Its pullback along Z_p↠ C_p, however, has a continuous integral lift ρ(x)=(I+pJ)^x =∑_j≥0binomxjp^jJ^j, x∈ Z_p. For each j, the polynomial binomxj is continuous on Z_p and takes values there: approximate x by nonnegative integers, where the binomial coefficient is integral, and use closedness of Z_p in Q…
Exact mathematical passage
```
\mathbb Z_p\twoheadrightarrow C_p
```

\mapstoUse in source · 7 passages

Search aliases: ↦ mapsto

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2069
the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-ax. Reduction of p^m-ax gives p^n-ap^m-nx. Thus for each target level n, every source level m≥ n+a maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce ident…
Exact mathematical passage
```
x\mapsto p^{n-a}x
```

\hookrightarrowUse in source · 1 passage

Search aliases: ↪ hookrightarrow

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2526
The evaluation map on degree-zero invariants is the inclusion pM↪ M. On tensoring with k its degree-zero map is zero. Evaluation on H^0(V)=(k^d)^G=k^d is an isomorphism. Naturality of evaluation defining β therefore proves H^0(β)=0. The long exact cohomology sequence of its cofiber now gives k^d in every degree: the groups in negative degrees come from H^i+1(U), those in positive degrees from H^i(V), and the two middle groups use the zero degree-zero comparison.
Exact mathematical passage
```
pM\hookrightarrow M
```

\partialUse in source · 1 passage

Search aliases: ∂ partial

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2338
[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^1(G,M) is finite and killed by |G|. In particular it is killed by p^e, where e=v_p(|G|), and the delay e is a valid uniform lifting bound.
Exact mathematical passage
```
 \boxed{0\longrightarrow M^G/p^nM^G\longrightarrow(M/p^nM)^G
 \xrightarrow{\partial_n}H^1(G,M)[p^n]\longrightarrow0.}
```

\partial_nUse in source · 1 passage

Search aliases: ∂_n partial

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2338
[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^1(G,M) is finite and killed by |G|. In particular it is killed by p^e, where e=v_p(|G|), and the delay e is a valid uniform lifting bound.
Exact mathematical passage
```
 \boxed{0\longrightarrow M^G/p^nM^G\longrightarrow(M/p^nM)^G
 \xrightarrow{\partial_n}H^1(G,M)[p^n]\longrightarrow0.}
```

\prod_{e\in J\setminus I}Use in source · 1 passage

Search aliases: ∏_e∈ J∖ I prod in setminus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1654
Fix a finite poset P and assign a positive integer n_e to every event e∈ P. For M∈Perf( Z) define X_M(I)=M, X_M(I⊆ J)=(∏_e∈ J∖ In_e)id_M. Empty products are 1. For nested ideals these products multiply exactly, so X_M is a strict chain-level diagram followed by the stable realization. When P=P_W, take identity transformation and coarsening functors and let the exchange cell for event e be multiplication by n_e. This realizes the diagram inside the typed two-axis grammar of [source reference]. Th…
Exact mathematical passage
```
 X_M(I)=M,\qquad
 X_M(I\subseteq J)=\left(\prod_{e\in J\setminus I}n_e\right)\id_M.
```

\prod_{e\in P}Use in source · 1 passage

Search aliases: ∏_e∈ P prod in

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1671
comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)≃ M⊗^ L_ ZC_ Z(N), N=∏_e∈ Pn_e. This is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vertices, whereas a cube's total cofiber measures their hig…
Exact mathematical passage
```
 \mathcal F_{X_M}(P)\simeq
 M\otimes^{\mathbf L}_{\mathbb Z}C_{\mathbb Z}(N),
 \qquad N=\prod_{e\in P}n_e.
```

\prod_{g\in G}Use in source · 1 passage

Search aliases: ∏_g∈ G prod in

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2349
More precisely, for d_G:M⟶∏_g∈ GM, d_G(m)=(gm-m)_g, the torsion subgroup of cokerd_G is canonically H^1(G,M). Therefore the sharp bound is computed by [source reference], and can be strictly smaller than e. The formula for c satisfies c(gh)=g c(h)+c(g), because this identity holds before division by p^n in the torsion-free module M. Changing m by p^n b changes c by g b-b. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can…
Exact mathematical passage
```
 d_G:M\longrightarrow\prod_{g\in G}M,\qquad
 d_G(m)=(gm-m)_g,
```

\prod_iUse in source · 1 passage

Search aliases: ∏_i prod

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3081
ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preser…
Exact mathematical passage
```
 \widehat A_i=\lambda_i I+p^a\widehat f_i(\widehat B),
 \qquad
 \rho(x_1,\ldots,x_r)=\prod_i\widehat A_i^{x_i}.
```

\prod_nUse in source · 1 passage

Search aliases: ∏_n prod

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2046
Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map 1-shift:∏_n M_n^i⟶∏_n M_n^i is surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the homotopy fiber is represented by that kernel complex. This is the degreewise Mittag--Leffler case of derived inverse limits [source reference].
Exact mathematical passage
```
 1-\mathrm{shift}:\prod_n M_n^i\longrightarrow\prod_n M_n^i
```

\prod_{p\in\mathcal P}Use in source · 5 passages

Search aliases: ∏_p∈ P prod in mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1863
Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bo…
Exact mathematical passage
```
 D=\prod_{p\in\mathcal P}p,\qquad B_n=\mathbb Z/D^n\mathbb Z\quad(n\geq1).
```

\sumUse in source · 8 passages

Search aliases: ∑ sum

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2296
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
 \qquad
 \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
```

\sum_gUse in source · 1 passage

Search aliases: ∑_g sum

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2849
vρ(g)pmodp^s (g∈ G), s>e. Then T=frac1|G|∑_g∈ Gσ(g)ρ(g)^-1 ∈ I+p^s-eM_d( Z_p) is invertible and satisfies σ(g)T=Tρ(g), σ(g)=Tρ(g)T^-1. Every summand is congruent to I modulo p^s. Its average is therefore I modulo p^s-e, so it is integral and invertible. For h∈ G, σ(h)Tρ(h)^-1 =frac1|G|∑_gσ(hg)ρ(hg)^-1=T, by reindexing the finite sum. This is the required intertwining identity.
Exact mathematical passage
```
 \sigma(h)T\rho(h)^{-1}
 =\frac1{|G|}\sum_g\sigma(hg)\rho(hg)^{-1}=T,
```

\sum_{g\in G}Use in source · 2 passages

Search aliases: ∑_g∈ G sum in

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2358
For completeness, let c:G→ M be a cocycle and b=∑_g∈ Gc(g). For u∈ G, ub=b-|G|c(u), |G|c(u)=u(-b)-(-b). Thus |G|[c]=0. The cocycle and coboundary modules are subquotients of finite free O-modules, so H^1 is finitely generated and hence finite when annihilated by this nonzero integer.
Exact mathematical passage
```
b=\sum_{g\in G}c(g)
```

\sum_{i=0}^{q-1}Use in source · 1 passage

Search aliases: ∑_i=0^q-1 sum

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2885
[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This criterion uses d^2 variables instead of all group-table cochains. Expand the noncommutat…
Exact mathematical passage
```
 c=\frac{\widetilde A^q-I}{p^n}\pmod p,
 \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
```

\sum_{i=1}^rUse in source · 1 passage

Search aliases: ∑_i=1^r sum

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2296
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
 \qquad
 \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
```

\sum_{j\geq0}Use in source · 1 passage

Search aliases: ∑_j≥0 sum geq

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2905
[A profinite lift need not preserve the finite quotient] Let p be odd and J∈ M_d( Z_p) be invertible modulo p. The action C_p→GL_d( Z/p^2) given by I+pJ has no integral lift as an action of C_p. Its pullback along Z_p↠ C_p, however, has a continuous integral lift ρ(x)=(I+pJ)^x =∑_j≥0binomxjp^jJ^j, x∈ Z_p. For each j, the polynomial binomxj is continuous on Z_p and takes values there: approximate x by nonnegative integers, where the binomial coefficient is integral, and use closedness of Z_p in Q…
Exact mathematical passage
```
 \rho(x)=(I+pJ)^x
 =\sum_{j\geq0}\binom{x}{j}p^jJ^j,
 \qquad x\in\mathbb Z_p.
```

\sum_{t=0}^{p-1}Use in source · 1 passage

Search aliases: ∑_t=0^p-1 sum

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2985
Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_p^2 put b(x)=lfloorfracxprfloor Jpmod p (0≤ x<p^2). Writing each x as its low …
Exact mathematical passage
```
 \sum_{t=0}^{p-1}(\delta u)(1,t)=0,
 \qquad\sum_{t=0}^{p-1}c(1,t)=J\ne0.
```

\sum_{t\in G}Use in source · 1 passage

Search aliases: ∑_t∈ G sum in

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2736
[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-e≥1, each I-b(g) is invertible.
Exact mathematical passage
```
 b(g)=\frac1N\sum_{t\in G}E(g,t),\qquad
 A'_g=(I-b(g))A_g.
```

\astUse in source · 1 passage

Search aliases: ∗ ast

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 3547
The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.
Exact mathematical passage
```
(T^\ast,R^\ast)
\in
\operatorname*{arg\,min}_{T,R}
\mathcal C\bigl(\Omega(T,R;P)\bigr),
```

\circ_{\K}Use in source · 1 passage

Search aliases: ∘_K ∘_ K circ K circ mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 269
Given (r,X)xrightarrowα(s,Y) and (s,Y)xrightarrowβ(t,Z), define β∘_ Kα := C_s→ r(β)∘α: X→ C_t→ r(Z).
Exact mathematical passage
```
\beta\circ_{\K}\alpha
:=
C_{s\to r}(\beta)\circ\alpha:
X\to C_{t\to r}(Z).
```

\capUse in source · 5 passages

Search aliases: ∩ cap

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1146
[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.
Exact mathematical passage
```
\begin{tikzcd}
 X_{I\cap J}\arrow[r]\arrow[d] &X_I\arrow[d]\\
 X_J\arrow[r]&X_{I\cup J}
\end{tikzcd}
```

\intUse in source · 2 passages

Search aliases: ∫ int

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 244
[Total Komorebi category] Given F: R^op→Cat, define the category K:=∫_ RF as follows.
Exact mathematical passage
```
\K:=\int_{\Rcat}F
```

\int_{\Rcat}Use in source · 2 passages

Search aliases: ∫_Rcat ∫_ R int Rcat int mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 244
[Total Komorebi category] Given F: R^op→Cat, define the category K:=∫_ RF as follows.
Exact mathematical passage
```
\K:=\int_{\Rcat}F
```

\sim_sUse in source · 2 passages

Search aliases: ∼_s sim

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 168
[Resolution monotonicity] Let r≤ s≤ t and X,Y∈ T_t. If X∼_sY, then X∼_rY.
Exact mathematical passage
```
X\sim_sY,
```

\equivUse in source · 9 passages

Search aliases: ≡ equiv

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2553
[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite free Z_p-representation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ…
Exact mathematical passage
```
 \boxed{\widetilde A^p-I\equiv p^2J\not\equiv0\pmod{p^3}.}
```

\leqUse in source · 11 passages

Search aliases: ≤ leq

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1954
rime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose an…
Exact mathematical passage
```
 \boxed{H^{-j}K_R(a_1,\ldots,a_t)\cong Q_h^{\oplus\binom tj}
 \quad(0\leq j\leq t),}
```

\geUse in source · 4 passages

Search aliases: ≥ ge

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 398
Let m=min{ν(X,Y),ν(Y,Z)}. For every n≤ m, we have X∼_nY and Y∼_nZ. Since ∼_n is an equivalence relation, X∼_nZ. Hence ν(X,Z)≥ m. Therefore 2^-ν(X,Z) ≤ 2^-m = max{2^-ν(X,Y),2^-ν(Y,Z)}. The identity and symmetry axioms are immediate, and separation gives positivity for distinct states. Without separation the same formula, with 2^-∞=0, is an ultrapseudometric.
Exact mathematical passage
```
\nu(X,Z)\ge m.
```

\subsetUse in source · 1 passage

Search aliases: ⊂ subset

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 978
coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its terminal object is the total residual Cofib(X_∅→ X_P), independent of the linear extension used to reach it; item exact functors preserve the entire di…
Exact mathematical passage
```
I\subset I\cup\{e\}
```

\bigoplusUse in source · 7 passages

Search aliases: ⊕ bigoplus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 723
[Finite split residual expansion] Fix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then W(s)≅ W_nf(s)⊕⊕_i=1^m E_i(s), where E_i(s) is the surrounding context applied to the source-side remainder of step i. In the stable case, E_i(s)≃ D_i[-1] for the step cofibers D_i defined below.
Exact mathematical passage
```
W(s)\cong W_{\mathrm{nf}}(s)\oplus\bigoplus_{i=1}^m E_i(s),
```

\bigoplus_{e\in P}Use in source · 1 passage

Search aliases: ⊕_e∈ P bigoplus in

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1201
[Independent events split, dependencies retain extensions] Under [source reference], if P is an antichain then F_X(P)≃⊕_e∈ Pd_e. For a general event poset, a direct-sum formula does not follow.
Exact mathematical passage
```
 \mathcal F_X(P)\simeq\bigoplus_{e\in P}d_e.
```

\bigoplus_{i=1}^mUse in source · 1 passage

Search aliases: ⊕_i=1^m bigoplus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 723
[Finite split residual expansion] Fix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then W(s)≅ W_nf(s)⊕⊕_i=1^m E_i(s), where E_i(s) is the surrounding context applied to the source-side remainder of step i. In the stable case, E_i(s)≃ D_i[-1] for the step cofibers D_i defined below.
Exact mathematical passage
```
W(s)\cong W_{\mathrm{nf}}(s)\oplus\bigoplus_{i=1}^m E_i(s),
```

\bigoplus_{i=1}^rUse in source · 1 passage

Search aliases: ⊕_i=1^r bigoplus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2296
[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and only if it lifts, for the same fixed map d, to precision n+h. No smaller nonnegative delay works uniformly for all n when h>0.
Exact mathematical passage
```
 Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
 \qquad
 \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
```

\bigoplus_{j=0}^{m-1}Use in source · 1 passage

Search aliases: ⊕_j=0^m-1 bigoplus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1680
[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other cohomology group vanishes. In particular the interaction vanishes exactly when g=1.
Exact mathematical passage
```
 K(a_1,\ldots,a_m):=
 \bigotimes_{i=1}^m C_{\mathbb Z}(a_i)
 \simeq\bigoplus_{j=0}^{m-1}
 C_{\mathbb Z}(g)[j]^{\oplus\binom{m-1}{j}}.
```

\bigoplus_{j=0}^{t-1}Use in source · 1 passage

Search aliases: ⊕_j=0^t-1 bigoplus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1949
[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology groups vanish. The complex is zero in D(R) exactly when at least one label is a unit modulo p. If h=n, all entries are zero in R. Otherwise choose …
Exact mathematical passage
```
 K_R(a_1,\ldots,a_t)\cong
 \bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\oplus\binom{t-1}{j}}.
```

\bigoplus_{k=1}^{r_i}Use in source · 1 passage

Search aliases: ⊕_k=1^r_i bigoplus

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1843
Changing bases in the source and target yields a chain isomorphism Cofib(f_i)≃ Z^b_i-r_i⊕ Z^a_i-r_i[1] ⊕⊕_k=1^r_iC_ Z(d_i,k). Unit diagonal entries contribute contractible complexes. Thus ϵ_i=1 means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the unit. A thick subcategory containing it is all of Perf( Z).
Exact mathematical passage
```
 \Cofib(f_i)\simeq
 \mathbb Z^{b_i-r_i}\oplus\mathbb Z^{a_i-r_i}[1]
 \oplus\bigoplus_{k=1}^{r_i}C_{\mathbb Z}(d_{i,k}).
```

\bigoplus_{p\in S}Use in source · 1 passage

Search aliases: ⊕_p∈ S bigoplus in

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2109
For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(widehat B); the field summands need not be perfect over widehat B. In the p-primary factor of B_n, inverting T gives zero if p∈ S and does nothing otherwise…
Exact mathematical passage
```
 \boxed{\operatorname{Fib}(\chi_M)\simeq
 \bigoplus_{p\in S}M\otimes^{\mathbf L}_{\widehat B}\mathbb Q_p.}
```

\bigotimesUse in source · 4 passages

Search aliases: ⊗ bigotimes

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1629
[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S, A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib(f_i). Equivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural…
Exact mathematical passage
```
 Y(S)=\bigotimes_{i=1}^m Z_i(S),\qquad
 Z_i(S)=\begin{cases}B_i&i\in S,\\A_i&i\notin S.\end{cases}
```

\otimes^{\mathbf L}Use in source · 1 passage

Search aliases: ⊗^ L otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1821
0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0,
Exact mathematical passage
```
\Cofib(f_i)\otimes^{\mathbf L}\Cofib(f_j)
```

\otimes^{\mathbf L}_{B_k}Use in source · 1 passage

Search aliases: ⊗^ L_B_k otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1920
orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C…
Exact mathematical passage
```
 C_{n,k}\bigl(N\otimes^{\mathbf L}_{B_n}C_{B_n}(a)\bigr)
 \simeq C_{n,k}(N)\otimes^{\mathbf L}_{B_k}C_{B_k}(a).
```

\otimes^{\mathbf L}_{B_m}Use in source · 1 passage

Search aliases: ⊗^ L_B_m otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1871
Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bo…
Exact mathematical passage
```
 C_{m,n}(M)=M\otimes^{\mathbf L}_{B_m}B_n,\qquad m\geq n.
```

\otimes^{\mathbf L}_{B_n}Use in source · 4 passages

Search aliases: ⊗^ L_B_n otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1900
For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For scalar unit templates take L= Z, so that the transformation functors are identities but the coarsening functors …
Exact mathematical passage
```
 A_{L,n}(M)=(L\otimes^{\mathbf L}_{\mathbb Z}B_n)\otimes^{\mathbf L}_{B_n}M.
```

\otimes^{\mathbf L}_RUse in source · 1 passage

Search aliases: ⊗^ L_R otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1925
_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step …
Exact mathematical passage
```
 N\otimes^{\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\qquad
 K_R(u,v)=C_R(u)\otimes^{\mathbf L}_R C_R(v).
```

\otimes^{\mathbf L}_{\widehat B}Use in source · 2 passages

Search aliases: ⊗^ L_widehat B otimes mathbf widehat

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2103
For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(widehat B); the field summands need not be perfect over widehat B. In the p-primary factor of B_n, inverting T gives zero if p∈ S and does nothing otherwise…
Exact mathematical passage
```
 \chi_M:M\otimes^{\mathbf L}_{\widehat B}\widehat B[1/T]
 \longrightarrow R\varprojlim_n
 \bigl(M\otimes^{\mathbf L}_{\widehat B}B_n[1/T]\bigr)
```

\otimes_{\mathbb Z}Use in source · 1 passage

Search aliases: ⊗_ Z otimes mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2194
ite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this c…
Exact mathematical passage
```
 \Lambda[1/D]\cong V,\qquad
 \Lambda\otimes_{\mathbb Z}\mathbb Z_p\cong L_p.
```

\otimes_{\mathbb Z_p}Use in source · 1 passage

Search aliases: ⊗_ Z_p otimes mathbb

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2185
[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence f…
Exact mathematical passage
```
 \theta_p:V\otimes_A\mathbb Q_p\xrightarrow{\sim}
 L_p\otimes_{\mathbb Z_p}\mathbb Q_p.
```

\otimes_AUse in source · 1 passage

Search aliases: ⊗_A otimes

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2185
[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence f…
Exact mathematical passage
```
 \theta_p:V\otimes_A\mathbb Q_p\xrightarrow{\sim}
 L_p\otimes_{\mathbb Z_p}\mathbb Q_p.
```

\bigotimes_iUse in source · 1 passage

Search aliases: ⊗_i bigotimes

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1649
For a commutative ring R and a∈ R, set C_R(a)=[RxrightarrowaR], with the source in cohomological degree -1 and the target in degree 0. The tensor product ⊗_i C_R(a_i) is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see [source reference]. We use them with an explicit sign convention and direct proofs below.
Exact mathematical passage
```
\bigotimes_i C_R(a_i)
```

\bigotimes_{i=1}^mUse in source · 3 passages

Search aliases: ⊗_i=1^m bigotimes

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1629
[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S, A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib(f_i). Equivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural…
Exact mathematical passage
```
 Y(S)=\bigotimes_{i=1}^m Z_i(S),\qquad
 Z_i(S)=\begin{cases}B_i&i\in S,\\A_i&i\notin S.\end{cases}
```

\otimes_OUse in source · 1 passage

Search aliases: ⊗_O otimes

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2256
For a map d:V→ W of lattices, put K=ker d, Q=cokerd, K_n=ker(d⊗_O O_n). The module K is saturated in V: if p^n v∈ K, then p^n d(v)=0 in the torsion-free module W, so v∈ K. It is therefore a finite free direct summand as an O-module, although a splitting is not canonical. Reduction supplies a natural injection α_n:K/p^nK→ K_n.
Exact mathematical passage
```
 K=\ker d,\qquad Q=\operatorname{coker}d,\qquad
 K_n=\ker(d\otimes_O O_n).
```

\otimes_O^{\mathbf L}Use in source · 3 passages

Search aliases: ⊗_O^ L otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2572
[Lifted-input exchange and a necessary obstruction to lifting] Let O→ R→ S be maps of commutative rings. Assume R and S are perfect as O-modules. For every N∈ D(O[G]), put M=N⊗_O^ LR. Then β_R,S,M:Γ_R(M)⊗_R^ LS xrightarrow∼Γ_S(M⊗_R^ LS). No perfectness of S over R is required. Consequently, if Cofib(β_R,S,M) is nonzero, there is no N∈ D(O[G]) together with an equivalence N⊗_O^ LR≃ M.
Exact mathematical passage
```
M=N\otimes_O^{\mathbf L}R
```

\otimes_RUse in source · 1 passage

Search aliases: ⊗_R otimes

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2583
For O= Z_p, R= Z/p^m, and S= Z/p^n with m≥ n≥1, this applies at every pair of levels. It applies in particular to reductions of a supplied integral G-lattice. The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite Γ_O(N)⊗_O^ LS · xrightarrowβ_O,R,N⊗_R S Γ_R(N⊗_O^ LR)⊗_R^ LS · xrightarrowβ_R,S,M Γ_S(N⊗_O^ LS) is β_O,S,N. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same …
Exact mathematical passage
```
 \begin{split}
 \Gamma_O(N)\otimes_O^{\mathbf L}S
 &\xrightarrow{\beta_{O,R,N}\otimes_R S}
 \Gamma_R(N\otimes_O^{\mathbf L}R)\otimes_R^{\mathbf L}S\\
 &\xrightarrow{\beta_{R,S,M}}
 \Gamma_S(N\otimes_O^{\mathbf L}S)
 \end{split}
```

\otimes_R^{\mathbf L}Use in source · 8 passages

Search aliases: ⊗_R^ L otimes mathbf

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2448
For a ring map R→ S there is a canonical comparison β_R,S,M:Γ_R(M)⊗_R^ LS ⟶Γ_S(M⊗_R^ LS). It is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to M. Its construction is natural in M and uses the actual coefficient map. No exchange weight is selected.
Exact mathematical passage
```
 \boxed{\beta_{R,S,M}:\Gamma_R(M)\otimes_R^{\mathbf L}S
 \longrightarrow\Gamma_S(M\otimes_R^{\mathbf L}S).}
```

\bigcapUse in source · 1 passage

Search aliases: ⋂ bigcap

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2321
Finally a compatible family of vectors modulo p^n has a unique lift to V, since V is finite free and complete. If every reduction solves the equation, its image under d lies in ⋂_n p^nW=0. This proves the inverse-limit assertion directly, without interchanging an arbitrary kernel and an unspecified completion.
Exact mathematical passage
```
\bigcap_n p^nW=0
```

\bigcap_nUse in source · 1 passage

Search aliases: ⋂_n bigcap

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2321
Finally a compatible family of vectors modulo p^n has a unique lift to V, since V is finite free and complete. If every reduction solves the equation, its image under d lies in ⋂_n p^nW=0. This proves the inverse-limit assertion directly, without interchanging an arbitrary kernel and an unspecified completion.
Exact mathematical passage
```
\bigcap_n p^nW=0
```

\bigcupUse in source · 4 passages

Search aliases: ⋃ bigcup

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 492
[Finitary iteration] Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage A. Then Φ preserves increasing unions of ω-chains. Then Z(Σ_0) = ⋃_n<ωΨ^n(∅).
Exact mathematical passage
```
\Zoo(\Seeds_0)
=
\bigcup_{n<\omega}\Psi^n(\varnothing).
```

\bigcup_{(i,j)\in\mathcal I}Use in source · 1 passage

Search aliases: ⋃_(i,j)∈ I bigcup in mathcal

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1792
[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j)∈ I, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive,…
Exact mathematical passage
```
 S_{\mathcal I}=\bigcup_{(i,j)\in\mathcal I}
 \{p:p\mid\gcd(n_i,n_j)\}.
```

\bigcup_{e\parallel f}Use in source · 1 passage

Search aliases: ⋃_eparallel f bigcup parallel

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 1731
For the fixed labelled poset define the finite prime set S_P=⋃_eparallel f {p prime:pmidgcd(n_e,n_f)}, where eparallel f means incomparable. Let J_P be the invariant thick closure of the square interaction residuals for the unit seed Z.
Exact mathematical passage
```
 S_P=\bigcup_{e\parallel f}
 \{p\text{ prime}:p\mid\gcd(n_e,n_f)\},
```

\bigcup_nUse in source · 1 passage

Search aliases: ⋃_n bigcup

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 500
For a constructor output from ⋃_n A_n, its finitely many inputs all belong to a common A_N. The output therefore already belongs to Φ(A_N). This proves preservation of increasing unions. The union on the right is a fixed point because Ψ preserves that union. It contains Σ_0 and is contained in every fixed point by induction on n, hence is least.
Exact mathematical passage
```
\bigcup_n A_n
```

\bigcup_{n<\omega}Use in source · 1 passage

Search aliases: ⋃_n<ω bigcup omega

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 492
[Finitary iteration] Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage A. Then Φ preserves increasing unions of ω-chains. Then Z(Σ_0) = ⋃_n<ωΨ^n(∅).
Exact mathematical passage
```
\Zoo(\Seeds_0)
=
\bigcup_{n<\omega}\Psi^n(\varnothing).
```

\langleUse in source · 1 passage

Search aliases: ⟨ langle

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2701
[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is inverti…
Exact mathematical passage
```
 G=C_p\times C_2=\langle\sigma,\tau\rangle,
 \qquad \sigma\mapsto I+pJ,\quad \tau\mapsto-I.
```

\rangleUse in source · 1 passage

Search aliases: ⟩ rangle

Link to this entry ·

Use in source · Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex, line 2701
[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is inverti…
Exact mathematical passage
```
 G=C_p\times C_2=\langle\sigma,\tau\rangle,
 \qquad \sigma\mapsto I+pJ,\quad \tau\mapsto-I.
```

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