Seeded Two-Axis Calculus

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38 \newcommand{\K}{\mathbf K}
39 \newcommand{\Rcat}{\mathbf R}
40 \newcommand{\Tcat}{\mathbf T}
41 \newcommand{\Cat}{\mathbf{Cat}}
42 \newcommand{\Ob}{\operatorname{Ob}}
43 \newcommand{\Hom}{\operatorname{Hom}}
44 \newcommand{\Lift}{\operatorname{Lift}}
45 \newcommand{\Cofib}{\operatorname{Cofib}}
46 \newcommand{\Recon}{\operatorname{Recon}}
47 \newcommand{\Obs}{\operatorname{Obs}}
48 \newcommand{\Zoo}{\mathcal Z}
49 \newcommand{\Seeds}{\Sigma}
50 \newcommand{\Res}{\mathsf{Res}}
51 \newcommand{\Trans}{\mathsf{Trans}}
52 \newcommand{\id}{\operatorname{id}}
53 \newcommand{\colim}{\operatorname*{colim}}
54 \newcommand{\status}[1]{\textbf{Status: #1.}}
55 
56 \title{\textbf{Komorebi Seeded Two-Axis Calculus}\\
57 \large Transformation, Resolution, Residual Generation, and Reconstruction\\[0.4em]
58 \normalsize Working Formalization v0.11.0}
59 \author{Komorebi P.}
60 \date{27 September 2026}
61 
62 \begin{document}
63 \maketitle
64 
65 \begin{abstract}
66 This manuscript develops a seed-generated calculus of transformations, resolution changes, and residuals of their exchange maps. Its categorical core uses indexed categories, coherent event diagrams, stable cofibers, and constructor-compatible quotients, while retaining history-sensitive filtrations. Arithmetic test beds include nonidentity coefficient towers, completion defects, lattice reconstruction, intrinsic derived exchanges, and a finite lifting horizon for a fixed finite group.
67 
68 Version 0.11.0 retains the actual profinite source. Continuous relation cohomology characterizes next-level lifts, and source inflation identifies when a finite-quotient obstruction disappears after that source resolution is refined. Compatible towers reconstruct continuously, and free pro-$p$ sources admit arbitrary coefficient lifts of their generator matrices.
69 
70 For continuous representations of $\mathbb Z_p^r$, 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 relations from torsion relations introduced by fixing a finite quotient. A mixed-variance groupoid records coefficient reduction and source inflation without identifying their directions.
71 
72 Written proofs, self-review, exact regression tests, and a usable rank-two certificate tool accompany the construction. No Lean certification, independent review, historical novelty, universal profinite lifting theorem, prescribed geometric deformation condition, automorphic existence, or Langlands reciprocity theorem is claimed.
73 \end{abstract}
74 \newpage
75 
76 \begingroup\small\tableofcontents\endgroup
77 \newpage
78 
79 \section{Purpose and status}
80 
81 The goal is to formalize the idea that a mathematical theory may contain two distinct ambient directions:
82 \[
83 \boxed{\text{transformation: change of representation}}
84 \qquad\text{and}\qquad
85 \boxed{\text{resolution: change of distinguishability}.}
86 \]
87 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.
88 
89 The central hypothesis of this note is that a large class of mathematical constructions can be organized by a two-axis calculus in which:
90 \begin{enumerate}
91     \item objects occur at resolutions;
92     \item each resolution has its own transformation category;
93     \item resolution changes induce functors between those categories;
94     \item mixed transformation-resolution squares can commute exactly, commute up to equivalence, or leave a structured residual;
95     \item residuals can themselves be promoted to new seeds;
96     \item compatible local or resolved data can sometimes be reconstructed into global objects.
97 \end{enumerate}
98 
99 \paragraph{Status convention.}
100 Results tagged ``proved here'' have written proofs under the stated hypotheses. They are not asserted to be Lean-checked. Results tagged ``standard machinery'' are direct applications of established category, order, or rewriting theory. Statements tagged ``program theorem'' are precise targets whose Komorebi-specific hypotheses remain to be supplied in applications. No claim of mathematical novelty is made merely because standard machinery is expressed in Komorebi language.
101 
102 \paragraph{Categorical regimes and size.}
103 The initial indexed construction uses small ordinary categories and a strict contravariant functor. Every section involving functorial cofibers, higher coherence, stable envelopes, or stable Verdier quotients uses stable $\infty$-categories, or a specified enhancement presenting them. These are small unless a section explicitly uses an unbounded ambient derived category in a fixed universe. All subcategories in closure constructions are full and replete. ``Stable'' means closed under finite limits and colimits, while ``thick'' additionally means closed under retracts. A stable seed zoo is not automatically idempotent-complete. Thick results are applied to a separately designated thick zoo or to its idempotent completion. Homotopy categories support the corresponding triangulated statements, not arbitrary canonical choices of cones. A fixed small presentation or larger ambient universe is understood whenever object collections are used as sets.
104 
105 \paragraph{What the two axes do not assert.}
106 An admissible transformation need not be invertible or information-preserving. A coarsening functor need not have an inverse. A factorization in a constructed fibration does not establish a canonical resolution system for every mathematical theory. These are separate semantic requirements, not consequences of the notation.
107 
108 \section{Resolution systems}
109 
110 \subsection{The resolution base}
111 
112 \begin{definition}[Resolution category]
113 A \emph{resolution category} is a small poset $(R,\le)$ regarded as a thin category $\Rcat$. An arrow $r\to s$ exists exactly when $r\le s$. We interpret $s$ as at least as fine as $r$.
114 \end{definition}
115 
116 The poset assumption is intentionally restrictive for the first formalization. A later version may replace $\Rcat$ by a general category, a site, a filtered category, or a multi-parameter resolution geometry.
117 
118 \begin{definition}[Indexed transformation system]
119 An \emph{indexed transformation system} over $\Rcat$ is a strict functor
120 \[
121 F:\Rcat^{op}\longrightarrow\Cat.
122 \]
123 For $r\in R$, write
124 \[
125 \Tcat_r:=F(r).
126 \]
127 For $r\le s$, write
128 \[
129 C_{s\to r}:=F(r\le s):\Tcat_s\longrightarrow\Tcat_r.
130 \]
131 We call $C_{s\to r}$ the \emph{coarsening functor}.
132 \end{definition}
133 
134 Strictness gives
135 \[
136 C_{r\to r}=\id_{\Tcat_r}
137 \]
138 and, for $r\le s\le t$,
139 \[
140 C_{t\to r}=C_{s\to r}\circ C_{t\to s}.
141 \]
142 A pseudofunctor version is more natural in some applications. The strict version isolates the main structure without coherence bookkeeping.
143 
144 \begin{definition}[Resolved object]
145 A \emph{resolved object} is a pair $(r,X)$ with $X\in\Ob(\Tcat_r)$.
146 \end{definition}
147 
148 \subsection{Resolution equivalence}
149 
150 Let $X,Y\in\Tcat_t$ and let $r\le t$.
151 
152 \begin{definition}[Indistinguishability at resolution $r$]
153 We write
154 \[
155 X\sim_r Y
156 \]
157 when
158 \[
159 C_{t\to r}(X)\cong C_{t\to r}(Y)
160 \]
161 in $\Tcat_r$.
162 \end{definition}
163 
164 For a fixed ambient fine level $t$, each $\sim_r$ is an equivalence relation on isomorphism classes of objects of $\Tcat_t$.
165 
166 \begin{theorem}[Resolution monotonicity]\label{thm:resolution-monotonicity}
167 Let $r\le s\le t$ and $X,Y\in\Tcat_t$. If
168 \[
169 X\sim_sY,
170 \]
171 then
172 \[
173 X\sim_rY.
174 \]
175 \end{theorem}
176 
177 \begin{proof}
178 An isomorphism
179 \[
180 C_{t\to s}(X)\cong C_{t\to s}(Y)
181 \]
182 is carried by the functor $C_{s\to r}$ to an isomorphism
183 \[
184 C_{s\to r}C_{t\to s}(X)
185 \cong
186 C_{s\to r}C_{t\to s}(Y).
187 \]
188 By functorial coherence this is
189 \[
190 C_{t\to r}(X)\cong C_{t\to r}(Y).
191 \]
192 Hence $X\sim_rY$.
193 \end{proof}
194 
195 \begin{corollary}[Nested distinguishability]\label{cor:nested}
196 For $r\le s\le t$, the equivalence relation $\sim_s$ refines $\sim_r$.
197 \end{corollary}
198 
199 This is the first exact form of the statement that coarsening may erase distinctions but cannot create distinctions that were absent at finer resolution.
200 
201 \subsection{Refinement as a fiber rather than a function}
202 
203 \begin{definition}[Lift set]
204 For $r\le s$ and $X\in\Tcat_r$, define the set of isomorphism classes of lifts
205 \[
206 \Lift_{r\to s}(X)
207 :=
208 \left\{
209 Y\in\Ob(\Tcat_s):C_{s\to r}(Y)\cong X
210 \right\}/\cong.
211 \]
212 \end{definition}
213 
214 Three basic cases occur:
215 \[
216 |\Lift_{r\to s}(X)|=0,
217 \qquad
218 |\Lift_{r\to s}(X)|=1,
219 \qquad
220 |\Lift_{r\to s}(X)|>1.
221 \]
222 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 $C_{s\to r}Y\cong X$ and all automorphisms. When these matter, use the groupoid of pairs $(Y,\theta)$ with $\theta:C_{s\to r}Y\xrightarrow{\sim}X$, and isomorphisms commuting with $\theta$. In the enhanced setting this is the homotopy fiber of the functor on cores.
223 
224 \begin{definition}[Resolution branching number]
225 When $\Lift_{r\to s}(X)$ is finite, define
226 \[
227 b_{r\to s}(X):=|\Lift_{r\to s}(X)|.
228 \]
229 When $b_{r\to s}(X)>0$, its logarithm
230 \[
231 H_{r\to s}(X):=\log b_{r\to s}(X)
232 \]
233 may be regarded as a crude \emph{resolution information} measuring how much ambiguity is hidden by coarsening.
234 \end{definition}
235 
236 The empty lift set is recorded as an obstruction, not as an ordinary finite entropy. This quantity is only a first model. In probabilistic, measurable, or derived settings one would replace cardinality by entropy, measure, dimension, rank, or a homotopical invariant.
237 
238 \section{The total two-axis category}
239 
240 \subsection{Grothendieck construction}
241 
242 \begin{construction}[Total Komorebi category]\label{cons:groth}
243 Given $F:\Rcat^{op}\to\Cat$, define the category
244 \[
245 \K:=\int_{\Rcat}F
246 \]
247 as follows.
248 
249 Its objects are pairs $(r,X)$ with $X\in\Tcat_r$.
250 
251 For $r\le s$, a morphism
252 \[
253 (r,X)\longrightarrow(s,Y)
254 \]
255 is a morphism
256 \[
257 \alpha:X\longrightarrow C_{s\to r}(Y)
258 \]
259 in $\Tcat_r$. If $r\nleq s$, there is no such morphism in this directed version.
260 \end{construction}
261 
262 Given
263 \[
264 (r,X)\xrightarrow{\alpha}(s,Y)
265 \quad\text{and}\quad
266 (s,Y)\xrightarrow{\beta}(t,Z),
267 \]
268 define
269 \[
270 \beta\circ_{\K}\alpha
271 :=
272 C_{s\to r}(\beta)\circ\alpha:
273 X\to C_{t\to r}(Z).
274 \]
275 
276 \begin{theorem}[Total category]\label{thm:total-category}
277 The construction above defines a category $\K$.
278 \end{theorem}
279 
280 \begin{proof}
281 Identity at $(r,X)$ is $\id_X:X\to C_{r\to r}(X)=X$. For associativity, let
282 \[
283 (r,X)\xrightarrow{\alpha}(s,Y)
284 \xrightarrow{\beta}(t,Z)
285 \xrightarrow{\gamma}(u,W).
286 \]
287 One composite gives
288 \[
289 C_{t\to r}(\gamma)\circ C_{s\to r}(\beta)\circ\alpha.
290 \]
291 The other gives the same expression by functoriality and
292 \[
293 C_{t\to r}=C_{s\to r}C_{t\to s}.
294 \]
295 Thus composition is associative.
296 \end{proof}
297 
298 Define
299 \[
300 \pi:\K\to\Rcat,
301 \qquad
302 \pi(r,X)=r.
303 \]
304 The fiber over $r$ is canonically equivalent to $\Tcat_r$.
305 
306 \subsection{Canonical factorization}
307 
308 For $r\le s$ and $Y\in\Tcat_s$, define the canonical morphism
309 \[
310 \kappa_{r,s,Y}:
311 (r,C_{s\to r}Y)\longrightarrow(s,Y)
312 \]
313 represented by the identity map
314 \[
315 \id_{C_{s\to r}Y}.
316 \]
317 
318 \begin{theorem}[Two-axis factorization]\label{thm:factorization}
319 Every morphism
320 \[
321 f:(r,X)\longrightarrow(s,Y)
322 \]
323 in $\K$ factors uniquely as
324 \[
325 (r,X)
326 \xrightarrow{f_v}
327 (r,C_{s\to r}Y)
328 \xrightarrow{\kappa_{r,s,Y}}
329 (s,Y),
330 \]
331 where $f_v$ lies entirely in the fiber $\Tcat_r$.
332 \end{theorem}
333 
334 \begin{proof}
335 By definition, $f$ is exactly a morphism
336 \[
337 \alpha:X\to C_{s\to r}Y
338 \]
339 in $\Tcat_r$. Take $f_v=\alpha$. Composition with $\kappa_{r,s,Y}$ is represented by
340 \[
341 \id_{C_{s\to r}Y}\circ\alpha=\alpha,
342 \]
343 so the composite is $f$. Any factorization through the fixed cartesian target $(r,C_{s\to r}Y)$ must use the same fiber morphism $\alpha$, hence is unique.
344 \end{proof}
345 
346 \begin{remark}
347 This is the formal version of the two-axis statement
348 \[
349 \boxed{
350 \text{mixed movement}
351 =
352 \text{within-resolution transformation}
353 +
354 \text{resolution movement}.
355 }
356 \]
357 Categorically, $\pi:\K\to\Rcat$ is the Grothendieck fibration associated with $F$. Indeed, a map $(u,Z)\to(s,Y)$ above $u\le r\le s$ is exactly a map $Z\to C_{s\to u}Y=C_{r\to u}C_{s\to r}Y$. This uniquely factors through $\kappa_{r,s,Y}$ as a morphism to $(r,C_{s\to r}Y)$ above $u\le r$, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object $Y$. It is not a procedure that selects a refinement of an arbitrary coarse object.
358 \end{remark}
359 
360 \section{Resolution geometry}
361 
362 The nested equivalence relations from \cref{cor:nested} induce a genuine geometry when the resolution set is linearly ordered.
363 
364 \begin{definition}[Separation depth]
365 Let $S$ be a set of states, with equivalence relations $\sim_n$ for every $n\in\mathbb N=\{0,1,\ldots\}$. Assume
366 \[
367 \sim_{n+1}\ \subseteq\ \sim_n,
368 \qquad \sim_0=S\times S.
369 \]
370 Define
371 \[
372 \nu(X,Y):=\sup\{n\in\mathbb N:X\sim_nY\}\in\mathbb N\cup\{\infty\}.
373 \]
374 The universal zeroth level makes this set nonempty. Assume separation, meaning that $X\sim_nY$ 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.
375 \end{definition}
376 
377 \begin{theorem}[Resolution ultrametric]\label{thm:ultrametric}
378 Define
379 \[
380 d_R(X,Y):=
381 \begin{cases}
382 0,&X=Y,\\
383 2^{-\nu(X,Y)},&X\ne Y.
384 \end{cases}
385 \]
386 Then $d_R$ is an ultrametric:
387 \[
388 d_R(X,Z)\le\max\{d_R(X,Y),d_R(Y,Z)\}.
389 \]
390 \end{theorem}
391 
392 \begin{proof}
393 Let
394 \[
395 m=\min\{\nu(X,Y),\nu(Y,Z)\}.
396 \]
397 For every $n\le m$, we have $X\sim_nY$ and $Y\sim_nZ$. Since $\sim_n$ is an equivalence relation, $X\sim_nZ$. Hence
398 \[
399 \nu(X,Z)\ge m.
400 \]
401 Therefore
402 \[
403 2^{-\nu(X,Z)}
404 \le
405 2^{-m}
406 =
407 \max\{2^{-\nu(X,Y)},2^{-\nu(Y,Z)}\}.
408 \]
409 The identity and symmetry axioms are immediate, and separation gives positivity for distinct states. Without separation the same formula, with $2^{-\infty}=0$, is an ultrapseudometric.
410 \end{proof}
411 
412 Thus the resolution axis can carry an intrinsic non-Archimedean geometry. Two objects are close when many levels of resolution are required before they can be distinguished.
413 
414 \begin{definition}[First distinguishing scale]
415 When finite, define
416 \[
417 \delta(X,Y):=\nu(X,Y)+1.
418 \]
419 This is the first resolution at which $X$ and $Y$ become distinguishable.
420 \end{definition}
421 
422 This quantity may be useful in $p$-adic, dyadic, symbolic, filtration, and machine-representation settings.
423 
424 \section{Seeds and generated zoos}
425 
426 \subsection{Seed signatures}
427 
428 Let $U$ be a set of isomorphism classes of resolved objects under consideration. Smallness is imposed only to use ordinary powerset fixed-point arguments.
429 
430 \begin{definition}[Seed signature]
431 A \emph{seed signature} on $U$ is data
432 \[
433 \mathfrak S=(\Seeds_0,\mathscr T,\mathscr C,\mathscr L,\mathscr D),
434 \]
435 where:
436 \begin{itemize}
437     \item $\Seeds_0\subseteq U$ is the initial seed set;
438     \item $\mathscr T$ is a collection of partial unary transformation constructors on $U$;
439     \item $\mathscr C$ is the collection of allowed coarsening constructors;
440     \item $\mathscr L$ assigns to an object its admissible refinement classes;
441     \item $\mathscr D$ is a collection of partial residual constructors.
442 \end{itemize}
443 \end{definition}
444 
445 The use of partial constructors is intentional. An operation need not be defined on every object or every resolution.
446 
447 \begin{definition}[One-step zoo operator]
448 For $A\subseteq U$, define
449 \[
450 \Phi(A):=
451 A
452 \cup \mathscr T[A]
453 \cup \mathscr C[A]
454 \cup \mathscr L[A]
455 \cup \mathscr D[A],
456 \]
457 where each term denotes all defined outputs of the corresponding constructors applied to inputs from $A$.
458 \end{definition}
459 
460 \begin{lemma}[Monotonicity]\label{lem:phi-monotone}
461 If $A\subseteq B$, then
462 \[
463 \Phi(A)\subseteq\Phi(B).
464 \]
465 \end{lemma}
466 
467 \begin{proof}
468 Every constructor output available from inputs in $A$ is also available from the same inputs viewed as elements of $B$.
469 \end{proof}
470 
471 \begin{theorem}[Seed zoo existence]\label{thm:zoo-existence}
472 There exists a least $\Phi$-closed set containing $\Seeds_0$. Equivalently, there is a least fixed point of the closure process above $\Seeds_0$.
473 \end{theorem}
474 
475 \begin{proof}
476 The powerset $\mathcal P(U)$ is a complete lattice. The map
477 \[
478 \Psi(A):=\Seeds_0\cup\Phi(A)
479 \]
480 is monotone by \cref{lem:phi-monotone}. By the Knaster--Tarski fixed-point theorem, $\Psi$ has a least fixed point. This is exactly the least constructor-closed set containing $\Seeds_0$.
481 \end{proof}
482 
483 \begin{definition}[Generated zoo]
484 The least fixed point in \cref{thm:zoo-existence} is denoted
485 \[
486 \Zoo(\Seeds_0).
487 \]
488 \end{definition}
489 
490 \begin{proposition}[Finitary iteration]\label{prop:finitary}
491 Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage $A$. Then $\Phi$ preserves increasing unions of $\omega$-chains. Then
492 \[
493 \Zoo(\Seeds_0)
494 =
495 \bigcup_{n<\omega}\Psi^n(\varnothing).
496 \]
497 \end{proposition}
498 
499 \begin{proof}
500 For a constructor output from $\bigcup_n A_n$, its finitely many inputs all belong to a common $A_N$. The output therefore already belongs to $\Phi(A_N)$. This proves preservation of increasing unions. The union on the right is a fixed point because $\Psi$ preserves that union. It contains $\Seeds_0$ and is contained in every fixed point by induction on $n$, hence is least.
501 \end{proof}
502 
503 This turns the seed calculus into an explicit generative process:
504 \[
505 \Seeds_0
506 \subseteq
507 \Seeds_1
508 \subseteq
509 \Seeds_2
510 \subseteq\cdots,
511 \qquad
512 \Seeds_{n+1}=\Psi(\Seeds_n).
513 \]
514 
515 \subsection{Zoo growth}
516 
517 \begin{definition}[Growth function]
518 When each finite stage contains finitely many isomorphism classes, define
519 \[
520 g_{\Seeds_0}(n):=|\Psi^n(\varnothing)|.
521 \]
522 \end{definition}
523 
524 The asymptotic class of $g(n)$ may distinguish finite, polynomial, exponential, and faster-growing presentations. It depends on the chosen generator signature and cost convention, not merely on the generated category. This is a proposed invariant of a generated mathematical grammar rather than a theorem of the present note.
525 
526 \section{Transformation-resolution exchange}
527 
528 \subsection{Resolution-indexed transformations}
529 
530 \begin{definition}[Lax resolution-compatible transformation]\label{def:laxtrans}
531 A \emph{lax resolution-compatible transformation} consists of endofunctors
532 \[
533 A_r:\Tcat_r\to\Tcat_r
534 \]
535 for every $r\in R$, together with natural transformations
536 \[
537 \eta^A_{s,r}:
538 A_rC_{s\to r}
539 \Longrightarrow
540 C_{s\to r}A_s
541 \qquad(r\le s),
542 \]
543 satisfying identity and composition coherence:
544 \[
545 \eta^A_{r,r}=\id
546 \]
547 and, for $r\le s\le t$,
548 \[
549 \eta^A_{t,r}
550 =
551 \bigl(C_{s\to r}\eta^A_{t,s}\bigr)
552 \circ
553 \bigl(\eta^A_{s,r}C_{t\to s}\bigr).
554 \]
555 If every $\eta^A_{s,r}$ is a natural isomorphism, $A$ is called \emph{invertibly resolution-compatible}. This is distinct from exactness of a functor between stable categories.
556 \end{definition}
557 
558 The square is
559 \[
560 \begin{tikzcd}[column sep=large,row sep=large]
561 A_rC_{s\to r}(X) \arrow[r,"\eta^A_{s,r,X}"] & C_{s\to r}A_s(X).
562 \end{tikzcd}
563 \]
564 It compares transforming after coarsening with coarsening after transforming.
565 
566 \begin{theorem}[Lift to the total category]\label{thm:lift-total}
567 Every lax resolution-compatible transformation $A$ induces an endofunctor
568 \[
569 \widehat A:\K\to\K
570 \]
571 defined on objects by
572 \[
573 \widehat A(r,X)=(r,A_rX).
574 \]
575 \end{theorem}
576 
577 \begin{proof}
578 A morphism $(r,X)\to(s,Y)$ in $\K$ is represented by
579 \[
580 \alpha:X\to C_{s\to r}Y.
581 \]
582 Define $\widehat A(\alpha)$ to be
583 \[
584 A_rX
585 \xrightarrow{A_r(\alpha)}
586 A_rC_{s\to r}Y
587 \xrightarrow{\eta^A_{s,r,Y}}
588 C_{s\to r}A_sY.
589 \]
590 Identity preservation follows from $\eta^A_{r,r}=\id$. Composition preservation is exactly the coherence condition in \cref{def:laxtrans}, together with naturality of $\eta^A$.
591 \end{proof}
592 
593 Thus coherent exchange data suffice to assemble the family into a functor of the total system. A converse requires the functor to lie over the resolution base and to have the specified fiberwise restrictions.
594 
595 \subsection{Mixed residuals}
596 
597 \begin{definition}[Stable mixed residual]
598 Assume each $\Tcat_r$ is a small stable $\infty$-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For $r\le s$ and $X\in\Tcat_s$, define
599 \[
600 \Delta^A_{s,r}(X)
601 :=
602 \Cofib\bigl(\eta^A_{s,r,X}\bigr)
603 \in\Tcat_r.
604 \]
605 \end{definition}
606 
607 Then there is a cofiber sequence
608 \[
609 A_rC_{s\to r}X
610 \longrightarrow
611 C_{s\to r}A_sX
612 \longrightarrow
613 \Delta^A_{s,r}(X).
614 \]
615 
616 \begin{proposition}[Residual vanishing criterion]\label{prop:residual-vanish}
617 In a stable fiber,
618 \[
619 \Delta^A_{s,r}(X)\simeq0
620 \]
621 if and only if
622 \[
623 \eta^A_{s,r,X}
624 \]
625 is an equivalence.
626 \end{proposition}
627 
628 \begin{proof}
629 In a stable category a morphism is an equivalence exactly when its cofiber is a zero object.
630 \end{proof}
631 
632 In an abelian fiber one may instead record
633 \[
634 \Delta^-_{s,r}(X)=\ker\eta^A_{s,r,X},
635 \qquad
636 \Delta^+_{s,r}(X)=\operatorname{coker}\eta^A_{s,r,X}.
637 \]
638 These vanish simultaneously exactly when the comparison is an isomorphism.
639 
640 \begin{definition}[Residual promotion]
641 If $X$ is a seed-generated object and $\Delta^A_{s,r}(X)$ is nonzero, the operation
642 \[
643 X\longmapsto\Delta^A_{s,r}(X)
644 \]
645 is called \emph{residual promotion} when the residual is added to the active seed set.
646 \end{definition}
647 
648 This creates the recursive pattern
649 \[
650 \text{seed}
651 \to
652 \text{mixed square}
653 \to
654 \text{residual}
655 \to
656 \text{new seed}
657 \to
658 \text{expanded zoo}.
659 \]
660 
661 \section{Normal forms for mixed words}
662 
663 To make the two-axis calculus computational, mixed transformation-resolution expressions should admit controlled rewriting.
664 
665 \subsection{Typed word system}
666 
667 Consider a typed alphabet containing transformation symbols $T$ and resolution symbols $C$. A mixed word is a composable expression such as
668 \[
669 W=T_1C_1T_2T_3C_2T_4.
670 \]
671 Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form
672 \[
673 TC\;\Rightarrow\;C\,T',
674 \]
675 where $T'$ is the correctly reindexed transformation at the new resolution.
676 
677 \begin{definition}[Inversion number]
678 For a word $W$, let $\iota(W)$ 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.
679 \end{definition}
680 
681 \begin{lemma}[Exchange termination]\label{lem:termination}
682 Every exchange rewrite
683 \[
684 TC\Rightarrow CT'
685 \]
686 reduces $\iota$ by one. Therefore every sequence of such rewrites terminates.
687 \end{lemma}
688 
689 \begin{proof}
690 The rewritten adjacent pair contributes one inversion before the rewrite and none afterward. The relative order of every other transformation-resolution pair is unchanged.
691 \end{proof}
692 
693 \begin{theorem}[Exact two-axis normal form]\label{thm:normal-form}
694 Assume the exchange rewriting system is locally confluent. Then every well-typed mixed word has a unique normal form up to the equality notion of the rewriting system. The normal form has all resolution operations grouped before all within-resolution transformations.
695 \end{theorem}
696 
697 \begin{proof}
698 Induct on the inversion number. Every rewrite lowers it, so normal forms are reached. If two paths start with different rewrites, local confluence gives joining paths from their first successors to a common word. Normalize that word. The induction hypothesis at each first successor identifies its normal form with the common descendant's normal form. Thus the two original paths have the same normal form. Since every adjacent inversion has a rule, a normal word has no inversion and therefore has all coarsening symbols to the left of all transformation symbols. This is the finite-height form of Newman's argument.
699 \end{proof}
700 
701 \status{proved here from standard rewriting theory under local confluence}
702 
703 The nontrivial application problem is to prove local confluence for the concrete labelled signature. This theorem concerns word syntax, not equality of the semantic comparison maps along different paths. The pure two-sort grammar is treated more explicitly in \cref{thm:event-ideals}.
704 
705 \subsection{Split residual expansion}
706 
707 Exact exchange is too restrictive for many applications. A first rigorous residual model can be obtained in an additive setting.
708 
709 \begin{definition}[Split residual exchange]
710 A chosen exchange morphism $\eta:TC\Rightarrow CT'$ is \emph{split residual} if it is naturally identified with the projection in a decomposition
711 \[
712 TC(X)\cong CT'(X)\oplus E_{T,C}(X).
713 \]
714 Here $E_{T,C}$ is the split \emph{source-side remainder}. It is not denoted by $\Delta$, because in the stable convention of this paper
715 \[
716 \Cofib(\eta_X)\simeq E_{T,C}(X)[1],
717 \qquad E_{T,C}(X)\simeq\Delta_{T,C}(X)[-1].
718 \]
719 \end{definition}
720 
721 \begin{theorem}[Finite split residual expansion]\label{thm:split-expansion}
722 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
723 \[
724 W(s)\cong W_{\mathrm{nf}}(s)\oplus\bigoplus_{i=1}^m E_i(s),
725 \]
726 where $E_i(s)$ is the surrounding context applied to the source-side remainder of step $i$. In the stable case, $E_i(s)\simeq D_i[-1]$ for the step cofibers $D_i$ defined below.
727 \end{theorem}
728 
729 \begin{proof}
730 At step $i$, the split exchange and additivity of the surrounding context give
731 $X_{i-1}\cong X_i\oplus E_i(s)$. Substituting these decompositions successively yields
732 $X_0\cong X_m\oplus E_m(s)\oplus\cdots\oplus E_1(s)$.
733 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.
734 \end{proof}
735 
736 This proof does not recursively normalize the residual branches. Such a recursive residual forest needs its own well-founded measure. Termination of the principal word alone does not prove termination of every newly generated residual word.
737 
738 \begin{definition}[Normalization path]
739 Let $\mathcal S$ be a stable $\infty$-category, or a stable enhancement with functorial cofibers. A \emph{normalization path} for a mixed word $W$ evaluated on $s$ is a finite chain
740 \[
741 X_0=W(s)\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m=W_{\mathrm{nf}}(s),
742 \]
743 where every $f_i$ is obtained by applying an exact surrounding context to one elementary exchange morphism. Write
744 \[
745 D_i:=\Cofib(f_i),
746 \qquad
747 D_{\mathrm{tot}}:=\Cofib(X_0\to X_m).
748 \]
749 \end{definition}
750 
751 \begin{theorem}[Stable residual filtration along a normalization path]\label{thm:stable-filtration}
752 For every finite normalization path in a stable enhancement there is a canonical finite filtration
753 \[
754 0=F_0\longrightarrow F_1\longrightarrow\cdots\longrightarrow F_m=D_{\mathrm{tot}}
755 \]
756 with cofiber sequences
757 \[
758 F_{i-1}\longrightarrow F_i\longrightarrow D_i.
759 \]
760 Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals $D_i$.
761 \end{theorem}
762 
763 \begin{proof}
764 Define
765 \[
766 F_i:=\Cofib(X_0\longrightarrow X_i).
767 \]
768 Then $F_0\simeq0$ and $F_m=D_{\mathrm{tot}}$. For the composable maps
769 \[
770 X_0\longrightarrow X_{i-1}\xrightarrow{f_i}X_i,
771 \]
772 the octahedral cofiber sequence gives
773 \[
774 \Cofib(X_0\to X_{i-1})
775 \longrightarrow
776 \Cofib(X_0\to X_i)
777 \longrightarrow
778 \Cofib(f_i).
779 \]
780 Thus
781 \[
782 F_{i-1}\longrightarrow F_i\longrightarrow D_i
783 \]
784 is a cofiber sequence. Since $f_i$ is obtained by placing an elementary exchange morphism inside an exact context, exactness identifies $D_i$ with the same context applied to the elementary exchange cofiber. Iterating gives the stated filtration.
785 \end{proof}
786 
787 \status{proved for a chosen normalization path in a stable enhancement}
788 
789 \begin{corollary}[Source-to-normal-form residual control]\label{cor:source-normal-control}
790 There is a cofiber sequence
791 \[
792 D_{\mathrm{tot}}[-1]\longrightarrow W(s)\longrightarrow W_{\mathrm{nf}}(s),
793 \]
794 and $D_{\mathrm{tot}}[-1]$ carries the shifted filtration with successive pieces $D_i[-1]$. Hence the failure of the original word to equal its normal form is an iterated extension of shifted elementary exchange residuals.
795 \end{corollary}
796 
797 The theorem is path-relative. If two normalization paths are related by coherent higher exchange cells, their total comparison maps may agree up to homotopy, but path-independence of the induced filtration is a separate problem. In an ordinary triangulated category without an enhancement, the octahedral axiom gives existence of such filtrations but not a canonical functorial choice.
798 
799 \begin{problem}[Order-independent residual filtration]\label{prob:path-independent}
800 Identify an intrinsic replacement for the family of path-relative filtrations. Mere higher coherence cannot force their ordered stages to be equivalent: \cref{prop:filtration-no-go} gives a counterexample. The residual diagram of \cref{thm:ideal-residual} is a replacement when a coherent event diagram is supplied. Additional hypotheses for equivalence of selected ordered filtrations must constrain the intermediate objects, not only the comparison maps.
801 \end{problem}
802 
803 \section{Normalization histories and intrinsic residual diagrams}\label{sec:histories}
804 
805 There are three different invariance questions. A word may have a unique normal-form \emph{object}, the normalization paths may induce the same \emph{comparison map}, and the cofiber of that map may carry several different \emph{ordered filtrations}. None of these statements should be silently substituted for the next.
806 
807 \subsection{A no-go theorem for ordered filtration independence}
808 
809 \begin{proposition}[Coherent diamonds do not determine one ordered filtration]\label{prop:filtration-no-go}
810 Even a strictly commuting bicartesian square of objects in a stable category can yield two normalization paths with the same total comparison map and inequivalent ordered residual filtrations.
811 \end{proposition}
812 
813 \begin{proof}
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
815 \[
816 \begin{tikzcd}[column sep=large,row sep=large]
817 0 \arrow[r] \arrow[d] & M \arrow[d,hook] \\
818 N \arrow[r,hook] & M\oplus N.
819 \end{tikzcd}
820 \]
821 The maps into $M\oplus N$ are the summand inclusions. This square is a pushout, hence also a pullback. Both composites are the unique map $0\to M\oplus N$.
822 
823 The residual filtrations of its two boundary paths are
824 \[
825 0\longrightarrow M\longrightarrow M\oplus N,
826 \qquad
827 0\longrightarrow N\longrightarrow M\oplus N.
828 \]
829 An equivalence of these objects of $\operatorname{Fun}([2],\operatorname{Perf}(k))$ would induce an equivalence $M\simeq N$ at the middle index. That is impossible because their degree-zero cohomology dimensions are $1$ and $2$. Nevertheless both total residuals are $M\oplus N$.
830 \end{proof}
831 
832 \begin{example}[The counterexample occurs inside the two-axis grammar]\label{ex:typed-diamond}
833 The preceding square is not merely an unrelated diagram. Let the resolution base be $0<1<2$, put
834 \[
835 \Tcat_0=\Tcat_1=\Tcat_2=\operatorname{Perf}(k)\times\operatorname{Perf}(k),
836 \]
837 and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections
838 \[
839 P_1(x,y)=(x,0),\qquad P_2(x,y)=(0,y).
840 \]
841 Use transformation families
842 \[
843 A_0=P_2,\quad A_1=A_2=\id,
844 \qquad
845 B_0=B_1=P_1,\quad B_2=\id.
846 \]
847 For $A$, the comparison across $1\to0$ is the inclusion $P_2\Rightarrow\id$, and across $2\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\Rightarrow\id$. The long comparisons are their composites. Units and transitivity therefore hold.
848 
849 On the seed $x=(k[0],k^2[0])$, the word
850 \[
851 W=A_0C_{1\to0}B_1C_{2\to1}
852 \]
853 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
854 \[
855 C_{1\to0}C_{2\to1}A_2B_2.
856 \]
857 Thus the complete paths have residual filtrations
858 \[
859 0\to P_1x\to x\xrightarrow{\id}x,
860 \qquad
861 0\to P_2x\to x\xrightarrow{\id}x.
862 \]
863 Their first nonzero stages are not equivalent. The transformations, contexts, and coarsenings are exact, and all comparison data are coherent. Consequently extra coherence alone cannot repair the general claim of ordered filtration independence.
864 \end{example}
865 
866 \status{general ordered-filtration independence refuted; the path-relative theorem remains valid}
867 
868 The correction is not to discard the filtration. It is to retain the collection of compatible intermediate stages in a diagram richer than a single linear order.
869 
870 \subsection{What suffices for independence of the total residual}
871 
872 The normal-form endpoint alone does not determine a residual. For example, the maps $\id_k$ and $0:k\to k$ have the same source and target, but cofibers $0$ and $k\oplus k[1]$. Comparison maps therefore belong to the data.
873 
874 \begin{theorem}[Semantic diamond criterion]\label{thm:semantic-diamond}
875 Let a rewrite graph have a height $h$ in $\mathbb N$ 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 $\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak
876 \[
877 x\xrightarrow{e}y,\qquad x\xrightarrow{f}z
878 \]
879 has joining paths $a:y\leadsto u$ and $b:z\leadsto u$ whose semantic composites satisfy
880 \[
881 [a]\,[e]=[b]\,[f]\quad\text{in the homotopy category}.
882 \]
883 Then every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.
884 \end{theorem}
885 
886 \begin{proof}
887 Induct on $h(x)$. A normal vertex has only the empty normalization path. Otherwise take two normalization paths with initial edges $e$ and $f$. Choose the joining paths in the hypothesis and any normalization path $t:u\leadsto n(x)$. Such a path exists, and its terminal vertex is $n(x)$ by uniqueness of normal forms.
888 
889 Since $h(y),h(z)<h(x)$, the induction hypothesis replaces the two original suffixes by $ta$ and $tb$ respectively without changing their homotopy classes. The full comparison classes are therefore
890 \[
891 [t][a][e]=[t][b][f].
892 \]
893 This proves equality for all normalization paths from $x$. Equality in the homotopy category gives a homotopy between the corresponding maps with fixed endpoints. A homotopy identifies the two arrows up to equivalence, and the cofiber functor preserves that equivalence.
894 \end{proof}
895 
896 \begin{remark}[Equivalence is not a chosen canonical identification]
897 The theorem asserts existence of equivalent total cofibers. A specified equivalence requires a specified homotopy between comparison maps. Independence of all such choices requires higher coherence. Neither conclusion identifies the intermediate ordered filtrations in \cref{prop:filtration-no-go}.
898 \end{remark}
899 
900 \subsection{The event poset of pure exchange}
901 
902 The pure exchange grammar has enough combinatorial structure to index all normalization histories without choosing their order. This subsection concerns only adjacent swaps $TC\Rightarrow CT$ that retain the identity of each symbol. More elaborate rules that duplicate, delete, or replace generators are not included.
903 
904 \begin{definition}[Exchange event poset]\label{def:event-poset}
905 Fix a finite word $W$ in two types of labelled symbols. Number its transformation symbols $T_1,\ldots,T_a$ and its coarsening symbols $C_1,\ldots,C_b$ in their original left-to-right order. An \emph{exchange event} is a pair
906 \[
907 (i,j)\in E(W)
908 \quad\Longleftrightarrow\quad
909 T_i\text{ occurs to the left of }C_j\text{ in }W.
910 \]
911 The event swaps the relative order of this pair exactly once. Define $P_W$ as the partial order generated by
912 \begin{align*}
913 (i,k)&\prec(i,j) &&\text{when }k<j\text{ and both events exist},\\
914 (h,j)&\prec(i,j) &&\text{when }h>i\text{ and both events exist}.
915 \end{align*}
916 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.
917 \end{definition}
918 
919 The first relation says that one transformation crosses the coarsenings from left to right. The second says that one coarsening crosses the transformations from right to left.
920 
921 \begin{theorem}[Normalization states are order ideals]\label{thm:event-ideals}
922 For the pure adjacent-swap grammar, the following correspondences are bijective:
923 \begin{enumerate}
924 \item words reachable from $W$ and ideals $I\in J(P_W)$;
925 \item enabled elementary swaps at a reachable word and minimal elements of $P_W\setminus I$;
926 \item complete normalization histories and linear extensions of $P_W$.
927 \end{enumerate}
928 The ideal attached to a reachable word is the set of events already performed. The original word corresponds to $\varnothing$, and the normal word to $P_W$.
929 \end{theorem}
930 
931 \begin{proof}
932 Every swap changes the order of exactly one transformation/coarsening pair, never to be reversed. The relative orders of the transformations among themselves and of the coarsenings among themselves remain fixed.
933 
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.
935 
936 That ideal determines the reachable word uniquely. It specifies whether each originally inverted pair has crossed, while pairs originally of the form $C_jT_i$ never change order. Together with the fixed orders inside each type, this determines every pairwise order of labelled symbols.
937 
938 Conversely, suppose an ideal $I$ has been performed and $(i,j)$ is a minimal event outside it. In the current word $T_i$ still precedes $C_j$. If another symbol lies between them, it is either some $C_k$ with $k<j$ or some $T_h$ with $h>i$. In the first case $(i,k)$ is an unperformed predecessor. In the second case $(h,j)$ is an unperformed predecessor. The relevant pair was already inverted in the original word, since swaps never turn an original $CT$ pair into $TC$. Both cases contradict minimality. Hence $T_i$ and $C_j$ are adjacent and the event is enabled.
939 
940 Any ideal can now be constructed by adding its events in an order compatible with $P_W$. This proves surjectivity and the statement about edges. A complete path adds each event exactly once, so it is exactly a linear extension. Conversely each linear extension prescribes enabled swaps and therefore a complete path.
941 \end{proof}
942 
943 \begin{lemma}[All histories are related by independent interchanges]\label{lem:linear-extension-swaps}
944 Any two linear extensions of a finite poset are connected by swaps of adjacent incomparable elements. Consequently any two pure normalization histories are related by commuting independent-event diamonds.
945 \end{lemma}
946 
947 \begin{proof}
948 Let $e$ be the first element of the second linear extension. It is minimal in the poset. In the first extension, every element appearing before $e$ is incomparable with it: it cannot precede $e$ in the poset by minimality, and it cannot follow $e$ in the poset while occurring before it in a linear extension. Move $e$ left across these incomparable elements. The two extensions now begin with the same element. Delete it and apply induction to the remaining finite poset.
949 
950 In the exchange poset, incomparable events share neither a transformation symbol nor a coarsening symbol. When both are enabled their adjacent swaps are disjoint. Performing them in either order gives the same word, namely the diamond for adjoining both events to the current ideal.
951 \end{proof}
952 
953 It follows that the category of rewrite paths modulo these diamond relations is the thin category $J(P_W)$. The conclusion concerns the pure syntax. In a typed application, labels after crossing must be defined by the specified reindexing rules, and their semantic exchange cells must satisfy the corresponding comparisons.
954 
955 \subsection{A diagram that retains the residual histories}
956 
957 \begin{definition}[Coherent exchange-event diagram]
958 For a finite event poset $P$, a coherent exchange-event diagram is a functor
959 \[
960 X:NJ(P)\longrightarrow\mathcal S
961 \]
962 into a stable $\infty$-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.
963 \end{definition}
964 
965 Strictly commuting exchange diagrams in a concrete enhanced model, such as complexes, supply examples. Merely checking squares in the homotopy category does not automatically supply all the higher data in this definition.
966 
967 \begin{theorem}[Ideal-indexed residual diagram]\label{thm:ideal-residual}
968 A coherent exchange-event diagram determines a functor
969 \[
970 \mathcal F_X:NJ(P)\longrightarrow\mathcal S,
971 \qquad
972 \mathcal F_X(I)=\Cofib(X_{\varnothing}\longrightarrow X_I),
973 \]
974 with the following properties:
975 \begin{enumerate}
976 \item $\mathcal F_X(\varnothing)\simeq0$;
977 \item along every maximal chain of ideals it restricts to the residual filtration of \cref{thm:stable-filtration};
978 \item for every cover $I\subset I\cup\{e\}$ there is a cofiber sequence
979 \[
980 \mathcal F_X(I)\longrightarrow\mathcal F_X(I\cup\{e\})
981 \longrightarrow\Cofib(X_I\to X_{I\cup\{e\}});
982 \]
983 \item its terminal object is the total residual $\Cofib(X_{\varnothing}\to X_P)$, independent of the linear extension used to reach it;
984 \item exact functors preserve the entire diagram, naturally in the specified coherent input.
985 \end{enumerate}
986 \end{theorem}
987 
988 \begin{proof}
989 The arrows from the initial ideal $\varnothing$ induce a natural transformation from the constant diagram at $X_{\varnothing}$ to $X$. Take its cofiber pointwise in the stable functor category. This constructs $\mathcal F_X$ together with its coherence, not just an assignment of objects.
990 
991 At $\varnothing$ the map is the identity, so its cofiber is zero. Restriction to a maximal chain gives exactly the pointwise formula defining the earlier path-relative filtration. For an inclusion of ideals, apply the cofiber sequence for the composable maps
992 \[
993 X_{\varnothing}\longrightarrow X_I\longrightarrow X_{I\cup\{e\}}.
994 \]
995 This gives the displayed cover cofiber sequence. At the terminal ideal the same formula gives the total residual. Finally an exact functor preserves these finite colimits pointwise and hence preserves the diagram and all its structure maps.
996 \end{proof}
997 
998 \status{proved relative to a specified coherent exchange-event diagram}
999 
1000 This is the appropriate replacement for a single path-independent ordered filtration. In \cref{prop:filtration-no-go}, $P$ has two incomparable events, so $J(P)$ is the four-object diamond. Its two chains remember different intermediate residuals without disagreeing about the total residual. The richer diagram preserves information that the proposed identification of ordered filtrations would have erased.
1001 
1002 \begin{corollary}[History-independent total obstruction]\label{cor:history-obstruction}
1003 Let $q:\mathcal S\to\mathcal O$ be exact. For a fixed coherent exchange-event diagram, the object
1004 \[
1005 q\mathcal F_X(P)
1006 \]
1007 is independent of the normalization history. Along every maximal chain it has the filtration of \cref{thm:obstruction-localization}. Therefore its nonvanishing forces a nonzero elementary quotient residual on \emph{each} normalization history. The particular event witnessing nonvanishing may depend on the history.
1008 \end{corollary}
1009 
1010 \begin{proof}
1011 The terminal object is fixed by \cref{thm:ideal-residual}. Apply \cref{thm:obstruction-localization} separately to the finite filtration supplied by each maximal chain.
1012 \end{proof}
1013 
1014 The converse is not asserted. Nonzero elementary residuals can participate in extensions whose total residual vanishes. Nor does the corollary exhibit one event that must witness the obstruction in every history.
1015 
1016 
1017 \section{Constructing semantic exchange diagrams}\label{sec:strict-semantics}
1018 
1019 The coherent diagram in \cref{thm:ideal-residual} was an input. We now construct it for a concrete enhanced realization of the typed pure grammar. This does not strictify arbitrary higher-categorical exchange data.
1020 
1021 \begin{definition}[Strict chain-natural realization]\label{def:strict-chain-realization}
1022 Fix a finite resolution chain $r_0\le\cdots\le r_b$. At each level choose a category $\mathcal E_j$ of complexes and a functor $L_j:\mathcal E_j\to\mathcal S_j$ into a stable $\infty$-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors
1023 \[
1024  C_j:\mathcal E_j\to\mathcal E_{j-1},\qquad
1025  T_{i,j}:\mathcal E_j\to\mathcal E_j,
1026 \]
1027 which induce exact functors in the chosen stable realizations, and strict natural transformations
1028 \[
1029  \chi_{i,j}:T_{i,j-1}C_j\Longrightarrow C_jT_{i,j}
1030  \qquad(1\le j\le b).
1031 \]
1032 The components of $\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in the ordinary strict 2-category of these models.
1033 \end{definition}
1034 
1035 Coarsening symbols occur in their fixed order $C_1,\ldots,C_b$. 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 $\mathcal E_b$ to $\mathcal E_0$. 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.
1036 
1037 \begin{theorem}[Strict semantic realization of the event diagram]\label{thm:strict-event-realization}
1038 For a typed pure word $W$ in a strict chain-natural realization and a seed $s\in\mathcal E_b$, evaluation gives a functor
1039 \[
1040  X_W:NJ(P_W)\longrightarrow\mathcal S_0.
1041 \]
1042 Its vertices evaluate the reachable words and its cover arrows evaluate the whiskered exchange maps. Thus the residual diagram in \cref{thm:ideal-residual} is constructed, rather than assumed, for this class of realizations.
1043 \end{theorem}
1044 
1045 \begin{proof}
1046 First work in the ordinary category of functors $\mathcal E_b\to\mathcal E_0$ and natural transformations. Assign to a rewrite edge its exchange transformation, whiskered by the surrounding word. Two simultaneously enabled swaps are disjoint. The two composites around their diamond are equal by the interchange law for natural transformations. This equality uses strict naturality, not invertibility of $\chi$.
1047 
1048 For ideals $I\subseteq J$, two paths adding the events in $J\setminus I$ are linear extensions of that interval's induced event order. The argument of \cref{lem:linear-extension-swaps} connects them by adjacent interchanges of incomparable events. Each interchange is one of the commuting diamonds just checked. Hence all such paths induce the same natural transformation. Evaluation therefore factors through the thin category $J(P_W)$.
1049 
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.
1051 \end{proof}
1052 
1053 \begin{remark}[Scope of coherence]
1054 Strict models are sufficient, not necessary. This theorem does not infer strict naturality from a collection of commuting squares in a homotopy category. Arbitrary weak exchange cells, additional rewrite relations, and higher reindexing coherence still need their own realization theorem. For the fixed pure grammar, however, no independent braid relation is needed: the only changes of history exchange disjoint events.
1055 \end{remark}
1056 
1057 \section{Interaction residuals and event reconstruction}\label{sec:interaction-reconstruction}
1058 
1059 An exchange square can commute coherently while changing the residual of one event after another event is performed. The appropriate measurement is a second cofiber. It measures failure of excision, not failure of commutativity.
1060 
1061 \subsection{The interaction of two exchanges}
1062 
1063 Let a specified coherent square in a stable $\infty$-category $\mathcal S$ be
1064 \[
1065 \begin{tikzcd}
1066  A \arrow[r,"u"] \arrow[d,"v"'] & B \arrow[d,"v'"]\\
1067  C \arrow[r,"u'"'] & D.
1068 \end{tikzcd}
1069 \]
1070 Write $E=\Cofib(u)$ and $E'=\Cofib(u')$. The square induces $E\to E'$.
1071 
1072 \begin{definition}[Interaction residual]\label{def:interaction-residual}
1073 The interaction residual of this specified square is
1074 \[
1075  \Theta=\Cofib\bigl(\Cofib(A\to B)\longrightarrow
1076                          \Cofib(C\to D)\bigr).
1077 \]
1078 The specified square, including its coherent homotopy, is part of the input.
1079 \end{definition}
1080 
1081 \begin{theorem}[Interaction is the total cofiber]\label{thm:interaction-cofiber}
1082 There are natural equivalences
1083 \[
1084  \Theta\simeq\Cofib\bigl(B\amalg_A C\longrightarrow D\bigr)
1085  \simeq\Cofib\bigl(\Cofib(A\to C)\longrightarrow
1086                          \Cofib(B\to D)\bigr).
1087 \]
1088 Consequently $\Theta\simeq0$ if and only if the square is a homotopy pushout, equivalently a homotopy pullback. An exact functor preserves these equivalences.
1089 \end{theorem}
1090 
1091 \begin{proof}
1092 Cofibers are pushouts along maps to zero. Interchanging the finite colimits in the two successive pushout constructions identifies the first expression with the cofiber of the map from the punctured-square colimit $B\amalg_A C$ to $D$. Performing the same construction in the other coordinate gives the second expression. These identifications arise from the universal properties of the same finite diagrams and are natural in the given square.
1093 
1094 A map in a stable category has zero cofiber exactly when it is an equivalence. Thus vanishing says precisely that the punctured-square colimit maps equivalently to $D$. Stability identifies pushout squares with pullback squares. Exact functors preserve zero objects and these finite colimits.
1095 \end{proof}
1096 
1097 In a chain model, interchanging coordinates includes the usual graded signs. The theorem is a natural equivalence, not an assertion that differently ordered cone matrices are literally identical. The cone conventions are recorded in \cite{StacksCones}; the finite-colimit argument supplies the enhanced statement used here.
1098 
1099 \subsection{An event's residual before and after unrelated events}
1100 
1101 Fix a finite poset $P$ and a coherent diagram $X:NJ(P)\to\mathcal S$. For $e\in P$, put
1102 \[
1103  D_e=\{p\in P:p<e\},\qquad
1104  \downarrow e=D_e\cup\{e\}.
1105 \]
1106 The eligible contexts for $e$ form the interval of ideals
1107 \[
1108  \mathcal A_e=\{I\in J(P):D_e\subseteq I,\ e\notin I\}.
1109 \]
1110 Its initial object is $D_e$. Define the residual functor on this interval by
1111 \[
1112  E_e(I)=\Cofib(X_I\to X_{I\cup\{e\}}).
1113 \]
1114 For incomparable events $e,f$ enabled at $I$, write $\Theta_{e,f}(I)$ for the interaction residual of their elementary square.
1115 
1116 \begin{theorem}[Criterion for context-independent event residuals]\label{thm:event-context-invariance}
1117 The following conditions are equivalent.
1118 \begin{enumerate}
1119 \item Every elementary independent-event square of $X$ is a homotopy pushout.
1120 \item Every $\Theta_{e,f}(I)$ vanishes.
1121 \item For each event $e$, the natural comparison
1122 \[
1123  \operatorname{const}_{\mathcal A_e}E_e(D_e)\longrightarrow E_e
1124 \]
1125 is an equivalence of diagrams.
1126 \end{enumerate}
1127 In this regime the successive cofibers of every normalization history are the event-labelled objects
1128 \[
1129  d_e:=\Cofib(X_{D_e}\to X_{\downarrow e}),
1130 \]
1131 listed in the order in which that history performs the events. Their labels and equivalence classes are independent of context, but their ordering and extension data need not disappear.
1132 \end{theorem}
1133 
1134 \begin{proof}
1135 The equivalence of (1) and (2) is \cref{thm:interaction-cofiber}. A cover in $\mathcal A_e$ adds an event $f$ incomparable with $e$. Indeed, predecessors of $e$ already lie in $I$, while a successor of $e$ cannot belong to an ideal omitting $e$. On this cover the map of event residuals has cofiber $\Theta_{e,f}(I)$. Condition (2) therefore makes every cover map an equivalence. Any inclusion in the finite interval factors into covers. The comparisons from its initial object consequently give the objectwise, hence diagrammatic, equivalence in (3).
1136 
1137 Conversely, (3) makes every cover comparison of event residuals an equivalence, so its cofiber vanishes. The final statement follows by restricting the residual diagram to a maximal chain and applying the same comparison to each enabled event.
1138 \end{proof}
1139 
1140 \subsection{Reconstructing the full diagram from principal ideals}
1141 
1142 The preceding criterion gives more than a common list of residuals. It makes the diagram recoverable from its values on the individual events together with their predecessors.
1143 
1144 \begin{lemma}[Elementary excision implies ideal excision]\label{lem:ideal-excision}
1145 If every elementary independent-event square of $X:NJ(P)\to\mathcal S$ is a homotopy pushout, then for all ideals $I,J$ the square
1146 \[
1147 \begin{tikzcd}
1148  X_{I\cap J}\arrow[r]\arrow[d] &X_I\arrow[d]\\
1149  X_J\arrow[r]&X_{I\cup J}
1150 \end{tikzcd}
1151 \]
1152 is a homotopy pushout.
1153 \end{lemma}
1154 
1155 \begin{proof}
1156 Elements of $I\setminus J$ and $J\setminus I$ are pairwise incomparable across the two sets. If $p\in I\setminus J$ satisfied $p\le q\in J\setminus I$, downward closure of $J$ would imply $p\in J$, a contradiction. The reverse comparison is excluded similarly.
1157 
1158 Choose linear extensions for both exclusive sets, beginning from $I\cap J$. Adjoining their successive initial segments gives a rectangular grid of ideals. Each cell adjoins one enabled event from each side and is an elementary independent-event square. It is a pushout by hypothesis. Repeated pasting of pushout squares gives the outer square displayed above. Empty exclusive sets give identity squares and cause no exception.
1159 \end{proof}
1160 
1161 \begin{theorem}[Principal-ideal reconstruction]\label{thm:principal-reconstruction}
1162 Let $P$ be finite and $\mathcal S$ stable. Write $\operatorname{Exc}_0(J(P),\mathcal S)$ for the full subcategory of coherent diagrams $F:NJ(P)\to\mathcal S$ such that $F(\varnothing)\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along
1163 \[
1164  j:P\longrightarrow J(P),\qquad e\longmapsto\downarrow e
1165 \]
1166 induces an equivalence
1167 \[
1168  \operatorname{Exc}_0(J(P),\mathcal S)
1169  \simeq\operatorname{Fun}(NP,\mathcal S).
1170 \]
1171 Its inverse sends $G:NP\to\mathcal S$ to
1172 \[
1173  (j_!G)(I)=\colim_{e\in I}G(e),
1174  \qquad (j_!G)(\varnothing)=0.
1175 \]
1176 All colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies
1177 \[
1178  \boxed{\mathcal F_X(I)\simeq
1179  \colim_{e\in I}\mathcal F_X(\downarrow e).}
1180 \]
1181 \end{theorem}
1182 
1183 \begin{proof}
1184 The displayed formula is the pointwise left Kan extension along $j$. It follows directly from the colimit universal property: maps from $j_!G$ to $F$ are the same as compatible maps $G(e)\to F(\downarrow e)$. For a principal ideal $\downarrow e$, its indexing poset has terminal object $e$, so $(j_!G)(\downarrow e)\simeq G(e)$.
1185 
1186 For arbitrary ideals $I,J$, every chain in $I\cup J$ 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 $G|_{I\cup J}$ are precisely compatible cocones on $G|_I$ and $G|_J$. The resulting colimit is the pushout of the two colimits over the colimit on $I\cap J$. Thus $j_!G$ is reduced and excisive.
1187 
1188 Now let $F$ be reduced and excisive. We prove the counit $j_!j^*F\to F$ 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 $e\in I$. Put $I'=I\setminus\{e\}$. Then
1189 \[
1190  I=I'\cup\downarrow e,\qquad I'\cap\downarrow e=D_e.
1191 \]
1192 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 equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.
1193 
1194 Finally, the total-cofiber construction is exact on square diagrams. Applied to the cofiber sequence of diagrams $\operatorname{const}X_{\varnothing}\to X\to\mathcal F_X$, it shows that the elementary total cofibers of $X$ and $\mathcal F_X$ agree, since a constant square has zero total cofiber. Thus $\mathcal F_X$ is reduced and excisive, and the first part applies.
1195 \end{proof}
1196 
1197 This is finite-poset reconstruction by left Kan extension and excision. It is a theorem about the specified event diagram, not the missing arithmetic globalization theorem K15. Its general categorical ingredients are standard, and no historical novelty claim is made.
1198 
1199 \begin{corollary}[Independent events split, dependencies retain extensions]\label{cor:independent-splitting}
1200 Under \cref{thm:principal-reconstruction}, if $P$ is an antichain then
1201 \[
1202  \mathcal F_X(P)\simeq\bigoplus_{e\in P}d_e.
1203 \]
1204 For a general event poset, a direct-sum formula does not follow.
1205 \end{corollary}
1206 
1207 \begin{proof}
1208 For an antichain, each principal ideal is a singleton, $D_e=\varnothing$, 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 $\operatorname{Perf}(\mathbb Z)$
1209 \[
1210  0\longrightarrow\mathbb Z/p\xrightarrow{\,p\,}\mathbb Z/p^2
1211 \]
1212 for a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\mathbb Z/p$ in degree zero, but the terminal object is not $\mathbb Z/p\oplus\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\mathbb Z\xrightarrow{p}\mathbb Z\xrightarrow{p}\mathbb Z$.
1213 \end{proof}
1214 
1215 The replacement for ordered-filtration independence is therefore now explicit: the event poset controls which attachments depend on earlier events, excision controls independence from unrelated events, and the principal-ideal diagram retains the extension data needed for reconstruction.
1216 
1217 
1218 \section{Obstruction theory}
1219 
1220 Residual and obstruction should not be identified. A residual is a structured discrepancy. An obstruction is a discrepancy that cannot be removed by the admissible calculus.
1221 
1222 \begin{definition}[Killable residual]
1223 Let $\delta\in\Zoo(\Seeds_0)$. Call $\delta$ \emph{killable} if there exists an admissible finite constructor word $W$ such that
1224 \[
1225 W(\delta)\simeq0.
1226 \]
1227 \end{definition}
1228 
1229 \begin{definition}[Persistent obstruction]
1230 A residual seed $\delta$ is a \emph{persistent obstruction} relative to a constructor zoo if it is not killable by any admissible finite word.
1231 \end{definition}
1232 
1233 This definition is intentionally operational. More refined applications should replace it by an obstruction class in a quotient, derived functor, cohomology group, or localization.
1234 
1235 A first taxonomy is:
1236 \[
1237 \boxed{
1238 \Omega=(\Omega_R,\Omega_T,\Omega_{TR},\Omega_G)
1239 }
1240 \]
1241 with:
1242 \begin{itemize}
1243     \item $\Omega_R$: no admissible refinement exists;
1244     \item $\Omega_T$: a required within-resolution transformation does not exist;
1245     \item $\Omega_{TR}$: exchange cannot be made coherent or its residual persists;
1246     \item $\Omega_G$: locally compatible data fail to reconstruct globally.
1247 \end{itemize}
1248 
1249 \subsection{Universal obstruction quotient in the stable calculus}
1250 
1251 Let $\mathcal Z$ be a small idempotent-complete stable $\infty$-category representing a thick seed zoo, and let $\mathscr M$ be a collection of exact constructor endofunctors of $\mathcal Z$ containing the identity and closed under composition.
1252 
1253 \begin{definition}[Operationally killable subcategory]
1254 Define
1255 \[
1256 \mathcal K_{\mathscr M}
1257 :=
1258 \{X\in\mathcal Z:\exists W\in\mathscr M,\;W(X)\simeq0\}.
1259 \]
1260 Let
1261 \[
1262 \mathcal N_{\mathscr M}
1263 :=
1264 \operatorname{Thick}(\mathcal K_{\mathscr M})
1265 \]
1266 be the smallest thick stable subcategory containing all operationally killable objects.
1267 \end{definition}
1268 
1269 The thick closure is necessary in general: two objects may be killed by unrelated constructor words with no single constructor word killing an extension of them.
1270 
1271 \begin{definition}[Stable obstruction category and obstruction class]\label{def:obstruction-quotient}
1272 The \emph{stable obstruction category} is the Verdier quotient
1273 \[
1274 \mathcal O_{\mathscr M}
1275 :=
1276 \mathcal Z/\mathcal N_{\mathscr M},
1277 \]
1278 with quotient functor
1279 \[
1280 q:\mathcal Z\longrightarrow\mathcal O_{\mathscr M}.
1281 \]
1282 For $X\in\mathcal Z$, define its \emph{obstruction class}
1283 \[
1284 \Obs_{\mathscr M}(X):=q(X).
1285 \]
1286 \end{definition}
1287 
1288 \begin{theorem}[Universal stable obstruction quotient]\label{thm:obstruction-quotient}
1289 The quotient $q$ is exact and satisfies:
1290 \begin{enumerate}
1291     \item every operationally killable object maps to zero;
1292     \item $q(X)\simeq0$ if and only if $X\in\mathcal N_{\mathscr M}$;
1293     \item for every exact functor $E:\mathcal Z\to\mathcal D$ into a stable $\infty$-category such that $E(K)\simeq0$ for every $K\in\mathcal K_{\mathscr M}$, there is an exact factorization, unique up to contractible choice together with its comparison to $E$,
1294     \[
1295     \mathcal Z\xrightarrow{q}\mathcal O_{\mathscr M}\xrightarrow{\bar E}\mathcal D.
1296     \]
1297 \end{enumerate}
1298 Thus $\mathcal O_{\mathscr M}$ is universal among exact targets in which all operationally killable residuals vanish.
1299 \end{theorem}
1300 
1301 \begin{proof}
1302 The Verdier quotient by the thick subcategory $\mathcal N_{\mathscr M}$ exists and the quotient functor is exact. Its kernel is exactly $\mathcal N_{\mathscr M}$. This gives (1)--(2). For (3), the kernel of an exact functor is thick. Since $E$ kills $\mathcal K_{\mathscr M}$, it kills the thick subcategory generated by that class, namely $\mathcal N_{\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\mathcal O_{\mathscr M}$ with the full subcategory of exact functors out of $\mathcal Z$ annihilating $\mathcal N_{\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\cite{Drew2015} for the stable localization and \cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.
1303 \end{proof}
1304 
1305 \status{proved in the stable regime using the standard Verdier quotient}
1306 
1307 A zero obstruction here means membership in the chosen thick kernel. It need not mean that one constructor word kills the object. Nor is $q(X)$ a chosen irreducible representative back inside $\mathcal Z$. A section or reflective localization would be extra data. The quotient also need not inherit the constructors automatically, as \cref{ex:quotient-collapse} will show.
1308 
1309 \begin{example}[Why thick closure is genuinely necessary]\label{ex:killable-not-thick}
1310 Let $\mathcal C$ be any nonzero stable category and set
1311 \[
1312 \mathcal Z=\mathcal C^3.
1313 \]
1314 Define exact idempotent endofunctors
1315 \[
1316 U(X_1,X_2,X_3)=(0,0,X_2\oplus X_3),
1317 \qquad
1318 V(X_1,X_2,X_3)=(0,0,X_1\oplus X_3).
1319 \]
1320 They satisfy
1321 \[
1322 U^2\simeq U,
1323 \qquad
1324 V^2\simeq V,
1325 \qquad
1326 U\circ V\simeq V,
1327 \qquad
1328 V\circ U\simeq U,
1329 \]
1330 so $\mathscr M=\{\id,U,V\}$ is closed under composition. Choose $0\not\simeq A\in\mathcal C$ and set
1331 \[
1332 X=(A,0,0),
1333 \qquad
1334 Y=(0,A,0).
1335 \]
1336 Then $U(X)\simeq0$ and $V(Y)\simeq0$, so both $X$ and $Y$ are operationally killable. However
1337 \[
1338 X\oplus Y=(A,A,0),
1339 \]
1340 and both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\oplus Y$. Thus $\mathcal K_{\mathscr M}$ need not be additive, much less thick.
1341 \end{example}
1342 
1343 This example shows that replacing $\mathcal K_{\mathscr M}$ by its thick closure in \cref{def:obstruction-quotient} is forced if the quotient is required to be universal for exact functors.
1344 
1345 \begin{definition}[Directed annihilator property]
1346 The constructor system $\mathscr M$ has the \emph{directed annihilator property} if for every $U,V\in\mathscr M$ there exist $P,Q,W\in\mathscr M$ and natural equivalences
1347 \[
1348 W\simeq P\circ U\simeq Q\circ V.
1349 \]
1350 Thus any two constructor words admit a common postcomposition. With multiplication written as composition, $P\circ U=Q\circ V$ is a common left multiple. The displayed equation, rather than a left/right name, fixes the convention.
1351 \end{definition}
1352 
1353 \begin{theorem}[When operational killability is already thick]\label{thm:killable-thick}
1354 If $\mathscr M$ consists of exact endofunctors and has the directed annihilator property, then $\mathcal K_{\mathscr M}$ is a thick stable subcategory. Hence
1355 \[
1356 \mathcal N_{\mathscr M}=\mathcal K_{\mathscr M}
1357 \]
1358 and
1359 \[
1360 \boxed{
1361 \Obs_{\mathscr M}(X)\simeq0
1362 \iff
1363 X\text{ is operationally killable}.}
1364 \]
1365 \end{theorem}
1366 
1367 \begin{proof}
1368 The zero object is killed by every exact constructor. If $U(X)\simeq0$, exactness gives $U(X[n])\simeq U(X)[n]\simeq0$ for every integer $n$, so the class is closed under both suspensions and desuspensions. It is closed under retracts because exact functors preserve retract diagrams.
1369 
1370 For cofibers, suppose $X$ is killed by $U$ and $Y$ is killed by $V$. Choose a common postcomposition
1371 \[
1372 W\simeq P\circ U\simeq Q\circ V.
1373 \]
1374 Then $W(X)\simeq0\simeq W(Y)$. For any morphism $f:X\to Y$, exactness gives
1375 \[
1376 W(\Cofib(f))\simeq\Cofib(Wf)\simeq\Cofib(0\to0)\simeq0.
1377 \]
1378 Thus the killable objects form a stable subcategory closed under retracts, hence a thick subcategory.
1379 \end{proof}
1380 
1381 \begin{example}[Idempotent projection model]
1382 Let $\mathcal Z=\mathcal A\times\mathcal B$ for stable categories $\mathcal A,\mathcal B$, and let
1383 \[
1384 L(a,b)=(a,0).
1385 \]
1386 With constructor monoid generated by the idempotent exact functor $L$, an object is operationally killable exactly when it lies in $0\times\mathcal B$. The obstruction quotient is equivalent to $\mathcal A$, and
1387 \[
1388 \Obs(a,b)\simeq a.
1389 \]
1390 Thus the quotient literally retains the component that cannot be annihilated by the admissible constructor.
1391 \end{example}
1392 
1393 \begin{theorem}[Functorial transport of obstruction]\label{thm:obstruction-transport}
1394 Let $(\mathcal Z_A,\mathscr M_A)$ and $(\mathcal Z_B,\mathscr M_B)$ be stable obstruction systems, and let
1395 \[
1396 \Lambda:\mathcal Z_A\longrightarrow\mathcal Z_B
1397 \]
1398 be exact. If
1399 \[
1400 \Lambda(\mathcal K_{\mathscr M_A})\subseteq\mathcal N_{\mathscr M_B},
1401 \]
1402 then $\Lambda$ induces an exact functor
1403 \[
1404 \bar\Lambda:\mathcal O_{\mathscr M_A}\longrightarrow\mathcal O_{\mathscr M_B}.
1405 \]
1406 If $\Lambda$ is an equivalence and carries $\mathcal N_{\mathscr M_A}$ onto $\mathcal N_{\mathscr M_B}$, then $\bar\Lambda$ is an equivalence. In that case obstruction vanishing is preserved and reflected.
1407 \end{theorem}
1408 
1409 \begin{proof}
1410 The hypothesis and exactness imply
1411 \[
1412 \Lambda(\mathcal N_{\mathscr M_A})\subseteq\mathcal N_{\mathscr M_B},
1413 \]
1414 because exact functors preserve the operations used to generate a thick subcategory. Therefore $q_B\Lambda$ kills $\mathcal N_{\mathscr M_A}$, so the universal property in \cref{thm:obstruction-quotient} gives $\bar\Lambda$. If $\Lambda$ is an equivalence identifying the two thick kernels, a quasi-inverse descends in the same way and yields a quasi-inverse to $\bar\Lambda$.
1415 \end{proof}
1416 
1417 This theorem gives a precise version of the intended local-global principle: a correspondence need not merely preserve local invariants; it can transport the quotient in which removable residual structure has been annihilated.
1418 
1419 \begin{theorem}[Obstruction localization along normalization]\label{thm:obstruction-localization}
1420 Let
1421 \[
1422 X_0\xrightarrow{f_1}X_1\xrightarrow{}\cdots\xrightarrow{f_m}X_m
1423 \]
1424 be a normalization path as in \cref{thm:stable-filtration}, with elementary transported residuals $D_i=\Cofib(f_i)$ and total residual
1425 \[
1426 D_{\mathrm{tot}}=\Cofib(X_0\to X_m).
1427 \]
1428 Let
1429 \[
1430 q:\mathcal Z\to\mathcal O_{\mathscr M}
1431 \]
1432 be the stable obstruction quotient. Then:
1433 \begin{enumerate}
1434     \item $q(D_{\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;
1435     \item if $q(D_i)\simeq0$ for every $i$, then
1436     \[
1437     q(D_{\mathrm{tot}})\simeq0;
1438     \]
1439     \item equivalently,
1440     \[
1441     q(D_{\mathrm{tot}})\not\simeq0
1442     \Longrightarrow
1443     \exists i,\;q(D_i)\not\simeq0.
1444     \]
1445 \end{enumerate}
1446 If the directed annihilator property of \cref{thm:killable-thick} holds, the phrase ``$q(D)\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''
1447 \end{theorem}
1448 
1449 \begin{proof}
1450 The residual filtration in \cref{thm:stable-filtration} consists of cofiber sequences
1451 \[
1452 F_{i-1}\longrightarrow F_i\longrightarrow D_i,
1453 \qquad
1454 F_0\simeq0,
1455 \qquad
1456 F_m\simeq D_{\mathrm{tot}}.
1457 \]
1458 Because the Verdier quotient functor $q$ is exact, applying $q$ gives cofiber sequences
1459 \[
1460 q(F_{i-1})\longrightarrow q(F_i)\longrightarrow q(D_i).
1461 \]
1462 This proves (1). If every $q(D_i)$ vanishes, then $q(F_{i-1})\to q(F_i)$ is an equivalence for every $i$. Starting from $q(F_0)\simeq0$ yields $q(F_m)\simeq0$, proving (2). Statement (3) is its contrapositive. Under the directed annihilator hypothesis, \cref{thm:killable-thick} identifies quotient-vanishing with operational killability.
1463 \end{proof}
1464 
1465 \status{proved from the stable residual filtration and exactness of the obstruction quotient}
1466 
1467 This result gives the calculus a precise diagnostic property: a persistent obstruction in a composite transformation-resolution word cannot appear from nowhere. Relative to a chosen normalization path, it must be visible in at least one elementary exchange residual after passage to the obstruction quotient.
1468 
1469 \subsection{When the obstruction quotient must retain the constructors}
1470 
1471 The universal property of $\mathcal Z/\mathcal N_{\mathscr M}$ concerns exact functors that kill a prescribed class. It does not automatically preserve the full transformation grammar. To make the quotient a new environment for that grammar, the kernel must be invariant under the constructors.
1472 
1473 \begin{example}[A constructor can fail to descend, and invariant closure can collapse]\label{ex:quotient-collapse}
1474 Return to \cref{ex:killable-not-thick}, with $\mathcal Z=\mathcal C^3$ and $\mathscr M=\{\id,U,V\}$. An object is killed by $U$ precisely when its second and third components vanish, and killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently
1475 \[
1476 \mathcal N_{\mathscr M}=\mathcal C\times\mathcal C\times0,
1477 \qquad
1478 \mathcal Z/\mathcal N_{\mathscr M}\simeq\mathcal C.
1479 \]
1480 Nevertheless
1481 \[
1482 U(0,A,0)=(0,0,A)
1483 \]
1484 for every $A\in\mathcal C$. The input lies in $\mathcal N_{\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\bar U$ on the quotient, then
1485 $qU(0,A,0)\simeq\bar Uq(0,A,0)\simeq0$, a contradiction for $A\not\simeq0$.
1486 
1487 Moreover, any constructor-invariant thick subcategory containing all killable objects contains both the first and second coordinate subcategories. Applying $U$ to the second coordinate forces it to contain the third. Closure under direct sums then gives all of $\mathcal Z$. Thus the constructor-compatible quotient in this example is the zero category.
1488 \end{example}
1489 
1490 \begin{definition}[Constructor-invariant thick saturation]\label{def:invariant-saturation}
1491 For a small stable constructor system $(\mathcal Z,\mathscr M)$ as above, put
1492 \[
1493 \mathcal J_{\mathscr M}
1494 :=\operatorname{Thick}\{W(K):W\in\mathscr M,
1495 \ K\in\mathcal K_{\mathscr M}\}.
1496 \]
1497 Its quotient is denoted
1498 \[
1499 q^{\mathrm{inv}}:\mathcal Z\longrightarrow
1500 \mathcal O^{\mathrm{inv}}_{\mathscr M}
1501 :=\mathcal Z/\mathcal J_{\mathscr M}.
1502 \]
1503 The identity constructor ensures $\mathcal N_{\mathscr M}\subseteq\mathcal J_{\mathscr M}$.
1504 \end{definition}
1505 
1506 \begin{theorem}[Universal constructor-compatible obstruction quotient]\label{thm:invariant-quotient}
1507 The subcategory $\mathcal J_{\mathscr M}$ is the least thick subcategory that contains $\mathcal K_{\mathscr M}$ and is preserved by every constructor. Each $T\in\mathscr M$ descends to an exact endofunctor of $\mathcal O^{\mathrm{inv}}_{\mathscr M}$. If the constructor action is supplied with coherent unit and composition data, those data descend as well.
1508 
1509 The quotient is universal among exact functors that annihilate every object $W(K)$ in the displayed generating family. In particular, every exact realization that annihilates killable objects and intertwines the given constructors with exact constructors of its target factors through $q^{\mathrm{inv}}$.
1510 \end{theorem}
1511 
1512 \begin{proof}
1513 Fix $T\in\mathscr M$. It takes a generator $W(K)$ to $(T\circ W)(K)$, again a generator because $\mathscr M$ is closed under composition. Exactness and preservation of retracts imply
1514 \[
1515 T(\mathcal J_{\mathscr M})\subseteq\mathcal J_{\mathscr M}.
1516 \]
1517 Conversely any invariant thick subcategory containing $\mathcal K_{\mathscr M}$ contains every $W(K)$ and therefore their thick closure. This proves minimality.
1518 
1519 Now $q^{\mathrm{inv}}T$ annihilates $\mathcal J_{\mathscr M}$, so the quotient universal property gives $\bar T$ with
1520 \[
1521 \bar T q^{\mathrm{inv}}\simeq q^{\mathrm{inv}}T.
1522 \]
1523 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.
1524 
1525 The last universal property follows directly from thick closure and Verdier localization. If $E$ is equivariant and kills $K$, then
1526 $E(WK)\simeq W'E(K)\simeq W'(0)\simeq0$. Thus it kills all the generators and factors through the quotient.
1527 \end{proof}
1528 
1529 \status{proved for small stable exact constructor systems}
1530 
1531 There is no general nontriviality conclusion. The zero category is a legitimate universal quotient. A useful application must show that the chosen admissible repairs do not annihilate everything after invariant thick closure.
1532 
1533 \begin{theorem}[Directed annihilators also give constructor invariance]\label{thm:directed-invariant}
1534 Under the directed annihilator hypothesis of \cref{thm:killable-thick},
1535 \[
1536 \mathcal J_{\mathscr M}=\mathcal N_{\mathscr M}
1537 =\mathcal K_{\mathscr M}.
1538 \]
1539 In particular the same quotient both supports the admitted constructors and detects operational killability exactly.
1540 \end{theorem}
1541 
1542 \begin{proof}
1543 The earlier theorem shows that $\mathcal K_{\mathscr M}$ is thick. It remains to prove invariance. Suppose $U(X)\simeq0$ and fix a constructor $T$. Directedness supplies constructors $P,Q,W$ with
1544 \[
1545 W\simeq P\circ U\simeq Q\circ T.
1546 \]
1547 Therefore
1548 \[
1549 Q(TX)\simeq P(UX)\simeq P(0)\simeq0.
1550 \]
1551 Thus $TX$ is killable. The killable subcategory is invariant and thick, so minimality of $\mathcal J_{\mathscr M}$ gives the asserted equalities.
1552 \end{proof}
1553 
1554 \begin{proposition}[A noncollapse certificate]\label{prop:noncollapse}
1555 Suppose $E:\mathcal Z\to\mathcal D$ is an exact equivariant realization into a stable constructor system, $E$ kills every operationally killable object, and $E(X)\not\simeq0$ for some $X$. Then
1556 \[
1557 q^{\mathrm{inv}}(X)\not\simeq0,
1558 \qquad
1559 \mathcal O^{\mathrm{inv}}_{\mathscr M}\not\simeq0.
1560 \]
1561 \end{proposition}
1562 
1563 \begin{proof}
1564 By \cref{thm:invariant-quotient}, write $E\simeq\bar E q^{\mathrm{inv}}$. Exactness of $\bar E$ would imply $E(X)\simeq0$ if $q^{\mathrm{inv}}(X)\simeq0$, a contradiction.
1565 \end{proof}
1566 
1567 For the projection example $\mathcal Z=\mathcal A\times\mathcal B$ and $L(a,b)=(a,0)$, projection to $\mathcal A$ is such a detector whenever $\mathcal A$ is nonzero. This gives a nontrivial model, in contrast with \cref{ex:quotient-collapse}. Admissibility matters: allowing the zero functor as a repair makes every object killable and necessarily collapses the quotient.
1568 
1569 The refined distinction is therefore
1570 \[
1571 \begin{aligned}
1572 \mathcal K_{\mathscr M}&:\ \text{objects killed by a single admitted word},\\
1573 \mathcal N_{\mathscr M}&:\ \text{their thick closure},\\
1574 \mathcal J_{\mathscr M}&:\ \text{their constructor-invariant thick closure}.
1575 \end{aligned}
1576 \]
1577 Only under additional hypotheses can these three classes be identified.
1578 
1579 
1580 \begin{problem}[Nonstable intrinsic obstruction]
1581 Construct an analogue of \cref{def:obstruction-quotient} before stable/additive completion, or prove that the stable quotient is the correct universal recipient for the applications of interest.
1582 \end{problem}
1583 
1584 \section{Universal excisive localization}\label{sec:excisive-localization}
1585 
1586 The constructor-invariant quotient can now be attached to a precise target property: making the event diagram excisive. This is different from declaring every residual removable.
1587 
1588 Let $\mathcal Z$ 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 $\mathcal Z$. Denote by $\mathscr H$ the family of interaction residuals of all their elementary independent-event squares. Put
1589 \[
1590  \mathcal J_{\square}
1591  =\operatorname{Thick}\{T(H):T\in\mathscr M,\ H\in\mathscr H\}.
1592 \]
1593 This notation does not assert that $H$ was operationally killable before localization.
1594 
1595 \begin{theorem}[Universal constructor-compatible excisive quotient]\label{thm:excisive-quotient}
1596 The stable Verdier quotient
1597 \[
1598  q_{\square}:\mathcal Z\longrightarrow\mathcal Q_{\square}
1599  :=\mathcal Z/\mathcal J_{\square}
1600 \]
1601 has the following properties.
1602 \begin{enumerate}
1603 \item The constructors descend, with their supplied coherent action data.
1604 \item Every diagram $q_{\square}X$, for $X\in\mathscr X$, has context-independent event residuals and satisfies principal-ideal reconstruction.
1605 \item Every exact equivariant functor $E:\mathcal Z\to\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\square}$, with the enhanced universal uniqueness of Verdier localization.
1606 \item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.
1607 \end{enumerate}
1608 The quotient is not automatically nonzero. If such an $E$ has $E(Y)\not\simeq0$, then $q_{\square}(Y)\not\simeq0$.
1609 \end{theorem}
1610 
1611 \begin{proof}
1612 The proof of \cref{thm:invariant-quotient} applies to any designated family of objects, not only killable objects. A constructor sends a generator $T(H)$ to another generator. Exactness then preserves its thick closure. The Verdier universal property therefore descends constructors and their already specified coherence.
1613 
1614 Since $q_{\square}(H)\simeq0$ and exact functors preserve interaction residuals, every prescribed elementary square in $q_{\square}X$ is a pushout. Apply \cref{thm:event-context-invariance,thm:principal-reconstruction} to obtain (2).
1615 
1616 If $EX$ is excisive, then $E(H)\simeq0$. Equivariance implies $E(T(H))\simeq T'(E(H))\simeq0$. Exactness annihilates the thick closure, so $E$ factors through the quotient. Conversely, the kernel of any exact equivariant localization making the diagrams excisive must contain $\mathscr H$ and its invariant thick closure. This proves the universal and minimality assertions. Finally a factorization $E\simeq\bar E q_{\square}$ would send $Y$ to zero if $q_{\square}(Y)$ were zero.
1617 \end{proof}
1618 
1619 The external input is stable Verdier localization, as in \cite{Drew2015,StacksVerdier}. The new use here is the explicitly chosen family of square interaction residuals. It does not supply an arithmetic interpretation for a quotient unless that interpretation is proved separately.
1620 
1621 \section{An integral seed model and prime-overlap calculus}\label{sec:integral-model}
1622 
1623 We now give a completely specified arithmetic instance. It uses perfect complexes over $\mathbb Z$, not Galois representations or automorphic representations. Its purpose is to make the coherence, residual, excision, noncollapse, and finite-generator obligations explicit in one model.
1624 
1625 \subsection{Tensor cubes and iterated residuals}
1626 
1627 \begin{theorem}[Tensor-cube residual formula]\label{thm:tensor-cube}
1628 Let $\mathcal S$ be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For $m\ge1$ and arrows $f_i:A_i\to B_i$, form the coherent $m$-cube
1629 \[
1630  Y(S)=\bigotimes_{i=1}^m Z_i(S),\qquad
1631  Z_i(S)=\begin{cases}B_i&i\in S,\\A_i&i\notin S.\end{cases}
1632 \]
1633 Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies
1634 \[
1635  \operatorname{tcofib}(Y)\simeq
1636  \bigotimes_{i=1}^m\Cofib(f_i).
1637 \]
1638 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.
1639 \end{theorem}
1640 
1641 \begin{proof}
1642 For $m=1$ this is the definition. Taking cofibers in the last coordinate replaces its factor $A_m\to B_m$ 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.
1643 \end{proof}
1644 
1645 For a commutative ring $R$ and $a\in R$, set
1646 \[
1647  C_R(a)=\bigl[R\xrightarrow{a}R\bigr],
1648 \]
1649 with the source in cohomological degree $-1$ and the target in degree $0$. The tensor product $\bigotimes_i C_R(a_i)$ is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see \cite{StacksKoszul}. We use them with an explicit sign convention and direct proofs below.
1650 
1651 \subsection{Scalar exchange diagrams}
1652 
1653 Fix a finite poset $P$ and assign a positive integer $n_e$ to every event $e\in P$. For $M\in\operatorname{Perf}(\mathbb Z)$ define
1654 \[
1655  X_M(I)=M,\qquad
1656  X_M(I\subseteq J)=\left(\prod_{e\in J\setminus I}n_e\right)\id_M.
1657 \]
1658 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 \cref{thm:strict-event-realization}. The object at every vertex is the same, but its comparison maps retain nontrivial arithmetic.
1659 
1660 For an enabled event $e$,
1661 \[
1662  E_e(I)\simeq M\otimes^{\mathbf L}_{\mathbb Z}C_{\mathbb Z}(n_e).
1663 \]
1664 The comparison after an unrelated event $f$ is multiplication by $n_f$ on that complex. Consequently
1665 \[
1666  \boxed{\Theta_{e,f}(I)\simeq
1667  M\otimes^{\mathbf L}_{\mathbb Z}
1668  C_{\mathbb Z}(n_e)\otimes^{\mathbf L}_{\mathbb Z}C_{\mathbb Z}(n_f).}
1669 \]
1670 The complexes $C_{\mathbb Z}(n)$ are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead
1671 \[
1672  \mathcal F_{X_M}(P)\simeq
1673  M\otimes^{\mathbf L}_{\mathbb Z}C_{\mathbb Z}(N),
1674  \qquad N=\prod_{e\in P}n_e.
1675 \]
1676 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 higher interaction. These are different constructions.
1677 
1678 \begin{theorem}[Integral interaction is supported at the gcd]\label{thm:integral-gcd}
1679 For positive integers $a_1,\ldots,a_m$, $m\ge1$, put $g=\gcd(a_1,\ldots,a_m)$. Then
1680 \[
1681  K(a_1,\ldots,a_m):=
1682  \bigotimes_{i=1}^m C_{\mathbb Z}(a_i)
1683  \simeq\bigoplus_{j=0}^{m-1}
1684  C_{\mathbb Z}(g)[j]^{\oplus\binom{m-1}{j}}.
1685 \]
1686 The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore
1687 \[
1688  H^{-j}K(a_1,\ldots,a_m)\cong
1689  (\mathbb Z/g)^{\oplus\binom{m-1}{j}}
1690  \quad(0\le j\le m-1),
1691 \]
1692 and every other cohomology group vanishes. In particular the interaction vanishes exactly when $g=1$.
1693 \end{theorem}
1694 
1695 \begin{proof}
1696 The Euclidean algorithm produces a unimodular matrix $U\in\operatorname{GL}_m(\mathbb Z)$ with
1697 \[
1698  (a_1,\ldots,a_m)U=(g,0,\ldots,0).
1699 \]
1700 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 $K(g,0,\ldots,0)$ to $K(a_1,\ldots,a_m)$.
1701 
1702 Each zero entry contributes $C_{\mathbb Z}(0)\cong\mathbb Z\oplus\mathbb Z[1]$ with zero differential. Distributing these $m-1$ factors gives the stated shifted copies, with the standard shift isomorphisms. Since $g>0$, the map $g:\mathbb Z\to\mathbb Z$ is injective. Thus $C_{\mathbb Z}(g)$ has only $H^0=\mathbb Z/g$, proving the cohomology formula and the vanishing criterion.
1703 \end{proof}
1704 
1705 For two events the entire calculation is visible:
1706 \[
1707  K(a,b)=\left[
1708  \mathbb Z\xrightarrow{\binom{-b}{a}}
1709  \mathbb Z^2\xrightarrow{(a\ \ b)}\mathbb Z
1710  \right],
1711  \qquad\text{degrees }-2,-1,0.
1712 \]
1713 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 $\mathbb Z/g$. Equivalently, if $au+bv=g$, then
1714 \[
1715  U=\begin{pmatrix}u&-b/g\\v&a/g\end{pmatrix},
1716  \qquad\det U=1,
1717  \qquad(a\ b)U=(g\ 0).
1718 \]
1719 
1720 Thus a strictly commuting scalar exchange square is a homotopy pushout on the seed $\mathbb Z$ exactly when the two exchange integers are coprime. Its commutativity alone says nothing about this condition.
1721 
1722 \begin{example}[Higher vanishing does not remove lower interactions]\label{ex:higher-not-lower}
1723 For $(6,10,15)$, all pair interactions are nonzero: their gcds are $2,3,5$. But the threefold interaction is zero because the joint gcd is $1$. Hence vanishing of a higher total cofiber does not imply vanishing of the lower square interactions. It cannot replace the square-by-square excision condition in \cref{thm:event-context-invariance}.
1724 \end{example}
1725 
1726 \subsection{A nontrivial universal quotient with an exact detector}
1727 
1728 Let $\mathscr M$ consist of tensoring by perfect complexes, closed under composition with its coherent tensor action. This is a constructor system, not the repair monoid defining operational killability in the earlier sections. We now kill only the designated interaction residuals.
1729 
1730 For the fixed labelled poset define the finite prime set
1731 \[
1732  S_P=\bigcup_{e\parallel f}
1733  \{p\text{ prime}:p\mid\gcd(n_e,n_f)\},
1734 \]
1735 where $e\parallel f$ means incomparable. Let $\mathcal J_P$ be the invariant thick closure of the square interaction residuals for the unit seed $\mathbb Z$.
1736 
1737 \begin{theorem}[Prime-overlap classification of the excisive kernel]\label{thm:prime-overlap-quotient}
1738 In $\operatorname{Perf}(\mathbb Z)$,
1739 \[
1740  \boxed{\mathcal J_P=
1741  \operatorname{Thick}\{\mathbb Z/p:p\in S_P\}.}
1742 \]
1743 Equivalently, $\mathcal J_P$ 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 $M$,
1744 \[
1745  q_P(M)\simeq0
1746  \quad\Longleftrightarrow\quad
1747  M\otimes^{\mathbf L}_{\mathbb Z}\mathbb Z[S_P^{-1}]\simeq0.
1748 \]
1749 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.
1750 \end{theorem}
1751 
1752 \begin{proof}
1753 First, every pair of incomparable events occurs in an elementary square. Take the ideal $D_e\cup D_f$. It contains neither $e$ nor $f$, contains their predecessors, and enables both. The square residual at the unit is $K(n_e,n_f)$.
1754 
1755 By \cref{thm:integral-gcd}, this complex has the same thick closure as $C_{\mathbb Z}(g)$ for $g=\gcd(n_e,n_f)$: it is a direct sum of that complex and its shift, and contains it as a retract. The thick closure of any family in $\operatorname{Perf}(\mathbb Z)$ is already a tensor ideal. Indeed $\operatorname{Perf}(\mathbb Z)=\operatorname{Thick}(\mathbb Z)$, so tensoring a fixed generator by a perfect complex builds its image using finite cones, shifts, and retracts from that same generator.
1756 
1757 For $p\mid g$, tensor $C_{\mathbb Z}(g)$ by $C_{\mathbb Z}(p)$. The gcd theorem identifies the result with $C_{\mathbb Z}(p)\oplus C_{\mathbb Z}(p)[1]$. Hence $\mathbb Z/p\simeq C_{\mathbb Z}(p)$ belongs to the thick closure. Conversely, the Chinese remainder decomposition of $\mathbb Z/g$ and the short exact sequences building $\mathbb Z/p^j$ from copies of $\mathbb Z/p$ put $C_{\mathbb Z}(g)\simeq\mathbb Z/g$ in the thick subcategory generated by its prime divisors. This proves the boxed equality.
1758 
1759 Having finite $S_P$-primary cohomology is preserved by finite triangles and retracts, by the long exact sequence in cohomology. Conversely, each such finite abelian cohomology group is built by finite extensions from the indicated $\mathbb Z/p$. A perfect complex is bounded, and its finite truncation triangles then build it from these groups. These truncations are perfect over $\mathbb Z$: every finitely generated abelian group has a finite free resolution of length at most one. This proves the cohomological characterization.
1760 
1761 Localization of abelian groups is exact. Thus tensoring with $\mathbb Z[S_P^{-1}]$ has cohomology obtained by localizing each cohomology group. A finitely generated abelian group localizes to zero exactly when it is finite with all prime divisors in $S_P$. The localization kernel is therefore $\mathcal J_P$, as is the kernel of the Verdier quotient by this thick subcategory. This proves the equivalence of vanishing tests on every source object.
1762 
1763 Finally $\mathbb Z[S_P^{-1}]$ is a nonzero ring, so localization detects that $q_P(\mathbb Z)\not\simeq0$. It is exact and intertwines tensor constructors with their localized counterparts. The universal excisive-quotient theorem supplies the remaining assertions. For general $M$, each square residual is its unit-seed counterpart tensored with $M$, so it lies in the same invariant kernel.
1764 \end{proof}
1765 
1766 The positivity hypothesis excludes a real collapse case. If two independent labels are both zero, then $K(0,0)\simeq\mathbb Z\oplus\mathbb Z[1]^{\oplus2}\oplus\mathbb Z[2]$ contains the unit as a retract. Killing that interaction forces the whole perfect category to vanish. Thus noncollapse is proved for the specified positive-label model, not for arbitrary scalar labels.
1767 
1768 No equivalence of the entire quotient category with a category of localized perfect complexes is needed for this theorem. The proved statement is equality of kernels, a nonzero exact detector, and the universal excisive property. This avoids inserting an unproved full-faithfulness or idempotent-completion claim.
1769 
1770 \begin{example}[Remove shared torsion without deleting the primary residuals]\label{ex:six-ten}
1771 Take two independent events labelled $6$ and $10$. Their interaction is
1772 \[
1773  K(6,10)\simeq\mathbb Z/2\oplus(\mathbb Z/2)[1].
1774 \]
1775 Here $S_P=\{2\}$. Inverting $2$ kills this interaction while retaining
1776 \[
1777  C_{\mathbb Z}(6)\otimes\mathbb Z[1/2]\simeq\mathbb Z/3,
1778  \qquad
1779  C_{\mathbb Z}(10)\otimes\mathbb Z[1/2]\simeq\mathbb Z/5.
1780 \]
1781 The total normalization residual is $C_{\mathbb Z}(60)$. After localization it is $\mathbb Z/15$, and principal-ideal reconstruction gives
1782 \[
1783  \mathbb Z/15\simeq\mathbb Z/3\oplus\mathbb Z/5.
1784 \]
1785 This is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\mathbb Z/6$ and $\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.
1786 \end{example}
1787 
1788 \subsection{A finite certificate for an infinite scalar grammar}
1789 
1790 \begin{corollary}[Finite-signature arithmetic certification]\label{cor:finite-signature}
1791 Fix finitely many positive integer exchange labels $n_1,\ldots,n_t$ and a specified symmetric relation $\mathcal I$ of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put
1792 \[
1793  S_{\mathcal I}=\bigcup_{(i,j)\in\mathcal I}
1794  \{p:p\mid\gcd(n_i,n_j)\}.
1795 \]
1796 Killing the finitely many unit-seed interactions $K(n_i,n_j)$ for $(i,j)\in\mathcal I$, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive, for every perfect input $M$. The quotient is nonzero and its kernel is the finite-prime torsion class supported on $S_{\mathcal I}$.
1797 \end{corollary}
1798 
1799 \begin{proof}
1800 There are only finitely many prescribed pair types. Every elementary square in every admitted diagram has one of these types, and its residual is the corresponding unit-seed Koszul complex tensored with $M$. All such residuals lie in the same invariant thick kernel. Apply \cref{thm:excisive-quotient,thm:prime-overlap-quotient}. The latter proof depends only on the finite family of gcds, so it applies unchanged to the signature-wide family.
1801 \end{proof}
1802 
1803 This is a genuine finite reduction of an unbounded word-length obligation, but only in the scalar integral regime with the stated independence relation. If a label can occur independently of itself, its primes belong to $S_{\mathcal I}$ and its primary torsion is erased. Declaring independence therefore changes the quotient's information content. Dependent events can share a prime without forcing its removal, as the $p$-then-$p$ chain in \cref{cor:independent-splitting} demonstrates.
1804 
1805 The construction resolves a concrete semantic and obstruction problem in arithmetic. It does not identify these scalar kernels with automorphic occupancy, a ramification obstruction, or a global Langlands parameter.
1806 
1807 \subsection{Beyond scalar exchanges: an integral matrix criterion}
1808 
1809 The noncollapse test extends to tensor cubes of finite integer matrices. This is a different concrete realization from identity functors with scalar exchange cells: its independent directions are tensor factors of actual two-term complexes.
1810 
1811 \begin{theorem}[Finite integral matrix collapse criterion]\label{thm:matrix-collapse}
1812 Let $f_i:\mathbb Z^{a_i}\to\mathbb Z^{b_i}$ 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 $d_{i,1},\ldots,d_{i,r_i}$. Put
1813 \[
1814  \epsilon_i=\begin{cases}
1815  1&a_i+b_i-2r_i>0,\\
1816  0&a_i+b_i-2r_i=0,
1817  \end{cases}
1818  \qquad
1819  P_i=\{p:p\mid d_{i,k}\text{ for some }k\}.
1820 \]
1821 Let $\mathcal J$ be the tensor-invariant thick subcategory generated by $\Cofib(f_i)\otimes^{\mathbf L}\Cofib(f_j)$ over the allowed independent pairs.
1822 \begin{enumerate}
1823 \item If an allowed pair has $\epsilon_i=\epsilon_j=1$, then $\mathcal J=\operatorname{Perf}(\mathbb Z)$ and its quotient is zero.
1824 \item Otherwise the quotient is nonzero and
1825 \[
1826  \mathcal J=\operatorname{Thick}\{\mathbb Z/p:p\in S\},
1827 \]
1828 where $S$ is the union over allowed pairs of
1829 \[
1830  S_{ij}=\begin{cases}
1831  P_i\cap P_j&\epsilon_i=\epsilon_j=0,\\
1832  P_j&\epsilon_i=1,\ \epsilon_j=0,\\
1833  P_i&\epsilon_i=0,\ \epsilon_j=1.
1834  \end{cases}
1835 \]
1836 Localization at $S$ is an exact nonzero detector with that kernel. It makes the prescribed tensor-cube squares excisive.
1837 \end{enumerate}
1838 A supplied Smith form is a finite certificate: its row and column changes must be unimodular and their matrix product must give the claimed diagonal form. The existence of Smith forms over $\mathbb Z$ follows from the Euclidean algorithm; the argument below needs only such a verified form.
1839 \end{theorem}
1840 
1841 \begin{proof}
1842 Changing bases in the source and target yields a chain isomorphism
1843 \[
1844  \Cofib(f_i)\simeq
1845  \mathbb Z^{b_i-r_i}\oplus\mathbb Z^{a_i-r_i}[1]
1846  \oplus\bigoplus_{k=1}^{r_i}C_{\mathbb Z}(d_{i,k}).
1847 \]
1848 Unit diagonal entries contribute contractible complexes. Thus $\epsilon_i=1$ 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 $\operatorname{Perf}(\mathbb Z)$.
1849 
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 \cref{thm:integral-gcd}. Taking finite direct sums gives exactly the listed sets $S_{ij}$. The argument of \cref{thm:prime-overlap-quotient} identifies the generated thick subcategory with finite $S$-primary cohomology and identifies its kernel with localization at $S$. The localized unit is nonzero. The tensor-cube formula and the excisive-quotient theorem finish the proof.
1851 \end{proof}
1852 
1853 This criterion shows why noncollapse cannot be inferred from commutativity. Two commuting tensor directions with free cofiber components can force total collapse when their interaction is declared zero. Pure torsion interactions, in contrast, admit the precise finite-prime localization just proved.
1854 
1855 
1856 \section{Arithmetic resolution towers and nilpotent interactions}\label{sec:arithmetic-towers}
1857 
1858 The integral model of \cref{sec:integral-model} has nontrivial exchange maps but identity coarsening functors. We now replace those coarsenings by derived reduction between different coefficient rings. Three questions must be kept separate: whether an exchange is canonically exact, what a specified noninvertible exchange cell leaves behind, and whether forcing that residual to vanish destroys the coefficient category.
1859 
1860 \subsection{Derived reduction is a genuine resolution operation}
1861 
1862 Fix a finite nonempty set $\mathcal P$ of primes and put
1863 \[
1864  D=\prod_{p\in\mathcal P}p,\qquad B_n=\mathbb Z/D^n\mathbb Z\quad(n\geq1).
1865 \]
1866 Write $B_{p,n}=\mathbb Z/p^n\mathbb Z$. The Chinese remainder isomorphisms
1867 \[
1868  B_n\cong\prod_{p\in\mathcal P}B_{p,n}
1869 \]
1870 are compatible with reduction. Use the enhanced category $\mathcal S_n=\operatorname{Perf}(B_n)$ and the coarsening
1871 \[
1872  C_{m,n}(M)=M\otimes^{\mathbf L}_{B_m}B_n,\qquad m\geq n.
1873 \]
1874 Derived tensor associativity supplies coherent equivalences $C_{l,n}\simeq C_{m,n}C_{l,m}$. 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 reduction. Preservation of perfect complexes under derived base change is standard \cite{StacksPerfect}; the pointwise cone model also shows exactness directly.
1875 
1876 \begin{theorem}[Nonidentity conservative resolution tower]\label{thm:adic-tower}
1877 The functors $C_{m,n}$ are exact and conservative on perfect complexes. They need not be faithful. In particular, for $m>n$, the nonzero endomorphism $D^n:B_m\to B_m$ becomes zero under $C_{m,n}$.
1878 \end{theorem}
1879 \begin{proof}
1880 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 $R=\mathbb Z/p^m$ is zero if its derived reduction to $\mathbb F_p$ is zero.
1881 
1882 A finite projective module over $R$ is free. Indeed, lift a basis modulo $(p)$ to obtain a surjection $R^r\to P$. 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.
1883 
1884 If a differential has a unit entry, elementary changes of bases split off the contractible two-term complex $R\xrightarrow{1}R$. The adjacent differentials have zero components into and out of that summand because their composites are zero. Repeating strictly decreases the total rank and terminates in a complex all of whose differential entries lie in $(p)$. Its reduction has zero differentials. If that reduction is acyclic, every remaining rank is zero. The original complex is contractible. Applied to a cone, this also proves that reduction reflects equivalences.
1885 
1886 Finally $D^n$ is not zero modulo $D^m$, but is zero modulo $D^n$. Since $\operatorname{Hom}_{D(B_m)}(B_m,B_m)=B_m$, this is genuine loss of morphism information, not just a change of representatives.
1887 \end{proof}
1888 
1889 Conservativity detects zero objects and equivalences. It does not recover every morphism, a preferred lift, or the full information in a fine presentation.
1890 
1891 \begin{example}[Why ordinary tensor and cohomology truncation are not substitutes]\label{ex:derived-not-naive}
1892 Over $R=\mathbb Z/p^2$, the exact sequence $0\to(p)\to R\to\mathbb F_p\to0$ ceases to be left exact after ordinary tensor with $\mathbb F_p$: the first map becomes zero. The module $\mathbb F_p$ has the infinite periodic free resolution with every differential multiplication by $p$. Tensoring that resolution with $\mathbb F_p$ gives zero differentials, hence $\operatorname{Tor}_i^R(\mathbb F_p,\mathbb F_p)=\mathbb F_p$ for every $i\geq0$.
1893 
1894 In contrast, $C_R(p)=[R\xrightarrow p R]$ is perfect, but has both $H^{-1}=\mathbb F_p$ and $H^0=\mathbb F_p$. It is not the module $\mathbb F_p$ placed in degree zero. We retain the actual two-term complex throughout the tower. Derived tensor is computed on these projective representatives, not on a truncated cohomology surrogate \cite{StacksDerivedTensor}.
1895 \end{example}
1896 
1897 \subsection{Coherent cells and their actual source of residuals}
1898 
1899 For a fixed perfect integral complex $L$, put
1900 \[
1901  A_{L,n}(M)=(L\otimes^{\mathbf L}_{\mathbb Z}B_n)\otimes^{\mathbf L}_{B_n}M.
1902 \]
1903 Base change gives a natural equivalence
1904 \[
1905  \theta_{L;m,n}:A_{L,n}C_{m,n}\simeq C_{m,n}A_{L,m}.
1906 \]
1907 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
1908 \[
1909  \eta_{L,b;m,n}=b^{m-n}\theta_{L;m,n}.
1910 \]
1911 For scalar unit templates take $L=\mathbb Z$, so that the transformation functors are identities but the coarsening functors are not.
1912 
1913 \begin{theorem}[Coherent weighted exchange and residual transport]\label{thm:weighted-adic-exchange}
1914 The cells $\eta_{L,b;m,n}$ are natural and coherent with composition of resolution changes. Their residual at $M\in\mathcal S_m$ is
1915 \[
1916  \Cofib(\eta_{L,b;m,n,M})\simeq
1917  A_{L,n}C_{m,n}(M)\otimes^{\mathbf L}_{B_n}C_{B_n}(b^{m-n}).
1918 \]
1919 For a fixed scalar $a$, subsequent base change transports its cone exactly:
1920 \[
1921  C_{n,k}\bigl(N\otimes^{\mathbf L}_{B_n}C_{B_n}(a)\bigr)
1922  \simeq C_{n,k}(N)\otimes^{\mathbf L}_{B_k}C_{B_k}(a).
1923 \]
1924 At a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction
1925 \[
1926  N\otimes^{\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\qquad
1927  K_R(u,v)=C_R(u)\otimes^{\mathbf L}_R C_R(v).
1928 \]
1929 These formulas also hold with compatible tensor templates absorbed into $N$.
1930 \end{theorem}
1931 \begin{proof}
1932 Naturality follows from central scalar multiplication. For $l\geq m\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\theta$ with $\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.
1933 
1934 After using $\theta$ to identify the target, the exchange is multiplication by $b^{m-n}$ on $A_{L,n}C_{m,n}M$. Exact tensor identifies its cone with the first formula. Associativity of derived tensor proves the second. The two-variable interaction formula follows by taking the two cofibers, as in \cref{thm:tensor-cube}. For a pure adjacent-swap word, tensor templates keep their relative order and all coarsenings combine into the same final reduction. Identifying those endpoints by the supplied tensor associativities leaves precisely the products of the scalar cells. This constructs the corresponding coherent scalar event diagram at the target level.
1935 \end{proof}
1936 
1937 \begin{remark}[Specified cells, not a claimed arithmetic correspondence]
1938 Nonzero residuals in this model come from the chosen noninvertible scalar cells. They are not defects of the canonical base-change equivalence, and they are not identified with a ramification or automorphy obstruction. The construction supplies a reproducible nonidentity resolution test bed with explicit semantics.
1939 \end{remark}
1940 
1941 \subsection{The finite-level Koszul calculation}
1942 
1943 \begin{theorem}[Prime-power interaction cohomology]\label{thm:prime-power-koszul}
1944 Let $R=\mathbb Z/p^n$, let $a_1,\ldots,a_t$ be positive integers with $t\geq1$, and put
1945 \[
1946  h=\min\{n,v_p(a_1),\ldots,v_p(a_t)\},\qquad Q_h=R/(p^h).
1947 \]
1948 Here $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism
1949 \[
1950  K_R(a_1,\ldots,a_t)\cong
1951  \bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\oplus\binom{t-1}{j}}.
1952 \]
1953 Consequently
1954 \[
1955  \boxed{H^{-j}K_R(a_1,\ldots,a_t)\cong Q_h^{\oplus\binom tj}
1956  \quad(0\leq j\leq t),}
1957 \]
1958 and all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.
1959 \end{theorem}
1960 \begin{proof}
1961 If $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \cref{thm:integral-gcd} and \cite{StacksKoszul}.
1962 
1963 Each zero entry contributes $R\oplus R[1]$. This proves the complex decomposition. Multiplication by $p^h$ on $R$ has cokernel $Q_h$ and kernel $p^{n-h}R$, 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 $H^0=Q_h\ne0$.
1964 \end{proof}
1965 
1966 For two events the cohomology ranks over $Q_h$ are $1,2,1$ in degrees $-2,-1,0$, rather than the $0,1,1$ pattern over the integral or complete domain. These extra classes are real finite-level cohomology. They may not be discarded while retaining the same complex. In particular, a cohomology group appearing here is not automatically a perfect $R$-module.
1967 
1968 \begin{theorem}[Nilpotent interaction forces excisive collapse]\label{thm:nilpotent-collapse}
1969 For any commutative ring $R$ and nilpotent $x\in R$, the thick subcategory generated by $C_R(x)$ is all of $\operatorname{Perf}(R)$. Therefore, over $R=\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.
1970 \end{theorem}
1971 \begin{proof}
1972 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 $\operatorname{Thick}(C_R(x))$. But
1973 \[
1974  C_R(x^N)=C_R(0)\cong R\oplus R[1].
1975 \]
1976 Retract closure therefore includes $R$, and $R$ thickly generates $\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \cite{StacksKoszul}.
1977 
1978 For two nonunit labels, \cref{thm:prime-power-koszul} makes their interaction a sum of $C_R(p^h)$ and its shift, with $1\leq h\leq n$. Thus it contains $C_R(p^h)$ as a retract. This element is nilpotent. Any exact functor annihilating the interaction annihilates the thick subcategory it generates, hence every perfect object.
1979 \end{proof}
1980 
1981 This is stronger than the zero-label counterexample over $\mathbb Z$. At finite $p$-power resolution, even nonzero labels such as $p$ and $p^2$ can force total collapse. Localizing $\mathbb Z/p^n$ at $p$ gives the zero ring. Noncollapse over the integral coefficients does not survive this base change automatically.
1982 
1983 \subsection{Several primes give a nonzero, but selective, quotient}
1984 
1985 Specify finitely many positive scalar labels and a collection $\mathcal I$ 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
1986 \[
1987  S=\{p\in\mathcal P:\text{some }(a,b)\in\mathcal I
1988                     \text{ satisfies }p\mid a\text{ and }p\mid b\},
1989  \qquad D_0=\prod_{p\in\mathcal P\setminus S}p.
1990 \]
1991 Let $\mathcal J_n$ be the tensor-invariant thick closure of those interactions in $\operatorname{Perf}(B_n)$. Positive lengths in \cref{thm:weighted-adic-exchange} do not change $S$.
1992 
1993 \begin{theorem}[Prime-component quotient and tower compatibility]\label{thm:finite-prime-quotient}
1994 Under the Chinese remainder equivalence,
1995 \[
1996  \boxed{\mathcal J_n=
1997  \prod_{p\in S}\operatorname{Perf}(\mathbb Z/p^n)
1998  \times\prod_{p\in\mathcal P\setminus S}0.}
1999 \]
2000 In particular there is an equivalence of stable categories
2001 \[
2002  \operatorname{Perf}(B_n)/\mathcal J_n
2003  \simeq \prod_{p\in\mathcal P\setminus S}\operatorname{Perf}(\mathbb Z/p^n)
2004  \simeq\operatorname{Perf}(\mathbb Z/D_0^n).
2005 \]
2006 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
2007 \[
2008  q_nC_{m,n}\simeq C^0_{m,n}q_m
2009 \]
2010 with the supplied base-change coherence. The quotient is nonzero exactly when $S\ne\mathcal P$. When $S=\mathcal P$, use the zero category and the zero ring $\mathbb Z/1$.
2011 \end{theorem}
2012 \begin{proof}
2013 At a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.
2014 
2015 Projection to the complementary factors is exact and has an exact fully faithful section. Every object is a direct sum of its selected and retained components. In the quotient the selected summand is zero, so that object becomes isomorphic to its retained summand. Projection and section therefore induce inverse equivalences. This proves the whole quotient assertion directly, without an unproved identification of an arbitrary Verdier quotient with localized perfect complexes.
2016 
2017 Tensor products and reduction respect the Chinese remainder idempotents. Projection consequently commutes with reduction, and the same tensor associativities give coherence for composite reductions. All prescribed square interactions vanish after projection, so the earlier excisive reconstruction results apply at every level.
2018 \end{proof}
2019 
2020 This repair does not retain the selected prime's finer torsion while deleting only one common exponent. It removes the entire selected prime component. Thus its minimality is a statement about exact excision, not a guarantee that it preserves all arithmetic information desired by a later application.
2021 
2022 
2023 \section{Completion defects and arithmetic reconstruction}\label{sec:completion-reconstruction}
2024 
2025 \subsection{A controlled reconstruction of a resolution tower}
2026 
2027 Write
2028 \[
2029  \widehat B=\varprojlim_n B_n\cong\prod_{p\in\mathcal P}\mathbb Z_p.
2030 \]
2031 The isomorphism follows by taking inverse limits of the compatible Chinese remainder decompositions. Here $\mathbb Z_p$ is the ring of compatible residue sequences modulo $p^n$, and $\mathbb Q_p=\mathbb Z_p[1/p]$.
2032 
2033 \begin{theorem}[Uniform based-complex tower reconstruction]\label{thm:based-tower-reconstruction}
2034 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 $m\geq n$. Then there is a unique based complex $\widehat M$ with those ranks over $\widehat B$ whose matrices reduce to the supplied ones.
2035 
2036 Compatible chain maps and specified compatible chain homotopies reconstruct coefficientwise. Moreover, as complexes of abelian groups,
2037 \[
2038  \widehat M\simeq R\varprojlim_n M_n.
2039 \]
2040 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.
2041 \end{theorem}
2042 \begin{proof}
2043 Every matrix entry is a compatible residue sequence and therefore determines a unique element of $\widehat B$. The equation $d^{i+1}d^i=0$ holds because it holds modulo every $D^n$ and the inverse-limit ring is separated. The same argument proves the equations for a reconstructed chain map or homotopy.
2044 
2045 Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map
2046 \[
2047  1-\mathrm{shift}:\prod_n M_n^i\longrightarrow\prod_n M_n^i
2048 \]
2049 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 \cite{StacksRlim}.
2050 
2051 All the listed finite constructions have matrices formed by finite sums and products, or by fixed block arrangements of entries. These operations commute with the coefficient inverse limit. This proves the final assertions.
2052 \end{proof}
2053 
2054 This theorem does not assert that arbitrary objects chosen independently at every level have a compatible system of bases, bounded ranks, or a coherent lift. It also does not claim an equivalence between all categories of perfect complexes and an inverse limit of categories. Those stronger effectivity statements require additional work. The precise input here is a compatible, uniformly finite, based tower.
2055 
2056 \begin{proposition}[Transient negative cohomology in the resolution direction]\label{prop:transient-koszul}
2057 Fix $p$ and an integer $a\geq1$. For $n\geq a$ put $R_n=\mathbb Z/p^n$. Then
2058 \[
2059  H^0C_{R_n}(p^a)\cong\mathbb Z/p^a,
2060  \qquad H^{-1}C_{R_n}(p^a)\cong\mathbb Z/p^a.
2061 \]
2062 Under reduction from $m$ to $n$ with $m\geq n\geq a$, 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
2063 \[
2064  C_{\mathbb Z_p}(p^a),\qquad
2065  H^{-1}=0,\quad H^0=\mathbb Z/p^a.
2066 \]
2067 \end{proposition}
2068 \begin{proof}
2069 Represent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\mathbb Z/p^a$ with it by $x\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\mathbb Z_p$ is injective, so its completed kernel vanishes.
2070 \end{proof}
2071 
2072 This calculation concerns the actual transition maps on homology, not merely the group appearing at each level. A list of isomorphism types $(\mathbb Z/p^a,\mathbb Z/p^a,\ldots)$ cannot distinguish the identity system from the pro-zero system. The two-axis data must retain those maps.
2073 
2074 \subsection{Completion and excisive repair do not generally commute}
2075 
2076 Let $S\subseteq\mathcal P$ be the interaction-prime set in \cref{thm:finite-prime-quotient}, and set $T=\prod_{p\in S}p$, with $T=1$ if $S$ is empty. Localizing the finite rings and then completing gives
2077 \[
2078  \varprojlim_n B_n[1/T]\cong\prod_{p\in\mathcal P\setminus S}\mathbb Z_p.
2079 \]
2080 Completing and then localizing instead gives
2081 \[
2082  \widehat B[1/T]\cong
2083  \prod_{p\in S}\mathbb Q_p\times
2084  \prod_{p\in\mathcal P\setminus S}\mathbb Z_p.
2085 \]
2086 The latter is also an exact realization that annihilates the specified scalar interactions, because each selected prime becomes invertible. There is a canonical comparison map between these rings induced by the reductions. Localization is the usual algebraic operation \cite{StacksLocalization}, not an additional choice of quotient semantics.
2087 
2088 \begin{theorem}[Explicit completion--repair defect]\label{thm:completion-repair-defect}
2089 The canonical map
2090 \[
2091  \chi:\widehat B[1/T]\longrightarrow\varprojlim_n B_n[1/T]
2092 \]
2093 is the projection that forgets the factors $\mathbb Q_p$ for $p\in S$. As $\widehat B$-modules it fits into the split exact sequence
2094 \[
2095  0\longrightarrow\prod_{p\in S}\mathbb Q_p
2096  \longrightarrow\widehat B[1/T]
2097  \xrightarrow{\chi}\prod_{p\in\mathcal P\setminus S}\mathbb Z_p
2098  \longrightarrow0.
2099 \]
2100 The splitting is a module splitting, not a claimed unital ring section.
2101 
2102 For every perfect $\widehat B$-complex $M$, the natural comparison
2103 \[
2104  \chi_M:M\otimes^{\mathbf L}_{\widehat B}\widehat B[1/T]
2105  \longrightarrow R\varprojlim_n
2106  \bigl(M\otimes^{\mathbf L}_{\widehat B}B_n[1/T]\bigr)
2107 \]
2108 has fiber
2109 \[
2110  \boxed{\operatorname{Fib}(\chi_M)\simeq
2111  \bigoplus_{p\in S}M\otimes^{\mathbf L}_{\widehat B}\mathbb Q_p.}
2112 \]
2113 Thus $\chi_M$ 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 $\widehat B$.
2114 \end{theorem}
2115 \begin{proof}
2116 In the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\in S$ and does nothing otherwise. In the $p$-primary factor of $\widehat B$, it gives $\mathbb Q_p$ if $p\in S$ and leaves $\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.
2117 
2118 For $M=\widehat B$, the derived inverse limit on the right is the ordinary one, since the retained residue-module tower is degreewise surjective. The asserted comparison and fiber are therefore exactly the split sequence. For a general perfect $M$, all functors in the claimed identity are exact and preserve finite sums and retracts. Homotopy inverse limits are exact in a stable derived category. The natural comparison identifies its right side with the retained summand of $M$: this holds on the unit by the just-computed inverse limit and hence on its thick closure, which is $\operatorname{Perf}(\widehat B)$. Tensoring the split sequence now identifies the fiber with the indicated finite sum. A morphism in a stable category is an equivalence precisely when its fiber is zero.
2119 \end{proof}
2120 
2121 \begin{example}[A nonzero generic fiber is lost at every finite repair]\label{ex:repair-before-completion}
2122 For $\mathcal P=\{p\}$ and two independent labels divisible by $p$, every finite excisive quotient is zero, whereas
2123 \[
2124  \bigl(\varprojlim_n\mathbb Z/p^n\bigr)[1/p]=\mathbb Q_p\ne0.
2125 \]
2126 For a nonzero finite-level quotient, take $D=30$ and labels $6,10$. Only the prime $2$ is shared, so the finite repair is
2127 \[
2128  B_n[1/2]\cong\mathbb Z/15^n.
2129 \]
2130 The two completed outputs are $\mathbb Q_2\times\mathbb Z_3\times\mathbb Z_5$ and $\mathbb Z_3\times\mathbb Z_5$, with fiber $\mathbb Q_2$. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite coarsening.
2131 \end{example}
2132 
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 $\mathbb Q_p$. 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 element is invertible has zero completion \cite{StacksDerivedCompletion}.
2134 
2135 \subsection{Reconstruction that retains the overlap instead of deleting it}
2136 
2137 There is a different operation from killing an interaction: retain the localized and completed descriptions together with their common generic information. In the present arithmetic setting this yields an actual pullback theorem.
2138 
2139 Put
2140 \[
2141  A=\mathbb Z[1/D],\qquad B=\prod_{p\in\mathcal P}\mathbb Z_p,
2142  \qquad G=\prod_{p\in\mathcal P}\mathbb Q_p.
2143 \]
2144 
2145 \begin{theorem}[Arithmetic fracture reconstruction]\label{thm:arithmetic-fracture}
2146 The diagonal maps and their difference give an exact sequence of abelian groups
2147 \[
2148  0\longrightarrow\mathbb Z
2149  \longrightarrow A\oplus B
2150  \xrightarrow{(a,b)\mapsto a-b}G
2151  \longrightarrow0.
2152 \]
2153 Consequently, for every perfect integral complex $M$ there is a natural homotopy pullback square in $D(\mathbb Z)$,
2154 \[
2155  \begin{tikzcd}
2156  M\arrow[r]\arrow[d]&M\otimes^{\mathbf L}_{\mathbb Z}A\arrow[d]\\
2157  M\otimes^{\mathbf L}_{\mathbb Z}B\arrow[r]&M\otimes^{\mathbf L}_{\mathbb Z}G.
2158  \end{tikzcd}
2159 \]
2160 Its bottom-left term is $R\varprojlim_n(M\otimes^{\mathbf L}_{\mathbb Z}B_n)$. In particular $M$ is zero if and only if its localized and completed terms are both zero.
2161 \end{theorem}
2162 \begin{proof}
2163 An element of $A$ integral at every $p\in\mathcal P$ is an integer: its possible denominator primes all lie in $\mathcal P$, and integrality removes each of them. This identifies the kernel with the diagonal copy of $\mathbb Z$.
2164 
2165 For surjectivity take $(y_p)\in G$. Choose $N$ large enough that $D^Ny_p\in\mathbb Z_p$ for every $p$. For each $p$, choose the residue of $D^Ny_p$ modulo $p^N$. The ordinary Chinese remainder theorem provides $z\in\mathbb Z$ with all these residues. Set $a=z/D^N\in A$. Then $a-y_p\in\mathbb Z_p$ for every $p$, so $(a,(a-y_p)_p)$ maps to $(y_p)_p$. This proves exactness.
2166 
2167 The exact sequence gives a cofiber sequence in the derived category. Derived tensor with $M$ is exact, so it gives the displayed homotopy pullback. For a bounded finite free representative of $M$, the tower of reductions has uniform ranks and compatible matrices. The based-tower theorem identifies its derived limit with $M\otimes B$. Finally if the localized and completed terms are zero, so is their generic base change, and the pullback is zero. The converse is immediate.
2168 \end{proof}
2169 
2170 This is the classical arithmetic-square mechanism, proved directly here using the Chinese remainder theorem and derived tensor. It is not claimed as a new local--global correspondence. It does show exactly which overlap data must accompany these arithmetic representations to reconstruct a supplied perfect complex.
2171 
2172 \begin{corollary}[Residuals reconstruct through the same arithmetic square]\label{cor:residual-fracture}
2173 Let $X$ be a finite coherent event diagram in $\operatorname{Perf}(\mathbb Z)$. Every edge residual, square interaction, and initial-to-terminal residual of $X$ satisfies \cref{thm:arithmetic-fracture}. Its terms are the corresponding residuals of the diagrams obtained by base change to $A,B,G$. Each such integral residual vanishes if and only if both its localized and completed versions vanish.
2174 \end{corollary}
2175 \begin{proof}
2176 Derived base change is exact and therefore preserves all the finite cofibers defining those residuals. Apply the arithmetic fracture theorem to the resulting perfect integral complex. Naturality gives the maps on the whole residual diagram, not merely equality of its cohomology dimensions.
2177 \end{proof}
2178 
2179 \subsection{Effective gluing of lattices from independent local input}
2180 
2181 The previous theorem starts with a global complex. The following finite-rank theorem instead constructs a global object from supplied local and localized data.
2182 
2183 \begin{theorem}[Effective arithmetic lattice gluing]\label{thm:lattice-gluing}
2184 Let $V$ be a finite free $A=\mathbb Z[1/D]$-module of rank $r$. For each $p\in\mathcal P$ let $L_p$ be a finite free $\mathbb Z_p$-module, with an isomorphism
2185 \[
2186  \theta_p:V\otimes_A\mathbb Q_p\xrightarrow{\sim}
2187  L_p\otimes_{\mathbb Z_p}\mathbb Q_p.
2188 \]
2189 Then
2190 \[
2191  \Lambda=\{v\in V:\theta_p(v)\in L_p\text{ for every }p\in\mathcal P\}
2192 \]
2193 is a finite free $\mathbb Z$-module of rank $r$, and the natural maps identify
2194 \[
2195  \Lambda[1/D]\cong V,\qquad
2196  \Lambda\otimes_{\mathbb Z}\mathbb Z_p\cong L_p.
2197 \]
2198 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.
2199 \end{theorem}
2200 \begin{proof}
2201 Choose a basis $V=A^r$, and let $E_p=\theta_p^{-1}(L_p)$ be the corresponding lattice in $\mathbb Q_p^r$. Bounds on the entries of $\theta_p$ and its inverse give a common $N\geq1$ such that, with $b=D^N$,
2202 \[
2203  b\mathbb Z_p^r\subseteq E_p\subseteq b^{-1}\mathbb Z_p^r
2204  \quad\text{for all }p.
2205 \]
2206 The intersection defining $\Lambda$ therefore satisfies
2207 \[
2208  b\mathbb Z^r\subseteq\Lambda\subseteq b^{-1}\mathbb Z^r.
2209 \]
2210 The right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\Lambda[1/D]=A^r$.
2211 
2212 Here is also an explicit finite construction proving the local identifications. Put $F_p=bE_p$, so
2213 \[
2214  b^2\mathbb Z_p^r\subseteq F_p\subseteq\mathbb Z_p^r.
2215 \]
2216 The Chinese remainder isomorphism identifies
2217 \[
2218  \mathbb Z^r/b^2\mathbb Z^r\cong
2219  \prod_{p\in\mathcal P}\mathbb Z_p^r/b^2\mathbb Z_p^r.
2220 \]
2221 Let $H\subseteq\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\mathbb Z_p^r$. Directly from the definitions, $H=b\Lambda$. The quotient $\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\mathbb Z_p$ retains exactly the $p$ component. Thus $H\otimes\mathbb Z_p$ is the inverse image of $F_p/b^2\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\Lambda\otimes\mathbb Z_p=E_p$, and $\theta_p$ identifies it with $L_p$.
2222 
2223 For completeness, tensoring the displayed subgroup inclusion is exact: $\mathbb Z_p$ is torsion-free over $\mathbb Z$, so it is flat; alternatively the assertion follows from a Smith diagonal presentation of the finite-index subgroup $H$. The latter proves the same statement using only integer bases.
2224 
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 $\Lambda$. A compatible $A$-linear map $V\to V'$ sends every $\theta_p$-integral element to a $\theta'_p$-integral element and hence restricts to $\Lambda\to\Lambda'$. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in $V$. These constructions are inverse on objects up to the specified identifications and on morphisms, proving the equivalence. The zero-rank case gives the zero module throughout.
2226 \end{proof}
2227 
2228 This is an effective local--global existence theorem for lattices, not for Galois representations or automorphic representations. The generic identifications are essential input. The construction does not derive them from matching coarse invariants. Nor does it assert gluing of arbitrary nonlinear structures, group actions, filtrations, or analytic conditions. Those would have to descend through the lattice construction with their own compatibility proofs.
2229 
2230 \paragraph{Consequences for the calculus.}
2231 The new example separates three operations that a universal slogan would conflate. Exact finite-level excision can erase an entire prime component. Completion can retain a generic fiber that all those finite repairs erased. Effective reconstruction instead retains the localized object, the completed objects, and their overlap identifications. The lattice theorem shows that this latter pattern can genuinely construct a global arithmetic object from local input, in a precisely specified class.
2232 
2233 \begin{corollary}[Compatible operators glue with the lattice]\label{cor:lattice-operators}
2234 In \cref{thm:lattice-gluing}, suppose $f_1,\ldots,f_s\in\operatorname{End}_A(V)$ and $g_{p,1},\ldots,g_{p,s}\in\operatorname{End}_{\mathbb Z_p}(L_p)$ satisfy the exact intertwining equations
2235 \[
2236  \theta_p(f_i\otimes1)=(g_{p,i}\otimes1)\theta_p.
2237 \]
2238 Then they induce unique integral endomorphisms of $\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.
2239 \end{corollary}
2240 \begin{proof}
2241 For $v\in\Lambda$, the displayed identity puts $\theta_p(f_i v)=g_{p,i}\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\in\Lambda$. Restriction is unique because $\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.
2242 \end{proof}
2243 
2244 This corollary glues an already compatible operator action. It does not construct a generic action, a continuous Galois action, or an automorphic realization from scalar or Frobenius invariants alone.
2245 
2246 
2247 \section{Intrinsic arithmetic exchange and certified lifting}\label{sec:intrinsic-exchange}
2248 
2249 The weighted comparisons of \cref{thm:weighted-adic-exchange} leave a concrete question open: can a nonzero defect arise from the arithmetic functors themselves? This section gives two answers and keeps their categorical regimes separate. Ordinary invariants against reduction have a first-cohomology defect. Fully derived invariants against a perfect coefficient change commute. Against a specified nonperfect coefficient change they can fail even as exact functors, with a non-scalar representation and no weight inserted into the comparison.
2250 
2251 Throughout the lattice part, fix a prime $p$, write $O=\mathbb Z_p$ and $O_n=O/p^nO$, and take $n\geq1$. 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.
2252 
2253 \subsection{A natural kernel--reduction comparison}
2254 
2255 For a map $d:V\to W$ of lattices, put
2256 \[
2257  K=\ker d,\qquad Q=\operatorname{coker}d,\qquad
2258  K_n=\ker(d\otimes_O O_n).
2259 \]
2260 The module $K$ is saturated in $V$: if $p^n v\in K$, then $p^n d(v)=0$ in the torsion-free module $W$, so $v\in K$. It is therefore a finite free direct summand as an $O$-module, although a splitting is not canonical. Reduction supplies a natural injection $\alpha_n:K/p^nK\to K_n$.
2261 
2262 \begin{theorem}[Intrinsic kernel exchange and its boundary]\label{thm:kernel-bockstein}
2263 There is a natural short exact sequence
2264 \[
2265  \boxed{0\longrightarrow K/p^nK\xrightarrow{\alpha_n}K_n
2266  \xrightarrow{\delta_n}Q[p^n]\longrightarrow0.}
2267 \]
2268 For $\bar v\in K_n$, choose a lift $v\in V$. Since $d(v)\in p^nW$, set
2269 \[
2270  \delta_n(\bar v)=\left[\frac{d(v)}{p^n}\right]\in Q.
2271 \]
2272 This formula is independent of the lift. For $m\geq n$, ordinary reduction $r_{m,n}:K_m\to K_n$ satisfies
2273 \[
2274  \boxed{\delta_n r_{m,n}=p^{m-n}\delta_m.}
2275 \]
2276 Consequently the cokernel of the natural comparison $\alpha_n$ 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 cokernel in degree zero.
2277 \end{theorem}
2278 \begin{proof}
2279 If $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\in K$. This proves exactness at $K_n$. Conversely, if $[w]\in Q[p^n]$, there is $v\in V$ with $p^n w=d(v)$, so $\bar v\in K_n$ maps to $[w]$. Saturation proves injectivity of $\alpha_n$.
2280 
2281 For the transition formula, use the same lift $v$ to represent an element first at level $m$ and then at level $n$. The quotients obey $d(v)/p^n=p^{m-n}d(v)/p^m$. Functoriality for a commuting square between two maps $d$ follows from this same formula. Finally the cone of an injection between degree-zero modules has just its cokernel in degree zero.
2282 \end{proof}
2283 
2284 This is a direct coefficient-boundary argument, not a new general homological-algebra principle. The modules on the two sides need not be perfect over $O_n$. The operation of taking a kernel is not an exact functor on arbitrary short exact sequences. Thus this theorem by itself does not instantiate the earlier exact-endofunctor assertions.
2285 
2286 \subsection{The exact amount of additional resolution needed}
2287 
2288 Choose Smith data for $d$ over the discrete valuation ring $O$. After invertible row and column operations its nonzero entries are $p^{a_1},\ldots,p^{a_r}$, with $a_i\geq0$. Units have been absorbed into the basis changes. Let
2289 \[
2290  h=\max\{a_1,\ldots,a_r,0\}.
2291 \]
2292 Equivalently, $h$ is the least integer with $p^h Q_{\mathrm{tors}}=0$. If $Q_{\mathrm{tors}}=0$, set $h=0$.
2293 
2294 \begin{theorem}[Residual profile and sharp finite-resolution lifting]\label{thm:precision-window}
2295 Let $v=\operatorname{rank}_O V$. Then
2296 \[
2297  Q[p^n]\cong\bigoplus_{i=1}^r O/p^{\min(a_i,n)}O,
2298  \qquad
2299  \ell_O(K_n)=n(v-r)+\sum_{i=1}^r\min(a_i,n).
2300 \]
2301 The splittings require chosen Smith bases and are not canonical. For every $n\geq1$,
2302 \[
2303  \boxed{\operatorname{im}(K_{n+h}\longrightarrow K_n)
2304        =\operatorname{im}(K/p^nK\longrightarrow K_n).}
2305 \]
2306 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 $h>0$.
2307 
2308 The residual system $(Q[p^n],p^{m-n})$ is pro-zero, and
2309 \[
2310  K\xrightarrow{\sim}\varprojlim_n K_n.
2311 \]
2312 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.
2313 \end{theorem}
2314 \begin{proof}
2315 In Smith coordinates a nonzero diagonal entry imposes $p^{a_i}x_i=0$ 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.
2316 
2317 By \cref{thm:kernel-bockstein}, the image of an element of $K_{n+h}$ has boundary $p^h\delta_{n+h}=0$, so lies in the image of $K/p^nK$. The reverse inclusion follows by reducing any exact lift at level $n+h$.
2318 
2319 If $c<h$, choose an element of exact order $p^h$ in $Q_{\mathrm{tors}}$ and take $n\geq h$. Surjectivity of $\delta_{n+c}$ 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.
2320 
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 $\bigcap_n p^nW=0$. This proves the inverse-limit assertion directly, without interchanging an arbitrary kernel and an unspecified completion.
2322 \end{proof}
2323 
2324 \begin{remark}[What a finite certificate must contain]
2325 For integer input matrices, a verified integer Smith decomposition gives the $p$-adic data by taking the valuations of its nonzero diagonal entries. Verification requires the two change-of-basis matrices to be unimodular and their product with $d$ to equal the displayed diagonal matrix. For arbitrary $p$-adic entries, a finite truncation alone does not certify a claimed zero diagonal entry or an exact rank. Such a claim requires an exact input presentation or a separate rank certificate. The theorem does not supply a universal numerical decision procedure from unbounded-precision oracles.
2326 \end{remark}
2327 
2328 \subsection{Symmetry provides the comparison without imposed weights}
2329 
2330 Let $G$ be a finite group acting $O$-linearly on a lattice $M$. The natural comparison is
2331 \[
2332  \alpha_{G,n}:M^G/p^nM^G\longrightarrow(M/p^nM)^G.
2333 \]
2334 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.
2335 
2336 \begin{theorem}[Cohomological identification of the symmetry residual]\label{thm:invariant-bockstein}
2337 There is a natural exact sequence
2338 \[
2339  \boxed{0\longrightarrow M^G/p^nM^G\longrightarrow(M/p^nM)^G
2340  \xrightarrow{\partial_n}H^1(G,M)[p^n]\longrightarrow0.}
2341 \]
2342 For a fixed residue class $\bar m$ and a lift $m\in M$, the obstruction is represented by the crossed homomorphism
2343 \[
2344  c_{\bar m}(g)=\frac{gm-m}{p^n}.
2345 \]
2346 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.
2347 
2348 More precisely, for
2349 \[
2350  d_G:M\longrightarrow\prod_{g\in G}M,\qquad
2351  d_G(m)=(gm-m)_g,
2352 \]
2353 the torsion subgroup of $\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \cref{thm:precision-window}, and can be strictly smaller than $e$.
2354 \end{theorem}
2355 \begin{proof}
2356 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 also be proved directly. If $p^n[c]=0$, there is $m\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.
2357 
2358 For completeness, let $c:G\to M$ be a cocycle and $b=\sum_{g\in G}c(g)$. For $u\in G$,
2359 \[
2360  ub=b-|G|c(u),\qquad |G|c(u)=u(-b)-(-b).
2361 \]
2362 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.
2363 
2364 Write $d^1(c)(g,u)=g c(u)-c(gu)+c(g)$. If a class $[c]$ in $\operatorname{coker}d_G$ 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.
2365 \end{proof}
2366 
2367 The long exact sequence, explicit boundary, and cyclic-group cohomology used here are classical. Our conventions agree with \cite[Chapter II, 1.9, 1.18--1.21]{MilneCFT2020}. The proof above is included to fix naturality, the coefficient regime, and the transition maps. If $p$ does not divide $|G|$, all these residuals vanish. Divisibility of the group order alone does not imply a nonzero residual: the trivial action on a lattice has $H^1(G,M)=0$.
2368 
2369 \subsection{Lifting transformations, not just invariant vectors}
2370 
2371 \begin{theorem}[Certified lifting of equivariant maps]\label{thm:intertwiner-lifting}
2372 Let $L$ and $M$ be $G$-lattices and put
2373 \[
2374  V=\operatorname{Hom}_O(L,M),\qquad
2375  (g\cdot f)=\rho_M(g)f\rho_L(g)^{-1}.
2376 \]
2377 Then
2378 \[
2379  \begin{split}
2380  0\longrightarrow&\operatorname{Hom}_{O[G]}(L,M)/p^n
2381  \longrightarrow\operatorname{Hom}_{O_n[G]}(L/p^nL,M/p^nM)\\
2382  \longrightarrow&H^1(G,V)[p^n]\longrightarrow0
2383  \end{split}
2384 \]
2385 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 \cref{thm:precision-window}, this is equivalent to lifting the same map equivariantly to precision $n+h$.
2386 
2387 If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.
2388 \end{theorem}
2389 \begin{proof}
2390 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 \cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\widetilde f$ the boundary cocycle is
2391 \[
2392  g\longmapsto\frac{\rho_M(g)\widetilde f\rho_L(g)^{-1}-\widetilde f}{p^n}.
2393 \]
2394 If this is $g\cdot b-b$, then $\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.
2395 \end{proof}
2396 
2397 The theorem concerns two already supplied lattices with an actual action of the same finite group. It does not construct a Galois representation, provide a finite quotient for an arbitrary profinite action, or produce an automorphic representation. It addresses a concrete transformation-lifting obligation once those inputs exist.
2398 
2399 \subsection{A non-scalar family with a sharp obstruction lifespan}
2400 
2401 Let $m\geq2$, and put
2402 \[
2403  N_m(t)=1+t+\cdots+t^{m-1},\qquad
2404  M_m=O[t]/(N_m(t)).
2405 \]
2406 This is free of rank $m-1$. Let a generator $\sigma$ of $C_m$ act by multiplication by $t$. Since $(t-1)N_m(t)=t^m-1$, this is a genuine cyclic-group action.
2407 
2408 \begin{theorem}[Norm-quotient lattice and transient invariants]\label{thm:norm-quotient}
2409 For this action,
2410 \[
2411  M_m^{C_m}=0,\qquad
2412  H^1(C_m,M_m)\cong O/mO.
2413 \]
2414 Consequently, with $a=v_p(m)$,
2415 \[
2416  (M_m/p^nM_m)^{C_m}\cong O/p^{\min(a,n)}O.
2417 \]
2418 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 $O$.
2419 \end{theorem}
2420 \begin{proof}
2421 The operator $\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.
2422 
2423 For a cyclic group, a crossed homomorphism is determined by $c(\sigma)=x$ with norm zero, and a coboundary has $x=(\sigma-1)y$. This gives $H^1=\ker N/(\sigma-1)M_m=O/mO$ directly, agreeing with \cite[Chapter II, 1.20]{MilneCFT2020}. The remaining assertions follow from the preceding theorems.
2424 \end{proof}
2425 
2426 For $p=m=3$, in the basis $1,t$,
2427 \[
2428  A=\begin{pmatrix}0&-1\\1&-1\end{pmatrix},\qquad
2429  A^3=I,\qquad \det(A-I)=3.
2430 \]
2431 The fixed vectors modulo $3^n$ are exactly
2432 \[
2433  \left\{b\,3^{n-1}\binom{1}{-1}:b\in\{0,1,2\}\right\}.
2434 \]
2435 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.
2436 
2437 \section{Fully derived exchange: a positive theorem and an exact counterexample}\label{sec:derived-invariant-exchange}
2438 
2439 \subsection{The comparison and its direction}
2440 
2441 For a commutative ring $R$, a finite group $G$, and $M\in D(R[G])$, define
2442 \[
2443  \Gamma_R(M)=\mathbf R\operatorname{Hom}_{R[G]}(R,M)\in D(R).
2444 \]
2445 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.
2446 
2447 For a ring map $R\to S$ there is a canonical comparison
2448 \[
2449  \boxed{\beta_{R,S,M}:\Gamma_R(M)\otimes_R^{\mathbf L}S
2450  \longrightarrow\Gamma_S(M\otimes_R^{\mathbf L}S).}
2451 \]
2452 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.
2453 
2454 This direction is reduction-after-invariants to invariants-after-reduction. It is the reverse of the optional $A_rC\to CA_s$ convention used for some earlier signatures. We use the direction supplied by the arithmetic comparison, not an unproved inverse. No earlier orientation-dependent normalization theorem is applied without changing its signature explicitly.
2455 
2456 A two-fiber exact model can be written entirely with endofunctors: set
2457 \[
2458  \mathcal K_R=D(R[G])\times D(R),\qquad
2459  A_R(M,N)=(0,\Gamma_R(M)),
2460 \]
2461 and let $C_{R,S}$ apply derived coefficient change in both components. The second component of $C_{R,S}A_R\to A_SC_{R,S}$ is precisely $\beta$. 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 making a claim that all outputs are perfect.
2462 
2463 \begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\label{thm:perfect-invariant-exchange}
2464 If $S$ is perfect as an $R$-module, then $\beta_{R,S,M}$ is an equivalence for every $M\in D(R[G])$. In particular, reduction $\mathbb Z_p\to\mathbb Z/p^n$ commutes with fully derived invariants.
2465 \end{theorem}
2466 \begin{proof}
2467 For a complex $P\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison
2468 \[
2469  \Gamma_R(M)\otimes_R^{\mathbf L}P
2470  \longrightarrow\Gamma_R(M\otimes_R^{\mathbf L}P).
2471 \]
2472 Both sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\operatorname{Perf}(R)=\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.
2473 
2474 Finally $\mathbb Z/p^n$ has the two-term finite free resolution $[\mathbb Z_p\xrightarrow{p^n}\mathbb Z_p]$. It is perfect over $\mathbb Z_p$. This uses the ordinary derived tensor construction, not termwise tensor of an arbitrary unbounded non-K-flat complex \cite{StacksDerivedTensor}.
2475 \end{proof}
2476 
2477 For a lattice the earlier nonzero $H^1[p^n]$ therefore records the effect of extracting degree-zero invariants, not a failure of this fully derived comparison over $\mathbb Z_p$. This positive theorem rules out a tempting but incorrect source of a new exact residual. It also gives an information-preserving way to handle that particular truncation defect: retain derived invariant data rather than require degree-zero invariants to commute with reduction.
2478 
2479 \subsection{A natural non-scalar defect at finite arithmetic resolution}
2480 
2481 The perfectness hypothesis cannot be removed. Fix an odd prime $p$, let
2482 \[
2483  R=\mathbb Z/p^2,\qquad k=\mathbb F_p,\qquad G=C_p,
2484 \]
2485 and choose a matrix $J\in M_d(\mathbb Z)$ whose reduction modulo $p$ is invertible. Put $M=R^d$ and let the chosen generator act by
2486 \[
2487  A=I+pJ\pmod{p^2}.
2488 \]
2489 For $d=2$, the choice $J=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ makes this action non-scalar. It mixes the two coordinates. The group action is input, but the exchange map remains the canonical $\beta$, not a scalar multiple of another comparison.
2490 
2491 \begin{theorem}[Unbounded intrinsic derived exchange defect]\label{thm:nonflat-invariant-exchange}
2492 For the preceding data, $A^p=I$ and the norm is $N=I+A+\cdots+A^{p-1}=pI$. Moreover,
2493 \[
2494  \Gamma_R(M)\simeq pM[0]\cong k^d[0].
2495 \]
2496 Let $U=\Gamma_R(M)\otimes_R^{\mathbf L}k$ and $V=\Gamma_k(M\otimes_R^{\mathbf L}k)$. Their cohomology is
2497 \[
2498  H^i(U)\cong\begin{cases}k^d&i\leq0,\\0&i>0,\end{cases}
2499  \qquad
2500  H^i(V)\cong\begin{cases}0&i<0,\\k^d&i\geq0.\end{cases}
2501 \]
2502 The natural comparison $\beta:U\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies
2503 \[
2504  \boxed{H^i\Cofib(\beta)\cong k^d\quad\text{for every }i\in\mathbb Z.}
2505 \]
2506 Both composites defining $\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.
2507 \end{theorem}
2508 \begin{proof}
2509 The binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so
2510 \[
2511  N=pI+p\frac{p(p-1)}2J=pI\pmod{p^2},
2512 \]
2513 using that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex
2514 \[
2515  M\xrightarrow{pJ}M\xrightarrow{pI}M
2516  \xrightarrow{pJ}M\xrightarrow{pI}\cdots
2517 \]
2518 in nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.
2519 
2520 Since $J$ is invertible over $R$, each displayed differential has image and kernel $pM$. All positive cohomology vanishes and $H^0=pM$. Inclusion of that kernel in degree zero is a quasi-isomorphism $pM[0]\to\Gamma_R(M)$.
2521 
2522 The $R$-module $k$ has an infinite free resolution, in nonpositive cochain degrees, with each differential multiplication by $p$. It is exact because $\ker(p:R\to R)=pR=\operatorname{im}(p)$. Tensoring this bounded-above free resolution with $k$ gives zero differentials. Since $pM\cong k^d$, this computes $H^i(U)$ as claimed.
2523 
2524 On the other side $M$ is free over $R$, so $M\otimes_R^{\mathbf L}k=k^d[0]$. Its action is trivial. In the cyclic cochain resolution over $k$, both $A-I$ and the norm are zero. This gives the displayed cohomology of $V$.
2525 
2526 The evaluation map on degree-zero invariants is the inclusion $pM\hookrightarrow M$. 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 $\beta$ therefore proves $H^0(\beta)=0$. 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.
2527 
2528 Exactness of derived coefficient change and derived invariants proves exactness of both composites. Their canonical natural transformation has a functorial cofiber in the stable enhancement. No illicit termwise tensor of the nonnegative invariant complex is used in computing $U$.
2529 \end{proof}
2530 
2531 \begin{example}[The two-dimensional $\mathbb Z/9$ instance]\label{ex:nine-invariants}
2532 Take
2533 \[
2534  R=\mathbb Z/9,\qquad
2535  A=\begin{pmatrix}1&3\\3&1\end{pmatrix},\qquad G=C_3.
2536 \]
2537 Then $A^3=I$ over $R$, $I+A+A^2=3I$, and $M^G=3M$. The invariant cochain complex is quasi-isomorphic to $(\mathbb F_3)^2$ 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 $(\mathbb F_3)^2$ in every degree. Ordinary tensor of the original nonnegative complex would incorrectly reproduce only the right-hand answer.
2538 \end{example}
2539 
2540 \begin{corollary}[Erasing this residual loses the coefficient unit]\label{cor:intrinsic-unit-loss}
2541 Let $\Delta=\Cofib(\beta)$ in \cref{thm:nonflat-invariant-exchange}. Any exact functor out of $D(k)$ that annihilates $\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.
2542 \end{corollary}
2543 \begin{proof}
2544 Every complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\Delta)=k^d\ne0$, the unit $k[0]$ is a retract of $\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\Delta$ annihilates $\operatorname{Thick}(k)=\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.
2545 \end{proof}
2546 
2547 Thus exactness alone does not make the comparison invertible, and existence of its cofiber does not make a quotient repair harmless. The arithmetic coefficient module $k$ is not perfect over $R$. Also $\Gamma_R(M)\simeq k^d$ is not perfect over $R$, and the resulting residual is unbounded over $k$. These facts explain exactly why neither the preceding perfect-base-change theorem nor a blanket perfect-fiber model applies.
2548 
2549 \subsection{Finite-level validity does not provide a finer action}
2550 
2551 \begin{theorem}[A one-step representation-lifting obstruction]\label{thm:one-step-group-lift}
2552 In \cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\mathbb Z/p^2)^d$ has no lift to a representation on $(\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\widetilde A$ obeys
2553 \[
2554  \boxed{\widetilde A^p-I\equiv p^2J\not\equiv0\pmod{p^3}.}
2555 \]
2556 This obstruction is unchanged by the choice of the lift. There is also no finite free $\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.
2557 \end{theorem}
2558 \begin{proof}
2559 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 $2\leq j<p$, 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 $p\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\sigma^p=1$ cannot hold at the next precision.
2560 
2561 A $\mathbb Z_p$-lift would reduce to such a level-three lift, which is impossible. Finally an invertible change of basis modulo $p^2$ lifts to an invertible matrix modulo $p^3$. Conjugating by that lift reduces an alleged representation lifting only up to isomorphism to the already refuted matrix-lifting problem.
2562 \end{proof}
2563 
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 \cref{thm:norm-quotient} genuinely exist over $\mathbb Z_p$ at all precisions, even though some of their finite-level invariant vectors do not lift. These are different failures.
2565 
2566 
2567 \subsection{Exact preservation on coefficient-lifted inputs}
2568 
2569 The preceding negative result has a complementary positive theorem. A coefficient quotient may be nonperfect over an intermediate ring, yet the canonical exchange can be exact on inputs that genuinely come from an earlier coefficient level.
2570 
2571 \begin{theorem}[Lifted-input exchange and a necessary obstruction to lifting]\label{thm:lifted-input-exchange}
2572 Let $O\to R\to S$ be maps of commutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\in D(O[G])$, put $M=N\otimes_O^{\mathbf L}R$. Then
2573 \[
2574  \boxed{\beta_{R,S,M}:\Gamma_R(M)\otimes_R^{\mathbf L}S
2575  \xrightarrow{\sim}\Gamma_S(M\otimes_R^{\mathbf L}S).}
2576 \]
2577 No perfectness of $S$ over $R$ is required. Consequently, if $\Cofib(\beta_{R,S,M})$ is nonzero, there is no $N\in D(O[G])$ together with an equivalence $N\otimes_O^{\mathbf L}R\simeq M$.
2578 
2579 For $O=\mathbb Z_p$, $R=\mathbb Z/p^m$, and $S=\mathbb Z/p^n$ with $m\geq n\geq1$, this applies at every pair of levels. It applies in particular to reductions of a supplied integral $G$-lattice.
2580 \end{theorem}
2581 \begin{proof}
2582 The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite
2583 \[
2584  \begin{split}
2585  \Gamma_O(N)\otimes_O^{\mathbf L}S
2586  &\xrightarrow{\beta_{O,R,N}\otimes_R S}
2587  \Gamma_R(N\otimes_O^{\mathbf L}R)\otimes_R^{\mathbf L}S\\
2588  &\xrightarrow{\beta_{R,S,M}}
2589  \Gamma_S(N\otimes_O^{\mathbf L}S)
2590  \end{split}
2591 \]
2592 is $\beta_{O,S,N}$. 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.
2593 
2594 By \cref{thm:perfect-invariant-exchange}, $\beta_{O,R,N}$ and $\beta_{O,S,N}$ are equivalences. Tensoring preserves an equivalence, so the two-out-of-three property proves that $\beta_{R,S,M}$ is an equivalence. Naturality makes the conclusion invariant under a specified equivalence between $M$ and a lifted object. The obstruction assertion is the contrapositive. For the arithmetic specialization, both quotients have two-term finite free resolutions over $\mathbb Z_p$.
2595 \end{proof}
2596 
2597 The cofiber of $\beta_{R,S,-}$ 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 \cref{thm:nonflat-invariant-exchange} lies outside that class. This is a functor-intrinsic necessary lifting obstruction, not merely a failed scalar equality.
2598 
2599 The converse is false in general: \cref{thm:blind-exchange} supplies a zero comparison cone for a nonliftable finite-group representation. The positive theorem preserves information on the class with supplied lifts. A sufficient criterion for a fixed finite source instead comes from the actual relation obstruction and the finite lifting horizon in \cref{thm:relation-H2,thm:finite-action-horizon}.
2600 
2601 \subsection{The resulting boundary of the arithmetic calculus}
2602 
2603 There are now four precisely separated regimes.
2604 \begin{enumerate}
2605 \item For a supplied integral lattice, ordinary invariants and reduction can differ. The residual is $H^1[p^n]$, and a finite amount of additional precision detects and repairs liftability of a specified vector or intertwiner without taking a destructive quotient.
2606 \item For a perfect coefficient change, fully derived invariants and reduction agree. Passing to derived data resolves the degree-zero truncation problem in this regime.
2607 \item A supplied coefficient lift over a common ring makes the intermediate comparison exact when the two coefficient quotients are perfect over that common ring. Nonvanishing of the comparison cone is therefore a necessary obstruction to such a lift.
2608 \item For the exhibited nonperfect change $\mathbb Z/p^2\to\mathbb F_p$, the fully derived canonical exchange itself has a nonzero, unbounded cofiber. Killing that cofiber destroys the target's perfect coefficient theory. Its input representation also fails to extend one coefficient level further.
2609 \end{enumerate}
2610 These statements distinguish intrinsic exchange from actual existence. The following section refutes sufficiency of the single comparison-cone test and proves a finite necessary-and-sufficient lifting criterion for a fixed finite group. That correction preserves the requested finite-level representation without killing the coefficient unit. It does not close unrestricted profinite-tower existence or the Langlands-facing globalization condition.
2611 
2612 
2613 \section{Sufficient lifting for a fixed finite group}\label{sec:finite-lifting}
2614 
2615 This section closes a specified version of the sufficient-lifting question left by K61. The source group $G$ is fixed and finite, the rank $d$ is fixed, the coefficient ring is $\mathbb Z_p$, and every lift must satisfy the same group multiplication table. We first exhibit a zero-exchange-defect representation that nevertheless does not lift. We then give a complete one-step obstruction and an explicit finite horizon after which convergent correction constructs an integral action. None of these assertions concerns arbitrary continuous representations of a changing system of finite Galois quotients.
2616 
2617 \paragraph{Attribution.}
2618 The one-step obstruction is the classical group-extension class in $H^2$, with the standard inhomogeneous conventions of \cite[Chapter II, 1.17--1.19]{MilneCFT2020}. Finite-group ultrametric stability and the quantitative bound used below are known: \cite[Lemma 7.3, Proposition 7.4, Example 7.5]{FournierFacio2026} supply the same valuation threshold, in a broader setting. We provide a direct matrix-averaging proof, translate it into an exact finite lifting criterion, and connect it to the present residual calculus. Historical novelty is not claimed for that bound or the deformation-theoretic mechanism.
2619 
2620 \subsection{The relation residual and its repair class}
2621 
2622 Put $R_n=\mathbb Z/p^n$, $k=\mathbb F_p$, and fix
2623 \[
2624  \rho_n:G\longrightarrow\operatorname{GL}_d(R_n),\qquad n\geq1.
2625 \]
2626 Let $V=\operatorname{End}_k(k^d)$ with action
2627 \[
2628  g\cdot X=\bar\rho(g)X\bar\rho(g)^{-1},
2629  \qquad \bar\rho=\rho_n\bmod p.
2630 \]
2631 Choose matrices $A_g\in\operatorname{GL}_d(R_{n+1})$ lifting $\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual
2632 \[
2633  c_A(g,h)=\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\pmod p\quad\in V.
2634 \]
2635 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 $2n\geq n+1$.
2636 
2637 For a normalized $1$-cochain $b:G\to V$, set
2638 \[
2639  (\delta b)(g,h)=b(g)+g\cdot b(h)-b(gh).
2640 \]
2641 For a normalized $2$-cochain set
2642 \[
2643  (\delta c)(g,h,t)=g\cdot c(h,t)-c(gh,t)+c(g,ht)-c(g,h).
2644 \]
2645 A cochain is normalized when it vanishes whenever an argument is the identity.
2646 
2647 \begin{theorem}[Complete one-step relation obstruction]\label{thm:relation-H2}
2648 The residual $c_A$ is a $2$-cocycle. Its cohomology class
2649 \[
2650  o_n(\rho_n)=[c_A]\in H^2(G,V)
2651 \]
2652 is independent of the chosen matrix lifts. Moreover,
2653 \[
2654  \boxed{o_n(\rho_n)=0
2655  \quad\Longleftrightarrow\quad
2656  \rho_n\text{ lifts to }\operatorname{GL}_d(R_{n+1}).}
2657 \]
2658 More precisely, all lifts with the fixed reduction have the form
2659 \[
2660  A_g^b=(I+p^n b(g))A_g,
2661 \]
2662 where representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is
2663 \[
2664  \boxed{\delta b=-c_A.}
2665 \]
2666 \end{theorem}
2667 \begin{proof}
2668 Associativity of the threefold product gives the exact identity
2669 \[
2670  (I+p^nc_A(g,h))(I+p^nc_A(gh,t))
2671  = (I+p^n g\cdot c_A(h,t))(I+p^nc_A(g,ht))
2672 \]
2673 in $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\delta c_A=0$.
2674 
2675 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
2676 \[
2677  c_{A^b}=c_A+\delta b.
2678 \]
2679 Thus the class is unchanged and the new matrices satisfy every group relation exactly if and only if $\delta b=-c_A$. 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.
2680 \end{proof}
2681 
2682 \begin{proposition}[Repair ambiguity, conjugacy, and finite certificates]\label{prop:repair-H1}
2683 If the one-step lift set is nonempty, its framed lifts form a torsor for $Z^1(G,V)$. Conjugation by matrices $I+p^nX$, $X\in V$, 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.
2684 
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 $\lambda$ on normalized $2$-cochains with
2686 \[
2687  \lambda\circ\delta=0,\qquad \lambda(c_A)\ne0
2688 \]
2689 is a negative certificate.
2690 \end{proposition}
2691 \begin{proof}
2692 If $b_0$ is one solution, every other solution is $b_0+z$ with $\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\cdot X$. 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 $\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.
2693 \end{proof}
2694 
2695 This is an actual obstruction to realizing the finer group action, not merely failure of a selected invariant to commute with coefficient change. The repair changes matrices within their prescribed residue classes and keeps the coefficient unit. It does not take a quotient of the coefficient theory.
2696 
2697 \subsection{A vanishing exchange cone can miss nonliftability}
2698 
2699 \begin{theorem}[A central sign hides the obstruction from derived invariants]\label{thm:blind-exchange}
2700 Let $p$ be odd, $R=\mathbb Z/p^2$, and let $J\in M_d(\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let
2701 \[
2702  G=C_p\times C_2=\langle\sigma,\tau\rangle,
2703  \qquad \sigma\mapsto I+pJ,\quad \tau\mapsto-I.
2704 \]
2705 For the canonical comparison of \cref{thm:lifted-input-exchange},
2706 \[
2707  \Gamma_R(M)\simeq0,
2708  \qquad \Gamma_k(M\otimes_R^{\mathbf L}k)\simeq0,
2709  \qquad \Cofib(\beta_{R,k,M})\simeq0.
2710 \]
2711 Nevertheless this representation does not lift to $\mathbb Z/p^3$, and hence does not lift as a finite free $\mathbb Z_p$-representation of this fixed group.
2712 \end{theorem}
2713 \begin{proof}
2714 The two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents
2715 \[
2716  e_+=(1+\tau)/2,\qquad e_-=(1-\tau)/2
2717 \]
2718 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 resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\beta$ therefore vanish.
2719 
2720 Any lift of this $G$-action would restrict to a lift of its $C_p$-action. By \cref{thm:one-step-group-lift}, each lift of $I+pJ$ has $p$th power $I+p^2J\ne I$ modulo $p^3$. This is impossible.
2721 \end{proof}
2722 
2723 The example refutes sufficiency of the single $\Gamma$-comparison test in the finite-group class. It does not merely leave the converse unproved. The loss is a loss of detection: a functor can see none of an input while the input still has inconsistent lifting relations. In contrast, conjugation by $-I$ is trivial on $\operatorname{End}_k(k^d)$, so the adjoint relation obstruction is not hidden by this sign. Twisting by the integral sign character also restores the earlier nonzero exchange example. Neither observation asserts that a finite family of such probe comparisons classifies all lifting obstructions.
2724 
2725 \subsection{Quadratic correction of an approximate action}
2726 
2727 Let $N=|G|$ and write $e=v_p(N)$. For a matrix $X$ over $\mathbb Q_p$, the assertion $v_p(X)\geq a$ means every entry lies in $p^a\mathbb Z_p$. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let $A:G\to\operatorname{GL}_d(\mathbb Z_p)$ be a normalized map, not necessarily a homomorphism. Define
2728 \[
2729  E(g,h)=A_gA_hA_{gh}^{-1}-I,
2730  \qquad \alpha_g(X)=A_gXA_g^{-1}.
2731 \]
2732 Suppose $E(g,h)\in p^mM_d(\mathbb Z_p)$ for every pair. The operators $\alpha_g$ are not assumed to give a group action. This distinction is essential before the repair has succeeded.
2733 
2734 \begin{lemma}[Averaged correction with controlled precision loss]\label{lem:quadratic-repair}
2735 If $m>e$, define in $M_d(\mathbb Q_p)$
2736 \[
2737  b(g)=\frac1N\sum_{t\in G}E(g,t),\qquad
2738  A'_g=(I-b(g))A_g.
2739 \]
2740 Then $b(g)\in p^{m-e}M_d(\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\operatorname{GL}_d(\mathbb Z_p)$. Its defect satisfies
2741 \[
2742  \boxed{A'_gA'_h(A'_{gh})^{-1}-I\in
2743  p^{2m-2e}M_d(\mathbb Z_p).}
2744 \]
2745 Also $A'_g-A_g\in p^{m-e}M_d(\mathbb Z_p)$.
2746 \end{lemma}
2747 \begin{proof}
2748 Division by $N=p^eu$, $u\in\mathbb Z_p^\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\geq1$, each $I-b(g)$ is invertible.
2749 
2750 Associativity gives an identity without assuming an exact adjoint action:
2751 \[
2752  (I+E(g,h))(I+E(gh,t))
2753  = (I+\alpha_g E(h,t))(I+E(g,ht)).
2754 \]
2755 Consequently
2756 \[
2757  E(g,h)+E(gh,t)-\alpha_g E(h,t)-E(g,ht)
2758  \in p^{2m}M_d(\mathbb Z_p).
2759 \]
2760 Sum over $t$ and divide by $N$. The permutation $t\mapsto ht$ identifies the last average with $b(g)$, giving
2761 \[
2762  E(g,h)-b(g)-\alpha_g b(h)+b(gh)
2763  \in p^{2m-e}M_d(\mathbb Z_p).
2764 \]
2765 Put $a=m-e>0$. The repaired multiplicativity product is exactly
2766 \[
2767  (I-b(g))(I-\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.
2768 \]
2769 Modulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\equiv I+b\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.
2770 \end{proof}
2771 
2772 This proof exhibits the residual as input to a repair operator. The correction is admissible because its denominators are absorbed by the defect's valuation. It requires no prior integral action and does not delete any module or coefficient information.
2773 
2774 \begin{theorem}[Convergent finite-group action repair]\label{thm:convergent-action-repair}
2775 If the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation
2776 \[
2777  \rho:G\longrightarrow\operatorname{GL}_d(\mathbb Z_p)
2778  \quad\text{with}\quad
2779  \rho(g)\equiv A_g\pmod{p^{m-e}}.
2780 \]
2781 After $j$ iterations the certified defect depth is
2782 \[
2783  m_j=2e+2^j(m-2e),
2784 \]
2785 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.
2786 \end{theorem}
2787 \begin{proof}
2788 Set $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\to\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.
2789 
2790 Matrix multiplication and inversion on $\operatorname{GL}_d(\mathbb Z_p)$ are continuous. The defects converge to zero because their depths are at least $m_j$. Passing to the limit gives $\rho(g)\rho(h)=\rho(gh)$ and $\rho(1)=I$.
2791 \end{proof}
2792 
2793 The recurrence is a quantitative certificate of convergence, not a claim that arbitrary small residuals can always be removed. At $m=2e$ this estimate does not improve the defect depth. We do not assert that the threshold is optimal for every group, rank, or representation.
2794 
2795 \subsection{A finite necessary-and-sufficient lifting horizon}
2796 
2797 \begin{theorem}[Finite horizon for fixed finite-group representations]\label{thm:finite-action-horizon}
2798 Let $G$ be a finite group, $d,n\geq1$, and $e=v_p(|G|)$. Set
2799 \[
2800  \boxed{h_G(n)=\max\{n+e,\,2e+1\}.}
2801 \]
2802 For every $\rho_n:G\to\operatorname{GL}_d(\mathbb Z/p^n)$, the following are equivalent:
2803 \begin{enumerate}
2804 \item There is a representation $\rho:G\to\operatorname{GL}_d(\mathbb Z_p)$ with reduction $\rho_n$.
2805 \item There is a representation $\rho_h:G\to\operatorname{GL}_d(\mathbb Z/p^{h_G(n)})$ with reduction $\rho_n$.
2806 \end{enumerate}
2807 The implication from (2) to (1) is constructive to arbitrary requested precision by \cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.
2808 \end{theorem}
2809 \begin{proof}
2810 Reduction proves (1) implies (2). Conversely, lift every matrix of $\rho_h$ 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 $h-e\geq n$, it returns precisely the specified $\rho_n$ on reduction.
2811 \end{proof}
2812 
2813 Thus a supplied high-precision action is a finite certificate for existence of an entire integral action. A level-$h$ witness may itself have an obstruction at its next level: the repair is allowed to alter the top $e$ digits while keeping the requested lower data. There is no claim that a positive certificate yields an exact lift of that witness at every digit.
2814 
2815 A terminating decision method follows. Begin with $\rho_n$ and use \cref{thm:relation-H2,prop:repair-H1} to enumerate all one-step solutions, retaining their actual framed matrices, until level $h_G(n)$. A surviving branch is a positive witness. If all branches die before that finite level, no integral lift exists. Every layer is finite, and Gaussian elimination provides checkable obstruction certificates at dead nodes. Choosing a single convenient branch is not sufficient for a negative verdict. Quotienting branches by conjugacy is optional and requires preserving their lifting data.
2816 
2817 For $p\nmid|G|$, 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.
2818 
2819 \begin{example}[A positive witness need not lift itself]\label{ex:horizon-witness}
2820 Take $p=2$, $G=C_2$, and rank one. The level-two representation with generator $3\pmod4$ has the level-three witness $3\pmod8$, since $3^2\equiv1\pmod8$. Here $e=1$ and $h_G(2)=3$. Nevertheless neither lift of $3\pmod8$ to $\mathbb Z/16$ has square one: both $3^2$ and $11^2$ are $9\pmod{16}$.
2821 
2822 Starting with the integer matrix $a_0=3$, the averaged repair becomes
2823 \[
2824  a_{j+1}=\frac{3-a_j^2}{2}\,a_j.
2825 \]
2826 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 $3\pmod4$. In the integral domain $\mathbb Z_2$, the equation $a^2=1$ has only the solutions $1,-1$. Thus the repaired limit is $-1$, which is $7\pmod8$, not the selected witness $3\pmod8$. This is exactly the distinction between preserving the requested seed and preserving every exploratory refinement.
2827 \end{example}
2828 
2829 \subsection{Controlled uniqueness and a finite catalogue}
2830 
2831 \begin{theorem}[Close integral actions are conjugate]\label{thm:close-action-conjugacy}
2832 Let $\rho,\sigma:G\to\operatorname{GL}_d(\mathbb Z_p)$ be representations satisfying
2833 \[
2834  \sigma(g)\equiv\rho(g)\pmod{p^s}\quad(g\in G),\qquad s>e.
2835 \]
2836 Then
2837 \[
2838  T=\frac1{|G|}\sum_{g\in G}\sigma(g)\rho(g)^{-1}
2839  \in I+p^{s-e}M_d(\mathbb Z_p)
2840 \]
2841 is invertible and satisfies
2842 \[
2843  \sigma(g)T=T\rho(g),\qquad
2844  \sigma(g)=T\rho(g)T^{-1}.
2845 \]
2846 \end{theorem}
2847 \begin{proof}
2848 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 $h\in G$,
2849 \[
2850  \sigma(h)T\rho(h)^{-1}
2851  =\frac1{|G|}\sum_g\sigma(hg)\rho(hg)^{-1}=T,
2852 \]
2853 by reindexing the finite sum. This is the required intertwining identity.
2854 \end{proof}
2855 
2856 This is the averaging mechanism underlying finite-precision rigidity of integral finite-group representations, closely related to the classical rigidity of integral finite-group lattices. The proof above is independent of any such external rigidity theorem. The strict inequality matters: for $p=2$, $G=C_2$, and rank one, the trivial and sign representations agree modulo $2$ but are not conjugate. Here $s=e=1$.
2857 
2858 \begin{corollary}[A finite catalogue of integral action classes]\label{cor:finite-action-catalogue}
2859 Let $s=e+1$ and $h=2e+1$. Denote by $\mathscr L_s$ the image of
2860 \[
2861  \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^h))
2862  \longrightarrow
2863  \operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z/p^s)).
2864 \]
2865 Reduction gives a bijection
2866 \[
2867  \boxed{
2868  \frac{\operatorname{Hom}(G,\operatorname{GL}_d(\mathbb Z_p))}
2869  {\operatorname{GL}_d(\mathbb Z_p)}
2870  \ \longleftrightarrow\
2871  \frac{\mathscr L_s}{\operatorname{GL}_d(\mathbb Z/p^s)}.}
2872 \]
2873 Consequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.
2874 \end{corollary}
2875 \begin{proof}
2876 The finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\operatorname{GL}_d(\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target is finite because it is a quotient of a finite set.
2877 \end{proof}
2878 
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.
2880 
2881 \subsection{A smaller certificate for cyclic groups}
2882 
2883 \begin{proposition}[Cyclic relation Jacobian]\label{prop:cyclic-jacobian}
2884 Let $G=C_q$ for an integer $q\geq1$, and let $A\in\operatorname{GL}_d(\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\widetilde A$ modulo $p^{n+1}$ and set
2885 \[
2886  c=\frac{\widetilde A^q-I}{p^n}\pmod p,
2887  \qquad L_{\bar A}(B)=\sum_{i=0}^{q-1}\bar A^iB\bar A^{q-1-i}.
2888 \]
2889 Then the lifts of the cyclic action to level $n+1$ are exactly the matrices
2890 \[
2891  \widetilde A+p^nB\quad\text{such that}\quad L_{\bar A}(B)=-c.
2892 \]
2893 If the affine solution set is nonempty, it has $p^{\dim_k\ker L_{\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.
2894 \end{proposition}
2895 \begin{proof}
2896 Expand the noncommutative product $(\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.
2897 \end{proof}
2898 
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.
2900 
2901 \subsection{Why a fixed finite quotient is a real restriction}
2902 
2903 \begin{proposition}[A profinite lift need not preserve the finite quotient]\label{prop:profinite-quotient-boundary}
2904 Let $p$ be odd and $J\in M_d(\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\to\operatorname{GL}_d(\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\mathbb Z_p\twoheadrightarrow C_p$, however, has a continuous integral lift
2905 \[
2906  \rho(x)=(I+pJ)^x
2907  =\sum_{j\geq0}\binom{x}{j}p^jJ^j,
2908  \qquad x\in\mathbb Z_p.
2909 \]
2910 \end{proposition}
2911 \begin{proof}
2912 For each $j$, the polynomial $\binom{x}{j}$ is continuous on $\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\mathbb Z_p$ in $\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\mathbb Z_p$, so continuity extends the identity $\rho(x+y)=\rho(x)\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.
2913 
2914 Modulo $p^2$ the series is $I+pxJ$, which depends only on $x\bmod p$ and gives the specified finite-level action. But $\rho(p)\equiv I+p^2J\ne I\pmod{p^3}$, so the integral lift does not factor through $C_p$. Nonexistence of an integral $C_p$-lift is \cref{thm:one-step-group-lift}.
2915 \end{proof}
2916 
2917 The same warning applies to finite quotients used in a Galois problem. A relation imposed by fixing one finite quotient may be absent from the full profinite source. Failure to lift through that fixed quotient therefore does not by itself obstruct a continuous lift with growing image. Conversely, the finite-group horizon depends on $v_p(|G|)$ and gives no uniform bound for arbitrary growing finite quotients.
2918 
2919 \subsection{What the new block establishes}
2920 
2921 There are now three distinct tests. A chosen functor's exchange cone is a necessary diagnostic on its detectable inputs, but the central-sign example shows it is not a complete lifting invariant. The class $o_n(\rho_n)$ is a complete one-step obstruction because its vanishing is proved equivalent to solving the actual multiplicativity equations. Finally, the finite horizon plus convergent correction turns a finite witness into a full $\mathbb Z_p$-action for a fixed finite group. This gives a positive reconstruction theorem without killing the residual in a Verdier quotient.
2922 
2923 In the two-axis vocabulary, the resolution fiber is the actual lift groupoid, the transformation direction includes changes of basis, the relation residual is a normalized cocycle, admissible repairs are cochains, and its class in $H^2$ is the obstruction. The averaging theorem then supplies a quantitative regime in which repeated repairs converge. These identifications recover and connect classical deformation and ultrametric-stability mathematics. They do not make the abstract obstruction formalism a proof of automorphic existence or Langlands reciprocity.
2924 
2925 
2926 \section{Continuous source resolution and abelian reconstruction}\label{sec:profinite-reconstruction}
2927 
2928 This section keeps the profinite source fixed while allowing its finite images to grow. It separates two independent resolution coordinates: coefficient precision and the finite quotient through which a particular observation factors. Its strongest constructive conclusion is a complete finite-matrix lifting criterion for continuous rank-two representations of $\mathbb Z_p^r$. In arbitrary rank the same criterion holds when a specified normalized generator is cyclic modulo $p$. No prescribed determinant, ramification type, Hodge condition, or automorphic realization is included in these statements.
2929 
2930 The continuous cohomology used below is the cohomology of continuous inhomogeneous cochains with a \emph{finite discrete} coefficient module. The finite-quotient description is classical \cite[Chapter II, Section 4, Proposition 4.2]{MilneCFT2020}; see also \cite{StacksContinuous}. The free pro-$p$ construction and cyclic-centralizer argument are given directly. The claims are written deductions under explicit hypotheses, not assertions of historical novelty.
2931 
2932 \subsection{The actual continuous obstruction}
2933 
2934 Fix a profinite group $\Gamma$, a prime $p$, $n,d\geq1$, and a continuous representation
2935 \[
2936  \rho_n:\Gamma\longrightarrow\operatorname{GL}_d(R_n),
2937  \qquad R_n=\mathbb Z/p^n.
2938 \]
2939 The target is finite and discrete. Give $V=M_d(\mathbb F_p)$ the adjoint action of $\bar\rho$. Choose a section of the reduction map on the finite set $\rho_n(\Gamma)$, taking the identity to the identity. Composing this section with $\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define
2940 \[
2941  c_A(g,h)=\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\pmod p.
2942 \]
2943 Here division is the identification of $p^nR_{n+1}$ with $\mathbb F_p$, as in \cref{thm:relation-H2}.
2944 
2945 \begin{theorem}[Complete continuous one-step obstruction]\label{thm:continuous-H2}
2946 The class
2947 \[
2948  o_n^{\mathrm{cts}}(\rho_n)=[c_A]\in H^2_{\mathrm{cts}}(\Gamma,V)
2949 \]
2950 is independent of the continuous choices. It vanishes if and only if $\rho_n$ has a \emph{continuous} lift to $\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is
2951 \[
2952  \delta b=-c_A,\qquad b:\Gamma\to V\text{ continuous and normalized},
2953 \]
2954 and the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\mathrm{cts}}(\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\mathrm{cts}}$ and $H^0_{\mathrm{cts}}$, respectively.
2955 \end{theorem}
2956 \begin{proof}
2957 The associativity and square-zero computations in \cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.
2958 \end{proof}
2959 
2960 \begin{theorem}[Source refinement and inflation]\label{thm:source-inflation}
2961 Choose an open normal $U\subseteq\ker\rho_n$ and compute $c_A$ on $Q=\Gamma/U$. Then
2962 \[
2963  o_n^{\mathrm{cts}}(\rho_n)=0
2964  \quad\Longleftrightarrow\quad
2965  \exists\, U'\triangleleft_{\mathrm{open}}\Gamma,\ U'\subseteq U:
2966  \operatorname{Inf}_{\Gamma/U}^{\Gamma/U'}[c_A]=0.
2967 \]
2968 Thus a continuous next-level lift exists exactly when a repair exists at \emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\geq0$,
2969 \[
2970  H^j_{\mathrm{cts}}(\Gamma,V)
2971  \cong\underset{U'\subseteq U}{\operatorname{colim}}
2972  H^j(\Gamma/U',V).
2973 \]
2974 \end{theorem}
2975 \begin{proof}
2976 Every continuous map $\Gamma^j\to V$ 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 through $(\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-quotient cochain complexes therefore have the continuous complex as their filtered union, and the same observation about witnesses proves the asserted cohomology colimit. In particular a class vanishes in that colimit exactly when it vanishes at some subsequent stage. Apply \cref{thm:continuous-H2}. This is the finite-module specialization of the cited classical inflation theorem.
2977 \end{proof}
2978 
2979 \begin{example}[An explicit obstruction killed by refining the source]\label{ex:inflation-carry}
2980 Let $p$ be odd, $J\in M_d(\mathbb Z)$ 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 $0\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is
2981 \[
2982  c(a,b)=\left\lfloor\frac{a+b}{p}\right\rfloor J\pmod p.
2983 \]
2984 The adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,
2985 \[
2986  \sum_{t=0}^{p-1}(\delta u)(1,t)=0,
2987  \qquad\sum_{t=0}^{p-1}c(1,t)=J\ne0.
2988 \]
2989 So the finite-quotient class is nonzero. On $C_{p^2}$ put
2990 \[
2991  b(x)=\left\lfloor\frac{x}{p}\right\rfloor J\pmod p
2992  \quad(0\leq x<p^2).
2993 \]
2994 Writing each $x$ as its low digit plus $p$ times its high digit shows
2995 $\delta b=-\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.
2996 \end{example}
2997 
2998 \subsection{Reconstructing a continuous tower}
2999 
3000 \begin{theorem}[Continuous reconstruction from unobstructed stages]\label{thm:continuous-tower}
3001 If $H^2_{\mathrm{cts}}(\Gamma,\operatorname{Ad}\bar\rho)=0$, every given $\rho_n$ with that residual action extends to a continuous representation
3002 \[
3003  \rho:\Gamma\longrightarrow\operatorname{GL}_d(\mathbb Z_p).
3004 \]
3005 Without the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If $\Gamma$ is topologically finitely generated, existence of \emph{some} extension of the fixed $\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.
3006 \end{theorem}
3007 \begin{proof}
3008 The residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \cref{thm:continuous-H2} permits recursive choice of compatible lifts $\rho_m$ for all $m\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because their reductions modulo $p$ are units. Each finite projection is continuous, so the map into the inverse-limit topology is continuous. The projections determine it uniquely.
3009 
3010 For the last assertion, choose $r$ topological generators. A continuous map into a finite group is determined by their images, since the equalizer of two such maps is closed and contains a dense subgroup when the images agree. Thus there are finitely many continuous representations at each level. Form the rooted tree of extensions of $\rho_n$, with edges given by one-step reduction. It is finitely branching. If every level is nonempty, the root has descendants at arbitrarily large depth. At least one child has the same property, since there are only finitely many children. Repeating gives an infinite compatible path. Apply the first limit argument. This compactness argument supplies no numerical stopping bound and no algorithm for computing $H^2_{\mathrm{cts}}$ of an arbitrary source.
3011 \end{proof}
3012 
3013 \begin{theorem}[Unrestricted coefficient lifting for a free pro-$p$ source]\label{thm:free-pro-p-lift}
3014 Let $F_r^{(p)}$ be the free pro-$p$ group on finitely many generators. Every continuous
3015 \[
3016  F_r^{(p)}\longrightarrow\operatorname{GL}_d(R_n)
3017 \]
3018 has a continuous lift to $\operatorname{GL}_d(\mathbb Z_p)$. One may choose arbitrary integral lifts of its generator matrices.
3019 \end{theorem}
3020 \begin{proof}
3021 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 a unique continuous map from this inverse limit. This proves the universal property needed here.
3022 
3023 The residual image $P\subseteq\operatorname{GL}_d(\mathbb F_p)$ is a $p$-group, being a finite quotient of a pro-$p$ group. Let $H$ be its inverse image in $\operatorname{GL}_d(\mathbb Z_p)$. Its quotient modulo $p^m$ is a finite $p$-group: its kernel over $P$ has the filtration by congruence subgroups with additive $M_d(\mathbb F_p)$ subquotients. Therefore $H$ is pro-$p$, including at $p=2$. Arbitrary integral lifts of the generator matrices lie in $H$. The just-proved universal property extends them to a continuous map. Reduction agrees with the input on the generators and therefore everywhere. This does not assert that the lift is faithful or has finite image.
3024 \end{proof}
3025 
3026 \subsection{Continuous powers and the cyclic-centralizer mechanism}
3027 
3028 \begin{lemma}[When an integral matrix has continuous procyclic powers]\label{lem:procyclic-powers}
3029 Let $A\in\operatorname{GL}_d(\mathbb Z_p)$. The map $m\mapsto A^m$ on $\mathbb Z$ extends uniquely to a continuous homomorphism $\mathbb Z_p\to\operatorname{GL}_d(\mathbb Z_p)$ 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 $\mathbb Z_p^r$.
3030 \end{lemma}
3031 \begin{proof}
3032 Necessity follows because a continuous image of $\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\in I+pM_d(\mathbb Z_p)$. For $B\in I+p^sM_d(\mathbb Z_p)$ with $s\geq1$, binomial expansion shows $B^p\in I+p^{s+1}M_d(\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\geq s+1$, so this estimate remains valid. It follows that
3033 \[
3034  A^{p^{t+j}}\in I+p^{j+1}M_d(\mathbb Z_p).
3035 \]
3036 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 continuously from the dense subgroup $\mathbb Z$ to $\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.
3037 \end{proof}
3038 
3039 \begin{lemma}[Cyclic centralizers over a coefficient ring]\label{lem:cyclic-centralizer}
3040 Let $R$ be a commutative ring, $B\in M_d(R)$, and suppose a vector $v$ makes
3041 \[
3042  Q=(v,Bv,\ldots,B^{d-1}v)\in\operatorname{GL}_d(R).
3043 \]
3044 Then every matrix commuting with $B$ is uniquely $f(B)$ for a polynomial $f\in R[t]$ of degree less than $d$. The coefficients of $f$ are the entries of $Q^{-1}Mv$. In particular, the centralizer is commutative.
3045 
3046 Over $R=\mathbb Z/p^s$, it is enough that $\bar B$ have a cyclic vector over $\mathbb F_p$. Every nonscalar $2\times2$ matrix over a field has such a vector.
3047 \end{lemma}
3048 \begin{proof}
3049 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 $p^s$.
3050 
3051 Finally, if a $2\times2$ 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 $\mathbb F_2$.
3052 \end{proof}
3053 
3054 \subsection{A complete normalized-commutator obstruction}
3055 
3056 Here the source is $\Gamma=\mathbb Z_p^r$, where $r\geq1$ is finite. Write the generator matrices of the given continuous representation $\rho_n$ as $A_1,\ldots,A_r$. They commute in $M_d(R_n)$ and each has $p$-power order modulo $p$.
3057 
3058 Choose $0\leq a<n$ and integer scalars $\lambda_i$ such that
3059 \[
3060  A_i\equiv\lambda_i I\pmod{p^a}\quad\text{for every }i.
3061 \]
3062 There is no restriction in this display when $a=0$. Define the \emph{normalized matrices}
3063 \[
3064  B_i=\frac{A_i-\lambda_i I}{p^a}
3065  \quad\in M_d(R_{n-a}).
3066 \]
3067 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
3068 \[
3069  \boxed{\Omega_{a,*}(\rho_n)=
3070  \bigl([B_*,B_i]\bigr)_{i=1}^r\in M_d(R_{n-a})^r,}
3071  \qquad [B,C]=BC-CB.
3072 \]
3073 
3074 \begin{theorem}[Normalized cyclic lifting criterion]\label{thm:normalized-cyclic-lift}
3075 Under the preceding hypotheses,
3076 \[
3077  \boxed{\rho_n\text{ has a continuous integral lift}
3078  \quad\Longleftrightarrow\quad\Omega_{a,*}(\rho_n)=0.}
3079 \]
3080 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
3081 \[
3082  \widehat A_i=\lambda_i I+p^a\widehat f_i(\widehat B),
3083  \qquad
3084  \rho(x_1,\ldots,x_r)=\prod_i\widehat A_i^{x_i}.
3085 \]
3086 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.
3087 \end{theorem}
3088 \begin{proof}
3089 Suppose first that $\widehat A_i$ are any commuting integral lifts. Then
3090 $\widehat B_i=(\widehat A_i-\lambda_iI)/p^a$ are integral. The identities
3091 \[
3092  0=[\widehat A_*,\widehat A_i]
3093    =p^{2a}[\widehat B_*,\widehat B_i]
3094 \]
3095 imply $[\widehat B_*,\widehat B_i]=0$, since matrices over $\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.
3096 
3097 Conversely, the cyclic-centralizer lemma supplies the unique polynomials $f_i$ from the vanishing commutators. Their integral polynomial evaluations commute exactly. Multiplication by $p^a$ shows that $\widehat A_i$ reduces to $A_i$ modulo $p^n$. Each is invertible because its reduction modulo $p$ is invertible. That reduction has $p$-power order by the input's continuity. The procyclic-powers lemma now supplies the continuous representation, whose reduction agrees on the topological generators and hence on all of $\mathbb Z_p^r$. In rank one the same argument is simply scalar lifting. Polynomial coefficients and a cyclic vector give a finite positive certificate; a nonzero entry of a displayed commutator gives a finite negative certificate under the verified cyclic hypothesis.
3098 \end{proof}
3099 
3100 \begin{remark}[What normalization reveals]
3101 The input relations only force
3102 \[
3103  p^{2a}[B_i,B_j]\equiv0\pmod{p^n}.
3104 \]
3105 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 matrix is claimed independently of these data.
3106 \end{remark}
3107 
3108 \begin{corollary}[A source-fixed finite horizon in the cyclic stratum]\label{cor:abelian-cyclic-horizon}
3109 Under the hypotheses of \cref{thm:normalized-cyclic-lift}, the following are equivalent:
3110 \[
3111  \rho_n\text{ lifts over }\mathbb Z_p,
3112  \qquad \rho_n\text{ extends over }R_{n+a},
3113  \qquad \Omega_{a,*}(\rho_n)=0.
3114 \]
3115 The extension remains a representation of the same $\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.
3116 \end{corollary}
3117 \begin{proof}
3118 An integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\lambda_iI$ 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
3119 $p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies
3120 $[B_*',B_i']=0$ modulo $p^{n-a}$, so $\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.
3121 \end{proof}
3122 
3123 \subsection{Rank two: a complete decision from the given precision}
3124 
3125 For $A\in M_2(R_n)$ define its scalar depth by
3126 \[
3127  a(A)=\max\{0\leq t\leq n:A\text{ is scalar modulo }p^t\}.
3128 \]
3129 Writing entries in $R_n$, this is the minimum of the capped valuations of
3130 $A_{12},A_{21},A_{11}-A_{22}$. It is unchanged by invertible conjugation. For a tuple put $a=\min_i a(A_i)$.
3131 
3132 \begin{theorem}[Complete rank-two profinite lifting test]\label{thm:rank-two-profinite}
3133 Let $p$ be any prime and let $\rho_n:\mathbb Z_p^r\to\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.
3134 \begin{enumerate}
3135 \item If $a=n$, all the $A_i$ are scalar and $\rho_n$ has a continuous integral lift.
3136 \item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then
3137 \[
3138  \boxed{\rho_n\text{ lifts over }\mathbb Z_p
3139  \quad\Longleftrightarrow\quad
3140  [B_*,B_i]=0\text{ in }M_2(R_{n-a})\ (1\leq i\leq r).}
3141 \]
3142 The test is finite matrix arithmetic on the \emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.
3143 \end{enumerate}
3144 \end{theorem}
3145 \begin{proof}
3146 If all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\mathbb F_p^\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.
3147 
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 \cref{lem:cyclic-centralizer}. Thus every hypothesis of \cref{thm:normalized-cyclic-lift} 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 $2\times2$ 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 be canonical.
3149 \end{proof}
3150 
3151 \begin{corollary}[A sharp uniform rank-two horizon]\label{cor:rank-two-sharp-horizon}
3152 For every $p,n,r$ as above,
3153 \[
3154  \boxed{\rho_n\text{ lifts continuously over }\mathbb Z_p
3155  \quad\Longleftrightarrow\quad
3156  \rho_n\text{ extends continuously to }R_{2n-1}.}
3157 \]
3158 For $n\geq2$ and $r\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\mathbb Z_p^r$ over $\mathbb F_p$ lift.
3159 \end{corollary}
3160 \begin{proof}
3161 For a nonscalar tuple, $a\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.
3162 
3163 For sharpness fix $n\geq2$ and put
3164 \[
3165  A=I+p^{n-1}E_{12},\qquad B=I+p^{n-1}E_{21}.
3166 \]
3167 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
3168 $\widetilde A=A+p^nP$, $\widetilde B=B+p^nQ$, and
3169 \[
3170  \widetilde A\widetilde B-\widetilde B\widetilde A
3171  \equiv p^{2n-2}
3172  \begin{pmatrix}1&0\\0&-1\end{pmatrix}
3173  \not\equiv0\pmod{p^{2n-1}}.
3174 \]
3175 Indeed every correction term has valuation at least $2n-1$. This proves nonextension at $2n-1$ for every prime, including $2$, and hence nonexistence of an integral lift. Add identity generators for $r>2$. No smaller uniform coefficient level can distinguish these inputs from extendable ones by mere existence of an extension.
3176 \end{proof}
3177 
3178 \begin{example}[The first hidden commutator]\label{ex:abelian-tangent}
3179 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, \cref{thm:rank-two-profinite} gives
3180 \[
3181  \rho_2\text{ lifts over }\mathbb Z_p
3182  \quad\Longleftrightarrow\quad
3183  [X_i,X_j]=0\text{ in }M_2(\mathbb F_p)\text{ for all }i,j.
3184 \]
3185 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 $\mathbb Z_p^2$. On the free pro-$p$ source the same pair \emph{does} lift, by \cref{thm:free-pro-p-lift}. The surviving obstruction is therefore an actual source relation, not finite-order torsion imposed by an incidental quotient.
3186 \end{example}
3187 
3188 \begin{example}[A positive repair that changes only unresolved digits]\label{ex:abelian-positive-repair}
3189 Work at $p=3$, $n=3$. Take
3190 \[
3191  A_1=\begin{pmatrix}1&3\\9&1\end{pmatrix},\qquad
3192  A_2=\begin{pmatrix}7&12\\9&7\end{pmatrix}\pmod{27}.
3193 \]
3194 They commute modulo $27$ and both are the identity modulo $3$. Choose $a=1$, $\lambda_1=\lambda_2=1$. Then
3195 \[
3196  B_1=\begin{pmatrix}0&1\\3&0\end{pmatrix},\qquad
3197  B_2=\begin{pmatrix}2&4\\3&2\end{pmatrix}\pmod9,
3198  \qquad B_2=2I+4B_1\pmod9.
3199 \]
3200 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 $\operatorname{diag}(-81,81)$. The criterion nevertheless passes. Retain $B_1$ integrally and replace $B_2$ by the exact polynomial $2I+4B_1$. This constructs
3201 \[
3202  \widehat A_1=\begin{pmatrix}1&3\\9&1\end{pmatrix},\qquad
3203  \widehat A_2=\begin{pmatrix}7&12\\36&7\end{pmatrix}.
3204 \]
3205 They commute exactly, remain units over $\mathbb Z_3$, and have the original reduction modulo $27$. Their continuous powers give the integral representation. This is an explicit non-destructive repair of a tuple, not a claim that its initial arbitrary integer representatives were already an exact action.
3206 \end{example}
3207 
3208 \subsection{The repaired scope of the two-axis calculus}
3209 
3210 There is a necessary variance distinction. Let $\mathcal Q_\Gamma$ have open normal subgroups as objects and an arrow $U\to U'$ when $U'\subseteq U$. Write $\mathscr P_d(n,U)$ for the groupoid of rank-$d$ representations of $\Gamma/U$ on $R_n^d$, with invertible intertwining matrices as arrows.
3211 
3212 \begin{proposition}[The source--coefficient diagram has mixed variance]\label{prop:mixed-source-variance}
3213 Reduction of coefficients and inflation of source define a functor
3214 \[
3215  \mathscr P_d:\mathbb N_{\geq1}^{op}\times\mathcal Q_\Gamma
3216  \longrightarrow\mathbf{Gpd}.
3217 \]
3218 Each source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.
3219 \end{proposition}
3220 \begin{proof}
3221 Coefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\Gamma/U'\to\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\Gamma/U'$ descends through $\Gamma/U$ only if the subgroup $U/U'$ acts trivially. This fails for general representations, as the explicit repaired action in \cref{ex:inflation-carry} shows at its higher coefficient level.
3222 \end{proof}
3223 
3224 Coefficient refinement and source refinement do different jobs. The former improves arithmetic accuracy. The latter permits more of the source to act, and can kill an inflated finite-quotient obstruction without deleting coefficient information. Neither permits violating a relation of the fixed source itself. In particular these two resolution coordinates cannot both be inserted into the earlier coarsening-only convention without recording their opposite variance or restricting to descending objects.
3225 
3226 The constructive abelian theorems avoid an unbounded quotient search altogether on their stated strata. Scalar-depth normalization reveals the missing relation, the cyclic centralizer supplies a finite polynomial certificate, and continuous matrix powers reconstruct the action. In rank two this yields a complete test at every coefficient precision and every finite source rank. In larger rank it requires the stated cyclic normalized generator. Free pro-$p$ sources furnish a contrasting unrestricted coefficient-lifting case because they impose no additional relations on the generators.
3227 
3228 These results do not imply a universal scalar-depth test in higher rank, an unconstrained theorem for all profinite groups, or a lift preserving extra arithmetic deformation conditions. The constructions make choices of higher matrix coefficients and may change determinants or other invariants not included among the equations. Continuous Galois representations with local geometric conditions and automorphic existence remain separate targets. The next substantive extension is a specified noncommutative profinite relation system with controlled centralizers or a verified deformation complex, rather than an assumption that every vanishing selected diagnostic gives a global object.
3229 
3230 
3231 \section{Local-to-global reconstruction}
3232 
3233 \subsection{A \v{C}ech torsor residual as a nonstable obstruction test bed}
3234 
3235 The stable Verdier quotient is not the only setting in which the residual/repair/obstruction pattern appears. A classical torsor gives a minimal nonstable model.
3236 
3237 \begin{theorem}[Torsor residual obstruction]\label{thm:torsor-obstruction}
3238 Let $\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\mathcal A$-torsor, and let $\{U_i\to U\}$ be a cover on which $P$ admits local sections $s_i\in P(U_i)$. On overlaps there are unique elements
3239 \[
3240 a_{ij}\in\mathcal A(U_{ij})
3241 \]
3242 such that
3243 \[
3244 s_j=s_i+a_{ij}.
3245 \]
3246 Then:
3247 \begin{enumerate}
3248     \item $a=(a_{ij})$ is a \v{C}ech $1$-cocycle;
3249     \item changing the local sections by a $0$-cochain $b=(b_i)$,
3250     \[
3251     s_i'=s_i+b_i,
3252     \]
3253     changes the residual by the \v{C}ech coboundary
3254     \[
3255     a'_{ij}=a_{ij}+b_j-b_i;
3256     \]
3257     \item the class
3258     \[
3259     \Omega(P;\{U_i\})=[a]\in\check H^1(\{U_i\},\mathcal A)
3260     \]
3261     is independent of the chosen local sections;
3262     \item $\Omega(P;\{U_i\})=0$ 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 $U$.
3263 \end{enumerate}
3264 \end{theorem}
3265 
3266 \begin{proof}
3267 On a triple overlap,
3268 \[
3269 s_k=s_j+a_{jk}=s_i+a_{ij}+a_{jk},
3270 \]
3271 while also $s_k=s_i+a_{ik}$. Freeness of the torsor action gives
3272 \[
3273 a_{ij}+a_{jk}=a_{ik},
3274 \]
3275 which is the \v{C}ech cocycle identity.
3276 
3277 If $s_i'=s_i+b_i$, then
3278 \[
3279 s_j'=s_j+b_j=s_i+a_{ij}+b_j
3280 =s_i'+(a_{ij}+b_j-b_i),
3281 \]
3282 so the residual changes by the coboundary $\delta b$. Hence its cohomology class is independent of the chosen local sections.
3283 
3284 If $[a]=0$, choose $b$ so that $a+\delta b=0$. The repaired sections $s_i'=s_i+b_i$ then agree on every overlap, hence glue by the sheaf property to a global section of $P$. Conversely, if a global section exists, its restrictions have zero transition residual. Comparing those restrictions with the original $s_i$ shows that $a$ differs from zero by a coboundary, so $[a]=0$.
3285 \end{proof}
3286 
3287 \status{proved directly from torsor descent and \v{C}ech cohomology}
3288 
3289 For the standard torsor interpretation see \cite{StacksTorsors}.
3290 
3291 In Komorebi language,
3292 \[
3293 \boxed{
3294 \text{overlap residual}
3295 \;/\;
3296 \text{local repair}
3297 \;=\;
3298 \text{global obstruction class}.}
3299 \]
3300 Here the residual is $a_{ij}$, admissible repair is the change $s_i\mapsto s_i+b_i$, repairable residual cocycles are coboundaries, and the obstruction is the cohomology class $[a]$. This local repair relation is not being identified with the monoid of exact endofunctors in the stable quotient construction. This example is important because it realizes the proposed obstruction pattern before stabilization and shows that the abstract vocabulary recovers an established local-to-global obstruction mechanism rather than manufacturing one artificially.
3301 It is specifically an obstruction to reconstructing a global \emph{section} (equivalently, to trivializing the torsor), not an obstruction to the existence of the torsor itself.
3302 
3303 \subsection{Descent data}
3304 
3305 Let $(\mathcal C,J)$ be a site and let $\mathscr K$ assign a Komorebi category to each object of the site, with restriction functors on overlaps.
3306 
3307 Given a cover $\{U_i\to U\}$, local objects $X_i\in\mathscr K(U_i)$ carry descent isomorphisms
3308 \[
3309 \phi_{ij}:X_i|_{U_{ij}}\xrightarrow{\sim}X_j|_{U_{ij}}
3310 \]
3311 satisfying the cocycle condition on triple overlaps.
3312 
3313 \begin{definition}[Effective descent]
3314 Fix a cover and its complete \v{C}ech diagram, including all iterated overlaps. Effective descent means that the natural functor from the global category to the category of descent data is an equivalence. Essential surjectivity supplies global objects. Full faithfulness supplies descent of morphisms. Uniqueness of object classes alone is insufficient.
3315 \end{definition}
3316 
3317 \begin{theorem}[Descent transfer lemma]\label{thm:descent-transfer}
3318 Let $A^\bullet,B^\bullet:\Delta\to\Cat_\infty$ be small \v{C}ech descent diagrams for a fixed cover. Suppose $\Lambda^\bullet:A^\bullet\to B^\bullet$ is a natural transformation that is an equivalence at every degree, including every overlap level. Then
3319 \[
3320 \lim_{[n]\in\Delta} A^n\simeq\lim_{[n]\in\Delta} B^n.
3321 \]
3322 For ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.
3323 \end{theorem}
3324 
3325 \begin{proof}
3326 Equivalences in a functor category are detected objectwise. Hence $\Lambda^\bullet$ is an equivalence of diagrams. Applying the limit functor preserves this equivalence. The ordinary category formulation is the same argument with coherent descent categories and their quasi-inverses.
3327 \end{proof}
3328 
3329 \begin{corollary}[Global reconstruction transfer]\label{cor:global-transfer}
3330 If the maps from the two global categories to these descent categories are equivalences, then the induced equivalence of descent categories yields an equivalence of global categories.
3331 \end{corollary}
3332 
3333 This is the abstract form of the principle:
3334 \[
3335 \boxed{
3336 \text{local equivalence}
3337 +
3338 \text{restriction compatibility}
3339 +
3340 \text{effective descent}
3341 \Longrightarrow
3342 \text{global equivalence}.
3343 }
3344 \]
3345 
3346 \section{Seeded local-global reciprocity blueprint}
3347 
3348 The previous section suggests a precise architecture for Langlands-facing work without claiming that the arithmetic hypotheses have been discharged.
3349 
3350 Let
3351 \[
3352 \mathbb K_{\mathrm{arith}}
3353 \qquad\text{and}\qquad
3354 \mathbb K_{\mathrm{aut}}
3355 \]
3356 be arithmetic and automorphic seeded two-axis calculi.
3357 
3358 A strong correspondence should include:
3359 \begin{align*}
3360 \Lambda(TX)&\simeq T'\Lambda(X),\\
3361 \Lambda(CX)&\simeq C'\Lambda(X),\\
3362 \Lambda(\Delta_{T,C}X)&\simeq\Delta_{T',C'}\Lambda(X),\\
3363 \bar\Lambda(\Obs X)&\simeq\Obs'\Lambda(X).
3364 \end{align*}
3365 
3366 Here $\bar\Lambda$ is the induced functor on the designated obstruction quotients. The last two requirements are stronger than matching ordinary local invariants. They demand that the correspondence preserve the structure of failed exchange and failed reconstruction.
3367 
3368 \begin{conjecture}[Seeded reciprocity principle]\label{conj:seeded-reciprocity}
3369 Suppose arithmetic and automorphic Komorebi systems satisfy:
3370 \begin{enumerate}
3371     \item local equivalence at every required place and resolution;
3372     \item compatibility with resolution restriction;
3373     \item preservation of the relevant local invariants;
3374     \item transport of mixed residual and obstruction data;
3375     \item effective descent on both sides;
3376     \item a global separation principle ensuring uniqueness from sufficiently rich local data.
3377 \end{enumerate}
3378 Then the induced local equivalence extends to an equivalence of the corresponding admissible global categories.
3379 \end{conjecture}
3380 
3381 Under categorical formulations strong enough to make items (1)--(5) an equivalence of descent systems, the conclusion follows from \cref{cor:global-transfer}. The research content lies in proving that the arithmetic and automorphic systems really instantiate those hypotheses.
3382 
3383 \paragraph{Interpretation.}
3384 This reframes a possible route to local-global reciprocity as:
3385 \[
3386 \boxed{
3387 \text{identify the globalization obstruction}
3388 \quad\text{and prove that local correspondence transports it.}
3389 }
3390 \]
3391 If
3392 \[
3393 \Lambda_*(\Omega_{\mathrm{arith}})=\Omega_{\mathrm{aut}},
3394 \]
3395 and $\Lambda_*$ 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 stated global conclusion. Neither a sheaf/stack model for arbitrary local Langlands data nor automorphic existence follows from the abstract categorical lemmas. Places of a number field are not automatically a covering family for an effective descent problem. Those arithmetic identifications remain obligations.
3396 
3397 \section{Toy models}
3398 
3399 \subsection{\texorpdfstring{$p$}{p}-adic congruence resolution}
3400 
3401 Let the resolution level be $n\ge1$ and take
3402 \[
3403 X_n=\mathbb Z/p^n\mathbb Z.
3404 \]
3405 For $m\le n$, coarsening is reduction
3406 \[
3407 C_{n\to m}:\mathbb Z/p^n\mathbb Z\to\mathbb Z/p^m\mathbb Z.
3408 \]
3409 For any polynomial $f\in\mathbb Z[x]$, the induced transformation satisfies
3410 \[
3411 C_{n\to m}(f(x))
3412 =
3413 f(C_{n\to m}(x)).
3414 \]
3415 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.
3416 This is a simple instance where arithmetic transformation is resolution-natural.
3417 
3418 By contrast, operations involving division by $p$ need not descend uniformly. Their domain of definition and lift multiplicity depend on valuation. This gives a natural source of nontrivial resolution obstruction.
3419 
3420 \subsection{Dyadic averaging}
3421 
3422 Let a fine state be
3423 \[
3424 x=(x_1,x_2)\in\mathbb R^2
3425 \]
3426 and let coarse resolution record only the average
3427 \[
3428 C(x)=\frac{x_1+x_2}{2}.
3429 \]
3430 Let the fine transformation be
3431 \[
3432 A_1(x_1,x_2)=(ax_1,bx_2)
3433 \]
3434 and the coarse transformation be multiplication by $c$:
3435 \[
3436 A_0(y)=cy.
3437 \]
3438 Then
3439 \[
3440 C(A_1x)-A_0(Cx)
3441 =
3442 \frac{(a-c)x_1+(b-c)x_2}{2}.
3443 \]
3444 Choosing
3445 \[
3446 c=\frac{a+b}{2}
3447 \]
3448 gives
3449 \[
3450 C(A_1x)-A_0(Cx)
3451 =
3452 \frac{a-b}{4}(x_1-x_2).
3453 \]
3454 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 exposing structure invisible at coarse resolution.
3455 
3456 \subsection{Exact symmetry under coarsening}
3457 
3458 For dyadic vectors, pairwise averaging commutes with reversal up to reversal of the coarse vector. This gives an exact exchange law and hence a zero mixed residual. The contrast with the previous example illustrates that residuals depend jointly on the transformation and the resolution operator.
3459 
3460 \section{Derived invariants for the zoo}
3461 
3462 The two-axis formalization suggests new quantities that can be defined without changing the underlying category theory.
3463 
3464 \subsection{Commutation depth}
3465 
3466 Fix $X\in\Tcat_s$. First record the exact commutation profile
3467 \[
3468 \mathcal E_A(X;s)=\{r\le s:\Delta^A_{s,r}(X)\simeq0\}.
3469 \]
3470 No interval or monotonicity property is assumed. For $R=\mathbb N$ and finite $s$, a well-defined consecutive coarsening depth is
3471 \[
3472 \kappa_A(X;s):=\max\{k\in\{0,\ldots,s\}:\Delta^A_{s,r}(X)\simeq0
3473 \text{ for all }s-k\le r\le s\}.
3474 \]
3475 The set is nonempty because $\Delta^A_{s,s}(X)\simeq0$. 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 identity level $r=s$ is always exact.
3476 
3477 \subsection{Residual complexity}
3478 
3479 When a numerical size function $\|\cdot\|$ is available on the fiber, define
3480 \[
3481 \mathcal E_A(s,r;X):=\|\Delta^A_{s,r}(X)\|.
3482 \]
3483 Possible choices include dimension, rank, norm, entropy, categorical amplitude, support size, or proof-complexity measures.
3484 
3485 \subsection{Resolution branching profile}
3486 
3487 For a chain $r_0<r_1<\cdots<r_n$ and chosen lifts $X_i\in\Tcat_{r_i}$ with $C_{r_{i+1}\to r_i}X_{i+1}\cong X_i$, record
3488 \[
3489 B_X=(b_{r_0\to r_1}(X_0),\ldots,b_{r_{n-1}\to r_n}(X_{n-1})).
3490 \]
3491 This profile measures how hidden multiplicity emerges under refinement.
3492 
3493 \subsection{Zoo growth spectrum}
3494 
3495 Instead of a single growth function, separate growth by generator type:
3496 \[
3497 g_T(n),\qquad g_R(n),\qquad g_\Delta(n).
3498 \]
3499 This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.
3500 
3501 \section{Candidate axioms for a foundational version}
3502 
3503 The present note derives its first results from ordinary category and order theory. A stronger foundational program would isolate Komorebi-specific axioms.
3504 
3505 \begin{axiom}[Two-axis presentation]
3506 Every admissible structured state is represented in a system
3507 \[
3508 \pi:\K\to\Rcat
3509 \]
3510 whose fibers encode transformations and whose base encodes resolution.
3511 \end{axiom}
3512 
3513 \begin{axiom}[Resolution compositionality]
3514 Coarsening is functorial:
3515 \[
3516 C_{t\to r}=C_{s\to r}C_{t\to s}.
3517 \]
3518 \end{axiom}
3519 
3520 \begin{axiom}[Exchange data]
3521 Every designated transformation family carries a coherent comparison
3522 \[
3523 A_rC_{s\to r}\Longrightarrow C_{s\to r}A_s.
3524 \]
3525 \end{axiom}
3526 
3527 \begin{axiom}[Residual closure]
3528 Every noninvertible exchange comparison generates an admissible residual object that may enter the seed zoo.
3529 \end{axiom}
3530 
3531 \begin{axiom}[Effective reconstruction, domain-specific]
3532 For a specified class of local systems, vanishing global obstruction is equivalent to effective global reconstruction.
3533 \end{axiom}
3534 
3535 These are axioms for a class of structures, not proposed additional axioms of ZFC. Two-axis existence alone imposes no substantive restriction: every category $\mathcal C$ has the degenerate presentation over the one-object resolution category with fiber $\mathcal C$. A foundational claim needs nontrivial, semantically specified resolutions and a theorem relating them to the intended mathematics.
3536 
3537 The last axiom should not be imposed universally without justification. In serious applications it should be a theorem of a sheaf, stack, patching, or descent theory.
3538 
3539 \section{Machine-native semantics}
3540 
3541 The seed calculus can be interpreted as a search grammar. A machine state may contain
3542 \[
3543 \mathsf S=
3544 (\Seeds,\mathscr T,\mathscr R,\Delta,\Omega,\mathsf{Cert}).
3545 \]
3546 For a target problem $P$, search can occur jointly over transformation and resolution:
3547 \[
3548 (T^\ast,R^\ast)
3549 \in
3550 \operatorname*{arg\,min}_{T,R}
3551 \mathcal C\bigl(\Omega(T,R;P)\bigr),
3552 \]
3553 where $\mathcal C$ 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.
3554 
3555 This changes the computational question from
3556 \[
3557 \text{``Which operation solves the problem?''}
3558 \]
3559 to
3560 \[
3561 \boxed{
3562 \text{``At which resolution and in which representation does the obstruction become simplest?''}
3563 }
3564 \]
3565 
3566 A generic discovery loop is:
3567 \[
3568 \boxed{
3569 \text{seed}
3570 \to
3571 \text{generate}
3572 \to
3573 \text{normalize}
3574 \to
3575 \text{measure residual}
3576 \to
3577 \text{promote residual}
3578 \to
3579 \text{expand zoo}
3580 \to
3581 \text{certify}.
3582 }
3583 \]
3584 
3585 Formal proof assistants can serve as the certificate layer without requiring the generative search process itself to be trusted.
3586 
3587 
3588 \section{Universal constructions: syntax and stable residual completion}\label{sec:universal}
3589 
3590 This section addresses the first foundational novelty test. The result is deliberately split into a syntactic theorem, which is standard double-category machinery, and a residual-completion theorem, which is a formal consequence of the free stable envelope of an additive category. The combination gives a rigorous universal construction for the additive/stable version of the seeded two-axis calculus. It does \emph{not} by itself establish that the construction is absent from the literature under another name.
3591 
3592 \subsection{Free seeded two-axis double syntax}
3593 
3594 \begin{definition}[Seeded two-axis double computad]\label{def:twoaxis-computad}
3595 A \emph{seeded two-axis double computad} $\mathbb S$ consists of a small double computad together with:
3596 \begin{enumerate}
3597     \item a distinguished set $\Seeds\subseteq \mathbb S_0$ of seed objects;
3598     \item a designated class of horizontal generating $1$-cells, interpreted as transformation generators;
3599     \item a designated class of vertical generating $1$-cells, interpreted as resolution generators;
3600     \item designated generating $2$-cells, interpreted as exchange or compatibility cells.
3601 \end{enumerate}
3602 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 $\mathbb S$ in a strict double category $\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the target and preservation of those designations.
3603 \end{definition}
3604 
3605 The label data are algebraically passive: they distinguish generators but impose no new equations. Hence the ordinary free-double-category construction applies unchanged.
3606 
3607 \begin{theorem}[Free seeded two-axis double category]\label{thm:free-double}
3608 For every small seeded two-axis double computad $\mathbb S$, there exists a strict double category $\mathbb D_{\mathbb S}$ and a canonical map of seeded double computads
3609 \[
3610 \iota_{\mathbb S}:\mathbb S\longrightarrow U\mathbb D_{\mathbb S}
3611 \]
3612 such that for every strict double category $\mathbb D$, restriction along $\iota_{\mathbb S}$ induces a natural bijection
3613 \[
3614 \operatorname{DblCat}(\mathbb D_{\mathbb S},\mathbb D)
3615 \cong
3616 \operatorname{DblCptd}(\mathbb S,U\mathbb D),
3617 \]
3618 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.
3619 \end{theorem}
3620 
3621 \begin{proof}
3622 Form horizontal and vertical words in the generating arrows. Form square terms from the generating squares and identity squares by every well-typed horizontal and vertical composition. Quotient the terms by the congruence generated by units, associativity, and interchange. These relations give a strict double category. Every interpretation of the generating data extends by evaluation of terms, and evaluation respects the congruence because the target satisfies the same axioms. Its values on generators force uniqueness. Labels merely restrict the allowed generator assignments when corresponding target labels have been specified.
3623 \end{proof}
3624 
3625 \status{proved by translation to standard free double-category machinery}
3626 
3627 This establishes a genuine universal property for the \emph{two-axis syntax}. It is not a novelty claim: free double categories and double computads are established constructions; see Fairbanks--Shulman~\cite{FairbanksShulman2026}.
3628 
3629 \subsection{Additive indexed systems}
3630 
3631 The residual layer becomes canonical after passing to stable categories. Let $\mathbf{Cat}^{\mathrm{add}}$ denote the $\infty$-category of small additive $\infty$-categories and additive functors, and let $\mathbf{Cat}^{\mathrm{st}}$ denote small stable $\infty$-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive $\infty$-category, not merely its homotopy category.
3632 
3633 \begin{definition}[Additive two-axis system]
3634 An \emph{additive two-axis system} over a small resolution category $\Rcat$ is a functor
3635 \[
3636 F:\Rcat^{op}\longrightarrow \mathbf{Cat}^{\mathrm{add}}.
3637 \]
3638 Thus every fiber $F(r)$ is additive and every coarsening functor $C_{s\to r}$ is additive.
3639 \end{definition}
3640 
3641 For a small additive category $\mathcal A$, write $\operatorname{St}^{\mathrm{add}}(\mathcal A)$ for its free stable envelope. We use the established adjunction
3642 \[
3643 \operatorname{St}^{\mathrm{add}}:
3644 \mathbf{Cat}^{\mathrm{add}}
3645 \rightleftarrows
3646 \mathbf{Cat}^{\mathrm{st}}:U,
3647 \]
3648 characterized by
3649 \[
3650 \operatorname{Fun}^{\mathrm{ex}}
3651 \bigl(\operatorname{St}^{\mathrm{add}}(\mathcal A),\mathcal C\bigr)
3652 \simeq
3653 \operatorname{Fun}^{\oplus}(\mathcal A,U\mathcal C)
3654 \]
3655 for every stable $\mathcal C$; see Saunier--Winges~\cite{SaunierWinges2026}.
3656 
3657 \begin{theorem}[Pointwise stable completion]\label{thm:pointwise-stable}
3658 Let $\Rcat$ be small. Postcomposition with $\operatorname{St}^{\mathrm{add}}$ defines a functor
3659 \[
3660 \operatorname{St}^{\mathrm{add}}_{\Rcat}:
3661 \operatorname{Fun}(\Rcat^{op},\mathbf{Cat}^{\mathrm{add}})
3662 \longrightarrow
3663 \operatorname{Fun}(\Rcat^{op},\mathbf{Cat}^{\mathrm{st}}),
3664 \qquad
3665 F\longmapsto \operatorname{St}^{\mathrm{add}}\circ F.
3666 \]
3667 It is left adjoint to pointwise forgetting
3668 \[
3669 U_{\Rcat}:
3670 \operatorname{Fun}(\Rcat^{op},\mathbf{Cat}^{\mathrm{st}})
3671 \longrightarrow
3672 \operatorname{Fun}(\Rcat^{op},\mathbf{Cat}^{\mathrm{add}}).
3673 \]
3674 Equivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence
3675 \[
3676 \operatorname{Map}
3677 \bigl(\operatorname{St}^{\mathrm{add}}_{\Rcat}F,G\bigr)
3678 \simeq
3679 \operatorname{Map}
3680 \bigl(F,U_{\Rcat}G\bigr).
3681 \]
3682 \end{theorem}
3683 
3684 \begin{proof}
3685 Any adjunction $L\dashv U$ induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation
3686 \[
3687 \alpha:L\circ F\Rightarrow G
3688 \]
3689 has pointwise adjoint mates
3690 \[
3691 \alpha_r^{\flat}:F(r)\longrightarrow U(G(r)).
3692 \]
3693 Naturality of the unit and counit implies that the family $\{\alpha_r^{\flat}\}_r$ is natural in $r$. The mate construction is invertible and respects higher coherence. Applying this to $L=\operatorname{St}^{\mathrm{add}}$ gives the result.
3694 \end{proof}
3695 
3696 \status{proved}
3697 
3698 This is the precise sense in which the stable version of the two-axis system has a \emph{free residual-complete envelope}: no extra choices are required to create finite cofibers in the fibers, and every additive map into a stable two-axis system extends essentially uniquely to an exact one.
3699 
3700 \subsection{Exchange data and canonical residuals}
3701 
3702 \begin{definition}[Coherent additive transformation family]
3703 Let $F:\Rcat^{op}\to\mathbf{Cat}^{\mathrm{add}}$. A coherent additive transformation family $A$ consists of additive endofunctors
3704 \[
3705 A_r:F(r)\to F(r)
3706 \]
3707 and, for each $r\le s$, a natural transformation
3708 \[
3709 \eta^A_{s,r}:A_rC_{s\to r}\Longrightarrow C_{s\to r}A_s,
3710 \]
3711 satisfying the identity and transitivity coherences for chains $r\le s\le t$.
3712 \end{definition}
3713 
3714 The universal property above is an equivalence of functor $\infty$-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend $\eta:f\Rightarrow g$ between additive functors $\mathcal A\to\mathcal B$, compose it with $j_{\mathcal B}:\mathcal B\to\operatorname{St}^{\mathrm{add}}(\mathcal B)$ and lift the resulting morphism through
3715 \[
3716 \operatorname{Fun}^{\mathrm{ex}}(\operatorname{St}^{\mathrm{add}}(\mathcal A),\operatorname{St}^{\mathrm{add}}(\mathcal B))
3717 \simeq\operatorname{Fun}^{\oplus}(\mathcal A,\operatorname{St}^{\mathrm{add}}(\mathcal B)).
3718 \]
3719 Full faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations
3720 \[
3721 \widehat A_r,
3722 \qquad
3723 \widehat C_{s\to r},
3724 \qquad
3725 \widehat\eta^A_{s,r}:
3726 \widehat A_r\widehat C_{s\to r}
3727 \Longrightarrow
3728 \widehat C_{s\to r}\widehat A_s.
3729 \]
3730 
3731 \begin{definition}[Universal stable mixed residual]\label{def:universal-residual}
3732 For $X\in\operatorname{St}^{\mathrm{add}}(F(s))$, define
3733 \[
3734 \Delta^A_{s,r}(X)
3735 :=
3736 \Cofib\!\left(
3737 \widehat\eta^A_{s,r,X}
3738 \right)
3739 \in
3740 \operatorname{St}^{\mathrm{add}}(F(r)).
3741 \]
3742 \end{definition}
3743 
3744 \begin{theorem}[Universal residual preservation]\label{thm:residual-preservation}
3745 Let $G:\Rcat^{op}\to\mathbf{Cat}^{\mathrm{st}}$ be a stable two-axis system and let
3746 \[
3747 \Phi:\operatorname{St}^{\mathrm{add}}_{\Rcat}F\longrightarrow G
3748 \]
3749 be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every $r\le s$ and every $X$ there is a canonical equivalence
3750 \[
3751 \Phi_r\bigl(\Delta^A_{s,r}(X)\bigr)
3752 \simeq
3753 \Cofib\!\left(
3754 \Phi_r(\widehat\eta^A_{s,r,X})
3755 \right).
3756 \]
3757 In particular, if the exchange data in $G$ are denoted by $\eta^B$, then
3758 \[
3759 \Phi_r\bigl(\Delta^A_{s,r}(X)\bigr)
3760 \simeq
3761 \Delta^B_{s,r}(\Phi_sX).
3762 \]
3763 \end{theorem}
3764 
3765 \begin{proof}
3766 Each $\Phi_r$ is exact, hence preserves finite colimits and therefore cofiber sequences. Apply $\Phi_r$ to the defining cofiber sequence
3767 \[
3768 \widehat A_r\widehat C_{s\to r}X
3769 \xrightarrow{\widehat\eta^A_{s,r,X}}
3770 \widehat C_{s\to r}\widehat A_sX
3771 \longrightarrow
3772 \Delta^A_{s,r}(X).
3773 \]
3774 The result is a cofiber sequence in $G(r)$. Compatibility of $\Phi$ with the transformation and exchange data identifies its first morphism with the target exchange morphism, giving the stated equivalence.
3775 \end{proof}
3776 
3777 \status{proved}
3778 
3779 Thus residual formation is not merely available in the stable completion: it is \emph{universal under exact realization}.
3780 
3781 \subsection{Residual promotion is forced by stable closure}
3782 
3783 \begin{definition}[Stable seeded zoo]
3784 Fix seed sets $\Seeds_r\subseteq\Ob(F(r))$. A \emph{stable seeded sub-zoo} of $\operatorname{St}^{\mathrm{add}}_{\Rcat}F$ is a family of full stable subcategories
3785 \[
3786 \mathcal Z_r\subseteq\operatorname{St}^{\mathrm{add}}(F(r))
3787 \]
3788 such that:
3789 \begin{enumerate}
3790     \item $\mathcal Z_r$ contains the images of all seeds in $\Seeds_r$;
3791     \item $\widehat A_r(\mathcal Z_r)\subseteq\mathcal Z_r$ for every designated transformation generator;
3792     \item $\widehat C_{s\to r}(\mathcal Z_s)\subseteq\mathcal Z_r$ for every $r\le s$.
3793 \end{enumerate}
3794 The \emph{minimal stable zoo} is the intersection of all such families.
3795 \end{definition}
3796 
3797 \begin{proposition}[Existence of the minimal stable zoo]\label{prop:min-stable-zoo}
3798 The minimal stable zoo exists.
3799 \end{proposition}
3800 
3801 \begin{proof}
3802 The family of entire fibers $\operatorname{St}^{\mathrm{add}}(F(r))$ is a stable seeded sub-zoo, so the defining class is nonempty. Arbitrary intersections of full stable subcategories are again full stable subcategories, and invariance under the designated exact functors is preserved under intersection. Hence the intersection is the least stable seeded sub-zoo.
3803 \end{proof}
3804 
3805 \begin{theorem}[Automatic residual promotion]\label{thm:auto-promotion}
3806 Let $\mathcal Z$ be the minimal stable zoo. If $X\in\mathcal Z_s$, then every mixed residual
3807 \[
3808 \Delta^A_{s,r}(X)
3809 \]
3810 belongs to $\mathcal Z_r$.
3811 \end{theorem}
3812 
3813 \begin{proof}
3814 By invariance,
3815 \[
3816 \widehat A_r\widehat C_{s\to r}X
3817 \quad\text{and}\quad
3818 \widehat C_{s\to r}\widehat A_sX
3819 \]
3820 are objects of $\mathcal Z_r$. Since $\mathcal Z_r$ is stable, it is closed under cofibers. Therefore the cofiber of the exchange morphism between these two objects, namely $\Delta^A_{s,r}(X)$, also lies in $\mathcal Z_r$.
3821 \end{proof}
3822 
3823 \status{proved}
3824 
3825 \begin{corollary}[Redundancy of residual promotion in the stable calculus]\label{cor:promotion-redundant}
3826 In the additive/stable two-axis calculus, ``promote every residual to a seed'' need not be imposed as an independent closure axiom. Once the zoo is stable and closed under transformation and coarsening, mixed residuals are already forced to lie in it.
3827 \end{corollary}
3828 
3829 This is a substantive simplification of the initial axiomatization. Residual promotion remains meaningful operationally for search and provenance, but mathematically it is redundant in the stable closure.
3830 
3831 \subsection{What has and has not been proved}
3832 
3833 Combining \cref{thm:free-double,thm:pointwise-stable,thm:residual-preservation,thm:auto-promotion} gives the following restricted universal construction.
3834 
3835 \begin{theorem}[Additive/stable seeded two-axis completion]\label{thm:additive-stable-completion}
3836 For a small additive indexed two-axis system with coherent additive transformation/exchange data and designated seeds, there is a canonical stable completion with the following properties:
3837 \begin{enumerate}
3838     \item the resolution diagram extends universally from additive fibers to stable fibers;
3839     \item all coarsening and transformation functors become exact;
3840     \item every exchange morphism has a canonical mixed residual given by a cofiber;
3841     \item exact realizations preserve these residuals;
3842     \item the least stable seed zoo exists and is automatically closed under mixed residual formation.
3843 \end{enumerate}
3844 The stable completion has the adjunction universal property for stable resolution systems receiving the original additive resolution diagram; with compatible exchange data, residual preservation is automatic for exact realizations.
3845 \end{theorem}
3846 
3847 \begin{proof}
3848 Items (1)--(2) are \cref{thm:pointwise-stable}; item (3) is \cref{def:universal-residual}; item (4) is \cref{thm:residual-preservation}; item (5) is \cref{prop:min-stable-zoo,thm:auto-promotion}. Initiality is the adjunction of \cref{thm:pointwise-stable}.
3849 \end{proof}
3850 
3851 \status{proved under the explicit additive/stable hypotheses}
3852 
3853 The stronger nonadditive claim remains open. In particular, this theorem does not construct a free \emph{arbitrary} residual-complete double category in which residuals satisfy a universal cofiber property without assuming an additive/stable target. It does, however, close the universal-construction problem for the regime in which the present mixed residual is actually defined as a cofiber.
3854 
3855 
3856 \subsection{Testing residual vanishing on orbit generators}
3857 
3858 The universal completion makes residuals available, but an application still needs a way to prove they vanish. Exactness gives a reduction to generators, with an important qualification: operation-orbit generators, rather than the original seed list alone, generally control the invariant zoo.
3859 
3860 \begin{lemma}[Residual formation is exact in its input]\label{lem:exact-residual-functor}
3861 Let $P,Q:\mathcal C\to\mathcal D$ be exact functors between small stable $\infty$-categories, and let $\eta:P\Rightarrow Q$ be a natural transformation. Then
3862 \[
3863 \Delta_\eta(X):=\Cofib(\eta_X)
3864 \]
3865 is an exact functor $\mathcal C\to\mathcal D$. If $q:\mathcal D\to\mathcal O$ is exact, the subcategory
3866 \[
3867 \{X:q\Delta_\eta(X)\simeq0\}
3868 \]
3869 is thick.
3870 \end{lemma}
3871 
3872 \begin{proof}
3873 Cofibers in the functor category are computed pointwise. The full subcategory of exact functors is closed under these finite colimits: a finite colimit of finite-colimit-preserving functors preserves finite colimits, and its value at zero is zero. Hence $\Delta_\eta$ is exact. The composite $q\Delta_\eta$ is exact as well. Its kernel is closed under shifts, cofibers, and retracts, which proves the last assertion.
3874 \end{proof}
3875 
3876 \begin{definition}[Operation orbit of the seeds]
3877 In a stable indexed system, let $\mathscr P$ be the designated exact transformations and coarsenings. For each $r$, let $\mathcal G_r$ contain every image at level $r$ of a seed under a finite well-typed word in $\mathscr P$, including the empty word. Define
3878 \[
3879 \mathcal Z_r^{\mathrm{thick}}:=\operatorname{Thick}(\mathcal G_r).
3880 \]
3881 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.
3882 \end{definition}
3883 
3884 \begin{proposition}[Orbit generation of the invariant thick zoo]\label{prop:orbit-zoo}
3885 The family $\{\mathcal Z_r^{\mathrm{thick}}\}_r$ is the least family of thick subcategories containing the seeds and preserved by all operations in $\mathscr P$.
3886 \end{proposition}
3887 
3888 \begin{proof}
3889 Every operation sends an orbit generator to an orbit generator at the target level. Exactness and preservation of retracts therefore imply that it sends the thick closure of the source orbit into the thick closure of the target orbit. Thus the displayed family is invariant. Any invariant thick family containing the seeds contains all finite operation words applied to them, hence contains all $\mathcal G_r$ and their thick closures. This proves minimality.
3890 \end{proof}
3891 
3892 \begin{theorem}[Orbit-generator test for an exchange obstruction]\label{thm:generator-test}
3893 Fix an exchange $\eta^A_{s,r}$ between exact functors, and an exact quotient $q_r$ of the target fiber. If
3894 \[
3895 q_r\Delta^A_{s,r}(g)\simeq0
3896 \quad\text{for every }g\in\mathcal G_s,
3897 \]
3898 then
3899 \[
3900 q_r\Delta^A_{s,r}(X)\simeq0
3901 \quad\text{for every }X\in\mathcal Z_s^{\mathrm{thick}}.
3902 \]
3903 The same assertion holds for $q_r=\id$, in which case the exchange is an equivalence on the entire generated thick zoo.
3904 \end{theorem}
3905 
3906 \begin{proof}
3907 By \cref{lem:exact-residual-functor}, the objects on which $q_r\Delta^A_{s,r}$ vanishes form a thick subcategory. By hypothesis it contains $\mathcal G_s$, hence it contains $\operatorname{Thick}(\mathcal G_s)$. Vanishing of the unquotiented cofiber is equivalent to invertibility of the exchange map.
3908 \end{proof}
3909 
3910 \status{proved by exactness and thick generation}
3911 
3912 This reduces a universal statement on a thick zoo to a statement on its operation orbit. It is not automatically a finite algorithm. The orbit may be infinite, and checking only the initial seeds does not suffice unless a separate argument propagates vanishing along every operation. In particular, this theorem does not turn finitely many numerical checks into proof of a universal arithmetic comparison.
3913 
3914 
3915 \section{Research program and theorem ledger}
3916 
3917 \begin{longtable}{>{\raggedright\arraybackslash}p{0.09\textwidth} >{\raggedright\arraybackslash}p{0.47\textwidth} >{\raggedright\arraybackslash}p{0.32\textwidth}}
3918 \toprule
3919 ID & Target & Current status\\
3920 \midrule
3921 \endhead
3922 K1 & Resolution monotonicity and nested equivalence relations & Proved in \cref{thm:resolution-monotonicity}\\
3923 K2 & Grothendieck total category & Proved in \cref{thm:total-category}\\
3924 K3 & Two-axis factorization & Proved in \cref{thm:factorization}\\
3925 K4 & Resolution ultrametric & Proved with a common state set, universal zeroth relation, nesting, and separation.\\
3926 K5 & Least seed zoo / closure existence & Proved in \cref{thm:zoo-existence}\\
3927 K6 & Finitary iterative zoo generation & Proved under continuity hypothesis in \cref{prop:finitary}\\
3928 K7 & Lifting coherent transformation families to the total category & Proved in \cref{thm:lift-total}\\
3929 K8 & Mixed residual vanishing criterion & Proved in stable fibers in \cref{prop:residual-vanish}\\
3930 K9 & Exact two-axis normal form & Proved under local confluence in \cref{thm:normal-form}\\
3931 K10 & Split residual expansion & Corrected source remainder shift and finite principal-path scope in \cref{thm:split-expansion}.\\
3932 K11 & Stable residual filtration without splitness & Proved for a chosen path. General ordered independence is refuted by K24.\\
3933 K12 & Stable obstruction quotient & Proved for the thick kernel. Constructor descent needs K28 or the directed hypothesis K29.\\
3934 K13 & Free seeded two-axis double syntax & Term-construction proof with typed labels in \cref{thm:free-double}.\\
3935 K14 & Universal residual-complete two-axis construction & Proved with additive/stable infinity-category and coherent exchange data. Nonadditive form open.\\
3936 K15 & Effective reconstruction theorem for a nontrivial arithmetic instance & Lattice case proved in K51; the intended reciprocity instantiation remains open.\\
3937 K16 & Descent transfer across a correspondence & Abstract lemma proved in \cref{thm:descent-transfer}\\
3938 K17 & Seeded local-global reciprocity instantiation & Program theorem, arithmetic hypotheses open\\
3939 K18 & Lean formalization of K1--K10 & Not yet implemented in this note\\
3940 K19 & Pointwise stable completion universal property & Proved in \cref{thm:pointwise-stable}\\
3941 K20 & Automatic residual promotion in stable zoos & Proved in \cref{thm:auto-promotion}\\
3942 K21 & Functorial transport of obstruction quotients & Proved in \cref{thm:obstruction-transport}\\
3943 K22 & Nonstable residual/repair obstruction test bed & Proved for abelian torsors in \cref{thm:torsor-obstruction}\\
3944 K23 & Localization of total obstruction & Proved pathwise in \cref{thm:obstruction-localization}.\\
3945 K24 & Ordered filtration independence & General claim refuted in \cref{prop:filtration-no-go,ex:typed-diamond}.\\
3946 K25 & Semantic diamond criterion & Proved in \cref{thm:semantic-diamond}; total cofibers equivalent, not uniquely identified without more data.\\
3947 K26 & Pure exchange event poset & States are ideals and histories are linear extensions, \cref{thm:event-ideals}.\\
3948 K27 & Ideal-indexed residual diagram & Proved for coherent input; K32 constructs strict chain-natural instances.\\
3949 K28 & Constructor-invariant obstruction quotient & Proved in \cref{thm:invariant-quotient}; can be zero, \cref{ex:quotient-collapse}.\\
3950 K29 & Directed operational quotient & Killability is thick and constructor-invariant, \cref{thm:directed-invariant}.\\
3951 K30 & Orbit-generator test for residual vanishing & Proved in \cref{thm:generator-test}; orbit need not be finite.\\
3952 K31 & Noncollapse certificate & Exact equivariant detector criterion, \cref{prop:noncollapse}.\\
3953 K32 & Strict semantic event realization & Proved for typed pure swaps and strict chain-natural cells, \cref{thm:strict-event-realization}.\\
3954 K33 & Interaction residual and excision & Total-cofiber criterion, \cref{thm:interaction-cofiber}.\\
3955 K34 & Context-independent event residuals & Equivalent to vanishing square interactions, \cref{thm:event-context-invariance}.\\
3956 K35 & Principal-ideal reconstruction & Reduced excisive diagrams classified by diagrams on $P$, \cref{thm:principal-reconstruction}.\\
3957 K36 & Universal excisive quotient & Constructor-compatible minimal thick kernel, \cref{thm:excisive-quotient}.\\
3958 K37 & Tensor-cube residual formula & Proved by finite-colimit interchange and exact tensor, \cref{thm:tensor-cube}.\\
3959 K38 & Integral gcd interaction formula & Explicit Koszul change of basis and cohomology, \cref{thm:integral-gcd}.\\
3960 K39 & Prime-overlap kernel & Exact kernel and noncollapse for positive scalar labels, \cref{thm:prime-overlap-quotient}.\\
3961 K40 & Finite-signature arithmetic certificate & All finite event diagrams in the stated scalar grammar, \cref{cor:finite-signature}.\\
3962 K41 & Integral matrix collapse criterion & Verified Smith data classify free collapse versus torsion support, \cref{thm:matrix-collapse}.\\
3963 K42 & Nonidentity arithmetic resolution tower & Derived reduction is exact and conservative, not faithful, \cref{thm:adic-tower}.\\
3964 K43 & Coherent weighted exchange cells & Explicit source of residuals and base-change transport, \cref{thm:weighted-adic-exchange}.\\
3965 K44 & Prime-power interaction cohomology & Full finite-level Koszul cohomology, \cref{thm:prime-power-koszul}.\\
3966 K45 & Nilpotent excision collapse & Nonunit pairs force the prime-power quotient to zero, \cref{thm:nilpotent-collapse}.\\
3967 K46 & Finite prime-component quotient & Exact quotient equivalence and tower coherence, \cref{thm:finite-prime-quotient}.\\
3968 K47 & Based tower reconstruction & Uniform bounded ranks and compatible matrices required, \cref{thm:based-tower-reconstruction}.\\
3969 K48 & Transient negative residual classes & Nonzero finite groups can form a pro-zero system, \cref{prop:transient-koszul}.\\
3970 K49 & Completion--repair comparison defect & Explicit generic-fiber formula and vanishing criterion, \cref{thm:completion-repair-defect}.\\
3971 K50 & Arithmetic fracture reconstruction & Perfect integral complexes and their residuals, \cref{thm:arithmetic-fracture,cor:residual-fracture}.\\
3972 K51 & Effective arithmetic lattice gluing & Existence and uniqueness from specified generic identifications, \cref{thm:lattice-gluing}.\\
3973 K52 & Compatible operator gluing & Preserves supplied integral polynomial relations, \cref{cor:lattice-operators}.\\
3974 K53 & Intrinsic kernel--reduction residual & Natural torsion boundary and transition law, \cref{thm:kernel-bockstein}.\\
3975 K54 & Sharp finite-resolution lifting window & Smith profile, minimal delay, and pro-zero residuals, \cref{thm:precision-window}.\\
3976 K55 & Finite-group invariant residual & Identified with $H^1[p^n]$, with group-order bound, \cref{thm:invariant-bockstein}.\\
3977 K56 & Equivariant morphism lifting & Obstruction and finite-precision isomorphism test for supplied lattices, \cref{thm:intertwiner-lifting}.\\
3978 K57 & Norm-quotient lattice family & Natural non-scalar actions and sharp transient invariants, \cref{thm:norm-quotient}.\\
3979 K58 & Perfect-base derived exchange & Exact commutation, not a new residual, \cref{thm:perfect-invariant-exchange}.\\
3980 K59 & Nonperfect-base intrinsic derived defect & Non-scalar action with unbounded cofiber and unit-loss boundary, \cref{thm:nonflat-invariant-exchange,cor:intrinsic-unit-loss}.\\
3981 K60 & One-step action-lifting obstruction & The finite cyclic action cannot lift from $p^2$ to $p^3$, \cref{thm:one-step-group-lift}.\\
3982 K61 & Exact exchange on coefficient-lifted inputs & Necessary diagnostic; converse refuted by K64, \cref{thm:lifted-input-exchange}.\\
3983 K62 & Complete one-step relation obstruction & $H^2$ class vanishes exactly for a next-level lift, \cref{thm:relation-H2}.\\
3984 K63 & Repair ambiguity and certificates & $H^1$, $H^0$, and exact finite linear witnesses, \cref{prop:repair-H1}.\\
3985 K64 & Blind derived-invariant comparison & Central sign refutes sufficiency of the single exchange cone, \cref{thm:blind-exchange}.\\
3986 K65 & Averaged quadratic repair & Exact associativity and explicit division loss, \cref{lem:quadratic-repair}.\\
3987 K66 & Convergent action repair & Fixed finite group and $m>2v_p(|G|)$, \cref{thm:convergent-action-repair}.\\
3988 K67 & Finite action-lifting horizon & Necessary and sufficient level $\max\{n+e,2e+1\}$, \cref{thm:finite-action-horizon}.\\
3989 K68 & Controlled integral conjugacy & Agreement beyond group-order valuation suffices, \cref{thm:close-action-conjugacy}.\\
3990 K69 & Finite integral action catalogue & Isomorphism classes, not an equivalence of groupoids, \cref{cor:finite-action-catalogue}.\\
3991 K70 & Cyclic relation Jacobian & Affine next-level certificate with $d^2$ variables, \cref{prop:cyclic-jacobian}.\\
3992 K71 & Fixed quotient versus profinite lifting & Explicit continuous lift with growing image, \cref{prop:profinite-quotient-boundary}.\\
3993 K72 & Continuous one-step obstruction & Finite discrete adjoint coefficients and continuous repair, \cref{thm:continuous-H2}.\\
3994 K73 & Source-inflation criterion & Continuous vanishing occurs on some finer quotient, \cref{thm:source-inflation}.\\
3995 K74 & Continuous tower reconstruction & Vanishing $H^2$ or a compatible tower; finite-generation compactness, \cref{thm:continuous-tower}.\\
3996 K75 & Free pro-$p$ coefficient lifting & Arbitrary integral generator lifts extend continuously, \cref{thm:free-pro-p-lift}.\\
3997 K76 & Procyclic matrix powers & Exact residual $p$-power-order criterion, including $p=2$, \cref{lem:procyclic-powers}.\\
3998 K77 & Cyclic-centralizer certificate & Finite polynomial description over a coefficient ring, \cref{lem:cyclic-centralizer}.\\
3999 K78 & Normalized cyclic lifting & Complete obstruction and explicit polynomial reconstruction, \cref{thm:normalized-cyclic-lift}.\\
4000 K79 & Cyclic-stratum horizon & Level $n+a$ suffices under the cyclic hypothesis, \cref{cor:abelian-cyclic-horizon}.\\
4001 K80 & Complete rank-two profinite test & All continuous $\mathbb Z_p^r$ inputs, no extra deformation conditions, \cref{thm:rank-two-profinite}.\\
4002 K81 & Sharp rank-two horizon & Level $2n-1$, optimal for $n,r\geq2$, \cref{cor:rank-two-sharp-horizon}.\\
4003 K82 & Mixed source--coefficient variance & Source inflation is not arbitrary source coarsening, \cref{prop:mixed-source-variance}.\\
4004 \bottomrule
4005 \end{longtable}
4006 
4007 \section{Immediate next formal steps}
4008 
4009 The general ordered-filtration independence target and the converse of the single derived-invariant test remain refuted. Fixed-finite-group lifting, continuous free pro-$p$ lifting, and rank-two lifting for $\mathbb Z_p^r$ now have constructive solutions. Higher-rank abelian lifting is covered on the specified cyclic normalized stratum. Arbitrary profinite relations, geometric local conditions, and automorphic globalization remain separate obligations.
4010 
4011 \begin{enumerate}
4012 \item \textbf{Beyond strict pure exchange.} Extend K32 to a specified non-strict enhanced realization or a rewrite signature with additional critical overlaps. Identify the missing coherent cells explicitly rather than assume that homotopy-category diamonds supply them.
4013 \item \textbf{Noncommutative and constrained lifting.} K72--K82 handle continuous source refinement and give complete rank-two $\mathbb Z_p^r$ reconstruction. Next specify a noncommutative profinite presentation or a noncyclic higher-rank centralizer. Construct its actual relation obstruction and a sufficient repair criterion. Prescribed determinants and geometric local conditions must be added as equations with their own compatibility checks. No uniform averaging bound over growing finite quotients is assumed.
4014 \item \textbf{Arithmetic existence beyond lattices.} K51 closes a concrete lattice case of K15. For the intended K17 application, construct the generic arithmetic or automorphic data and the required identifications, rather than importing them as gluing hypotheses. A globally glued lattice with supplied operators is not an automorphic representation.
4015 \item \textbf{Broader coefficient rings.} Determine which parts of the prime-support kernel and matrix criterion persist over a specified Dedekind ring or a higher-dimensional ring. A replacement for the integral Smith decomposition must be proved, not silently imported.
4016 \item \textbf{Lean core.} Formalize the finite event-poset correspondence, strict naturality argument, closure statements, and integral certificate checks. This manuscript still has no Lean proof package.
4017 \item \textbf{Beyond stability.} Develop another precisely typed nonadditive obstruction example beyond torsor sections. Do not identify a nonabelian obstruction set with a cofiber object without a comparison theorem.
4018 \end{enumerate}
4019 
4020 \section{Brainstorm: stronger directions}
4021 
4022 The formalism above suggests several deeper lines of investigation.
4023 
4024 \subsection{Resolution completion}
4025 A resolution tower may admit an inverse limit
4026 \[
4027 X_\infty\simeq\varprojlim_r X_r.
4028 \]
4029 One can ask whether the seed zoo commutes with completion:
4030 \[
4031 \Zoo(X_\infty)
4032 \stackrel{?}{\simeq}
4033 \varprojlim_r\Zoo(X_r).
4034 \]
4035 This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.
4036 
4037 \subsection{Transformation completion}
4038 Repeated transformations may generate an orbit category. Completing this orbit under limits, colimits, idempotent splitting, or derived closure may produce a canonical transformation hull.
4039 
4040 \subsection{Bi-completion}
4041 There are then two completion operations:
4042 \[
4043 \widehat{(-)}_T,
4044 \qquad
4045 \widehat{(-)}_R.
4046 \]
4047 After both completions and the comparison have been defined in a common category, the mixed question is whether
4048 \[
4049 \widehat{\widehat X_T}_R
4050 \simeq
4051 \widehat{\widehat X_R}_T.
4052 \]
4053 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 \cref{thm:completion-repair-defect}: even compatible finite-level repairs can fail to commute with completion, with a nonzero generic-fiber defect. An unconditional bi-completion claim is therefore false.
4054 
4055 \subsection{Resolution spectrum}
4056 For a fixed $X\in\Tcat_s$ and a specified detection predicate $P_r$ at each level $r\le s$, define:
4057 \[
4058 \operatorname{Spec}_R(P;X)
4059 =
4060 \{r\le s:P_r(C_{s\to r}X)\}.
4061 \]
4062 This could behave like a support or spectrum for distinguishability.
4063 
4064 \subsection{Residual genealogy}
4065 Residual promotion creates a directed graph
4066 \[
4067 s\to\Delta_1\to\Delta_2\to\cdots.
4068 \]
4069 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.
4070 
4071 \subsection{Seed minimality}
4072 Given a generated zoo $\Zoo$, ask for the smallest seed set $\Seeds$ with
4073 \[
4074 \Zoo(\Seeds)=\Zoo.
4075 \]
4076 This is a mathematical compression problem. Nonuniqueness of minimal seeds could itself be meaningful.
4077 
4078 \subsection{Dual zoos}
4079 If the calculus carries a duality $D$, study
4080 \[
4081 D\Zoo(\Seeds)
4082 \quad\text{versus}\quad
4083 \Zoo(D\Seeds).
4084 \]
4085 The residual of dualization against zoo generation measures whether the generative grammar is duality-complete.
4086 
4087 \subsection{Proof-resolution}
4088 A theorem itself may have resolutions. At coarse resolution one stores only a proposition and certificate. At finer resolution one exposes lemmas, dependencies, computational witnesses, and failed branches. This creates a resolution geometry on proofs and may be useful for machine reasoning and proof compression.
4089 
4090 \subsection{World-resolution}
4091 A foundational extension can assign resolution to models or ``worlds'' themselves. Morphisms between worlds then have two axes: change of world and change of resolution within a world. This may connect the current calculus to the earlier compatibility/residual/obstruction maps between models of a foundational theory.
4092 
4093 \section{Provisional philosophical interpretation}
4094 
4095 The mathematics does not require the following interpretation, but the two-axis formalism sharpens it.
4096 
4097 A structure may be characterized simultaneously by:
4098 \begin{itemize}
4099     \item what transformations it admits;
4100     \item what resolutions distinguish its internal states;
4101     \item what survives both kinds of movement;
4102     \item what residuals appear when the two directions fail to commute;
4103     \item what obstructions prevent refinement, transformation, or global reconstruction.
4104 \end{itemize}
4105 
4106 This suggests the strengthened ontological phrase:
4107 \[
4108 \boxed{
4109 \text{to remain identifiable is to remain coherent under both transformation and resolution.}
4110 }
4111 \]
4112 The mathematical content of the project is not this sentence itself. The mathematical content is the attempt to make ``coherent,'' ``transformation,'' ``resolution,'' ``residual,'' and ``reconstruction'' precise enough that such a sentence can be tested rather than merely asserted.
4113 
4114 \section{Conclusion}
4115 
4116 The first formalization supports a concrete picture:
4117 \[
4118 \boxed{
4119 \begin{array}{c}
4120 \text{seeds generate states},\\
4121 \text{transformations generate representations},\\
4122 \text{resolutions generate distinguishability},\\
4123 \text{exchange maps relate the axes},\\
4124 \text{residuals measure failed exchange},\\
4125 \text{residuals become new seeds},\\
4126 \text{obstructions classify persistent failure},\\
4127 \text{descent reconstructs compatible local structure}.
4128 \end{array}
4129 }
4130 \]
4131 The retained categorical core supplies explicit constructions under fixed hypotheses. Stable completion provides functorial cofibers, exact realizations preserve residuals, and the invariant stable zoo contains them automatically. Each normalization path carries a residual filtration, but the filtration is genuinely history-sensitive. The corrected intrinsic object is an ideal-indexed residual diagram for a specified coherent exchange system, whose terminal residual is independent of schedule.
4132 
4133 Likewise, a thick obstruction quotient is not automatically a new environment for the original constructors. Their action on the quotient requires an invariant kernel. The invariant saturation gives the universal repair, but its possible collapse is a substantive constraint on admissibility. Directed common postcomposition makes operational killability itself invariant and thick, while exact equivariant detectors certify noncollapse. Finally, thick orbit generation reduces exchange vanishing to a well-defined generator test without promising that the test is finite.
4134 
4135 The next layer is now explicit. Strict chain-natural cells construct semantic exchange diagrams. Their second cofibers measure excision, and vanishing of those interaction objects makes event residuals independent of unrelated prior events. Principal-ideal reconstruction retains exactly the poset diagram needed to assemble the whole residual history. In the integral models, common prime support identifies what must be removed, while free cofiber components identify a genuine collapse risk. These are concrete semantic and arithmetic results, not a conclusion drawn solely from the breadth of the vocabulary.
4136 
4137 The arithmetic tower sharpens that boundary further. Finite resolution can turn a nonzero nilpotent interaction into a generator of the whole perfect category. Several prime components permit nontrivial excisive quotients, but deleting a component before completion loses a generic fiber. Retaining the localized and completed branches with their overlap instead gives a proved arithmetic fracture square and an effective lattice-gluing theorem. These use established algebraic mechanisms to make the calculus concrete.
4138 
4139 The intrinsic exchange layer now distinguishes degree-zero information loss from a genuine failure between fully derived exact functors. Residual torsion gives a sharp finite lifting window for fixed integral constraints. A nonperfect coefficient change can instead expose an unbounded defect and a genuine missing representation lift. Exact comparison is restored on supplied coefficient-lifted inputs under the stated perfectness hypotheses, without taking a quotient that erases the coefficient unit.
4140 
4141 The sufficient-lifting layer now adds an explicit distinction between detection and construction. The adjoint relation cocycle is a complete next-level obstruction, whereas derived invariants can be blind to nonliftability. For a fixed finite group, quantitative averaging converts a finite high-precision witness into an integral representation and gives controlled uniqueness up to conjugacy. The stated threshold is known in ultrametric stability, and the direct proof here makes its relation to residual repair explicit. It does not justify substituting fixed finite-group existence for a continuous Galois or automorphic existence problem.
4142 
4143 The continuous-source layer now separates coefficient precision from source inflation. A finite-quotient obstruction can disappear when more of the source is allowed to act, but an actual commutativity relation cannot be discarded. Scalar-depth normalization exposes that relation, and a cyclic centralizer constructs compatible integral matrices from finite polynomial data. In rank two the cyclic condition is automatic after normalization, yielding a complete test and the sharp $2n-1$ horizon. This proves reconstruction without eliminating the coefficient category, while the mixed-variance diagram states exactly how the two resolution coordinates interact.
4144 
4145 The results do not establish a new foundation or a reciprocity theorem. The remaining arithmetic content requires actual generic parameters, compatible actions, and the specific existence theorem demanded by the intended correspondence. Neither a torsor trivialization, a reconstructed event diagram, nor a glued integral lattice is an automorphic realization by definition.
4146 
4147 
4148 \begin{thebibliography}{99}
4149 \addcontentsline{toc}{section}{References}
4150 \bibitem{FairbanksShulman2026}
4151 A.~D. Fairbanks and M.~Shulman,
4152 \emph{Doubly Weak Double Categories},
4153 arXiv:2506.23651. Background reference for double computads.
4154 The paper is background for double computads. The present free strict syntax result is proved directly by terms and relations in \cref{thm:free-double}; no unchecked theorem-number identification is needed.
4155 
4156 \bibitem{SaunierWinges2026}
4157 V.~Saunier and C.~Winges,
4158 \emph{On exact categories and their stable envelopes},
4159 arXiv:2502.03408v2, 28 November 2025.
4160 Definition 3.1 and the following universal property give the free stable envelope $\operatorname{St}^{\mathrm{add}}$ and its adjunction with the forgetful functor from stable to additive categories.
4161 \url{https://arxiv.org/abs/2502.03408v2}.
4162 
4163 \bibitem{StacksVerdier}
4164 The Stacks Project Authors,
4165 \emph{Quotients of triangulated categories}, Section 13.6, Tag 05RA, especially Lemmas 13.6.2, 13.6.8, and 13.6.9, accessed 26 September 2026.
4166 The quotient construction, kernel description, and exact universal property are used in \cref{thm:obstruction-quotient,thm:obstruction-transport}; Section 13.3, Tag 05QK, records the octahedral axiom TR4 underlying the homotopy-category form of \cref{thm:stable-filtration}. Functorial cofibers require the stable enhancement fixed in the text.
4167 \bibitem{Drew2015}
4168 B.~Drew,
4169 \emph{Verdier quotients of stable quasi-categories are localizations},
4170 arXiv:1511.08287v1, 26 November 2015, Definition 1.1 and Theorem 1.3.
4171 \url{https://arxiv.org/abs/1511.08287v1}.
4172 
4173 \bibitem{StacksTorsors}
4174 The Stacks Project Authors,
4175 \emph{First cohomology and torsors}, Section 21.4, Tag 03AG, accessed 26 September 2026.
4176 
4177 \bibitem{StacksCones}
4178 The Stacks Project Authors,
4179 \emph{Cones and termwise split sequences}, Section 13.9, Tag 014D,
4180 Definition 13.9.1 and Lemma 13.9.2, checked 26 September 2026.
4181 \url{https://stacks.math.columbia.edu/tag/014D}.
4182 
4183 \bibitem{StacksKoszul}
4184 The Stacks Project Authors,
4185 \emph{The Koszul complex}, Section 15.29, Tag 0621,
4186 Definitions 15.29.1--2 and Lemmas 15.29.4, 15.29.7--8, 15.29.12,
4187 checked 26 September 2026.
4188 The cited section uses homological grading. This manuscript puts the exterior
4189 $k$-th term in cohomological degree $-k$ and states all differentials explicitly.
4190 \url{https://stacks.math.columbia.edu/tag/0621}.
4191 
4192 \bibitem{StacksPerfect}
4193 The Stacks Project Authors,
4194 \emph{Perfect complexes}, Section 15.76, Tag 0656,
4195 Definition 15.76.1 and Lemma 15.76.9, checked 27 September 2026.
4196 \url{https://stacks.math.columbia.edu/tag/0656}.
4197 
4198 \bibitem{StacksDerivedTensor}
4199 The Stacks Project Authors,
4200 \emph{Derived tensor product}, Section 15.60, Tag 06XY,
4201 checked 27 September 2026. K-flat models and the exact derived tensor construction.
4202 \url{https://stacks.math.columbia.edu/tag/06XY}.
4203 
4204 \bibitem{StacksRlim}
4205 The Stacks Project Authors,
4206 \emph{Rlim of abelian groups}, Section 15.88, Tag 07KV,
4207 checked 27 September 2026. The product model and the Mittag--Leffler case of derived inverse limits.
4208 \url{https://stacks.math.columbia.edu/tag/07KV}.
4209 
4210 \bibitem{StacksLocalization}
4211 The Stacks Project Authors,
4212 \emph{Localization}, Section 10.9, Tag 00CM,
4213 checked 27 September 2026. The universal property and exactness on modules.
4214 \url{https://stacks.math.columbia.edu/tag/00CM}.
4215 
4216 \bibitem{StacksDerivedCompletion}
4217 The Stacks Project Authors,
4218 \emph{Derived Completion}, Section 15.93, Tag 091N,
4219 Lemma 15.93.12, checked 27 September 2026. Inverting a defining element before derived completion gives zero.
4220 \url{https://stacks.math.columbia.edu/tag/091N}.
4221 
4222 \bibitem{MilneCFT2020}
4223 J.~S. Milne, \emph{Class Field Theory}, version 4.03, 6 August 2020.
4224 Chapter II, 1.9 (coefficient long exact sequence), 1.18--1.21 (crossed homomorphisms, cyclic $H^1$, and explicit boundaries), and 3.4 (cyclic periodic resolution). Primary source checked 27 September 2026, including PDF pages 70, 75, and 90, corresponding to printed pages 61, 66, and 81.
4225 \url{https://www.jmilne.org/math/CourseNotes/CFT.pdf}.
4226 
4227 \bibitem{FournierFacio2026}
4228 F.~Fournier-Facio, \emph{Ultrametric analogues of Ulam stability of groups},
4229 arXiv:2105.00516v5, version posted 22 July 2026, manuscript dated 23 July 2026.
4230 Lemma 7.3, Proposition 7.4, and Example 7.5, printed pages 54--55. Exact version and displayed hypotheses checked 27 September 2026.
4231 \url{https://arxiv.org/abs/2105.00516v5}.
4232 
4233 
4234 \bibitem{StacksContinuous}
4235 The Stacks Project Authors,
4236 \emph{Tate's continuous cohomology}, Section 59.58, Tag 0DVG.
4237 \url{https://stacks.math.columbia.edu/tag/0DVG}.
4238 Continuous inhomogeneous-cochain conventions checked on 27 September 2026. The new continuous obstruction results use finite discrete coefficient modules, not an identification of arbitrary topological cohomology theories.
4239 
4240 \end{thebibliography}
4241 
4242 \appendix
4243 \addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
4244 \section{Compact definition sheet}
4245 
4246 \begin{align*}
4247 \text{Resolution system:}\quad
4248 &F:\Rcat^{op}\to\Cat.\\[0.4em]
4249 \text{Fiber:}\quad
4250 &\Tcat_r=F(r).\\[0.4em]
4251 \text{Coarsening:}\quad
4252 &C_{s\to r}:\Tcat_s\to\Tcat_r.\\[0.4em]
4253 \text{Total category:}\quad
4254 &\K=\int_{\Rcat}F.\\[0.4em]
4255 \text{Resolution equivalence:}\quad
4256 &X\sim_rY\iff C_{t\to r}X\cong C_{t\to r}Y.\\[0.4em]
4257 \text{Lift set:}\quad
4258 &\Lift_{r\to s}(X)=\{Y:C_{s\to r}Y\cong X\}/\cong.\\[0.4em]
4259 \text{Exchange map:}\quad
4260 &\eta^A_{s,r}:A_rC_{s\to r}\Rightarrow C_{s\to r}A_s.\\[0.4em]
4261 \text{Stable residual:}\quad
4262 &\Delta^A_{s,r}(X)=\Cofib(\eta^A_{s,r,X}).\\[0.4em]
4263 \text{Stable obstruction quotient:}\quad
4264 &\mathcal O_{\mathscr M}=\mathcal Z/\operatorname{Thick}(\mathcal K_{\mathscr M}).\\[0.4em]
4265 \text{Invariant obstruction quotient:}\quad
4266 &\mathcal O^{\mathrm{inv}}_{\mathscr M}=\mathcal Z/\mathcal J_{\mathscr M}.\\[0.4em]
4267 \text{Event residual diagram:}\quad
4268 &\mathcal F_X(I)=\Cofib(X_{\varnothing}\to X_I).\\[0.4em]
4269 \text{Obstruction class:}\quad
4270 &\Obs_{\mathscr M}(X)=q(X).\\[0.4em]
4271 \text{Seed zoo:}\quad
4272 &\Zoo(\Seeds_0)=\mu A.\bigl(\Seeds_0\cup\Phi(A)\bigr).\\[0.4em]
4273 \text{Resolution ultrametric:}\quad
4274 &d_R(X,Y)=2^{-\nu(X,Y)},\qquad 2^{-\infty}=0.\\[0.4em]
4275 \text{Reconstruction target:}\quad
4276 &\Omega_G=0\iff\text{the specified descent datum is effective}.
4277 \end{align*}
4278 
4279 
4280 \section{Preserved audit and verification record for v0.6.0}\label{app:audit}
4281 
4282 This section is a historical audit record. The current continuation is recorded in \cref{app:audit-v0100}; the intervening version records retain their original scopes.
4283 
4284 \subsection{Coverage and evidence}
4285 The v0.5.0 LaTeX source was read in full. This release preserves the running manuscript and revises its statements and proofs rather than replacing it with a shortened summary. The review is an authoring-session self-review, not an independent referee report. The exact source snapshot, change inventory, executable tests, and output are included in the companion package.
4286 
4287 The primary audit verdict for the former package is \emph{correct only after stated restrictions and corrections}. In particular, the ordered-filtration ambition fails in general, whereas the retained path-relative theorem survives. Stable quotient existence survives, but it alone does not prove constructor descent or noncollapse. No Lean verification is claimed.
4288 
4289 \subsection{Mathematical corrections}
4290 \begin{enumerate}
4291 \item \textbf{Resolution:} fixed a common state set and a universal zeroth equivalence relation in the ultrametric construction. Distinguished isomorphism classes of lifts from framed lift groupoids. Excluded $\log 0$ from ordinary branching information.
4292 \item \textbf{Exchange:} distinguished invertible comparison maps from exact functors. Required complete local exchange rules for the syntactic normal-form statement. Replaced the split cofiber identification by the correct shift $\Delta\simeq E[1]$. Removed an unsupported recursive termination claim for residual branches.
4293 \item \textbf{Coherence:} separated a normal object, its comparison map, and a filtration. Noninvertible exchange cells are extended using the full functor-category stable-envelope universal property, not just a mapping-space adjunction.
4294 \item \textbf{Quotients:} distinguished killability, thick killability, and constructor-invariant thick killability. Stated universal uniqueness in the enhanced categorical sense. Corrected the postcomposition orientation in the directed condition.
4295 \item \textbf{Descent and scope:} included all overlap degrees and descent of morphisms. Required reflection as well as preservation of obstruction vanishing. Corrected the quotient functor's type. Kept arithmetic globalization and unrestricted reciprocity outside the proved scope.
4296 \item \textbf{Derived quantities:} replaced an identity-dominated commutation-depth formula and typed the branching profile along an actual lift chain. Marked zoo growth and search costs as presentation-dependent. No minimum or decision procedure is asserted by an optimization display.
4297 \end{enumerate}
4298 
4299 \subsection{Exact finite regression}
4300 The companion script \texttt{checks/check\_models.py} uses integer matrices, rational Gaussian elimination, and finite labelled words. It contains no floating-point tests. The recorded run uses Python 3.13.5 and checks every two-sort word of length $0$ through $10$, including the empty word and words containing only one sort.
4301 
4302 \begin{center}
4303 \begin{tabular}{lr}
4304 \toprule
4305 Quantity & Checked count\\
4306 \midrule
4307 Initial words & 2,047\\
4308 Reachable word states, summed over initial words & 82,499\\
4309 Elementary rewrite edges, summed over initial words & 168,454\\
4310 Largest event poset & 25 events\\
4311 Products in the three-constructor example & 9\\
4312 Binary supports in the three-coordinate example & 8\\
4313 Exact rank checks for cone models & 6\\
4314 \bottomrule
4315 \end{tabular}
4316 \end{center}
4317 
4318 The word test constructs reachable labelled words and order ideals independently, then compares their states and edges. The constructor test confirms the non-thick killability and invariant-collapse examples, together with a nonzero directed projection model. The cone tests compare the ordered dimensions $(0,1,3)$ and $(0,2,3)$, verify the split projection's shifted cofiber, and distinguish the identity and zero maps with equal endpoints.
4319 
4320 All these finite checks passed. They test the stated finite models and implementation paths. The universal statements rest on their written proofs, not on extrapolation from the finite enumeration. The package's result record stores the exact script hash and bounds. Build and visual-review results are recorded separately from the mathematical test status.
4321 
4322 \subsection{Source verification boundary}
4323 The stable-envelope input was checked in Saunier--Winges, arXiv:2502.03408v2, Definition 3.1 and the subsequent functor-category universal property. The stable localization input was checked in Drew, arXiv:1511.08287v1, Definition 1.1 and Theorem 1.3. The quotient kernel and triangulated formulations were checked in the cited Stacks sections. The free strict syntax theorem now has its own term-construction proof. This was an exact-dependency check, not a comprehensive novelty search.
4324 
4325 The manuscript does not certify a new axiom of set theory, a classification of all mathematical structures, a physical law, or a Langlands reciprocity theorem. Its new claims concern the explicitly defined exchange grammar and stable constructor systems.
4326 
4327 \clearpage
4328 \section{Audit and verification record for v0.7.0}\label{app:audit-v070}
4329 
4330 \subsection{Scope, proof status, and provenance}
4331 The source base is the supplied v0.6.0 manuscript and its complete package. This release preserves that manuscript, integrates new proof blocks, and updates the abstract, theorem ledger, and next obligations. It does not claim a fresh independent audit of every retained earlier theorem. The new proofs received an authoring-session self-review. No independent referee review or Lean compilation was performed.
4332 
4333 The three executed routes were strict semantic realization, interaction/excision reconstruction, and integral quotient classification. They produced K32--K41 under their written hypotheses. The arithmetic test bed uses perfect complexes over $\mathbb Z$ and explicit finite integer data. It does not discharge a Langlands-facing existence hypothesis.
4334 
4335 \subsection{Adversarial boundaries}
4336 \begin{enumerate}
4337 \item Strict naturality supplies the event diagram only for the specified pure grammar. Weak cells and additional rewrite relations retain separate coherence obligations.
4338 \item A square's commutativity is not its excision. The second cofiber vanishes precisely for a homotopy pushout, not for every commuting square.
4339 \item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the $\mathbb Z/p^2$ chain is an explicit counterexample.
4340 \item The total normalization residual $C(N)$ and a cube's total cofiber $K(n_1,\ldots,n_m)$ are different objects. The $(6,10,15)$ example refutes an inference from highest interaction vanishing to all lower interactions vanishing.
4341 \item The integral noncollapse theorem uses positive scalar labels. Two zero labels give an interaction with a free unit summand and collapse the entire quotient. The matrix criterion identifies the analogous free-cofiber failure.
4342 \item Equality of the kernel with that of localization is proved. No unproved equivalence of complete quotient categories or idempotent completions is substituted for it.
4343 \end{enumerate}
4344 
4345 \subsection{Exact finite checks}
4346 Both regression scripts ran successfully on Python 3.13.5. They use exact integer identities and finite-field elimination, with no random sampling or numerical tolerance. The inherited check was rerun with every two-sort word through length 10: 2,047 initial words, 82,499 states, and 168,454 edges. Its constructor and filtration counterexamples also passed.
4347 
4348 The new check covers the following explicitly bounded populations.
4349 \begin{center}
4350 \begin{tabular}{p{0.72\textwidth}r}
4351 \toprule
4352 Check population & Count\\
4353 \midrule
4354 Positive label vectors, arity 1--4, entries 1--6 & 1,554\\
4355 Integral exterior-power chain-map identities & 5,910\\
4356 Integral differential-square identities & 4,356\\
4357 Cohomology dimensions over $\mathbb F_2,\mathbb F_3,\mathbb F_5,\mathbb F_7$ & 29,856\\
4358 Initial scalar two-sort words through length 8 & 511\\
4359 Ideal states in those scalar diagrams & 6,917\\
4360 Cover arrows and eligible event contexts & 10,660\\
4361 Elementary independent-event squares & 4,931\\
4362 Explicit localized context-map inverses & 5,019\\
4363 Ideal-pair combinatorial excision checks, word length at most 6 & 5,289\\
4364 Ordered pairs from an explicit 11-matrix test family & 121\\
4365 Matrix-pair cohomology dimensions over five finite fields & 1,815\\
4366 \bottomrule
4367 \end{tabular}
4368 \end{center}
4369 
4370 The matrix family includes identity, zero, rank-deficient, triangular, and torsion examples. Its 121 pairs are exhaustive for that listed family, not for all integer matrices. Nine pairs have two free cone components. Additional signed/zero-vector checks, the excluded all-zero pair, and the named $(6,10)$, $(6,10,15)$, and nonsplit prime-chain examples are recorded separately. No primary case within the stated bounds was skipped.
4371 
4372 The integral normal-form checks verify unimodular exterior-power chain isomorphisms. Finite-field ranks provide a distinct check on signs and degree dimensions. These are reproducible implementation checks. They neither instantiate every stable category nor replace the written universal proofs.
4373 
4374 \subsection{Sources and remaining review}
4375 The exact source check used Stacks Tags 014D, 0621, and 05RA, and Drew's arXiv:1511.08287v1, Definition 1.1 and Theorem 1.3. The source record separates the homological Koszul convention from the cohomological convention used here. Direct finite-colimit, lattice, and integer arguments are supplied for the new deductions. This was not a comprehensive novelty search. The companion audit contains the dependency map, script hashes, source record, build logs, and visual-review record.
4376 
4377 Independent mathematical review remains outstanding. K15, K17, K18, arbitrary higher-coherent exchange signatures, and the nonadditive universal construction remain open within this project.
4378 
4379 \clearpage
4380 
4381 \clearpage
4382 \section{Audit and verification record for v0.8.0}\label{app:audit-v080}
4383 
4384 \subsection{Scope and proof boundary}
4385 The supplied v0.7.0 PDF, source, and full package are preserved unchanged. Their hashes are recorded in the release. This continuation adds K42--K52, updates the restricted lattice status of K15, and retains the earlier open reciprocity obligations. The new proofs have received an authoring-session self-review, not an independent referee review or Lean certification. No fresh complete audit of all inherited theorems is claimed.
4386 
4387 The three executed mathematical routes were derived coefficient towers, excision versus completion, and arithmetic reconstruction with retained overlaps. The first produces explicit finite-level interactions. The second refutes unconditional commutation of repair and completion and proves a generic-fiber criterion. The third proves a fracture square and effective lattice gluing, with compatible operator descent.
4388 
4389 \subsection{Adversarial checks}
4390 \begin{enumerate}
4391 \item Derived reduction is exact, but naive tensor on arbitrary modules is not. The infinite periodic resolution over $\mathbb Z/p^2$ prevents replacing a perfect two-term cone by its degree-zero cohomology.
4392 \item Canonical tensor/base-change exchanges have zero residual. Nonzero scalar cells are specified additional data, not an unexplained consequence of coarsening.
4393 \item Prime-power interactions can generate the whole perfect category even for nonzero labels. A multi-prime quotient removes whole components and can still collapse if every prime is selected.
4394 \item Coherent finite reductions do not imply commutation with an infinite inverse limit. The field factors in K49 exhibit the precise missing information.
4395 \item The based tower theorem requires common bounded ranks and compatible matrices. Arbitrary incompatible level choices are not claimed to lift.
4396 \item The fracture square reconstructs a supplied complex. K51 separately proves effective existence for independent lattice data, but requires the generic identifications explicitly. Operator gluing requires exact intertwining, not only matching characteristic polynomials.
4397 \item Generic-fiber defects are computed in $D(\widehat B)$, not asserted to be perfect objects over the completed ring. Their cofiber is the $[1]$ shift of the stated fiber.
4398 \end{enumerate}
4399 
4400 \subsection{Reproducible finite checks}
4401 All three check scripts passed in this session with Python 3.13.5. The inherited two scripts reproduce their earlier ranges. The new script additionally uses SymPy 1.14.0 for integer Hermite normal forms; each resulting lattice basis is checked by its exact finite residues and index. There is no random sampling or numerical tolerance.
4402 
4403 The new suite checks all 1,080 ordered coefficient pairs in the eight listed rings $\mathbb Z/q$ with $q\in\{2,3,4,5,8,9,16,25\}$. 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 $\mathbb F_2^2$ and $\mathbb F_3^2$, 2,430 operator-preservation instances, 6,100 rank-one rational lattice membership tests, and 972 principal-part reconstruction witnesses. Exact bounds and nonclaims are in the JSON report.
4404 
4405 These are finite regression checks, not computational proofs of universal categorical statements or assertions about every $p$-adic input. The limiting and gluing theorems have separate written proofs. The arithmetic-square argument and nilpotent-cone argument use classical mechanisms; no historical novelty is inferred from their packaging here.
4406 
4407 \subsection{Build and source verification}
4408 The standalone source is regenerated from the preserved base and the two new proof fragments by \texttt{audit/integrate\_v080.py}. The source diff, run reports, source checks, build log, and rendered-page audit accompany the release. The first general web search returned irrelevant results and was not used as evidence; the cited primary Stacks Project pages were then opened directly. No external source text or font files are bundled. The lack of a Lean executable in this runtime is recorded as an environment limitation, not a mathematical failure.
4409 
4410 \clearpage
4411 \section{Audit and verification record for v0.9.0}\label{app:audit-v090}
4412 
4413 \subsection{Authority and evidence status}
4414 The supplied v0.8.0 PDF, LaTeX source, and full package are preserved unchanged in the new package. Their hashes identify the base. The continuation adds K53--K61, with written proofs and an authoring-session self-review. It is not a new independent audit of every inherited theorem. There is no Lean proof package and no claim of historical novelty.
4415 
4416 Three routes were executed. The kernel and invariants route produces explicit coefficient boundaries and a sharp lifting window. The fully derived regular-base route proves exact commutation, ruling out a putative defect in that regime. The finite nonperfect-base route produces a genuinely nonzero canonical exchange and a separate one-step group-action obstruction. Transitivity of the canonical comparison then gives a positive theorem on coefficient-lifted inputs.
4417 
4418 \subsection{Proof boundaries checked}
4419 \begin{enumerate}
4420 \item The boundary uses torsion-freeness of the source lattices and of the target of the linear constraint. The residual at precision $n$ is an annihilator group, with transition multiplication by $p^{m-n}$, not an arbitrarily identified constant system.
4421 \item Sharp lifting delay concerns solutions for the same fixed integral constraints. It does not grant a lift of constraints or group actions that were supplied only at finite level.
4422 \item Ordinary invariants are not an exact functor. Their cokernel residual is not silently substituted into an exact-functor theorem.
4423 \item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.
4424 \item The $\mathbb Z/p^2$ example assumes $p$ odd and $J$ invertible modulo $p$. Explicit tests outside both hypotheses have nonzero positive cohomology and do not satisfy its asserted quasi-isomorphism.
4425 \item The source of the derived comparison is computed from a bounded-above free resolution of its degree-zero invariant module. Tensoring the original nonnegative invariant complex termwise would give the wrong answer.
4426 \item The vanishing degree-zero comparison follows from naturality of evaluation. The cofiber's cohomology in all integer degrees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.
4427 \item Nonzero derived exchange obstructs a coefficient lift. Its converse is not proved. The separate matrix relation calculation rules out the specified lift from level two to level three.
4428 \item Killing the unbounded target residual kills every perfect $k$-complex. To conclude that it kills all of $D(k)$, the exact functor must also preserve coproducts.
4429 \end{enumerate}
4430 
4431 \subsection{Exact regression record}
4432 All four check scripts passed in this session. The three inherited scripts retain their source hashes and were rerun at their documented ranges. The new deterministic suite uses exact integers, residue enumeration, and verified integer Smith changes, with no random sampling or tolerance-based acceptance.
4433 
4434 \begin{center}
4435 \begin{tabular}{lr}
4436 \toprule
4437 New check family & Count\\
4438 \midrule
4439 All $2\times2$ matrices with entries in $[-2,2]$ & 625\\
4440 Matrix/precision cases for kernel and boundary & 3,125\\
4441 Vectors in those kernel enumerations & 108,750\\
4442 Boundary elements checked individually & 8,342\\
4443 Finite lifting-window cases & 2,500\\
4444 Cases witnessing sharpness of the delay & 512\\
4445 Boundary transition elements checked & 6,462\\
4446 Norm-quotient Smith certificates, $2\leq m\leq12$ & 11\\
4447 Equivariant-map representation pairs & 25\\
4448 Explicit integral isomorphism lifts & 61\\
4449 Invertible $2\times2$ action matrices over $\mathbb F_p$, $p=3,5,7$ & 2,544\\
4450 All next-level lifts of the three selected non-scalar matrices & 3,107\\
4451 Independent full-bar-resolution fixtures & 4\\
4452 \bottomrule
4453 \end{tabular}
4454 \end{center}
4455 
4456 The kernel levels are $(p,n)=(2,1),(2,2),(2,3),(3,1),(3,2)$. The sharpness and lifting loops use the precisely computed valuation bounds. No matrix in that family is skipped. The full group-action enumeration is exactly $\operatorname{GL}_2(\mathbb F_3)$, $\operatorname{GL}_2(\mathbb F_5)$, and $\operatorname{GL}_2(\mathbb F_7)$. The next-level lifts are all matrices $A+p^2B$ for the chosen swap $J$ and $B\in M_2(\mathbb F_p)$ at those three primes.
4457 
4458 As an orthogonal check, the full inhomogeneous cyclic-group cochain differentials, rather than the small periodic resolution, are constructed in degrees zero through two for four fixtures. Twelve verified Smith certificates compute their finite cohomology sizes, and eight chain composites vanish modulo the coefficient modulus. This agrees with the periodic computations. It is an independent implementation check within one authoring session, not independent mathematical review.
4459 
4460 The infinite-degree counterexample and the general exactness, lifting, and completion assertions have separate proofs. The test suite does not infer an unbounded derived tensor product from a bounded truncation. The JSON report records exact families, environment, command, counts, and script hash.
4461 
4462 \subsection{Sources, build, and unresolved obligations}
4463 The source check uses Milne's \emph{Class Field Theory}, version 4.03, Chapter II, for finite-group cohomology, boundary maps, and the cyclic resolution, and the official Stacks Project derived-tensor section. The relevant primary PDF pages were also inspected visually. An initial broad search returned irrelevant pages and was discarded in favor of direct primary sources. No external full-text source or font files are bundled. This was a dependency check, not an exhaustive novelty search.
4464 
4465 The final source is reproducibly integrated from the exact v0.8.0 source and the new proof fragments. The package includes the source diff, final LaTeX log, static reference check, rendering/layout record, checksum inventory, and run logs. The current build and proofread outcomes are recorded in the companion audit. K17, K18, arbitrary higher-coherent rewriting, and a sufficient intrinsic arithmetic-lifting criterion remain open. None is closed by the number of new ledger entries.
4466 
4467 
4468 \clearpage
4469 \section{Audit and verification record for v0.10.0}\label{app:audit-v0100}
4470 
4471 \subsection{Source authority and evidence status}
4472 The source base is the supplied v0.9.0 manuscript and full package. Their SHA-256 hashes are recorded in \texttt{audit/base\_sha256.json}, and the three original files are preserved in \texttt{previous/}. This continuation adds K62--K71. All inherited theorem, lemma, proposition, corollary, and proof blocks are preserved verbatim. Current prose and the ledger now distinguish the refuted exchange-converse claim from the proved fixed-finite-group lifting criterion. Historical version and audit records remain historical.
4473 
4474 The new block has written proofs and an authoring-session self-review, not independent review or Lean certification. It does not assert that all earlier mathematics received a fresh audit. The Lean and Lake executables were not available on the runtime path. This is a tool limitation, not a mathematical obstruction. No historical novelty is claimed for the deformation-theoretic or finite-group stability results.
4475 
4476 \subsection{Executed routes and decisive outcomes}
4477 Three routes were pursued. First, the normalized multiplicativity defect gives a complete next-level obstruction in $H^2(G,\operatorname{Ad}\bar\rho)$, with explicit linear repair and nonexistence certificates. Second, a central sign factor makes the derived-invariant test blind to an action that still cannot lift. This refutes sufficiency of that one diagnostic. Third, finite averaging gives a convergent correction whenever defect depth exceeds twice the valuation of the group order. Its finite-horizon corollary supplies a necessary-and-sufficient existence test for a fixed finite group.
4478 
4479 The corresponding ultrametric stability threshold is established in \cite[Lemma 7.3]{FournierFacio2026}. The source was checked at the exact v5 version, with the displayed hypotheses and proof, not from its abstract. The direct matrix proof here is independent of that citation and supplies the implementation contract. The normalized group-cohomology conventions were checked against \cite[Chapter II, 1.17--1.19]{MilneCFT2020}. The new source record states the translation and the limits of the overlap check. An initial broad search returned irrelevant material, which was discarded. No inference of novelty is drawn from that search.
4480 
4481 \subsection{Proof boundaries examined}
4482 \begin{enumerate}
4483 \item The ideal $p^nR_{n+1}$ is square-zero even at $n=1$. Matrix corrections use left multiplication, which fixes the signs in the cocycle equation and strict-conjugacy action.
4484 \item The next-level obstruction is a class, not the chosen raw cocycle. The affine equation is necessary and sufficient for the entire fixed group multiplication table. A failed choice of representative matrices is not by itself an obstruction.
4485 \item The approximate conjugations $\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.
4486 \item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\pmod8$ for $C_2$ verifies that this distinction is real.
4487 \item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.
4488 \item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.
4489 \item The profinite example changes the source from $C_p$ to $\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying a fixed-finite-quotient nonlifting verdict indiscriminately to a Galois representation.
4490 \end{enumerate}
4491 
4492 \subsection{Finite computation contract}
4493 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.
4494 
4495 The cyclic suite enumerates all 367 valid rank-two actions and all 40,987 next-level generator matrices in its six explicitly listed $(p,n,q)$ cases. The averaging suite performs 330 corrections in 110 deterministic fixtures, including primes dividing and not dividing the group order. Every use of division tracks finite working precision, and the exact associativity identity is checked before the predicted new defect depth. There are also 36 rank-one horizon comparisons, 98 conjugacy-average checks, and all 3,107 next-level lifts of three selected non-scalar nonliftable matrices. The sign and profinite power calculations are supplementary finite checks.
4496 
4497 These are bounded tests of the implementation. They do not establish $p$-adic convergence, exhaustive classification for arbitrary groups, or historical novelty. Those conclusions either have their written proofs above or are not claimed. Full iterators, exclusions, counts, interpreter version, wall times, command lines, hashes before and after execution, and report hashes are stored in the package. Each suite has a 40-second subprocess timeout. No memory cap was set. Numerical-library thread limits are cooperative.
4498 
4499 \subsection{Manuscript and continuation}
4500 The standalone source is reproduced by \texttt{audit/integrate\_v0100.py}. The source diff, inheritance checks, LaTeX log, rendering record, and checksum manifest are included. No external source text or font files are bundled. Final proofreading checks the new formulas, hypotheses, labels, references, and surrounding integration prose.
4501 
4502 The sufficient existence question is closed here only for a fixed finite source group over $\mathbb Z_p$. Continuous lifting with growing finite images, geometric local conditions, and automorphic realization remain separate obligations. A useful continuation must address those actual hypotheses rather than merely rename the already-classical finite-group obstruction or averaging argument.
4503 
4504 \clearpage
4505 \section{Audit and verification record for v0.11.0}\label{app:audit-v0110}
4506 
4507 \subsection{Source authority and evidence status}
4508 The supplied v0.10.0 PDF, source, and full package are preserved unchanged in \texttt{previous/}. Their SHA-256 hashes are recorded in \texttt{audit/base\_sha256.json}. The new continuation adds K72--K82 and the proof fragment \texttt{proofs/profinite\_lifting.tex}. All inherited theorem, lemma, proposition, corollary, and proof blocks remain verbatim and in their original relative order. Current summary prose, the theorem ledger, and next-step statements are updated. Historical audit and version entries remain historical.
4509 
4510 The new results have written proofs and an authoring-session self-review. They have not received an independent mathematical review or Lean certification. Lean and Lake are not available on the runtime path. No claim of historical novelty is made for continuous cohomology, free pro-$p$ groups, or cyclic centralizers, nor is novelty inferred for their combination from a limited search.
4511 
4512 \subsection{Executed routes and mathematical boundaries}
4513 The continuous-obstruction route retains the actual profinite source and proves the finite-quotient inflation criterion, including an explicit carry cocycle killed by source refinement. It does not infer that all finite-quotient obstructions eventually vanish. The free-source route constructs lifts without source relations beyond those of a free pro-$p$ group. The abelian route retains commutativity, exposes its normalized residual, and gives a complete criterion on the cyclic normalized stratum, including every rank-two input. The example with $E_{12},E_{21}$ proves the uniform $2n-1$ horizon sharp for $n,r\geq2$.
4514 
4515 The self-review checked the following distinctions.
4516 \begin{enumerate}
4517 \item Continuous $H^2$ here has a finite discrete coefficient module. No statement about arbitrary topological coefficient theories is substituted for that definition.
4518 \item A continuous repair cochain may require a finer finite source quotient. It is not required to factor through the initial quotient.
4519 \item The cohomology-vanishing theorem gives a sufficient condition and the finitely branching tree gives a compactness criterion. Neither supplies a general numerical horizon for all profinite groups.
4520 \item In the cyclic test, the normalized generator must be cyclic modulo $p$. Rank two makes that condition automatic after minimal scalar-depth normalization. No such assertion is made for arbitrary higher rank.
4521 \item Cancellation of $p^{2a}$ takes place in torsion-free integral matrices for necessity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.
4522 \item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\mathbb Z_p$, not necessarily unimodular over $\mathbb Z$.
4523 \item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis.
4524 \item Source inflation and coefficient reduction have opposite variance. A preliminary draft description treating both as unrestricted coarsenings was replaced by \cref{prop:mixed-source-variance}; reversing inflation would require descent through the smaller quotient.
4525 \end{enumerate}
4526 
4527 \subsection{Exact computations and usable certificates}
4528 All six regression suites passed with Python 3.13.5. The inherited five suites were rerun at their previously recorded bounds. The new standard-library suite uses exact integer and modular arithmetic with no random sampling or floating tolerance.
4529 
4530 The source-inflation checks cover 495 cocycle triples, all 3,107 refined-quotient pairs for $p=3,5,7$, and 83 repaired matrix values. The centralizer checks cover 1,017 cyclic matrices and 982,827 candidate centralizer pairs in the explicitly listed finite rings and companion-matrix family. Complete rank-two tables contain all 10, 33, 832, and 29,889 valid continuous two-generator actions over $\mathbb F_2$, $\mathbb F_3$, $\mathbb Z/4$, and $\mathbb Z/9$, respectively. Each positive case is checked against an exactly commuting integral polynomial certificate.
4531 
4532 An orthogonal enumeration of all 1,048,576 ordered pairs of admissible generator matrices modulo $8$ gives 59,392 commuting pairs. Their 664 distinct reductions modulo $4$ agree exactly with the positive set from the normalized criterion. All 4,096 rank-two tangent triples over $\mathbb F_2$ are also checked. The sharpness suite includes 24 prime/precision families and all 72,353 correction pairs in three stated finite fixtures. Higher-rank checks use 54 constructed three-generator fixtures, not an exhaustive higher-rank population, with 162 recovered polynomial certificates and 648 finite continuity checks. Exact ranges and nonclaims are in \texttt{checks/profinite\_results.json}.
4533 
4534 The included command-line tool \texttt{tools/lift\_abelian\_rank2.py} validates the matrix input and returns either exact integral generators or a nonzero normalized-commutator certificate. It applies only to the unconstrained rank-two $\mathbb Z_p^r$ problem. Its positive and negative examples are preserved in \texttt{examples/}. The universal lifting and convergence statements follow from the written proofs, not from these bounded computations.
4535 
4536 \subsection{Sources, build, and reproducibility}
4537 The direct-limit cohomology input was checked in Milne's \emph{Class Field Theory}, version 4.03, Chapter II, Section 4, Proposition 4.2, printed page 87. That PDF page was inspected visually. The continuous-cochain differential was checked on the official Stacks Project Tag 0DVG. Broad searches returning unrelated material were not treated as evidence. This is a dependency check, not an exhaustive novelty survey. No external full-text paper or font file is bundled.
4538 
4539 The source is regenerated from the exact preserved base and the new proof and audit fragments by \texttt{audit/integrate\_v0110.py}. The package includes the source diff, final LaTeX log, source-preservation checks, run manifests, visual-review record, and checksums. The runner removes stale result files before execution and records each script's before/after hash. A shell inspection command initially rejected a leading-hyphen \texttt{printf} argument; it was rerun without that argument and had not modified source. The companion audit records final build outcomes and the precise review scope.
4540 
4541 
4542 \clearpage
4543 \section{Version ledger}
4544 
4545 \begingroup\footnotesize\raggedright
4546 \begin{description}[leftmargin=8em,style=nextline]
4547 \item[v0.11.0] Proves continuous one-step obstruction and source-inflation criteria, continuous tower reconstruction, and free pro-$p$ coefficient lifting. Gives a normalized cyclic-centralizer lifting criterion for $\mathbb Z_p^r$, a complete rank-two test, and the sharp uniform horizon $2n-1$. Correctly separates source inflation from coefficient reduction by mixed variance. Adds exact positive and negative examples, a certificate CLI, and a sixth regression suite. Preserves the entire v0.10.0 package. Written proofs and self-review only.
4548 \item[v0.10.0] Adds a complete $H^2$ next-level obstruction, cochain repair certificates, and a central-sign counterexample to the sufficiency of derived-invariant exchange. Gives a direct averaging proof of known finite-group ultrametric stability, an exact finite lifting horizon, controlled conjugacy, a finite class catalogue, a cyclic Jacobian certificate, and an explicit profinite scope boundary. Five exact suites pass. Preserves every inherited theorem/proof block and the full v0.9.0 package. Written proofs and self-review only.
4549 \item[v0.9.0] Adds intrinsic kernel and invariant comparisons, explicit cohomological residuals, sharp precision-window lifting, equivariant morphism correction, and norm-quotient examples. Proves perfect-base derived exchange and exhibits a non-scalar cyclic action with an unbounded canonical exchange defect over singular finite coefficients. Proves the one-step failure to lift that action and exact exchange for supplied coefficient-lifted inputs. Preserves the full v0.8.0 package. All four finite suites pass, including independent full-bar checks. Written proofs and self-review only.
4550 \item[v0.8.0] Adds nonidentity derived arithmetic coarsenings, coherent weighted exchanges, the prime-power Koszul formula, and nilpotent excision collapse. Classifies finite prime-component quotients and their coherence. Proves uniform based-tower reconstruction, computes transient negative classes, and identifies the generic-fiber defect of completion versus repair. Proves arithmetic fracture reconstruction, effective lattice gluing, and compatible operator descent. Preserves the v0.7.0 package and adds exact finite tests. Written proofs and self-review only.
4551 \item[v0.7.0] Constructs strict chain-natural event diagrams. Defines interaction residuals and proves the total-cofiber and context-independence criteria. Proves principal-ideal reconstruction for reduced excisive event diagrams. Constructs the universal constructor-compatible excisive quotient. Gives tensor-cube, integral gcd, prime-overlap kernel, finite-signature, and integral matrix collapse theorems. Adds explicit nonsplitting, higher/lower interaction, and zero-label collapse boundaries, together with exact finite regression and a preserved full v0.6.0 package. Written proofs and self-review only.
4552 \item[v0.6.0] Audits size, variance, categorical regimes, lift multiplicity, cofiber shifts, exactness, descent, and quotient interpretation. Refutes general ordered filtration independence, including inside the typed two-axis grammar. Proves the semantic diamond criterion and the order-ideal description of pure exchange histories. Constructs the coherent ideal-indexed residual diagram. Proves constructor-invariant quotient and directed invariance theorems, records a quotient-collapse counterexample, and adds exact orbit-generator tests and a noncollapse certificate. Includes finite exact regression checks. Written proofs only, not Lean verification.
4553 \item[v0.5.0] Combines the stable residual filtration with the Verdier obstruction quotient to prove obstruction localization along any chosen normalization path: if all elementary exchange residuals vanish in the obstruction quotient, so does the total normalization residual; contrapositively, a persistent total obstruction forces at least one persistent elementary exchange obstruction. Under directed annihilators the statement upgrades from quotient-vanishing to operational killability.
4554 \item[v0.4.0] Adds a direct nonstable obstruction test bed: local sections of an abelian torsor produce overlap residuals, local changes of trivialization act as repairs, and the resulting \v{C}ech $H^1$ class vanishes exactly when the local sections can be repaired to reconstruct a global section. This closes the first descent test-bed obligation and shows that the residual/repair/obstruction pattern recovers a classical cohomological mechanism.
4555 \item[v0.3.0] Proves the stable residual filtration along a chosen normalization path by iterated octahedral cofiber sequences. Constructs the universal stable obstruction quotient by the thick closure of operationally killable residuals, proves exact detection under a directed common-annihilator hypothesis, and proves functorial transport of obstruction quotients under exact correspondences. K11 and K12 are closed in these stable forms; path-independent filtration and nonstable obstruction remain open.
4556 \item[v0.2.0] Adds the free seeded two-axis double-computad universal property and proves the pointwise free stable completion theorem for additive indexed systems. Mixed residuals become canonical cofibers, exact realizations preserve them, and residual promotion is shown to be automatic in the minimal stable zoo. K13 is closed at the strict syntactic level and K14 is closed under additive/stable hypotheses; the nonadditive residual universal construction remains open.
4557 \item[v0.1.0] Initial continuous formalization. Introduces indexed resolution categories, total Grothendieck category, factorization, resolution ultrametric, seed zoo closure, exchange transformations, stable residuals, exact normal forms, split residual expansion, obstruction taxonomy, descent transfer, and a seeded local-global reciprocity blueprint.
4558 \end{description}
4559 \endgroup
4560 
4561 \end{document}