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Stratified Universality

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51   pdftitle={Stratified Universality for Resolved Representations},
52   pdfsubject={Profiled lane transport, semantic quotient calculus, carrier and observer change, typed obstruction theory, and certified descent},
53   pdfkeywords={resolved representation, stratified universality, lane transport, semantic quotient, carrier change, observer base change, task-relative minimality, address holonomy, exact loss, obstruction, certified descent}
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186 \renewcommand{\KomorebiTitle}{Stratified Universality for Resolved Representations}
187 \renewcommand{\KomorebiSubtitle}{Profiled lane transport, semantic quotients, carrier change, and certified descent}
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197 \begin{document}
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199 \KomorebiOpeningNote{\begin{tcolorbox}[width=0.88\textwidth,colback=KomorebiPaleGray,colframe=KomorebiGray,boxrule=0.7pt,arc=1mm]
200 \centering\small
201 \textbf{Mathematical base: version 2.0.0 (23 September 2026).} This edition retains the finite atlas and finite profiled and holonomy-twisted lane theorems, with the local corrections recorded in the publication note. The mathematical base adds a formal calculus of task-relative semantic quotients, exact bridge loss, carrier factorization, observer base change, admissibility-relative obstruction, and enrichment lifting. Inherited claims retain their original proof boundaries. New claims are separated in Part IV and in the expanded theorem concordance.
202 \end{tcolorbox}
203 
204 Universality is asserted only relative to declared grammars, strata, bridges, preservation profiles, normalization targets, and descent doctrines.}
205 
206 
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208 
209 \hypersetup{pageanchor=true}
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211 \pagenumbering{roman}
212 \thispagestyle{plain}
213 \begin{abstract}
214 Komorebi Math is formulated as a relative classifying and transport theory of resolved representations. A resolved theory records typed operations, packet decompositions, refinement laws, residuals, certificates, and only those analytic, tensor, metric, or arithmetic structures included in a declared preservation profile. Its realizations are organized into normalized strata. Reversible changes of presentation form representation groupoids, while quotients, completions, reductions, and other information-losing constructions remain in a directed category with explicit loss passports.
215 
216 The first theorem layer establishes the additive--Karoubi resolved envelope, finite profiled atlas assembly, and componentwise invariant descent. The finite profiled Universal Lane Transport Theorem identifies every morphism between complete finite split packet systems with its full lane matrix. Composition becomes block contraction, refinement is functorial, and selectors split retained transport from a named residual. Under finite address holonomy, the same calculus descends as sections of a matrix local system. Monodromy orbits give the finest ordinary global lane coarsening, and pair orbits carry the compatible selector algebra.
217 
218 The second theorem layer formalizes task-relative semantics. For a declared continuation grammar and observation family, the behavioral quotient is the coarsest exact representation preserving every future observation. In module-valued slices, task combination, common coarsening, relative residuals, bridge loss, essential-image failure, and task weakening fit into canonical exact sequences. Carrier replacement is possible exactly on a kernel-factorization locus. Set-valued residual equality and linear residual span are separated by an explicit relation module. Convex value packets, provenance states, additive route packets, and weighted-linear states therefore remain distinct unless certified comparison maps are supplied.
219 
220 Finally, obstruction is indexed by the admissible realization class. Enlarging that class can reduce an optimized defect, while zero infimum may still fail to be attained. Rich structure descends under forgetting, but reconstructing it from a bare bridge is a typed lifting problem. The resulting theory is finite, profiled, task-relative, and auditable. It does not assert one global coordinate system, one observer-independent minimum, equivalence of all carriers, or effectivity of an unrestricted heterogeneous stack.
221 \end{abstract}
222 
223 \begin{center}
224 \small\textbf{Keywords:} resolved representation, stratified universality, lane transport, semantic quotient, behavioral core, carrier change, observer base change, exact loss, address holonomy, admissibility-relative obstruction, certified descent.
225 \end{center}
226 
227 \begin{claimbox}
228 \textbf{Central claim.}
229 Komorebi Math is a stratified classifying and transport theory of resolved representations. Within a declared profile, it separates exact representation change from directional information loss. Within a declared semantic slice, it constructs the coarsest representation preserving all future task observations. Changes of carrier, observer, geometry, completion, or enrichment are legal only through certified comparison or lifting data.
230 \end{claimbox}
231 
232 \begin{boundarybox}
233 \textbf{Claim boundary.}
234 The paper does not assert that every mathematical object has a finite Komorebi presentation, that all strata or carriers are equivalent, that every bridge is reversible, that arbitrary presentations admit one observer-independent minimum, or that the unrestricted heterogeneous atlas is an effective stack. Internal proof status, exact computation, classical import, historical novelty, and independent peer validation remain distinct classifications.
235 \end{boundarybox}
236 
237 \clearpage
238 \begin{claimbox}
239 \textbf{Publication edition and source availability.}
240 This publication edition preserves the theorem labels, concordance component identities, and claim boundaries of the 23 September 2026 version 2.0.0, with five local amendments recorded explicitly: orbit supports are distinguished from nonzero selector action; observer updates have finite branching, finite raw-history aggregation, and declared empty-fiber conventions; the compactness proof of escape uses an attained positive minimum; the coinduction and finite-certification proofs make their fixed-point implications explicit; and metric statements declare positive dimension, determinant normalization, and inverse edge traversal. The original sources and inherited records remain included unchanged. Its companion visual guide and Lean guide are reading and evidence guides to that mathematical base.
241 
242 The complete release includes the Markdown (\texttt{.md}) files, \LaTeX{} sources, figure generators, data, exact validators, and inherited validation reports. Inherited reports retain their original dates and scope. Written formal mathematics, exact finite computation, publication checks, and Lean certification are separate forms of evidence. No Lean source files or Lean build project are present in the supplied version 2.0.0 source tree. Separate related Lean developments, where identified, require their own source and theorem correspondence. This publication edition does not claim a fresh Lean build or a new independent proof audit of the complete manuscript.
243 \end{claimbox}
244 \clearpage
245 
246 \tableofcontents
247 \clearpage
248 \pagenumbering{arabic}
249 
250 \section{Introduction}
251 
252 \subsection{The assembly problem}
253 
254 The source manuscripts already contain typed quiver and module semantics, seed presentations, exact packet coordinates, refinement and residual calculus, nonlinear packet composition, finite holonomy, relative obstruction objects, operator-algebraic recognition, Galois descent, and several exact sequence and arithmetic realizations \cite{P1,RT,P3,GAL,SEQ}. The central problem is therefore no longer to reconstruct these foundations independently. It is to determine when they assemble into one finite, typed, coherent classifying object.
255 
256 The proposed architecture has six layers:
257 \[
258 \boxed{
259 \begin{gathered}
260 \text{resolved syntax}\longrightarrow\text{normalized strata}
261 \longrightarrow\text{typed overlaps}\longrightarrow\text{coherence}\cr
262 \longrightarrow\text{effective descent}
263 \longrightarrow\text{compatible complete invariants}.
264 \end{gathered}}
265 \]
266 Each arrow carries a proof obligation. A semantic analogy is not a bridge. A bridge is not reversible merely because formulas exist in both directions. Local completeness is not global completeness until the local isomorphisms themselves satisfy descent.
267 
268 \subsection{Main theorem and classification domain}
269 
270 Three theorem levels are kept distinct. First, the finite profiled Universal Lane Transport Theorem is an internal theorem about complete finite split packet systems in an additive idempotent-complete category. Its holonomy-twisted extension retains exact analysis, synthesis, composition, refinement, selectors, and residuals when chart loops permute lane addresses. The global coordinate object is then a matrix local system, and ordinary lane projectors descend at the monodromy-orbit resolution.
271 
272 Second, the finite atlas assembly theorem is conditional in general. Given profile-preserving normalized strata, typed reversible overlaps, coherent comparison cells, effective descent, and compatible native invariants, the descended invariant reflects isomorphism. The manuscript verifies these hypotheses on a finite certified atlas whose reversible core has five connected semantic components.
273 
274 Third, the semantic quotient theorem works inside a fixed doctrine and declared observation category. It constructs the maximal continuation-stable invisible core and the associated task-relative quotient. In module-valued slices, semantic combination and bridge transport admit canonical exact sequences. Carrier and observer changes are separate factorization problems, while changes of admissible geometry or enrichment are typed lifting problems.
275 
276 The statements are strongest precisely because their targets are explicit. Lane transport concerns declared finite resolutions. Atlas classification concerns the declared normalized finite atlas. Behavioral minimality concerns one semantic slice. None of these statements classifies arbitrary presentations, identifies all carrier algebras, or promotes unverified doctrine changes to equivalences.
277 
278 \subsection{Contributions}
279 
280 The paper makes fourteen consolidated contributions.
281 \begin{enumerate}
282 \item It provides a source-controlled theorem concordance and an explicit profile-indexed additive--Karoubi resolved envelope.
283 \item It separates representation groupoids from the directed transformation category and assigns every certified bridge a preservation and information-loss passport.
284 \item It proves a finite profiled atlas assembly theorem and distinguishes holonomy, relative cokernel, Brauer, and higher-coherence defects.
285 \item It establishes normalized effective-descent atlases and complete native invariants on the five-component certified semantic union.
286 \item It proves exact bridge comparison and higher coherence, including the nonlinear--algebraic--holonomic Fuss--Catalan triangle.
287 \item It proves the finite profiled Universal Lane Transport Theorem with exact block transport, refinement, selectors, residuals, and separately typed fold and kernel clauses.
288 \item It removes the trivial-address-holonomy restriction from transport by proving matrix-local-system descent, orbit coarsening, pair-orbit selectors, and orbital path composition.
289 \item It defines the task-relative behavioral quotient as the coarsest exact representation preserving all declared future observations.
290 \item It identifies the behavioral kernel as the maximal continuation-stable invisible core and gives a greatest-fixed-point computation with finite-dimensional termination.
291 \item It proves the semantic diamond, relative semantic residual, task-change exact sequence, bridge-loss exact sequence, and extension law for composite loss in module-valued slices.
292 \item It proves the carrier factorization gate and the universal joint completion preserving two incomparable carrier semantics.
293 \item It separates Set-valued continuation minimization from linear residual minimization by an exact relation-module sequence and an explicit polyphase family with infinite discrete state and finite Hankel rank.
294 \item It proves carrier-indexed observer aggregation and observer-conditioned convex packet transport, while isolating cancellation, idempotence, singular-resurrection, and provenance boundaries.
295 \item It formalizes admissibility monotonicity, completion/gap/escape stratification, a sharp convex-versus-quadratic example, and the forgetful-descent versus enrichment-lifting principle.
296 \end{enumerate}
297 
298 \subsection{Proof status and organization}
299 
300 Classical inputs, internally proved theorems, exact finite verification, conditional statements, framework definitions, and conjectures are separately marked. The theorem concordance is source control for this distinction. Part I gives the common resolved foundation and finite assembly theorem. Part II develops the certified native strata. Part III gives global synthesis and finite profiled and holonomy-twisted lane transport. Part IV develops task-relative semantic quotients, carrier and observer change, exact loss, admissibility-relative obstruction, and doctrine lifting. The final sections state scope and conclusion. Detailed development, claim-boundary, and validation records are retained in the appendices.
301 
302 \input{block14_section.tex}
303 
304 \part{Common foundations and finite assembly}
305 \section{Resolved theories, packet coordinates, and preservation profiles}
306 \label{sec:resolved-foundations}
307 
308 This section expands the common formal core used by every later atlas construction.  The purpose is not to force every domain into one category.  It is to separate four layers that are often conflated:
309 \[
310 \boxed{
311 \text{typed syntax}
312 \longrightarrow
313 \text{resolved coordinates}
314 \longrightarrow
315 \text{profile-preserving realization}
316 \longrightarrow
317 \text{domain semantics}.}
318 \]
319 The first two layers are algebraic.  The third records which algebraic or universal constructions survive a realization.  The fourth supplies domain-specific hypotheses such as completeness, positivity, convergence, integrality, measurability, or analytic continuation.  A cross-domain statement is a theorem only when all four layers are present.
320 
321 \subsection{Typed resolved signatures}
322 \label{subsec:typed-resolved-signatures}
323 
324 \begin{definition}[Finite resolved signature]
325 \label{def:finite-resolved-signature}
326 Fix a commutative coefficient ring \(\kk\).  A finite resolved signature is a tuple
327 \[
328 \mathbb K_{\rm fin}
329 =
330 (Q,I,\mathsf P,\mathsf{Ref},\mathsf{Comp},\mathsf{Cert})
331 \]
332 with the following data.
333 \begin{enumerate}[label=\textup{(\roman*)}]
334 \item \(Q=(Q_0,Q_1,s,t)\) is a finite quiver.  Its vertices are sorts and its arrows are typed primitive operations.
335 \item \(I\triangleleft \kk Q\) is a finitely generated two-sided ideal generated by \(\kk\)-linear combinations of paths with common source and common target.
336 \item \(\mathsf P\) is a finite list of packet declarations.  A declaration on a sort \(X\) consists of endomorphism words \(p_{X,r,i}\) and the relations
337 \[
338 p_{X,r,i}p_{X,r,j}=\delta_{ij}p_{X,r,i},
339 \qquad
340 \sum_i p_{X,r,i}=1_X.
341 \]
342 \item \(\mathsf{Ref}\) is a finite list of refinement declarations.  For \(s\succeq r\), an address map
343 \[
344 \rho_{sr}:I_{X,s}\twoheadrightarrow I_{X,r}
345 \]
346 is accompanied by the equations
347 \[
348 p_{X,r,i}
349 =
350 \sum_{\rho_{sr}(j)=i}p_{X,s,j}.
351 \]
352 \item \(\mathsf{Comp}\) is a list of optional completion tags.  A tag names a doctrine to be added later, such as Banach completion, a \(C^*\)-involution, a multiplier completion, or a monoidal structure.  A tag is not itself an existence theorem.
353 \item \(\mathsf{Cert}\) is a list of named residual and certificate types.  Each certificate type specifies its source object, target object, and the equations or inequalities it is intended to record.
354 \end{enumerate}
355 \end{definition}
356 
357 The finite signature is the presentation-level object.  It records legal words and declared equations.  It does not identify a preferred realization, a preferred coordinate basis, or a preferred semantic domain.
358 
359 \begin{definition}[Realization of a finite resolved signature]
360 \label{def:resolved-realization}
361 Let \(\mathcal C\) be a \(\kk\)-linear category.  A realization of \(\mathbb K_{\rm fin}\) in \(\mathcal C\) assigns an object \(M_X\) to every vertex \(X\in Q_0\) and a morphism
362 \[
363 M_a:M_{s(a)}\longrightarrow M_{t(a)}
364 \]
365 to every arrow \(a\in Q_1\), such that all relations in \(I\), all packet equations, and all declared refinement equations hold.  Completion and certificate tags are interpreted only when the realization profile explicitly includes them.
366 \end{definition}
367 
368 \begin{theorem}[Typed realization and module equivalence, imported]
369 \label{thm:typed-realization-module-equivalence}
370 Let
371 \[
372 \Lambda_{\mathbb K}:=\kk Q/I.
373 \]
374 In \(\kk\)-modules, the category of typed realizations of the linear grammar \((Q,I)\) is equivalent to the category of unital left \(\Lambda_{\mathbb K}\)-modules.  Under the equivalence, the vertex idempotent \(e_X\) acts as the projector onto the component of sort \(X\), and each arrow acts in its declared source-target block.
375 \end{theorem}
376 
377 \begin{proof}
378 A typed realization defines the action of a path by composition and then extends \(\kk\)-linearly.  Every element of \(I\) acts by zero, so the action factors through \(\Lambda_{\mathbb K}\).  Conversely, for a unital module \(M\), the pairwise orthogonal vertex idempotents give
379 \[
380 M=\bigoplus_{X\in Q_0}e_XM,
381 \]
382 and an arrow \(a:X\to Y\) maps \(e_XM\) into \(e_YM\).  These constructions are inverse up to their canonical natural isomorphisms.  This is the typed-module theorem of Paper I \cite{P1}.
383 \end{proof}
384 
385 \begin{remark}[Presentation is not classification]
386 \label{rem:presentation-not-classification}
387 The theorem fixes the semantics of one declared grammar.  It does not say that the grammar is unique, minimal, or recoverable from a finite list of observations.  Distinct quivers with relations can present equivalent module categories, and one realization can annihilate additional syntax through the kernel of its action.
388 \end{remark}
389 
390 \begin{example}[Two-sort transfer grammar]
391 \label{ex:two-sort-transfer}
392 Take two sorts \(X,Y\), one arrow \(a:X\to Y\), and no nontrivial relation.  A realization is a single typed map
393 \[
394 M_X\xrightarrow{M_a}M_Y.
395 \]
396 After collapsing to \(M_X\oplus M_Y\), the vertex idempotents are coordinate packets and \(M_a\) is one off-diagonal transfer block.  The smallest typed grammar therefore already contains the distinction between object packets and transfer channels.
397 \end{example}
398 
399 \subsection{Packet objects, analysis, and exact transfer coordinates}
400 \label{subsec:packet-coordinate-core}
401 
402 \begin{definition}[Finite packet system]
403 \label{def:finite-packet-system-expanded}
404 Let \(X\) be an object of a preadditive category.  A finite packet system on \(X\) is a family
405 \[
406 \mathcal P=(p_i)_{i\in I}\subseteq\operatorname{End}(X)
407 \]
408 satisfying
409 \[
410 p_i^2=p_i,
411 \qquad
412 p_ip_j=0\quad(i\ne j),
413 \qquad
414 \sum_{i\in I}p_i=1_X.
415 \]
416 No self-adjointness or metric orthogonality is assumed unless stated separately.
417 \end{definition}
418 
419 Assume now that \(\mathcal C\) is additive and idempotent complete.  Choose a splitting
420 \[
421 X_i\xrightarrow{\iota_i}X\xrightarrow{\pi_i}X_i,
422 \qquad
423 \iota_i\pi_i=p_i,
424 \qquad
425 \pi_i\iota_i=1_{X_i}.
426 \]
427 The packet equations imply
428 \[
429 \pi_i\iota_j=0\quad(i\ne j).
430 \]
431 
432 \begin{definition}[Packet analysis and synthesis]
433 \label{def:packet-analysis-synthesis-expanded}
434 The packet analysis and synthesis maps are
435 \[
436 \Analysis_{\mathcal P}
437 :=
438 (\pi_i)_{i\in I}:
439 X\longrightarrow\bigoplus_{i\in I}X_i,
440 \]
441 \[
442 \Synthesis_{\mathcal P}
443 :=
444 (\iota_i)_{i\in I}:
445 \bigoplus_{i\in I}X_i\longrightarrow X.
446 \]
447 \end{definition}
448 
449 \begin{theorem}[Exact packet-coordinate isomorphism, imported]
450 \label{thm:exact-packet-coordinate-expanded}
451 For every finite packet system in an additive idempotent-complete category,
452 \[
453 \boxed{
454 \Synthesis_{\mathcal P}\Analysis_{\mathcal P}=1_X,
455 \qquad
456 \Analysis_{\mathcal P}\Synthesis_{\mathcal P}
457 =1_{\oplus_iX_i}.}
458 \]
459 Consequently,
460 \[
461 X\cong\bigoplus_{i\in I}X_i.
462 \]
463 \end{theorem}
464 
465 \begin{proof}
466 The first identity is
467 \[
468 \Synthesis_{\mathcal P}\Analysis_{\mathcal P}
469 =
470 \sum_i\iota_i\pi_i
471 =
472 \sum_i p_i
473 =
474 1_X.
475 \]
476 The \((i,j)\)-entry of the second composite is \(\pi_i\iota_j\), which equals \(1_{X_i}\) for \(i=j\) and zero otherwise.  This is the exact packet-coordinate theorem of Paper I \cite{P1}.
477 \end{proof}
478 
479 \begin{corollary}[Packetwise equality]
480 \label{cor:packetwise-equality-expanded}
481 For parallel morphisms \(f,g:Z\to X\),
482 \[
483 f=g
484 \quad\Longleftrightarrow\quad
485 \pi_i f=\pi_i g\text{ for every }i.
486 \]
487 For parallel morphisms \(f,g:X\to Z\),
488 \[
489 f=g
490 \quad\Longleftrightarrow\quad
491 f\iota_i=g\iota_i\text{ for every }i.
492 \]
493 \end{corollary}
494 
495 \begin{proof}
496 One direction is immediate.  In the other direction, insert the reconstruction identity on the appropriate side and sum the packetwise equalities.
497 \end{proof}
498 
499 \begin{corollary}[Universal mapping property of packet coordinates]
500 \label{cor:packet-universal-expanded}
501 For every object \(Z\), restriction to the packet objects gives natural bijections
502 \[
503 \operatorname{Hom}(X,Z)
504 \cong
505 \prod_i\operatorname{Hom}(X_i,Z),
506 \qquad
507 \operatorname{Hom}(Z,X)
508 \cong
509 \prod_i\operatorname{Hom}(Z,X_i),
510 \]
511 where the products are finite.  Packetwise maps therefore assemble uniquely into a global map.
512 \end{corollary}
513 
514 \begin{definition}[Mixed transfer blocks]
515 \label{def:mixed-transfer-blocks-expanded}
516 Let \(\mathcal P=(p_i)\) be a packet system on \(X\), let \(\mathcal Q=(q_a)\) be a packet system on \(Y\), and choose splittings \((\iota_i,\pi_i)\) and \((\jmath_a,\varpi_a)\).  For \(f:X\to Y\), define the packet-to-packet map
517 \[
518 f_{ai}:=\varpi_a f\iota_i:X_i\longrightarrow Y_a.
519 \]
520 Its ambient corner is
521 \[
522 q_afp_i=\jmath_a f_{ai}\pi_i.
523 \]
524 \end{definition}
525 
526 \begin{theorem}[Exact transfer reconstruction and block composition, imported]
527 \label{thm:transfer-reconstruction-expanded}
528 Every morphism \(f:X\to Y\) has the exact expansion
529 \[
530 \boxed{
531 f
532 =
533 \sum_{a,i}q_afp_i
534 =
535 \sum_{a,i}\jmath_a f_{ai}\pi_i.}
536 \]
537 If \(g:Y\to Z\) is resolved by a packet system \(\mathcal R=(r_c)\), then
538 \[
539 \boxed{
540 (gf)_{ci}
541 =
542 \sum_a g_{ca}f_{ai}.}
543 \]
544 \end{theorem}
545 
546 \begin{proof}
547 Insert the finite packet identities on both sides of \(f\):
548 \[
549 f
550 =
551 \left(\sum_aq_a\right)f\left(\sum_ip_i\right).
552 \]
553 For composition, insert \(1_Y=\sum_aq_a\) between \(g\) and \(f\).  This is the transfer-block reconstruction and multiplication law of Paper I \cite{P1}.
554 \end{proof}
555 
556 \begin{corollary}[Path expansion]
557 \label{cor:packet-path-expansion}
558 For a composable string
559 \[
560 X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}\cdots\xrightarrow{f_m}X_m
561 \]
562 with packet systems on every \(X_j\), each source-target block of \(f_m\cdots f_1\) is the sum over all compatible intermediate packet paths of the corresponding block products.
563 \end{corollary}
564 
565 \begin{example}[Exact oblique packets]
566 \label{ex:oblique-packets}
567 Over any field, set
568 \[
569 p_1=
570 \begin{pmatrix}
571 1&1\\0&0
572 \end{pmatrix},
573 \qquad
574 p_2=
575 \begin{pmatrix}
576 0&-1\\0&1
577 \end{pmatrix}.
578 \]
579 Then \(p_1^2=p_1\), \(p_2^2=p_2\), \(p_1p_2=p_2p_1=0\), and \(p_1+p_2=I\).  The decomposition of \(\kk^2\) is exact, although the packets are not orthogonal for the standard inner product.  This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.
580 \end{example}
581 
582 \subsection{Refinement, selected support, and named residuals}
583 \label{subsec:refinement-residual-calculus}
584 
585 \begin{definition}[Finite refinement]
586 \label{def:finite-refinement-expanded}
587 A packet system \(\mathcal P^s=(p^s_j)_{j\in I_s}\) refines \(\mathcal P^r=(p^r_i)_{i\in I_r}\) when there is a surjection
588 \[
589 \rho_{sr}:I_s\twoheadrightarrow I_r
590 \]
591 such that
592 \[
593 p^r_i
594 =
595 \sum_{\rho_{sr}(j)=i}p^s_j.
596 \]
597 The map \(\rho_{sr}\) records the address of each fine packet inside its coarse ancestor.
598 \end{definition}
599 
600 \begin{proposition}[Aggregation and transitivity]
601 \label{prop:refinement-aggregation-transitivity}
602 A finite aggregation of pairwise annihilating packets is again a packet system.  If \(t\succeq s\succeq r\), then refinement is transitive with address map
603 \[
604 \rho_{tr}=\rho_{sr}\circ\rho_{ts}.
605 \]
606 \end{proposition}
607 
608 \begin{proof}
609 For an aggregated packet, idempotence follows because all cross terms vanish.  Distinct aggregates annihilate each other, and the total sum remains the identity.  Substituting the \(t\)-level expansion into the \(s\)-level expansion gives the composite address law.
610 \end{proof}
611 
612 \begin{theorem}[Exact refinement law, imported]
613 \label{thm:exact-refinement-expanded}
614 Let \(p^r_i=\sum_{\rho_{sr}(j)=i}p^s_j\) and \(q^u_a=\sum_{\sigma_{vu}(b)=a}q^v_b\).  For every morphism \(f:X\to Y\),
615 \[
616 \boxed{
617 q^u_afp^r_i
618 =
619 \sum_{\sigma_{vu}(b)=a}
620 \sum_{\rho_{sr}(j)=i}
621 q^v_bfp^s_j.}
622 \]
623 Coarsening the fine block product of two composable maps gives the coarse block product.
624 \end{theorem}
625 
626 \begin{proof}
627 Insert the two refinement identities on the left and right of \(f\).  For composition, expand both maps at the fine level, multiply, and sum over descendant intermediate packets.  The fine intermediate projectors sum to the coarse intermediate projector.  This is the exact refinement law of Resolution Theory \cite{RT}.
628 \end{proof}
629 
630 \begin{definition}[Support compression and residual]
631 \label{def:support-compression-expanded}
632 For packet systems \(\mathcal P\) on \(X\) and \(\mathcal Q\) on \(Y\), let \(E\subseteq I_{\mathcal Q}\times I_{\mathcal P}\).  Define
633 \[
634 C_E(f):=\sum_{(a,i)\in E}q_afp_i,
635 \qquad
636 R_E(f):=f-C_E(f).
637 \]
638 The first term is the retained transfer support and the second is its named residual.
639 \end{definition}
640 
641 \begin{proposition}[Boolean algebra of block selectors]
642 \label{prop:block-selector-algebra}
643 As endomorphisms of the \(\kk\)-module \(\operatorname{Hom}(X,Y)\), the selectors satisfy
644 \[
645 C_E^2=C_E,
646 \qquad
647 C_EC_F=C_{E\cap F},
648 \qquad
649 R_E=I-C_E.
650 \]
651 In particular,
652 \[
653 C_ER_E=R_EC_E=0,
654 \qquad
655 f=C_E(f)+R_E(f).
656 \]
657 \end{proposition}
658 
659 \begin{proof}
660 Applying \(C_E\) to a block \(q_bfp_j\) retains it exactly when \((b,j)\in E\).  Pairwise annihilation removes every mismatched source or target index.  The formulas follow block by block.
661 \end{proof}
662 
663 \begin{definition}[Relational support product]
664 \label{def:relational-support-product}
665 For \(E\subseteq I_Y\times I_X\) and \(F\subseteq I_Z\times I_Y\), define
666 \[
667 F\circ E
668 :=
669 \{(c,i):\text{there exists }a\in I_Y
670 \text{ with }(c,a)\in F\text{ and }(a,i)\in E\}.
671 \]
672 \end{definition}
673 
674 \begin{proposition}[Support propagation under composition]
675 \label{prop:support-propagation}
676 If \(f=C_E(f)\) and \(g=C_F(g)\), then
677 \[
678 gf=C_{F\circ E}(gf).
679 \]
680 Thus exact zero patterns propagate by relational composition.
681 \end{proposition}
682 
683 \begin{proof}
684 Every block of \(gf\) is a sum \(\sum_a g_{ca}f_{ai}\).  A nonzero summand can occur only along an intermediate address \(a\) belonging to the relational product.
685 \end{proof}
686 
687 \begin{proposition}[Exact residual product identity]
688 \label{prop:exact-residual-product}
689 Let
690 \[
691 f_E=C_E(f),\quad r_f=R_E(f),
692 \qquad
693 g_F=C_F(g),\quad r_g=R_F(g).
694 \]
695 Then
696 \[
697 \boxed{
698 gf-g_Ff_E
699 =
700 r_gf_E+g_Fr_f+r_gr_f.}
701 \]
702 If the morphisms lie in a normed operator algebra with a submultiplicative norm, then
703 \[
704 \|gf-g_Ff_E\|
705 \le
706 \|r_g\|\,\|f_E\|
707 +\|g_F\|\,\|r_f\|
708 +\|r_g\|\,\|r_f\|.
709 \]
710 \end{proposition}
711 
712 \begin{proof}
713 Substitute \(g=g_F+r_g\) and \(f=f_E+r_f\) and expand.  The norm estimate is immediate from the triangle inequality and submultiplicativity.
714 \end{proof}
715 
716 The identity is a basic certificate-propagation law.  It explains exactly which residual terms enter after composing two compressed transfers.  No asymptotic claim is required.
717 
718 \begin{theorem}[Hilbert--Schmidt optimality, imported]
719 \label{thm:hs-optimality-expanded}
720 Assume \(X,Y\) are Hilbert spaces, the packet idempotents are orthogonal projectors, and \(f:X\to Y\) is Hilbert--Schmidt.  Then \(C_E\) is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and
721 \[
722 \|R_E(f)\|_{\mathrm{HS}}^2
723 =
724 \sum_{(a,i)\notin E}\|q_afp_i\|_{\mathrm{HS}}^2.
725 \]
726 Consequently, \(C_E(f)\) is the unique best Hilbert--Schmidt approximation to \(f\) among operators supported on \(E\).
727 \end{theorem}
728 
729 \begin{boundary}[Algebraic exactness is not metric optimality]
730 \label{bnd:algebraic-vs-metric}
731 For arbitrary idempotent packets, reconstruction and residual decomposition remain exact.  Orthogonal energy identities and best-approximation statements require a Hilbert structure, orthogonal projectors, and the stated operator class.  Those hypotheses are part of the theorem rather than consequences of the packet notation.
732 \end{boundary}
733 
734 \subsection{Preservation profiles and proof-support certificates}
735 \label{subsec:profile-calculus}
736 
737 A realization can preserve one part of the resolved calculus and forget another.  The preservation profile is therefore a mathematical interface, not informal metadata.
738 
739 \begin{definition}[Constructor universe]
740 \label{def:constructor-universe}
741 Fix a declared universe \(\mathfrak C\) of constructors and universal properties.  Typical entries are
742 \[
743 \begin{gathered}
744 0,\ 1,\ \text{finite sums},\ \text{composition},\ \text{scalar action},\ \text{biproducts},\\
745 \text{kernels},\ \text{cokernels},\ \text{quotients},\ \text{tensor products},\\
746 \text{packet idempotents},\ \text{transfer blocks},\ \text{refinements},\ \text{named residuals},\\
747 \text{finite limits},\ \text{finite colimits},\ \text{topological completions},\ \text{adjoints}.
748 \end{gathered}
749 \]
750 The list may be enlarged, but every new entry must specify the exact identity or universal property to be preserved.
751 \end{definition}
752 
753 \begin{definition}[Verified preservation profile]
754 \label{def:verified-preservation-profile-expanded}
755 A verified preservation profile \(\Pi_F\) for a functor \(F:\mathcal C\to\mathcal D\) consists of a subset of \(\mathfrak C\) together with a proof or declared comparison isomorphism showing that \(F\) preserves each listed constructor.  We write \(c\in\Pi_F\) only after the corresponding preservation claim has been verified.
756 \end{definition}
757 
758 \begin{definition}[Proof-support certificate]
759 \label{def:proof-support-certificate}
760 A proof-support certificate for a finite proof \(\mathcal P\) is a finite subset
761 \[
762 \operatorname{csupp}(\mathcal P)\subseteq\mathfrak C
763 \]
764 containing every constructor and universal property used in the proof.  The certificate need not be minimal.  It must be sufficient and auditable.
765 \end{definition}
766 
767 \begin{proposition}[Common profile and composition]
768 \label{prop:common-profile-composition}
769 Let \(F:\mathcal C\to\mathcal D\) and \(G:\mathcal D\to\mathcal E\).  Every constructor preserved by both functors, with compatible chosen comparison data, is preserved by \(GF\).  For a finite diagram of realizations and bridges, the intersection
770 \[
771 \Pi_{\rm common}:=\bigcap_F\Pi_F
772 \]
773 is therefore a valid common profile for every proof transported through the whole diagram.
774 \end{proposition}
775 
776 \begin{proof}
777 Preservation equations and comparison isomorphisms compose.  Universal properties remain universal after applying the second functor whenever both preservation witnesses are present.  Intersecting the verified lists retains only constructors supported at every stage.
778 \end{proof}
779 
780 \begin{theorem}[Equational realization principle, imported]
781 \label{thm:equational-realization-expanded}
782 Let \(F:\mathcal C\to\mathcal D\) be additive between preadditive categories.  Every equality of morphism expressions built from declared generators by finite sums, additive inverses, identities, zero maps, and composition is preserved by \(F\).  Scalar multiplication is preserved when the categories are enriched over the same coefficient ring and \(F\) is linear.  If a proof additionally invokes a chosen biproduct, kernel, cokernel, quotient, tensor product, completion, or another universal construction, the proof transports only when that constructor belongs to \(\Pi_F\).
783 \end{theorem}
784 
785 \begin{proof}
786 Functoriality preserves identities and composition.  Additivity preserves zero, sums, and additive inverses.  Linearity preserves the declared scalar action.  Structural induction on the morphism expression transports every equational step.  When a step is characterized by a universal property, the verified profile witness identifies the image construction with the corresponding target construction.  This is the equational realization principle of Paper I \cite{P1}.
787 \end{proof}
788 
789 \begin{proposition}[Profile-safe theorem transport]
790 \label{prop:profile-safe-theorem-transport}
791 Let \(\mathcal P\) prove a statement \(T\) in the resolved syntax, and let \(F\) be a realization or bridge.  Suppose
792 \[
793 \operatorname{csupp}(\mathcal P)\subseteq\Pi_F.
794 \]
795 After every additional domain hypothesis used by \(T\) has been verified in the target, \(F(T)\) is a valid target theorem.  If the inclusion fails, the source proof does not by itself establish the target statement.
796 \end{proposition}
797 
798 \begin{proof}
799 Apply \cref{thm:equational-realization-expanded} to every equational and universal-construction step in the proof certificate.  The remaining assumptions are precisely the domain-specific hypotheses, which are checked separately.
800 \end{proof}
801 
802 \begin{definition}[Semantic realization certificate]
803 \label{def:semantic-realization-certificate}
804 A semantic realization certificate for transporting a theorem consists of four fields:
805 \begin{enumerate}[label=\textup{(C\arabic*)}]
806 \item an explicit functor or intertwining construction;
807 \item its verified preservation profile;
808 \item a proof-support certificate for the source theorem together with the inclusion into the profile;
809 \item verification of every target-specific hypothesis and a restatement of the conclusion in native target objects.
810 \end{enumerate}
811 \end{definition}
812 
813 \begin{table}[htbp]
814 \centering
815 \caption{Typical preservation gates.  The right column names the extra obligation rather than asserting it automatically.}
816 \label{tab:preservation-gates}
817 \small
818 \begin{tabularx}{\textwidth}{@{}L{0.22\textwidth}L{0.25\textwidth}Y@{}}
819 \toprule
820 Proof step & Required profile entry & Additional target obligation\\
821 \midrule
822 Finite block identity & finite sums, composition, packet idempotents & verify that the declared packet equations hold\\
823 Kernel or cokernel argument & the corresponding universal property & check the target category has the selected construction\\
824 Quotient descent & quotient preservation & identify the target quotient and, topologically, its quotient topology\\
825 Infinite reconstruction & declared convergence/completion doctrine & specify norm, strong, weak, or unconditional convergence\\
826 Tensor reconstruction & tensor preservation and coherence & verify associativity, unit, and any symmetry or dagger structure used\\
827 Positivity or best approximation & adjoint, positivity, Hilbert structure & verify orthogonality and the relevant operator ideal\\
828 Arithmetic congruence & integral or local realization data & verify valuation, denominator, and reduction hypotheses\\
829 \bottomrule
830 \end{tabularx}
831 \end{table}
832 
833 \begin{figure}[htbp]
834 \centering
835 \includegraphics[width=0.98\textwidth]{profile_transport_gate.pdf}
836 \caption{The theorem-transport gate.  A shared vocabulary is not enough: the constructors used by the proof must lie inside the verified profile, and target-specific hypotheses must still be checked.}
837 \label{fig:profile-transport-gate}
838 \end{figure}
839 
840 \begin{boundary}[Terminology is not semantic transport]
841 \label{bnd:terminology-not-transport}
842 Calling a frequency band, residue class, ramification sector, or latent channel a ``packet'' is only terminological reuse until the packet idempotents, reconstruction law, transfer blocks, and required preservation maps are supplied.  Failure of a profile gate is an explicit open obligation, not permission to infer the desired theorem by analogy.
843 \end{boundary}
844 
845 \subsection{The finite additive-Karoubi resolved envelope}
846 \label{subsec:additive-karoubi-envelope-expanded}
847 
848 The full profile-indexed envelope remains a multi-doctrine problem.  The finite linear, additive, and split-idempotent core can be constructed exactly.
849 
850 \begin{definition}[Linear grammar category]
851 \label{def:linear-grammar-category}
852 For a finite linear resolved presentation \(\mathbb K_{\rm lin}=(Q,I,\mathsf P,\mathsf{Ref})\), let
853 \[
854 \mathcal A_{\mathbb K}:=\kk Q/I.
855 \]
856 This is the \(\kk\)-linear category whose objects are the vertices of \(Q\) and whose morphisms are \(\kk\)-linear combinations of typed paths modulo the declared relations.
857 \end{definition}
858 
859 \begin{definition}[Finite additive hull]
860 \label{def:additive-hull-expanded}
861 The category \(\Add(\mathcal A_{\mathbb K})\) has finite tuples
862 \[
863 A=(A_1,\dots,A_m)
864 \]
865 as objects.  A morphism \(A\to B=(B_1,\dots,B_n)\) is an \(n\times m\) matrix with entries in \(\mathcal A_{\mathbb K}(A_j,B_i)\).  Composition is matrix multiplication using composition and addition in \(\mathcal A_{\mathbb K}\).
866 \end{definition}
867 
868 \begin{definition}[Karoubi envelope]
869 \label{def:karoubi-envelope-expanded}
870 The Karoubi envelope \(\Kar(\Add(\mathcal A_{\mathbb K}))\) has pairs \((A,e)\), where \(e:A\to A\) is idempotent, as objects.  A morphism
871 \[
872 f:(A,e)\longrightarrow(B,d)
873 \]
874 is a morphism \(f:A\to B\) satisfying
875 \[
876 f=dfe.
877 \]
878 The identity on \((A,e)\) is \(e\).
879 \end{definition}
880 
881 \begin{theorem}[Finite additive-Karoubi resolved envelope]
882 \label{thm:additive-karoubi-envelope}
883 The category
884 \[
885 \boxed{
886 \Syn_{\mathrm{ak}}(\mathbb K_{\rm lin})
887 :=
888 \Kar\!\bigl(\Add(\kk Q/I)\bigr)}
889 \]
890 is small, additive, idempotent complete, and \(\kk\)-linear.  For every additive idempotent-complete \(\kk\)-linear category \(\mathcal C\), restriction to the generating quiver induces an equivalence
891 \[
892 \Fun_{\kk}^{\oplus}
893 \bigl(\Syn_{\mathrm{ak}}(\mathbb K_{\rm lin}),\mathcal C\bigr)
894 \simeq
895 \Mod_{\mathrm{ak}}(\mathbb K_{\rm lin},\mathcal C),
896 \]
897 where the right side is the category of typed realizations satisfying all declared relations and carrying every declared packet idempotent to a split idempotent.
898 \end{theorem}
899 
900 \begin{proof}
901 There are four universal stages.
902 
903 \emph{Free typed stage.}  The free \(\kk\)-linear category on \(Q\) is universal for assigning objects to vertices and morphisms to arrows.
904 
905 \emph{Relation stage.}  Quotienting by \(I\) imposes exactly the declared linear, packet, and refinement equations.  A functor out of \(\kk Q/I\) is therefore the same as a typed realization satisfying those equations.
906 
907 \emph{Additive stage.}  Every \(\kk\)-linear functor from \(\kk Q/I\) into an additive category extends, uniquely up to canonical natural isomorphism, to an additive functor from the finite additive hull by sending a tuple to the corresponding biproduct and a matrix to its induced biproduct map.
908 
909 \emph{Idempotent-splitting stage.}  Every additive functor into an idempotent-complete category extends, again uniquely up to canonical natural isomorphism, across the Karoubi envelope by sending \((A,e)\) to the chosen image of the split idempotent \(F(e)\).  Morphisms satisfying \(f=dfe\) descend to the corresponding split summands.
910 
911 Composing these universal properties yields the displayed equivalence.  Smallness is preserved because only finite tuples and idempotents in hom sets of a small category are adjoined.
912 \end{proof}
913 
914 \begin{proposition}[Fully faithful grammar embedding]
915 \label{prop:grammar-embedding}
916 The canonical composite
917 \[
918 \mathcal A_{\mathbb K}
919 \longrightarrow
920 \Add(\mathcal A_{\mathbb K})
921 \longrightarrow
922 \Kar(\Add(\mathcal A_{\mathbb K}))
923 \]
924 is fully faithful.  Thus the envelope adds finite biproducts and split summands without identifying two previously distinct grammar morphisms.
925 \end{proposition}
926 
927 \begin{proof}
928 The first functor sends an object to the one-term tuple and preserves its hom set exactly.  The second sends \(A\) to \((A,1_A)\); its morphism condition is \(f=1_Bf1_A\), which imposes no additional relation.
929 \end{proof}
930 
931 \begin{corollary}[Declared packets become literal objects]
932 \label{cor:packet-objects-literal}
933 Every declared idempotent \(p:X\to X\) defines an object \((X,p)\) of \(\Syn_{\mathrm{ak}}(\mathbb K_{\rm lin})\).  A complete packet family \((p_i)\) gives the canonical isomorphism
934 \[
935 (X,1_X)\cong\bigoplus_i(X,p_i).
936 \]
937 \end{corollary}
938 
939 \begin{corollary}[Linear resolved strata as model categories]
940 \label{cor:linear-strata-model-categories}
941 Every additive split-idempotent realization stratum of \(\mathbb K_{\rm lin}\) is a category of profile-preserving functors out of \(\Syn_{\mathrm{ak}}(\mathbb K_{\rm lin})\).
942 \end{corollary}
943 
944 \begin{example}[One operator and primary packets]
945 \label{ex:single-operator-envelope}
946 Take one sort, one endomorphism \(T\), and one relation \(\mu(T)=0\).  The linear grammar algebra is \(\kk[x]/(\mu)\).  When
947 \[
948 \mu=\mu_1\cdots\mu_r
949 \]
950 is a product of pairwise comaximal factors, the Chinese remainder idempotents define packet objects for the primary summands.  The envelope splits those idempotents formally.  Over a realization where the same factorization remains valid, the packet objects become the corresponding invariant primary sectors.
951 \end{example}
952 
953 \begin{remark}[Universal does not mean presentation-minimal]
954 \label{rem:universal-not-minimal}
955 The envelope is universal for realizations of the declared presentation.  It does not provide a canonical smallest quiver, decide equivalence of arbitrary presentations, or identify a globally minimal packet system.  Those are recognition or minimization problems inside particular strata.
956 \end{remark}
957 
958 \subsection{Profile-indexed extensions and the exact boundary of Foundation layer}
959 \label{subsec:profile-indexed-extension-boundary}
960 
961 \begin{definition}[Profile-indexed resolved envelope]
962 \label{def:profile-indexed-envelope}
963 For a preservation profile \(\Pi\), a profile-indexed resolved envelope \(\Syn_\Pi(\mathbb K)\), when it exists, is a universal source satisfying
964 \[
965 \Mod_\Pi(\mathbb K,\mathcal C)
966 \simeq
967 \Fun_\Pi(\Syn_\Pi(\mathbb K),\mathcal C)
968 \]
969 for every target category \(\mathcal C\) carrying the structures named by \(\Pi\).  The equivalence must preserve the chosen comparison data, not only the underlying functors.
970 \end{definition}
971 
972 \begin{table}[htbp]
973 \centering
974 \caption{Profile extensions beyond the proved additive-Karoubi core.}
975 \label{tab:profile-extensions}
976 \small
977 \begin{tabularx}{\textwidth}{@{}L{0.19\textwidth}L{0.28\textwidth}Y@{}}
978 \toprule
979 Doctrine & Additional data required & Boundary\\
980 \midrule
981 Nonlinear / operadic & colored finite-arity operations, symmetric actions, substitution, coherence & available locally in the nonlinear packet calculus, not freely present in every stratum\\
982 Banach / topological & topology, boundedness, convergence mode, completion and quotient-topology rules & algebraic packet sums do not choose a topology\\
983 \(C^*\) / \(W^*\) & involution, norm, positivity, closure, multiplier or weak completion & faithful corner realization does not automatically preserve an independent tensor product\\
984 Countable packets & strong, norm, weak, or unconditional convergence with boundedness controls & finite identities do not determine the infinite convergence doctrine\\
985 Monoidal / Tannakian & tensor product, unit, coherence, exact faithful fiber functor when reconstruction is claimed & ordinary faithful representation is weaker than monoidal reconstruction\\
986 Arithmetic / local & integral models, reductions, valuations, completeness, residue and lift data & denominator and reduction hypotheses remain domain-specific\\
987 \bottomrule
988 \end{tabularx}
989 \end{table}
990 
991 \begin{figure}[htbp]
992 \centering
993 \includegraphics[width=0.99\textwidth]{foundation_envelope_stack.pdf}
994 \caption{The proved common envelope and the optional profile branches.  The branches should be added only for the strata that actually preserve them.}
995 \label{fig:foundation-envelope-stack}
996 \end{figure}
997 
998 \begin{boundary}[No single overpowered envelope]
999 \label{bnd:no-overpowered-envelope}
1000 The existence of \(\Syn_{\mathrm{ak}}\) does not prove the existence of one universal category simultaneously free, complete, \(C^*\), \(W^*\), nonlinear-operadic, monoidal, and arithmetically integral in a way preserved by every target.  The mathematically safer target is a family of profile-indexed envelopes or a doctrine-valued construction with explicit forgetful and comparison maps.
1001 \end{boundary}
1002 
1003 \begin{resultbox}
1004 \textbf{Section result.}
1005 The foundational language is now explicit at four levels: typed grammar semantics, exact finite packet and residual calculus, proof-support-controlled theorem transport, and the universal additive-Karoubi envelope.  The nonlinear, analytic, operator-algebraic, monoidal, and arithmetic envelopes remain separate profile obligations rather than hidden assumptions.
1006 \end{resultbox}
1007 
1008 \section{Strata, normalized charts, typed bridges, and obstruction objects}
1009 \label{sec:strata-bridges-obstructions}
1010 
1011 The common envelope developed in \cref{sec:resolved-foundations} is deliberately weaker than most concrete mathematical domains.  The next layer records how that envelope is realized, normalized, compared, and sometimes only partially remembered.  The central organizing rule is that a chart is not specified by a list of examples.  It is specified by a category of models, a preservation profile, a semantic functor, a normalized classification target, and explicit overlap data.
1012 
1013 \subsection{Stratum passports and normalized model categories}
1014 \label{subsec:stratum-passports}
1015 
1016 \begin{definition}[Profiled stratum passport]
1017 \label{def:profiled-stratum-passport}
1018 A profiled Komorebi stratum is a record
1019 \[
1020 \mathbf S_\alpha
1021 =
1022 (\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha,
1023  \mathcal N_\alpha,\Sem_\alpha,\mathcal I_\alpha),
1024 \]
1025 with the following fields.
1026 \begin{enumerate}[label=\textup{(\roman*)}]
1027 \item \(\mathcal C_\alpha\) is the ambient semantic category in which the realized objects live.
1028 \item \(\Pi_\alpha\) is the verified preservation profile of the realization.
1029 \item \(\mathcal M_\alpha\) is a full replete category of admissible models of the declared resolved syntax.
1030 \item \(\mathcal N_\alpha\subseteq\mathcal M_\alpha\) is a full replete normalized subcategory on which the intended inverse or classification theorem is stated.
1031 \item
1032 \[
1033 \Sem_\alpha:\mathcal M_\alpha\longrightarrow\mathcal O_\alpha
1034 \]
1035 is the native semantics functor.
1036 \item
1037 \[
1038 \mathcal I_\alpha:\Core(\mathcal N_\alpha)\longrightarrow\mathcal D_\alpha
1039 \]
1040 is a declared invariant functor, when one has been constructed.  Its codomain \(\mathcal D_\alpha\) and classification target are part of the record.
1041 \end{enumerate}
1042 The original four-field stratum
1043 \((\mathcal C_\alpha,\Pi_\alpha,\mathcal M_\alpha,\Sem_\alpha)\)
1044 remains the underlying realization datum.  The normalized and invariant fields are additional assembly data rather than automatic consequences.
1045 \end{definition}
1046 
1047 \begin{definition}[Normalization witness]
1048 \label{def:normalization-witness}
1049 A normalization witness on a full subcategory \(\mathcal U_\alpha\subseteq\mathcal M_\alpha\) consists of a functor
1050 \[
1051 N_\alpha:\mathcal U_\alpha\longrightarrow\mathcal N_\alpha
1052 \]
1053 and a natural isomorphism
1054 \[
1055 \nu_\alpha:\Sem_\alpha N_\alpha
1056 \Longrightarrow
1057 \Sem_\alpha|_{\mathcal U_\alpha}.
1058 \]
1059 It is \emph{classification-normalizing} when every object of \(\mathcal U_\alpha\) is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic in \(\mathcal N_\alpha\).  A normalization may be partial, domain-restricted, or defined only after auxiliary choices.  Those restrictions belong to the passport.
1060 \end{definition}
1061 
1062 \begin{definition}[Complete invariant on a declared target]
1063 \label{def:complete-invariant-target}
1064 An invariant \(\mathcal I_\alpha\) is complete on \(\mathcal N_\alpha\) when
1065 \[
1066 \mathcal I_\alpha(X)\cong\mathcal I_\alpha(Y)
1067 \quad\Longleftrightarrow\quad
1068 X\cong Y
1069 \]
1070 for all \(X,Y\in\mathcal N_\alpha\).  This statement classifies the objects of \(\mathcal N_\alpha\) up to the isomorphisms allowed there.  It does not classify arbitrary presentations outside that subcategory.
1071 \end{definition}
1072 
1073 \begin{proposition}[Semantic conservativity of a normalized chart]
1074 \label{prop:normalized-semantic-conservativity}
1075 If \(N_\alpha\) is classification-normalizing, then
1076 \[
1077 \Sem_\alpha|_{\mathcal N_\alpha}
1078 \]
1079 reflects isomorphisms.  Consequently, a complete invariant on \(\mathcal N_\alpha\) can be transported through a reversible overlap only after the overlap is shown to preserve the normalization and reflect the relevant isomorphisms.
1080 \end{proposition}
1081 
1082 \begin{proof}
1083 Let \(X,Y\in\mathcal N_\alpha\) and suppose their semantic objects are isomorphic.  The second clause of \cref{def:normalization-witness} gives \(X\cong Y\).  The final statement follows because completeness is an assertion about isomorphism classes, not merely about equality of numerical outputs.
1084 \end{proof}
1085 
1086 \begin{figure}[htbp]
1087 \centering
1088 \includegraphics[width=0.98\textwidth]{stratum_passport.pdf}
1089 \caption{A stratum passport separates the ambient category, admissible models, preservation profile, semantics, normalization, and invariant target.  Omitting any one of these fields makes cross-stratum classification claims ambiguous.}
1090 \label{fig:stratum-passport}
1091 \end{figure}
1092 
1093 The present program uses the following principal strata.  The normalized fields are intentionally narrower than the ambient strata.  Several rows have exact slices but no canonical normalization for the whole broad class \cite{SEQ,GAL,FMH,NLO}.
1094 
1095 \begin{landscape}
1096 \scriptsize
1097 \begin{longtable}{@{}L{0.095\linewidth}L{0.15\linewidth}L{0.135\linewidth}L{0.165\linewidth}L{0.19\linewidth}L{0.165\linewidth}@{}}
1098 \caption{Current stratum passports at the level supported by the audited Library.  A dash means that no whole-stratum normalization theorem is asserted.}
1099 \label{tab:stratum-passports}\\
1100 \toprule
1101 Stratum & Native objects & Native morphisms & Normalized or exact slice & Exact coordinate or invariant already available & Remaining boundary\\
1102 \midrule
1103 \endfirsthead
1104 \toprule
1105 Stratum & Native objects & Native morphisms & Normalized or exact slice & Exact coordinate or invariant already available & Remaining boundary\\
1106 \midrule
1107 \endhead
1108 Finite linear and C-finite & finite recurrences, rational OGFs, finite linear states & intertwiners, recurrence changes, similarity & minimal recurrence, reduced OGF, reachable-observable state & Hankel rank, reduced denominator degree, minimal state dimension, torsion--Fourier subsequence profile & no classification of arbitrary nonminimal or mixed-grammar presentations\\
1109 Holonomic & P-recursive sequences, D-finite functions, differential modules & coefficient/differential intertwiners & fixed operator convention and selected local data & effective P-recursive--D-finite translation; finite annihilator modules & minimal multisection operators, singularity orbits, monodromy aliasing, and cancellation signatures\\
1110 Digital & DFAO, finite digit kernels, regular linear representations & state maps and radix-compatible intertwiners & minimal DFAO for fixed radix and reading direction & unique minimal DFAO up to state renaming; exact finite \(k\)-kernel compilation & no all-moduli packet profile or general linear-state minimality for \(k\)-regular systems\\
1111 Algebraic & selected algebraic branches and packet polynomial systems & branch-preserving polynomial maps & a declared branch with a fixed defining system & exact packet polynomial realization and synthesis & minimal system dimension, branch-collision data, and effective packet-degree bounds\\
1112 Nonlinear/tree & packet jets, colored operations, rooted-tree systems & operadic substitutions and smooth intertwiners & marked polynomial fixed points; positive simply generated tree families & packet Fa\`a di Bruno law, diagonal profile compression, SCC cyclic core, arity-gcd support and dominant-orbit signature & no global nonlinear normal form, mixed-direction universal compression, or coordinate-free minimal system dimension\\
1113 Product and multiplicative & filtered Euler products, prime-power seed systems & exponent and local-factor maps & fixed Euler basis or declared multiplicative class & unique Euler exponents in the fixed basis; exact reconstruction from prime-power seeds & no canonical factorization across arbitrary product grammars and no general additive--multiplicative equivalence\\
1114 Local, Mahler, and polyphase & Newton--Mahler coordinates, affine branches, functional masks, local states & affine restriction, refinement, Cartier and polyphase maps & fixed local chart and fixed modulus/radix conventions & unique Mahler coordinates, exact affine-polyphase transfer, exact Cartier automata for rational masks & no canonical global local normal form and no unrestricted bridge between functional and Newton--Mahler theories\\
1115 Galois and Tannakian & semilinear packets, fixed-field charts, tensor representations & equivariant maps, induction/restriction, tensor morphisms & finite separable semisimple descent or neutral Tannakian fiber functor & Hilbert 90 descent, Brauer obstruction, subgroup/fixed-field classification, tensor reconstruction & modular and wild cases, and monoidal reconstruction without a verified fiber functor\\
1116 \bottomrule
1117 \end{longtable}
1118 \end{landscape}
1119 
1120 \begin{boundary}[Multiple valid placements]
1121 \label{bnd:multiple-stratum-placements}
1122 A semantic object may occur in several strata.  Catalan numbers, for example, may be algebraic, hypergeometric, P-recursive, and tree-generated.  The atlas records these as distinct presentations with certified bridges.  It does not force one placement to erase the others.
1123 \end{boundary}
1124 
1125 \subsection{Representation groupoids, gauge levels, and the directed transformation category}
1126 \label{subsec:groupoid-directed}
1127 
1128 For every stratum, the invertible part is the maximal subgroupoid
1129 \[
1130 \mathfrak{Rep}_\alpha(\bbK)
1131 =
1132 \Core(\mathcal M_\alpha).
1133 \]
1134 This core records changes of presentation that preserve all structure required by the overlap.  Quotients, truncations, completions, specializations, differential annihilator maps, and compression maps generally remain outside it.
1135 
1136 \begin{definition}[Finite coordinate action groupoid]
1137 \label{def:coordinate-action-groupoid}
1138 Fix a dimension vector \(\mathbf d=(d_X)_{X\in\mathsf S}\).  Let
1139 \[
1140 G_{\mathbf d}=\prod_{X\in\mathsf S}GL_{d_X}(\kk)
1141 \]
1142 act on the finite-dimensional representation space \(\Rep_{\mathbf d}(\bbK)\) by base change.  The action groupoid
1143 \[
1144 [\Rep_{\mathbf d}(\bbK)/G_{\mathbf d}]
1145 \]
1146 has representations as objects and coordinate changes carrying one representation to another as morphisms.
1147 \end{definition}
1148 
1149 \begin{proposition}[Coordinate groupoid realizes isomorphism]
1150 \label{prop:coordinate-groupoid-isomorphism}
1151 Two objects of \(\Rep_{\mathbf d}(\bbK)\) are isomorphic as typed realizations exactly when they lie in the same \(G_{\mathbf d}\)-orbit.  The stabilizer of a point is its coordinate automorphism group.
1152 \end{proposition}
1153 
1154 \begin{proof}
1155 A typed isomorphism is a family of invertible linear maps on the sort spaces intertwining every generating operation.  This is precisely an element of \(G_{\mathbf d}\) carrying one coordinate tuple to the other.  Elements fixing the tuple are exactly the typed automorphisms.
1156 \end{proof}
1157 
1158 Three symmetry levels must remain separate.
1159 
1160 \begin{table}[htbp]
1161 \centering
1162 \caption{Three noninterchangeable symmetry notions.}
1163 \label{tab:symmetry-levels}
1164 \small
1165 \begin{tabularx}{\textwidth}{@{}L{0.18\textwidth}L{0.26\textwidth}Y L{0.22\textwidth}@{}}
1166 \toprule
1167 Symmetry level & Acting object & What it changes or preserves & Extra hypothesis\\
1168 \midrule
1169 Gauge symmetry & coordinate-change group \(G_{\mathbf d}\) or a local gauge family & changes presentation coordinates while preserving the represented object & an explicitly typed action on the chart\\
1170 Intrinsic symmetry & \(\Aut_{\mathcal M_\alpha}(M)\) & automorphisms of one resolved object inside the stratum & none beyond the native category\\
1171 Reconstructed symmetry & \(\Aut^{\otimes}(\omega)\) & tensor automorphisms of a fiber functor and the corresponding affine group scheme & faithful exact tensor fiber functor\\
1172 \bottomrule
1173 \end{tabularx}
1174 \end{table}
1175 
1176 \begin{definition}[Typed directed transformation category]
1177 \label{def:directed-transformation-category}
1178 The directed transformation category \(\mathfrak{Dir}(\bbK)\) has pairs \((\alpha,M)\), with \(M\in\mathcal M_\alpha\), as objects.  A morphism
1179 \[
1180 (\alpha,M)\longrightarrow(\beta,N)
1181 \]
1182 is an admitted bridge instance equipped with a bridge passport as in \cref{def:bridge-passport}.  Composition is defined only when the codomain profile of the first bridge satisfies the domain requirements of the second.  The union of the representation groupoids embeds as the wide invertible subcategory.
1183 \end{definition}
1184 
1185 \begin{boundary}[Invertibility is evidence, not typography]
1186 \label{bnd:invertibility-evidence}
1187 Drawing a two-headed arrow does not make a bridge reversible.  A bridge enters the representation groupoid only after an inverse, quasi-inverse, or recognition theorem has been proved on a named overlap category.
1188 \end{boundary}
1189 
1190 \subsection{Bridge passports, bridge classes, and information-loss profiles}
1191 \label{subsec:bridge-passports}
1192 
1193 \begin{definition}[Bridge passport]
1194 \label{def:bridge-passport}
1195 A certified bridge from stratum \(\alpha\) to stratum \(\beta\) is a record
1196 \[
1197 \mathfrak b
1198 =
1199 (\mathcal U_\mathfrak b,\mathcal V_\mathfrak b,
1200  B_\mathfrak b,\tau_\mathfrak b,
1201  \Pi^{\mathrm{pres}}_\mathfrak b,
1202  \Pi^{\mathrm{refl}}_\mathfrak b,
1203  \Lambda_\mathfrak b,
1204  \nu_\mathfrak b,\chi_\mathfrak b),
1205 \]
1206 where:
1207 \begin{enumerate}[label=\textup{(\roman*)}]
1208 \item \(\mathcal U_\mathfrak b\subseteq\mathcal M_\alpha\) and \(\mathcal V_\mathfrak b\subseteq\mathcal M_\beta\) are the declared domain and codomain subcategories;
1209 \item \(B_\mathfrak b:\mathcal U_\mathfrak b\to\mathcal V_\mathfrak b\) is the functor;
1210 \item \(\tau_\mathfrak b\) is its bridge class;
1211 \item \(\Pi^{\mathrm{pres}}_\mathfrak b\) records preserved constructors and universal properties;
1212 \item \(\Pi^{\mathrm{refl}}_\mathfrak b\) records properties or isomorphisms reflected by the bridge;
1213 \item \(\Lambda_\mathfrak b\) is the information-loss profile;
1214 \item \(\nu_\mathfrak b\) is the semantic comparison map;
1215 \item \(\chi_\mathfrak b\) records normalization, residual, and certificate compatibility.
1216 \end{enumerate}
1217 A passport may state that no inverse or no certificate transport is known.  Missing data remain missing rather than being inferred from analogy.
1218 \end{definition}
1219 
1220 The permitted bridge classes follow the work program in the controlling handoff.
1221 
1222 \begin{table}[htbp]
1223 \centering
1224 \caption{Bridge classes used by the concrete atlas.}
1225 \label{tab:bridge-classes}
1226 \small
1227 \begin{tabularx}{\textwidth}{@{}L{0.22\textwidth}L{0.20\textwidth}Y@{}}
1228 \toprule
1229 Class & Required evidence & Interpretation\\
1230 \midrule
1231 Equivalence & quasi-inverse and coherent unit/counit & reversible overlap of presentation groupoids\\
1232 Fully faithful inclusion & bijections on hom-sets and full image description & no morphism loss inside the image, but possibly a smaller chart\\
1233 Conservative forgetful functor & reflected isomorphisms & object equivalence remains detectable although structure may be forgotten\\
1234 Information-losing directional functor & explicit semantic and loss statement & quotient, projection, annihilator, reduction, or compression\\
1235 Constructor & proof that source data generate a target object & one-way realization, often with nonunique inverse data\\
1236 Completion & topology and universal completion property & adds limits or closes a domain; reversibility requires a recognition theorem\\
1237 Partially defined bridge & exact domain predicate & valid only on a named locus, branch, modulus, or regularity class\\
1238 No certified bridge & none & no arrow is inserted into the trusted atlas\\
1239 \bottomrule
1240 \end{tabularx}
1241 \end{table}
1242 
1243 \begin{definition}[Information-loss profile]
1244 \label{def:information-loss-profile}
1245 For a functor \(B:\mathcal U\to\mathcal V\), its information-loss profile contains at least:
1246 \begin{align*}
1247 \operatorname{MorKer}_B(X,Y)
1248 &=
1249 \{(f,g)\in\Hom_{\mathcal U}(X,Y)^2:Bf=Bg\},\\
1250 \operatorname{Coll}_B
1251 &=
1252 \{(X,Y)\in\operatorname{Ob}(\mathcal U)^2:BX\cong BY\},\\
1253 \operatorname{EssIm}(B)
1254 &\subseteq\mathcal V,
1255 \end{align*}
1256 together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by \(B\).  The first component records morphism collapse, the second object collapse, and the third which target objects actually arise.
1257 \end{definition}
1258 
1259 \begin{definition}[Faithfulness, conservativity, and reversibility]
1260 \label{def:lossless-bridge}
1261 A bridge is:
1262 \begin{enumerate}[label=\textup{(\roman*)}]
1263 \item \emph{morphism-faithful} when every \(\operatorname{MorKer}_B(X,Y)\) is diagonal;
1264 \item \emph{conservative} when \(Bf\) invertible implies that \(f\) is invertible;
1265 \item \emph{classification-faithful} or \emph{essentially injective on objects} when \(BX\cong BY\) implies \(X\cong Y\) on the declared domain;
1266 \item \emph{lossless on its image} when it is fully faithful;
1267 \item \emph{reversible} when it is an equivalence onto the declared overlap.
1268 \end{enumerate}
1269 These conditions answer different questions and should not be conflated.
1270 \end{definition}
1271 
1272 \begin{proposition}[Composition law for bridge loss]
1273 \label{prop:bridge-loss-composition}
1274 Let
1275 \[
1276 \mathcal U\xrightarrow{B}\mathcal V\xrightarrow{C}\mathcal W
1277 \]
1278 be certified bridges.
1279 \begin{enumerate}[label=\textup{(\roman*)}]
1280 \item Every constructor or universal property preserved by both \(B\) and \(C\) is preserved by \(CB\).
1281 \item
1282 \[
1283 \operatorname{MorKer}_B(X,Y)
1284 \subseteq
1285 \operatorname{MorKer}_{CB}(X,Y).
1286 \]
1287 \item
1288 \[
1289 \operatorname{Coll}_B
1290 \subseteq
1291 \operatorname{Coll}_{CB}.
1292 \]
1293 \item
1294 \[
1295 \operatorname{EssIm}(CB)
1296 \subseteq
1297 \operatorname{EssIm}(C).
1298 \]
1299 \item If both bridges are conservative, then their composite is conservative.  If both are classification-faithful, then their composite is classification-faithful.  If both are equivalences, then their composite is an equivalence.
1300 \end{enumerate}
1301 Thus information forgotten at an earlier stage cannot be recovered by merely composing more forgetful bridges, although a later bridge may attach auxiliary data through a separate constructor.
1302 \end{proposition}
1303 
1304 \begin{proof}
1305 Functoriality proves the preservation claim.  If \(Bf=Bg\), then \(CBf=CBg\), proving the second inclusion.  If \(BX\cong BY\), applying \(C\) gives \(CBX\cong CBY\), proving the third.  Every object in the essential image of \(CB\) is visibly in the essential image of \(C\).  For conservativity, if \(CBf\) is invertible then \(Bf\) is invertible because \(C\) is conservative, and then \(f\) is invertible because \(B\) is conservative.  For classification-faithfulness, \(CBX\cong CBY\) implies \(BX\cong BY\) and then \(X\cong Y\).  Composition of equivalences is standard.
1306 \end{proof}
1307 
1308 \begin{figure}[htbp]
1309 \centering
1310 \includegraphics[width=0.98\textwidth]{bridge_passport_and_loss.pdf}
1311 \caption{A bridge passport separates reversible equivalence from directional transformation and records the dimensions of information loss.  The reversible core is a theorem-level subcategory, not an informal interpretation of the whole bridge.}
1312 \label{fig:bridge-passport-loss}
1313 \end{figure}
1314 
1315 Three typing cautions are especially important.
1316 
1317 \begin{enumerate}
1318 \item A selected algebraic branch can produce a D-finite annihilator, but the operator may forget the selected branch and its minimal polynomial.  The bridge is directional unless that data are retained.
1319 \item A rational functional-Mahler mask over \(\mathbf F_p\) produces a Cartier representation and finite automaton under the stated hypotheses, but this does not identify it with a Newton--Mahler interpolation chart \cite{FMH,SEQ}.
1320 \item A faithful Hilbert-space corner representation realizes packet projectors and transfers, but it does not automatically preserve a tensor product strongly enough for Tannakian reconstruction \cite{RT,GAL}.
1321 \end{enumerate}
1322 
1323 \subsection{Overlap categories and bridge-law certificates}
1324 \label{subsec:overlap-categories}
1325 
1326 \begin{definition}[Typed overlap]
1327 \label{def:typed-overlap}
1328 A reversible overlap between strata \(\alpha\) and \(\beta\) consists of full replete subcategories
1329 \[
1330 \mathcal M_{\alpha\beta}\subseteq\mathcal M_\alpha,
1331 \qquad
1332 \mathcal M_{\beta\alpha}\subseteq\mathcal M_\beta,
1333 \]
1334 an equivalence
1335 \[
1336 \Phi_{\alpha\beta}:\mathcal M_{\alpha\beta}
1337 \overset{\sim}{\longrightarrow}
1338 \mathcal M_{\beta\alpha},
1339 \]
1340 and a semantic natural isomorphism
1341 \[
1342 \sigma_{\alpha\beta}:
1343 J_{\alpha\beta}\Sem_\alpha
1344 \Longrightarrow
1345 \Sem_\beta\Phi_{\alpha\beta}
1346 \]
1347 for an explicitly declared comparison functor \(J_{\alpha\beta}\) between the observable doctrines.  An overlap is \emph{normalized} when the equivalence restricts to the selected normalized subcategories.
1348 \end{definition}
1349 
1350 \begin{definition}[Bridge-law certificate]
1351 \label{def:bridge-law-certificate}
1352 A bridge-law certificate for \(B:\mathcal U\to\mathcal V\) records commuting or naturally commuting diagrams for each structure used downstream:
1353 \begin{enumerate}[label=\textup{(\roman*)}]
1354 \item native semantics;
1355 \item packet analysis and synthesis;
1356 \item refinement and coarsening maps;
1357 \item selected-support compression and named residuals;
1358 \item proof or computation certificates;
1359 \item normalization and invariant comparison maps.
1360 \end{enumerate}
1361 For example, packet naturality requires comparison maps under which
1362 \[
1363 B(p_{Y,b}fp_{X,a})
1364 \cong
1365 p'_{B(Y),b}\,B(f)\,p'_{B(X),a},
1366 \]
1367 and refinement naturality requires the image of each coarse packet to equal the declared sum of its refined descendants.
1368 \end{definition}
1369 
1370 \begin{proposition}[Bridge-law transport]
1371 \label{prop:bridge-law-transport}
1372 Let \(\mathcal P\) be a proof whose constructor support lies in the preserved part of a bridge-law certificate.  Then the image proof is valid in the target chart.  If the proof concludes equality of normalized invariants, that equality descends across the bridge only when the normalization and invariant comparison maps are also included in the certificate.
1373 \end{proposition}
1374 
1375 \begin{proof}
1376 The first statement is the equational realization principle of Paper I applied with the bridge-specific preservation profile.  The second statement follows because normalization and invariant functors are additional constructors in the proof diagram; they cannot be transported when their comparison cells are absent \cite{P1}.
1377 \end{proof}
1378 
1379 On triple overlaps, pairwise equivalences are not enough.  One needs comparison cells
1380 \[
1381 \eta_{\alpha\beta\gamma}:
1382 \Phi_{\beta\gamma}\Phi_{\alpha\beta}
1383 \Longrightarrow
1384 \Phi_{\alpha\gamma}.
1385 \]
1386 For four charts, there are two composites from
1387 \(
1388 \Phi_{\gamma\delta}\Phi_{\beta\gamma}\Phi_{\alpha\beta}
1389 \)
1390 to \(\Phi_{\alpha\delta}\).  Their ratio is the higher coherence defect.
1391 
1392 \begin{definition}[Quadruple-overlap coherence defect]
1393 \label{def:quadruple-coherence-defect}
1394 Write horizontal composition of natural transformations as \(\ast\).  Define
1395 \[
1396 \Omega_{\alpha\beta\gamma\delta}
1397 =
1398 \Bigl[
1399 \eta_{\alpha\gamma\delta}
1400 \circ
1401 (\id\ast\eta_{\alpha\beta\gamma})
1402 \Bigr]
1403 \circ
1404 \Bigl[
1405 \eta_{\alpha\beta\delta}
1406 \circ
1407 (\eta_{\beta\gamma\delta}\ast\id)
1408 \Bigr]^{-1}.
1409 \]
1410 It is a natural automorphism of \(\Phi_{\alpha\delta}\) on the quadruple overlap.  The atlas is coherent there exactly when
1411 \[
1412 \Omega_{\alpha\beta\gamma\delta}=\id.
1413 \]
1414 \end{definition}
1415 
1416 \begin{proposition}[Two elementary coherence mechanisms]
1417 \label{prop:elementary-coherence-mechanisms}
1418 The coherence defect vanishes in either of the following situations.
1419 \begin{enumerate}[label=\textup{(\roman*)}]
1420 \item All overlap functors compose strictly and every comparison cell is the identity.
1421 \item Between each pair of relevant normalized objects there is at most one admissible comparison isomorphism compatible with semantics.
1422 \end{enumerate}
1423 \end{proposition}
1424 
1425 \begin{proof}
1426 The first case is immediate.  In the second case, the two sides of the coherence equation are admissible comparison isomorphisms with the same source and target.  Uniqueness forces them to agree.  This is the mechanism used later in the normalized C-finite three-chart atlas.
1427 \end{proof}
1428 
1429 \begin{boundary}[Prestack versus stack]
1430 \label{bnd:prestack-versus-stack}
1431 Typed overlap functors and coherent comparison cells define atlas or prestack data.  The word \emph{stack} additionally requires effective descent: compatible local objects and morphisms must arise from global ones in the declared category.
1432 \end{boundary}
1433 
1434 \subsection{Reversible cores inside directional bridges}
1435 \label{subsec:reversible-core-directional}
1436 
1437 Directional maps can contain a canonical reversible overlap without being globally invertible.  Split packet analysis is the basic model.
1438 
1439 \begin{theorem}[Split bridge rectification]
1440 \label{thm:split-bridge-rectification}
1441 Let \(\mathcal A\) be an additive idempotent-complete category and suppose
1442 \[
1443 X\xrightarrow{A}Y\xrightarrow{L}X
1444 \]
1445 satisfies \(LA=\id_X\).  Put \(E=AL\).  Then:
1446 \begin{enumerate}[label=\textup{(\roman*)}]
1447 \item \(E^2=E\);
1448 \item \(Y\cong\im E\oplus\im(1-E)\);
1449 \item \(A:X\to\im E\) is an isomorphism with inverse induced by \(L\);
1450 \item \(L(1-E)=0\), so \(\im(1-E)\) is a named residual invisible to the decoder;
1451 \item the directional two-chart system becomes reversible after restricting the packet chart to the consistent core \(\im E\).
1452 \end{enumerate}
1453 In a module or vector-space realization, \(\im(1-E)=\ker L\).
1454 \end{theorem}
1455 
1456 \begin{proof}
1457 The identity \(E^2=A(LA)L=AL=E\) proves the first statement.  Idempotent completeness splits \(E\), producing the direct sum.  The image of \(A\) lies in \(\im E\), and \(L A=\id_X\) while \(A L=E\) acts as the identity on \(\im E\), so the restricted maps are inverse.  Finally,
1458 \[
1459 L(1-E)=L-LAL=L-L=0.
1460 \]
1461 For modules, every \(y\in\ker L\) satisfies \((1-E)y=y\), while every vector in \(\im(1-E)\) is killed by \(L\).
1462 \end{proof}
1463 
1464 \begin{corollary}[Exact meaning of a decoded packet core]
1465 \label{cor:decoded-packet-core}
1466 A left inverse proves exact recovery on the consistent packet core.  It does not prove that every ambient packet vector is consistent, that the residual is small, or that the packet representation is compressed.  Those are separate geometric, analytic, or coding claims.
1467 \end{corollary}
1468 
1469 The integral Mahler packet theorem is a concrete instance: the analysis matrix has an integer left inverse, the consistency projector is \(AL\), and the residual is the complementary packet component.  The same decoder survives every base change, while rank loss of individual residue channels remains visible after reduction \cite{CT4}.
1470 
1471 \subsection{Finite holonomy and the maximal path-flat core}
1472 \label{subsec:maximal-flat-core}
1473 
1474 Finite graph transport gives the sharpest reversible gluing model in the Library.  Let \(\Gamma=(V,E)\) be a finite connected graph.  Attach a finite-dimensional \(\kk\)-vector space \(F_v\) to each vertex and an isomorphism \(T_e:F_u\to F_v\) to each oriented edge \(e:u\to v\), with \(T_{\bar e}=T_e^{-1}\).  Choose a root \(o\) and spanning tree \(S\).  If \(G_v:F_o\to F_v\) is tree transport, then every non-tree edge \(e:u\to v\) has fundamental holonomy
1475 \[
1476 H_e=G_v^{-1}T_eG_u\in GL(F_o).
1477 \]
1478 Define
1479 \[
1480 C_o
1481 =
1482 \bigcap_{e\notin S}\ker(H_e-I)
1483 =
1484 \Fix\langle H_e:e\notin S\rangle,
1485 \qquad
1486 C_v=G_vC_o.
1487 \]
1488 
1489 \begin{theorem}[Maximal flat core and global sections]
1490 \label{thm:maximal-flat-core}
1491 Under the preceding hypotheses \cite{P3,CT4}:
1492 \begin{enumerate}[label=\textup{(\roman*)}]
1493 \item evaluation at the root identifies the space of global parallel sections with \(C_o\);
1494 \item \((C_v)_v\) is path-flat, meaning that all path transports with the same endpoints agree on the core;
1495 \item it is the unique greatest path-flat subfamily;
1496 \item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies;
1497 \item for
1498 \[
1499 \Delta_S:F_o\longrightarrow\bigoplus_{e\notin S}F_o,
1500 \qquad
1501 \Delta_Sx=((H_e-I)x)_{e\notin S},
1502 \]
1503 one has
1504 \[
1505 C_o=\ker\Delta_S,
1506 \qquad
1507 \dim C_o=\dim F_o-\rank\Delta_S;
1508 \]
1509 \item the full fibers glue to one global frame exactly when every fundamental holonomy is the identity;
1510 \item a gauge change conjugates the holonomies and transports the core, so \(\dim C_o\) is gauge-invariant.
1511 \end{enumerate}
1512 \end{theorem}
1513 
1514 \begin{proof}
1515 For \(x\in C_o\), set \(x_v=G_vx\).  Tree edges satisfy the parallel relation by construction, and a non-tree edge satisfies it because \(H_ex=x\).  Conversely, any global section is determined along the tree by its root value; every remaining edge forces that value to be fixed by the corresponding \(H_e\).  This proves the section classification.  Closed paths are words in fundamental cycles, which gives path-flatness and tree independence.  Any path-flat family must be fixed by every fundamental cycle and is therefore contained in \((C_v)_v\).  The rank formula is rank--nullity.  Full flatness and gauge covariance follow from spanning-tree normalization and conjugation of loop products.
1516 \end{proof}
1517 
1518 \begin{corollary}[Global section versus global frame]
1519 \label{cor:global-section-versus-frame}
1520 Nonidentity holonomy can leave a nonzero maximal core.  Therefore the existence of some global compatible vectors is weaker than the existence of one global frame on the full fibers.  The former is measured by \(\dim C_o\); the latter requires \(C_o=F_o\).
1521 \end{corollary}
1522 
1523 \begin{figure}[htbp]
1524 \centering
1525 \includegraphics[width=0.96\textwidth]{flat_core_holonomy.pdf}
1526 \caption{A spanning tree removes local gauge redundancy.  Each remaining edge contributes a fundamental holonomy.  Their common fixed space is the maximal path-flat core; the stacked defect map \(\Delta_S\) measures the lost dimension.}
1527 \label{fig:flat-core-holonomy}
1528 \end{figure}
1529 
1530 \subsection{Relative cokernel obstructions and loss under coarsening}
1531 \label{subsec:relative-cokernel-obstruction}
1532 
1533 Holonomy addresses compatibility of invertible transports.  Correction problems have a different canonical object.
1534 
1535 \begin{definition}[Linear correction datum]
1536 \label{def:block02-linear-correction}
1537 A linear correction datum is a map
1538 \[
1539 D:U\longrightarrow V
1540 \]
1541 and a defect \(r\in V\).  A correction is \(u\in U\) satisfying \(Du=-r\).  The relative obstruction class is
1542 \[
1543 \Ob_D(r)=[r]\in\coker D=V/\im D.
1544 \]
1545 \end{definition}
1546 
1547 \begin{theorem}[Cokernel criterion, universality, and naturality]
1548 \label{thm:block02-cokernel-naturality}
1549 For a correction datum \((D,r)\) \cite{P3}:
1550 \begin{enumerate}[label=\textup{(\roman*)}]
1551 \item a correction exists if and only if \(\Ob_D(r)=0\);
1552 \item every linear obstruction test \(\varphi:V\to W\) satisfying \(\varphi D=0\) factors uniquely through \(V\to\coker D\);
1553 \item for every commuting square
1554 \[
1555 \begin{array}{ccc}
1556 U & \xrightarrow{D} & V\\
1557 \downarrow a & & \downarrow b\\
1558 U' & \xrightarrow{D'} & V',
1559 \end{array}
1560 \qquad bD=D'a,
1561 \]
1562 there is a unique induced map
1563 \[
1564 \overline b:\coker D\longrightarrow\coker D'
1565 \]
1566 with
1567 \[
1568 \overline b\,\Ob_D(r)=\Ob_{D'}(br).
1569 \]
1570 \end{enumerate}
1571 \end{theorem}
1572 
1573 \begin{proof}
1574 The class vanishes exactly when \(r\in\im D\), which is equivalent to solvability of \(Du=-r\).  A map annihilating \(\im D\) factors uniquely through the quotient.  In the square, the quotient map for \(D'\), composed with \(b\), vanishes on \(\im D\), so the same universal property produces \(\overline b\).
1575 \end{proof}
1576 
1577 \begin{proposition}[Packet coordinates on the obstruction quotient]
1578 \label{prop:block02-obstruction-packets}
1579 Suppose \((Q_a)\) is a complete finite family of pairwise annihilating idempotents on \(V\) and
1580 \[
1581 Q_a(\im D)\subseteq\im D
1582 \]
1583 for every \(a\).  Then the \(Q_a\) descend to quotient idempotents \(\overline Q_a\) on \(\coker D\),
1584 \[
1585 \coker D
1586 \cong
1587 \bigoplus_a\overline Q_a(\coker D),
1588 \qquad
1589 \overline Q_a\Ob_D(r)=[Q_ar].
1590 \]
1591 If \(U\) and \(V\) already split into matching packet summands and \(D\) is block diagonal, this becomes
1592 \[
1593 \coker D\cong\bigoplus_a\coker D_a.
1594 \]
1595 Off-diagonal correction blocks can couple packet defects, so the block-diagonal hypothesis cannot be omitted.
1596 \end{proposition}
1597 
1598 \begin{proof}
1599 The invariance of \(\im D\) makes \([v]\mapsto[Q_av]\) well defined.  Idempotence, annihilation, and reconstruction descend to the quotient.  In the block-diagonal case the image of \(D\) is the direct sum of the images of the restrictions.
1600 \end{proof}
1601 
1602 A coarse representation can erase a nonzero fine obstruction.  The strengthened Paper III gives the exact bookkeeping device.
1603 
1604 \begin{theorem}[Relative obstruction exact sequence]
1605 \label{thm:block02-relative-obstruction-sequence}
1606 Suppose the diagram
1607 \[
1608 \begin{array}{ccccccccc}
1609 0&\to&U_0&\to&U_f&\to&U_c&\to&0\\
1610  &&\downarrow D_0&&\downarrow D_f&&\downarrow D_c\\
1611 0&\to&V_0&\to&V_f&\to&V_c&\to&0
1612 \end{array}
1613 \]
1614 commutes and has exact rows.  Then there is a natural exact sequence
1615 \[
1616 \begin{aligned}
1617 0&\longrightarrow\ker D_0
1618 \longrightarrow\ker D_f
1619 \longrightarrow\ker D_c
1620 \xrightarrow{\delta}\coker D_0\\
1621 &\longrightarrow\coker D_f
1622 \longrightarrow\coker D_c
1623 \longrightarrow0.
1624 \end{aligned}
1625 \]
1626 If a fine obstruction maps to zero in \(\coker D_c\), then it comes from \(\coker D_0\), the obstruction sector removed by coarsening.  Coarse vanishing therefore does not imply fine vanishing.
1627 \end{theorem}
1628 
1629 \begin{proof}
1630 This is the direct kernel--cokernel sequence proved in the strengthened Paper III.  The connecting map lifts an element of \(\ker D_c\) to \(U_f\), applies \(D_f\), identifies the result in \(V_0\), and takes its class modulo \(\im D_0\).  Exactness verifies independence of the lift and identifies each successive kernel with the preceding image \cite{P3,MIG}.
1631 \end{proof}
1632 
1633 \begin{example}[A fine obstruction erased by summation]
1634 \label{ex:fine-obstruction-erased}
1635 Take all correction spaces to be zero,
1636 \[
1637 V_f=\kk^2,
1638 \qquad
1639 V_c=\kk,
1640 \qquad
1641 p(x,y)=x+y.
1642 \]
1643 The fine defect \((1,-1)\) is nonzero in \(\coker D_f=V_f\), but its coarse image is zero.  It lies entirely in
1644 \[
1645 V_0=\ker p=\Span\{(1,-1)\}.
1646 \]
1647 This is cancellation by coarsening, not correction of the fine defect.
1648 \end{example}
1649 
1650 \begin{definition}[Mapping cone]
1651 \label{def:block02-mapping-cone}
1652 For a cochain map \(D:C^\bullet\to E^\bullet\), define
1653 \[
1654 \Cone(D)^n=E^n\oplus C^{n+1},
1655 \qquad
1656  d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
1657 \]
1658 \end{definition}
1659 
1660 \begin{proposition}[Cohomological correction criterion]
1661 \label{prop:block02-cone-obstruction}
1662 For a cocycle \(e\in E^n\), the class of \((e,0)\) in \(H^n(\Cone(D))\) vanishes exactly when \([e]\) lies in the image of \(H^n(D)\).  When the complexes are concentrated in degree zero,
1663 \[
1664 H^0(\Cone(D))\cong\coker D.
1665 \]
1666 \end{proposition}
1667 
1668 \begin{proof}
1669 The class of \((e,0)\) is a boundary exactly when there are \(e'\in E^{n-1}\) and \(c\in C^n\) with
1670 \[
1671 e=d_Ee'+Dc,
1672 \qquad
1673 d_Cc=0.
1674 \]
1675 This is equivalent to \([e]=H^n(D)[c]\).  Concentration in degree zero reduces the cone quotient to \(E^0/\im D\).
1676 \end{proof}
1677 
1678 \begin{boundary}[The correction map is problem data]
1679 \label{bnd:correction-map-problem-data}
1680 The cokernel is canonical relative to \(D\).  The theory does not yet supply one choice-free rule that constructs the correct \(D\) for every mathematical problem and every stratum.  Calling \([r]\in\coker D\) canonical must not hide this dependence.
1681 \end{boundary}
1682 
1683 \subsection{Semilinear effectivity, projective descent, and the Brauer class}
1684 \label{subsec:brauer-descent}
1685 
1686 Galois descent gives a second exact local-to-global model.  Its linear and projective cases separate effectivity from obstruction sharply.
1687 
1688 \begin{theorem}[Effective vector descent by Hilbert 90]
1689 \label{thm:block02-hilbert90}
1690 Let \(L/K\) be finite Galois with group \(G\).  Suppose
1691 \[
1692 a_{\sigma\tau}=a_\sigma\,\sigma(a_\tau),
1693 \qquad
1694 a_\sigma\in GL_n(L),
1695 \]
1696 defines a semilinear packet cocycle.  Then there is \(B\in GL_n(L)\) such that
1697 \[
1698 a_\sigma=B^{-1}\sigma(B).
1699 \]
1700 The twisted fixed packet
1701 \[
1702 W=(L^n)^{G,\star}
1703 \]
1704 is an \(n\)-dimensional \(K\)-vector space and
1705 \[
1706 W\otimes_KL\xrightarrow{\sim}L^n.
1707 \]
1708 Thus every finite linear semilinear atlas of this form is effective \cite{GAL}.
1709 \end{theorem}
1710 
1711 \begin{theorem}[Brauer obstruction to projective packet descent]
1712 \label{thm:block02-brauer}
1713 Projective semilinear descent data define a class
1714 \[
1715 \alpha\in H^1(G,PGL_n(L)).
1716 \]
1717 The connecting map for
1718 \[
1719 1\longrightarrow L^\times
1720 \longrightarrow GL_n(L)
1721 \longrightarrow PGL_n(L)
1722 \longrightarrow1
1723 \]
1724 produces
1725 \[
1726 \delta(\alpha)
1727 \in
1728 H^2(G,L^\times)
1729 =
1730 \Br(L/K).
1731 \]
1732 The projective atlas lifts to linear semilinear descent data exactly when \(\delta(\alpha)=0\).  When the class is nonzero, the descended object is naturally a module over a central simple \(K\)-algebra split by \(L\), rather than an ordinary \(K\)-vector space \cite{GAL}.
1733 \end{theorem}
1734 
1735 \begin{proof}
1736 This is the nonabelian cohomology exact sequence together with nonabelian Hilbert 90.  Vanishing of the connecting class gives a \(GL_n(L)\)-valued cocycle; \cref{thm:block02-hilbert90} then trivializes it and constructs the descended vector space.  A nonzero class is the classical relative Brauer obstruction.
1737 \end{proof}
1738 
1739 \begin{corollary}[Projective compatibility is not linear effectivity]
1740 \label{cor:projective-not-linear-effectivity}
1741 A projective atlas may satisfy every visible overlap equation in \(PGL_n\) and still have no global vector-space realization.  Projective cocycle coherence and vanishing Brauer class are distinct conditions.
1742 \end{corollary}
1743 
1744 The tensor reconstruction problem lies beyond ordinary corner realization.
1745 
1746 \begin{theorem}[Tannakian boundary]
1747 \label{thm:block02-tannakian-boundary}
1748 Let \(\mathcal T\) be a neutral Tannakian category over \(k\) and
1749 \[
1750 \omega:\mathcal T\longrightarrow\operatorname{Vect}_k
1751 \]
1752 a faithful exact tensor fiber functor.  Then
1753 \[
1754 \mathcal G=\Aut^{\otimes}(\omega)
1755 \]
1756 is an affine group scheme and
1757 \[
1758 \mathcal T\simeq\operatorname{Rep}_k(\mathcal G)
1759 \]
1760 as tensor categories.  A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary \cite{GAL,RT}.
1761 \end{theorem}
1762 
1763 \subsection{Higher coherence defects and typed obstruction comparison}
1764 \label{subsec:obstruction-comparison}
1765 
1766 The four obstruction families now admit a precise comparison.  Each starts from different data and answers a different vanishing question.
1767 
1768 \begin{landscape}
1769 \small
1770 \begin{longtable}{@{}L{0.14\linewidth}L{0.20\linewidth}L{0.20\linewidth}L{0.20\linewidth}L{0.18\linewidth}@{}}
1771 \caption{Typed obstruction comparison.}
1772 \label{tab:obstruction-comparison}\\
1773 \toprule
1774 Failure mode & Required datum & Obstruction object & Vanishing consequence & Nonzero meaning\\
1775 \midrule
1776 \endfirsthead
1777 \toprule
1778 Failure mode & Required datum & Obstruction object & Vanishing consequence & Nonzero meaning\\
1779 \midrule
1780 \endhead
1781 Finite transport holonomy & connected graph, invertible edge transports, chosen structure group & fundamental cycle holonomies or the conjugacy class of \(\pi_1(\Gamma)\to G\) & one global frame on the full fibers; on a fixed subspace, path-independent transport & path dependence; possibly a smaller nonzero fixed core\\
1782 Relative correction failure & correction map \(D:U\to V\) and defect \(r\) & \([r]\in\coker D\), or a mapping-cone cohomology class & an allowed correction exists & the defect is outside the declared correction image\\
1783 Projective Galois descent & projective semilinear cocycle over \(L/K\) & \(\delta(\alpha)\in H^2(G,L^\times)=\Br(L/K)\) & lift to linear descent, then effective vector descent by Hilbert 90 & descent to a twisted central-simple-algebra object rather than an ordinary vector space\\
1784 Higher atlas coherence & pairwise equivalences and comparison \(2\)-cells & \(\Omega_{\alpha\beta\gamma\delta}\), a natural automorphism on a quadruple overlap & pseudofunctor coherence of the atlas & pairwise bridges use incompatible conventions or incomplete comparison data\\
1785 \bottomrule
1786 \end{longtable}
1787 \end{landscape}
1788 
1789 \begin{proposition}[Typed comparison principle for obstruction objects]
1790 \label{prop:typed-obstruction-comparison}
1791 An obstruction object transports from one problem to another only after a compatibility morphism for the underlying data has been supplied:
1792 \begin{enumerate}[label=\textup{(\roman*)}]
1793 \item commuting squares of correction maps induce maps on cokernels;
1794 \item group homomorphisms or intertwiners compatible with edge transports induce maps on holonomy data and fixed cores;
1795 \item morphisms of the relevant central extensions induce maps on cohomological descent obstructions;
1796 \item modifications of atlas pseudofunctors transport higher coherence defects.
1797 \end{enumerate}
1798 Absent such typed data, there is no canonical reduction of one obstruction family to another.
1799 \end{proposition}
1800 
1801 \begin{proof}
1802 Each item follows by applying the functorial construction defining the obstruction: quotient universality for cokernels, conjugation or intertwining for path products, functoriality of cohomology for Brauer classes, and naturality of horizontal and vertical composition for coherence cells.  Since the input categories and codomains differ, no comparison exists without an explicit map between the inputs.
1803 \end{proof}
1804 
1805 \begin{figure}[htbp]
1806 \centering
1807 \includegraphics[width=0.99\textwidth]{obstruction_comparison.pdf}
1808 \caption{Four obstruction pipelines.  The word \emph{obstruction} names a role, not one universal scalar.  The required input datum determines both the obstruction object and the exact consequence of vanishing.}
1809 \label{fig:obstruction-comparison}
1810 \end{figure}
1811 
1812 \begin{resultbox}
1813 \textbf{Section result.}
1814 The atlas layer now has explicit stratum passports, normalized subcategories, representation groupoids, a directed transformation category, bridge classes, information-loss profiles, overlap and bridge-law certificates, and a concrete higher-coherence defect.  The four obstruction theories are stated with their own data and vanishing criteria.  Finite holonomy supplies a maximal path-flat core, split bridges expose reversible consistent cores inside directional maps, relative cokernels separate correction from coarsening loss, and Galois theory separates Hilbert-90 effectivity from Brauer and Tannakian boundaries.  The general heterogeneous atlas is still not declared effective until its actual bridge passports and coherence cells are verified.
1815 \end{resultbox}
1816 
1817 
1818 \section{The theorem concordance and edition control}
1819 
1820 \subsection{Audit result}
1821 
1822 The concordance contains 56 components. Thirty-eight are proved internally, eight are classical theorems imported and reinterpreted, two are proved conditionally, two are definition or framework rows, and six remain conjectural. The new foundations row records the finite profiled Universal Lane Transport Theorem. The conjectural entries remain concentrated in the cross-stratum assembly layer. The local packet, obstruction, Galois, sequence, and Mahler components are substantially more complete than an older open-question ledger suggests.
1823 
1824 \input{generated/status_summary_table.tex}
1825 \input{generated/domain_summary_table.tex}
1826 
1827 \subsection{Material supersession decisions}
1828 
1829 Three corrections are mathematically consequential.
1830 
1831 \paragraph{Relative obstruction theory.}
1832 Older Paper III editions leave a broad universal packet-obstruction principle conjectural. The strengthened edition replaces the finite-projector formulation by the relative class \([r]\in\coker D\), with vanishing, naturality, packet decomposition, quotient projectors, relative exact sequences, and mapping-cone formulations proved for declared correction data \cite{P3,MIG}. The unresolved problem is not the quotient construction itself. It is a choice-free mechanism assigning the appropriate correction datum to arbitrary problems and strata.
1833 
1834 \paragraph{Paper I edition policy.}
1835 The complete symbolic core remains the theorem-ID and proof-status spine. The expanded clarified edition strengthens exposition and hypotheses without replacing the stable record identifiers \cite{P1}.
1836 
1837 \paragraph{Core-Transfer status.}
1838 Addendum 04 replaces the isolated \(m=6,d=3\) boundary by the exact composite-modulus rank law, sharp integral packet minimum, universal decoder, and all-degree stabilization \cite{CT4}. The former example is now a sharp negative control inside a general theorem.
1839 
1840 \subsection{Existing bridges and exact invariant slices}
1841 
1842 The strongest source edition already contains the following exact or effective bridges on declared domains:
1843 
1844 \begin{itemize}
1845 \item constant recurrence \(\leftrightarrow\) rational ordinary generating function \(\leftrightarrow\) finite linear state;
1846 \item P-recursive \(\leftrightarrow\) D-finite coefficient translation;
1847 \item finite digit realization \(\leftrightarrow\) finite \(k\)-kernel;
1848 \item rational functional-Mahler mask \(\to\) Cartier representation \(\to\) finite automaton;
1849 \item finite-mask Cartier transitions \(\leftrightarrow\) carry matrices;
1850 \item selected algebraic branch \(\to\) packet polynomial system;
1851 \item invertible transform \(\leftrightarrow\) transported packet resolution;
1852 \item Euler-product exponent coordinates \(\leftrightarrow\) logarithmic-derivative recurrence data;
1853 \item finite modular state \(\to\) local Newton-Mahler interpolation;
1854 \item local Mahler coordinates \(\leftrightarrow\) affine polyphase branches;
1855 \item finite resolved \(C^*\)-system \(\to\) faithful Hilbert packet realization.
1856 \end{itemize}
1857 
1858 The persistent C-finite stratum has the strongest complete transfer invariant. Active spectral and torsion-phase data determine the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank drops, and periodic packet profile of every arithmetic subsequence \cite{SEQ}. Restricted exact coordinate systems also exist for fixed-radix automata, filtered Euler products, multiplicative prime-power seeds, local Mahler charts, finite semisimple Galois descent, and finite seed lattices. None of these restricted results is a single all-stratum classifier.
1859 
1860 \begin{figure}[htbp]
1861 \centering
1862 \includegraphics[width=0.98\textwidth]{dependency_graph.pdf}
1863 \caption{The corrected dependency graph. The theorem concordance and local results feed the assembly theorem, but the heterogeneous conclusion still depends on explicit atlas construction, coherence, effectivity, and isomorphism descent.}
1864 \label{fig:dependency-graph}
1865 \end{figure}
1866 
1867 \section{Finite profiled resolved atlas assembly}
1868 
1869 \subsection{Finite atlases and descent data}
1870 
1871 \begin{definition}[Finite profiled resolved atlas]
1872 Fix a small resolved theory \(\bbK\). A finite profiled resolved atlas consists of:
1873 \begin{enumerate}[label=\textup{(A\arabic*)}]
1874 \item a finite indexing category \(\check{\mathcal A}\), retained through the overlap degrees needed for the cocycle law;
1875 \item a pseudofunctor
1876 \[
1877 \mathfrak R:\check{\mathcal A}^{\mathrm{op}}\to\Cat;
1878 \]
1879 \item a declared preservation profile \(\Pi_U\) on every chart and overlap category \(\mathfrak R(U)\);
1880 \item coherent equivalences on the reversible overlaps and separately typed directional transformations;
1881 \item a pseudonatural semantics
1882 \[
1883 \Sem:\mathfrak R\Rightarrow\mathfrak O
1884 \]
1885 into a common observable doctrine;
1886 \item optional pseudonatural invariant families with explicitly typed comparison cells.
1887 \end{enumerate}
1888 \end{definition}
1889 
1890 \begin{definition}[Descent category]
1891 The category \(\Desc(\mathfrak R)\) has pseudocones over \(\mathfrak R\) as objects and modifications as morphisms. Concretely, an object is a compatible family
1892 \[
1893 (M_\alpha,\theta_{\alpha\beta})
1894 \]
1895 with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.
1896 \end{definition}
1897 
1898 \subsection{The assembly theorem}
1899 
1900 \begin{theorem}[Finite profiled resolved atlas assembly]
1901 \label{thm:atlas-assembly}
1902 Let
1903 \[
1904 \mathfrak R:\check{\mathcal A}^{\mathrm{op}}\to\Cat
1905 \]
1906 be a finite profiled resolved atlas. Assume the chart and overlap categories are small, or locally small in one fixed universe, the reversible transitions and comparison cells satisfy the pseudofunctor coherence equations, and every construction used below belongs to the verified profile of every participating chart and bridge. Then:
1907 
1908 \begin{enumerate}[label=\textup{(\roman*)}]
1909 \item \(\Desc(\mathfrak R)\) exists and has the universal property of the finite pseudolimit of \(\mathfrak R\).
1910 \item There is a canonical equivalence
1911 \[
1912 \Core\bigl(\Desc(\mathfrak R)\bigr)
1913 \simeq
1914 \Desc\bigl(\mathfrak R^{\simeq}\bigr).
1915 \]
1916 \item Every pseudonatural transformation \(T:\mathfrak R\Rightarrow\mathfrak S\) induces a functor
1917 \[
1918 \Desc(T):\Desc(\mathfrak R)\to\Desc(\mathfrak S).
1919 \]
1920 If every \(T_U\) is an equivalence, then \(\Desc(T)\) is an equivalence. A pointwise noninvertible \(T\) remains directional.
1921 \item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in \(\Desc(\mathfrak R)\).
1922 \item A coherent pseudonatural invariant family \(I:\mathfrak R\Rightarrow\mathfrak V\) induces
1923 \[
1924 \Desc(I):\Desc(\mathfrak R)\to\Desc(\mathfrak V).
1925 \]
1926 \item If every local invariant is complete on its normalized chart and the groupoid of local invariant-compatible isomorphisms has effective descent, then \(\Desc(I)\) is complete on the corresponding global normalized category.
1927 \end{enumerate}
1928 
1929 If the semantic pseudonatural transformation is effective for objects and morphisms, then \(\Desc(\Sem)\) identifies descent data with the declared global semantic objects.
1930 \end{theorem}
1931 
1932 \begin{proof}
1933 Objects and morphisms of \(\Desc(\mathfrak R)\) are finite compatible families. Identities and composition are defined componentwise. The pseudofunctor coherence equations ensure that componentwise composition remains compatible. This explicit construction has the universal property of pseudocones, proving (i).
1934 
1935 A descent morphism is invertible exactly when each component is invertible. The componentwise inverses satisfy the overlap equations obtained by inverting the original equations. This proves (ii).
1936 
1937 A pseudonatural transformation acts on chart objects, overlap comparisons, and chart morphisms componentwise. Its coherence cells supply the target cocycle law. Pointwise quasi-inverses and their modifications descend, proving (iii).
1938 
1939 For (iv), the equational realization principle transports the proof to each chart because every constructor used by the proof is preserved. Bridge naturality makes the local equalities compatible on overlaps. Equality in the descent category is componentwise. The argument creates no missing analytic or arithmetic hypotheses.
1940 
1941 Statement (v) is the same pseudonatural-transformation construction applied to the invariant family. For (vi), an isomorphism between descended invariants gives local invariant matches. Local completeness supplies local object isomorphisms. The added isomorphism-effectivity hypothesis is exactly what glues those local isomorphisms to a global one. Without it, a holonomy class, a relative cokernel class, a Brauer class, or a higher coherence defect may obstruct the gluing.
1942 
1943 The final assertion is the definition of semantic effectivity applied to the induced descent functor.
1944 \end{proof}
1945 
1946 \begin{corollary}[No automatic promotion of directional bridges]
1947 A directional bridge contributes a global functor only when it is pseudonatural with respect to the chosen atlas. It contributes an equivalence only when the chartwise bridge is an equivalence on the named overlap. No quotient, completion, or compression becomes reversible merely because it appears in the same diagram.
1948 \end{corollary}
1949 
1950 \begin{corollary}[Global theorem transport]
1951 A theorem proved from packet reconstruction, refinement, residual calculus, or another preserved constructor descends globally only when every chart and bridge used by the proof carries that constructor in its verified preservation profile.
1952 \end{corollary}
1953 
1954 \begin{remark}[Exact status]
1955 \Cref{thm:atlas-assembly} is a proved conditional theorem. It does not prove that the full proposed Komorebi atlas satisfies the hypotheses. The remaining work is the verification of the actual profiles, overlap categories, naturality cells, higher coherence, and effectivity.
1956 \end{remark}
1957 
1958 \part{Certified native strata}
1959 
1960 \section{Verified finite Mahler atlas models}
1961 
1962 Two exact finite models already realize substantial pieces of the assembly theorem. They remain inside the affine-Mahler and packet-analysis setting, but they verify nontrivial coherence and effectivity mechanisms.
1963 
1964 \subsection{Affine-Mahler chart groupoid}
1965 
1966 Fix \(d\ge0\). Over characteristic zero, the affine Mahler coefficient maps \(A_{m,s}^{(d)}\) are invertible and satisfy
1967 \[
1968 A_{n,t}^{(d)}A_{m,s}^{(d)}=A_{mn,mt+s}^{(d)}.
1969 \]
1970 For coprime \(m,n\), the two paths around every declared Chinese-remainder diamond are equal \cite{CT4}. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with trivial holonomy on those diamonds.
1971 
1972 \begin{proposition}[Rational affine-Mahler chart groupoid]
1973 Every finite characteristic-zero diagram generated by the invertible affine Mahler charts and the declared CRT comparison cells defines a strict finite groupoid atlas. Its path transport depends only on the represented affine substitution, and every declared CRT cycle has identity holonomy.
1974 \end{proposition}
1975 
1976 \subsection{Integral split packet descent}
1977 
1978 Let
1979 \[
1980 A=P_{m,S}^{(d)}:\mathbb Z^{d+1}\to\mathbb Z^{|S|(d+1)}
1981 \]
1982 be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse \(L\) exists,
1983 \[
1984 LA=I,
1985 \qquad
1986 \Pi=AL,
1987 \qquad
1988 \Pi^2=\Pi,
1989 \]
1990 and
1991 \[
1992 \mathbb Z^{|S|(d+1)}=\im A\oplus\ker L.
1993 \]
1994 Thus the source chart and the normalized packet core \(\im A\) are equivalent, while \(\ker L\) is a named residual rather than discarded information.
1995 
1996 Addendum 04 proves that the prefix
1997 \[
1998 S_*=\{0,1,\dots,L_{\mathbb Z}(m,d)-1\}
1999 \]
2000 with
2001 \[
2002 L_{\mathbb Z}(m,d)=\min(d+1,B(m)),
2003 \qquad
2004 B(m)=\max\bigl(\{1\}\cup\{p^{v_p(m)}:p\mid m\}\bigr),
2005 \]
2006 always admits an integer decoder and is cardinality-minimal \cite{CT4}.
2007 
2008 \begin{proposition}[Integral split Mahler packet descent]
2009 The analysis-decoder pair above gives effective object and morphism descent between the source chart and the normalized packet-core chart. The split identity survives every base change. The residual summand \(\ker L\) is functorial under maps intertwining both \(A\) and \(L\).
2010 \end{proposition}
2011 
2012 \begin{proof}
2013 The identities \(LA=I\) and \(\Pi=AL\) show that \(A\) identifies the source with \(\im A\), with inverse \(L|_{\im A}\). For any packet vector \(y\),
2014 \[
2015 y=ALy+(y-ALy),
2016 \]
2017 where \(ALy\in\im A\) and \(y-ALy\in\ker L\). The intersection is zero because \(Ax\in\ker L\) implies \(x=LAx=0\). Tensoring \(LA=I\) with any coefficient ring preserves the split identity. Intertwining maps preserve the image and kernel decompositions.
2018 \end{proof}
2019 
2020 \begin{figure}[htbp]
2021 \centering
2022 \includegraphics[width=0.98\textwidth]{mahler_models.pdf}
2023 \caption{Two verified finite atlas mechanisms. The first is reversible and coherent. The second is split analysis-synthesis with a retained residual.}
2024 \label{fig:mahler-models}
2025 \end{figure}
2026 
2027 \section{The normalized C-finite three-chart atlas}
2028 
2029 This section completes the first cross-presentation model in the theorem dependency graph. It does not combine unrelated broad strata, but it does prove a genuine three-chart effective descent theorem with nontrivial morphisms on the state side.
2030 
2031 \subsection{Persistent C-finite semantics}
2032 
2033 Fix a field \(K\). A nonzero C-finite sequence \(a=(a_n)_{n\ge0}\) is \emph{persistent} when its minimal characteristic polynomial has nonzero constant term. Equivalently, its active characteristic roots are nonzero, or its minimal state transition is invertible \cite{SEQ}.
2034 
2035 Let \(\Seq_K^{\mathrm{pers}}\) be the discrete groupoid of persistent C-finite sequences over \(K\).
2036 
2037 \subsection{The recurrence chart}
2038 
2039 An object of \(\Rec_K^{\mathrm{min}}\) is a pair
2040 \[
2041 (\chi,\mathbf a_0),
2042 \]
2043 where
2044 \[
2045 \chi(x)=x^d-c_{d-1}x^{d-1}-\cdots-c_1x-c_0,
2046 \qquad c_0\ne0,
2047 \]
2048 is the monic minimal recurrence polynomial and \(\mathbf a_0=(a_0,\dots,a_{d-1})\) is the initial vector. It generates the sequence
2049 \[
2050 a_{n+d}=c_{d-1}a_{n+d-1}+\cdots+c_1a_{n+1}+c_0a_n.
2051 \]
2052 Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus \(\Rec_K^{\mathrm{min}}\) is a discrete groupoid.
2053 
2054 \subsection{The rational OGF chart}
2055 
2056 An object of \(\Rat_K^{\mathrm{red}}\) is a reduced rational power series
2057 \[
2058 A(z)=\frac{P(z)}{Q(z)},
2059 \qquad
2060 Q(0)=1,
2061 \qquad
2062 \gcd(P,Q)=1,
2063 \qquad
2064 \deg P<\deg Q,
2065 \]
2066 whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so \(\Rat_K^{\mathrm{red}}\) is discrete.
2067 
2068 The recurrence-to-OGF bridge is explicit. From
2069 \[
2070 \chi(x)=x^d-c_{d-1}x^{d-1}-\cdots-c_0
2071 \]
2072 set
2073 \[
2074 Q(z)=z^d\chi(z^{-1})
2075 =1-c_{d-1}z-\cdots-c_0z^d.
2076 \]
2077 The numerator is the degree-\(<d\) truncation of \(Q(z)\sum_{n\ge0}a_nz^n\). Conversely, clearing \(Q\) recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.
2078 
2079 \subsection{The minimal state chart}
2080 
2081 An object of \(\State_K^{\mathrm{min}}\) is a triple
2082 \[
2083 (C,v,\lambda),
2084 \qquad
2085 C\in M_d(K),\quad v\in K^d,\quad \lambda\in(K^d)^*,
2086 \]
2087 with \(C\) invertible, producing
2088 \[
2089 a_n=\lambda C^nv,
2090 \]
2091 and satisfying reachability and observability. A morphism
2092 \[
2093 S:(C,v,\lambda)\longrightarrow(C',v',\lambda')
2094 \]
2095 is a similarity satisfying
2096 \[
2097 C'=SCS^{-1},
2098 \qquad
2099 v'=Sv,
2100 \qquad
2101 \lambda'=\lambda S^{-1}.
2102 \]
2103 The semantic functor sends a state to its coefficient sequence.
2104 
2105 The state-to-OGF bridge is the resolvent
2106 \[
2107 A(z)=\lambda(I-zC)^{-1}v.
2108 \]
2109 The reduced denominator has degree equal to the minimal state dimension, which equals the Hankel rank. Conversely, a reduced rational OGF has a companion realization. Reducedness makes the companion realization reachable, while the companion output is observable. Any two minimal realizations of the same sequence are uniquely similar through their common shift action on the Hankel column space \cite{SEQ}.
2110 
2111 \subsection{Effective descent across the three charts}
2112 
2113 \begin{theorem}[Normalized C-finite three-chart atlas]
2114 \label{thm:cfinite-three-chart}
2115 For every field \(K\), the semantic functors fit into equivalences of groupoids
2116 \[
2117 \Rec_K^{\mathrm{min}}
2118 \simeq
2119 \Rat_K^{\mathrm{red}}
2120 \simeq
2121 \State_K^{\mathrm{min}}
2122 \simeq
2123 \Seq_K^{\mathrm{pers}}.
2124 \]
2125 The recurrence-rational bridge is a strict inverse pair. The rational-state bridge is an equivalence with companion quasi-inverse. The comparison isomorphism on the state-side composite is the unique similarity between minimal realizations. Consequently:
2126 
2127 \begin{enumerate}[label=\textup{(\roman*)}]
2128 \item the three charts and their pairwise overlaps form a coherent finite profiled atlas;
2129 \item the triple-overlap comparison satisfies the cocycle law by uniqueness of minimal similarity;
2130 \item descent for objects and morphisms is effective;
2131 \item the descent category is equivalent to \(\Seq_K^{\mathrm{pers}}\).
2132 \end{enumerate}
2133 \end{theorem}
2134 
2135 \begin{proof}
2136 The equivalence between normalized minimal recurrences and reduced rational ordinary generating functions follows from the explicit reciprocal-denominator formula and clearing of the denominator. These constructions are mutually inverse.
2137 
2138 A minimal state gives a rational resolvent. Conversely, a reduced rational series gives a companion state. Minimal linear dimension equals Hankel rank, and the reduced denominator degree has the same value. Hence the companion state is minimal. Any two minimal states representing the same sequence identify with the forward shift on the Hankel column space and are therefore similar. The similarity is unique because a reachable state is spanned by \(C^nv\), so an intertwiner fixing the input vector is determined on a spanning set.
2139 
2140 Choose the recurrence-to-state bridge to be the companion realization obtained through the rational chart. Then one triangular composite is strict. On the state side, the composite through the rational and recurrence charts returns the companion realization of the same sequence. The unique minimal similarity supplies the comparison isomorphism. Any two composites around a triple overlap are comparison isomorphisms between the same minimal realizations, so uniqueness forces the cocycle law.
2141 
2142 Each semantic functor is an equivalence to the same discrete groupoid of persistent sequences. Compatible local objects and morphisms therefore descend uniquely up to the stated similarity, proving effectivity and the final equivalence.
2143 \end{proof}
2144 
2145 \begin{figure}[htbp]
2146 \centering
2147 \includegraphics[width=0.82\textwidth]{cfinite_triangle.pdf}
2148 \caption{The normalized C-finite atlas. The recurrence and rational charts are canonical normal forms. The state chart has nontrivial presentation coordinates, but minimality makes every semantic identification unique up to similarity.}
2149 \label{fig:cfinite-triangle}
2150 \end{figure}
2151 
2152 \subsection{Descent of the minimality and torsion-Fourier invariants}
2153 
2154 For a persistent C-finite sequence, the minimal dimension
2155 \[
2156 \rho(a)=\linrank(a)=\rank H(a)
2157 \]
2158 is simultaneously the minimal recurrence order, the reduced denominator degree, and the dimension of a minimal reachable-observable state.
2159 
2160 Over characteristic zero, write the unique active expansion over an algebraic closure as
2161 \[
2162 a_n=\sum_{\lambda\in\Lambda}P_\lambda(n)\lambda^n.
2163 \]
2164 For each sampling modulus \(N\) and address \(j\), the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of \(a_{Nm+j}\) \cite{SEQ}.
2165 
2166 \begin{corollary}[Cross-presentation torsion-Fourier descent]
2167 The minimal dimension and the complete arithmetic-subsequence packet-complexity profile are independent of whether the input is supplied as a normalized minimal recurrence, a reduced rational OGF, or a minimal state realization. The exact compiler outputs agree under every bridge in \cref{thm:cfinite-three-chart}.
2168 \end{corollary}
2169 
2170 \begin{proof}
2171 All three charts represent the same sequence. The compiler theorem accepts any of the three presentation types and computes quantities determined by the active modal decomposition or, equivalently, by exact reachable-observable reduction of the jump state. The bridge functors preserve the coefficient sequence, so the outputs agree.
2172 \end{proof}
2173 
2174 The phrase ``complete invariant'' here has a declared target. The torsion-Fourier signature is complete for the minimal recurrence and complexity profile of every arithmetic subsequence. It is not asserted to classify arbitrary sequence presentations or all sequence phenomena.
2175 
2176 \subsection{Exact examples}
2177 
2178 \begin{table}[htbp]
2179 \centering
2180 \caption{Three exact C-finite examples across the normalized atlas.}
2181 \label{tab:cfinite-examples}
2182 \small
2183 \begin{tabularx}{\textwidth}{@{}L{0.18\textwidth}L{0.19\textwidth}L{0.19\textwidth}L{0.15\textwidth}Y@{}}
2184 \toprule
2185 Sequence & Minimal recurrence & Reduced OGF & Minimal dimension & Packet behavior\\
2186 \midrule
2187 Fibonacci \(F_n\) & \(x^2-x-1\) & \(z/(1-z-z^2)\) & \(2\) & The root ratio is not torsion. Every arithmetic subsequence retains rank \(2\).\\
2188 \(1+(-1)^n\) & \(x^2-1\) & \(2/(1-z^2)\) & \(2\) & At \(N=2\), the even packet is constant of rank \(1\) and the odd packet is zero.\\
2189 \(n+1)\) & \((x-1)^2\) & \(1/(1-z)^2\) & \(2\) & Every packet remains a nonzero affine polynomial in the packet index and retains rank \(2\).\\
2190 \bottomrule
2191 \end{tabularx}
2192 \end{table}
2193 
2194 \section{A first heterogeneous effective-descent atlas}
2195 \label{sec:periodic-heterogeneous-atlas}
2196 
2197 The previous sections separated the formal assembly theorem from the nonformal work of declaring and verifying an actual atlas.  This section discharges that burden for a deliberately restricted but genuinely heterogeneous example.  The charts do not merely use different coordinates inside one linear category.  They belong to three of the principal strata:
2198 \[
2199 \text{finite linear and C-finite}
2200 \qquad
2201 \text{digital and automatic}
2202 \qquad
2203 \text{functional-Mahler and Cartier}.
2204 \]
2205 The common semantic objects are unit purely periodic sequences over a prime field.  The restriction is strong enough to make every bridge explicit, every normalized fiber rigid, every nonempty triple overlap coherent, and descent effective for both objects and morphisms.  At the same time it is broad enough to contain repeated-root recurrences in characteristic \(p\), nontrivial minimal automata, rational functional-Mahler masks, and redundant Cartier compiler presentations.
2206 
2207 \subsection{The restricted semantic doctrine}
2208 
2209 Fix a prime \(p\).  Let \(\Per_p^{\times,\mathrm{per}}\) denote the discrete category whose objects are sequences
2210 \[
2211 a=(a_n)_{n\geq 0}\in \mathbb F_p^{\mathbb N}
2212 \]
2213 satisfying
2214 \[
2215 a_0=1
2216 \qquad\text{and}\qquad
2217  a_{n+T}=a_n
2218 \quad(n\geq 0)
2219 \]
2220 for some \(T\geq 1\).  The least such positive integer is the \emph{minimal period} \(T(a)\).  Write
2221 \[
2222 w(a)=(a_0,\ldots,a_{T(a)-1})
2223 \]
2224 for the primitive period word and
2225 \[
2226 W_a(z)=\sum_{r=0}^{T(a)-1}a_rz^r.
2227 \]
2228 Then
2229 \begin{equation}
2230 \label{eq:periodic-ogf-unreduced}
2231 F_a(z)=\sum_{n\geq 0}a_nz^n
2232 =
2233 \frac{W_a(z)}{1-z^{T(a)}}.
2234 \end{equation}
2235 The fraction in \cref{eq:periodic-ogf-unreduced} need not be reduced.  Cancellation is part of the normalized rational chart rather than an error.  For example, over \(\mathbb F_3\), the period word \(12\) has
2236 \[
2237 \frac{1+2z}{1-z^2}=\frac{1}{1+z}.
2238 \]
2239 
2240 \begin{boundary}[Why the unit condition is retained]
2241 The functional-Mahler bridge uses the formal quotient
2242 \[
2243 R_a(z)=\frac{F_a(z)}{F_a(z^p)}.
2244 \]
2245 It is regular at the origin exactly when \(F_a(0)\neq 0\).  The normalization \(a_0=1\) removes scalar ambiguity and gives \(R_a(0)=1\).  Purely periodic sequences with \(a_0=0\) remain valid C-finite and automatic objects, but they need not lie in this regular unit-mask overlap.  They are not discarded; they lie outside the selected heterogeneous atlas.
2246 \end{boundary}
2247 
2248 \begin{proposition}[Finite-field persistence equals pure periodicity]
2249 \label{prop:persistent-periodic}
2250 For a sequence over \(\mathbb F_p\), the following are equivalent.
2251 \begin{enumerate}[label=\textup{(\roman*)}]
2252 \item The sequence is C-finite and its minimal recurrence has nonzero constant term.
2253 \item The sequence has a finite-dimensional state realization
2254 \[
2255 a_n=\lambda C^nv
2256 \]
2257 with \(C\in GL_d(\mathbb F_p)\).
2258 \item The sequence is purely periodic from \(n=0\).
2259 \end{enumerate}
2260 If \(m_a(x)\) is the monic minimal recurrence polynomial and \(T=T(a)\), then
2261 \begin{equation}
2262 \label{eq:minimal-period-annihilator}
2263 m_a(x)\mid x^T-1,
2264 \qquad
2265 T=\min\{N\geq 1:m_a(x)\mid x^N-1\}.
2266 \end{equation}
2267 \end{proposition}
2268 
2269 \begin{proof}
2270 A recurrence with nonzero constant term has an invertible companion matrix, giving (i) \(\Rightarrow\) (ii).  The orbit of \(v\) under an invertible map of the finite set \(\mathbb F_p^d\) is a cycle containing its initial state, so (ii) \(\Rightarrow\) (iii).  A sequence of period \(T\) is annihilated by \(E^T-I\), where \(E\) is the forward shift, giving (iii) \(\Rightarrow\) (i).  The minimal annihilator divides every annihilator, hence \(m_a\mid x^T-1\).  Conversely, if \(m_a\mid x^N-1\), then \(E^N-I\) annihilates the sequence.  Minimality of \(T\) proves \cref{eq:minimal-period-annihilator}.
2271 \end{proof}
2272 
2273 Let
2274 \[
2275 F_a(z)=\frac{P_a(z)}{Q_a(z)}
2276 \]
2277 be reduced with
2278 \[
2279 P_a(0)=Q_a(0)=1.
2280 \]
2281 If \(d=\deg m_a\), then
2282 \[
2283 Q_a(z)=z^dm_a(z^{-1}),
2284 \]
2285 with the evident coefficient normalization.  Taking reciprocal polynomials in \cref{eq:minimal-period-annihilator} gives
2286 \begin{equation}
2287 \label{eq:minimal-period-denominator}
2288 Q_a(z)\mid 1-z^T,
2289 \qquad
2290 T=\min\{N\geq 1:Q_a(z)\mid 1-z^N\}.
2291 \end{equation}
2292 Thus pure periodicity is visible both in the linear annihilator and in the reduced rational denominator.
2293 
2294 \subsection{Four normalized charts}
2295 
2296 The reversible core of the selected atlas has four charts.  Each chart carries its own native objects and morphisms, but every normalized fiber over a fixed sequence will be shown to be contractible.
2297 
2298 \subsubsection{The minimal recurrence chart}
2299 
2300 An object of \(\Rec_p^{\times,\mathrm{per}}\) is the normalized minimal recurrence polynomial
2301 \[
2302 m(x)=x^d+c_1x^{d-1}+\cdots+c_d,
2303 \qquad c_d\neq 0,
2304 \]
2305 together with the initial vector \((a_0,\ldots,a_{d-1})\), subject to \(a_0=1\) and the requirement that the generated sequence is purely periodic.  Minimality means that no monic recurrence of lower degree annihilates the sequence.  Morphisms are recurrence-presentation isomorphisms preserving the coefficient sequence.  Because the minimal annihilator and initial data are unique, every endomorphism is the identity.
2306 
2307 \subsubsection{The reduced rational chart}
2308 
2309 An object of \(\Rat_p^{\times,\mathrm{per}}\) is a reduced fraction
2310 \[
2311 F(z)=\frac{P(z)}{Q(z)}\in\mathbb F_p(z),
2312 \qquad
2313 P(0)=Q(0)=1,
2314 \]
2315 whose coefficient sequence is purely periodic.  Equivalently, \(Q\mid 1-z^T\) for some \(T\) and the reduced coefficient sequence has no transient prefix.  Morphisms preserve the rational function.  Reducedness and the constant-term normalization make this chart rigid.
2316 
2317 The recurrence-rational bridge is the strict inverse pair already used in the C-finite atlas: multiply the generating series by the reciprocal recurrence denominator, or recover the recurrence by clearing the reduced denominator.  On the present finite-field subcategory, the additional condition \(Q\mid1-z^T\) records pure periodicity.
2318 
2319 \subsubsection{The canonical digital chart}
2320 
2321 For \(0\leq d<p\), define the Cartier section of a sequence by
2322 \[
2323 (\Lambda_da)_n=a_{pn+d}.
2324 \]
2325 The \(p\)-kernel is
2326 \begin{equation}
2327 \label{eq:p-kernel}
2328 \mathcal K_p(a)
2329 =
2330 \left\{
2331 \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a:
2332 \ell\geq 0,\ 0\leq d_i<p
2333 \right\}.
2334 \end{equation}
2335 We use least-significant-digit-first words.  Every finite digit word is allowed.  Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state \(b\) is \(b_0\) and \((\Lambda_0b)_0=b_0\).
2336 
2337 The canonical Moore automaton has state set \(\mathcal K_p(a)\), start state \(a\), transition
2338 \[
2339 \delta_d(b)=\Lambda_db,
2340 \]
2341 and output
2342 \[
2343 o(b)=b_0.
2344 \]
2345 After reading \(d_0,\ldots,d_{\ell-1}\), its state is
2346 \[
2347 \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}a
2348 =
2349 \bigl(n\longmapsto a_{p^\ell n+r}\bigr),
2350 \qquad
2351 r=\sum_{j=0}^{\ell-1}d_jp^j.
2352 \]
2353 
2354 \begin{proposition}[Finite periodic kernel and exact state model]
2355 \label{prop:periodic-kernel}
2356 Let \(a\in\Per_p^{\times,\mathrm{per}}\) have minimal period \(T\) and period word \(w\).  Every kernel state has the form
2357 \begin{equation}
2358 \label{eq:periodic-kernel-state}
2359 b_{u,r}(n)=w_{un+r\bmod T},
2360 \qquad
2361 u\in U_p(T),\quad r\in\mathbb Z/T\mathbb Z,
2362 \end{equation}
2363 where
2364 \[
2365 U_p(T)=\{p^e\bmod T:e\geq 0\}.
2366 \]
2367 The digit transition on pair representatives is
2368 \begin{equation}
2369 \label{eq:pair-transition}
2370 (u,r)\xrightarrow{d}(pu,r+ud)\pmod T,
2371 \end{equation}
2372 and the output is \(w_r\).  Quotienting pair representatives by equality of the sequences in \cref{eq:periodic-kernel-state} gives the canonical minimal least-significant-digit-first \(p\)-DFAO for \(a\).
2373 \end{proposition}
2374 
2375 \begin{proof}
2376 After \(\ell\) digits, the resulting subsequence is \(n\mapsto a_{p^\ell n+r}\), which depends only on \((p^\ell\bmod T,r\bmod T)\).  Applying \(\Lambda_d\) gives
2377 \[
2378 \Lambda_db_{u,r}(n)
2379 =b_{u,r}(pn+d)
2380 =w_{(pu)n+(r+ud)},
2381 \]
2382 proving \cref{eq:pair-transition}.  The pair set is finite, so the kernel is finite.  Every kernel state is reachable by construction.  If two kernel sequences differ, they differ at some \(n\); reading the base-\(p\) digits of that \(n\) distinguishes the states by their outputs.  Hence the kernel automaton is minimal.  Any other reachable minimal automaton producing the same sequence is uniquely isomorphic to it by the usual behavioral quotient.
2383 \end{proof}
2384 
2385 The pair model gives a useful finite bound.  Write
2386 \[
2387 T=p^\nu M,
2388 \qquad
2389 \gcd(M,p)=1,
2390 \]
2391 and set \(\operatorname{ord}_1(p)=1\).
2392 
2393 \begin{lemma}[Exact multiplier-orbit size]
2394 \label{lem:multiplier-orbit-size}
2395 The set \(U_p(T)\) has cardinality
2396 \begin{equation}
2397 \label{eq:u-size}
2398 |U_p(T)|
2399 =
2400 \nu+\operatorname{ord}_M(p).
2401 \end{equation}
2402 Consequently,
2403 \begin{equation}
2404 \label{eq:kernel-state-bound}
2405 |\mathcal K_p(a)|
2406 \leq
2407 T\bigl(\nu+\operatorname{ord}_M(p)\bigr).
2408 \end{equation}
2409 \end{lemma}
2410 
2411 \begin{proof}
2412 For \(0\leq e<\nu\), the residues \(p^e\bmod p^\nu\) have distinct \(p\)-adic valuations.  For \(e\geq\nu\), the residue is zero modulo \(p^\nu\), while modulo \(M\) it runs through one multiplicative cycle of length \(\operatorname{ord}_M(p)\).  The early and eventual residues cannot coincide.  This proves \cref{eq:u-size}.  The kernel is a quotient of the pair set \(U_p(T)\times\mathbb Z/T\mathbb Z\), giving \cref{eq:kernel-state-bound}.
2413 \end{proof}
2414 
2415 Let \(\Aut_p^{\times,\mathrm{per}}\) be the groupoid of reachable minimal least-significant-digit-first \(p\)-DFAOs whose output sequence lies in \(\Per_p^{\times,\mathrm{per}}\).  Morphisms are automaton isomorphisms preserving the start state, transitions, and outputs.  Reachability forces every automorphism to be the identity.
2416 
2417 \subsubsection{The rational functional-Mahler mask chart}
2418 
2419 For a unit formal series \(F(0)=1\), define its local functional-Mahler mask by
2420 \begin{equation}
2421 \label{eq:unique-local-mask}
2422 \mathcal M_p(F)(z)
2423 =
2424 \frac{F(z)}{F(z^p)}.
2425 \end{equation}
2426 Then
2427 \begin{equation}
2428 \label{eq:functional-mahler-equation}
2429 F(z)=\mathcal M_p(F)(z)F(z^p).
2430 \end{equation}
2431 Iterating \cref{eq:functional-mahler-equation} gives the coefficientwise product
2432 \begin{equation}
2433 \label{eq:mask-product}
2434 F(z)
2435 =
2436 \prod_{j\geq0}\mathcal M_p(F)(z^{p^j}).
2437 \end{equation}
2438 Only finitely many factors contribute to each fixed coefficient.
2439 
2440 If \(F=P/Q\) is reduced and unit normalized, then
2441 \begin{equation}
2442 \label{eq:rational-mask-formula}
2443 R_F(z)
2444 =
2445 \frac{P(z)Q(z^p)}{Q(z)P(z^p)}.
2446 \end{equation}
2447 Thus every object of \(\Rat_p^{\times,\mathrm{per}}\) has a rational mask.  Reduce \cref{eq:rational-mask-formula} to
2448 \[
2449 R_F(z)=\frac{A(z)}{B(z)},
2450 \qquad
2451 A(0)=B(0)=1,
2452 \qquad
2453 \gcd(A,B)=1.
2454 \]
2455 
2456 Define \(\Mask_p^{\times,\mathrm{per}}\) to be the groupoid of such reduced rational masks whose normalized product solution \cref{eq:mask-product} has purely periodic coefficients.  Morphisms preserve the reduced mask.
2457 
2458 \begin{theorem}[Unit rational functional-Mahler equivalence]
2459 \label{thm:unit-rational-mask-equivalence}
2460 The assignments
2461 \[
2462 F\longmapsto \frac{F(z)}{F(z^p)}
2463 \qquad\text{and}\qquad
2464 R\longmapsto \prod_{j\geq0}R(z^{p^j})
2465 \]
2466 are inverse equivalences
2467 \[
2468 \Rat_p^{\times,\mathrm{per}}
2469 \simeq
2470 \Mask_p^{\times,\mathrm{per}}.
2471 \]
2472 The mask is unique for the represented series.
2473 \end{theorem}
2474 
2475 \begin{proof}
2476 Starting with \(F\), the finite product through level \(N\) telescopes:
2477 \[
2478 \prod_{j=0}^{N}\frac{F(z^{p^j})}{F(z^{p^{j+1}})}
2479 =
2480 \frac{F(z)}{F(z^{p^{N+1}})}.
2481 \]
2482 As \(N\to\infty\), the denominator converges coefficientwise to \(F(0)=1\), proving that the product recovers \(F\).  Conversely, the product solution satisfies \(F(z)=R(z)F(z^p)\), so \(R=F/F(z^p)\).  Reduced normalization makes the rational mask unique.
2483 \end{proof}
2484 
2485 \begin{boundary}[Functional Mahler is not Newton-Mahler]
2486 The mask equation in this section is
2487 \[
2488 F(z)=R(z)F(z^p).
2489 \]
2490 It is a digit-section and automaton construction.  It is not the Newton-Mahler interpolation expansion
2491 \[
2492 f(x)=\sum_rc_r\binom{x}{r}.
2493 \]
2494 Both occur elsewhere in the Library, but this atlas uses only the functional-Mahler meaning.
2495 \end{boundary}
2496 
2497 \begin{figure}[htbp]
2498 \centering
2499 \includegraphics[width=0.99\textwidth]{periodic_heterogeneous_atlas.pdf}
2500 \caption{The first heterogeneous atlas proved effective in this manuscript.  Four normalized reversible charts span three strata.  The raw Cartier compiler is retained as a directional lane because its numerator coordinates need not be minimal.}
2501 \label{fig:periodic-heterogeneous-atlas}
2502 \end{figure}
2503 
2504 \subsection{Explicit bridge passports}
2505 
2506 The reversible bridge core is generated by three explicit constructions:
2507 \[
2508 \Rec_p^{\times,\mathrm{per}}
2509 \leftrightarrow
2510 \Rat_p^{\times,\mathrm{per}},
2511 \qquad
2512 \Rat_p^{\times,\mathrm{per}}
2513 \leftrightarrow
2514 \Aut_p^{\times,\mathrm{per}},
2515 \qquad
2516 \Rat_p^{\times,\mathrm{per}}
2517 \leftrightarrow
2518 \Mask_p^{\times,\mathrm{per}}.
2519 \]
2520 The remaining reversible bridges are their normalized composites.  A separate constructor and reachable-restriction lane connects masks to the digital normal form through the Cartier compiler.
2521 
2522 \begin{landscape}
2523 \begin{longtable}{@{}L{0.13\linewidth}L{0.14\linewidth}L{0.14\linewidth}L{0.14\linewidth}L{0.15\linewidth}L{0.21\linewidth}@{}}
2524 \caption{Bridge passports for the selected periodic atlas.}\label{tab:periodic-bridge-passports}\\
2525 \toprule
2526 Bridge & Exact class & Domain & Preserved data & Reflected data & Information loss or boundary\\
2527 \midrule
2528 \endfirsthead
2529 \toprule
2530 Bridge & Exact class & Domain & Preserved data & Reflected data & Information loss or boundary\\
2531 \midrule
2532 \endhead
2533 \midrule
2534 \multicolumn{6}{r}{\small Continued on next page}\\
2535 \endfoot
2536 \bottomrule
2537 \endlastfoot
2538 Recurrence \(\leftrightarrow\) reduced OGF & equivalence & minimal invertible recurrence and reduced purely periodic rational series & coefficients, minimal annihilator, initial data, period, Hankel rank & equality and isomorphism & none after reduced normalization\\[4pt]
2539 Reduced OGF \(\leftrightarrow\) minimal DFAO & equivalence on the periodic essential image & unit purely periodic series and reachable minimal LSD-first \(p\)-DFAOs & coefficient semantics, minimal period, canonical \(p\)-kernel, output behavior & equality of represented sequences & a nonminimal residue-pair automaton is quotiented before entering the normalized chart\\[4pt]
2540 Reduced OGF \(\leftrightarrow\) rational mask & equivalence on the unit-periodic essential image & \(F(0)=1\), periodic rational \(F\), reduced regular mask \(R\) & exact series, functional equation, reduced rational mask, minimal period & equality of series and masks & undefined for nonunit series; unrestricted rational masks may have nonperiodic products\\[4pt]
2541 Mask \(\to\) Cartier numerator representation & constructor & reduced \(A/B\) over \(\mathbb F_p\) & exact coefficient semantics, digit matrices, theorem degree bound & not generally reflective & introduces basis and denominator coordinates and may contain unreachable ambient states\\[4pt]
2542 Ambient Cartier compiler \(\to\) minimal DFAO & reachable restriction & full finite numerator-state compiler & every reachable coefficient section and every future digit behavior & equality of the represented \(p\)-kernel states & removes only ambient states outside the orbit of the initial numerator\\[4pt]
2543 Unrestricted automatic \(\dashrightarrow\) rational mask & partial bridge only & automatic sequences whose unit inverse mask is rational and whose product lies in the declared periodic class & coefficient semantics on the domain of definition & no unrestricted reflection theorem & no claim that every automatic sequence has a rational periodic mask\\
2544 \end{longtable}
2545 \end{landscape}
2546 
2547 \subsubsection{The rational-to-digital construction}
2548 
2549 For a periodic word \(w\) of length \(T\), construct the finite pair automaton from \cref{eq:pair-transition}, then quotient by equality of the induced subsequences.  This gives the canonical minimal DFAO directly.  Conversely, a minimal DFAO in the selected chart is promised to produce a purely periodic sequence.  Its minimal period can be recovered exactly by enumerating \(N=1,2,\ldots\), constructing the finite-state transducer for addition by \(N\), and testing equivalence between the original output and its \(N\)-shift.  Equivalence of finite deterministic output automata is a finite product-graph calculation.  The promise of pure periodicity guarantees termination.  Once \(T\) is known, \cref{eq:periodic-ogf-unreduced} gives the rational series, which is then reduced.
2550 
2551 This inverse is effective but not claimed to be complexity optimal.  The point is effectivity, not a sharp automata algorithm.
2552 
2553 \subsubsection{The rational-mask compiler lane}
2554 
2555 Let
2556 \[
2557 R(z)=\frac{A(z)}{B(z)},
2558 \qquad
2559 A(0)=B(0)=1,
2560 \]
2561 with
2562 \[
2563 a=\deg A,
2564 \qquad
2565 b=\deg B,
2566 \qquad
2567 c=a+(p-2)b,
2568 \]
2569 and define
2570 \begin{equation}
2571 \label{eq:cartier-degree-bound-block03}
2572 D=\max\left\{b,\left\lceil\frac{c}{p-1}\right\rceil\right\}.
2573 \end{equation}
2574 The rational-mask Cartier theorem gives an invariant numerator space of degree at most \(D\) under
2575 \[
2576 P\longmapsto\Lambda_d(PAB^{p-2}),
2577 \qquad 0\leq d<p.
2578 \]
2579 The resulting linear state vector belongs to \(\mathbb F_p^{D+1}\), so the raw output automaton has at most
2580 \begin{equation}
2581 \label{eq:compiler-state-bound-block03}
2582 p^{D+1}
2583 \end{equation}
2584 states \cite{FMH}.  It computes the coefficient sequence exactly.
2585 
2586 \begin{proposition}[Compiler reachability lands in the canonical digital chart]
2587 \label{prop:compiler-minimal-core}
2588 For a mask in \(\Mask_p^{\times,\mathrm{per}}\), the reachable part of the fixed-denominator Cartier compiler is uniquely isomorphic to the canonical \(p\)-kernel automaton of \cref{prop:periodic-kernel}.  In particular, the subsequent quotient by equality of future output behavior is trivial.
2589 \end{proposition}
2590 
2591 \begin{proof}
2592 The fixed-denominator intertwining map sends the numerator reached by a digit word \(w\) to the Cartier section \(\Lambda_wF\).  It is injective, as proved in the more general \cref{thm:fixed-denominator-cartier-realization,thm:reachable-numerator-core}.  Hence distinct reachable numerator states represent distinct section series and are distinguished by some future coefficient.  The reachable compiler state set is therefore exactly the canonical \(p\)-kernel state set, with matching transitions and outputs.
2593 \end{proof}
2594 
2595 \begin{figure}[htbp]
2596 \centering
2597 \includegraphics[width=0.99\textwidth]{cartier_compiler_reduction.pdf}
2598 \caption{The raw mask compiler is a directional constructor.  Reachability removes the ambient states outside the numerator orbit.  Injectivity of the fixed-denominator realization then shows that the remaining states are already the unique minimal digital normal form.}
2599 \label{fig:cartier-compiler-reduction}
2600 \end{figure}
2601 
2602 \subsection{Triple-overlap and quadruple coherence}
2603 
2604 There are four nonempty triple overlaps:
2605 \[
2606 (\Rec,\Rat,\Aut),
2607 \qquad
2608 (\Rec,\Rat,\Mask),
2609 \qquad
2610 (\Rec,\Aut,\Mask),
2611 \qquad
2612 (\Rat,\Aut,\Mask).
2613 \]
2614 All four charts meet on the same unit periodic semantic objects.
2615 
2616 \begin{lemma}[Rigidity of normalized fibers]
2617 \label{lem:periodic-fiber-rigidity}
2618 Fix \(a\in\Per_p^{\times,\mathrm{per}}\).  In each of the four normalized charts, the groupoid of presentations of \(a\) is contractible.
2619 \end{lemma}
2620 
2621 \begin{proof}
2622 The minimal recurrence with its initial data, the reduced rational function, and the reduced mask are unique, so their fibers are terminal groupoids.  A normalized digital presentation is a reachable minimal automaton.  Any automorphism fixes the start state and commutes with every transition.  Reachability then forces it to fix every state.  Hence its automorphism group is trivial, and any two minimal presentations are joined by the unique behavioral isomorphism.
2623 \end{proof}
2624 
2625 \begin{theorem}[Coherence of the periodic heterogeneous atlas]
2626 \label{thm:periodic-atlas-coherence}
2627 Choose every pairwise bridge by the explicit constructions above and choose every comparison \(2\)-cell to be the unique normalized isomorphism over the common sequence.  Then:
2628 \begin{enumerate}[label=\textup{(\roman*)}]
2629 \item every one of the four triple-overlap comparison diagrams commutes;
2630 \item the two composites of comparison cells on the four-chart overlap agree;
2631 \item the quadruple-overlap coherence defect \(\Omega\) is the identity.
2632 \end{enumerate}
2633 \end{theorem}
2634 
2635 \begin{proof}
2636 Every comparison in question is an isomorphism between normalized presentations of one fixed semantic sequence.  By \cref{lem:periodic-fiber-rigidity}, there is exactly one such isomorphism.  Any two composites with the same source and target must therefore agree.  This proves all triple laws and the quadruple law simultaneously.
2637 \end{proof}
2638 
2639 \begin{figure}[htbp]
2640 \centering
2641 \includegraphics[width=0.86\textwidth]{periodic_coherence_tetrahedron.pdf}
2642 \caption{The coherence tetrahedron.  Contractibility of every normalized semantic fiber makes each comparison cell unique, so all triple and quadruple coherence equations hold automatically after the bridge constructions have been typed.}
2643 \label{fig:periodic-coherence}
2644 \end{figure}
2645 
2646 \subsection{Effective descent for objects and morphisms}
2647 
2648 Let \(\mathfrak R_p^{\mathrm{per}}\) denote the four-chart pseudofunctor defined by the preceding passports and comparison cells.
2649 
2650 \begin{theorem}[First heterogeneous effective-descent theorem]
2651 \label{thm:first-heterogeneous-descent}
2652 The semantic pseudonatural transformation induces an equivalence
2653 \begin{equation}
2654 \label{eq:first-heterogeneous-descent}
2655 \Desc(\mathfrak R_p^{\mathrm{per}})
2656 \simeq
2657 \Per_p^{\times,\mathrm{per}}.
2658 \end{equation}
2659 In particular, descent is effective for both objects and morphisms.
2660 \end{theorem}
2661 
2662 \begin{proof}
2663 Given \(a\in\Per_p^{\times,\mathrm{per}}\), form its minimal recurrence, reduced rational OGF, canonical minimal \(p\)-kernel automaton, and reduced rational mask.  The explicit bridges supply a descent object.  This defines a functor
2664 \[
2665 N:\Per_p^{\times,\mathrm{per}}
2666 \longrightarrow
2667 \Desc(\mathfrak R_p^{\mathrm{per}}).
2668 \]
2669 Conversely, any descent object has local presentations whose semantic images agree on every overlap.  Since the semantic target is discrete, they determine one sequence \(a\).  Normalization and \cref{lem:periodic-fiber-rigidity} identify the local presentations with those constructed by \(N(a)\).  Thus \(N\) is essentially surjective and fully faithful.
2670 
2671 For morphisms, a compatible local family lies over one equality of semantic sequences.  Each normalized fiber has at most one morphism, so the local family glues uniquely.  Identities, composition, equality, and invertibility are therefore detected componentwise and descend.
2672 \end{proof}
2673 
2674 \begin{corollary}[A verified instance of the assembly theorem]
2675 The atlas \(\mathfrak R_p^{\mathrm{per}}\) satisfies realization, semantic adequacy, typed overlaps, higher coherence, effective descent, and preservation for the finite algebraic constructions used in this section.  Therefore \cref{thm:atlas-assembly} applies without a remaining categorical hypothesis on this restricted domain.
2676 \end{corollary}
2677 
2678 The conclusion is intentionally local to the selected atlas.  It does not imply effective descent for arbitrary automatic sequences, unrestricted rational masks, general local analytic objects, or the full eight-stratum registry.
2679 
2680 \subsection{A compatible complete invariant family}
2681 
2682 For \(a\in\Per_p^{\times,\mathrm{per}}\), define the periodic resolved invariant package
2683 \begin{equation}
2684 \label{eq:periodic-invariant-package}
2685 \mathcal J_p(a)
2686 =
2687 \bigl(
2688 T(a),
2689  w(a),
2690  m_a(x),
2691  P_a(z)/Q_a(z),
2692  \mathcal K_p(a),
2693  \mathcal A_p(a),
2694  R_a(z)
2695 \bigr),
2696 \end{equation}
2697 where \(\mathcal A_p(a)\) is the canonical minimal digital automaton.  Its comparison identities include
2698 \begin{equation}
2699 \label{eq:periodic-invariant-identities}
2700 \begin{aligned}
2701 \deg m_a
2702 &=\deg Q_a
2703 =\rank H(a)
2704 =\dim \State_{\min}(a),\\
2705 T(a)
2706 &=\min\{N:m_a\mid x^N-1\}
2707 =\min\{N:Q_a\mid 1-z^N\},\\
2708 |\mathcal A_p(a)|
2709 &=|\mathcal K_p(a)|
2710 \leq T(a)\bigl(\nu_p(T(a))+\operatorname{ord}_{M}(p)\bigr),\\
2711 R_a(z)
2712 &=\frac{F_a(z)}{F_a(z^p)},
2713 \qquad T(a)=p^{\nu_p(T(a))}M,
2714 \quad \gcd(M,p)=1.
2715 \end{aligned}
2716 \end{equation}
2717 
2718 \begin{theorem}[Invariant descent and global completeness on the periodic atlas]
2719 \label{thm:periodic-invariant-completeness}
2720 The native invariants in the recurrence, rational, digital, and mask charts admit coherent comparison maps to \(\mathcal J_p\).  These maps descend to
2721 \[
2722 \mathcal J_p:
2723 \Desc(\mathfrak R_p^{\mathrm{per}})
2724 \longrightarrow
2725 \mathbf{Inv}_p^{\mathrm{per}}.
2726 \]
2727 The descended invariant is complete on the declared semantic category.
2728 \end{theorem}
2729 
2730 \begin{proof}
2731 Every component of \(\mathcal J_p(a)\) is reconstructed from the coefficient sequence by the explicit normalization procedures above.  Hence every pairwise bridge preserves the package and the comparison cells are identities after semantic identification.  Triple coherence follows from \cref{thm:periodic-atlas-coherence}.  If \(\mathcal J_p(a)=\mathcal J_p(b)\), then the primitive period words agree, so \(a=b\).  Under \cref{eq:first-heterogeneous-descent}, equality of semantic objects is exactly isomorphism in the descent category.
2732 \end{proof}
2733 
2734 The completeness claim has a narrow classification target: unit purely periodic sequences over \(\mathbb F_p\).  It is not a complete invariant for all C-finite, all automatic, or all rational functional-Mahler sequences.
2735 
2736 \subsection{Exact examples and a normalization boundary}
2737 
2738 \input{generated/block03_examples_table.tex}
2739 
2740 \paragraph{Even indicator over \(\mathbb F_2\).}
2741 For the word \(10\),
2742 \[
2743 T=2,
2744 \qquad
2745 m(x)=x^2+1=(x+1)^2,
2746 \qquad
2747 F(z)=\frac{1}{1+z^2},
2748 \qquad
2749 R(z)=1+z^2.
2750 \]
2751 The repeated recurrence root is compatible with pure periodicity because the field has characteristic two.  The canonical \(2\)-kernel has three states: the original sequence, the constant-one sequence, and the constant-zero sequence.  The rational-mask compiler gives the safe ambient bound \(2^3=8\) before reachability restriction.
2752 
2753 \paragraph{Alternating unit over \(\mathbb F_3\).}
2754 For the word \(12\),
2755 \[
2756 T=2,
2757 \qquad
2758 m(x)=x+1,
2759 \qquad
2760 F(z)=\frac{1}{1+z},
2761 \qquad
2762 R(z)=\frac{1+z^3}{1+z}=1+2z+z^2.
2763 \]
2764 The recurrence order is one although the minimal period is two.  The \(3\)-kernel consists of the sequence and its negative, giving two minimal digital states.  The compiler bound is nine.
2765 
2766 \paragraph{Period-three unit word over \(\mathbb F_2\).}
2767 For the word \(110\),
2768 \[
2769 T=3,
2770 \qquad
2771 m(x)=x^2+x+1,
2772 \qquad
2773 F(z)=\frac{1}{1+z+z^2},
2774 \qquad
2775 R(z)=1+z+z^2.
2776 \]
2777 The reachable pair model has six representatives, while equality of subsequence behavior collapses them to three kernel states.  This is a concrete instance of the directional quotient in \cref{prop:compiler-minimal-core}.
2778 
2779 \begin{figure}[htbp]
2780 \centering
2781 \includegraphics[width=0.99\textwidth]{periodic_example_automata.pdf}
2782 \caption{Three exact examples.  A state label \(s\mid o\) gives the state identifier and output.  The compiler bounds are ambient theorem bounds before reachability restriction, not claims of minimal state count.}
2783 \label{fig:periodic-example-automata}
2784 \end{figure}
2785 
2786 \paragraph{Negative control: a nonunit periodic sequence.}
2787 Over \(\mathbb F_2\), the word \(01\) gives
2788 \[
2789 F(z)=\frac{z}{1+z^2}.
2790 \]
2791 Then
2792 \[
2793 \frac{F(z)}{F(z^2)}
2794 =
2795 \frac{1+z^2}{z},
2796 \]
2797 which is Laurent rather than regular at the origin.  The sequence remains purely periodic, C-finite, and automatic, but it is not an object of \(\Mask_2^{\times,\mathrm{per}}\).  This verifies that the unit overlap condition is structural rather than cosmetic.
2798 
2799 \subsection{Exact finite verification}
2800 
2801 \input{generated/block03_validation_table.tex}
2802 
2803 The supplied verifier enumerates every primitive unit-periodic word over \(\mathbb F_2\) through period eight and over \(\mathbb F_3\) through period six.  For each object it checks, using exact prime-field and polynomial arithmetic:
2804 \begin{enumerate}
2805 \item Berlekamp-Massey recovery of the minimal recurrence;
2806 \item divisibility by \(x^T-1\) and \(1-z^T\);
2807 \item recovery of the least period from both normalized linear charts;
2808 \item the reduced rational functional-Mahler identity;
2809 \item the exact reachable pair model and its behavioral quotient;
2810 \item the multiplier-orbit formula \cref{eq:u-size};
2811 \item coefficient output of the minimal digit automaton;
2812 \item the Cartier compiler state bound \cref{eq:compiler-state-bound-block03}.
2813 \end{enumerate}
2814 The finite computation supports the implementation and examples.  The symbolic theorems above do not depend on the enumeration.
2815 
2816 \begin{resultbox}
2817 \textbf{Section result.}
2818 The manuscript now contains a fully declared finite atlas spanning three genuinely different strata: minimal finite-linear recurrences, minimal digital automata, and rational functional-Mahler masks.  Every reversible bridge has a complete passport.  The raw Cartier compiler is retained as a directional constructor with an exact loss statement.  All four triple overlaps and the quadruple coherence equation are proved through rigidity of normalized fibers.  Objects and morphisms descend effectively, and the compatible periodic invariant package is globally complete on the declared unit-periodic semantic category.  This is the first theorem in the manuscript that satisfies the finite heterogeneous completion criterion rather than only the one-stratum cross-presentation criterion.
2819 \end{resultbox}
2820 
2821 
2822 \section{Rational-mask digital descent beyond pure periodicity}
2823 \label{sec:rational-mask-digital-atlas}
2824 
2825 Periodic atlas verified a four-chart atlas on unit purely periodic sequences.  That domain made the linear, rational, digital, and functional-Mahler charts meet simultaneously, but it left open the larger overlap between rational functional-Mahler masks and automatic sequences.  This section removes the periodicity hypothesis while keeping the normalization and bridge standards unchanged.
2826 
2827 Fix a prime \(p\).  Let
2828 \[
2829 \RMah_p^{\times}
2830 =
2831 \left\{
2832 F(z)\in 1+z\mathbb F_p[[z]]:
2833 \frac{F(z)}{F(z^p)}\in\mathbb F_p(z)
2834 \right\}
2835 \]
2836 be the discrete category of unit formal series with rational canonical functional-Mahler mask.  This category contains the periodic objects of \cref{sec:periodic-heterogeneous-atlas}, but for odd \(p\) it also contains genuinely nonrational and non-C-finite automatic series.
2837 
2838 \subsection{Frobenius-radical characterization}
2839 
2840 For a unit formal series, define its canonical local mask by
2841 \begin{equation}
2842 \label{eq:block04-canonical-mask}
2843 R_F(z)
2844 =
2845 \frac{F(z)}{F(z^p)}.
2846 \end{equation}
2847 The coefficientwise infinite product
2848 \begin{equation}
2849 \label{eq:block04-mask-product}
2850 \mathcal M_p[R](z)
2851 =
2852 \prod_{j\geq 0}R\!\left(z^{p^j}\right)
2853 \end{equation}
2854 is well defined for every \(R\in1+z\mathbb F_p[[z]]\), because only finitely many factors contribute to a fixed coefficient.
2855 
2856 \begin{theorem}[Frobenius-radical characterization]
2857 \label{thm:frobenius-radical-characterization}
2858 Let \(F\in1+z\mathbb F_p[[z]]\).  Then
2859 \begin{equation}
2860 \label{eq:block04-frobenius-mask}
2861 R_F(z)=F(z)^{1-p}.
2862 \end{equation}
2863 Consequently, the following are equivalent.
2864 \begin{enumerate}[label=\textup{(\roman*)}]
2865 \item The canonical mask \(R_F\) is rational.
2866 \item The power \(F^{p-1}\) belongs to \(\mathbb F_p(z)\).
2867 \item There are coprime polynomials \(A,B\in\mathbb F_p[z]\) with
2868 \[
2869 A(0)=B(0)=1
2870 \]
2871 and
2872 \begin{equation}
2873 \label{eq:block04-rational-mask-equation}
2874 B(z)F(z)=A(z)F(z^p).
2875 \end{equation}
2876 \end{enumerate}
2877 For the reduced mask \(R_F=A/B\), one has
2878 \begin{equation}
2879 \label{eq:block04-radical-equation}
2880 F(z)^{p-1}=\frac{B(z)}{A(z)}.
2881 \end{equation}
2882 The ratio and product constructions are strict inverses:
2883 \[
2884 F\longmapsto R_F,
2885 \qquad
2886 R\longmapsto\mathcal M_p[R].
2887 \]
2888 In particular, every object of \(\RMah_p^{\times}\) is algebraic over \(\mathbb F_p(z)\) of degree at most \(p-1\).
2889 \end{theorem}
2890 
2891 \begin{proof}
2892 In characteristic \(p\), Frobenius gives
2893 \[
2894 F(z^p)=F(z)^p.
2895 \]
2896 Substitution into \cref{eq:block04-canonical-mask} gives \cref{eq:block04-frobenius-mask}.  Thus \(R_F\) is rational exactly when \(F^{p-1}=R_F^{-1}\) is rational.  Writing the reduced rational mask as \(A/B\) gives \cref{eq:block04-rational-mask-equation} and \cref{eq:block04-radical-equation}.
2897 
2898 Conversely, a reduced rational mask \(A/B\) with unit constant terms defines a unique formal solution of \cref{eq:block04-rational-mask-equation}: the product in \cref{eq:block04-mask-product}.  Telescoping gives
2899 \[
2900 \frac{\mathcal M_p[R](z)}{\mathcal M_p[R](z^p)}=R(z),
2901 \]
2902 while uniqueness follows by comparing coefficients recursively.  Hence the ratio and product constructions are strict inverses.  Finally, \cref{eq:block04-radical-equation} is an algebraic equation of degree \(p-1\).
2903 \end{proof}
2904 
2905 \begin{boundary}[Functional Mahler versus Newton-Mahler]
2906 The word ``Mahler'' continues to name two distinct constructions.  This section concerns the functional equation
2907 \[
2908 F(z)=R(z)F(z^p).
2909 \]
2910 It does not identify rational masks with Newton-Mahler expansions
2911 \[
2912 f(x)=\sum_{r\geq0}c_r\binom xr
2913 \]
2914 or with \(p\)-adic analytic interpolation.  Any bridge to that local analytic chart still requires its own passport and proof.
2915 \end{boundary}
2916 
2917 \subsection{The rational-collapse obstruction}
2918 
2919 Let \(R=A/B\) be reduced and normalized.  For each monic irreducible polynomial \(\pi\in\mathbb F_p[z]\), define
2920 \[
2921 \omega_{\pi}(R)
2922 =
2923 \bigl(v_{\pi}(B)-v_{\pi}(A)\bigr)\bmod(p-1).
2924 \]
2925 Include the valuation at infinity
2926 \[
2927 \omega_{\infty}(R)
2928 =
2929 \bigl(\deg A-\deg B\bigr)\bmod(p-1).
2930 \]
2931 The finite-support family
2932 \begin{equation}
2933 \label{eq:block04-collapse-obstruction}
2934 \omega_{p-1}(R)
2935 =
2936 \bigl(\omega_v(R)\bigr)_v
2937 \end{equation}
2938 is the \emph{rational-collapse obstruction}.
2939 
2940 \begin{theorem}[Exact rational-collapse criterion]
2941 \label{thm:rational-collapse-criterion}
2942 Let \(F=\mathcal M_p[A/B]\).  The following are equivalent.
2943 \begin{enumerate}[label=\textup{(\roman*)}]
2944 \item \(F\in\mathbb F_p(z)\).
2945 \item \(B/A\) is a \((p-1)\)-st power in \(\mathbb F_p(z)\).
2946 \item \(\omega_{p-1}(A/B)=0\).
2947 \end{enumerate}
2948 When these conditions hold, there is a unique rational function \(G\in\mathbb F_p(z)\) with \(G(0)=1\) and
2949 \[
2950 G^{p-1}=B/A,
2951 \]
2952 and \(F=G\).
2953 
2954 If \(p=2\), every rational mask collapses to a rational series.  If \(p>2\) and \(\omega_{p-1}(A/B)\neq0\), then \(F\) is nonrational, non-C-finite, and not eventually periodic.
2955 \end{theorem}
2956 
2957 \begin{proof}
2958 The equivalence of (i) and (ii) follows from \cref{eq:block04-radical-equation}.  If \(B/A=G^{p-1}\), every valuation of \(B/A\) is divisible by \(p-1\), so (ii) implies (iii).
2959 
2960 Conversely, suppose all valuation residues vanish.  Since \(A\) and \(B\) are coprime, every irreducible exponent in each polynomial is divisible by \(p-1\).  Factoring into monic irreducibles therefore gives
2961 \[
2962 A=C^{p-1},
2963 \qquad
2964 B=D^{p-1}
2965 \]
2966 for polynomials \(C,D\) after a constant normalization.  The unit conditions \(A(0)=B(0)=1\) remove the remaining scalar obstruction.  Thus \(B/A=(D/C)^{p-1}\).  Multiplying \(D/C\) by its inverse constant term does not change its \((p-1)\)-st power, and gives the unique root with constant term one.  The formal solution is unique, so \(F=G\).
2967 
2968 For \(p=2\), the exponent \(p-1\) is one, hence the obstruction group is trivial and \(F=B/A\).  Over a finite field, a rational coefficient series is C-finite and eventually periodic, while an eventually periodic sequence has a rational generating function.  Therefore a nonzero obstruction forces all three failures stated in the theorem.
2969 \end{proof}
2970 
2971 \begin{figure}[htbp]
2972 \centering
2973 \includegraphics[width=0.98\textwidth]{rational_collapse_obstruction.pdf}
2974 \caption{The rational-collapse obstruction.  A rational mask always gives an automatic series, but it gives a rational or C-finite series exactly on the vanishing divisor-residue locus.}
2975 \label{fig:rational-collapse-obstruction}
2976 \end{figure}
2977 
2978 \subsection{Denominator stabilization and the canonical Cartier module}
2979 
2980 Write
2981 \[
2982 a=\deg A,
2983 \qquad
2984 b=\deg B,
2985 \qquad
2986 H(z)=A(z)B(z)^{p-2},
2987 \]
2988 and define
2989 \begin{equation}
2990 \label{eq:block04-degree-bound}
2991 D(A,B)
2992 =
2993 \max\left\{
2994 b,
2995 \left\lceil\frac{a+(p-2)b}{p-1}\right\rceil
2996 \right\}.
2997 \end{equation}
2998 Let \(\mathcal P_D=\mathbb F_p[z]_{\leq D}\).  For \(0\leq d<p\), define
2999 \begin{equation}
3000 \label{eq:block04-numerator-transition}
3001 T_d(P)=\Lambda_d(PH),
3002 \end{equation}
3003 where \(\Lambda_d\) selects the coefficients in degrees congruent to \(d\) modulo \(p\).
3004 
3005 \begin{theorem}[Fixed-denominator Cartier realization]
3006 \label{thm:fixed-denominator-cartier-realization}
3007 Let \(F=\mathcal M_p[A/B]\) and \(D=D(A,B)\).  Then:
3008 \begin{enumerate}[label=\textup{(\roman*)}]
3009 \item every \(T_d\) preserves \(\mathcal P_D\);
3010 \item the linear map
3011 \begin{equation}
3012 \label{eq:block04-theta-map}
3013 \Theta_{A,B}:\mathcal P_D\longrightarrow\mathbb F_p[[z]],
3014 \qquad
3015 \Theta_{A,B}(P)=\frac{P}{B}F,
3016 \end{equation}
3017 is injective;
3018 \item the transitions intertwine exactly:
3019 \begin{equation}
3020 \label{eq:block04-intertwining}
3021 \Lambda_d\Theta_{A,B}(P)
3022 =
3023 \Theta_{A,B}(T_d(P));
3024 \end{equation}
3025 \item starting from \(P_0=B\), the coefficient of index
3026 \[
3027 n=d_0+d_1p+\cdots+d_{\ell-1}p^{\ell-1}
3028 \]
3029 is
3030 \begin{equation}
3031 \label{eq:block04-compiler-output}
3032 [z^n]F
3033 =
3034 \epsilon_0\!\left(
3035 \Theta_{A,B}
3036 \bigl(T_{d_{\ell-1}}\cdots T_{d_0}B\bigr)
3037 \right),
3038 \end{equation}
3039 where \(\epsilon_0(G)=[z^0]G\).
3040 \end{enumerate}
3041 \end{theorem}
3042 
3043 \begin{proof}
3044 If \(\deg P\leq D\), then
3045 \[
3046 \deg(PH)
3047 \leq
3048 D+a+(p-2)b.
3049 \]
3050 Cartier selection divides the retained degrees by \(p\).  The definition of \(D\) gives
3051 \[
3052 \deg T_d(P)
3053 \leq
3054 \left\lfloor\frac{D+a+(p-2)b}{p}\right\rfloor
3055 \leq D.
3056 \]
3057 This proves (i).
3058 
3059 The series \(F/B\) is a unit in \(\mathbb F_p[[z]]\).  Multiplication by a unit is injective, so \(\Theta_{A,B}\) is injective.  The rational-mask Cartier identity imported from \cite{FMH} is precisely \cref{eq:block04-intertwining}.  Iterating it from \(\Theta(B)=F\) and taking constant terms proves \cref{eq:block04-compiler-output}.
3060 \end{proof}
3061 
3062 For a digit word \(w=d_0\cdots d_{\ell-1}\), read least significant digit first and write
3063 \[
3064 \Lambda_w
3065 =
3066 \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_0}.
3067 \]
3068 Define the Cartier kernel and its linear span by
3069 \begin{equation}
3070 \label{eq:block04-kernel-span}
3071 \mathcal K_p(F)
3072 =
3073 \{\Lambda_wF:w\in\{0,\ldots,p-1\}^{*}\},
3074 \qquad
3075 \mathcal V_p(F)
3076 =
3077 \operatorname{span}_{\mathbb F_p}\mathcal K_p(F).
3078 \end{equation}
3079 
3080 \begin{definition}[Canonical Cartier module and Cartier rank]
3081 The canonical Cartier module of \(F\) is
3082 \begin{equation}
3083 \label{eq:block04-cartier-module}
3084 \mathsf C_p(F)
3085 =
3086 \left(
3087 \mathcal V_p(F),
3088 (\Lambda_d|_{\mathcal V_p(F)})_{0\leq d<p},
3089 F,
3090 \epsilon_0
3091 \right).
3092 \end{equation}
3093 Its dimension
3094 \begin{equation}
3095 \label{eq:block04-cartier-rank}
3096 r_p(F)=\dim_{\mathbb F_p}\mathcal V_p(F)
3097 \end{equation}
3098 is the \emph{Cartier rank}.
3099 \end{definition}
3100 
3101 \begin{theorem}[Minimality of the canonical Cartier module]
3102 \label{thm:canonical-cartier-minimality}
3103 For every \(F\in\RMah_p^{\times}\), the module \(\mathsf C_p(F)\) is finite dimensional, reachable, and observable.  It is therefore a minimal digit-linear realization of the coefficient function.  Every other reachable-observable digit-linear realization of the same coefficient function is uniquely similar to \(\mathsf C_p(F)\).
3104 \end{theorem}
3105 
3106 \begin{proof}
3107 By \cref{thm:fixed-denominator-cartier-realization}, every section \(\Lambda_wF\) lies in the \((D+1)\)-dimensional image of \(\Theta_{A,B}\).  Hence \(\mathcal V_p(F)\) is finite dimensional.  Reachability is built into its definition as the span of the orbit of \(F\).
3108 
3109 For observability, let \(G\in\mathcal V_p(F)\) satisfy
3110 \[
3111 \epsilon_0(\Lambda_wG)=0
3112 \]
3113 for every digit word \(w\).  For any \(n\geq0\), choose the base-\(p\) digits of \(n\).  The section identity gives
3114 \[
3115 [z^n]G=\epsilon_0(\Lambda_wG)=0.
3116 \]
3117 Thus \(G=0\), so the observation family separates points.
3118 
3119 For uniqueness, let \((V,(L_d),v,\lambda)\) be another reachable-observable realization of the same digit function.  Send the reachable vector \(L_wv\) to \(\Lambda_wF\).  If a linear combination of reachable vectors vanishes, applying every continuation and then \(\lambda\) shows that the corresponding combination of Cartier sections has every coefficient zero.  Observability makes the assignment well defined.  Reachability makes it surjective, and the symmetric construction gives an inverse.  The intertwining, initial-vector, and output identities are immediate.  Any such similarity is determined on the reachable spanning set, so it is unique.
3120 \end{proof}
3121 
3122 \subsection{The exact reachable compiler core}
3123 
3124 Let \(\mathcal R(A,B)\subseteq\mathcal P_D\) be the finite set of numerator states reachable from \(B\) under the transitions \(T_d\), and let
3125 \[
3126 W(A,B)=\operatorname{span}_{\mathbb F_p}\mathcal R(A,B).
3127 \]
3128 
3129 \begin{theorem}[Reachable numerator core]
3130 \label{thm:reachable-numerator-core}
3131 The map \(\Theta_{A,B}\) restricts to
3132 \begin{equation}
3133 \label{eq:block04-reachable-bijections}
3134 \mathcal R(A,B)
3135 \overset{\sim}{\longrightarrow}
3136 \mathcal K_p(F)
3137 \end{equation}
3138 and to a linear isomorphism
3139 \begin{equation}
3140 \label{eq:block04-reachable-span-isomorphism}
3141 W(A,B)
3142 \overset{\sim}{\longrightarrow}
3143 \mathcal V_p(F).
3144 \end{equation}
3145 The reachable deterministic output automaton of the fixed-denominator compiler is already minimal.  No nontrivial output-equivalence quotient remains after reachability restriction.
3146 \end{theorem}
3147 
3148 \begin{proof}
3149 For every word \(w\), repeated use of \cref{eq:block04-intertwining} gives
3150 \[
3151 \Theta_{A,B}(T_wB)=\Lambda_wF.
3152 \]
3153 Thus \(\Theta_{A,B}\) maps the reachable numerator set onto the Cartier kernel.  Injectivity of \(\Theta_{A,B}\) makes the map bijective, and taking linear spans gives \cref{eq:block04-reachable-span-isomorphism}.
3154 
3155 If two reachable numerators \(P\neq Q\) had identical future output behavior, the two series \(\Theta(P)\) and \(\Theta(Q)\) would have identical coefficients and would therefore be equal.  Injectivity would force \(P=Q\), a contradiction.  Hence all reachable states are behaviorally distinct.  Reachability and distinguishability prove minimality.
3156 \end{proof}
3157 
3158 The preceding theorem sharpens the generic reduction language used in Periodic atlas.  A behavioral quotient remains a safe general construction for arbitrary output automata, but it is trivial for the normalized fixed-denominator numerator compiler.
3159 
3160 Let
3161 \begin{equation}
3162 \label{eq:block04-kernel-size}
3163 \kappa_p(F)=|\mathcal K_p(F)|.
3164 \end{equation}
3165 
3166 \begin{corollary}[Exact state-complexity hierarchy]
3167 \label{cor:cartier-state-hierarchy}
3168 For a reduced rational mask \(A/B\),
3169 \begin{equation}
3170 \label{eq:block04-state-hierarchy}
3171 r_p(F)
3172 \leq
3173 \kappa_p(F)
3174 \leq
3175 p^{r_p(F)}
3176 \leq
3177 p^{D(A,B)+1}.
3178 \end{equation}
3179 Moreover,
3180 \[
3181 r_p(F)=\dim W(A,B),
3182 \qquad
3183 \kappa_p(F)=|\mathcal R(A,B)|.
3184 \]
3185 \end{corollary}
3186 
3187 \begin{proof}
3188 The Cartier kernel spans \(\mathcal V_p(F)\), so its cardinality is at least the dimension.  It is a subset of the \(p^{r_p(F)}\) vectors in \(\mathcal V_p(F)\).  Finally, \(\mathcal V_p(F)\) embeds into the \((D+1)\)-dimensional compiler space by \cref{thm:reachable-numerator-core}.
3189 \end{proof}
3190 
3191 \begin{figure}[htbp]
3192 \centering
3193 \includegraphics[width=0.99\textwidth]{cartier_state_hierarchy.pdf}
3194 \caption{The exact reachable core of the fixed-denominator compiler.  The linear rank and deterministic state count are different invariants, but both are bounded by the explicit mask degree profile.}
3195 \label{fig:cartier-state-hierarchy}
3196 \end{figure}
3197 
3198 \subsection{Three normalized charts}
3199 
3200 The rational-mask domain has three normalized charts.
3201 
3202 \paragraph{Reduced mask chart.}
3203 An object of \(\Mask_p^{\mathrm{rat}}\) is a reduced rational function
3204 \[
3205 R(z)=\frac{A(z)}{B(z)},
3206 \qquad
3207 A(0)=B(0)=1.
3208 \]
3209 Morphisms preserve the rational function.  The semantics is \(R\mapsto\mathcal M_p[R]\).  The inverse chart map is the canonical ratio \(F\mapsto F/F(z^p)\).
3210 
3211 \paragraph{Minimal Cartier-module chart.}
3212 An object of \(\mathbf{Cart}_p^{\mathrm{min,rat}}\) is a finite-dimensional tuple
3213 \[
3214 (V,(L_d)_{d=0}^{p-1},v,\lambda)
3215 \]
3216 that is reachable and observable, whose digit coefficient function belongs to \(\RMah_p^{\times}\).  Morphisms are similarities preserving every \(L_d\), the initial vector, and the output functional.  Its normalized representative is \(\mathsf C_p(F)\).
3217 
3218 \paragraph{Minimal digital chart.}
3219 An object of \(\mathbf{Aut}_p^{\mathrm{min,rat}}\) is a reachable minimal least-significant-digit-first \(p\)-DFAO whose coefficient series belongs to \(\RMah_p^{\times}\).  Its canonical representative has state set \(\mathcal K_p(F)\), transition \(G\mapsto\Lambda_dG\), start state \(F\), and output \(G\mapsto G(0)\).
3220 
3221 \begin{table}[htbp]
3222 \centering
3223 \caption{Bridge passports for the rational-mask digital atlas.}
3224 \label{tab:block04-bridge-passports}
3225 \small
3226 \begin{tabularx}{\textwidth}{@{}L{0.19\textwidth}L{0.18\textwidth}L{0.16\textwidth}Y@{}}
3227 \toprule
3228 Bridge & Class & Preserved data & Exact boundary\\
3229 \midrule
3230 Mask \(\leftrightarrow\) series & strict inverse pair & every coefficient and the canonical mask & the mask must be rational and unit normalized\\
3231 Series \(\leftrightarrow\) Cartier module & normalized equivalence & all Cartier sections, linear relations, initial vector, output & the module is restricted to reachable-observable realizations\\
3232 Series \(\leftrightarrow\) minimal DFAO & normalized equivalence & the complete \(p\)-kernel and coefficient function & the automaton uses the fixed radix and LSD-first convention\\
3233 Mask \(\to\) ambient compiler & directional constructor & exact coefficients and fixed-denominator transitions & adds unreachable ambient numerator states\\
3234 Ambient compiler \(\to\) normalized core & reachable restriction & all reachable series and all future outputs & discards only states outside the orbit of \(B\)\\
3235 \bottomrule
3236 \end{tabularx}
3237 \end{table}
3238 
3239 \begin{theorem}[Rational-mask three-chart equivalence]
3240 \label{thm:rational-mask-three-chart-equivalence}
3241 The semantic functors induce equivalences of groupoids
3242 \begin{equation}
3243 \label{eq:block04-three-chart-equivalence}
3244 \Mask_p^{\mathrm{rat}}
3245 \simeq
3246 \mathbf{Cart}_p^{\mathrm{min,rat}}
3247 \simeq
3248 \mathbf{Aut}_p^{\mathrm{min,rat}}
3249 \simeq
3250 \RMah_p^{\times}.
3251 \end{equation}
3252 The mask-series bridge is a strict inverse pair.  The Cartier and digital bridges are normalized equivalences with unique comparison isomorphisms.
3253 \end{theorem}
3254 
3255 \begin{proof}
3256 The strict mask-series equivalence is \cref{thm:frobenius-radical-characterization}.  The canonical Cartier construction gives an object of the second chart, and \cref{thm:canonical-cartier-minimality} gives a unique similarity from every other minimal realization of the same series.  The canonical \(p\)-kernel automaton is finite by \cref{thm:fixed-denominator-cartier-realization}.  It is reachable by construction.  Distinct kernel series differ in some coefficient, and the digit word for that coefficient distinguishes their future outputs, so the automaton is minimal.  The usual behavioral map gives the unique isomorphism from any other reachable minimal DFAO for the same series.
3257 \end{proof}
3258 
3259 \begin{figure}[htbp]
3260 \centering
3261 \includegraphics[width=0.98\textwidth]{rational_mask_digital_atlas.pdf}
3262 \caption{The Rational-mask atlas atlas.  The normalized reversible core contains a rational mask, a minimal Cartier module, and a minimal digital automaton.  The full polynomial compiler remains a larger directional presentation.}
3263 \label{fig:rational-mask-digital-atlas}
3264 \end{figure}
3265 
3266 \subsection{Coherence and effective descent}
3267 
3268 \begin{lemma}[Rigidity of rational-mask normalized fibers]
3269 \label{lem:rational-mask-fiber-rigidity}
3270 Fix \(F\in\RMah_p^{\times}\).  The groupoid of normalized presentations of \(F\) in each of the three charts is contractible.
3271 \end{lemma}
3272 
3273 \begin{proof}
3274 The reduced canonical mask is unique.  Any automorphism of a reachable Cartier module that fixes the initial vector and commutes with all digit operators fixes a spanning set, hence is the identity.  Any automorphism of a reachable deterministic automaton that fixes the start state and commutes with every digit transition fixes every state, hence is the identity.  Existence and uniqueness of isomorphisms between two normalized realizations follow from \cref{thm:canonical-cartier-minimality,thm:rational-mask-three-chart-equivalence}.
3275 \end{proof}
3276 
3277 Let \(\mathfrak R_p^{\mathrm{rm}}\) denote the three-chart pseudofunctor defined by the passports above.
3278 
3279 \begin{theorem}[Rational-mask coherence and effective descent]
3280 \label{thm:rational-mask-effective-descent}
3281 Choose every pairwise bridge by the canonical constructions and every comparison cell by the unique normalized isomorphism over the common series.  Then the triple-overlap coherence law holds, and
3282 \begin{equation}
3283 \label{eq:block04-effective-descent}
3284 \Desc(\mathfrak R_p^{\mathrm{rm}})
3285 \simeq
3286 \RMah_p^{\times}.
3287 \end{equation}
3288 Descent is effective for objects and morphisms.
3289 \end{theorem}
3290 
3291 \begin{proof}
3292 On the only nonempty triple overlap, both composites are isomorphisms between normalized presentations of the same series.  By \cref{lem:rational-mask-fiber-rigidity}, the isomorphism is unique, so the comparison diagram commutes.
3293 
3294 Given \(F\in\RMah_p^{\times}\), its reduced mask, canonical Cartier module, and minimal \(p\)-kernel automaton form descent data.  Conversely, compatible local objects have one common semantic series.  Fiber rigidity identifies them uniquely with its normalized presentations.  The same argument applies to morphisms.  Hence the normalization functor is fully faithful and essentially surjective.
3295 \end{proof}
3296 
3297 This theorem is strictly broader than \cref{thm:first-heterogeneous-descent}: no periodicity or C-finiteness hypothesis is imposed.  It is also strictly narrower than an atlas of all automatic sequences, because an automatic series may have a nonrational canonical mask.
3298 
3299 \subsection{Compatible invariants and collapse stratification}
3300 
3301 For \(F\in\RMah_p^{\times}\), define
3302 \begin{equation}
3303 \label{eq:block04-invariant-package}
3304 \mathcal J_p^{\mathrm{rm}}(F)
3305 =
3306 \left(
3307 R_F,
3308 \omega_{p-1}(R_F),
3309 D(R_F),
3310 \mathcal V_p(F),
3311 r_p(F),
3312 \mathcal K_p(F),
3313 \kappa_p(F),
3314 \mathcal A_p(F)
3315 \right).
3316 \end{equation}
3317 
3318 \begin{theorem}[Invariant descent on the rational-mask atlas]
3319 \label{thm:rational-mask-invariant-descent}
3320 The native mask, Cartier-module, and automaton invariants admit coherent comparison maps to \(\mathcal J_p^{\mathrm{rm}}\).  They descend to the category in \cref{eq:block04-effective-descent}.  The descended invariant is complete on \(\RMah_p^{\times}\).
3321 \end{theorem}
3322 
3323 \begin{proof}
3324 The fixed-denominator intertwining theorem identifies the reachable numerator span with \(\mathcal V_p(F)\) and the reachable numerator set with \(\mathcal K_p(F)\).  Hence all components of the package agree under the bridges.  Coherence follows from \cref{thm:rational-mask-effective-descent}.  If two packages agree, their reduced canonical masks agree.  The coefficientwise mask product then reconstructs the same formal series, so the semantic objects are equal.
3325 \end{proof}
3326 
3327 The obstruction \(\omega_{p-1}\) divides the semantic category into two exact loci:
3328 \[
3329 \RMah_p^{\times}
3330 =
3331 \RMah_{p,\mathrm{col}}^{\times}
3332 \sqcup
3333 \RMah_{p,\mathrm{ncol}}^{\times}.
3334 \]
3335 The collapse locus consists of rational, hence ultimately periodic, series.  The noncollapse locus consists of non-C-finite automatic series.  The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside \(\RMah_{p,\mathrm{col}}^{\times}\).
3336 
3337 \subsection{Exact examples and strict boundaries}
3338 
3339 \input{generated/block04_examples_table.tex}
3340 
3341 \paragraph{Cantor-digit indicator.}
3342 Over \(\mathbb F_3\), let
3343 \[
3344 R(z)=1+z.
3345 \]
3346 Then
3347 \[
3348 F(z)=\prod_{j\geq0}(1+z^{3^j})
3349 \]
3350 has coefficient one exactly when the ternary expansion of the index uses only the digits zero and one.  The functional equation is
3351 \[
3352 F(z)=(1+z)F(z^3),
3353 \qquad
3354 F(z)^2=\frac1{1+z}.
3355 \]
3356 The factor \(1+z\) has odd divisor multiplicity, so the collapse obstruction is nonzero.  Thus the sequence is automatic but not C-finite or eventually periodic.  The canonical automaton has two states, while the Cartier rank is one.
3357 
3358 \paragraph{A genuine denominator example.}
3359 Over \(\mathbb F_3\), take
3360 \[
3361 R(z)=\frac1{1+z}.
3362 \]
3363 Then
3364 \[
3365 F(z)^2=1+z.
3366 \]
3367 The reduced denominator remains fixed under every Cartier section.  The ambient compiler has dimension two, its reachable numerator set is
3368 \[
3369 \{0,1,2,1+z\},
3370 \]
3371 and its reachable span has dimension two.  Hence
3372 \[
3373 r_3(F)=2,
3374 \qquad
3375 \kappa_3(F)=4.
3376 \]
3377 The nonzero collapse obstruction again proves that the automatic coefficient sequence is not eventually periodic.
3378 
3379 \paragraph{Collapse back to the periodic atlas.}
3380 For
3381 \[
3382 R(z)=(1+z)^2=1+2z+z^2
3383 \]
3384 over \(\mathbb F_3\), one has
3385 \[
3386 F(z)^2=\frac1{(1+z)^2},
3387 \qquad
3388 F(z)=\frac1{1+z}.
3389 \]
3390 The coefficients alternate between one and two.  This object lies in the rational-collapse locus and in the unit-periodic atlas of Periodic atlas.
3391 
3392 \paragraph{Characteristic-two transient.}
3393 Over \(\mathbb F_2\), choose
3394 \[
3395 R(z)=\frac1{1+z+z^2}.
3396 \]
3397 Since \(p-1=1\),
3398 \[
3399 F(z)=1+z+z^2.
3400 \]
3401 Its coefficient sequence is \(1,1,1,0,0,\ldots\).  It is ultimately periodic but not purely periodic.  This is the simplest exact witness that Rational-mask atlas extends the semantic domain even in characteristic two.
3402 
3403 \begin{figure}[htbp]
3404 \centering
3405 \includegraphics[width=0.99\textwidth]{rational_mask_nonperiodic_examples.pdf}
3406 \caption{Two exact nonperiodic objects in the rational-mask atlas.  State labels give the reachable numerator polynomial followed by its output.}
3407 \label{fig:rational-mask-nonperiodic-examples}
3408 \end{figure}
3409 
3410 \paragraph{Negative control: automatic does not imply rational mask.}
3411 Let \(t_n\in\mathbb F_2\) be the parity of the binary digit sum of \(n\), and put
3412 \[
3413 u_n=1+t_n.
3414 \]
3415 The sequence \(u\) is unit and \(2\)-automatic.  It is not eventually periodic.  Indeed, eventual period \(T\) would propagate backwards through the identities
3416 \[
3417 t_{2^k+n}=1+t_n
3418 \qquad
3419 (0\leq n<2^k)
3420 \]
3421 and force a pure period \(T\).  Writing \(T=2^{\nu}q\) with \(q\) odd, the binary digit count of \(2^m-T\) has parity
3422 \[
3423 m-\nu-s_2(q-1)
3424 \]
3425 for all sufficiently large \(m\).  Choosing the parity of \(m\) makes
3426 \[
3427 t_{2^m-T}\neq t_{2^m},
3428 \]
3429 contradicting pure periodicity.
3430 
3431 Let \(U(z)=\sum_{n\geq0}u_nz^n\).  If its canonical mask were rational, then \cref{thm:rational-collapse-criterion} with \(p=2\) would make \(U\) rational, hence eventually periodic.  Therefore \(U/U(z^2)\) is not rational.  The rational-mask atlas is a proper normalized subatlas of the unit automatic stratum.
3432 
3433 \subsection{Exact finite verification}
3434 
3435 \input{generated/block04_validation_table.tex}
3436 
3437 The Rational-mask atlas verifier enumerates every reduced unit mask with
3438 \[
3439 \deg A,\deg B\leq3\quad(p=2),
3440 \qquad
3441 \deg A,\deg B\leq2\quad(p=3),
3442 \qquad
3443 \deg A,\deg B\leq1\quad(p=5).
3444 \]
3445 For every mask it checks, using exact finite-field and polynomial arithmetic:
3446 \begin{enumerate}
3447 \item the invariant degree bound on the full ambient numerator space;
3448 \item the functional-equation coefficient recursion against compiler output;
3449 \item transition closure and fixed-denominator intertwining;
3450 \item distinguishability of every pair of reachable numerator states;
3451 \item the identities
3452 \[
3453 W(A,B)\cong\mathcal V_p(F),
3454 \qquad
3455 \mathcal R(A,B)\cong\mathcal K_p(F);
3456 \]
3457 \item the state hierarchy in \cref{eq:block04-state-hierarchy};
3458 \item characteristic-two collapse and the named collapse or noncollapse examples.
3459 \end{enumerate}
3460 The finite calculation validates the implementation on the recorded ranges.  It is not used as a substitute for the unbounded proofs.
3461 
3462 \begin{resultbox}
3463 \textbf{Section result.}
3464 The verified heterogeneous atlas now extends beyond pure periodicity.  Every unit rational functional-Mahler mask over \(\mathbb F_p\) has a finite canonical Cartier module and a finite canonical \(p\)-kernel automaton.  The three normalized charts satisfy coherence and effective descent.  The fixed-denominator compiler has an exact reachable core: reachable numerator states are already the minimal digital states, and their span is the minimal Cartier module.  The rational-collapse obstruction separates the ultimately periodic sublocus from genuinely non-C-finite automatic series.  The resulting invariant package is globally complete on the declared rational-mask semantic category, but no claim is made for automatic sequences whose canonical mask is nonrational.
3465 \end{resultbox}
3466 
3467 \section{Fixed-radix regular normalization and the automatic finite-orbit boundary}
3468 \label{sec:block05-regular}
3469 
3470 Rational-mask atlas closes a substantial automatic slice, but its semantic domain is still defined by the existence of a rational functional-Mahler mask.  The next digital question is more basic and more general: what is the canonical normalized representation of a sequence whose complete fixed-radix section orbit spans a finite-dimensional vector space, even when that orbit itself is infinite?
3471 
3472 This section answers that question over a field.  It separates two invariants that coincide only in special cases:
3473 \[
3474 \text{linear section rank}
3475 \qquad\text{and}\qquad
3476 \text{cardinality of the section orbit}.
3477 \]
3478 The first controls regularity and minimal digit-linear dimension.  The second controls automaticity and minimal deterministic-state cardinality.  Their exact relationship is the automatic--regular boundary developed below.
3479 
3480 \begin{figure}[htbp]
3481 \centering
3482 \includegraphics[width=0.99\textwidth]{regular_fixed_radix_atlas.pdf}
3483 \caption{The fixed-radix regular atlas.  The canonical section module and its strict shortlex matrix form exist throughout the regular locus.  The minimal DFAO chart appears exactly on the finite-orbit sublocus.}
3484 \label{fig:regular-fixed-radix-atlas}
3485 \end{figure}
3486 
3487 \subsection{Section operators and the canonical kernel module}
3488 
3489 Fix an integer radix \(k\geq2\) and a field \(K\).  Let
3490 \[
3491 \Sigma_k=\{0,1,\ldots,k-1\}.
3492 \]
3493 Words are read from least significant digit to most significant digit.  For
3494 \[
3495 w=d_0d_1\cdots d_{\ell-1}\in\Sigma_k^*,
3496 \]
3497 define
3498 \[
3499 |w|=\ell,
3500 \qquad
3501 [w]_k=\sum_{j=0}^{\ell-1}d_jk^j.
3502 \]
3503 The empty word \(\varepsilon\) has length zero and value zero.
3504 
3505 For a sequence \(a=(a_n)_{n\geq0}\in K^{\mathbb N}\), define the digit-section operators
3506 \[
3507 (\Lambda_d a)(n)=a(kn+d),
3508 \qquad 0\leq d<k.
3509 \]
3510 For a word \(w=d_0\cdots d_{\ell-1}\), put
3511 \[
3512 \Lambda_w
3513 =
3514 \Lambda_{d_{\ell-1}}\cdots\Lambda_{d_1}\Lambda_{d_0}.
3515 \]
3516 Then
3517 \begin{equation}
3518 (\Lambda_w a)(n)
3519 =
3520  a\bigl(k^{|w|}n+[w]_k\bigr).
3521 \label{eq:block05-word-section}
3522 \end{equation}
3523 
3524 \begin{definition}[Fixed-radix kernel and section module]
3525 The \(k\)-kernel and its linear span are
3526 \[
3527 \mathcal K_k(a)
3528 =
3529 \{\Lambda_w a:w\in\Sigma_k^*\},
3530 \qquad
3531 \mathcal V_k(a)
3532 =
3533 \Span_K\mathcal K_k(a).
3534 \]
3535 The sequence is \emph{\(k\)-regular over \(K\)} when \(\mathcal V_k(a)\) is finite dimensional.  Its regular rank is
3536 \[
3537 r_k(a)=\dim_K\mathcal V_k(a).
3538 \]
3539 It is \emph{\(k\)-automatic in the present reading convention} when \(\mathcal K_k(a)\) is finite.  In that case set
3540 \[
3541 \kappa_k(a)=|\mathcal K_k(a)|.
3542 \]
3543 \end{definition}
3544 
3545 The terminology records two different finiteness statements.  Regularity asks for a finite-dimensional linear hull.  Automaticity asks for a finite orbit under the section semigroup itself.
3546 
3547 \begin{definition}[Digit-linear realization]
3548 A digit-linear realization of \(a\) is a tuple
3549 \[
3550 \mathsf L=(W,A_0,\ldots,A_{k-1},v,\lambda),
3551 \]
3552 where \(W\) is a finite-dimensional \(K\)-vector space, \(A_d\in\End_K(W)\), \(v\in W\), and \(\lambda\in W^*\), such that for every word \(w=d_0\cdots d_{\ell-1}\),
3553 \begin{equation}
3554 a_{[w]_k}
3555 =
3556 \lambda A_wv,
3557 \qquad
3558 A_w=A_{d_{\ell-1}}\cdots A_{d_0}.
3559 \label{eq:block05-digit-linear-semantics}
3560 \end{equation}
3561 Because different words can represent the same integer after terminal high zeros are appended, equation \eqref{eq:block05-digit-linear-semantics} includes the required leading-zero consistency.
3562 \end{definition}
3563 
3564 \begin{theorem}[Section-module realization theorem]
3565 \label{thm:block05-section-module-realization}
3566 For a sequence \(a\in K^{\mathbb N}\), the following are equivalent:
3567 \begin{enumerate}
3568 \item \(a\) is \(k\)-regular over \(K\);
3569 \item \(a\) admits a finite-dimensional digit-linear realization.
3570 \end{enumerate}
3571 When these conditions hold, the tuple
3572 \begin{equation}
3573 \mathsf C_k(a)
3574 =
3575 \bigl(
3576 \mathcal V_k(a),
3577 \Lambda_0,\ldots,\Lambda_{k-1},
3578  a,
3579 \epsilon_0
3580 \bigr),
3581 \qquad
3582 \epsilon_0(f)=f(0),
3583 \label{eq:block05-canonical-section-module}
3584 \end{equation}
3585 is a canonical realization of dimension \(r_k(a)\).
3586 \end{theorem}
3587 
3588 \begin{proof}
3589 Assume first that \(\mathcal V_k(a)\) is finite dimensional.  It is invariant under every \(\Lambda_d\) by construction.  Choose any basis, represent the section operators by matrices, take the initial vector to be \(a\), and take output to be evaluation at zero.  For every word \(w\),
3590 \[
3591 \epsilon_0(\Lambda_wa)
3592 =(\Lambda_wa)(0)
3593 =a_{[w]_k},
3594 \]
3595 so the resulting realization satisfies \eqref{eq:block05-digit-linear-semantics}.
3596 
3597 Conversely, suppose \(a\) has a realization \((W,A_d,v,\lambda)\).  For \(x\in W\), define its represented behavior by
3598 \[
3599 \mathcal B(x)(n)
3600 =
3601 \lambda A_{\operatorname{dig}_k(n)}x,
3602 \]
3603 where \(A_{\operatorname{dig}_k(n)}\) is the product associated with the canonical least-significant-digit-first expansion of \(n\).  Equation \eqref{eq:block05-digit-linear-semantics} and concatenation of digit words give
3604 \[
3605 \Lambda_w a
3606 =
3607 \mathcal B(A_wv).
3608 \]
3609 Hence every kernel section belongs to \(\mathcal B(W)\), so
3610 \[
3611 \dim_K\mathcal V_k(a)
3612 \leq
3613 \dim_K W.
3614 \]
3615 Thus \(a\) is regular.  The realization in \eqref{eq:block05-canonical-section-module} has dimension exactly \(r_k(a)\).
3616 \end{proof}
3617 
3618 \begin{proposition}[Reachability and observability]
3619 \label{prop:block05-reach-observe}
3620 The canonical section realization \(\mathsf C_k(a)\) is linearly reachable and observable:
3621 \[
3622 \Span_K\{\Lambda_wa:w\in\Sigma_k^*\}
3623 =
3624 \mathcal V_k(a),
3625 \]
3626 and
3627 \[
3628 \bigcap_{w\in\Sigma_k^*}
3629 \ker(\epsilon_0\Lambda_w)
3630 =
3631 \{0\}.
3632 \]
3633 \end{proposition}
3634 
3635 \begin{proof}
3636 Reachability is the definition of \(\mathcal V_k(a)\).  If \(f\in\mathcal V_k(a)\) satisfies \((\epsilon_0\Lambda_w)(f)=0\) for every word \(w\), then
3637 \[
3638 f([w]_k)=0
3639 \]
3640 for every word.  Every nonnegative integer has such an expansion, so \(f=0\).
3641 \end{proof}
3642 
3643 \subsection{Word-Hankel rank and minimal digit-linear dimension}
3644 
3645 The canonical module can be read directly from a Hankel-type matrix indexed by digit words.
3646 
3647 \begin{definition}[Word-Hankel matrix]
3648 The fixed-radix word-Hankel matrix of \(a\) is the infinite matrix
3649 \begin{equation}
3650 H_a(u,v)
3651 =
3652  a_{[u]_k+k^{|u|}[v]_k}
3653 =
3654 (\Lambda_u a)([v]_k),
3655 \qquad
3656 u,v\in\Sigma_k^*.
3657 \label{eq:block05-word-hankel}
3658 \end{equation}
3659 \end{definition}
3660 
3661 \begin{theorem}[Regular rank theorem]
3662 \label{thm:block05-regular-rank}
3663 Let \(a\) be \(k\)-regular over \(K\).  Then
3664 \begin{equation}
3665 \boxed{
3666  r_k(a)
3667  =
3668  \rank_K H_a
3669  =
3670  \min\{\dim_K W:W\text{ supports a digit-linear realization of }a\}.
3671 }
3672 \label{eq:block05-rank-identity}
3673 \end{equation}
3674 Moreover:
3675 \begin{enumerate}
3676 \item there exist finite word sets \(U,V\subset\Sigma_k^*\), each of cardinality \(r_k(a)\), for which the minor \(H_a[U,V]\) is nonsingular;
3677 \item every reachable-observable realization of dimension \(r_k(a)\) is isomorphic to \(\mathsf C_k(a)\);
3678 \item any two minimal reachable-observable realizations are related by a unique similarity intertwining all digit operators, the initial vector, and the output functional.
3679 \end{enumerate}
3680 \end{theorem}
3681 
3682 \begin{proof}
3683 The row indexed by \(u\) in \(H_a\) is the behavior of the section \(\Lambda_u a\) evaluated on all continuation words.  By observability in \cref{prop:block05-reach-observe}, evaluation on all words is injective on \(\mathcal V_k(a)\).  Therefore the row span of \(H_a\) is naturally isomorphic to \(\mathcal V_k(a)\), and
3684 \[
3685 \rank_K H_a=r_k(a).
3686 \]
3687 
3688 For an arbitrary realization \((W,A_d,v_{\mathrm{in}},\lambda)\), concatenation gives the factorization
3689 \[
3690 H_a(u,v)
3691 =
3692 \lambda A_vA_uv_{\mathrm{in}}.
3693 \]
3694 Thus \(H_a\) factors through \(W\), so \(\rank H_a\leq\dim W\).  The canonical realization has dimension \(r_k(a)\), proving the minimality formula.
3695 
3696 An infinite matrix of finite rank \(r_k(a)\) has a nonsingular finite \(r_k(a)\times r_k(a)\) minor, which gives the first assertion.  For the remaining assertions, define the behavior map of any realization by
3697 \[
3698 \mathcal B_W(x)(n)=\lambda A_{\operatorname{dig}_k(n)}x.
3699 \]
3700 Observability makes this map injective.  Reachability and equality of the represented sequence make its image equal to \(\mathcal V_k(a)\).  In minimal dimension it is therefore an isomorphism intertwining the digit operators, initial vector, and output.  Composing the two behavior isomorphisms gives the unique similarity between any two minimal reachable-observable realizations.
3701 \end{proof}
3702 
3703 \begin{figure}[htbp]
3704 \centering
3705 \includegraphics[width=0.99\textwidth]{regular_hankel_rank.pdf}
3706 \caption{Three exact meanings of the regular rank.  The finite nonsingular minor is a certificate rather than an approximation: it proves a lower bound that is attained by the canonical section module.}
3707 \label{fig:regular-hankel-rank}
3708 \end{figure}
3709 
3710 \begin{remark}[Finite certification]
3711 The theorem is infinite in statement but finite in certificate.  A proposed rank \(r\) can be certified by:
3712 \begin{enumerate}
3713 \item an explicit \(r\)-dimensional section-stable span containing \(a\), which proves \(r_k(a)\leq r\);
3714 \item an explicit nonsingular \(r\times r\) word-Hankel minor, which proves \(r_k(a)\geq r\).
3715 \end{enumerate}
3716 When both are supplied, the regular rank is exact.
3717 \end{remark}
3718 
3719 \subsection{Strict shortlex normalization}
3720 
3721 Minimal realizations are unique up to similarity, but the stratified atlas needs a literal normalized chart rather than an unnamed similarity class.  The actual section functions supply a canonical basis choice.
3722 
3723 Order \(\Sigma_k^*\) by shortlex order: shorter words precede longer words, and words of equal length are ordered lexicographically.  Starting from the empty list, scan the words in this order and retain a word precisely when its section is not in the span of the previously retained sections.
3724 
3725 \begin{definition}[Greedy shortlex kernel basis]
3726 For a regular sequence \(a\), let
3727 \[
3728 U_a=(u_1,\ldots,u_{r_k(a)})
3729 \]
3730 be the unique ordered list produced by the greedy shortlex rule, and put
3731 \[
3732 b_i=\Lambda_{u_i}a.
3733 \]
3734 For the zero sequence, the list is empty.
3735 \end{definition}
3736 
3737 Because membership in a linear span is intrinsic, the resulting ordered basis depends only on the represented sequence, the field, the radix, and the fixed word order.
3738 
3739 \begin{construction}[Strict shortlex realization]
3740 \label{con:block05-shortlex}
3741 Write
3742 \[
3743 \Lambda_db_j
3744 =
3745 \sum_{i=1}^{r_k(a)}(M_d(a))_{ij}b_i,
3746 \qquad
3747  a=\sum_i v_i(a)b_i,
3748 \qquad
3749 \ell_i(a)=b_i(0).
3750 \]
3751 The strict shortlex record is
3752 \begin{equation}
3753 \mathsf{SL}_{k,K}(a)
3754 =
3755 \bigl(
3756 U_a,
3757 M_0(a),\ldots,M_{k-1}(a),
3758  v(a),
3759 \ell(a)
3760 \bigr).
3761 \label{eq:block05-shortlex-record}
3762 \end{equation}
3763 \end{construction}
3764 
3765 \begin{theorem}[Strict fixed-radix normal form]
3766 \label{thm:block05-shortlex-normal-form}
3767 The record \(\mathsf{SL}_{k,K}(a)\) is uniquely determined by \(a\) and reconstructs \(a\) exactly through
3768 \[
3769 a_{[w]_k}
3770 =
3771 \ell(a)M_w(a)v(a).
3772 \]
3773 It has dimension \(r_k(a)\).  Two regular sequences have the same strict shortlex record if and only if they are equal.
3774 \end{theorem}
3775 
3776 \begin{proof}
3777 The greedy word list is uniquely determined by the actual section functions.  Once the ordered basis is fixed, the coefficients of every digit section, the initial vector, and the output functional are unique.  The reconstruction identity is the coordinate form of the canonical section realization.  Equality of records therefore implies equality of every coefficient, while equality of sequences gives the same sections and hence the same greedy record.
3778 \end{proof}
3779 
3780 \begin{theorem}[Fixed-radix regular normalized atlas]
3781 \label{thm:block05-regular-atlas}
3782 Let \(\mathbf{Reg}_k(K)\) be the discrete semantic category of \(K\)-valued \(k\)-regular sequences.  Let \(\mathbf{Sec}_k^{\mathrm{can}}(K)\) be the category of canonical section-module records \eqref{eq:block05-canonical-section-module}, and let \(\mathbf{Lin}_{k,K}^{\mathrm{sl}}\) be the category of strict shortlex records \eqref{eq:block05-shortlex-record}, with morphisms required to preserve the represented sequence.  Then
3783 \begin{equation}
3784 \boxed{
3785 \mathbf{Reg}_k(K)
3786 \simeq
3787 \mathbf{Sec}_k^{\mathrm{can}}(K)
3788 \simeq
3789 \mathbf{Lin}_{k,K}^{\mathrm{sl}}.
3790 }
3791 \label{eq:block05-regular-atlas-equivalence}
3792 \end{equation}
3793 Every normalized fiber over a fixed sequence is contractible.  Consequently the associated finite presentation atlas has effective object and morphism descent.
3794 \end{theorem}
3795 
3796 \begin{proof}
3797 A sequence determines its canonical section module and its strict shortlex record functorially.  Conversely, either record reconstructs every coefficient.  The two composites are identities because the section module is defined from the represented sequence and the shortlex basis is intrinsic.  The uniqueness in \cref{thm:block05-shortlex-normal-form} leaves no nontrivial automorphism in a normalized fiber.  Compatible local objects and morphisms therefore glue uniquely.
3798 \end{proof}
3799 
3800 \begin{definition}[Fixed-radix regular invariant]
3801 The normalized fixed-radix invariant is
3802 \begin{equation}
3803 \mathcal J_{k,K}^{\mathrm{reg}}(a)
3804 =
3805 \bigl(
3806  r_k(a),
3807  \mathsf C_k(a),
3808  \mathsf{SL}_{k,K}(a),
3809  [H_a[U,V]]
3810 \bigr),
3811 \label{eq:block05-regular-invariant}
3812 \end{equation}
3813 where \([H_a[U,V]]\) denotes any recorded nonsingular rank certificate together with its word addresses.  The strict normal form makes this package complete on \(\mathbf{Reg}_k(K)\).
3814 \end{definition}
3815 
3816 The finite Hankel minor is not canonical unless an ordering rule is imposed on candidate column sets.  It is therefore treated as a certificate attached to the canonical invariant, not as the invariant itself.
3817 
3818 \subsection{The automatic finite-orbit subatlas}
3819 
3820 The automatic locus is not a second unrelated theory.  It is the locus on which the orbit of the distinguished initial vector inside the canonical regular module is finite.
3821 
3822 \begin{theorem}[Kernel automaton theorem]
3823 \label{thm:block05-kernel-automaton}
3824 Assume \(\mathcal K_k(a)\) is finite.  Define a deterministic automaton with output by
3825 \[
3826 \text{state set }\mathcal K_k(a),
3827 \qquad
3828 f\xrightarrow{d}\Lambda_df,
3829 \qquad
3830 \text{initial state }a,
3831 \qquad
3832 \text{output }f(0).
3833 \]
3834 Then:
3835 \begin{enumerate}
3836 \item the automaton represents \(a\) under least-significant-digit-first reading;
3837 \item every state is reachable;
3838 \item distinct states are behaviorally distinguishable;
3839 \item it is the unique minimal DFAO up to state renaming.
3840 \end{enumerate}
3841 \end{theorem}
3842 
3843 \begin{proof}
3844 After reading a word \(w\), the state is \(\Lambda_wa\), whose output is \(a_{[w]_k}\).  Reachability is immediate from the definition of the kernel.  If \(f\neq g\), then there is an integer \(n\) with \(f(n)\neq g(n)\).  Reading a digit word for \(n\) from either state distinguishes them.  Thus no two states can be merged, and the reachable automaton is minimal.  The usual residual-behavior argument then gives uniqueness up to renaming.
3845 \end{proof}
3846 
3847 \begin{theorem}[Automatic--regular finite-orbit equivalence]
3848 \label{thm:block05-automatic-regular-equivalence}
3849 For a \(k\)-regular sequence \(a\), the following are equivalent:
3850 \begin{enumerate}
3851 \item \(a\) is \(k\)-automatic;
3852 \item the orbit \(\{\Lambda_wa:w\in\Sigma_k^*\}\) of the initial vector in \(\mathsf C_k(a)\) is finite;
3853 \item the orbit of \(v(a)\) under the strict shortlex digit matrices is finite.
3854 \end{enumerate}
3855 On this locus, the minimal DFAO is recovered as the orbit itself.  Hence
3856 \begin{equation}
3857 \mathbf{Aut}_k(K)
3858 \simeq
3859 \mathbf{FinOrbSec}_k(K)
3860 \simeq
3861 \mathbf{FinOrbLin}_{k,K}^{\mathrm{sl}}
3862 \simeq
3863 \mathbf{DFAO}_{k,K}^{\min}.
3864 \label{eq:block05-automatic-finite-orbit-atlas}
3865 \end{equation}
3866 If \(K=\mathbb F_q\) is finite, every \(k\)-regular \(K\)-valued sequence is automatic.
3867 \end{theorem}
3868 
3869 \begin{proof}
3870 The first three conditions are the same orbit statement written in semantic, intrinsic-module, and coordinate form.  The kernel automaton in \cref{thm:block05-kernel-automaton} recovers the deterministic chart, while its state-to-section map recovers the finite orbit inside the canonical module.  These constructions are inverse on normalized objects.  If \(K=\mathbb F_q\) and \(r_k(a)<\infty\), the vector space \(\mathcal V_k(a)\) has only \(q^{r_k(a)}\) elements, so the section orbit is automatically finite.
3871 \end{proof}
3872 
3873 \begin{corollary}[State-rank hierarchy]
3874 \label{cor:block05-state-rank}
3875 Let \(a\) be automatic with values in \(\mathbb F_q\).  Then
3876 \begin{equation}
3877 \boxed{
3878  r_k(a)
3879  \leq
3880  \kappa_k(a)
3881  \leq
3882  q^{r_k(a)}.
3883 }
3884 \label{eq:block05-state-rank-bound}
3885 \end{equation}
3886 The first inequality can be strict: behavior-distinct kernel states may be linearly dependent.
3887 \end{corollary}
3888 
3889 \begin{proof}
3890 The kernel states span \(\mathcal V_k(a)\), so their number is at least its dimension.  They form a subset of the \(q^{r_k(a)}\)-element vector space \(\mathcal V_k(a)\), proving the upper bound.
3891 \end{proof}
3892 
3893 \begin{figure}[htbp]
3894 \centering
3895 \includegraphics[width=0.99\textwidth]{automatic_regular_rank_distribution.pdf}
3896 \caption{Exact bounded enumeration of the automatic overlap.  Bubble labels count distinct minimized automata.  Points above the diagonal are finite kernel orbits whose state behaviors are distinct but linearly dependent.}
3897 \label{fig:automatic-regular-rank-distribution}
3898 \end{figure}
3899 
3900 The corollary explains why deterministic state minimization and linear-state minimization must remain distinct coordinates.  The minimal automaton records the finite orbit as a set with digit action.  The regular rank records the dimension of its linear span.  Neither number alone reconstructs the other.
3901 
3902 \subsection{Exact regular examples beyond automaticity}
3903 
3904 Over an infinite field, finite linear rank does not force a finite section orbit.  The following families give exact, unbounded witnesses.
3905 
3906 \begin{theorem}[Polynomial regular rank in characteristic zero]
3907 \label{thm:block05-polynomial-rank}
3908 Let \(K\) have characteristic zero and let \(P\in K[x]\) be nonzero of degree \(m\).  For the sequence \(a_n=P(n)\),
3909 \[
3910 r_k(a)=m+1.
3911 \]
3912 If \(m\geq1\), its \(k\)-kernel is infinite.
3913 \end{theorem}
3914 
3915 \begin{proof}
3916 Every section has the form
3917 \[
3918 (\Lambda_wa)(n)
3919 =
3920 P(k^{|w|}n+[w]_k),
3921 \]
3922 which is a polynomial in \(n\) of degree at most \(m\).  Hence the section span has dimension at most \(m+1\).
3923 
3924 Choose \(\ell\) with \(k^\ell>m\) and consider the \(m+1\) sections
3925 \[
3926 P(k^\ell n+r),
3927 \qquad 0\leq r\leq m.
3928 \]
3929 When expanded in the basis \(1,n,\ldots,n^m\), their coefficient functions as functions of \(r\) have successive degrees \(m,m-1,\ldots,0\), with nonzero leading coefficients.  Evaluation at the distinct points \(0,1,\ldots,m\) therefore gives an invertible generalized Vandermonde matrix.  These sections are linearly independent, proving rank \(m+1\).
3930 
3931 For \(m\geq1\), the sections \(P(k^\ell n+r)\) are distinct as \(r\) varies through arbitrarily large complete residue ranges, so the kernel orbit is infinite.
3932 \end{proof}
3933 
3934 \begin{corollary}
3935 For \(a_n=n\), one has \(r_k(a)=2\) and exactly \(k^\ell\) distinct sections at depth \(\ell\).  For \(a_n=n^2\), one has \(r_k(a)=3\) and again exactly \(k^\ell\) distinct sections at depth \(\ell\).
3936 \end{corollary}
3937 
3938 \begin{theorem}[Digit-sum regularity]
3939 \label{thm:block05-digit-sum}
3940 Let \(s_k(n)\) be the sum of the base-\(k\) digits of \(n\), regarded as a sequence over a characteristic-zero field.  Then
3941 \[
3942 r_k(s_k)=2,
3943 \]
3944 and
3945 \begin{equation}
3946 \mathcal K_k(s_k)
3947 =
3948 \{s_k+c:c\in\mathbb Z_{\geq0}\}.
3949 \label{eq:block05-digit-sum-kernel}
3950 \end{equation}
3951 At exact depth \(\ell\), the number of distinct sections is
3952 \[
3953 (k-1)\ell+1.
3954 \]
3955 Thus \(s_k\) is \(k\)-regular but not \(k\)-automatic over a characteristic-zero field.
3956 \end{theorem}
3957 
3958 \begin{proof}
3959 The digit identity
3960 \[
3961 s_k(kn+d)=s_k(n)+d
3962 \]
3963 gives
3964 \[
3965 \Lambda_d s_k=s_k+d.
3966 \]
3967 Iterating shows that every section is \(s_k+c\), so the span is contained in \(\Span\{1,s_k\}\).  The two sequences are independent, giving rank two.  At depth \(\ell\), the additive constant is the digit sum of an \(\ell\)-digit residue.  Every integer from \(0\) through \((k-1)\ell\) occurs, giving exactly \((k-1)\ell+1\) states.
3968 \end{proof}
3969 
3970 \input{generated/block05_examples_table.tex}
3971 
3972 \begin{figure}[htbp]
3973 \centering
3974 \includegraphics[width=0.99\textwidth]{regular_infinite_orbit_examples.pdf}
3975 \caption{Fixed section rank does not bound orbit cardinality over an infinite field.  The Thue--Morse orbit is finite, while \(n\), \(n^2\), and the digit sum have finite linear rank but unboundedly many distinct sections.}
3976 \label{fig:regular-infinite-orbit}
3977 \end{figure}
3978 
3979 \subsection{Exact finite verification}
3980 
3981 \input{generated/block05_validation_table.tex}
3982 
3983 The Fixed-radix regular atlas verifier performs two complementary exact audits.
3984 
3985 First, it exhaustively enumerates bounded deterministic automata in three configurations:
3986 \[
3987 (k,q,s_{\max})=(2,2,3),\qquad(2,3,3),\qquad(3,2,2).
3988 \]
3989 Every candidate is restricted to reachable states, minimized by behavioral equivalence, and filtered by the terminal-high-zero consistency required by integer semantics.  For every resulting normalized automaton the verifier checks:
3990 \begin{enumerate}
3991 \item reachability and pairwise behavioral distinguishability;
3992 \item construction of the canonical linear quotient from continuation-output rows;
3993 \item transition and output equivariance of the state embedding;
3994 \item the inequalities in \cref{eq:block05-state-rank-bound};
3995 \item equality of word-Hankel rank and regular rank;
3996 \item exact equality between the finite linear orbit and the minimized automaton state set;
3997 \item coefficient agreement on a fixed exact test range.
3998 \end{enumerate}
3999 
4000 Second, it checks unbounded symbolic families using exact integer and rational matrices.  Polynomial sequences \(n^m\) are checked for bases two and three through degree six, and digit sums are checked in bases two, three, and four, including their exact depth-orbit growth formulas.
4001 
4002 The finite audit records 89,055 exact checks and no failures.  It supports the implementation, examples, and bounded state-rank distribution.  The unbounded theorems above are proved symbolically and do not depend on the enumeration.
4003 
4004 \begin{boundarybox}
4005 \textbf{Exact boundary of Fixed-radix regular atlas.}
4006 The normalization theorem is fixed-radix and field-valued.  It does not yet give one integral normal form valid simultaneously modulo every prime power, a semiring minimality theorem, or a canonical all-radix comparison invariant.  Over a general commutative ring, the section module can have torsion, fail to be free, and admit inequivalent minimal generating systems.  Saturation, Smith/Fitting data, prime-power reduction towers, and local completion therefore remain separate arithmetic obligations rather than being hidden inside the field theorem.
4007 \end{boundarybox}
4008 
4009 \begin{resultbox}
4010 \textbf{Section result.}
4011 Every field-valued fixed-radix regular sequence has a canonical section module, a minimal reachable-observable digit-linear realization, an exact word-Hankel rank, and a strict shortlex normal form.  These charts satisfy effective normalized descent and yield a complete invariant on the declared fixed-radix regular category.  The automatic locus is exactly the finite-orbit locus inside the same module, where the minimal DFAO is recovered as the orbit of the distinguished initial vector.  Over finite fields regularity and automaticity coincide, while over characteristic-zero fields polynomial and digit-sum examples have finite rank and infinite section orbit.  This closes fixed-radix linear-state minimality over fields, not the all-moduli or integral regular-sequence problem.
4012 \end{resultbox}
4013 
4014 
4015 \section{Integral saturation, Smith--Fitting defects, and prime-power effectivity}
4016 \label{sec:block06-integral-prime-power}
4017 
4018 Fixed-radix regular atlas proves a complete fixed-radix normalization theorem over a field.  The next question is not obtained by replacing the coefficient field with \(\mathbb Z\) symbol by symbol.  Over \(\mathbb Z\), two canonical lattices appear inside the same rational section space.  Their finite-index difference is invisible over \(\mathbb Q\), but it controls reduction modulo every integer, the appearance of ghost coordinates, and the distinction between an effective \(p\)-adic lift and an effective integral lift.
4019 
4020 Throughout this section, \(k\geq2\) is fixed and
4021 \[
4022 a:\mathbb N\longrightarrow\mathbb Z
4023 \]
4024 is an integer-valued sequence whose rational section space
4025 \[
4026 V_{\mathbb Q}(a)
4027 =
4028 \operatorname{span}_{\mathbb Q}\mathcal K_k(a)
4029 \subseteq
4030 \mathbb Q^{\mathbb N}
4031 \]
4032 has finite dimension
4033 \[
4034 \rho_k(a)=\dim_{\mathbb Q}V_{\mathbb Q}(a).
4035 \]
4036 Thus \(a\) is \(k\)-regular after extension of scalars to \(\mathbb Q\).
4037 
4038 \subsection{The raw section lattice and its canonical saturation}
4039 
4040 \begin{definition}[Raw and saturated section lattices]
4041 \label{def:block06-lattices}
4042 The \emph{raw section lattice} and the \emph{saturated section lattice} are
4043 \[
4044 M_k(a)
4045 =
4046 \operatorname{span}_{\mathbb Z}\mathcal K_k(a),
4047 \qquad
4048 L_k(a)
4049 =
4050 V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}.
4051 \]
4052 The intersection defining \(L_k(a)\) is taken inside \(\mathbb Q^{\mathbb N}\).
4053 \end{definition}
4054 
4055 The raw lattice is generated by the actual radix sections.  The saturated lattice contains every integer-valued sequence lying in their rational span.  These lattices agree over \(\mathbb Q\), but need not agree over \(\mathbb Z\).
4056 
4057 \begin{lemma}[Finite coordinate detection]
4058 \label{lem:block06-coordinate-detection}
4059 Let \(V\subseteq\mathbb Q^{\mathbb N}\) be a vector space of finite dimension \(\rho\).  There exist indices
4060 \[
4061 0\leq n_1<\cdots<n_\rho
4062 \]
4063 such that the evaluation map
4064 \[
4065 E_{\mathbf n}:V\longrightarrow\mathbb Q^\rho,
4066 \qquad
4067 f\longmapsto
4068 \bigl(f(n_1),\ldots,f(n_\rho)\bigr)
4069 \]
4070 is an isomorphism.
4071 \end{lemma}
4072 
4073 \begin{proof}
4074 The coordinate functionals on \(\mathbb Q^{\mathbb N}\) separate points of \(V\).  Choose \(n_1\) for which some element of \(V\) has nonzero value.  If \(n_1,\ldots,n_j\) have been chosen and the joint evaluation map has rank \(j<\rho\), its kernel contains a nonzero \(f\).  Choose \(n_{j+1}\) with \(f(n_{j+1})\neq0\).  The rank increases.  After \(\rho\) steps, the evaluation map has full rank.
4075 \end{proof}
4076 
4077 \begin{theorem}[Canonical finite saturated section lattice]
4078 \label{thm:block06-saturation}
4079 The groups \(M_k(a)\) and \(L_k(a)\) are free abelian of rank \(\rho_k(a)\).  They are stable under every digit section \(\Lambda_d\), and
4080 \[
4081 M_k(a)\subseteq L_k(a)
4082 \]
4083 has finite index.  Moreover, \(L_k(a)\) is the unique maximal subgroup of \(V_{\mathbb Q}(a)\) whose elements are integer-valued.
4084 \end{theorem}
4085 
4086 \begin{proof}
4087 Choose a coordinate-detection set \(\mathbf n\) by \cref{lem:block06-coordinate-detection}.  Evaluation embeds \(L_k(a)\) into \(\mathbb Z^\rho\), so \(L_k(a)\) is a finitely generated free abelian group of rank at most \(\rho\).  The raw lattice \(M_k(a)\) is contained in \(L_k(a)\), and its rational span is all of \(V_{\mathbb Q}(a)\).  Hence \(M_k(a)\) has rank \(\rho\), and therefore \(L_k(a)\) also has rank \(\rho\).  A subgroup of a finitely generated free abelian group is finitely generated and free, so \(M_k(a)\) is free of rank \(\rho\).  Equal-rank lattice inclusion has finite index.
4088 
4089 Every \(\Lambda_d\) preserves \(V_{\mathbb Q}(a)\) and preserves integer-valued sequences.  Hence it preserves \(L_k(a)\).  It also maps each radix section to another radix section, so it preserves \(M_k(a)\).  Finally, any integer-valued subgroup of \(V_{\mathbb Q}(a)\) lies in \(V_{\mathbb Q}(a)\cap\mathbb Z^{\mathbb N}=L_k(a)\).
4090 \end{proof}
4091 
4092 \begin{definition}[Integral saturation defect]
4093 \label{def:block06-defect}
4094 The finite abelian group
4095 \[
4096 C_k(a)
4097 =
4098 L_k(a)/M_k(a)
4099 \]
4100 is the \emph{integral saturation defect}.  Its order
4101 \[
4102 \delta_k(a)
4103 =
4104 [L_k(a):M_k(a)]
4105 \]
4106 is the \emph{saturation index}.
4107 \end{definition}
4108 
4109 By the Smith normal form, there are uniquely determined integers
4110 \[
4111 1\leq d_1(a)\mid d_2(a)\mid\cdots\mid d_{\rho}(a)
4112 \]
4113 such that
4114 \[
4115 C_k(a)
4116 \cong
4117 \bigoplus_{i=1}^{\rho}\mathbb Z/d_i(a)\mathbb Z.
4118 \]
4119 Factors with \(d_i=1\) are harmless but are retained so that the list has length \(\rho\).
4120 
4121 \begin{figure}[htbp]
4122 \centering
4123 \includegraphics[width=.93\textwidth]{saturation_reduction_defect.pdf}
4124 \caption{The rational section space contains the raw section lattice and its canonical integer-valued saturation.  The finite quotient is invisible over \(\mathbb Q\), but it controls every modular reduction.}
4125 \label{fig:block06-saturation-defect}
4126 \end{figure}
4127 
4128 \begin{example}[A rank-two nontrivial defect]
4129 \label{ex:block06-double-digitsum}
4130 Let \(s_k(n)\) be the base-\(k\) digit-sum sequence and let \(a=2s_k\).  Since
4131 \[
4132 \Lambda_d s_k=s_k+d,
4133 \]
4134 the rational section space is spanned by \(s_k\) and the constant sequence \(1\).  The actual sections of \(2s_k\) generate
4135 \[
4136 M_k(2s_k)
4137 =
4138 2\mathbb Z\,s_k\oplus2\mathbb Z\,1,
4139 \]
4140 whereas
4141 \[
4142 L_k(2s_k)
4143 =
4144 \mathbb Z\,s_k\oplus\mathbb Z\,1.
4145 \]
4146 Thus
4147 \[
4148 C_k(2s_k)
4149 \cong
4150 (\mathbb Z/2\mathbb Z)^2.
4151 \]
4152 The rational regular rank is two, but reduction modulo two loses the entire semantic lattice.
4153 \end{example}
4154 
4155 \subsection{The exact reduction sequence}
4156 
4157 For an integer \(m\geq2\), the raw coefficient reduction is
4158 \[
4159 M_k(a)/mM_k(a),
4160 \]
4161 while the semantic reduction inside the saturated lattice is
4162 \[
4163 \overline M_{k,m}(a)
4164 =
4165 \operatorname{im}
4166 \left(
4167 M_k(a)/mM_k(a)
4168 \longrightarrow
4169 L_k(a)/mL_k(a)
4170 \right).
4171 \]
4172 These need not agree.
4173 
4174 \begin{definition}[Ghost and missing reduction modules]
4175 \label{def:block06-ghost-missing}
4176 Define
4177 \[
4178 G_{k,m}(a)
4179 =
4180 \ker
4181 \left(
4182 M_k(a)/mM_k(a)
4183 \longrightarrow
4184 L_k(a)/mL_k(a)
4185 \right)
4186 \]
4187 and
4188 \[
4189 Q_{k,m}(a)
4190 =
4191 \operatorname{coker}
4192 \left(
4193 M_k(a)/mM_k(a)
4194 \longrightarrow
4195 L_k(a)/mL_k(a)
4196 \right).
4197 \]
4198 The first records raw residue coordinates that represent the zero semantic sequence.  The second records saturated residue classes not reached by raw sections.
4199 \end{definition}
4200 
4201 \begin{theorem}[Smith--Tor reduction sequence]
4202 \label{thm:block06-tor-sequence}
4203 For every \(m\geq2\), tensoring
4204 \[
4205 0\longrightarrow M_k(a)\longrightarrow L_k(a)\longrightarrow C_k(a)\longrightarrow0
4206 \]
4207 with \(\mathbb Z/m\mathbb Z\) gives a canonical digit-equivariant exact sequence
4208 \[
4209 0
4210 \longrightarrow
4211 C_k(a)[m]
4212 \longrightarrow
4213 M_k(a)/mM_k(a)
4214 \longrightarrow
4215 L_k(a)/mL_k(a)
4216 \longrightarrow
4217 C_k(a)/mC_k(a)
4218 \longrightarrow
4219 0.
4220 \]
4221 Consequently,
4222 \[
4223 G_{k,m}(a)\cong C_k(a)[m],
4224 \qquad
4225 Q_{k,m}(a)\cong C_k(a)/mC_k(a).
4226 \]
4227 \end{theorem}
4228 
4229 \begin{proof}
4230 Both \(M_k(a)\) and \(L_k(a)\) are free abelian.  The long exact Tor sequence therefore begins
4231 \[
4232 0\longrightarrow
4233 \operatorname{Tor}_1^{\mathbb Z}
4234 (C_k(a),\mathbb Z/m\mathbb Z)
4235 \longrightarrow
4236 M_k(a)\otimes\mathbb Z/m\mathbb Z
4237 \longrightarrow
4238 L_k(a)\otimes\mathbb Z/m\mathbb Z.
4239 \]
4240 For any abelian group \(C\),
4241 \[
4242 \operatorname{Tor}_1^{\mathbb Z}(C,\mathbb Z/m\mathbb Z)
4243 \cong C[m],
4244 \qquad
4245 C\otimes\mathbb Z/m\mathbb Z
4246 \cong C/mC.
4247 \]
4248 The digit-section maps preserve the original short exact sequence, so naturality of Tor makes the five-term sequence digit-equivariant.
4249 \end{proof}
4250 
4251 \begin{corollary}[Exact Smith formulas]
4252 \label{cor:block06-smith-formulas}
4253 Let \(g_i(m)=\gcd(m,d_i(a))\).  Then
4254 \[
4255 G_{k,m}(a)
4256 \cong
4257 Q_{k,m}(a)
4258 \cong
4259 \bigoplus_{i=1}^{\rho}
4260 \mathbb Z/g_i(m)\mathbb Z,
4261 \]
4262 and
4263 \[
4264 \left|\overline M_{k,m}(a)\right|
4265 =
4266 \prod_{i=1}^{\rho}
4267 \frac{m}{g_i(m)}.
4268 \]
4269 The reduction map is an isomorphism if and only if
4270 \[
4271 \gcd\bigl(m,d_\rho(a)\bigr)=1.
4272 \]
4273 Equivalently, every prime dividing a reduction defect divides the saturation index.
4274 \end{corollary}
4275 
4276 \begin{proof}
4277 Choose Smith bases in which the inclusion \(M_k(a)\hookrightarrow L_k(a)\) is the diagonal matrix
4278 \[
4279 D=\operatorname{diag}(d_1,\ldots,d_\rho).
4280 \]
4281 Modulo \(m\), the kernel and cokernel of multiplication by \(d_i\) are both cyclic of order \(\gcd(m,d_i)\), and the image has order \(m/\gcd(m,d_i)\).  The map is invertible in every coordinate exactly when every \(d_i\) is a unit modulo \(m\), which is equivalent to \(\gcd(m,d_\rho)=1\).
4282 \end{proof}
4283 
4284 \begin{proposition}[Fitting-ideal certificate]
4285 \label{prop:block06-fitting}
4286 For \(0\leq j<\rho\),
4287 \[
4288 \operatorname{Fitt}_j(C_k(a))
4289 =
4290 \left(
4291 d_1(a)d_2(a)\cdots d_{\rho-j}(a)
4292 \right),
4293 \]
4294 and
4295 \[
4296 \operatorname{Fitt}_j(C_k(a))=\mathbb Z
4297 \qquad
4298 (j\geq\rho).
4299 \]
4300 Thus the Smith profile and the Fitting filtration contain the same reduction-defect data at different resolutions.
4301 \end{proposition}
4302 
4303 \begin{proof}
4304 Use the diagonal presentation \(D:\mathbb Z^\rho\to\mathbb Z^\rho\).  The \(j\)-th Fitting ideal is generated by the minors of size \(\rho-j\).  Since \(d_1\mid\cdots\mid d_\rho\), the product of the first \(\rho-j\) diagonal entries divides every other such minor and is itself one of the minors.
4305 \end{proof}
4306 
4307 \begin{boundary}[What the defect classifies]
4308 \label{bnd:block06-defect-boundary}
4309 The group \(C_k(a)\) classifies the finite-index failure of the raw section lattice to be saturated.  It does not classify the whole integral digit module, the sequence, or its section orbit.  Two nonisomorphic digit modules can have the same Smith profile.
4310 \end{boundary}
4311 
4312 \subsection{Digit equivariance and the prime-power tower}
4313 
4314 Choose Smith bases for \(M_k(a)\) and \(L_k(a)\), and let
4315 \[
4316 D=\operatorname{diag}(d_1,\ldots,d_\rho)
4317 \]
4318 be the inclusion matrix.  Let \(A_d\) and \(B_d\) be the integer matrices of \(\Lambda_d\) on the raw and saturated lattices.
4319 
4320 \begin{proposition}[Intertwining across saturation]
4321 \label{prop:block06-intertwining}
4322 For every digit \(d\),
4323 \[
4324 B_dD=DA_d.
4325 \]
4326 Consequently, the raw modules, semantic images, ghost modules, and missing modules at every modulus form compatible digit modules.
4327 \end{proposition}
4328 
4329 \begin{proof}
4330 The inclusion \(M_k(a)\hookrightarrow L_k(a)\) commutes with \(\Lambda_d\).  The displayed matrix identity is that commutative square in the chosen bases.  Reduction modulo \(m\), passage to kernel, image, and cokernel preserve the induced digit action.
4331 \end{proof}
4332 
4333 Fix a prime \(p\) and write
4334 \[
4335 \nu_i=\nu_{p,i}(a)=v_p(d_i(a)).
4336 \]
4337 
4338 \begin{theorem}[Prime-power defect profile]
4339 \label{thm:block06-prime-power-profile}
4340 For \(r\geq1\),
4341 \[
4342 G_{k,p^r}(a)
4343 \cong
4344 Q_{k,p^r}(a)
4345 \cong
4346 \bigoplus_{i=1}^{\rho}
4347 \mathbb Z/p^{\min(r,\nu_i)}\mathbb Z,
4348 \]
4349 and
4350 \[
4351 \left|\overline M_{k,p^r}(a)\right|
4352 =
4353 p^{\sum_{i=1}^{\rho}(r-\min(r,\nu_i))}.
4354 \]
4355 The minimal number of generators of \(\overline M_{k,p^r}(a)\) as a \(\mathbb Z/p^r\mathbb Z\)-module is
4356 \[
4357 \mu_{p^r}(a)
4358 =
4359 \#\{i:\nu_i<r\}.
4360 \]
4361 In particular, \(p\) is a good reduction prime exactly when every \(\nu_i=0\).
4362 \end{theorem}
4363 
4364 \begin{proof}
4365 Apply \cref{cor:block06-smith-formulas} to \(m=p^r\).  The \(i\)-th semantic image is the ideal
4366 \[
4367 p^{\min(r,\nu_i)}\mathbb Z/p^r\mathbb Z.
4368 \]
4369 It is nonzero and cyclic exactly when \(\nu_i<r\).  Over the local ring \(\mathbb Z/p^r\mathbb Z\), the minimal number of generators of a direct sum of cyclic ideals is the number of nonzero summands.
4370 \end{proof}
4371 
4372 \begin{figure}[htbp]
4373 \centering
4374 \includegraphics[width=.94\textwidth]{prime_power_regular_tower.pdf}
4375 \caption{The raw residue tower maps onto the semantic prime-power tower.  Ghost coordinates occur at finite depth when the raw lattice is not \(p\)-saturated.  The compatible semantic tower converges to the embedded \(p\)-adic raw lattice.}
4376 \label{fig:block06-prime-power-tower}
4377 \end{figure}
4378 
4379 \begin{definition}[Prime-power section automaton]
4380 \label{def:block06-prime-power-automaton}
4381 The semantic \(p^r\)-kernel is
4382 \[
4383 \mathcal K_{k,p^r}^{\mathrm{sem}}(a)
4384 =
4385 \left\{
4386 \Lambda_w a\bmod p^rL_k(a):
4387 w\in\Sigma_k^*
4388 \right\}
4389 \subseteq
4390 \overline M_{k,p^r}(a).
4391 \]
4392 Its cardinality is denoted
4393 \[
4394 \kappa_{k,p^r}^{\mathrm{sem}}(a).
4395 \]
4396 \end{definition}
4397 
4398 \begin{theorem}[Finite prime-power automata]
4399 \label{thm:block06-prime-power-automata}
4400 For every \(r\geq1\), the semantic \(p^r\)-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence \(a\bmod p^r\).  Moreover,
4401 \[
4402 \kappa_{k,p^r}^{\mathrm{sem}}(a)
4403 \leq
4404 \left|\overline M_{k,p^r}(a)\right|
4405 =
4406 p^{\sum_i(r-\min(r,\nu_i))}.
4407 \]
4408 Reduction induces surjective digit-equivariant maps
4409 \[
4410 \mathcal K_{k,p^{r+1}}^{\mathrm{sem}}(a)
4411 \twoheadrightarrow
4412 \mathcal K_{k,p^r}^{\mathrm{sem}}(a).
4413 \]
4414 \end{theorem}
4415 
4416 \begin{proof}
4417 The semantic section module is finite, so its orbit under the digit maps is finite.  As in the field-valued proof of \cref{thm:block05-kernel-automaton}, every orbit state is reachable.  Distinct section sequences have a continuation on which they differ, so they are behaviorally distinguishable.  This gives the minimal deterministic automaton.  The cardinality bound is the ambient module bound.  The same section word at level \(r+1\) reduces to the corresponding section word at level \(r\), so the transition map on orbit states is surjective.
4418 \end{proof}
4419 
4420 \begin{corollary}[Good-prime transfer]
4421 \label{cor:block06-good-prime}
4422 If \(p\nmid\delta_k(a)\), then for every \(r\geq1\),
4423 \[
4424 M_k(a)/p^rM_k(a)
4425 \overset{\sim}{\longrightarrow}
4426 \overline M_{k,p^r}(a)
4427 =
4428 L_k(a)/p^rL_k(a).
4429 \]
4430 At a good prime, the raw and saturated prime-power towers agree exactly.
4431 \end{corollary}
4432 
4433 \input{generated/block06_defect_profiles_table.tex}
4434 
4435 \subsection{p-adic completion and the disappearance of finite ghosts}
4436 
4437 Define
4438 \[
4439 M_{k,p}(a)=M_k(a)\otimes_{\mathbb Z}\mathbb Z_p,
4440 \qquad
4441 L_{k,p}(a)=L_k(a)\otimes_{\mathbb Z}\mathbb Z_p.
4442 \]
4443 
4444 \begin{theorem}[\(p\)-adic saturation sequence]
4445 \label{thm:block06-padic-saturation}
4446 There is a digit-equivariant short exact sequence
4447 \[
4448 0
4449 \longrightarrow
4450 M_{k,p}(a)
4451 \longrightarrow
4452 L_{k,p}(a)
4453 \longrightarrow
4454 C_k(a)_{(p)}
4455 \longrightarrow
4456 0,
4457 \]
4458 where \(C_k(a)_{(p)}\) is the \(p\)-primary subgroup of \(C_k(a)\).  In Smith coordinates,
4459 \[
4460 M_{k,p}(a)
4461 =
4462 \bigoplus_{i=1}^{\rho}p^{\nu_i}\mathbb Z_p e_i
4463 \subseteq
4464 \bigoplus_{i=1}^{\rho}\mathbb Z_p e_i
4465 =
4466 L_{k,p}(a).
4467 \]
4468 \end{theorem}
4469 
4470 \begin{proof}
4471 The \(\mathbb Z\)-module \(\mathbb Z_p\) is torsion free and hence flat.  Tensor the integral short exact sequence with \(\mathbb Z_p\).  For a finite abelian group \(C\),
4472 \[
4473 C\otimes_{\mathbb Z}\mathbb Z_p\cong C_{(p)}.
4474 \]
4475 The Smith-coordinate formula follows by tensoring the diagonal inclusion.
4476 \end{proof}
4477 
4478 \begin{theorem}[Raw and semantic inverse limits]
4479 \label{thm:block06-inverse-limits}
4480 There are canonical digit-equivariant identifications
4481 \[
4482 \varprojlim_r M_k(a)/p^rM_k(a)
4483 \cong
4484 M_{k,p}(a)
4485 \]
4486 and
4487 \[
4488 \varprojlim_r \overline M_{k,p^r}(a)
4489 \cong
4490 M_{k,p}(a)
4491 \subseteq
4492 L_{k,p}(a).
4493 \]
4494 Furthermore,
4495 \[
4496 \varprojlim_r G_{k,p^r}(a)=0,
4497 \qquad
4498 \varprojlim_r Q_{k,p^r}(a)
4499 \cong
4500 C_k(a)_{(p)}.
4501 \]
4502 \end{theorem}
4503 
4504 \begin{proof}
4505 The first identification is the definition of the \(p\)-adic completion of a finite free lattice.  In Smith coordinate \(i\), the semantic image at level \(r\) is
4506 \[
4507 p^{\min(r,\nu_i)}\mathbb Z/p^r\mathbb Z.
4508 \]
4509 Its inverse limit is \(p^{\nu_i}\mathbb Z_p\), giving the second identification.
4510 
4511 A compatible ghost tower represents an element \(x\in\mathbb Z_p\) satisfying \(p^{\nu_i}x=0\).  Since \(\mathbb Z_p\) is torsion free, \(x=0\).  Hence the inverse limit of the ghost modules vanishes.  The missing quotient stabilizes coordinatewise at \(\mathbb Z/p^{\nu_i}\mathbb Z\), whose product is the \(p\)-primary Smith quotient.
4512 \end{proof}
4513 
4514 The finite ghosts are therefore real finite-level defects, but they do not survive as nonzero points of the completed raw lattice.  By contrast, the missing saturated quotient survives exactly as the \(p\)-primary saturation defect.
4515 
4516 \subsection{Orbit closure and three levels of effectivity}
4517 
4518 Let
4519 \[
4520 \mathcal O_k(a)=\{\Lambda_w a:w\in\Sigma_k^*\}\subseteq M_k(a)
4521 \]
4522 be the actual section orbit, and let
4523 \[
4524 \mathcal O^{\mathrm{raw}}_{k,p^r}(a)
4525 \]
4526 be its image in \(M_k(a)/p^rM_k(a)\).
4527 
4528 \begin{theorem}[Raw orbit-closure descent]
4529 \label{thm:block06-orbit-closure}
4530 The reduction maps
4531 \[
4532 \mathcal O^{\mathrm{raw}}_{k,p^{r+1}}(a)
4533 \twoheadrightarrow
4534 \mathcal O^{\mathrm{raw}}_{k,p^r}(a)
4535 \]
4536 are surjective, and
4537 \[
4538 \overline{\mathcal O_k(a)}^{\,p}
4539 \overset{\sim}{\longrightarrow}
4540 \varprojlim_r
4541 \mathcal O^{\mathrm{raw}}_{k,p^r}(a),
4542 \]
4543 where the closure is taken in \(M_{k,p}(a)\).
4544 \end{theorem}
4545 
4546 \begin{proof}
4547 A section word at level \(r\) is lifted by the same word at level \(r+1\), proving surjectivity.  Every \(p\)-adic limit of orbit points gives a compatible tower.  Conversely, choose for each compatible residue state an orbit representative \(x_r\).  Compatibility gives
4548 \[
4549 x_{r+1}-x_r\in p^rM_k(a),
4550 \]
4551 so \((x_r)\) is Cauchy in \(M_{k,p}(a)\).  Its limit lies in the orbit closure and maps to the prescribed tower.
4552 \end{proof}
4553 
4554 \begin{definition}[Three effectivity obstructions]
4555 \label{def:block06-effectivity}
4556 Let \(x\in M_{k,p}(a)\).
4557 \begin{enumerate}
4558 \item The \emph{\(p\)-adic effectivity condition} is simply \(x\in M_{k,p}(a)\); every compatible full semantic module tower satisfies it by \cref{thm:block06-inverse-limits}.
4559 \item The \emph{integral lattice obstruction} is
4560 \[
4561 \epsilon_{\mathrm{int}}(x)
4562 =
4563 [x]\in M_{k,p}(a)/M_k(a).
4564 \]
4565 It vanishes exactly when \(x\) is represented by one integral lattice element.
4566 \item The \emph{section-orbit obstruction} is membership in
4567 \[
4568 M_k(a)\setminus\mathcal O_k(a).
4569 \]
4570 Even after integral effectivity holds, the lift need not be an actual radix section.
4571 \end{enumerate}
4572 \end{definition}
4573 
4574 \begin{boundary}[Semantic orbit towers]
4575 \label{bnd:block06-semantic-orbit}
4576 The inverse limit of the full semantic modules is always \(M_{k,p}(a)\).  For the smaller semantic orbit sets, surjectivity of reduction remains automatic, but lifting to an actual orbit closure can be obscured by finite ghost fibers.  The unconditional orbit-closure theorem is therefore stated for the raw orbit tower.  A semantic-orbit effectivity theorem requires a compatible choice of raw lifts or an equivalent fiberwise Mittag--Leffler condition.
4577 \end{boundary}
4578 
4579 \begin{example}[Digit sum separates all three levels]
4580 \label{ex:block06-digitsum-effectivity}
4581 For the base-\(k\) digit sum \(s_k\),
4582 \[
4583 \mathcal O_k(s_k)
4584 =
4585 \{s_k+c:c\in\mathbb Z_{\geq0}\},
4586 \]
4587 and
4588 \[
4589 M_k(s_k)=L_k(s_k)=\mathbb Zs_k\oplus\mathbb Z1.
4590 \]
4591 Hence
4592 \[
4593 \overline{\mathcal O_k(s_k)}^{\,p}
4594 =
4595 \{s_k+c:c\in\mathbb Z_p\}.
4596 \]
4597 The partial sums
4598 \[
4599 c_N=\sum_{j=0}^{N}p^{2j}
4600 \]
4601 are nonnegative integers and therefore produce actual section states \(s_k+c_N\).  They converge in \(\mathbb Z_p\) to
4602 \[
4603 c_\infty
4604 =
4605 \frac{1}{1-p^2},
4606 \]
4607 which is not an ordinary integer.  Thus \(s_k+c_\infty\) is \(p\)-adically effective and lies in the orbit closure, but its integral obstruction is nonzero.
4608 
4609 By contrast, \(s_k-1\) is an integral lattice point and lies in the \(p\)-adic orbit closure because \(-1\in\mathbb Z_p\) is a limit of nonnegative integers.  It is not an actual section, since every actual offset is nonnegative.  Integral effectivity therefore does not imply section-orbit effectivity.
4610 \end{example}
4611 
4612 \begin{figure}[htbp]
4613 \centering
4614 \includegraphics[width=.93\textwidth]{digit_sum_padic_effectivity.pdf}
4615 \caption{For the digit-sum sequence, actual section states form a nonnegative integral ray.  Its \(p\)-adic closure contains all \(p\)-adic offsets, including nonintegral limits and integral points that are not actual sections.}
4616 \label{fig:block06-digitsum-effectivity}
4617 \end{figure}
4618 
4619 \subsection{All-moduli completion and the integral effectivity obstruction}
4620 
4621 Let
4622 \[
4623 \widehat{M_k(a)}
4624 =
4625 \varprojlim_m M_k(a)/mM_k(a)
4626 \]
4627 be the profinite completion over divisibility of moduli.
4628 
4629 \begin{theorem}[Profinite saturation sequence]
4630 \label{thm:block06-profinite}
4631 There are canonical digit-equivariant identifications
4632 \[
4633 \widehat{M_k(a)}
4634 \cong
4635 \prod_p M_{k,p}(a),
4636 \qquad
4637 \widehat{L_k(a)}
4638 \cong
4639 \prod_p L_{k,p}(a),
4640 \]
4641 and a short exact sequence
4642 \[
4643 0
4644 \longrightarrow
4645 \widehat{M_k(a)}
4646 \longrightarrow
4647 \widehat{L_k(a)}
4648 \longrightarrow
4649 C_k(a)
4650 \longrightarrow
4651 0.
4652 \]
4653 A compatible all-moduli raw tower has an ordinary integral lift exactly when its class in
4654 \[
4655 \widehat{M_k(a)}/M_k(a)
4656 \]
4657 vanishes.
4658 \end{theorem}
4659 
4660 \begin{proof}
4661 Finite free abelian groups have profinite completion equal to the product of their \(p\)-adic completions.  In Smith coordinates, the inclusion is multiplication by \(d_i\) in each profinite coordinate.  Its cokernel is
4662 \[
4663 \widehat{\mathbb Z}/d_i\widehat{\mathbb Z}
4664 \cong
4665 \mathbb Z/d_i\mathbb Z.
4666 \]
4667 Taking the direct sum over \(i\) gives \(C_k(a)\).  The final assertion is the definition of belonging to the diagonal image of the integral lattice inside its profinite completion.
4668 \end{proof}
4669 
4670 \begin{corollary}[Local-global separation]
4671 \label{cor:block06-local-global}
4672 Every compatible prime-power tower is effective in a \(p\)-adic completion.  A compatible family over all primes is effective in the profinite completion.  Neither statement alone implies an ordinary integral lift.  Integral effectivity is the separate diagonal condition
4673 \[
4674 \widehat{x}\in M_k(a)\subseteq\widehat{M_k(a)}.
4675 \]
4676 \end{corollary}
4677 
4678 This is the precise sense in which prime-power effectivity and integral effectivity differ.  Completion solves the Cauchy problem in the declared topology.  It does not force the resulting point to lie in the original discrete lattice.
4679 
4680 \subsection{Canonical integral record and directional reduction atlas}
4681 
4682 \begin{definition}[Canonical integral regular record]
4683 \label{def:block06-integral-record}
4684 The canonical integral record of \(a\) is
4685 \[
4686 \mathcal I_k(a)
4687 =
4688 \left(
4689 V_{\mathbb Q}(a),
4690 M_k(a),
4691 L_k(a),
4692 C_k(a),
4693 (\Lambda_d)_{0\leq d<k},
4694 a,
4695 \operatorname{ev}_0
4696 \right).
4697 \]
4698 Its Smith--Fitting profile is
4699 \[
4700 \mathsf{SF}_k(a)
4701 =
4702 \left(
4703 d_1,\ldots,d_\rho;
4704 \operatorname{Fitt}_0(C_k(a)),\ldots,
4705 \operatorname{Fitt}_{\rho-1}(C_k(a))
4706 \right).
4707 \]
4708 \end{definition}
4709 
4710 The record is canonical but not a single field-valued normal form.  It retains both the raw generative lattice and the maximal integer-valued lattice, together with the exact quotient between them.
4711 
4712 \begin{theorem}[Integral normalization and directional local reduction]
4713 \label{thm:block06-integral-normalization}
4714 For fixed \(k\), the assignment
4715 \[
4716 a\longmapsto\mathcal I_k(a)
4717 \]
4718 is functorial for digit-module morphisms preserving the distinguished sequence and evaluation.  It is uniquely determined by the semantic sequence.  For every prime \(p\), reduction defines a directional bridge
4719 \[
4720 \mathcal I_k(a)
4721 \longmapsto
4722 \left(
4723 \overline M_{k,p^r}(a),
4724 \mathcal K_{k,p^r}^{\mathrm{sem}}(a)
4725 \right)_{r\geq1}
4726 \]
4727 whose exact information-loss passport is:
4728 \begin{enumerate}
4729 \item finite-depth ghost kernel \(C_k(a)[p^r]\);
4730 \item finite-depth missing quotient \(C_k(a)/p^rC_k(a)\);
4731 \item completed essential image \(M_{k,p}(a)\subseteq L_{k,p}(a)\);
4732 \item integral effectivity obstruction \(M_{k,p}(a)/M_k(a)\);
4733 \item actual-section obstruction inside the integral lattice.
4734 \end{enumerate}
4735 \end{theorem}
4736 
4737 \begin{proof}
4738 Every component of \(\mathcal I_k(a)\) is defined from the semantic sequence and its digit sections.  A compatible morphism carries actual sections to actual sections and integer-valued rational combinations to integer-valued rational combinations, so it induces maps on \(M_k(a)\), \(L_k(a)\), and the quotient.  The local reduction statements are \cref{thm:block06-tor-sequence,thm:block06-inverse-limits,def:block06-effectivity}.
4739 \end{proof}
4740 
4741 \begin{resultbox}
4742 \textbf{Section result.}
4743 For every integer-valued fixed-radix sequence that is regular over \(\mathbb Q\), the actual sections generate a canonical finite free lattice \(M_k(a)\), and the integer-valued part of the rational section space is a canonical finite free saturation \(L_k(a)\).  Their Smith--Fitting quotient controls every modular reduction exactly.  At each prime power, the coefficient sequence has a finite minimal semantic section automaton.  The compatible tower reconstructs the \(p\)-adic raw lattice, while finite ghost coordinates vanish in the inverse limit and the missing saturated quotient survives as the \(p\)-primary defect.  \(p\)-adic, profinite, integral-lattice, and actual-section effectivity are distinct and are separated by explicit obstruction quotients.
4744 \end{resultbox}
4745 
4746 \input{generated/block06_validation_table.tex}
4747 
4748 \begin{boundarybox}
4749 \textbf{Exact boundary of Integral prime-power atlas.}
4750 The results classify saturation and reduction defects for integer-valued sequences that are already known to have finite rational section rank.  They do not prove finite-data recognition of that hypothesis, a minimal realization theorem over arbitrary rings or semirings, a base-independent integral normal form, or compatibility across different radices.  The prime-power tower is a directional local reduction system, not an equivalence between integral and modular presentations.
4751 \end{boundarybox}
4752 
4753 
4754 \section{Filtered Euler products, prime-power seeds, and the divisor--ghost bridge}
4755 \label{sec:block07-product-multiplicative}
4756 
4757 The next native-invariant layer has two different index monoids.  Ordinary generating functions use the additive monoid of coefficient addresses: Cauchy multiplication combines indices by addition.  Arithmetic functions use the multiplicative monoid of positive integers: Dirichlet convolution combines divisor addresses by multiplication.  These structures meet exactly in selected local and logarithmic-derivative coordinates, but they are not globally interchangeable.  This section constructs the fixed-basis filtered-product normal form, the prime-power normal form for multiplicative arithmetic functions, and one rigorously typed cross-stratum bridge between them.
4758 
4759 \subsection{Filtered products and coefficient stabilization}
4760 
4761 \begin{definition}[Coefficientwise filtered product]
4762 Let $R$ be a commutative ring.  A filtered factor family is a sequence
4763 \[
4764 F_m(z)\in 1+z^{w(m)}R[[z]],
4765 \qquad
4766 w(m)\longrightarrow\infty.
4767 \]
4768 Its coefficientwise product is
4769 \[
4770 \prod_{m\geq1}F_m(z),
4771 \]
4772 where the coefficient of $z^N$ is computed from the finitely many factors satisfying $w(m)\leq N$.
4773 \end{definition}
4774 
4775 \begin{proposition}[Filtered product completion]
4776 Every filtered factor family has a unique coefficientwise product in $1+zR[[z]]$.  For every $N$, the truncation modulo $z^{N+1}$ is equal to the product of the finitely many factors with $w(m)\leq N$.
4777 \end{proposition}
4778 
4779 \begin{proof}
4780 Fix $N$.  Every factor with $w(m)>N$ is congruent to $1$ modulo $z^{N+1}$, so only finitely many factors can affect the coefficient prefix through degree $N$.  These finite products are compatible as $N$ increases.  Their inverse limit in the $z$-adic filtration is the required series.  Uniqueness follows because every coefficient is fixed at a finite stage.
4781 \end{proof}
4782 
4783 The proposition supplies existence and exact finite-prefix evaluation.  It does not assert that an arbitrary factor family is structurally compressed.  A finite grammar requires an effective rule for the factors and a proof of stabilization; otherwise the factor list can merely store the coefficient data in another form.
4784 
4785 \subsection{Canonical Euler coordinates}
4786 
4787 Let
4788 \[
4789 \mathcal U_{\mathbb Z}=1+z\mathbb Z[[z]]
4790 \]
4791 be the multiplicative group of integral unit series, and let
4792 \[
4793 \mathcal E_{\mathbb Z}=\mathbb Z^{\mathbb N_{\geq1}}
4794 \]
4795 be the additive group of integer exponent sequences.
4796 
4797 \begin{definition}[Euler-coordinate map]
4798 For $b=(b_m)_{m\geq1}\in\mathcal E_{\mathbb Z}$, define
4799 \[
4800 \mathcal E(b)(z)
4801 =
4802 \prod_{m\geq1}(1-z^m)^{-b_m}.
4803 \]
4804 The product is coefficientwise filtered because the $m$th factor is congruent to $1$ modulo $z^m$.
4805 \end{definition}
4806 
4807 \begin{theorem}[Integral Euler-coordinate classification]
4808 \label{thm:block07-euler-classification}
4809 The map
4810 \[
4811 \mathcal E:
4812 (\mathcal E_{\mathbb Z},+)
4813 \longrightarrow
4814 (\mathcal U_{\mathbb Z},\cdot)
4815 \]
4816 is a group isomorphism.  Consequently, every $A(z)\in1+z\mathbb Z[[z]]$ has a unique representation
4817 \[
4818 \boxed{
4819 A(z)=\prod_{m\geq1}(1-z^m)^{-b_m},
4820 \qquad b_m\in\mathbb Z.
4821 }
4822 \]
4823 \end{theorem}
4824 
4825 \begin{proof}
4826 Filtered multiplication gives
4827 \[
4828 \mathcal E(b+b')=\mathcal E(b)\mathcal E(b').
4829 \]
4830 To prove surjectivity and uniqueness, construct the exponents recursively.  Suppose $b_1,\ldots,b_{n-1}$ have been chosen so that
4831 \[
4832 A(z)\prod_{m<n}(1-z^m)^{b_m}
4833 \equiv 1+r_nz^n\pmod{z^{n+1}}.
4834 \]
4835 Because
4836 \[
4837 (1-z^n)^{-r_n}\equiv1+r_nz^n\pmod{z^{n+1}},
4838 \]
4839 the unique choice $b_n=r_n$ cancels the degree-$n$ residual.  Every $r_n$ is integral because all preceding operations occur in $\mathbb Z[[z]]$.  Induction produces one exponent sequence with the required product.  If two exponent sequences differed first at $n$, their quotient would have nonzero coefficient $(b_n-b_n')z^n$ modulo $z^{n+1}$, contradicting equality.  Hence the representation is unique.
4840 \end{proof}
4841 
4842 \begin{corollary}[Finite-prefix triangularity]
4843 \label{cor:block07-euler-prefix}
4844 For each $N$, the coefficient prefix
4845 \[
4846 (a_1,\ldots,a_N)
4847 \quad\text{of}\quad
4848 A(z)=1+\sum_{n\geq1}a_nz^n
4849 \]
4850 and the exponent prefix $(b_1,\ldots,b_N)$ determine one another by integral triangular recursions.  No exponent $b_m$ with $m>N$ affects the coefficient prefix through degree $N$.
4851 \end{corollary}
4852 
4853 \subsection{Ghost coordinates and logarithmic-derivative recurrence}
4854 
4855 \begin{definition}[Euler ghost coordinates]
4856 For $b\in\mathcal E_{\mathbb Z}$, define
4857 \[
4858 c_n(b)=\sum_{d\mid n}d\,b_d,
4859 \qquad n\geq1.
4860 \]
4861 Let $\mathcal G_{\mathbb Z}$ be the set of integer sequences $c=(c_n)$ satisfying
4862 \[
4863 \frac1n\sum_{d\mid n}\mu\!\left(\frac nd\right)c_d\in\mathbb Z
4864 \qquad(n\geq1).
4865 \]
4866 \end{definition}
4867 
4868 \begin{theorem}[Ghost-coordinate equivalence]
4869 \label{thm:block07-ghost-equivalence}
4870 The divisor ghost map
4871 \[
4872 \mathcal W:
4873 \mathcal E_{\mathbb Z}
4874 \longrightarrow
4875 \mathcal G_{\mathbb Z},
4876 \qquad
4877 b\longmapsto c(b),
4878 \]
4879 is an additive group isomorphism.  Its inverse is
4880 \[
4881 \boxed{
4882 b_n
4883 =
4884 \frac1n\sum_{d\mid n}
4885 \mu\!\left(\frac nd\right)c_d.
4886 }
4887 \]
4888 If $A(z)=\mathcal E(b)(z)$, then
4889 \[
4890 \boxed{
4891 \frac{zA'(z)}{A(z)}
4892 =
4893 \sum_{n\geq1}c_nz^n.
4894 }
4895 \]
4896 \end{theorem}
4897 
4898 \begin{proof}
4899 Formal logarithmic differentiation gives
4900 \[
4901 \frac{zA'}A
4902 =
4903 \sum_{m\geq1}b_m\frac{mz^m}{1-z^m}
4904 =
4905 \sum_{m\geq1}\sum_{r\geq1}m b_m z^{mr}.
4906 \]
4907 The coefficient at $z^n$ is $\sum_{m\mid n}m b_m=c_n$.  M\"obius inversion gives the displayed inverse formula.  The definition of $\mathcal G_{\mathbb Z}$ is exactly the integrality condition for that inverse.
4908 \end{proof}
4909 
4910 \begin{theorem}[Exact coefficient recurrence]
4911 \label{thm:block07-euler-recurrence}
4912 Let
4913 \[
4914 A(z)=1+\sum_{n\geq1}a_nz^n
4915 \]
4916 and let $c_n$ be its Euler ghost coordinates.  Then
4917 \[
4918 \boxed{
4919 na_n
4920 =
4921 \sum_{j=1}^{n}c_j a_{n-j}
4922 \qquad(n\geq1),
4923 }
4924 \]
4925 with $a_0=1$.  Conversely, any $c\in\mathcal G_{\mathbb Z}$ and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.
4926 \end{theorem}
4927 
4928 \begin{proof}
4929 The logarithmic-derivative identity is equivalent to
4930 \[
4931 zA'(z)=\left(\sum_{j\geq1}c_jz^j\right)A(z).
4932 \]
4933 Comparing coefficients gives the recurrence.  Conversely, the recurrence uniquely determines the coefficient sequence over $\mathbb Q$.  Since $c\in\mathcal G_{\mathbb Z}$ reconstructs an integral exponent sequence, the resulting series is the Euler product from \cref{thm:block07-euler-classification} and therefore has integral coefficients.
4934 \end{proof}
4935 
4936 \begin{figure}[htbp]
4937 \centering
4938 \includegraphics[width=0.99\textwidth]{euler_coordinate_atlas.pdf}
4939 \caption{The fixed Euler basis gives three exact coordinate charts: integral exponents, coefficients, and divisor ghosts.  Every finite prefix is triangular, while the ghost chart is restricted by its explicit M\"obius divisibility conditions.}
4940 \label{fig:block07-euler-atlas}
4941 \end{figure}
4942 
4943 \begin{example}[Partition and power families]
4944 The following fixed-basis exponent functions give familiar filtered products:
4945 \[
4946 \begin{array}{c|c|c}
4947  b_n & A_b(z) & c_n\\
4948 \hline
4949 1 & \displaystyle\prod_{n\geq1}(1-z^n)^{-1} & \sigma_1(n)\\[4pt]
4950 n & \displaystyle\prod_{n\geq1}(1-z^n)^{-n} & \sigma_2(n)\\[4pt]
4951 n^r & \displaystyle\prod_{n\geq1}(1-z^n)^{-n^r} & \sigma_{r+1}(n).
4952 \end{array}
4953 \]
4954 The first row is the ordinary partition series, and the second is the plane-partition series.  More generally, the power family converts the exponent rule $n^r$ into the divisor-sum ghost $\sigma_{r+1}$.
4955 \end{example}
4956 
4957 \begin{example}[Distinct parts and normalization]
4958 The product
4959 \[
4960 \prod_{m\geq1}(1+z^m)
4961 =
4962 \prod_{m\geq1}\frac{1-z^{2m}}{1-z^m}
4963 \]
4964 has normalized Euler exponent
4965 \[
4966 b_n=
4967 \begin{cases}
4968 1,&n\text{ odd},\\
4969 0,&n\text{ even}.
4970 \end{cases}
4971 \]
4972 Thus the distinct-part and odd-part products become the same object after normalization in the fixed Euler basis.  This is an exact equality inside the declared product grammar, not a claim that arbitrary product grammars possess one preferred factorization.
4973 \end{example}
4974 
4975 \begin{boundary}[Coordinate universality versus compact structural realization]
4976 \label{bnd:block07-storage-universality}
4977 Theorem~\ref{thm:block07-euler-classification} gives a unique coordinate sequence for every integral unit series.  It does not make every such series finitely describable.  If the exponent sequence is merely an arbitrary list encoding all coefficients, the representation is storage universality.  A compact seed grammar additionally requires an effective finite rule for the exponent sequence or factor family.
4978 \end{boundary}
4979 
4980 \subsection{Multiplicative arithmetic functions as prime-power descent}
4981 
4982 For arithmetic functions $f,g:\mathbb N_{\geq1}\to\mathbb Z$, write
4983 \[
4984 (f\mathbin{\ast_D}g)(n)
4985 =
4986 \sum_{d\mid n}f(d)g(n/d)
4987 \]
4988 for Dirichlet convolution.  Its identity is $\varepsilon(1)=1$ and $\varepsilon(n)=0$ for $n>1$.
4989 
4990 \begin{definition}[Integral multiplicative category]
4991 Let $\mathbf{Mult}_{\mathbb Z}$ be the discrete category whose objects are multiplicative arithmetic functions $f:\mathbb N_{\geq1}\to\mathbb Z$ satisfying
4992 \[
4993 f(1)=1,
4994 \qquad
4995 f(mn)=f(m)f(n)
4996 \quad\text{when }(m,n)=1.
4997 \]
4998 The monoidal operation is Dirichlet convolution.
4999 \end{definition}
5000 
5001 \begin{proposition}[Dirichlet inverse and multiplicative closure]
5002 If $f(1)=1$, then $f$ has a unique integral Dirichlet inverse.  If $f$ and $g$ are multiplicative, then $f\mathbin{\ast_D}g$ and $f^{-1_D}$ are multiplicative.
5003 \end{proposition}
5004 
5005 \begin{proof}
5006 The coefficient equation at $n$ is triangular:
5007 \[
5008 g(1)=1,
5009 \qquad
5010 g(n)=-\sum_{\substack{d\mid n\\d<n}}f(n/d)g(d).
5011 \]
5012 Hence the inverse exists integrally and uniquely.  The standard coprime-divisor factorization of a divisor of $mn$ proves closure of multiplicative functions under convolution; uniqueness of the inverse then proves multiplicativity of $f^{-1_D}$.
5013 \end{proof}
5014 
5015 \begin{definition}[Prime-power seed series]
5016 For a multiplicative function $f$ and a prime $p$, define
5017 \[
5018 F_{f,p}(T)
5019 =
5020 \sum_{e\geq0}f(p^e)T^e
5021 \in1+T\mathbb Z[[T]].
5022 \]
5023 \end{definition}
5024 
5025 \begin{theorem}[Prime-power descent equivalence]
5026 \label{thm:block07-prime-power-descent}
5027 The assignment
5028 \[
5029 f\longmapsto\bigl(F_{f,p}\bigr)_p
5030 \]
5031 is a group isomorphism
5032 \[
5033 \boxed{
5034 (\mathbf{Mult}_{\mathbb Z},\mathbin{\ast_D})
5035 \simeq
5036 \prod_{p\text{ prime}}
5037 (1+T\mathbb Z[[T]],\cdot).
5038 }
5039 \]
5040 The inverse reconstructs
5041 \[
5042 \boxed{
5043 f(n)=\prod_{p^e\parallel n}f(p^e).}
5044 \]
5045 Under this correspondence,
5046 \[
5047 F_{f\ast_D g,p}(T)
5048 =
5049 F_{f,p}(T)F_{g,p}(T).
5050 \]
5051 \end{theorem}
5052 
5053 \begin{proof}
5054 Unique factorization writes every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula.  Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product.  For $n=p^e$, Dirichlet convolution becomes
5055 \[
5056 (f\mathbin{\ast_D}g)(p^e)
5057 =
5058 \sum_{r=0}^{e}f(p^r)g(p^{e-r}),
5059 \]
5060 which is exactly the coefficient of $T^e$ in the local Cauchy product.
5061 \end{proof}
5062 
5063 \subsection{Prime-Euler and local ghost coordinates}
5064 
5065 Apply the Euler-coordinate theorem independently at each prime:
5066 \[
5067 F_{f,p}(T)
5068 =
5069 \prod_{m\geq1}(1-T^m)^{-\beta_{p,m}(f)}.
5070 \]
5071 Define local ghosts
5072 \[
5073 \gamma_{p,e}(f)
5074 =
5075 \sum_{d\mid e}d\,\beta_{p,d}(f).
5076 \]
5077 
5078 \begin{theorem}[Complete prime-Euler coordinates]
5079 \label{thm:block07-prime-euler}
5080 For integral multiplicative functions, the following normalized data are equivalent and complete:
5081 \[
5082  f,
5083 \qquad
5084 \bigl(F_{f,p}\bigr)_p,
5085 \qquad
5086 \bigl(\beta_{p,m}(f)\bigr)_{p,m},
5087 \qquad
5088 \bigl(\gamma_{p,e}(f)\bigr)_{p,e}.
5089 \]
5090 Dirichlet convolution is linearized by the last two charts:
5091 \[
5092 \beta_{p,m}(f\mathbin{\ast_D}g)
5093 =
5094 \beta_{p,m}(f)+\beta_{p,m}(g),
5095 \]
5096 \[
5097 \gamma_{p,e}(f\mathbin{\ast_D}g)
5098 =
5099 \gamma_{p,e}(f)+\gamma_{p,e}(g).
5100 \]
5101 The ghost chart consists exactly of the prime-indexed sequences satisfying the local M\"obius divisibility conditions
5102 \[
5103 \frac1e\sum_{d\mid e}\mu\!\left(\frac ed\right)\gamma_{p,d}\in\mathbb Z.
5104 \]
5105 \end{theorem}
5106 
5107 \begin{proof}
5108 Theorem~\ref{thm:block07-euler-classification} and Theorem~\ref{thm:block07-ghost-equivalence} apply to each local unit series $F_{f,p}$.  The prime-power descent theorem then reconstructs the global function.  Local multiplication corresponds to Dirichlet convolution, while Euler exponents and ghosts add under multiplication.
5109 \end{proof}
5110 
5111 \begin{theorem}[Valuation logarithmic derivative]
5112 \label{thm:block07-valuation-derivative}
5113 Fix a prime $p$.  Let $\Gamma_{f,p}$ be the arithmetic function supported on powers of $p$ with
5114 \[
5115 \Gamma_{f,p}(1)=0,
5116 \qquad
5117 \Gamma_{f,p}(p^e)=\gamma_{p,e}(f)
5118 \quad(e\geq1).
5119 \]
5120 Then
5121 \[
5122 \boxed{
5123 (v_p\cdot f)(n)
5124 =
5125 (\Gamma_{f,p}\mathbin{\ast_D}f)(n)
5126 \qquad(n\geq1).
5127 }
5128 \]
5129 Equivalently,
5130 \[
5131 \frac{T F_{f,p}'(T)}{F_{f,p}(T)}
5132 =
5133 \sum_{e\geq1}\gamma_{p,e}(f)T^e.
5134 \]
5135 \end{theorem}
5136 
5137 \begin{proof}
5138 The local logarithmic-derivative recurrence gives
5139 \[
5140 e f(p^e)
5141 =
5142 \sum_{j=1}^{e}\gamma_{p,j}(f)f(p^{e-j}).
5143 \]
5144 Write $n=p^e u$ with $(u,p)=1$ and multiply by $f(u)$.  Since $\Gamma_{f,p}$ is supported on $p$-powers, the resulting sum is exactly the Dirichlet convolution at $n$.
5145 \end{proof}
5146 
5147 \begin{proposition}[Multiplicative residue-packet law]
5148 For $N\geq1$, let $P_{N,r}f$ be the arithmetic function equal to $f(n)$ when $n\equiv r\pmod N$ and zero otherwise.  Then
5149 \[
5150 \boxed{
5151 P_{N,k}(f\mathbin{\ast_D}g)
5152 =
5153 \sum_{rs\equiv k\, (N)}
5154 (P_{N,r}f)\mathbin{\ast_D}(P_{N,s}g).
5155 }
5156 \]
5157 Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid $\mathbb Z/N\mathbb Z$, including its zero divisors.
5158 \end{proposition}
5159 
5160 \begin{proof}
5161 In each divisor pair $n=d(n/d)$, the residue of $n$ modulo $N$ is the product of the two divisor residues.  Group the divisor sum by those two residues.
5162 \end{proof}
5163 
5164 \begin{figure}[htbp]
5165 \centering
5166 \includegraphics[width=0.99\textwidth]{multiplicative_prime_power_atlas.pdf}
5167 \caption{A multiplicative arithmetic function is effective descent data over its prime-power local series.  Local Euler exponents and local ghosts turn Dirichlet convolution into addition.}
5168 \label{fig:block07-multiplicative-atlas}
5169 \end{figure}
5170 
5171 \input{generated/block07_named_local_table.tex}
5172 
5173 \subsection{The divisor--ghost cross-stratum bridge}
5174 
5175 The Euler ghost transform can itself be written in the Dirichlet-convolution language.  Let $\mathbf{1}(n)=1$, let $\operatorname{id}(n)=n$, and let pointwise multiplication be denoted by a centered dot.
5176 
5177 \begin{definition}[Divisor--ghost transform]
5178 For an arithmetic exponent function $b:\mathbb N_{\geq1}\to\mathbb Z$, define
5179 \[
5180 \mathfrak G(b)
5181 =
5182 \mathbf{1}\mathbin{\ast_D}(\operatorname{id}\cdot b),
5183 \]
5184 so that
5185 \[
5186 \mathfrak G(b)(n)
5187 =
5188 \sum_{d\mid n}d\,b(d).
5189 \]
5190 \end{definition}
5191 
5192 \begin{theorem}[Exact divisor--ghost bridge]
5193 \label{thm:block07-divisor-ghost}
5194 The map $b\mapsto c=\mathfrak G(b)$ is injective, with inverse
5195 \[
5196 \boxed{
5197 b(n)
5198 =
5199 \frac1n
5200 (\mu\mathbin{\ast_D}c)(n).
5201 }
5202 \]
5203 Its integral image consists exactly of the arithmetic functions $c$ satisfying
5204 \[
5205 n\mid(\mu\mathbin{\ast_D}c)(n)
5206 \qquad(n\geq1).
5207 \]
5208 For the Euler product
5209 \[
5210 A_b(z)=\prod_{n\geq1}(1-z^n)^{-b(n)},
5211 \]
5212 one has
5213 \[
5214 \frac{zA_b'(z)}{A_b(z)}
5215 =
5216 \sum_{n\geq1}c(n)z^n.
5217 \]
5218 \end{theorem}
5219 
5220 \begin{proof}
5221 The first identity is the convolution form of the divisor sum.  Since $\mu\mathbin{\ast_D}\mathbf{1}=\varepsilon$,
5222 \[
5223 \mu\mathbin{\ast_D}c
5224 =
5225 \operatorname{id}\cdot b.
5226 \]
5227 Division by $n$ gives the inverse and its exact integrality condition.  The logarithmic-derivative identity is Theorem~\ref{thm:block07-ghost-equivalence} with the exponent sequence regarded as an arithmetic function.
5228 \end{proof}
5229 
5230 \begin{theorem}[Multiplicativity is preserved and reflected]
5231 \label{thm:block07-multiplicative-ghost}
5232 Assume $b(1)=1$ and $c=\mathfrak G(b)$.  Then
5233 \[
5234 \boxed{
5235 b\text{ is multiplicative}
5236 \quad\Longleftrightarrow\quad
5237 c\text{ is multiplicative}.}
5238 \]
5239 On this locus, both $b$ and $c$ are reconstructed from prime-power seeds.  For every prime $p$ and $e\geq1$,
5240 \[
5241 \boxed{
5242 c(p^e)=\sum_{j=0}^{e}p^j b(p^j),}
5243 \]
5244 \[
5245 \boxed{
5246 b(p^e)=\frac{c(p^e)-c(p^{e-1})}{p^e}.}
5247 \]
5248 \end{theorem}
5249 
5250 \begin{proof}
5251 If $b$ is multiplicative, then $\operatorname{id}\cdot b$ is multiplicative, and Dirichlet convolution with $\mathbf{1}$ preserves multiplicativity.  Conversely, if $c$ is multiplicative, then $\mu\mathbin{\ast_D}c$ is multiplicative.  Dividing pointwise by the completely multiplicative function $\operatorname{id}$ preserves multiplicativity over $\mathbb Q$; the integral image condition ensures that the resulting values are integers.  The prime-power formulas follow by listing the divisors of $p^e$ and subtracting consecutive sums.
5252 \end{proof}
5253 
5254 \begin{figure}[htbp]
5255 \centering
5256 \includegraphics[width=0.99\textwidth]{divisor_ghost_bridge.pdf}
5257 \caption{The divisor--ghost transform gives a reversible invariant bridge on the multiplicative exponent locus.  It connects a fixed-basis ordinary Euler product to multiplicative prime-power data without identifying the two global convolution algebras.}
5258 \label{fig:block07-divisor-ghost}
5259 \end{figure}
5260 
5261 \subsection{A restricted product--multiplicative atlas}
5262 
5263 \begin{definition}[Multiplicative Euler-overlap category]
5264 Let $\mathbf{EulMult}_{\mathbb Z}$ be the discrete category of multiplicative exponent functions
5265 \[
5266 b:\mathbb N_{\geq1}\to\mathbb Z,
5267 \qquad b(1)=1.
5268 \]
5269 Associate to $b$ the four normalized charts
5270 \[
5271 \mathsf B(b)=b,
5272 \qquad
5273 \mathsf A(b)=A_b(z),
5274 \qquad
5275 \mathsf C(b)=\mathfrak G(b),
5276 \qquad
5277 \mathsf P(b)=\bigl(b(p^e),c(p^e)\bigr)_{p,e}.
5278 \]
5279 \end{definition}
5280 
5281 \begin{theorem}[Effective descent on the divisor--ghost overlap]
5282 \label{thm:block07-product-multiplicative-descent}
5283 The exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas
5284 \[
5285 \mathfrak R_{\mathrm{DG}}.
5286 \]
5287 Every bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and
5288 \[
5289 \boxed{
5290 \Desc(\mathfrak R_{\mathrm{DG}})
5291 \simeq
5292 \mathbf{EulMult}_{\mathbb Z}.
5293 }
5294 \]
5295 Descent is effective for objects and morphisms.
5296 \end{theorem}
5297 
5298 \begin{proof}
5299 The Euler-product chart is uniquely inverted by Theorem~\ref{thm:block07-euler-classification}.  The ghost chart is uniquely inverted by Theorem~\ref{thm:block07-divisor-ghost}.  The prime-power chart is uniquely inverted by multiplicativity and unique factorization.  Hence every normalized fiber over $b$ is a singleton.  All comparison cells and all multiple-overlap coherence laws are therefore forced to be identities.  A compatible local family glues to its common exponent function, and morphism effectivity is automatic because the chart categories are discrete.
5300 \end{proof}
5301 
5302 \begin{theorem}[Complete descended prime-power invariant]
5303 \label{thm:block07-complete-product-invariant}
5304 The prime-power package
5305 \[
5306 \boxed{
5307 \mathcal J_{\mathrm{DG}}(b)
5308 =
5309 \left(
5310  b(p^e),
5311  c(p^e)
5312 \right)_{p\text{ prime},\ e\geq0}
5313 }
5314 \]
5315 subject to
5316 \[
5317 p^e b(p^e)=c(p^e)-c(p^{e-1})
5318 \quad(e\geq1)
5319 \]
5320 is complete on $\mathbf{EulMult}_{\mathbb Z}$.  It reconstructs $b$, $c$, the Euler product $A_b$, and the coefficient recurrence of $A_b$.
5321 \end{theorem}
5322 
5323 \begin{proof}
5324 Prime-power values reconstruct the multiplicative functions $b$ and $c$.  Theorem~\ref{thm:block07-euler-classification} reconstructs $A_b$, and Theorem~\ref{thm:block07-euler-recurrence} reconstructs its coefficient sequence from $c$.  Therefore equal invariant packages imply equality in every chart.
5325 \end{proof}
5326 
5327 \begin{example}[Power bridge]
5328 For $b(n)=n^r$, which is completely multiplicative, the divisor ghost is
5329 \[
5330 c(n)=\sum_{d\mid n}d^{r+1}=\sigma_{r+1}(n).
5331 \]
5332 At a prime power,
5333 \[
5334 \sigma_{r+1}(p^e)
5335 =
5336 1+p^{r+1}+\cdots+p^{e(r+1)}.
5337 \]
5338 Thus one simple multiplicative exponent rule simultaneously produces a filtered ordinary product, a multiplicative divisor-sum ghost, and a one-dimensional geometric local rule at every prime.
5339 \end{example}
5340 
5341 \subsection{Why the bridge is restricted}
5342 
5343 \begin{proposition}[The multiplicative exponent locus is not a product subgroup]
5344 The set of multiplicative Euler exponent functions is not closed under the addition of exponent coordinates.  Consequently, multiplication of the corresponding ordinary Euler products does not define an internal group law on $\mathbf{EulMult}_{\mathbb Z}$.
5345 \end{proposition}
5346 
5347 \begin{proof}
5348 The constant function $b(n)=1$ is multiplicative.  But $b+b$ has value $2$ at $1$ and therefore is not multiplicative under the normalization $f(1)=1$.  Since ordinary Euler-product multiplication adds exponents, the multiplicative exponent locus is not closed under that operation.
5349 \end{proof}
5350 
5351 \begin{proposition}[Euler coefficients are not multiplicative semantics]
5352 The coefficient sequence of $A_b(z)$ need not be a multiplicative arithmetic function even when $b$ is multiplicative.  For example, $b(n)=1$ gives the partition numbers, and
5353 \[
5354 p(6)=11\neq p(2)p(3)=6.
5355 \]
5356 \end{proposition}
5357 
5358 \begin{boundary}[Prime-power data require multiplicativity]
5359 Two arbitrary arithmetic functions can agree on every prime power and differ on a composite supported at more than one prime.  Prime-power seeds are complete only after the multiplicativity axiom has been declared.  A compact local seed rule also does not remove the computational cost of factoring the input integer.
5360 \end{boundary}
5361 
5362 \begin{figure}[htbp]
5363 \centering
5364 \includegraphics[width=0.98\textwidth]{two_monoid_boundary.pdf}
5365 \caption{Cauchy and Dirichlet convolution use different global address monoids.  They overlap exactly after localization at one valuation coordinate, but no unrestricted algebra equivalence follows.}
5366 \label{fig:block07-two-monoids}
5367 \end{figure}
5368 
5369 \begin{boundary}[No universal additive--multiplicative equivalence]
5370 \label{bnd:block07-no-global-convolution-equivalence}
5371 The divisor--ghost transform is an exact invariant bridge on a named essential image.  It is not an algebra isomorphism between ordinary Cauchy convolution and Dirichlet convolution, does not identify the coefficients of an Euler product with a multiplicative function, and does not produce a canonical factorization across arbitrary product grammars.  Any larger bridge must name its transform, kernel, essential image, and information-loss profile.
5372 \end{boundary}
5373 
5374 \subsection{Exact finite verification}
5375 
5376 \input{generated/block07_validation_table.tex}
5377 
5378 The Product-multiplicative atlas verifier performs four exact audits.  First, it exhaustively checks the coefficient, exponent, and ghost triangular maps for every exponent prefix in $\{-2,-1,0,1,2\}^6$ and every unit coefficient prefix in $\{-2,-1,0,1,2\}^5$.  Second, it constructs deterministic prime-power systems, reconstructs global multiplicative functions, and verifies local Euler inversion, local logarithmic derivatives, and Dirichlet convolution.  Third, it checks the divisor--ghost bridge and the power family $b(n)=n^r$.  Fourth, it verifies the named local Euler coordinates printed in \cref{tab:block07-named-local}.  The recorded 595,964 exact checks have no failures.
5379 
5380 \begin{boundarybox}
5381 \textbf{Exact boundary of Product-multiplicative atlas.}
5382 The fixed Euler basis gives a canonical coordinate system for integral unit series, and prime-power seed series give a canonical coordinate system for integral multiplicative functions.  The divisor--ghost transform produces one complete cross-stratum overlap on multiplicative exponent functions.  The block does not prove compactness for arbitrary exponent data, a minimal product compiler, closure of the multiplicative exponent slice under ordinary product multiplication, a global Cauchy--Dirichlet algebra isomorphism, or a canonical factorization across unrestricted product grammars.
5383 \end{boundarybox}
5384 
5385 \begin{resultbox}
5386 \textbf{Section result.}
5387 Integral unit series admit unique fixed-basis Euler exponents and unique divisor ghosts, with exact logarithmic-derivative recurrence and finite-prefix triangularity.  Integral multiplicative functions are effective descent from prime-power local series; local Euler exponents and local ghosts linearize Dirichlet convolution and form complete invariants.  On the multiplicative exponent locus, the divisor--ghost transform gives a reversible four-chart atlas linking exponent functions, ordinary filtered Euler products, multiplicative ghosts, and prime-power seeds.  This closes one explicit product--multiplicative invariant bridge while preserving the global distinction between additive and multiplicative index monoids.
5388 \end{resultbox}
5389 
5390 
5391 
5392 \section{Local Newton--Mahler coordinates, affine polyphase descent, and precision ladders}
5393 \label{sec:block08-local}
5394 
5395 Local Mahler-polyphase atlas develops the local stratum promised by the work program.  Its purpose is not to merge every use of the word ``Mahler.''  It freezes one precise local semantic doctrine, constructs its normalized coefficient and polyphase charts, types the characteristic-zero, integral, and residue-level affine channels separately, and identifies the exact data needed to pass from finite modular dynamics to certified local analytic packets.
5396 
5397 \subsection{Two Mahler languages and the declared local doctrine}
5398 \label{subsec:block08-two-languages}
5399 
5400 Two constructions in the Library use the same historical name but have different variables and different semantics.
5401 
5402 \begin{enumerate}
5403 \item A \emph{functional-Mahler equation}
5404 \[
5405  F(z)=R(z)F(z^p)
5406 \]
5407 is an identity for a formal coefficient series in the variable \(z\).  Cartier operators act on the coefficient index.  the periodic and rational-mask atlases use this channel over finite fields.
5408 \item A \emph{Newton--Mahler expansion}
5409 \[
5410  f(x)=\sum_{r\geq0}c_r\binom xr
5411 \]
5412 is a coordinate expansion for a function of a \(p\)-adic input variable \(x\).  Affine substitutions \(x=Mq+s\) produce polyphase branches of the input domain.
5413 \end{enumerate}
5414 
5415 This section treats the second channel.  Fix a prime \(p\).  Let
5416 \[
5417  \mathcal C_p=C(\mathbb Z_p,\mathbb Z_p)
5418 \]
5419 with the uniform norm, and let
5420 \[
5421  c_0(\mathbb Z_p)
5422  =
5423  \left\{c=(c_r)_{r\geq0}:c_r\in\mathbb Z_p,\ c_r\longrightarrow0\right\}
5424 \]
5425 with the coefficient sup norm.
5426 
5427 \begin{theorem}[Mahler coordinate theorem]
5428 \label{thm:block08-mahler-coordinate}
5429 The synthesis map
5430 \[
5431  \mathcal M_p:c_0(\mathbb Z_p)\longrightarrow\mathcal C_p,
5432  \qquad
5433  c\longmapsto
5434  \left(x\longmapsto\sum_{r\geq0}c_r\binom xr\right),
5435 \]
5436 is an isometric isomorphism of \(\mathbb Z_p\)-modules.  Its inverse is
5437 \[
5438  \boxed{c_r=\Delta^r f(0).}
5439 \]
5440 In particular, the Newton--Mahler coefficient sequence is a canonical complete coordinate for a continuous \(\mathbb Z_p\)-valued function.
5441 \end{theorem}
5442 
5443 \begin{proof}
5444 This is the classical Mahler coordinate theorem \cite{Mahler1958}.  Finite Newton interpolation gives the coefficient identity on nonnegative integers.  Continuity implies that the forward-difference coefficients tend to zero, so the binomial series converges uniformly on \(\mathbb Z_p\).  Conversely, every coefficient sequence tending to zero gives a uniformly convergent continuous function.  Uniqueness follows from the finite-difference formula.
5445 \end{proof}
5446 
5447 \begin{definition}[Exact valuation profile and certified class]
5448 \label{def:block08-weight-class}
5449 For \(f=\mathcal M_p(c)\), define
5450 \[
5451  \nu_f(r)=v_p(c_r)\in\mathbb Z_{\geq0}\cup\{\infty\}.
5452 \]
5453 If \(w:\mathbb N\to\mathbb Z_{\geq0}\) is nondecreasing and tends to infinity, put
5454 \[
5455  \mathcal M_w
5456  =
5457  \{c\in c_0(\mathbb Z_p):v_p(c_r)\geq w(r)\},
5458  \qquad
5459  \mathcal C_w=\mathcal M_p(\mathcal M_w).
5460 \]
5461 The function \(w\) is a declared precision certificate.  The class \(\mathcal C_w\) is not asserted to be a characterization of every possible notion of local analyticity.
5462 \end{definition}
5463 
5464 \begin{figure}[htbp]
5465 \centering
5466 \includegraphics[width=.98\textwidth]{local_mahler_atlas.pdf}
5467 \caption{The normalized local atlas.  Continuous semantics, canonical Mahler coefficients, complete clopen branch packets, and branchwise Mahler coefficients are equivalent charts.  A single continuous branch remains directional.}
5468 \label{fig:block08-local-atlas}
5469 \end{figure}
5470 
5471 \subsection{Complete clopen packets and effective local descent}
5472 \label{subsec:block08-clopen-descent}
5473 
5474 For \(a\geq1\) and \(0\leq s<p^a\), define the branch
5475 \[
5476  \mathcal B_{a,s}f(q)=f(p^a q+s),
5477  \qquad q\in\mathbb Z_p,
5478 \]
5479 and the complete packet
5480 \[
5481  \mathcal B_a f=\bigl(\mathcal B_{a,s}f\bigr)_{0\leq s<p^a}.
5482 \]
5483 
5484 \begin{theorem}[Complete clopen packet equivalence]
5485 \label{thm:block08-clopen-equivalence}
5486 The analysis map
5487 \[
5488  \mathcal B_a:\mathcal C_p
5489  \longrightarrow
5490  \prod_{s=0}^{p^a-1}\mathcal C_p
5491 \]
5492 is an isometric isomorphism.  Its inverse is the piecewise synthesis
5493 \[
5494  \boxed{
5495  (\mathcal S_a g)(x)
5496  =g_s\!\left(\frac{x-s}{p^a}\right)
5497  \quad\text{for }x\in s+p^a\mathbb Z_p.}
5498 \]
5499 Thus the clopen residue cover of \(\mathbb Z_p\) has effective descent for both objects and equality morphisms.
5500 \end{theorem}
5501 
5502 \begin{proof}
5503 The cosets \(s+p^a\mathbb Z_p\) are pairwise disjoint, clopen, and cover \(\mathbb Z_p\).  Restriction followed by the displayed synthesis is the identity.  Conversely, synthesis followed by restriction recovers every component.  Continuity is checked separately on each clopen component, and the uniform norm is the maximum of the branch norms.
5504 \end{proof}
5505 
5506 \begin{theorem}[Strict nested-branch law]
5507 \label{thm:block08-nested-branches}
5508 For \(a,b\geq1\), \(0\leq s<p^a\), and \(0\leq t<p^b\),
5509 \[
5510  \boxed{(f_{a,s})_{b,t}=f_{a+b,s+p^a t}.}
5511 \]
5512 All multiple-overlap comparison cells for the nested \(p\)-power branch atlas are identities.  Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.
5513 \end{theorem}
5514 
5515 \begin{proof}
5516 Direct substitution gives
5517 \[
5518  f_{a,s}(p^bq+t)
5519  =f(p^a(p^bq+t)+s)
5520  =f(p^{a+b}q+s+p^a t).
5521 \]
5522 The address is uniquely determined in the interval \(0\leq s+p^a t<p^{a+b}\).
5523 \end{proof}
5524 
5525 A single branch \(\mathcal B_{a,s}\) is not injective on \(\mathcal C_p\) when \(a\geq1\): any nonzero continuous function supported on another clopen coset lies in its kernel.  The complete packet is therefore the reversible object.  This observation will be contrasted below with the finite-degree characteristic-zero polynomial chart, where one affine branch is invertible.
5526 
5527 \subsection{The affine Mahler kernel and mixed-radix coherence}
5528 \label{subsec:block08-affine-kernel}
5529 
5530 For \(M\geq1\), \(s\geq0\), and \(r,k\geq0\), define
5531 \[
5532  \Phi_M(z)=(1+z)^M-1
5533 \]
5534 and
5535 \[
5536  \boxed{
5537  \Theta_{M,s}(r,k)
5538  =[z^r](1+z)^s\Phi_M(z)^k.}
5539 \]
5540 Equivalently,
5541 \[
5542  \boxed{
5543  \Theta_{M,s}(r,k)
5544  =\sum_{j=0}^{k}(-1)^{k-j}\binom kj\binom{Mj+s}{r}.}
5545 \]
5546 The support is finite:
5547 \[
5548  \Theta_{M,s}(r,k)=0
5549  \quad\text{unless}\quad
5550  k\leq r\leq Mk+s.
5551 \]
5552 
5553 \begin{theorem}[Affine Newton--Mahler transfer]
5554 \label{thm:block08-affine-transfer}
5555 Let
5556 \[
5557  f(x)=\sum_{r\geq0}c_r\binom xr
5558 \]
5559 be a convergent Mahler expansion and put \(g_s(q)=f(Mq+s)\).  Then
5560 \[
5561  g_s(q)=\sum_{k\geq0}d_{s,k}\binom qk,
5562 \]
5563 where
5564 \[
5565  \boxed{
5566  d_{s,k}
5567  =\sum_{r=k}^{Mk+s}c_r\Theta_{M,s}(r,k).}
5568 \]
5569 Every output coefficient depends on a finite source window.
5570 \end{theorem}
5571 
5572 \begin{proof}
5573 Expand
5574 \[
5575  (1+z)^{Mq+s}
5576  =(1+z)^s\bigl(1+\Phi_M(z)\bigr)^q
5577  =\sum_{k\geq0}\binom qk(1+z)^s\Phi_M(z)^k.
5578 \]
5579 Taking \([z^r]\) gives the affine expansion of \(\binom{Mq+s}{r}\).  Substitution into the Mahler series gives the formula.  The finite support of \(\Theta_{M,s}\) makes each coefficient sum finite.
5580 \end{proof}
5581 
5582 Let \(\mathcal T_{M,s}\) denote this coefficient operator.  Conjugating the clopen packet theorem by \(\mathcal M_p\) gives the following normalized coefficient atlas.
5583 
5584 \begin{corollary}[Complete coefficient-packet equivalence]
5585 \label{cor:block08-coefficient-packet}
5586 For \(M=p^a\), the map
5587 \[
5588  \mathcal T_a:c_0(\mathbb Z_p)
5589  \longrightarrow
5590  \prod_{s=0}^{p^a-1}c_0(\mathbb Z_p),
5591  \qquad
5592  c\longmapsto(\mathcal T_{p^a,s}c)_s,
5593 \]
5594 is a topological isomorphism.  It is the coefficient conjugate of complete clopen analysis.
5595 \end{corollary}
5596 
5597 \begin{theorem}[Mixed-radix kernel composition]
5598 \label{thm:block08-kernel-composition}
5599 For \(M,N\geq1\),
5600 \[
5601  \boxed{
5602  \sum_{k\geq0}
5603  \Theta_{M,s}(r,k)\Theta_{N,t}(k,\ell)
5604  =
5605  \Theta_{MN,Mt+s}(r,\ell).}
5606 \]
5607 Equivalently,
5608 \[
5609  \boxed{
5610  \mathcal T_{N,t}\mathcal T_{M,s}
5611  =
5612  \mathcal T_{MN,Mt+s}.}
5613 \]
5614 When \(\gcd(M,N)=1\), the two Chinese-remainder refinement paths to a common address agree exactly.  Their holonomy is the identity.
5615 \end{theorem}
5616 
5617 \begin{proof}
5618 Apply the affine substitution \(x=Mq+s\), then \(q=Ny+t\).  The composite is \(x=MNy+Mt+s\).  Uniqueness of Mahler coefficients yields the kernel identity.  The Chinese-remainder diamond is the same equality written along two factorizations of one affine address.
5619 \end{proof}
5620 
5621 \begin{resultbox}
5622 \textbf{Normalized local atlas.}
5623 For every fixed prime \(p\) and depth \(a\), the following normalized charts are equivalent:
5624 \[
5625  \boxed{
5626  \mathbf{Loc}^{\mathrm{cts}}_p
5627  \simeq
5628  \mathbf{Mah}^{c_0}_p
5629  \simeq
5630  \mathbf{Branch}^{\mathrm{cts}}_{p,a}
5631  \simeq
5632  \mathbf{BranchMah}^{c_0}_{p,a}.}
5633 \]
5634 The first chart contains continuous functions on \(\mathbb Z_p\), the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding branchwise coefficient packet.  Every normalized fiber is contractible.  The descent category is therefore equivalent to the discrete semantic category \(\mathbf{Loc}^{\mathrm{cts}}_p\).
5635 \end{resultbox}
5636 
5637 \subsection{Finite-degree bridge ladder and collective recovery}
5638 \label{subsec:block08-finite-degree}
5639 
5640 The same integer kernel matrix has different categorical status after changing the coefficient doctrine.  Fix \(d\geq0\) and write
5641 \[
5642  \mathcal M_d(R)=R^{d+1}
5643 \]
5644 for the formal truncated Mahler coefficient module.  Let
5645 \[
5646  A_{M,s}^{(d)}
5647  =\bigl[\Theta_{M,s}(r,k)\bigr]_{0\leq k,r\leq d},
5648 \]
5649 where the target degree \(k\) is the row index.
5650 
5651 \begin{theorem}[Typed affine-channel ladder]
5652 \label{thm:block08-channel-ladder}
5653 For every \(M\geq1\), \(s\geq0\), and \(d\geq0\):
5654 \begin{enumerate}
5655 \item The matrix \(A_{M,s}^{(d)}\) is upper triangular with diagonal
5656 \[
5657  1,M,M^2,\ldots,M^d.
5658 \]
5659 Hence
5660 \[
5661  \boxed{\det A_{M,s}^{(d)}=M^{d(d+1)/2}.}
5662 \]
5663 Every characteristic-zero polynomial chart is invertible.
5664 \item On the integral lattice \(\mathbb Z^{d+1}\), one channel is injective.  If \(M>1\) and \(d\geq1\), its image has index \(M^{d(d+1)/2}\), so the channel has no integer linear left inverse.
5665 \item Let \(p\) be prime and put \(\beta_p=p^{v_p(M)}\).  Reduction modulo \(p\) has exact rank
5666 \[
5667  \boxed{
5668  \rank_{\mathbb F_p}\overline A_{M,s}^{(d)}
5669  =\left\lfloor\frac d{\beta_p}\right\rfloor+1.}
5670 \]
5671 A local channel loses residual rank exactly at primes dividing \(M\).
5672 \item A complete neighboring packet can recover the source integrally.  The canonical prefix of \(\min(M,d+1)\) standard shifts has an integer left inverse, valid after base change to every commutative ring.
5673 \end{enumerate}
5674 \end{theorem}
5675 
5676 \begin{proof}
5677 The diagonal follows from the leading term of \(\binom{Mq+s}{r}\), or from the generating identity for \(\Theta_{M,s}\).  The determinant gives the integral index statement.  Modulo \(p\), the first nonzero source degree in target row \(k\) is \(\beta_pk\), which gives the rank formula.  For the prefix packet, choose from channel \(s\) and target row \(k\) the equation indexed by \(n=Mk+s\).  The selected square coefficient matrix is unit lower triangular, so its inverse is integral and remains a left inverse after every base change.
5678 \end{proof}
5679 
5680 \begin{theorem}[Sharp standard packet size]
5681 \label{thm:block08-sharp-packet}
5682 Define
5683 \[
5684  B(M)
5685  =
5686  \max\left(\{1\}\cup\{p^{v_p(M)}:p\mid M\}\right).
5687 \]
5688 The least number of standard affine channels whose stacked map has an integer left inverse is
5689 \[
5690  \boxed{L_{\mathbb Z}(M,d)=\min(d+1,B(M)).}
5691 \]
5692 The prefix \(\{0,1,\ldots,L_{\mathbb Z}(M,d)-1\}\) attains the minimum.
5693 
5694 For a fixed modulus \(N\geq2\), set
5695 \[
5696  B_N(M)
5697  =
5698  \max\left(\{1\}\cup
5699  \{p^{v_p(M)}:p\mid\gcd(M,N)\}\right).
5700 \]
5701 The least packet size for a left inverse over \(\mathbb Z/N\mathbb Z\) is
5702 \[
5703  \boxed{L_N(M,d)=\min(d+1,B_N(M)).}
5704 \]
5705 The exponents \(v_p(N)\) do not change the minimum; the first residue layer already detects the obstruction.
5706 \end{theorem}
5707 
5708 \begin{proof}
5709 This is the primary-rank and Smith/Hermite criterion of Core-Transfer Addendum 04.  At a prime \(p\mid M\), selected shifts are visible only through their residues modulo \(p^{v_p(M)}\).  A prefix of the stated length has full rank at every obstructing prime, hence a unimodular-row left inverse.  Any smaller packet fails at a prime attaining the largest relevant primary component.  A nonzero first-layer kernel vector lifts to a kernel vector at every higher prime-power depth.
5710 \end{proof}
5711 
5712 \begin{figure}[htbp]
5713 \centering
5714 \includegraphics[width=.93\textwidth]{local_channel_ladder.pdf}
5715 \caption{The same affine Mahler kernel has four distinct bridge passports.  Characteristic-zero invertibility, integral injection, modular rank, and complete-packet splitting must not be conflated.}
5716 \label{fig:block08-channel-ladder}
5717 \end{figure}
5718 
5719 \begin{figure}[htbp]
5720 \centering
5721 \includegraphics[width=.94\textwidth]{polyphase_kernel_sparsity.pdf}
5722 \caption{The finite triangular support of the affine kernel and its Frobenius-sparse residue pattern at a prime-power radix.}
5723 \label{fig:block08-kernel-sparsity}
5724 \end{figure}
5725 
5726 \subsection{Valuation ledgers and Frobenius sparsity}
5727 \label{subsec:block08-valuations}
5728 
5729 The coefficient transform carries a precise lower-bound calculus.
5730 
5731 \begin{theorem}[Exact branchwise valuation ledger]
5732 \label{thm:block08-valuation-ledger}
5733 For every refined coefficient,
5734 \[
5735  \boxed{
5736  v_p(d_{s,k})
5737  \geq
5738  \min_{k\leq r\leq Mk+s}
5739  \left(v_p(c_r)+v_p(\Theta_{M,s}(r,k))\right).}
5740 \]
5741 If \(v_p(c_r)\geq w(r)\) for a nondecreasing weight \(w\), then
5742 \[
5743  \boxed{v_p(d_{s,k})\geq w(k).}
5744 \]
5745 Thus every declared monotone Mahler precision ladder survives affine polyphase refinement.
5746 \end{theorem}
5747 
5748 \begin{proof}
5749 The transfer sum is finite.  Apply the non-Archimedean triangle inequality.  Since \(r\geq k\), monotonicity gives \(w(r)\geq w(k)\), and every kernel entry is integral.
5750 \end{proof}
5751 
5752 Define the min-plus certificate transform
5753 \[
5754  \boxed{
5755  (\mathfrak W_{M,s}w)(k)
5756  =
5757  \min_{k\leq r\leq Mk+s}
5758  \left(w(r)+v_p(\Theta_{M,s}(r,k))\right).}
5759 \]
5760 This transform is a certified lower bound.  It need not be the exact output valuation, because cancellation can increase the valuation beyond the termwise minimum.
5761 
5762 \begin{theorem}[Prime-power Frobenius sparsification]
5763 \label{thm:block08-frobenius-sparsity}
5764 For \(M=p^a\),
5765 \[
5766  \boxed{
5767  \Theta_{p^a,s}(r,k)
5768  \equiv
5769  \binom{s}{r-p^ak}
5770  \pmod p,}
5771 \]
5772 where the binomial coefficient is zero outside \(0\leq r-p^ak\leq s\).  Therefore
5773 \[
5774  \boxed{
5775  d_{s,k}
5776  \equiv
5777  \sum_{u=0}^{s}\binom su c_{p^ak+u}
5778  \pmod p.}
5779 \]
5780 In particular,
5781 \[
5782  \boxed{d_{0,k}\equiv c_{p^ak}\pmod p.}
5783 \]
5784 \end{theorem}
5785 
5786 \begin{proof}
5787 In characteristic \(p\),
5788 \[
5789  (1+z)^{p^a}-1\equiv z^{p^a}.
5790 \]
5791 Substitute this into the generating definition of \(\Theta_{p^a,s}\) and compare coefficients.
5792 \end{proof}
5793 
5794 \subsection{Carmichael--Mahler state constructors and precision amplification}
5795 \label{subsec:block08-carmichael}
5796 
5797 Finite modular state dynamics provide a directional constructor into the local Mahler chart.  Let
5798 \[
5799  v_{n+1}=Cv_n,
5800  \qquad
5801  a_n=w^{\mathsf T}C^nv_0,
5802 \]
5803 with \(C\in M_d(\mathbb Z_p)\) invertible modulo \(p\).
5804 
5805 \begin{theorem}[Matrix Carmichael--Mahler constructor]
5806 \label{thm:block08-matrix-carmichael}
5807 Assume
5808 \[
5809  C^H=I+p^LB,
5810  \qquad L\geq1.
5811 \]
5812 For every address \(j\),
5813 \[
5814  \boxed{
5815  a_{Hx+j}
5816  =
5817  \sum_{r\geq0}
5818  p^{Lr}\binom xr
5819  w^{\mathsf T}C^jB^rv_0,}
5820 \]
5821 where the series converges for \(x\in\mathbb Z_p\).  Its Mahler coefficients satisfy
5822 \[
5823  \boxed{v_p(c_{j,r})\geq Lr,}
5824 \]
5825 and
5826 \[
5827  \boxed{\Delta_H^r a_n\equiv0\pmod{p^{Lr}}.}
5828 \]
5829 \end{theorem}
5830 
5831 \begin{proof}
5832 Use
5833 \[
5834  C^{Hx+j}=C^j(I+p^LB)^x
5835 \]
5836 and expand the principal-unit matrix power by the convergent binomial series.  One forward difference in \(x\) contributes the factor \(p^LB\).
5837 \end{proof}
5838 
5839 \begin{theorem}[Principal-unit precision amplification]
5840 \label{thm:block08-precision-amplification}
5841 If \(X=I+p^LB\), then for every \(a\geq0\) there exists \(B_a\in M_d(\mathbb Z_p)\) such that
5842 \[
5843  \boxed{X^{p^a}=I+p^{L+a}B_a.}
5844 \]
5845 Consequently, for \(H_a=Hp^a\), every branch \(x\mapsto a_{H_ax+j}\) has Mahler coefficients satisfying
5846 \[
5847  \boxed{v_p(c_{a,j,r})\geq(L+a)r.}
5848 \]
5849 Each additional \(p\)-power refinement contributes at least one further digit of precision per Mahler depth.
5850 \end{theorem}
5851 
5852 \begin{proof}
5853 Induct on \(a\).  If \(X_a=I+p^{L+a}B_a\), the binomial expansion of \(X_a^p\) has linear term divisible by \(p^{L+a+1}\), and every higher term has at least the same valuation.  Apply \cref{thm:block08-matrix-carmichael} at the refined jump.
5854 \end{proof}
5855 
5856 \begin{example}[Refined dyadic Fibonacci packets]
5857 For the Fibonacci companion matrix
5858 \[
5859  C=\begin{pmatrix}1&1\\1&0\end{pmatrix},
5860 \]
5861 one has
5862 \[
5863  C^{12}
5864  =I+8\begin{pmatrix}29&18\\18&11\end{pmatrix}.
5865 \]
5866 Hence
5867 \[
5868  \boxed{
5869  \Delta_{12\cdot2^a}^{\,r}F_n
5870  \equiv0\pmod{2^{(a+3)r}}.}
5871 \]
5872 \end{example}
5873 
5874 \begin{figure}[htbp]
5875 \centering
5876 \includegraphics[width=.96\textwidth]{local_precision_tree.pdf}
5877 \caption{A principal-unit state packet acquires one additional guaranteed \(p\)-adic digit per Mahler depth at each \(p\)-power refinement.}
5878 \label{fig:block08-precision-tree}
5879 \end{figure}
5880 
5881 The state-to-local construction is directional.  A local branch can forget unreachable state directions, nonminimal coordinates, or modes annihilated by the output functional.  Recovering a state presentation requires a separate recognition or minimality theorem and is not part of the Carmichael--Mahler constructor itself.
5882 
5883 \subsection{The exact boundary with functional-Mahler masks}
5884 \label{subsec:block08-functional-boundary}
5885 
5886 \begin{proposition}[No canonical unrestricted bridge between the two Mahler channels]
5887 \label{prop:block08-no-unrestricted-mahler-bridge}
5888 Functional-Mahler mask data over \(\mathbb F_p\) does not determine a unique Newton--Mahler chart over \(\mathbb Z_p\), and a continuous Newton--Mahler function does not in general determine a rational functional-Mahler mask.
5889 \end{proposition}
5890 
5891 \begin{proof}
5892 A residue sequence \((\bar a_n)\) admits many integral lifts.  If \((a_n)\) is one lift, then \((a_n+pn)\) is another with the same reduction modulo \(p\).  Their Newton--Mahler coefficients differ by the nonzero coefficient sequence of \(pn\).  Thus the residue-series or functional-mask presentation cannot select one local interpolation without a declared lifting rule.
5893 
5894 Conversely, a continuous function \(f:\mathbb Z_p\to\mathbb Z_p\) canonically determines its Newton--Mahler coefficients, but rationality of an ordinary generating series or of the mask \(F(z)/F(z^p)\) is an additional property of a separate coefficient-series presentation.  Continuity alone does not supply it.
5895 \end{proof}
5896 
5897 \begin{figure}[htbp]
5898 \centering
5899 \includegraphics[width=.95\textwidth]{two_mahler_languages.pdf}
5900 \caption{Functional-Mahler and Newton--Mahler constructions require different variables, coefficient doctrines, and normalization data.  A bridge exists only after those extra choices are declared.}
5901 \label{fig:block08-two-mahler-languages}
5902 \end{figure}
5903 
5904 \begin{boundarybox}
5905 \textbf{Exact local boundary.}
5906 Local Mahler-polyphase atlas proves a complete normalized atlas for continuous \(\mathbb Z_p\)-valued functions and their complete \(p^a\)-branch packets.  It proves exact affine coefficient transport, mixed-radix coherence, finite-degree characteristic-zero invertibility, integral split recovery, modular rank loss, valuation preservation, and principal-unit precision amplification under the printed hypotheses.  It does not identify every continuous function as locally analytic, compute exact higher valuations of every \(\Theta_{p^a,s}(r,k)\), choose an optimal mixed-radix refinement tree, or construct the combined spectral--local invariant requested by the broader program.
5907 \end{boundarybox}
5908 
5909 \subsection{Exact examples and finite verification}
5910 \label{subsec:block08-verification}
5911 
5912 For \(f(x)=x^2\), the source Mahler vector is
5913 \[
5914  (c_0,c_1,c_2)=(0,1,2).
5915 \]
5916 The two dyadic branches are
5917 \[
5918  f(2q)=4\binom q1+8\binom q2,
5919 \]
5920 and
5921 \[
5922  f(2q+1)=1+8\binom q1+8\binom q2.
5923 \]
5924 This example exhibits exact finite-window coefficient transport.
5925 
5926 For a principal unit \(u=1+p\),
5927 \[
5928  u^x=\sum_{r\geq0}p^r\binom xr.
5929 \]
5930 The branch \(x=pq\) has coefficients
5931 \[
5932  \bigl(u^p-1\bigr)^k,
5933 \]
5934 whose valuation is at least \(2k\).  The branch therefore improves the original slope-one certificate.
5935 
5936 At \(p=2\), \(M=2\), and \(d=3\), each single reduced channel has rank two, while the two-channel packet has rank four.  This is the smallest visible instance of collective recovery.
5937 
5938 \input{generated/block08_examples_table.tex}
5939 \input{generated/block08_validation_table.tex}
5940 
5941 The verifier checks the affine generating identity, finite polynomial transport, triangular determinants, mixed-radix composition, Chinese-remainder diamonds, nested branch addresses, prime-power Frobenius sparsity, exact local ranks, packet minimality, integer decoders, composite primary minima, valuation ledgers, scalar principal-unit amplification, and the refined Fibonacci matrix congruences.  It also includes a negative control showing that equal residue data can have distinct Newton--Mahler lifts.
5942 
5943 \begin{resultbox}
5944 \textbf{Local Mahler--polyphase result.}
5945 The local continuous, Newton--Mahler coefficient, complete clopen branch, and branch-coefficient presentations form an effective normalized atlas.  Affine polyphase kernels compose strictly and transport declared valuation ladders.  The finite-degree restriction exposes a four-level bridge ladder: reversible characteristic-zero charts, nonsplit integral channels, rank-deficient residue channels, and split neighboring packets.  Carmichael--Mahler transfer embeds finite principal-unit state dynamics into the local atlas with a certified precision slope that improves under \(p\)-power refinement.  Functional-Mahler masks remain a distinct typed channel unless coefficient lifts and interpolation data are supplied.
5946 \end{resultbox}
5947 
5948 
5949 \input{block09_section.tex}
5950 
5951 \input{block10_section.tex}
5952 
5953 \input{block11_section.tex}
5954 
5955 \input{block12_section.tex}
5956 
5957 \part{Certified global synthesis and scope}
5958 
5959 \input{block13_section.tex}
5960 
5961 \input{block15_section.tex}
5962 
5963 \input{block16_section_final.tex}
5964 
5965 \part{Relative semantic quotients and changes of doctrine}
5966 
5967 \input{block17_semantic_quotients_final.tex}
5968 
5969 \input{block18_carrier_observer_change_final.tex}
5970 
5971 \input{block19_admissibility_lifting_final.tex}
5972 
5973 \input{publication_scope_section.tex}
5974 \input{publication_conclusion.tex}
5975 
5976 \appendix
5977 
5978 \section{Theorem concordance and proof-status index}
5979 \label{app:full-concordance}
5980 
5981 The machine-readable concordance in the source package preserves, for all 75 components, the selected strongest source, exact theorem coordinate, hypotheses, proof class, imported input, and remaining obligation.  The publication PDF retains the compact index below.  This avoids duplicating the full theorem prose while preserving a complete auditable record in \nolinkurl{Komorebi_Theorem_Concordance_Final_2026-10-06.json} and \nolinkurl{final_generated/concordance_entries.tex}.
5982 
5983 \input{final_generated/compact_concordance.tex}
5984 
5985 \clearpage
5986 \section{Edition-control ledger}
5987 \input{final_generated/edition_ledger.tex}
5988 
5989 \clearpage
5990 \section{Publication audit and development provenance}
5991 \label{app:block15-publication-audit}
5992 
5993 The consolidated edition retains a compact theorem-suite map and a development-provenance ledger so that editorial reordering does not erase proof dependencies or claim boundaries. The tables in this appendix are inherited records: their page counts and validation results refer to the original artifacts and the dated version 2.0.0 mathematical base. The final-edition build and publication checks are recorded separately in the release validation report.
5994 
5995 \input{generated/block15_publication_audit_table.tex}
5996 
5997 \IfFileExists{generated/block16_publication_audit_table.tex}{\input{generated/block16_publication_audit_table.tex}}{}
5998 
5999 \input{generated/formal_foundations_v2_publication_audit_table.tex}
6000 
6001 \input{generated/block14_development_ledger.tex}
6002 
6003 \clearpage
6004 \input{appendix_open_program.tex}
6005 
6006 \clearpage
6007 \input{appendix_claim_ledger_final.tex}
6008 
6009 \clearpage
6010 \section{Validation record}
6011 
6012 The validation summarized below concerns internal package integrity and exact finite verification. It is not an external authenticity or peer-review certificate. The thematic verifiers record 39,805 checks for the unit-periodic atlas, 11,391 for rational masks, 89,055 for fixed-radix regular systems, 697,958 for integral saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute \(\BlockThirteenBridgeChecks\) bridge assertions and \(\BlockThirteenInvariantChecks\) weighted Fuss and tagged-invariant assertions. They include \(\BlockThirteenRegressionChecks\) authenticated replay assertions already counted in the preceding records and contribute \(\BlockThirteenNewChecks\) new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is \(\BlockThirteenCumulative\) exact assertions with no failures; the raw executed total, including replay, is \(\BlockThirteenCumulativeExecuted\). The Block 15 verifier separately executes \(\BlockFifteenValidationAssertions\) exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes \(\BlockSixteenValidationAssertions\) exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit projectors, and theorem-source contracts, with \(\BlockSixteenValidationFailures\) recorded failures. Detailed tables and machine-readable JSON records are included in the source package. The Part IV semantic-diamond audit additionally executes exact rational linear-algebra trials for quotient dimensions, factorization, relative residuals, and compatibility sequences. Its role is implementation validation; the universal statements are proved in the text.
6013 
6014 \input{generated/validation_table.tex}
6015 
6016 \input{generated/block17_validation_table.tex}
6017 
6018 A legacy all-project unit-test script was attempted twice during an earlier consolidation stage, including a 600-second bounded run. The completed tests were passing when the second run timed out, but that historical script did not finish within the bound. This statement does not qualify or replace the named thematic and Block 15 verifier records. No pass is claimed for that separate legacy all-project invocation.
6019 
6020 \clearpage
6021 \begin{thebibliography}{99}
6022 
6023 \bibitem{P1}
6024 Komorebi P., \emph{Projector-Resolved Seed Calculus I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates}, Complete Symbolic Core Edition and Expanded Clarified Edition, 2026.
6025 
6026 \bibitem{RT}
6027 Komorebi P., \emph{Resolution Theory: Projector Corner Representations, Refinement Naturality, and Nonlinear Transfer}, final journal manuscript, 2026.
6028 
6029 \bibitem{P3}
6030 Komorebi P., \emph{Nonlinear Packet Calculus, Obstruction Complexes, and Holonomy}, strengthened Annals edition, 2026.
6031 
6032 \bibitem{MIG}
6033 Komorebi P., \emph{Paper III Migration and Expansion Map}, 2026.
6034 
6035 \bibitem{CT4}
6036 Komorebi P., \emph{Affine Mahler Packets and Periodic Test Sets: Core-Transfer Theorem Addendum 04}, 2026.
6037 
6038 \bibitem{GAL}
6039 Komorebi P., \emph{Resolution Theory: Projector Geometry, Galois Descent, and Multiscale Realizations}, expanded journal manuscript, 2026.
6040 
6041 \bibitem{SEQ}
6042 Komorebi P., \emph{Projector-Seed Calculus for Integer Sequences: A Unified Realization Theory with Spectral, Local, Adelic, and Polyphase Transfer}, 2026.
6043 
6044 \bibitem{FMH}
6045 Komorebi P., \emph{Custom Generating Functions III: Cartier Automata, Prime-Power Unit Kernels, Mixed-Radix q-Carries, and Cyclotomic Collapse}, research report 0.3, 2026.
6046 
6047 \bibitem{NLO}
6048 Komorebi P., \emph{Nonlinear Transfer Operads and Projector-Resolved Jet Calculus}, 2026.
6049 
6050 \bibitem{ABS}
6051 Komorebi P., \emph{Abstraction as a Resolved Practice: Mathematical Objects, Imagination, Bridges, and Human Reachability}, 2026.
6052 
6053 \bibitem{Ore1933}
6054 O. Ore, ``Theory of non-commutative polynomials,'' \emph{Annals of Mathematics} \textbf{34} (1933), 480--508.
6055 
6056 \bibitem{Stanley1980}
6057 R. P. Stanley, ``Differentiably finite power series,'' \emph{European Journal of Combinatorics} \textbf{1} (1980), 175--188.
6058 
6059 \bibitem{BeukersHeckman1989}
6060 F. Beukers and G. Heckman, ``Monodromy for the hypergeometric function \({}_nF_{n-1}\),'' \emph{Inventiones Mathematicae} \textbf{95} (1989), 325--354.
6061 
6062 \bibitem{Mahler1958}
6063 K. Mahler, ``An interpolation series for continuous functions of a \(p\)-adic variable,'' \emph{Journal f\"ur die reine und angewandte Mathematik} \textbf{199} (1958), 23--34.
6064 
6065 \bibitem{FlajoletSedgewick2009}
6066 P. Flajolet and R. Sedgewick, \emph{Analytic Combinatorics}, Cambridge University Press, 2009.
6067 
6068 \bibitem{SerreLocalFields}
6069 J.-P. Serre, \emph{Local Fields}, Springer, 1979.
6070 
6071 \bibitem{SerreGaloisCohomology}
6072 J.-P. Serre, \emph{Galois Cohomology}, Springer, 1997.
6073 
6074 \bibitem{DeligneMilne}
6075 P. Deligne and J. S. Milne, ``Tannakian Categories,'' in \emph{Hodge Cycles, Motives, and Shimura Varieties}, Lecture Notes in Mathematics 900, Springer, 1982.
6076 
6077 
6078 \bibitem{Nerode1958}
6079 A. Nerode, ``Linear automaton transformations,'' \emph{Proceedings of the American Mathematical Society} \textbf{9} (1958), 541--544.
6080 
6081 \bibitem{CarlylePaz1971}
6082 J. W. Carlyle and A. Paz, ``Realizations by stochastic finite automata,'' \emph{Journal of Computer and System Sciences} \textbf{5} (1971), 26--40.
6083 
6084 \bibitem{LiRabusseauPrecup2018}
6085 T. Li, G. Rabusseau, and D. Precup, ``Nonlinear weighted finite automata,'' in \emph{Proceedings of AISTATS 2018}, PMLR 84, 679--688, 2018.
6086 
6087 \bibitem{Litvinov2007}
6088 G. L. Litvinov, ``The Maslov dequantization, idempotent and tropical mathematics: a brief introduction,'' \emph{Journal of Mathematical Sciences} \textbf{140} (2007), 426--444.
6089 
6090 \bibitem{John1948}
6091 F. John, ``Extremum problems with inequalities as subsidiary conditions,'' in \emph{Studies and Essays Presented to R. Courant on His 60th Birthday}, Interscience, 1948, 187--204.
6092 
6093 \bibitem{AndoShih1998}
6094 T. Ando and M.-H. Shih, ``Simultaneous contractibility,'' \emph{SIAM Journal on Matrix Analysis and Applications} \textbf{19} (1998), 487--498.
6095 
6096 \bibitem{Schneider2014}
6097 R. Schneider, \emph{Convex Bodies: The Brunn--Minkowski Theory}, second edition, Cambridge University Press, 2014.
6098 
6099 \bibitem{PC}
6100 Komorebi P., \emph{Projector Calculus: Resolution, Coordinate Atlases, Packet Arithmetic, Transfer Algebra, Residual Decomposition, and Infinite Completion}, final current edition, 2026.
6101 
6102 \end{thebibliography}
6103 
6104 \end{document}
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