Stratified Universality

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1 \clearpage
2 \section{Holonomy-twisted lane transport and orbit descent}
3 \label{sec:block16-holonomy-twisted-lane-transport}
4 
5 The finite atlas theorem in \cref{thm:block15-lane-descent} imposed trivial
6 address holonomy in order to obtain one globally fixed set of lane labels.  That
7 hypothesis is sufficient for ordinary globally addressed lanes, but it is
8 stronger than the hypothesis needed for transport itself.  When addresses are
9 permuted around atlas loops, the local lane matrices still glue.  They glue as
10 sections of a finite matrix local system rather than as one matrix with a fixed
11 global row and column labeling.
12 
13 This section isolates that distinction.  The local analysis, synthesis,
14 composition, refinement, selector, and residual laws survive arbitrary finite
15 address holonomy.  The holonomy orbits then give the finest ordinary global
16 coarsening obtainable by summing the original lane projectors.  Thus nontrivial
17 holonomy is an obstruction to untwisted labels, not an obstruction to the
18 finite transport calculus.
19 
20 \subsection{Holonomy-profiled lane data}
21 \label{subsec:block16-holonomy-profiled-data}
22 
23 Let \(\mathscr G\) be a finite connected groupoid of charts.  For an arrow
24 \(\gamma:U\to V\), let
25 \[
26  F_\gamma:\mathcal C_U\longrightarrow\mathcal C_V
27 \]
28 be an additive equivalence between additive idempotent-complete categories.
29 The transition functors carry specified identity and composition comparisons.
30 To keep the formulas readable, those comparison isomorphisms are suppressed;
31 every displayed cocycle equation is understood after inserting the declared
32 comparisons.
33 
34 A descent object
35 \[
36  A=(A_U,\theta^A_\gamma)
37 \]
38 consists of objects \(A_U\in\mathcal C_U\) and isomorphisms
39 \[
40  \theta^A_\gamma:F_\gamma(A_U)\xrightarrow{\sim}A_V
41 \]
42 satisfying the identity and cocycle equations.  A descended morphism
43 \(f:X\to Y\) is a compatible family \((f_U)_U\) satisfying
44 \[
45  \theta^Y_\gamma F_\gamma(f_U)=f_V\theta^X_\gamma.
46 \]
47 
48 \begin{definition}[Holonomy-profiled finite lane datum]
49 \label{def:block16-holonomy-profiled-lane-datum}
50 Fix a finite resolution shape \(\mathsf R\).  A holonomy-profiled finite lane
51 datum on a descent object \(A\) consists of the following information.
52 
53 For every chart \(U\) and resolution \(r\in\mathsf R\), the object \(A_U\)
54 has a finite profiled lane system
55 \[
56  \left(I_{A,U,r},p^{r}_{A,U,i},A^r_{U,i},
57  \iota^r_{A,U,i},\pi^r_{A,U,i}\right)_{i\in I_{A,U,r}}
58 \]
59 in the sense of \cref{def:block15-finite-profiled-lane-system}.  For every
60 arrow \(\gamma:U\to V\), there is an address bijection
61 \[
62  \lambda^A_{\gamma,r}:I_{A,U,r}\xrightarrow{\sim}I_{A,V,r}
63 \]
64 and, for every address \(i\), a splitting comparison
65 \[
66  \beta^A_{\gamma,r,i}:
67  F_\gamma(A^r_{U,i})\xrightarrow{\sim}
68  A^r_{V,\lambda^A_{\gamma,r}(i)}.
69 \]
70 They satisfy the address and splitting cocycle laws and the two compatibility
71 identities
72 \[
73  \theta^A_\gamma F_\gamma(\iota^r_{A,U,i})
74  =
75  \iota^r_{A,V,\lambda^A_{\gamma,r}(i)}
76  \beta^A_{\gamma,r,i},
77 \]
78 \[
79  \beta^A_{\gamma,r,i}F_\gamma(\pi^r_{A,U,i})
80  =
81  \pi^r_{A,V,\lambda^A_{\gamma,r}(i)}\theta^A_\gamma.
82 \]
83 For \(s\succeq r\), the refinement maps are equivariant:
84 \[
85  \rho^{A,V}_{sr}\lambda^A_{\gamma,s}
86  =
87  \lambda^A_{\gamma,r}\rho^{A,U}_{sr}.
88 \]
89 No global address set and no global trivialization of the address bijections is
90 assumed.
91 \end{definition}
92 
93 \begin{definition}[Matrix local system]
94 \label{def:block16-matrix-local-system}
95 Let \(X,Y\) carry holonomy-profiled lane data.  At source resolution \(r\)
96 and target resolution \(u\), define the matrix group over a chart \(U\) by
97 \[
98  \mathscr M^{u,r}_{Y,X}(U)
99  :=
100  \bigoplus_{a\in I_{Y,U,u}}
101  \bigoplus_{i\in I_{X,U,r}}
102  \Hom_{\mathcal C_U}(X^r_{U,i},Y^u_{U,a}).
103 \]
104 For \(h=(h_{ai})\in\mathscr M^{u,r}_{Y,X}(U)\), define transport along
105 \(\gamma:U\to V\) by
106 \[
107  \left(\gamma_*h\right)_{
108  \lambda^Y_{\gamma,u}(a),\lambda^X_{\gamma,r}(i)}
109  :=
110  \beta^Y_{\gamma,u,a}
111  F_\gamma(h_{ai})
112  \left(\beta^X_{\gamma,r,i}\right)^{-1}.
113 \]
114 The group of compatible sections is
115 \[
116  \Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right)
117  :=
118  \left\{(h_U)_U:\gamma_*h_U=h_V
119  \text{ for every }\gamma:U\to V\right\}.
120 \]
121 \end{definition}
122 
123 \subsection{The holonomy-twisted transport theorem}
124 \label{subsec:block16-main-theorem}
125 
126 \begin{theorem}[Holonomy-Twisted Universal Lane Transport Theorem]
127 \label{thm:block16-holonomy-twisted-lane-transport}
128 Let \(X,Y,Z\) be descent objects over \(\mathscr G\), each equipped with a
129 holonomy-profiled finite lane datum over the same finite resolution shape.
130 Then the following statements hold.
131 
132 \begin{enumerate}[label=\textup{(H\arabic*)}]
133 \item \textbf{Matrix local system.}
134 The maps \(\gamma_*\) of
135 \cref{def:block16-matrix-local-system} are additive isomorphisms and satisfy
136 the identity and cocycle laws.  Hence
137 \(\mathscr M^{u,r}_{Y,X}\) is a local system of abelian groups with finitely many matrix positions in every chart.
138 
139 \item \textbf{Exact twisted analysis and synthesis.}
140 Lane analysis induces an isomorphism
141 \[
142  \mathfrak A^{u,r}_{Y,X}:
143  \Hom_{\Desc(\mathscr G)}(X,Y)
144  \xrightarrow{\sim}
145  \Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right),
146  \qquad
147  f\longmapsto
148  \left(\bigl([f_U]^{u,r}_{ai}\bigr)_{a,i}\right)_U.
149 \]
150 Its inverse is chartwise lane synthesis:
151 \[
152  (h_U)_U\longmapsto
153  \left(
154  \sum_{a,i}
155  \iota^u_{Y,U,a}h_{U,ai}\pi^r_{X,U,i}
156  \right)_U.
157 \]
158 Thus nontrivial address holonomy does not destroy the complete matrix record.
159 It changes the global type of that record from a constantly labeled matrix to
160 a compatible section of a matrix local system.
161 
162 \item \textbf{Twisted matrix composition.}
163 For
164 \[
165  h\in\Gamma(\mathscr G,\mathscr M^{u,r}_{Y,X}),
166  \qquad
167  k\in\Gamma(\mathscr G,\mathscr M^{v,u}_{Z,Y}),
168 \]
169 define chartwise contraction by
170 \[
171  (k\star h)_{U,ci}
172  :=
173  \sum_{a\in I_{Y,U,u}}k_{U,ca}h_{U,ai}.
174 \]
175 Then \(k\star h\) is a compatible section of
176 \(\mathscr M^{v,r}_{Z,X}\), and
177 \[
178  \mathfrak A^{v,r}_{Z,X}(gf)
179  =
180  \mathfrak A^{v,u}_{Z,Y}(g)
181  \star
182  \mathfrak A^{u,r}_{Y,X}(f).
183 \]
184 The diagonal identity matrices are compatible sections.  More precisely, fix
185 one declared resolution \(r_A\) for each lane-decorated descent object \(A\).
186 The category with the same objects, morphism groups
187 \[
188  \Gamma\!\left(\mathscr G,
189  \mathscr M^{r_B,r_A}_{B,A}\right),
190 \]
191 and composition \(\star\) is isomorphic, by analysis and synthesis, to the
192 corresponding category of lane-decorated descent objects and their underlying
193 descended morphisms.
194 
195 \item \textbf{Equivariant refinement.}
196 Every certified fine-to-coarse coordinate change from
197 \cref{prop:block15-certified-coarsening} commutes with the transition maps
198 \(\gamma_*\).  It therefore defines an isomorphism of matrix local systems.
199 These isomorphisms compose strictly over chains of declared resolutions and
200 intertwine twisted matrix contraction.
201 
202 \item \textbf{Holonomy-compatible selectors and residuals.}
203 Let
204 \[
205  E_U\subseteq I_{Y,U,u}\times I_{X,U,r}
206 \]
207 be a family satisfying
208 \[
209  E_V=
210  (\lambda^Y_{\gamma,u}\times\lambda^X_{\gamma,r})(E_U)
211 \]
212 for every arrow \(\gamma:U\to V\).  Then the local selectors
213 \[
214  C_{E_U}(h_U)_{ai}
215  =
216  \begin{cases}
217  h_{U,ai},&(a,i)\in E_U,\\
218  0,&(a,i)\notin E_U
219  \end{cases}
220 \]
221 assemble to an idempotent endomorphism \(C_E\) of the section group.  For two
222 compatible support families \(E,F\),
223 \[
224  C_EC_F=C_{E\cap F},
225  \qquad
226  R_E:=1-C_E,
227  \qquad
228  h=C_Eh+R_Eh.
229 \]
230 Hence named residuals remain exact under arbitrary finite address holonomy,
231 provided the retained support itself is holonomy compatible.
232 \end{enumerate}
233 \end{theorem}
234 
235 \begin{proof}
236 For (H1), additivity and invertibility follow from the corresponding
237 properties of \(F_\gamma\) and the splitting comparisons \(\beta\).  The
238 address and splitting cocycle laws give
239 \((\delta\gamma)_*=\delta_*\gamma_*\) after the declared transition
240 comparisons.
241 
242 For (H2), let \(f=(f_U)_U\) be descended.  For a transition
243 \(\gamma:U\to V\), the transported \((a,i)\)-block is
244 \[
245  \begin{aligned}
246  &\beta^Y_{\gamma,u,a}
247  F_\gamma\!\left(
248  \pi^u_{Y,U,a}f_U\iota^r_{X,U,i}
249  \right)
250  \left(\beta^X_{\gamma,r,i}\right)^{-1}\\
251  &\quad=
252  \pi^u_{Y,V,\lambda^Y(a)}
253  \theta^Y_\gamma F_\gamma(f_U)(\theta^X_\gamma)^{-1}
254  \iota^r_{X,V,\lambda^X(i)}\\
255  &\quad=
256  \pi^u_{Y,V,\lambda^Y(a)}
257  f_V
258  \iota^r_{X,V,\lambda^X(i)},
259  \end{aligned}
260 \]
261 which is the corresponding block of \(f_V\).  Thus analysis produces a
262 compatible section.
263 
264 Conversely, let \(h=(h_U)_U\) be a compatible section and synthesize
265 \[
266  f_U:=\sum_{a,i}\iota^u_{Y,U,a}h_{U,ai}\pi^r_{X,U,i}.
267 \]
268 Transporting this sum, applying the splitting comparison identities, and
269 reindexing through the address bijections gives \(f_V\).  Thus \((f_U)_U\) is
270 a descended morphism.  Chartwise analysis and synthesis are mutually inverse
271 by \cref{thm:block15-universal-lane-transport}; therefore their restrictions
272 to compatible families are mutually inverse.
273 
274 For (H3), the local path law gives the displayed contraction formula in every
275 chart.  Applying \(\gamma_*\) to a contracted block distributes over the finite
276 sum and carries each product to the product of the transported blocks.  Since
277 \(\lambda^Y_{\gamma,u}\) is a bijection, reindexing the intermediate address
278 shows
279 \[
280  \gamma_*(k_U\star h_U)
281  =
282  (\gamma_*k_U)\star(\gamma_*h_U)
283  =
284  k_V\star h_V.
285 \]
286 The identity section is compatible by the splitting cocycle.  The equivalence
287 then follows from (H2).
288 
289 For (H4), the local fine-to-coarse coordinate map is analysis after synthesis.
290 Both operations commute with transition by (H2), so their composite does as
291 well.  The strict composition and contraction identities are the corresponding
292 chartwise identities from
293 \cref{cor:block15-functorial-coarsening,thm:block15-matrix-category-equivalence}.
294 
295 For (H5), support compatibility implies
296 \(\gamma_*C_{E_U}=C_{E_V}\gamma_*\), so the selectors preserve compatible
297 sections.  Every Boolean and residual identity is then checked chartwise.
298 \end{proof}
299 
300 
301 \begin{corollary}[Base-fiber invariant normal form]
302 \label{cor:block16-base-fiber-invariant-normal-form}
303 Fix a base chart \(U_0\).  Evaluation at \(U_0\) induces an isomorphism
304 \[
305  \Gamma\!\left(\mathscr G,\mathscr M^{u,r}_{Y,X}\right)
306  \xrightarrow{\sim}
307  \mathscr M^{u,r}_{Y,X}(U_0)^{\Aut_{\mathscr G}(U_0)},
308 \]
309 where the right-hand side is the subgroup fixed by every loop-transport map.
310 Consequently,
311 \[
312  \Hom_{\Desc(\mathscr G)}(X,Y)
313  \cong
314  \mathscr M^{u,r}_{Y,X}(U_0)^{\Aut_{\mathscr G}(U_0)}.
315 \]
316 After choosing one arrow \(\tau_U:U_0\to U\) for each chart, the inverse sends
317 a fixed base matrix \(h_0\) to the section
318 \(h_U=(\tau_U)_*h_0\).  The result is independent of the chosen arrows because
319 two choices differ by loop transport at \(U_0\).
320 \end{corollary}
321 
322 \begin{proof}
323 A compatible section evaluates to a loop-fixed element at \(U_0\).  Conversely,
324 for an arrow \(\gamma:U\to V\), the two arrows
325 \(\gamma\tau_U\) and \(\tau_V\) differ by a loop at \(U_0\); loop invariance
326 therefore gives \(\gamma_*h_U=h_V\).  Evaluation and this reconstruction are
327 mutually inverse.  Compose with the analysis isomorphism in (H2).
328 \end{proof}
329 
330 \begin{remark}[What has and has not been removed]
331 \label{rem:block16-removed-triviality-only}
332 The theorem removes only the global address trivializations
333 \(\mu_U^{A,r}\) from \cref{thm:block15-lane-descent}.  It does not remove the
334 address cocycle, refinement naturality, projector compatibility, splitting
335 comparisons, or additive preservation hypotheses.  Without those data, even
336 the matrix transition maps are undefined or fail their cocycle law.
337 \end{remark}
338 
339 \subsection{Monodromy orbits and the finest untwisted coarsening}
340 \label{subsec:block16-orbit-descent}
341 
342 Fix a base chart \(U_0\).  Every loop \(\gamma\in\Aut_{\mathscr G}(U_0)\)
343 acts on an address set by \(i\mapsto\lambda^A_{\gamma,r}(i)\).  Write
344 \[
345  H_A^r
346  :=
347  \operatorname{im}\!\left(
348  \Aut_{\mathscr G}(U_0)
349  \longrightarrow
350  \operatorname{Sym}(I_{A,U_0,r})
351  \right)
352 \]
353 for the resulting finite monodromy group.  For a source-target pair, the same
354 loop acts diagonally by
355 \[
356  (a,i)\longmapsto
357  \left(\lambda^Y_{\gamma,u}(a),
358  \lambda^X_{\gamma,r}(i)\right).
359 \]
360 Its image in the symmetric group of
361 \(I_{Y,U_0,u}\times I_{X,U_0,r}\) is denoted
362 \(H_{Y,X}^{u,r}\).  This is the simultaneous monodromy image; it need not be
363 the direct product of the two separate monodromy groups.
364 
365 \begin{theorem}[Orbit descent and the finest globally labeled coarsening]
366 \label{thm:block16-orbit-descent}
367 Under the hypotheses of
368 \cref{thm:block16-holonomy-twisted-lane-transport}, the following statements
369 hold.
370 
371 \begin{enumerate}[label=\textup{(O\arabic*)}]
372 \item For every orbit \(\mathcal O\subseteq I_{A,U_0,r}\) of \(H_A^r\), path
373 transport defines a path-independent orbit subset
374 \(\mathcal O_U\subseteq I_{A,U,r}\) in every chart.  The local projector sums
375 \[
376  P^r_{A,U,\mathcal O}
377  :=
378  \sum_{i\in\mathcal O_U}p^r_{A,U,i}
379 \]
380 descend to a split idempotent on \(A\).  As \(\mathcal O\) ranges over the
381 monodromy orbits, these projectors are pairwise annihilating and sum to
382 \(1_A\).
383 
384 \item The orbit projectors form the finest ordinary globally labeled
385 coarsening whose local projectors are sums of the original lane projectors.
386 More precisely, any globally labeled projector of the form
387 \[
388  Q_U=\sum_{i\in S_U}p^r_{A,U,i}
389 \]
390 with \(S_V=\lambda^A_{\gamma,r}(S_U)\) and with the same global label preserved
391 around every loop has \(S_{U_0}\) equal to a union of \(H_A^r\)-orbits.
392 
393 \item A particular local lane descends as an individually labeled global lane
394 exactly when its address is fixed by \(H_A^r\).  All original lane labels
395 descend simultaneously exactly when the monodromy action on the address set is
396 trivial.
397 
398 \item Holonomy-compatible support families in (H5) are in bijection with
399 unions of \(H_{Y,X}^{u,r}\)-orbits in the base pair-address set.  The minimal
400 nonempty compatible supports are therefore the single pair orbits.  Their
401 associated selectors need not act nontrivially on the section group: a
402 single-orbit selector is nonzero exactly when that orbit supports a nonzero
403 compatible section.
404 
405 \item If \(s\succeq r\), equivariance of the refinement map sends every fine
406 monodromy orbit onto a coarse monodromy orbit.  For a coarse orbit
407 \(\mathcal O\),
408 \[
409  P^r_{A,\mathcal O}
410  =
411  \sum_{\substack{\mathcal Q\text{ a fine orbit}\\
412  \mathcal Q\subseteq(\rho^A_{sr})^{-1}(\mathcal O)}}
413  P^s_{A,\mathcal Q}.
414 \]
415 Thus orbit descent and projector-sum refinement commute exactly.
416 \end{enumerate}
417 \end{theorem}
418 
419 \begin{proof}
420 Choose a path from \(U_0\) to \(U\) and transport \(\mathcal O\) along its
421 address bijection.  Two path choices differ by a loop at \(U_0\), which
422 preserves \(\mathcal O\) setwise.  Hence \(\mathcal O_U\) is path independent.
423 Projector compatibility gives
424 \[
425  \theta^A_\gamma F_\gamma(P^r_{A,U,\mathcal O})
426  =
427  P^r_{A,V,\mathcal O}\theta^A_\gamma,
428 \]
429 so the orbit sums descend.  Their local splittings are the finite direct sums
430 \(\bigoplus_{i\in\mathcal O_U}A^r_{U,i}\); the direct sums of the \(\beta\)-maps,
431 with the induced permutation of summands, provide their descent data.  This
432 proves (O1).
433 
434 For (O2), a globally fixed label must return to itself after every loop, so
435 \(S_{U_0}\) is invariant under \(H_A^r\).  Every invariant subset of a finite
436 group action is a union of orbits.  Therefore every such globally labeled
437 coarsening is coarser than the orbit partition.  Statement (O3) is the
438 singleton case.
439 
440 The same path-independence argument on the pair-address set shows that a
441 compatible support family is determined by an invariant base subset, and every
442 invariant base subset is a union of pair orbits.  A selector acts as the
443 identity on compatible sections supported in its orbit and vanishes on the
444 complementary support.  It is therefore nonzero precisely when the orbit
445 supports a nonzero compatible section.  This proves (O4).
446 
447 Finally, equivariance of \(\rho^A_{sr}\) implies that the image of a fine orbit
448 is the orbit of the image of any one of its points.  Hence the preimage of a
449 coarse orbit is a union of fine orbits.  Summing the local projector-refinement
450 identity over all addresses in the coarse orbit gives the formula in (O5).
451 \end{proof}
452 
453 \begin{remark}[Object orbits and matrix orbits differ]
454 \label{rem:block16-object-vs-pair-orbits}
455 The orbit lanes on an object are controlled by \(H_A^r\backslash I_{A,r}\).
456 Transport selectors are controlled by the generally finer orbit set
457 \[
458  H_{Y,X}^{u,r}\backslash
459  \left(I_{Y,u}\times I_{X,r}\right).
460 \]
461 Even when an object has only one transitive address orbit, its endomorphism
462 matrices can have several pair orbits.  Holonomy can therefore erase individual
463 object labels while retaining a nontrivial global transport algebra.
464 \end{remark}
465 
466 \subsection{Permutation modules and the orbital path algebra}
467 \label{subsec:block16-permutation-orbital-calculus}
468 
469 The one-chart groupoid with automorphism group \(H\) gives a concrete exact
470 model.  It also identifies the algebra carried by the pair-orbit selectors.
471 
472 \begin{theorem}[Permutation-module orbital lane calculus]
473 \label{thm:block16-permutation-orbital-calculus}
474 Let \(\kk\) be a commutative ring, let a finite group \(H\) act on finite sets
475 \(I,J,K\), and let \(\kk[I],\kk[J],\kk[K]\) be the corresponding permutation
476 modules.  For an orbit
477 \(\mathcal O\subseteq J\times I\) under the diagonal action, let
478 \(B_{\mathcal O}:\kk[I]\to\kk[J]\) be the matrix whose \((j,i)\)-entry is one
479 when \((j,i)\in\mathcal O\) and zero otherwise.  Then
480 \[
481  \Hom_{\kk H}(\kk[I],\kk[J])
482  =
483  \bigoplus_{\mathcal O\in H\backslash(J\times I)}
484  \kk B_{\mathcal O}.
485 \]
486 In particular, an equivariant matrix is exactly a matrix constant on every
487 pair orbit.
488 
489 Let
490 \[
491  \mathcal P\subseteq J\times I,
492  \qquad
493  \mathcal Q\subseteq K\times J,
494  \qquad
495  \mathcal R\subseteq K\times I
496 \]
497 be diagonal \(H\)-orbits.  For any \((k,i)\in\mathcal R\), define
498 \[
499  c^{\mathcal R}_{\mathcal Q,\mathcal P}
500  :=
501  \#\left\{
502  j\in J:(k,j)\in\mathcal Q,
503  (j,i)\in\mathcal P
504  \right\}.
505 \]
506 This integer is independent of the chosen \((k,i)\in\mathcal R\), and
507 \[
508  B_{\mathcal Q}B_{\mathcal P}
509  =
510  \sum_{\mathcal R\in H\backslash(K\times I)}
511  c^{\mathcal R}_{\mathcal Q,\mathcal P}B_{\mathcal R}.
512 \]
513 Thus the lane path law descends to an exact orbit basis with nonnegative
514 integer structure constants.
515 \end{theorem}
516 
517 \begin{proof}
518 Write an arbitrary \(\kk\)-linear map as a matrix \(A=(A_{ji})\).  The
519 intertwining equations for the permutation actions are equivalent to
520 \[
521  A_{h j,h i}=A_{j i}
522  \qquad(h\in H).
523 \]
524 Hence the entries are constant on diagonal orbits.  The orbit-indicator
525 matrices have disjoint supports, so they are linearly independent over every
526 commutative ring and span all invariant matrices.
527 
528 The \((k,i)\)-entry of
529 \(B_{\mathcal Q}B_{\mathcal P}\) is the displayed cardinality.  If
530 \((k',i')=(hk,hi)\), then \(j\mapsto hj\) is a bijection between the two sets
531 being counted.  The entry is therefore constant on \(\mathcal R\), which gives
532 the orbit-basis expansion.
533 \end{proof}
534 
535 \begin{corollary}[Orbital endomorphism algebra]
536 \label{cor:block16-orbital-endomorphism-algebra}
537 For one finite \(H\)-set \(I\), the pair-orbit matrices form a basis of
538 \(\End_{\kk H}(\kk[I])\).  Matrix multiplication in this basis is exactly the
539 finite compatible-path contraction of
540 \cref{thm:block16-holonomy-twisted-lane-transport}, with the preceding
541 intersection numbers as structure constants.
542 \end{corollary}
543 
544 \begin{example}[The smallest nontrivial holonomy]
545 \label{ex:block16-c2-swap}
546 Let \(H=C_2=\{1,\sigma\}\) act on \(I=\{1,2\}\) by swapping the two addresses.
547 On \(X=\kk^2\), let
548 \[
549  S=
550  \begin{pmatrix}0&1\\1&0\end{pmatrix},
551  \qquad
552  p_1=
553  \begin{pmatrix}1&0\\0&0\end{pmatrix},
554  \qquad
555  p_2=
556  \begin{pmatrix}0&0\\0&1\end{pmatrix}.
557 \]
558 The individual projectors do not descend because
559 \(Sp_1S^{-1}=p_2\), while their orbit sum \(p_1+p_2=1\) does descend.  The
560 pair-address set has two orbits,
561 \[
562  \Delta=\{(1,1),(2,2)\},
563  \qquad
564  \Omega=\{(1,2),(2,1)\}.
565 \]
566 Accordingly,
567 \[
568  \End_{\kk C_2}(\kk^2)
569  =
570  \left\{
571  \begin{pmatrix}a&b\\b&a\end{pmatrix}:a,b\in\kk
572  \right\}
573  =
574  \kk B_\Delta\oplus\kk B_\Omega.
575 \]
576 The singleton endpoint selector \(\{(1,1)\}\) is not holonomy compatible and
577 sends the equivariant identity matrix to \(p_1\), which is not equivariant.
578 The orbit selectors \(C_\Delta\) and \(C_\Omega\) are compatible and split every
579 equivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.
580 \end{example}
581 
582 \subsection{Cyclotomic orbit lanes over the rational field}
583 \label{subsec:block16-cyclotomic-orbit-lanes}
584 
585 The split cyclic group algebra supplies an arithmetic realization in which the
586 address monodromy and its orbit coarsening are explicit.
587 
588 \begin{proposition}[Rational descent of split cyclic Fourier lanes]
589 \label{prop:block16-rational-cyclotomic-orbit-lanes}
590 Let \(N\ge1\), let \(C_N=\langle g\rangle\), and let
591 \(K=\mathbb Q(\zeta_N)\).  In \(K[C_N]\), define
592 \[
593  e_j
594  :=
595  \frac1N\sum_{\ell=0}^{N-1}
596  \zeta_N^{-j\ell}g^\ell,
597  \qquad j\in\mathbb Z/N\mathbb Z.
598 \]
599 These are the primitive split Fourier idempotents.  For
600 \(\sigma_a\in\Gal(K/\mathbb Q)\), where
601 \(\sigma_a(\zeta_N)=\zeta_N^a\),
602 \[
603  \sigma_a(e_j)=e_{aj}.
604 \]
605 The Galois orbits are indexed by divisors \(q\mid N\), with
606 \[
607  \mathcal O_q
608  =
609  \left\{j\in\mathbb Z/N\mathbb Z:
610  \frac{N}{\gcd(N,j)}=q\right\}.
611 \]
612 Their orbit sums
613 \[
614  E_q:=\sum_{j\in\mathcal O_q}e_j
615 \]
616 belong to \(\mathbb Q[C_N]\) and satisfy
617 \[
618  E_qE_{q'}=\delta_{q,q'}E_q,
619  \qquad
620  \sum_{q\mid N}E_q=1.
621 \]
622 More explicitly,
623 \[
624  \boxed{
625  E_q
626  =
627  \frac1N\sum_{\ell=0}^{N-1}c_q(\ell)g^\ell,}
628 \]
629 where
630 \[
631  c_q(\ell)
632  :=
633  \sum_{\substack{1\le a\le q\\\gcd(a,q)=1}}
634  e^{2\pi i a\ell/q}
635  =
636  \sum_{d\mid\gcd(q,\ell)}d\,\mu(q/d)
637  \in\mathbb Z
638 \]
639 is the Ramanujan sum.  The projectors \((E_q)_{q\mid N}\) form the finest
640 \(\mathbb Q\)-rational coarsening obtainable by summing the split Fourier
641 idempotents, and
642 \[
643  \mathbb Q[C_N]E_q\cong\mathbb Q(\zeta_q).
644 \]
645 \end{proposition}
646 
647 \begin{proof}
648 Character orthogonality gives the primitive idempotent relations for the
649 \(e_j\).  Applying \(\sigma_a\) to their coefficients gives
650 \(\sigma_a(e_j)=e_{aj}\).  The unit group
651 \((\mathbb Z/N\mathbb Z)^\times\) acts transitively on elements of
652 \(\mathbb Z/N\mathbb Z\) having the same additive order, so its orbits are the
653 sets \(\mathcal O_q\).
654 
655 Write every \(j\in\mathcal O_q\) uniquely as
656 \(j=(N/q)a\) with \(a\in(\mathbb Z/q\mathbb Z)^\times\).  The coefficient of
657 \(g^\ell\) in \(E_q\) is then
658 \[
659  \frac1N
660  \sum_{a\in(\mathbb Z/q\mathbb Z)^\times}
661  \zeta_q^{-a\ell}
662  =
663  \frac{c_q(\ell)}{N},
664 \]
665 where the last equality uses the integrality and evenness of the Ramanujan
666 sum.  Hence \(E_q\in\mathbb Q[C_N]\).  Orthogonality and completeness follow
667 by grouping the primitive \(e_j\) by their disjoint Galois orbits.  Any rational
668 sum of split primitive idempotents must be Galois invariant, so its support is a
669 union of these orbits; this proves fineness.  Finally,
670 \[
671  \mathbb Q[C_N]
672  \cong
673  \mathbb Q[x]/(x^N-1)
674  \cong
675  \prod_{q\mid N}\mathbb Q[x]/(\Phi_q(x))
676  \cong
677  \prod_{q\mid N}\mathbb Q(\zeta_q),
678 \]
679 and \(E_q\) is the identity projector of the \(q\)-th factor.
680 \end{proof}
681 
682 \begin{boundary}[Cyclic character lanes are not coefficient-residue lanes]
683 \label{bnd:block16-cyclic-character-residue-distinction}
684 The preceding proposition concerns primitive character idempotents in the
685 split group algebra \(K[C_N]\) and their rational Galois descent.  The
686 coefficient-residue projectors \(\mathcal P_{N,j}\) on formal series from
687 \cref{thm:block15-factorial-residue-realization} are defined coefficientwise
688 over every commutative ring and already have fixed residue labels.  A chosen
689 rotation representation relates the two constructions after scalar extension,
690 but their descent data are not identical and should not be conflated.  Nothing
691 in orbit descent turns the factorial step-product into an additive idempotent
692 splitting.
693 \end{boundary}
694 
695 \subsection{Exact finite validation and scope}
696 \label{subsec:block16-validation-scope}
697 
698 The accompanying verifier tests finite permutation actions, pair-orbit bases,
699 equivariant matrix composition, orbit-compatible and deliberately incompatible
700 selectors, equivariant refinement, and the exact Ramanujan-sum projectors in
701 \(\mathbb Q[C_N]\).  It uses exact integer arithmetic after clearing the common cyclotomic
702 denominators.  These tests are
703 an implementation regression for the displayed formulas, not a substitute for
704 the proofs and not an external validation record.
705 
706 \IfFileExists{generated/block16_validation_table.tex}{%
707 \input{generated/block16_validation_table.tex}%
708 }{%
709 \begin{table}[htbp]
710 \centering
711 \caption{Exact finite regression for selected holonomy-twisted lane clauses.
712 Run the Block 16 verifier to generate the complete table.}
713 \label{tab:block16-validation-placeholder}
714 \begin{tabular}{@{}ll@{}}
715 \toprule
716 Record & Status\\
717 \midrule
718 Machine-readable validation & Run \texttt{verify\_block16\_holonomy\_lane\_transport.py}\\
719 \bottomrule
720 \end{tabular}
721 \end{table}%
722 }
723 
724 \begin{resultbox}
725 \textbf{Holonomy-twisted lane transport result.}
726 Finite nontrivial address holonomy does not invalidate exact lane transport.
727 The complete global coordinate object is a matrix local system, and descended
728 morphisms are exactly its compatible sections.  Composition remains finite
729 path contraction, complete resolution changes remain isomorphisms, and every
730 holonomy-compatible selector retains an exact named residual.  Monodromy
731 orbits give the finest ordinary globally labeled coarsening of object lanes,
732 while simultaneous pair orbits give the atomic global transport selectors.  In
733 permutation modules these pair orbits form an exact basis of the intertwiner
734 space with integer path-count structure constants.
735 \end{resultbox}
736 
737 \begin{boundary}[Finite twisted scope]
738 \label{bnd:block16-finite-twisted-scope}
739 The theorem remains finite and profiled.  It assumes invertible chart
740 transitions, finite address sets, coherent address and splitting cocycles, and
741 profile preservation of the operations used.  It does not prove convergence
742 for countably infinite lane sums, descent across noninvertible bridges, higher
743 stack effectivity, or a canonical orbit system independent of the initially
744 declared lane presentation.  Orbit count is not identified with dimension,
745 rank, entropy, information, or complexity.  The permutation-module and
746 cyclotomic formulas have classical algebraic antecedents; their role here is to
747 provide exact native realizations of the twisted lane doctrine.
748 \end{boundary}

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