Stratified Universality

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• block13_bridge_matrix.tex
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• block13_qary_examples_table.tex
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1 \section{Global invariant descent on the certified finite atlas}
2 \label{sec:block13-global-invariant}
3 
4 Global synthesis atlas performs the assembly step requested after the native stratum invariants.  The preceding sections do not justify one unrestricted equivalence among all eight broad strata.  They do, however, supply several finite connected systems of normalized charts for which the semantic object, overlap categories, bridge directions, information loss, coherence, and local completeness are explicit.  This section freezes those systems into one authoritative certified atlas, constructs the invariant comparison maps, verifies their higher coherence, and proves the strongest global completeness theorem supported by the assembled results.
5 
6 The global object in this section is deliberately a finite disjoint union of connected semantic components.  Each component contains nontrivial reversible bridges across different kinds of presentation.  The component tag is retained as part of the invariant.  No bridge is invented between two components merely because their formulas use similar words or symbols.  In particular, the functional-Mahler coefficient component and the continuous Newton--Mahler interpolation component remain distinct even though both belong to the broad local stratum.
7 
8 \subsection{The authoritative bridge matrix}
9 \label{subsec:block13-bridge-matrix}
10 
11 Write
12 \[
13  \mathsf E,
14  \quad
15  \mathsf{FF},
16  \quad
17  \mathsf D,
18  \quad
19  \mathsf P,
20  \quad
21  \varnothing
22 \]
23 for an equivalence on a named normalized overlap, a fully faithful inclusion, an information-losing directional functor, a partial or hypothesis-dependent bridge, and the absence of a certified broad bridge, respectively.  The matrix below records only relations already established in the preceding sections or in the audited source concordance.  A symbol carrying a subscript refers to the named overlap rather than to the whole broad stratum.
24 
25 \input{generated/block13_bridge_matrix.tex}
26 
27 The matrix has three immediate consequences.
28 
29 \begin{enumerate}
30 \item The reversible core is strictly smaller than the directed transformation category.
31 \item A bridge may become an equivalence after restricting to a normalized overlap without becoming invertible on the ambient stratum.
32 \item Empty cells are mathematical data.  They prevent later arguments from silently transporting an invariant through an uncertified route.
33 \end{enumerate}
34 
35 The entries \(E_{\mathrm{per}}\), \(E_{\mathrm{rm}}\), and \(E_{\mathrm{mark}}\) refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap.  In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate from neutral Tannakian reconstruction.
36 
37 The full object-level bridge ledger, including every normalized source, target, overlap, invariant comparison, and loss or gate, is retained as a machine-generated source artifact.  The printed matrix and the component table below contain the proof-relevant summary; a relation absent from the ledger is not used in the descent nerve.
38 
39 \subsection{Certified connected components}
40 \label{subsec:block13-components}
41 
42 \begin{definition}[Certified Global synthesis atlas atlas]
43 \label{def:block13-certified-atlas}
44 Fix a prime \(p\), a characteristic-zero field \(K\), and a finite Galois extension \(L/K\) for the Galois component.  The certified atlas \(\mathfrak A_{13}\) is the finite coproduct of the following five connected profiled atlases.
45 
46 \begin{enumerate}[label=\textup{(C\arabic*)}]
47 \item \(\mathfrak A_{\mathrm{DRLM},p}\), the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts;
48 \item \(\mathfrak A_{\mathrm{Loc},p}\), the continuous Newton--Mahler and complete clopen-branch component;
49 \item \(\mathfrak A_{\mathrm{Fuss}}\), the marked Fuss--Catalan nonlinear--algebraic--holonomic component;
50 \item \(\mathfrak A_{\mathrm{EM}}\), the fixed-Euler-basis and prime-uniform multiplicative component;
51 \item \(\mathfrak A_{\mathrm{Gal},L/K}\), the finite semisimple Galois component for the fixed extension \(L/K\).
52 \end{enumerate}
53 
54 The reversible arrows are exactly the normalized equivalences printed below.  Directional constructions, reductions, forgetful maps, and obstructed lifts are attached as satellites but do not enter the maximal subgroupoid used for invariant descent.
55 \end{definition}
56 
57 \input{generated/block13_component_table.tex}
58 
59 \begin{figure}[htbp]
60 \centering
61 \includegraphics[width=.99\textwidth]{global_certified_atlas.pdf}
62 \caption{The certified finite atlas.  Each box is a nontrivial connected system of normalized charts.  Dashed external arrows are directional, partial, or obstructed satellites and are not promoted to equivalences.}
63 \label{fig:block13-certified-atlas}
64 \end{figure}
65 
66 The semantic category of the total atlas is
67 \[
68  \mathcal O_{13}
69  =
70  \mathcal O_{\mathrm{DRLM}}
71  \sqcup
72  \mathcal O_{\mathrm{Loc}}
73  \sqcup
74  \mathcal O_{\mathrm{Fuss}}
75  \sqcup
76  \mathcal O_{\mathrm{EM}}
77  \sqcup
78  \mathcal O_{\mathrm{Gal}}.
79 \]
80 The coproduct symbol is literal.  It does not assert that a finite-field regular sequence, a continuous \(p\)-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyped category.  The tag records which semantic doctrine is in force.
81 
82 \subsection{Normalizer-induced higher coherence}
83 \label{subsec:block13-normalizer-coherence}
84 
85 Every connected component in \cref{def:block13-certified-atlas} has a canonical semantic normalizer.  For a chart \(\mathcal C_{\lambda,a}\) in component \(\lambda\), write
86 \[
87  N_{\lambda,a}:\mathcal C_{\lambda,a}\longrightarrow\mathcal O_\lambda
88 \]
89 for the semantic normalization functor and choose its canonical inverse chart
90 \[
91  Q_{\lambda,a}:\mathcal O_\lambda\longrightarrow\mathcal C_{\lambda,a}.
92 \]
93 The certified bridge from chart \(a\) to chart \(b\) is
94 \[
95  \Phi_{ab}=Q_{\lambda,b}N_{\lambda,a}.
96 \]
97 
98 \begin{theorem}[Normalizer-induced coherence]
99 \label{thm:block13-normalizer-coherence}
100 Assume that every normalized fiber of \(N_{\lambda,a}\) is contractible.  Then the comparison isomorphism
101 \[
102  \eta_{abc}:
103  \Phi_{bc}\Phi_{ab}
104  \Longrightarrow
105  \Phi_{ac}
106 \]
107 is unique.  Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the normalized chart.
108 \end{theorem}
109 
110 \begin{proof}
111 Both \(\Phi_{bc}\Phi_{ab}(X)\) and \(\Phi_{ac}(X)\) are normalized presentations of the same semantic object \(N_{\lambda,a}(X)\).  Contractibility of the normalized fiber gives a unique isomorphism between them.  Any two composites occurring in a coherence diagram are isomorphisms between the same two normalized presentations, so uniqueness makes them equal.  The cycle statement follows by applying the same argument to the starting chart and itself.
112 \end{proof}
113 
114 \begin{figure}[htbp]
115 \centering
116 \includegraphics[width=.93\textwidth]{normalizer_induced_coherence.pdf}
117 \caption{Every reversible chart in one certified component normalizes to the same semantic object.  The unique normalized comparison forces all finite bridge cycles to commute.}
118 \label{fig:block13-normalizer-coherence}
119 \end{figure}
120 
121 This theorem is not a substitute for constructing the normalizer.  It applies here because the preceding sections have already supplied unique reduced masks, reachable-observable Cartier modules, minimal automata, canonical section modules, marked fixed-point systems, selected algebraic branches, fixed Euler exponents, prime-power local factors, fixed subgroups, fixed fields, and faithful Reynolds projectors on the declared domains.
122 
123 \subsection{Invariant descent theorem}
124 \label{subsec:block13-invariant-descent}
125 
126 For every chart \(\mathcal C_{\lambda,a}\), let
127 \[
128  I_{\lambda,a}:\mathcal C_{\lambda,a}\longrightarrow\mathcal V_{\lambda,a}
129 \]
130 be its native normalized invariant.  A bridge comparison is a natural isomorphism
131 \[
132  \kappa_{ab}:
133  J_{ab}I_{\lambda,a}
134  \Longrightarrow
135  I_{\lambda,b}\Phi_{ab},
136 \]
137 where \(J_{ab}\) transports invariant coordinates between their chart-specific target categories.
138 
139 \begin{definition}[Coherent invariant family]
140 The family \((I_{\lambda,a},\kappa_{ab})\) is coherent when the two induced comparison maps from \(a\) to \(c\) agree on every triple overlap and the corresponding quadruple-overlap equation holds.
141 \end{definition}
142 
143 \begin{theorem}[Certified finite invariant descent]
144 \label{thm:block13-certified-invariant-descent}
145 Let \(\mathfrak A_{13}\) be the atlas of \cref{def:block13-certified-atlas}.  Suppose that, in every connected component:
146 \begin{enumerate}
147 \item the local invariant is complete on every normalized chart;
148 \item the bridge comparisons are coherent;
149 \item invariant-compatible local isomorphisms satisfy effective descent.
150 \end{enumerate}
151 Then the local invariants induce a functor
152 \[
153  \mathcal I_{13}:
154  \Desc(\mathfrak A_{13})
155  \longrightarrow
156  \Desc(\mathfrak V_{13})
157 \]
158 that reflects isomorphisms.  If each semantic component is discrete, equality of descended invariants is equivalent to equality of the represented semantic objects.
159 \end{theorem}
160 
161 \begin{proof}
162 The local comparison maps form a pseudonatural invariant family, so \cref{thm:atlas-assembly} gives the descended functor.  Equality or isomorphism of the descended invariant gives compatible local invariant matches.  Local completeness supplies local object isomorphisms.  Effective isomorphism descent glues those local isomorphisms to an isomorphism in the descent category.  On a discrete semantic component, every isomorphism is equality.
163 \end{proof}
164 
165 \begin{figure}[htbp]
166 \centering
167 \includegraphics[width=.95\textwidth]{global_invariant_descent.pdf}
168 \caption{The global invariant theorem has four separate requirements.  Local completeness alone is insufficient without comparison maps, higher coherence, and effective descent of the resulting isomorphisms.}
169 \label{fig:block13-invariant-descent}
170 \end{figure}
171 
172 \subsection{Digital--regular--linear--functional-Mahler component}
173 \label{subsec:block13-drlm}
174 
175 Fix the prime \(p\) and let
176 \[
177  \mathcal O_{\mathrm{DRLM}}
178  =\mathbf{Reg}_p(\mathbb F_p)
179 \]
180 be the discrete semantic category of \(p\)-regular sequences over \(\mathbb F_p\).  By the finite-field fixed-radix theorem, every object is also \(p\)-automatic.  Its globally defined normalized charts are the canonical section module, the strict shortlex pointed digit-linear record, and the minimal least-significant-digit-first \(p\)-DFAO.
181 
182 Two full replete subcategories carry additional charts.
183 \begin{enumerate}
184 \item On the rational-mask locus \(\RMah_p^{\times}\), the reduced functional-Mahler mask and the minimal reachable-observable Cartier module are equivalent to the digital charts.
185 \item On the unit purely periodic locus, the minimal recurrence, reduced rational OGF, minimal finite linear state, primitive period word, and rational-mask chart are all equivalent.
186 \end{enumerate}
187 
188 For \(a\in\mathcal O_{\mathrm{DRLM}}\), define
189 \[
190  \mathcal J_{\mathrm{DRLM},p}(a)
191  =
192  \left(
193  \mathsf{SL}_{p,\mathbb F_p}(a),
194  \mathcal P_{\mathrm{rm}}(a),
195  \mathcal P_{\mathrm{per}}(a)
196  \right),
197 \]
198 where \(\mathsf{SL}_{p,\mathbb F_p}(a)\) is the strict shortlex record, \(\mathcal P_{\mathrm{rm}}(a)\) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and \(\mathcal P_{\mathrm{per}}(a)\) is empty off the purely periodic locus and otherwise contains the primitive period word, exact period, minimal recurrence, reduced OGF, and periodic packet profile.
199 
200 \begin{theorem}[Digital--regular--linear--Mahler comparison theorem]
201 \label{thm:block13-drlm-comparison}
202 The invariant \(\mathcal J_{\mathrm{DRLM},p}\) is preserved by every reversible bridge in \(\mathfrak A_{\mathrm{DRLM},p}\).  Its comparison cells satisfy all higher coherence equations.  The strict shortlex coordinate reconstructs the represented regular sequence, so the descended invariant is complete on \(\mathbf{Reg}_p(\mathbb F_p)\).  The mask and periodic packages are compatible exact refinements on their respective subloci.
203 \end{theorem}
204 
205 \begin{proof}
206 The canonical section module, strict shortlex record, and minimal orbit automaton normalize to the same coefficient sequence by the Fixed-radix regular atlas effective-descent theorem.  On the rational-mask locus, fixed-denominator Cartier intertwining identifies the reachable numerator span with the canonical section module and the reachable numerator set with the minimal automaton states.  On the periodic locus, the recurrence, rational, state, automaton, and mask charts normalize to the same primitive coefficient word.  Every comparison therefore factors through the same semantic normalizer, and \cref{thm:block13-normalizer-coherence} applies.  Completeness follows from the strict shortlex reconstruction formula.
207 \end{proof}
208 
209 \input{generated/block13_periodic_examples.tex}
210 
211 The table gives representative objects for which recurrence order, regular rank, automaton size, and mask-compiler dimension are different coordinates of the same normalized semantic sequence.  Equality of one displayed scalar alone is not a complete invariant; the complete records are the chart objects described above.
212 
213 The inclusions of the rational-mask and periodic loci are not essentially surjective.  Over a finite field every regular sequence is automatic, but many automatic sequences have no rational canonical mask, and many rational-mask sequences are not periodic.  The optional fields in \(\mathcal J_{\mathrm{DRLM},p}\) record these exact domain boundaries rather than forcing one chart to exist everywhere.
214 
215 \subsection{Continuous Newton--Mahler component}
216 \label{subsec:block13-loc}
217 
218 Let
219 \[
220  \mathcal O_{\mathrm{Loc}}
221  =C(\mathbb Z_p,\mathbb Z_p)
222 \]
223 with equality morphisms.  Its canonical coordinate is the Mahler sequence
224 \[
225  \mathcal M_p(f)
226  =
227  (c_r(f))_{r\ge0},
228  \qquad
229  c_r(f)=\Delta^r f(0),
230  \qquad
231  c_r(f)\longrightarrow0.
232 \]
233 At every declared depth \(p^a\), the complete clopen branch packet
234 \[
235  \bigl(f(p^aq+s)\bigr)_{0\le s<p^a}
236 \]
237 and the corresponding branchwise Mahler coefficients are reversible by explicit piecewise synthesis.  Affine branch kernels and mixed-radix refinements are comparison charts of the same semantic function.
238 
239 \begin{theorem}[Continuous local invariant comparison]
240 \label{thm:block13-local-comparison}
241 The Mahler coefficient sequence
242 \[
243  \mathcal J_{\mathrm{Loc},p}(f)
244  =
245  (c_r(f))_{r\ge0}
246 \]
247 is a complete invariant of \(\mathfrak A_{\mathrm{Loc},p}\).  Complete branch packets, branchwise Mahler coordinates, affine coefficient kernels, and nested refinements admit coherent comparison maps to \(\mathcal J_{\mathrm{Loc},p}\).
248 \end{theorem}
249 
250 \begin{proof}
251 Mahler analysis and synthesis are mutually inverse on \(C(\mathbb Z_p,\mathbb Z_p)\).  Complete clopen analysis and piecewise synthesis are mutually inverse at each depth, and the nested branch law is strict.  Conjugating those maps by Mahler analysis gives the affine coefficient comparisons, whose mixed-radix composition was proved in Local Mahler-polyphase atlas.  All normalized charts therefore return the same function and the same Mahler sequence, so \cref{thm:block13-normalizer-coherence} gives higher coherence.
252 \end{proof}
253 
254 Functional-Mahler equations for coefficient series are not included in this component.  A finite-state or functional-Mahler object enters the Newton--Mahler component only through a separately declared lift and interpolation constructor carrying its own precision and information-loss certificate.
255 
256 \subsection{A genuine nonlinear--algebraic--holonomic overlap}
257 \label{subsec:block13-fuss}
258 
259 Let \(K\) be a characteristic-zero field.  For an integer \(q\geq2\) and \(\alpha\in K^\times\), let
260 \[
261  U_{q,\alpha}(z)\in zK[[z]]
262 \]
263 be the unique selected branch satisfying
264 \begin{equation}
265  U=z+\alpha U^q.
266  \label{eq:block13-fuss-tree}
267 \end{equation}
268 The derivative of
269 \[
270  P_{q,\alpha}(z,Y)=Y-z-\alpha Y^q
271 \]
272 with respect to \(Y\) is one at \((0,0)\), so the branch is simple and selected by \(U(0)=0\).
273 
274 Put
275 \[
276  x=z^{q-1},
277  \qquad
278  U_{q,\alpha}(z)=zC_{q,\alpha}(x).
279 \]
280 Then
281 \begin{equation}
282  C=1+\alpha xC^q.
283  \label{eq:block13-fuss-compressed}
284 \end{equation}
285 Lagrange inversion gives
286 \begin{equation}
287  [x^n]C_{q,\alpha}(x)
288  =
289  c_n(q,\alpha)
290  =
291  \frac{\alpha^n}{(q-1)n+1}
292  \binom{qn}{n}.
293  \label{eq:block13-fuss-coefficients}
294 \end{equation}
295 Define
296 \[
297  A_q(n)
298  =(n+1)\prod_{r=2}^{q}\bigl((q-1)n+r\bigr),
299 \]
300 \[
301  B_q(n)
302  =\prod_{r=1}^{q}(qn+r).
303 \]
304 Then the coefficient sequence satisfies the first-order polynomial recurrence
305 \begin{equation}
306  \boxed{
307  A_q(n)c_{n+1}
308  =
309  \alpha B_q(n)c_n.}
310  \label{eq:block13-fuss-recurrence}
311 \end{equation}
312 After dividing the polynomial gcd of \(A_q\) and \(B_q\), this is the normalized first-order tail recurrence with initial value \(c_0=1\).
313 
314 \begin{theorem}[Fuss--Catalan three-stratum equivalence]
315 \label{thm:block13-fuss-equivalence}
316 The following normalized marked presentations determine one another exactly:
317 \begin{enumerate}
318 \item the polynomial tree system \(U=z+\alpha U^q\);
319 \item the selected algebraic branch \((P_{q,\alpha},U(0)=0)\);
320 \item the compressed P-recursive coefficient chart \((\cref{eq:block13-fuss-recurrence},c_0=1)\) together with the support embedding \(U(z)=zC(z^{q-1})\).
321 \end{enumerate}
322 Every normalized fiber is contractible.  All triangle comparisons return the same parameter pair \((q,\alpha)\).
323 \end{theorem}
324 
325 \begin{proof}
326 The marked tree equation is the selected algebraic equation.  Simplicity at the origin gives its unique formal branch.  Substitution \(U=zC(z^{q-1})\) gives \cref{eq:block13-fuss-compressed}.  Lagrange inversion gives \cref{eq:block13-fuss-coefficients}, and taking the ratio of consecutive coefficients gives \cref{eq:block13-fuss-recurrence}.  Conversely, the recurrence and initial value reconstruct \(C\), the support embedding reconstructs \(U\), and the reconstructed series satisfies the selected algebraic equation.  A normalized automorphism fixes \(q\), \(\alpha\), and the unique branch, hence is the identity.
327 \end{proof}
328 
329 \begin{figure}[htbp]
330 \centering
331 \includegraphics[width=.87\textwidth]{fuss_catalan_three_stratum.pdf}
332 \caption{The marked Fuss--Catalan family gives a genuine overlap among the nonlinear tree, selected algebraic, and holonomic recurrence strata.  Every path returns the same marked pair \((q,\alpha)\).}
333 \label{fig:block13-fuss-triangle}
334 \end{figure}
335 
336 The tree arity period is \(q-1\).  For packet modulus \(N\), set
337 \[
338  g_N=\gcd(N,q-1).
339 \]
340 The nonzero coefficient addresses of \(U_{q,\alpha}\) are exactly
341 \[
342  j\equiv1\pmod{g_N},
343 \]
344 and the complete dominant singularity orbit has size
345 \[
346  \frac{q-1}{g_N}
347 \]
348 under the positive analytic hypotheses of \cref{thm:block11-qary-phase}.  These packet signatures agree in the tree and algebraic charts because both are computed from the same selected branch.
349 
350 Define
351 \[
352  \mathcal J_{\mathrm{Fuss}}(q,\alpha)
353  =
354  \left(
355  q,
356  \alpha,
357  P_{q,\alpha},
358  \{c_n(q,\alpha)\}_{n\geq0},
359  \{g_N\}_{N\geq1},
360  \left\{\frac{q-1}{g_N}\right\}_{N\geq1}
361  \right).
362 \]
363 
364 \begin{corollary}[Complete cross-stratum invariant on the marked Fuss family]
365 \label{cor:block13-fuss-invariant}
366 The nonlinear, algebraic, holonomic, packet-support, and dominant-orbit invariants admit coherent comparison maps to \(\mathcal J_{\mathrm{Fuss}}\).  The package is complete on the declared marked family.
367 \end{corollary}
368 
369 \begin{proof}
370 Coherence follows from \cref{thm:block13-fuss-equivalence,thm:block13-normalizer-coherence}.  The first two entries reconstruct the marked fixed-point equation and therefore its unique selected branch.
371 \end{proof}
372 
373 \input{generated/block13_qary_examples_table.tex}
374 
375 The examples show that the finite invariant retains the weight parameter as well as the arity.  Support period alone recovers only \(q-1\); the coefficient \([z^q]U=\alpha\) supplies the second coordinate.
376 
377 This is not a classification of arbitrary algebraic branches, arbitrary P-recursive sequences, or unmarked nonlinear fixed points.  It is the first explicit complete invariant on a connected overlap of those three strata.
378 
379 \subsection{Euler--multiplicative component}
380 \label{subsec:block13-em}
381 
382 Let
383 \[
384  A_b(z)=\prod_{m\geq1}(1-z^m)^{-b_m}
385 \]
386 be an integral unit series in the fixed Euler basis, and let
387 \[
388  c_n=\sum_{d\mid n}d\,b_d
389 \]
390 be its divisor-ghost sequence.  Define the prime-uniform multiplicative function
391 \[
392  f_b(n)=\prod_{p^e\parallel n}[z^e]A_b(z).
393 \]
394 Every prime has the same local factor \(A_b(T)\).  By \cref{thm:block07-euler-classification,thm:block07-product-multiplicative-descent}, the coefficient, exponent, ghost, and prime-uniform multiplicative charts are reversible on this declared locus.
395 
396 \begin{theorem}[Euler--multiplicative invariant comparison]
397 \label{thm:block13-em-comparison}
398 The exponent sequence
399 \[
400  \mathcal J_{\mathrm{EM}}(A_b)=b
401 \]
402 is a complete invariant of the certified Euler--multiplicative component.  In every chart it is recovered by the same M\"obius formula
403 \[
404  b_n
405  =
406  \frac1n
407  \sum_{d\mid n}\mu(n/d)c_d,
408 \]
409 and Dirichlet convolution corresponds to exponent addition.
410 \end{theorem}
411 
412 \begin{proof}
413 Unique Euler factorization reconstructs \(A_b\) from \(b\).  Prime-power reconstruction then reconstructs \(f_b\).  The logarithmic-derivative identity gives \(c\), and M\"obius inversion recovers \(b\).  Every bridge therefore factors through the same exponent normalizer, so higher coherence follows from \cref{thm:block13-normalizer-coherence}.
414 \end{proof}
415 
416 The theorem remains confined to the fixed Euler basis and the prime-uniform multiplicative essential image.  It does not identify arbitrary Cauchy-product grammars with arbitrary Dirichlet-convolution systems.
417 
418 \subsection{Finite Galois component}
419 \label{subsec:block13-gal}
420 
421 Fix a finite Galois extension \(L/K\) in the semisimple setting.  For each subgroup \(H\leq G\), define
422 \[
423  \mathcal J_{\mathrm{Gal}}(H)
424  =
425  \left(
426  H,
427  L^H,
428  P_H,
429  \operatorname{Core}_G(H),
430  \operatorname{Ncl}_G(H),
431  \{J\backslash G/H\}_{J\leq G}
432  \right).
433 \]
434 
435 \begin{theorem}[Finite Galois invariant comparison]
436 \label{thm:block13-gal-comparison}
437 The subgroup, fixed-field, and faithful Reynolds-projector charts have coherent comparison maps to \(\mathcal J_{\mathrm{Gal}}\).  The invariant is complete on the normalized subgroup atlas for the fixed extension \(L/K\).
438 \end{theorem}
439 
440 \begin{proof}
441 The first three entries determine one another by \cref{thm:block12-normalized-galois-chart}.  Core, normal closure, and double-coset packets are functorial constructions from the subgroup.  The subgroup coordinate itself is complete, and the strict subgroup normalizer forces coherence.
442 \end{proof}
443 
444 The Brauer class and Tannakian group scheme are not added to this complete subgroup invariant as if they were automatic coordinates.  They belong to separately typed projective and tensor extensions whose hypotheses and obstructions were stated in Galois-Tannakian atlas.
445 
446 \subsection{The global tagged invariant}
447 \label{subsec:block13-global-tagged}
448 
449 Let
450 \[
451  \Lambda_{13}
452  =
453  \{\mathrm{DRLM},\mathrm{Loc},\mathrm{Fuss},\mathrm{EM},\mathrm{Gal}\}
454 \]
455 be the component-tag set.  Define
456 \[
457  \mathcal J_{13}(X)
458  =
459  \bigl(\lambda(X),\mathcal J_{\lambda(X)}(X)\bigr).
460 \]
461 
462 \begin{theorem}[Certified global completeness theorem]
463 \label{thm:block13-global-completeness}
464 The invariant \(\mathcal J_{13}\) descends across every reversible bridge in \(\mathfrak A_{13}\) and is complete on
465 \[
466  \mathcal O_{13}
467  =
468  \mathcal O_{\mathrm{DRLM}}
469  \sqcup
470  \mathcal O_{\mathrm{Loc}}
471  \sqcup
472  \mathcal O_{\mathrm{Fuss}}
473  \sqcup
474  \mathcal O_{\mathrm{EM}}
475  \sqcup
476  \mathcal O_{\mathrm{Gal}}.
477 \]
478 More precisely,
479 \[
480  \boxed{
481  \mathcal J_{13}(X)=\mathcal J_{13}(Y)
482  \quad\Longleftrightarrow\quad
483  X=Y
484  }
485 \]
486 inside the declared discrete semantic category.  The associated invariant functor reflects isomorphisms on the descent category.
487 \end{theorem}
488 
489 \begin{proof}
490 Within each component, completeness is \cref{thm:block13-drlm-comparison,thm:block13-local-comparison,cor:block13-fuss-invariant,thm:block13-em-comparison,thm:block13-gal-comparison}.  Their comparison maps are coherent by \cref{thm:block13-normalizer-coherence}.  The component tag prevents an object in one semantic doctrine from being identified with an object in another.  Apply \cref{thm:block13-certified-invariant-descent} componentwise and then take the finite coproduct.
491 \end{proof}
492 
493 \begin{remark}[What is genuinely global]
494 The theorem is not the tautology that every object is classified by itself.  Each component invariant is computed in one native chart and transported through several nonidentical presentation categories.  The global assertion is that these transported coordinates agree coherently and classify the descent object.  The coproduct tag is nevertheless essential: the current Library does not provide reversible bridges joining all five components.
495 \end{remark}
496 
497 \subsection{Directional satellites and exact loss records}
498 \label{subsec:block13-directional-satellites}
499 
500 The certified reversible core is surrounded by directional bridges that remain useful for theorem transport and computation.  Their failure to enter \(\mathcal J_{13}\) is controlled by explicit loss data.
501 
502 \begin{table}[htbp]
503 \centering
504 \caption{Directional satellites of the certified global invariant atlas.}
505 \label{tab:block13-directional-satellites}
506 \small
507 \begin{tabularx}{\textwidth}{@{}L{0.23\textwidth}L{0.19\textwidth}Y@{}}
508 \toprule
509 Directional bridge & Retained structure & Exact loss or obstruction\\
510 \midrule
511 Integral regular lattice \(\to\) residue section module & digit transitions and residue semantics & Smith--Tor ghost kernel and missing quotient\\
512 General regular sequence \(\dashrightarrow\) rational mask & coefficient semantics on the essential image & rational-mask existence and collapse obstruction\\
513 Algebraic branch \(\to\) holonomic annihilator & D-finite coefficient semantics & selected branch and minimal polynomial may be forgotten\\
514 Marked nonlinear system \(\to\) semantic fixed point & the represented fixed point & generator labels, arities, and tree grammar may be forgotten\\
515 Finite-state dynamics \(\to\) Newton--Mahler chart & congruence branch values and precision ladder & depends on a chosen lift and principal-unit hypothesis\\
516 Projective Galois cocycle \(\dashrightarrow\) linear descent & projective action & relative Brauer class\\
517 Faithful corner realization \(\dashrightarrow\) Tannakian symmetry & additive packet and operator equations & exact tensor, dual, unit, and fiber-functor data are absent\\
518 \bottomrule
519 \end{tabularx}
520 \end{table}
521 
522 \begin{proposition}[No completeness transport through an unreflected directional bridge]
523 \label{prop:block13-no-directional-completeness}
524 Let \(T:\mathcal C\to\mathcal D\) be a directional bridge.  A complete invariant on \(\mathcal D\) pulls back to a complete invariant on \(\mathcal C\) only if \(T\) reflects isomorphisms on the declared source subcategory.  In particular, any nontrivial object-collapse relation or obstruction fiber prevents automatic completeness transport.
525 \end{proposition}
526 
527 \begin{proof}
528 If \(T\) does not reflect isomorphisms, there exist nonisomorphic \(X,Y\) with \(T(X)\cong T(Y)\).  Every invariant depending only on the target object then takes the same value on \(X\) and \(Y\), so its pullback cannot be complete.
529 \end{proof}
530 
531 \subsection{Exact finite verification}
532 \label{subsec:block13-verification}
533 
534 Two complementary exact suites audit Global synthesis atlas.  The first authenticates the frozen preceding verification records and checks the broad bridge matrix, periodic and product--multiplicative overlap replays, directional-loss controls, finite \v{C}ech holonomy models, and the five completeness gates.  The second checks the weighted \(q\)-ary tree--algebraic--holonomic overlap, its coefficient and packet equations, parameter recovery, detailed bridge ledger, and tagged component separation.
535 
536 \input{generated/block13_validation_table.tex}
537 
538 \input{generated/block13_combined_validation_table.tex}
539 
540 The two suites execute \(\BlockThirteenChecks\) exact assertions with no failures.  Of these, \(\BlockThirteenRegressionChecks\) authenticate or replay assertions already represented in the preceding cumulative record.  The global synthesis therefore contributes \(\BlockThirteenNewChecks\) nonduplicative comparison, coherence, packet, and loss-control assertions.  The blockwise cumulative record is
541 \[
542  \boxed{\BlockThirteenCumulative}
543 \]
544 exact assertions with no recorded failures.  Counting every regression assertion executed during the Global synthesis atlas replay gives \(\BlockThirteenCumulativeExecuted\) total executed assertions.  Neither number is a count of independent theorems.
545 
546 The bridge-synthesis suite replays primitive unit-periodic words over finite prime fields, compares their recurrence, OGF, automaton, section-module, and functional-Mahler coordinates, verifies the prime-uniform Euler--multiplicative overlap, and tests finite nontrivial \v{C}ech cycle models.  The weighted \(q\)-ary suite checks the fixed-point identity, Lagrange coefficients, first-order P-recursive recurrence, support phases, exact packet synthesis, address-carry equations, and all triangle normalizers over a declared finite parameter grid.  Both machine-readable records are included in the source package.
547 
548 \begin{boundarybox}
549 \textbf{Exact boundary of Global synthesis atlas.}
550 The global invariant theorem is proved for the five-component certified finite atlas in \cref{def:block13-certified-atlas}.  It does not assert a complete invariant for every object in the eight broad strata, a reversible bridge between the five semantic components, effective descent for every directional satellite, cross-radix or all-ring normalization, modular or wild Galois effectivity, nonneutral Tannakian reconstruction, or classification of unmarked general holonomic, algebraic, and nonlinear systems.
551 \end{boundarybox}
552 
553 \begin{resultbox}
554 \textbf{Section result.}
555 The manuscript now contains an authoritative broad-stratum bridge matrix, a finite certified atlas with five nontrivial connected components, explicit invariant comparison maps, a normalizer-induced higher-coherence theorem, and a complete tagged global invariant on the declared semantic union.  The new Fuss--Catalan triangle gives a genuine nonlinear--algebraic--holonomic overlap.  The digital--regular--functional-Mahler, continuous Newton--Mahler, Euler--multiplicative, and finite Galois components consolidate the exact cross-presentation results of the preceding stratum sections without conflating the two Mahler theories.  Directional bridges remain outside the reversible core and retain their precise loss or obstruction records.  This closes the finite global-invariant milestone on a substantial atlas, not the unrestricted eight-stratum stack.
556 \end{resultbox}

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