Realization Calculus

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131 \title[Tame and wild packet geometry]{Tame and Wild Packet Geometry for Local Langlands}
132 \author[Komorebi P.]{Komorebi P.}
133 \subjclass[2020]{11F70, 20C33, 22E50, 11S37, 20C25}
134 \keywords{local Langlands correspondence, tame induction, Heisenberg representations,
135 wild inertia, quadratic refinements, holonomy, arithmetic packet realization}
136 \date{}
137 
138 \pdfstringdefDisableCommands{\let\unskip\relax}
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140   pdftitle={Tame and Wild Packet Geometry for Local Langlands},
141   pdfauthor={Komorebi P.},
142   pdfsubject={Tame Frobenius packets, wild Heisenberg envelopes, finite Weil phase, matrix holonomy, and restricted local bridges}
143 }
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175 \begin{abstract}
176 Let \(K\) be a nonarchimedean local field. We resolve several finite layers that occur inside the local Langlands correspondence for general linear groups. Frobenius semisimple Weil Deligne parameters first split into irreducible Weil atoms and finite monodromy chains. In the tame case, character projectors show that every irreducible atom is one transitive Frobenius orbit, and Mackey theory gives an exact greatest common divisor and least common multiple rule for tensor packets. Regular finite field characters produce primitive cuspidal projectors and finite Gauss phases. For wild data, projective intertwiners recover an alternating commutator form. Its radical free quotient determines a finite Heisenberg representation, and maximal isotropic averaging gives an exact selector. We prove a restricted one break bridge: an equivariant symplectic isometry between a declared Galois envelope and a type side quotient induces the Heisenberg comparison uniquely up to scalar, while one cyclic phase condition removes the remaining ambiguity. Odd residue characteristic is governed by finite Weil phases. Residue characteristic two requires a quadratic refinement and its Arf invariant. A matrix holonomy theorem then gives the exact gluing criterion on a finite transport graph. The general arithmetic packet realization is stated as Conjecture 66. Detailed ordinary proofs give restricted arithmetic realizations in all odd prime-power ranks and in all dyadic ranks for the stated increasing-break families. The complete comparison of pair local factors remains a conditional input for the same arithmetic candidate.
177 \end{abstract}
178 \makeatletter\@setabstract\makeatother
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188 
189 \section{Introduction}\label{sec:introduction}
190 
191 The local Langlands correspondence for \(\GL_n(K)\) is an established theorem. It matches irreducible smooth representations of \(\GL_n(K)\) with Frobenius semisimple Weil Deligne representations of dimension \(n\). Its proof and normalization draw on global methods, harmonic analysis, types, local factors, and the geometry of Shimura varieties \cite{Henniart2000,HarrisTaylor2001,BushnellHenniart2006}. This chapter asks a narrower structural question. Which parts of the local correspondence admit finite packet coordinates whose projectors, transfers, phases, and gluing laws can be proved directly?
192 
193 The answer has a sharp division. The tame layers are explicit. A tame irreducible Weil representation is induced from a character of an unramified extension. Its inertia restriction is one Frobenius orbit. Regular characters of elliptic finite field tori give cuspidal blocks through the Deligne and Lusztig construction \cite{DeligneLusztig1976}. Their tensor products are controlled by a greatest common divisor and least common multiple decomposition. The corresponding cycle orientation and finite Gauss phases can be computed exactly. These features are closely related to the tame lifting and essentially tame correspondence of Bushnell and Henniart \cite{BushnellHenniart1996,BushnellHenniart1999,BushnellHenniart2005ET1,BushnellHenniart2005ET2,BushnellHenniart2010ET3}.
194 
195 Wild data are different. The restriction to wild inertia need not split into cyclic characters. On the type side, simple character theory produces finite symplectic quotients and Heisenberg representations \cite{BushnellKutzko1993}. Canonical beta extensions fix a phase ambiguity in the extension to the normalizer \cite{Blondel2012,Dotto2022}. On the Galois side, higher ramification gives a canonical filtration and strong numerical constraints \cite{BushnellHenniart2017}. The missing construction is not another finite matrix calculation. It is a choice free passage from an arbitrary wild inertia atom to a finite symplectic envelope that agrees with the type side quotient.
196 
197 We therefore separate three levels of assertion. First, this chapter proves finite algebraic theorems about tame orbits, Heisenberg groups, twisted group algebras, finite Weil phases, and transport graphs. Second, it proves a restricted one break bridge once a normal wild datum and an equivariant symplectic identification are declared. Third, it records the exact additional input needed to pass from that restricted bridge to an independent construction of all wild atoms.
198 
199 The finite Heisenberg theory used here is classical. We use the Stone von Neumann theorem, finite Weil representations, and standard projective representation theory \cite{Gerardin1977,Neuhauser2002,Prasad2009,Karpilovsky1985,GurevichHadani2007}. The contribution is the way these results are arranged into one comparison diagram and the matrix holonomy theorem that controls finite gluing. In residue characteristic two, the alternating form does not determine the phase. A quadratic refinement and its Arf invariant are required. The underlying quadratic form theory is standard \cite{MilnorHusemoller1973}.
200 
201 Local factors provide the final test. Henniart showed that pair epsilon factors characterize the local correspondence in the required range \cite{Henniart1993}. The tame packet calculations below recover exact local polynomials and finite Gauss phases for the declared tame atoms. They do not supply the complete pair factor comparison for arbitrary wild atoms. This boundary is retained throughout this chapter.
202 
203 \begin{figure}[t]
204 \centering
205 \includegraphics[width=.98\textwidth]{01_architecture.pdf}
206 \caption{The finite theorem layers and the remaining wild boundary.}
207 \label{fig:architecture}
208 \end{figure}
209 
210 Arithmetic realization and identification of an actual candidate are
211 separate problems. The realization endpoint is
212 \cref{conj:arithmetic_realization}. The restricted constructions are developed
213 in \cref{sec:arithmetic_progress} and the detailed construction appendices.
214 They have ordinary mathematical proofs with retained internal reviews.
215 Their formal coverage is stated separately in the formal companion.
216 
217 The accompanying sources include detailed proofs, construction notes,
218 proof reviews and research records.
219 These documents distinguish established results, conditional results, and
220 the remaining obligations associated with Conjecture 66.
221 
222 This chapter is organized as follows. Weil Deligne chains appear in \cref{sec:wd}. Tame Frobenius packets, finite field cuspidal projectors, tensor refinement, and Gauss phases are developed in \cref{sec:tame,sec:finitefield,sec:tensor,sec:gauss}. Finite Heisenberg groups and projective commutator envelopes appear in \cref{sec:heisenberg,sec:twisted}. The restricted one break Galois envelope and its comparison with the type side are treated in \cref{sec:onebreak,sec:typeside,sec:bridge}. Odd and dyadic phase data are separated in \cref{sec:oddphase,sec:dyadic}. Matrix holonomy and breakwise reconstruction appear in \cref{sec:holonomy,sec:breakwise}. The boundary for local factors appears in \cref{sec:boundary}. Examples and scope
223 follow in \cref{sec:examples,sec:scope}. Section~\ref{sec:arithmetic_progress}
224 and the appendices develop the restricted arithmetic constructions. The
225 separate progress paper reconciles subsequent routes, corrections and
226 remaining obligations without replacing these source records.
227 
228 \section{Weil Deligne chains}\label{sec:wd}
229 
230 Let \(W_K\) be the Weil group of \(K\), let \(I_K\) be inertia, let \(P_K\) be wild inertia, and choose a geometric Frobenius lift \(\Phi\). A Weil Deligne representation is a pair \((r,N)\) with
231 \[
232 r(w)Nr(w)^{-1}=\abs{w}_K N.
233 \]
234 
235 \begin{standardfact}[Indecomposable monodromy blocks]\label{fact:spdecomp}
236 Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks
237 \[
238 \operatorname{Sp}_m(\tau),
239 \]
240 where \(\tau\) is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length \(m\).
241 \end{standardfact}
242 
243 This standard decomposition reduces the construction of a local correspondence to two tasks. One must understand irreducible Weil atoms, then one must attach the correct generalized Steinberg chain. The second task is formal once the first is known.
244 
245 \begin{definition}[Monodromy packet chain]\label{def:monodromy_chain}
246 Let \(V_0,\ldots,V_{m-1}\) be copies of the space of \(\tau\). Put
247 \[
248 V=\bigoplus_{j=0}^{m-1}V_j.
249 \]
250 A monodromy packet chain is a nilpotent operator \(N\) with
251 \[
252 N(V_j)\subseteq V_{j-1}
253 \]
254 for \(j>0\), with \(N(V_0)=0\), together with a Frobenius operator whose conjugation scales \(N\) by \(q^{-1}\).
255 \end{definition}
256 
257 \begin{proposition}[Exact block reconstruction]\label{prop:monodromy_reconstruction}
258 Let \(P_j\) be the projector onto \(V_j\). Then
259 \[
260 N=\sum_{j=1}^{m-1}P_{j-1}NP_j,
261 \qquad
262 N^m=0.
263 \]
264 Every power \(N^r\) is the sum of the packet paths of length \(r\) along the chain.
265 \end{proposition}
266 
267 \begin{proof}
268 The projectors resolve the identity. Since \(N\) maps only from \(V_j\) to \(V_{j-1}\), every block except the displayed nearest neighbor blocks vanishes. A product of \(r\) nonzero blocks moves down exactly \(r\) levels. No path of length \(m\) exists, so \(N^m=0\).
269 \end{proof}
270 
271 \begin{proposition}[Scalar covariance model]\label{prop:scalar_covariance}
272 Let
273 \[
274 F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
275 \]
276 and let \(N_{j-1,j}=1\). Then
277 \[
278 FNF^{-1}=q^{-1}N.
279 \]
280 \end{proposition}
281 
282 \begin{proof}
283 The only nonzero entry in column \(j\) and row \(j-1\) is multiplied by \(q^{j-1}q^{-j}=q^{-1}\) under conjugation.
284 \end{proof}
285 
286 \begin{figure}[t]
287 \centering
288 \includegraphics[width=.9\textwidth]{02_weil_deligne_chain.pdf}
289 \caption{A Weil Deligne chain is reconstructed from one atom and finitely many nearest neighbor transfers.}
290 \label{fig:wdchain}
291 \end{figure}
292 
293 \begin{theorem}[Formal atomic reconstruction]\label{thm:formal_atomic}
294 Assume that every irreducible Weil representation \(\tau\) has been assigned a supercuspidal representation \(\sigma_\tau\), compatibly with isomorphism and unramified twist. Then the rules
295 \[
296 \operatorname{Sp}_m(\tau)
297 \longmapsto
298 \operatorname{St}_m(\sigma_\tau)
299 \]
300 and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
301 \end{theorem}
302 
303 \begin{proof}
304 Decompose the parameter into indecomposable monodromy blocks. Replace each Weil atom by its assigned supercuspidal atom and each monodromy chain by the corresponding generalized Steinberg block. Order the resulting essentially square integrable factors in Langlands order and take the unique irreducible quotient. The multiset of indecomposable Weil Deligne blocks is intrinsic, so the result does not depend on the chosen display.
305 \end{proof}
306 
307 \begin{remark}\label{rem:atomic_boundary}
308 \Cref{thm:formal_atomic} is a reconstruction theorem. It does not produce the atomic assignment \(\tau\mapsto\sigma_\tau\), and it does not prove the required local factor identities.
309 \end{remark}
310 
311 \section{Tame Frobenius packets}\label{sec:tame}
312 
313 Assume that \(P_K\) acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let \(t\) generate a quotient of order \(M\), and put \(\zeta_M=\e^{2\pi i/M}\). For \(j\in\ZZ/M\ZZ\), define
314 \[
315 P_j
316 =
317 \frac1M
318 \sum_{a=0}^{M-1}
319 \zeta_M^{-ja}r(t)^a.
320 \]
321 
322 \begin{proposition}[Tame character projectors]\label{prop:tame_projectors}
323 The operators \(P_j\) are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by
324 \[
325 \Phi P_j\Phi^{-1}=P_{qj}.
326 \]
327 \end{proposition}
328 
329 \begin{proof}
330 The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of \cref{sec:wd}, the relation is \(\Phi^{-1}t\Phi=t^q\). Equivalently, \(\Phi t\Phi^{-1}=t^{q^{-1}}\), with inverse taken modulo \(M\) since \(\gcd(q,M)=1\). Conjugating the Fourier sum and changing its summation index therefore gives \(\Phi P_j\Phi^{-1}=P_{qj}\). The inverse Frobenius convention reverses the label direction while preserving the same orbits.
331 \end{proof}
332 
333 \begin{theorem}[Tame orbit induction]\label{thm:tame_induction}
334 Let \(\tau\) be an irreducible complex representation of \(W_K\) with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length \(f\). If \(K_f/K\) is the unramified extension of degree \(f\), there is a character \(\xi\) of \(W_{K_f}\) such that
335 \[
336 \boxed{
337 \tau\cong\Ind_{W_{K_f}}^{W_K}\xi.
338 }
339 \]
340 \end{theorem}
341 
342 \begin{proof}
343 The nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under \(W_K\), so irreducibility forces one orbit. Choose one packet \(V_\theta\). Its stabilizer is \(W_{K_f}=\langle I_K,\Phi^f\rangle\). A nonzero proper stable subspace of \(V_\theta\) would generate a nonzero proper \(W_K\) stable subspace after summing its Frobenius translates. Thus \(V_\theta\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \(\theta\), and the remaining action is generated by \(\Phi^f\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.
344 \end{proof}
345 
346 \begin{corollary}[Dimension and regularity]\label{cor:tame_dimension}
347 The dimension of \(\tau\) equals the Frobenius orbit length of its tame inertia character. The inducing character is regular, meaning that no smaller positive power of Frobenius fixes it.
348 \end{corollary}
349 
350 \begin{proof}
351 The induced representation has one line for every Frobenius translate of the chosen character. A shorter orbit would produce a proper stabilizer and contradict the minimality of \(f\).
352 \end{proof}
353 
354 
355 \begin{proposition}[Dual and self dual tame packets]\label{prop:tame_dual}
356 For a tame atom
357 \[
358 \tau_\xi=\Ind_{W_{K_f}}^{W_K}\xi,
359 \]
360 we have
361 \[
362 \tau_\xi^\vee
363 \cong
364 \Ind_{W_{K_f}}^{W_K}\xi^{-1}.
365 \]
366 The atom is self dual exactly when
367 \[
368 \xi^{-1}=\xi^{\Phi^j}
369 \]
370 for some \(j\in\ZZ/f\ZZ\).
371 \end{proposition}
372 
373 \begin{proof}
374 Duality commutes with finite index induction, giving the first statement. Two regular induced tame atoms are isomorphic exactly when their inducing characters lie in the same Frobenius orbit. Applying this to \(\xi\) and \(\xi^{-1}\) gives the criterion.
375 \end{proof}
376 
377 \begin{proposition}[Unramified twist]\label{prop:tame_twist}
378 Let \(\nu\) be an unramified character of \(W_K\). Then
379 \[
380 \tau_\xi\otimes\nu
381 \cong
382 \Ind_{W_{K_f}}^{W_K}
383 \left(
384 \xi\,\nu|_{W_{K_f}}
385 \right).
386 \]
387 The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar \(a\), the twisted cycle has closing scalar
388 \[
389 a\nu(\Phi)^f.
390 \]
391 \end{proposition}
392 
393 \begin{proof}
394 The tensor identity follows from the projection formula for induction. Since \(\nu\) is trivial on inertia, it does not change any tame character packet. It multiplies every Frobenius step by \(\nu(\Phi)\), so a closed cycle of length \(f\) gains the factor \(\nu(\Phi)^f\).
395 \end{proof}
396 
397 \begin{figure}[t]
398 \centering
399 \includegraphics[width=.62\textwidth]{03_tame_frobenius_orbit.pdf}
400 \caption{A tame irreducible atom is one transitive Frobenius orbit.}
401 \label{fig:tameorbit}
402 \end{figure}
403 
404 \subsection{Regular orbit counts}
405 
406 Characters of \(\FF_{q^n}^{\times}\) are indexed by exponents modulo \(q^n-1\), and Frobenius acts by multiplication by \(q\).
407 
408 \begin{proposition}[Möbius orbit count]\label{prop:mobius_count}
409 The number of regular Frobenius orbits of length \(n\) is
410 \[
411 \boxed{
412 c_n(q)
413 =
414 \frac1n
415 \sum_{d\mid n}
416 \mu(d)
417 \bigl(q^{n/d}-1\bigr).
418 }
419 \]
420 \end{proposition}
421 
422 \begin{proof}
423 For every divisor \(r\mid n\), the characters fixed by \(\Frob^r\) are exactly those descending to \(\FF_{q^r}^{\times}\), and there are \(q^r-1\) of them. Möbius inversion gives the number of characters of exact orbit length \(n\). Division by \(n\) gives the number of orbits.
424 \end{proof}
425 
426 \begin{example}[Degree four over \(\FF_2\)]\label{ex:degree_four}
427 The regular exponent orbits modulo fifteen are
428 \[
429 (1,2,4,8),
430 \qquad
431 (3,6,12,9),
432 \qquad
433 (7,14,13,11).
434 \]
435 The formula gives
436 \[
437 \frac14\bigl((2^4-1)-(2^2-1)\bigr)=3.
438 \]
439 \end{example}
440 
441 \begin{figure}[t]
442 \centering
443 \includegraphics[width=.76\textwidth]{04_regular_orbit_counts.pdf}
444 \caption{Regular Frobenius orbit counts for three residue field sizes.}
445 \label{fig:orbitcounts}
446 \end{figure}
447 
448 \section{Finite field cuspidal projectors}\label{sec:finitefield}
449 
450 Let
451 \[
452 G_n=\GL_n(\FF_q),
453 \qquad
454 T_n\cong\FF_{q^n}^{\times}
455 \]
456 with \(T_n\) an elliptic maximal torus.
457 
458 \begin{standardfact}[Regular elliptic torus characters]\label{fact:deligne_lusztig}
459 A regular character \(\theta:T_n\to\CC^{\times}\) determines an irreducible cuspidal representation \(\kappa_\theta\) of \(G_n\). Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of \(G_n\) occurs in this way.
460 \end{standardfact}
461 
462 This is the finite field input from Deligne and Lusztig theory \cite{DeligneLusztig1976}. We use standard finite group character theory for the central idempotents \cite{Serre1977,Isaacs1976}. Once \(\kappa_\theta\) is known, its block is selected by a central idempotent.
463 
464 \begin{proposition}[Primitive cuspidal idempotent]\label{prop:cuspidal_idempotent}
465 Let \(\chi_\theta\) be the irreducible character of \(\kappa_\theta\), and let \(d_\theta=\chi_\theta(1)\). Then
466 \[
467 e_\theta
468 =
469 \frac{d_\theta}{\abs{G_n}}
470 \sum_{g\in G_n}
471 \chi_\theta(g^{-1})g
472 \]
473 is the primitive central idempotent selecting the \(\kappa_\theta\) block in \(\CC[G_n]\).
474 \end{proposition}
475 
476 \begin{proof}
477 On an irreducible representation of character \(\psi\), the central element acts by a scalar. Taking traces shows that the scalar is
478 \[
479 \frac{d_\theta}{\abs{G_n}\psi(1)}
480 \sum_{g\in G_n}
481 \chi_\theta(g^{-1})\psi(g).
482 \]
483 Character orthogonality makes this scalar one for \(\psi=\chi_\theta\) and zero otherwise. The element is therefore central, idempotent, and primitive.
484 \end{proof}
485 
486 \begin{proposition}[Parabolic annihilation]\label{prop:parabolic_annihilation}
487 Let \(P=LU\) be a proper parabolic subgroup of \(G_n\), and put
488 \[
489 e_U=\frac1{\abs U}\sum_{u\in U}u.
490 \]
491 If \(\kappa_\theta\) is cuspidal, then
492 \[
493 \boxed{
494 e_Ue_\theta=e_\theta e_U=0.
495 }
496 \]
497 \end{proposition}
498 
499 \begin{proof}
500 The operator \(e_U\) projects onto the subspace of \(U\) fixed vectors. A cuspidal representation has no nonzero fixed vector under the unipotent radical of a proper parabolic. Thus \(e_U\) acts by zero on the \(\kappa_\theta\) block. Since \(e_\theta\) is central, both products vanish.
501 \end{proof}
502 
503 \begin{figure}[t]
504 \centering
505 \includegraphics[width=.92\textwidth]{05_cuspidal_projector.pdf}
506 \caption{A regular torus character determines a cuspidal block and its primitive central projector.}
507 \label{fig:cuspidal}
508 \end{figure}
509 
510 \subsection{Depth zero reconstruction and cycle orientation}
511 
512 The depth zero construction inflates a finite cuspidal representation to \(\GL_n(\OO_K)\), extends it to the standard compact modulo center subgroup, and then compactly induces it to \(\GL_n(K)\). This is classical type theory \cite{BushnellKutzko1993,BushnellHenniart2006}.
513 
514 The induced tame Weil atom has an elementary cycle invariant. In an orbit basis \(e_0,\ldots,e_{n-1}\), write
515 \[
516 \tau(\Phi)e_j=e_{j+1}
517 \]
518 for \(j<n-1\), and
519 \[
520 \tau(\Phi)e_{n-1}=ae_0.
521 \]
522 
523 \begin{proposition}[Frobenius cycle polynomial]\label{prop:frobenius_cycle}
524 The Frobenius transfer matrix has characteristic polynomial
525 \[
526 X^n-a
527 \]
528 and determinant
529 \[
530 \boxed{
531 \det\tau(\Phi)=(-1)^{n-1}a.
532 }
533 \]
534 \end{proposition}
535 
536 \begin{proof}
537 The matrix is the companion matrix of \(X^n-a\). Its only nonzero determinant term is the closed \(n\) cycle. That cycle has sign \((-1)^{n-1}\), and the closing edge contributes \(a\).
538 \end{proof}
539 
540 \begin{corollary}[Unramified orientation character]\label{cor:orientation}
541 The sign \((-1)^{n-1}\) is the orientation of the closed Frobenius packet cycle. It is unchanged by rescaling the individual orbit basis vectors.
542 \end{corollary}
543 
544 \begin{proof}
545 A basis rescaling conjugates the transfer matrix by a diagonal matrix, so the determinant and characteristic polynomial remain unchanged.
546 \end{proof}
547 
548 \section{Tensor refinement of tame atoms}\label{sec:tensor}
549 
550 Let
551 \[
552 \tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi,
553 \qquad
554 \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
555 \]
556 where \(K_n/K\) and \(K_m/K\) are unramified. Put
557 \[
558 d=\gcd(n,m),
559 \qquad
560 L=\lcm(n,m).
561 \]
562 
563 \begin{theorem}[Greatest common divisor and least common multiple packets]\label{thm:gcd_lcm}
564 There is a decomposition
565 \[
566 \boxed{
567 \tau_\xi\otimes\tau_\eta
568 \cong
569 \bigoplus_{i=0}^{d-1}
570 \Ind_{W_{K_L}}^{W_K}
571 \left(
572 \xi_{L,n}\eta_{L,m}^{\Phi^i}
573 \right),
574 }
575 \]
576 where \(\xi_{L,n}\) and \(\eta_{L,m}\) are the norm lifts to \(K_L\).
577 \end{theorem}
578 
579 \begin{proof}
580 Apply Mackey decomposition \cite{Mackey1952}. The double cosets
581 \[
582 W_{K_n}\backslash W_K/W_{K_m}
583 \]
584 are the orbits of simultaneous translation on \(\ZZ/n\ZZ\times\ZZ/m\ZZ\). There are \(d\) orbits, each of length \(L\). The stabilizer of a point in each orbit is \(W_{K_L}\). The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the double cosets gives the decomposition.
585 \end{proof}
586 
587 \begin{corollary}[Packet count and dimension]\label{cor:gcd_lcm_count}
588 The tensor product has \(d\) induced interaction packets, each with carrier dimension and Frobenius coset-cycle length \(L\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because
589 \[
590 dL=nm.
591 \]
592 \end{corollary}
593 \begin{proof}
594 The double coset calculation in \cref{thm:gcd_lcm} gives exactly \(d\) summands, and every stabilizer is \(W_{K_L}\), so every induced packet has dimension \(L\). The identity \(dL=nm\) follows from the definitions of greatest common divisor and least common multiple.
595 \end{proof}
596 
597 
598 \begin{figure}[t]
599 \centering
600 \includegraphics[width=.82\textwidth]{06_gcd_lcm_refinement.pdf}
601 \caption{Tensor refinement gives as many packets as the greatest common divisor, each with carrier dimension equal to the least common multiple.}
602 \label{fig:gcdlcm}
603 \end{figure}
604 
605 \subsection{Exponent coordinates}
606 
607 Represent the residue characters by exponents
608 \[
609 a\pmod{q^n-1},
610 \qquad
611 b\pmod{q^m-1}.
612 \]
613 
614 \begin{proposition}[Refined exponent formula]\label{prop:refined_exponent}
615 The \(i\)th interaction packet on \(\FF_{q^L}^{\times}\) has exponent
616 \[
617 \boxed{
618 c_i
619 =
620 a\frac{q^L-1}{q^n-1}
621 +bq^i\frac{q^L-1}{q^m-1}
622 \pmod{q^L-1}.
623 }
624 \]
625 It is trivial on tame inertia exactly when \(c_i=0\).
626 \end{proposition}
627 
628 \begin{proof}
629 Norm lifting a character exponent from \(\FF_{q^n}^{\times}\) to \(\FF_{q^L}^{\times}\) multiplies the exponent by \((q^L-1)/(q^n-1)\). The Frobenius shift multiplies the second exponent by \(q^i\). Multiplication of characters adds exponents. Triviality is equivalent to the resulting exponent being zero modulo \(q^L-1\).
630 \end{proof}
631 
632 \subsection{Tame local polynomials}
633 
634 Let \(V\) be one interaction packet. If its inertia character is nontrivial, then \(V^{I_K}=0\). If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.
635 
636 \begin{proposition}[Local polynomial of one tame packet]\label{prop:tame_local_polynomial}
637 Suppose an interaction packet has orbit length \(r\) and closing scalar \(a\). Its local polynomial is
638 \[
639 \det\left(I-u\Phi\mid V^{I_K}\right)
640 =
641 \begin{cases}
642 1-a u^r,&\text{if the inertia character is trivial},\\
643 1,&\text{otherwise}.
644 \end{cases}
645 \]
646 \end{proposition}
647 
648 \begin{proof}
649 For nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of \(X^r-a\). Substituting \(X=u^{-1}\) and multiplying by \(u^r\) gives \(1-a u^r\).
650 \end{proof}
651 
652 \begin{corollary}[Tame tensor local factor]\label{cor:tame_tensor_factor}
653 The local factor of the tame tensor product is the product of
654 \[
655 (1-a_i u^{r_i})^{-1}
656 \]
657 over precisely those refined packets whose inertia exponent vanishes.
658 \end{corollary}
659 
660 \begin{proof}
661 The invariant space is the direct sum of the invariant refined packets, and determinant is multiplicative on direct sums.
662 \end{proof}
663 
664 \begin{proposition}[Trivial tensor packet criterion]\label{prop:trivial_tensor_packet}
665 For every refined interaction packet, let \(r_i\) be its cycle length and \(a_i\) its closing scalar. The dimension of the \(W_K\) fixed space in \(\tau_\xi\otimes\tau_\eta\) equals the number of packets satisfying both
666 \[
667 c_i=0
668 \qquad\text{and}\qquad
669 a_i=1.
670 \]
671 \end{proposition}
672 
673 \begin{proof}
674 A nonzero fixed vector must lie in an inertia trivial packet, so \(c_i=0\). On that packet, Frobenius is a companion cycle with characteristic polynomial \(X^{r_i}-a_i\). Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when \(a_i=1\), and it is zero otherwise. Summing over the refined packets proves the formula.
675 \end{proof}
676 
677 \begin{corollary}[Tame Hom test]\label{cor:tame_hom}
678 For irreducible tame atoms,
679 \[
680 \dim\Hom_{W_K}(\tau_\xi^\vee,\tau_\eta)
681 \]
682 is zero or one, and it is one exactly when the refined tensor packet test in \cref{prop:trivial_tensor_packet} finds one trivial packet.
683 \end{corollary}
684 
685 \begin{proof}
686 The Hom space is the fixed space of \(\tau_\xi\otimes\tau_\eta\). Schur's lemma gives dimension zero or one, and \cref{prop:trivial_tensor_packet} gives the packet criterion.
687 \end{proof}
688 
689 \subsection{Tame conductor}
690 
691 \begin{proposition}[Conductor of a nontrivial tame atom]\label{prop:tame_conductor}
692 Let \(\tau\) be an irreducible tame Weil representation of dimension \(n>1\). Then
693 \[
694 \Swan(\tau)=0,
695 \qquad
696 \dim\tau^{I_K}=0,
697 \qquad
698 \Art(\tau)=n.
699 \]
700 \end{proposition}
701 
702 \begin{proof}
703 Wild inertia acts trivially, so the Swan conductor is zero. An inertia fixed vector would generate a nonzero Frobenius stable sum of trivial character packets. Irreducibility would then make inertia act trivially on all of \(\tau\). An irreducible representation of the quotient generated by Frobenius would be one dimensional, contrary to \(n>1\). The Artin conductor is therefore
704 \[
705 \dim\tau-\dim\tau^{I_K}+\Swan(\tau)=n.
706 \]
707 \end{proof}
708 
709 \section{Finite Gauss phases}\label{sec:gauss}
710 
711 Let \(k=\FF_q\), let \(\psi:k\to\CC^{\times}\) be a nontrivial additive character, and let \(\chi:k^{\times}\to\CC^{\times}\) be a multiplicative character. Define
712 \[
713 G(\chi,\psi)
714 =
715 \sum_{x\in k^{\times}}\chi(x)\psi(x).
716 \]
717 The standard theory of Gauss and Jacobi sums will be used in the form recorded in \cite{DavenportHasse1935,BerndtEvansWilliams1998}.
718 
719 \begin{proposition}[Gauss magnitude]\label{prop:gauss_magnitude}
720 If \(\chi\) is nontrivial, then
721 \[
722 \boxed{
723 \abs{G(\chi,\psi)}=q^{1/2}.
724 }
725 \]
726 \end{proposition}
727 
728 \begin{proof}
729 Compute
730 \[
731 \abs{G(\chi,\psi)}^2
732 =
733 \sum_{x,y\in k^{\times}}
734 \chi(xy^{-1})\psi(x-y).
735 \]
736 Write \(x=ty\). The inner sum over \(y\) equals \(q-1\) when \(t=1\) and equals \(-1\) otherwise. Since \(\sum_{t\in k^{\times}}\chi(t)=0\), the result is \(q\).
737 \end{proof}
738 
739 \begin{definition}[Normalized Gauss phase]\label{def:gauss_phase}
740 For nontrivial \(\chi\), define
741 \[
742 \gamma(\chi,\psi)
743 =
744 q^{-1/2}G(\chi,\psi).
745 \]
746 Then \(\abs{\gamma(\chi,\psi)}=1\).
747 \end{definition}
748 
749 \begin{proposition}[Duality of Gauss phases]\label{prop:gauss_duality}
750 For nontrivial \(\chi\),
751 \[
752 \boxed{
753 \gamma(\chi,\psi)\gamma(\chi^{-1},\psi)
754 =
755 \chi(-1).
756 }
757 \]
758 \end{proposition}
759 
760 \begin{proof}
761 The standard Gauss identity gives
762 \[
763 G(\chi,\psi)G(\chi^{-1},\psi)=\chi(-1)q.
764 \]
765 Divide by \(q\).
766 \end{proof}
767 
768 \begin{standardfact}[Hasse and Davenport lifting]\label{fact:hasse_davenport}
769 Let \(k_r/k\) be the extension of degree \(r\). Lift \(\chi\) by the norm and \(\psi\) by the trace. Then the lifted Gauss sum satisfies the Hasse and Davenport product relation, with a sign determined by the chosen additive character normalization.
770 \end{standardfact}
771 
772 \begin{figure}[t]
773 \centering
774 \includegraphics[width=.53\textwidth]{07_gauss_phase_circle.pdf}
775 \caption{Normalized finite Gauss sums lie on the unit circle.}
776 \label{fig:gausscircle}
777 \end{figure}
778 
779 \begin{remark}\label{rem:gauss_boundary}
780 Finite Gauss phases reproduce the finite field part of tame and depth zero normalization. They do not by themselves give the full epsilon factor of a general wild pair.
781 \end{remark}
782 
783 \section{Finite Heisenberg packets}\label{sec:heisenberg}
784 
785 Let \(p\) be odd. Let \(X\) be an \(r\) dimensional vector space over \(\FF_p\), and put
786 \[
787 V=X\oplus X^*.
788 \]
789 Define the alternating form
790 \[
791 b\bigl((x,\xi),(y,\eta)\bigr)
792 =
793 \eta(x)-\xi(y).
794 \]
795 Let \(Z=\FF_p\), and define the Heisenberg group
796 \[
797 \frakH(V)=V\times Z
798 \]
799 with multiplication
800 \[
801 (v,z)(w,t)
802 =
803 \left(v+w,z+t+\frac12 b(v,w)\right).
804 \]
805 Fix the additive character
806 \[
807 \psi(z)=\e^{2\pi iz/p}.
808 \]
809 
810 \begin{definition}[Schrödinger operators]\label{def:schrodinger}
811 On \(\calH=\CC^X\), define
812 \[
813 \bigl(\rho(x,\xi,z)f\bigr)(u)
814 =
815 \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
816 \]
817 \end{definition}
818 
819 \begin{theorem}[Finite Schrödinger representation]\label{thm:schrodinger_rep}
820 The map \(\rho:\frakH(V)\to\GL(\calH)\) is a representation. It is irreducible, its center acts by \(\psi\), and
821 \[
822 \dim\calH=p^r=\sqrt{\abs V}.
823 \]
824 \end{theorem}
825 
826 \begin{proof}
827 A direct calculation gives
828 \[
829 \rho(v,z)\rho(w,t)
830 =
831 \rho\left(v+w,z+t+\frac12b(v,w)\right).
832 \]
833 Translations by \(X\) and multiplications by characters from \(X^*\) generate the full matrix algebra on \(\CC^X\). Indeed, the character multipliers give all diagonal projectors by finite Fourier inversion, and translations move each coordinate line to every other coordinate line. The commutant is therefore scalar, so the representation is irreducible. The dimension is \(\abs X=p^r\).
834 \end{proof}
835 
836 \begin{proposition}[Character support]\label{prop:heisenberg_trace}
837 The character of the Schrödinger representation satisfies
838 \[
839 \boxed{
840 \Tr\rho(v,z)
841 =
842 \begin{cases}
843 p^r\psi(z),&v=0,\\
844 0,&v\ne0.
845 \end{cases}
846 }
847 \]
848 \end{proposition}
849 
850 \begin{proof}
851 If the translation component \(x\) is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If \(x=0\) but \(\xi\ne0\), the trace is
852 \[
853 \psi(z)\sum_{u\in X}\psi(\xi(u))=0.
854 \]
855 When \(x=\xi=0\), the operator is the scalar \(\psi(z)\).
856 \end{proof}
857 
858 \begin{standardfact}[Finite Stone von Neumann theorem]\label{fact:stone_von_neumann}
859 For a finite Heisenberg group with nondegenerate commutator form and a fixed nontrivial central character, there is one irreducible representation up to isomorphism. Its dimension is the square root of the order of the symplectic quotient.
860 \end{standardfact}
861 
862 The standard fact follows from the twisted group algebra argument in \cref{sec:twisted}. It is also the finite form used in the classical Weil representation literature \cite{Gerardin1977,Prasad2009}.
863 
864 \subsection{Maximal isotropic selectors}
865 
866 Let \(L\subseteq V\) be maximal isotropic. Choose a subgroup lift \(\widetilde L\subseteq\frakH(V)\) on which the central coordinate is zero.
867 
868 \begin{proposition}[Isotropic averaging projector]\label{prop:isotropic_selector}
869 The operator
870 \[
871 P_L
872 =
873 \frac1{\abs L}
874 \sum_{\ell\in\widetilde L}
875 \rho(\ell)
876 \]
877 is an orthogonal projector. In the Schrödinger model with \(L=X\), it has rank one and selects the constant functions.
878 \end{proposition}
879 
880 \begin{proof}
881 The lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For \(L=X\), the average is the translation average
882 \[
883 (P_Lf)(u)
884 =
885 \frac1{\abs X}
886 \sum_{x\in X}f(u+x),
887 \]
888 which is the constant function with value equal to the mean of \(f\). Its range is one dimensional.
889 \end{proof}
890 
891 \begin{corollary}[Primitive line selector]\label{cor:primitive_selector}
892 Every maximal isotropic subgroup selects a one dimensional fixed line in the irreducible Heisenberg representation.
893 \end{corollary}
894 
895 \begin{proof}
896 The projector rank equals its trace. The identity element contributes \(p^r\), while every nonidentity element of the lifted isotropic subgroup is noncentral and has trace zero by \cref{prop:heisenberg_trace}. Since \(\abs L=p^r\), we get
897 \[
898 \Tr P_L
899 =
900 \frac{p^r}{p^r}=1.
901 \]
902 Thus the projector has rank one.
903 \end{proof}
904 
905 \begin{figure}[t]
906 \centering
907 \includegraphics[width=.9\textwidth]{08_heisenberg_geometry.pdf}
908 \caption{The finite Heisenberg packet and its maximal isotropic selector.}
909 \label{fig:heisenberg}
910 \end{figure}
911 
912 \section{Twisted group algebras and commutator envelopes}\label{sec:twisted}
913 
914 Let \(A\) be a finite abelian group, and let
915 \[
916 c:A\times A\to\CC^{\times}
917 \]
918 be a normalized two cocycle. The twisted group algebra \(\CC^c[A]\) has basis \((u_a)_{a\in A}\) and multiplication
919 \[
920 u_au_b=c(a,b)u_{a+b}.
921 \]
922 Define the commutator bicharacter
923 \[
924 b_c(a,b)=c(a,b)c(b,a)^{-1}.
925 \]
926 
927 \begin{proposition}[Scalar choice invariance]\label{prop:cocycle_choice}
928 Replacing \(u_a\) by \(\lambda_au_a\), with \(\lambda_a\in\CC^{\times}\), changes \(c\) by a coboundary and leaves \(b_c\) unchanged.
929 \end{proposition}
930 
931 \begin{proof}
932 The new cocycle is
933 \[
934 c'(a,b)
935 =
936 \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
937 \]
938 The scalar coboundary factor is symmetric in \(a\) and \(b\), so it cancels from \(c'(a,b)c'(b,a)^{-1}\).
939 \end{proof}
940 
941 \begin{proposition}[Alternating commutator form]\label{prop:commutator_form}
942 The map \(b_c\) is an alternating bicharacter. Its radical
943 \[
944 R=\{a\in A:b_c(a,x)=1\text{ for every }x\in A\}
945 \]
946 is a subgroup, and \(b_c\) descends to a nondegenerate alternating bicharacter on \(A/R\).
947 \end{proposition}
948 
949 \begin{proof}
950 Associativity of the twisted product gives multiplicativity of scalar commutators in each variable. We have \(b_c(a,a)=1\) and \(b_c(b,a)=b_c(a,b)^{-1}\). The radical is the kernel of the homomorphism from \(A\) to its character group defined by \(a\mapsto b_c(a,\cdot)\). The induced pairing on the quotient is nondegenerate by construction.
951 \end{proof}
952 
953 \begin{theorem}[Nondegenerate twisted group algebra]\label{thm:twisted_matrix}
954 Assume that \(b_c\) is nondegenerate. Then \(\abs A\) is a square, and
955 \[
956 \boxed{
957 \CC^c[A]
958 \cong
959 M_{\sqrt{\abs A}}(\CC).
960 }
961 \]
962 Consequently, \(\CC^c[A]\) has one irreducible module up to isomorphism, of dimension \(\sqrt{\abs A}\).
963 \end{theorem}
964 
965 \begin{proof}
966 Let
967 \[
968 z=\sum_{a\in A}z_au_a
969 \]
970 be central. Commuting with \(u_x\) gives
971 \[
972 z_ab_c(x,a)=z_a
973 \]
974 for every \(x\). Nondegeneracy forces \(z_a=0\) for every nonzero \(a\). Thus the center is \(\CC\). To prove semisimplicity directly, choose a linear projection \(P\) onto a submodule of a finite dimensional \(\CC^c[A]\)-module and form
975 \[
976 \overline P=\frac{1}{\abs A}\sum_{a\in A}u_a P u_a^{-1}.
977 \]
978 Every summand maps into the submodule and is the identity there. Conjugation by any \(u_x\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \(\overline P\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \(\CC\) permits only one simple factor. A finite dimensional central simple algebra over \(\CC\) is a full matrix algebra. Its dimension is \(\abs A\), so the matrix size is \(\sqrt{\abs A}\).
979 \end{proof}
980 
981 \begin{corollary}[Radical free envelope]\label{cor:radical_free}
982 For an arbitrary scalar projective action of \(A\), the quotient
983 \[
984 \boxed{
985 A^{\mathrm{symp}}=A/\rad b_c
986 }
987 \]
988 is the maximal quotient detected by noncommuting projective intertwiners. If \(A\) is an elementary abelian \(p\) group, then \(A^{\mathrm{symp}}\) is a symplectic \(\FF_p\) space.
989 \end{corollary}
990 
991 \begin{proof}
992 Every radical element commutes with all projective generators, so it contributes only central data. Quotienting by the radical leaves the nondegenerate commutator form. In the elementary abelian case, multiplicative values in the group of \(p\)th roots can be written as an alternating bilinear form over \(\FF_p\).
993 \end{proof}
994 
995 \begin{figure}[t]
996 \centering
997 \includegraphics[width=.94\textwidth]{09_commutator_envelope.pdf}
998 \caption{Projective intertwiners determine a radical free symplectic envelope.}
999 \label{fig:commutator}
1000 \end{figure}
1001 
1002 \begin{remark}\label{rem:projective_scope}
1003 The envelope depends only on the projective representation. It does not by itself prove that a given wild inertia representation carries a canonical finite quotient with the required projective action.
1004 \end{remark}
1005 
1006 \section{A restricted one break Galois envelope}\label{sec:onebreak}
1007 
1008 We now isolate a finite Clifford setting in which the quotient projective action is defined on a multiplicity space. This avoids assigning a quotient action directly to arbitrary representatives of \(G/N\).
1009 
1010 \begin{definition}[One break Clifford datum]\label{def:one_break_datum}
1011 A one break Clifford datum consists of finite groups and representations
1012 \[
1013 N\triangleleft G,
1014 \qquad
1015 A=G/N,
1016 \qquad
1017 \Pi:G\to\GL(\calX),
1018 \qquad
1019 \rho:N\to\GL(W),
1020 \]
1021 with the following properties.
1022 
1023 \begin{enumerate}
1024 \item The quotient \(A\) is a finite elementary abelian \(p\) group.
1025 \item The representations \(\Pi\) and \(\rho\) are irreducible.
1026 \item The isomorphism class of \(\rho\) is fixed by \(G\).
1027 \item The restriction of \(\Pi\) to \(N\) is \(\rho\) isotypic.
1028 \item The quotient records one declared ramification break.
1029 \end{enumerate}
1030 
1031 The multiplicity space is
1032 \[
1033 M_\rho=\Hom_N(W,\calX).
1034 \]
1035 \end{definition}
1036 
1037 The isotypic condition gives an \(N\) equivariant evaluation isomorphism
1038 \[
1039 W\otimes M_\rho
1040 \longrightarrow
1041 \calX.
1042 \]
1043 Choose a section \(s:A\to G\). For every \(a\in A\), choose an intertwiner \(I_a\) from \(\rho\) to its conjugate by \(s(a)\). Under the evaluation isomorphism, there is a unique operator \(T_a\) on \(M_\rho\) such that
1044 \[
1045 \Pi(s(a))=I_a\otimes T_a.
1046 \]
1047 The operators \(T_a\) form a projective representation of \(A\).
1048 
1049 \begin{proposition}[Clifford multiplicity action]\label{prop:clifford_action}
1050 The projective equivalence class of the action \((T_a)_{a\in A}\) is independent of the section, the intertwiners, and the chosen isotypic evaluation isomorphism. The projective action is irreducible.
1051 \end{proposition}
1052 
1053 \begin{proof}
1054 Changing an intertwiner \(I_a\) by a scalar changes \(T_a\) by the inverse scalar. Changing the isotypic evaluation isomorphism conjugates all \(T_a\) by one element of \(\GL(M_\rho)\). Changing the section multiplies \(s(a)\) by an element of \(N\). That factor is absorbed by the \(\rho\) action on \(W\), leaving a scalar change on the multiplicity operator by Schur's lemma. Thus all choices give projectively equivalent actions.
1055 
1056 If a nonzero proper subspace of \(M_\rho\) were stable under all projective operators \(T_a\), then its tensor product with \(W\) would give a nonzero proper \(G\) stable subspace of \(\calX\). This contradicts irreducibility of \(\Pi\).
1057 \end{proof}
1058 
1059 \begin{definition}[Clifford cocycle class]\label{def:clifford_class}
1060 Choose scalars \(c(a,b)\) such that
1061 \[
1062 T_aT_b=c(a,b)T_{a+b}.
1063 \]
1064 The cohomology class
1065 \[
1066 [c_\rho]\in H^2(A,\CC^{\times})
1067 \]
1068 is the Clifford cocycle class of the datum.
1069 \end{definition}
1070 
1071 \begin{proposition}[Intrinsic commutator pairing]\label{prop:one_break_form}
1072 The pairing
1073 \[
1074 b_\rho(a,b)
1075 =
1076 c(a,b)c(b,a)^{-1}
1077 \]
1078 is independent of every choice in \cref{def:one_break_datum}. It is an alternating bicharacter
1079 \[
1080 b_\rho:A\times A\to\mu_p.
1081 \]
1082 \end{proposition}
1083 
1084 \begin{proof}
1085 The projective equivalence class is intrinsic by \cref{prop:clifford_action}. Conjugation does not change scalar commutators, and scalar changes alter the cocycle by a coboundary. The commutator pairing is therefore unchanged by \cref{prop:cocycle_choice}. Its alternating and bilinear properties follow from \cref{prop:commutator_form}. Since every element of \(A\) has order \(p\), every value has order dividing \(p\).
1086 \end{proof}
1087 
1088 \begin{definition}[One break symplectic envelope]\label{def:one_break_envelope}
1089 The one break symplectic envelope is
1090 \[
1091 \boxed{
1092 V_\rho=A/\rad b_\rho.
1093 }
1094 \]
1095 \end{definition}
1096 
1097 \begin{theorem}[Nondegenerate one break packet]\label{thm:one_break_matrix}
1098 If \(b_\rho\) is nondegenerate, then
1099 \[
1100 \CC^{c_\rho}[A]
1101 \cong
1102 \End(M_\rho),
1103 \qquad
1104 \dim M_\rho=\sqrt{\abs A}.
1105 \]
1106 \end{theorem}
1107 
1108 \begin{proof}
1109 The projective action gives a nonzero representation of the twisted group algebra on \(M_\rho\). The algebra is a full matrix algebra by \cref{thm:twisted_matrix}. Its unique irreducible module has dimension \(\sqrt{\abs A}\). The projective multiplicity action is irreducible by \cref{prop:clifford_action}, so it is that module.
1110 \end{proof}
1111 
1112 \begin{proposition}[Isomorphism invariance]\label{prop:one_break_invariance}
1113 The cocycle class and the equivariant isometry class of \(V_\rho\) depend only on the isomorphism class of the one break Clifford datum.
1114 \end{proposition}
1115 
1116 \begin{proof}
1117 An isomorphism of data transports the restriction, the multiplicity space, and its projective quotient action. It therefore transports the cocycle class, the commutator pairing, its radical, and the quotient.
1118 \end{proof}
1119 
1120 \begin{proposition}[Tame stabilizer action]\label{prop:tame_stabilizer}
1121 Suppose a finite group \(C\) acts by automorphisms of the one break Clifford datum. Then \(C\) acts on \(V_\rho\) by symplectic automorphisms.
1122 \end{proposition}
1123 
1124 \begin{proof}
1125 The automorphism action transports the intrinsic projective multiplicity representation. It therefore preserves the scalar commutator pairing and its radical. The induced action on the quotient is symplectic.
1126 \end{proof}
1127 
1128 \begin{figure}[t]
1129 \centering
1130 \includegraphics[width=.96\textwidth]{10_one_break_envelope.pdf}
1131 \caption{The restricted one break construction is made on the Clifford multiplicity space.}
1132 \label{fig:onebreak}
1133 \end{figure}
1134 
1135 \begin{corollary}[Heisenberg representation attached to one break]\label{cor:one_break_heisenberg}
1136 Fix a nontrivial central character. The envelope \(V_\rho\) determines one irreducible finite Heisenberg representation up to isomorphism.
1137 \end{corollary}
1138 
1139 \begin{proof}
1140 Apply the finite Stone von Neumann theorem to the nondegenerate symplectic quotient.
1141 \end{proof}
1142 
1143 \begin{remark}\label{rem:clifford_scope}
1144 The group cohomology class records the extension obstruction for the multiplicity action. General background on projective representations and group cohomology can be found in \cite{Karpilovsky1985,Brown1982}. The theorem remains restricted because the finite quotient and the one break isotypic datum are assumed rather than constructed from arbitrary wild inertia.
1145 \end{remark}
1146 
1147 \section{The type side symplectic quotient}\label{sec:typeside}
1148 
1149 Let \(\theta\) be a maximal simple character for \(\GL_n(K)\), with compact open subgroups
1150 \[
1151 H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
1152 \]
1153 The quotient
1154 \[
1155 V_\theta=J_\theta^1/H_\theta^1
1156 \]
1157 is regarded as a finite vector space over \(\FF_p\), where \(p\) is the residue characteristic. Fix a nontrivial additive character \(\psi_0:\FF_p\to\mu_p\subset\CC^\times\), where \(\mu_p\) is the group of \(p\)-th roots of unity. The simple-character commutator takes values in \(\mu_p\). Since \(\psi_0\) is injective, the equation
1158 \[
1159 \psi_0\bigl(h_\theta(\bar x,\bar y)\bigr)
1160 =
1161 \theta([x,y])
1162 \]
1163 uniquely defines the \(\FF_p\)-valued commutator pairing \(h_\theta\). We use the same identification of \(\mu_p\) with the additive group of \(\FF_p\) on the Galois side when comparing the two symplectic spaces.
1164 
1165 \begin{standardfact}[Type side Heisenberg quotient]\label{fact:type_heisenberg}
1166 The pairing \(h_\theta\) is nondegenerate and alternating. There is one irreducible representation \(\eta_\theta\) of \(J_\theta^1\) containing \(\theta\), and its dimension is
1167 \[
1168 \sqrt{\abs{J_\theta^1/H_\theta^1}}.
1169 \]
1170 \end{standardfact}
1171 
1172 This is part of the simple character and Heisenberg representation theory used in the construction of types \cite{BushnellKutzko1993}. The extension of \(\eta_\theta\) from \(J_\theta^1\) to \(J_\theta\) is not unique before a normalization is chosen.
1173 
1174 \begin{standardfact}[Canonical beta extension]\label{fact:canonical_beta}
1175 The canonical beta extension is obtained from a primary extension together with finite symplectic and tame orientation corrections. The construction is functorial in the declared simple character data and removes the remaining extension ambiguity.
1176 \end{standardfact}
1177 
1178 The normalization and its relation to finite Weil representations are developed in \cite{Blondel2012,Dotto2022}. This chapter uses this theorem as type side input. It does not rederive simple character theory.
1179 
1180 \begin{proposition}[Type side projective commutator]\label{prop:type_commutator}
1181 The projective action of \(J_\theta/J_\theta^1\) on the Heisenberg representation preserves the commutator form on \(V_\theta\). Any two choices of projective lifts give the same alternating form.
1182 \end{proposition}
1183 
1184 \begin{proof}
1185 Conjugation by \(J_\theta\) preserves \(\theta\), so it preserves the commutator pairing defined from \(\theta([x,y])\). Two choices of projective lifts differ by scalars. The scalar commutator is therefore unchanged by \cref{prop:cocycle_choice}.
1186 \end{proof}
1187 
1188 \section{The restricted Galois to type bridge}\label{sec:bridge}
1189 
1190 The full bridge would construct an intrinsic Galois envelope for every wild atom and identify it canonically with \(V_\theta\). We prove the comparison after the required symplectic identification has been supplied.
1191 
1192 \begin{definition}[Bridge datum]\label{def:bridge_datum}
1193 A restricted bridge datum consists of
1194 
1195 \begin{enumerate}
1196 \item a one break Galois envelope \((V_\rho,b_\rho)\)
1197 \item a type side quotient \((V_\theta,h_\theta)\)
1198 \item an equivariant symplectic isometry
1199 \[
1200 \iota:V_\rho\longrightarrow V_\theta
1201 \]
1202 for a finite cyclic stabilizer \(C=\langle c\rangle\)
1203 \item the same nontrivial central character on the two Heisenberg groups
1204 \item when \(p=2\), compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting \(\iota\).
1205 \end{enumerate}
1206 \end{definition}
1207 
1208 \begin{theorem}[Restricted Heisenberg bridge]\label{thm:restricted_bridge}
1209 A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map
1210 \[
1211 U:\calH_\rho\longrightarrow\calH_\theta
1212 \]
1213 such that
1214 \[
1215 U\omega_\rho(h)
1216 =
1217 \omega_\theta(\iota_H(h))U
1218 \]
1219 for every Heisenberg element \(h\), and every other such map is a scalar multiple of \(U\).
1220 \end{theorem}
1221 
1222 \begin{proof}
1223 When \(p\) is odd, the symplectic isometry and the common central character induce the standard isomorphism \(\iota_H\) of the two Heisenberg groups. When \(p=2\), the required lift of \(\iota\) is part of the bridge datum because the alternating form alone does not determine the central extension. Pulling the type side representation back through \(\iota_H\) gives an irreducible Heisenberg representation with the same central character as the Galois side representation. The finite Stone von Neumann theorem gives an isomorphism. Schur's lemma gives uniqueness up to scalar.
1224 \end{proof}
1225 
1226 The stabilizer actions need not agree after choosing \(U\). Their discrepancy is scalar because both actions implement the same automorphisms of the irreducible Heisenberg representation.
1227 
1228 \begin{proposition}[Cyclic discrepancy character]\label{prop:cyclic_discrepancy}
1229 Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group \(C\). Then there is a character
1230 \[
1231 \delta:C\to\CC^{\times}
1232 \]
1233 such that
1234 \[
1235 U\Omega_\rho(g)
1236 =
1237 \delta(g)\Omega_\theta(g)U
1238 \]
1239 for every \(g\in C\).
1240 \end{proposition}
1241 
1242 \begin{proof}
1243 For each \(g\), the operator
1244 \[
1245 \Omega_\theta(g)^{-1}U\Omega_\rho(g)U^{-1}
1246 \]
1247 commutes with the type side Heisenberg representation. It is therefore scalar. Since both stabilizer actions are honest representations, multiplication of group elements shows that the scalars form a character.
1248 \end{proof}
1249 
1250 \begin{lemma}[Linearization over a cyclic stabilizer]\label{lem:cyclic_linearization}
1251 Every finite dimensional projective representation of a finite cyclic group over \(\CC\) is projectively equivalent to an honest representation.
1252 \end{lemma}
1253 
1254 \begin{proof}
1255 Let \(C=\langle c\rangle\) have order \(m\), and let \(T\) be a lift of the projective generator. Since \(c^m=1\), the operator \(T^m\) is scalar. Choose \(\lambda\in\CC^{\times}\) with \(\lambda^mT^m=I\). Replacing \(T\) by \(\lambda T\) gives an honest cyclic representation.
1256 \end{proof}
1257 
1258 \begin{theorem}[One phase closes the cyclic bridge]\label{thm:one_phase_bridge}
1259 In a restricted bridge datum, choose linearizations of the cyclic stabilizer actions. If the two actions have the same scalar phase on the generator \(c\), then the Heisenberg intertwiner can be chosen to intertwine the full semidirect product action.
1260 \end{theorem}
1261 
1262 \begin{proof}
1263 By \cref{prop:cyclic_discrepancy}, the discrepancy is a character \(\delta\). Equality of the generator phase gives \(\delta(c)=1\). Since \(c\) generates \(C\), the character is trivial. The chosen map \(U\) then intertwines every stabilizer element as well as the Heisenberg group.
1264 \end{proof}
1265 
1266 \begin{figure}[t]
1267 \centering
1268 \includegraphics[width=.94\textwidth]{11_restricted_bridge.pdf}
1269 \caption{An equivariant symplectic isometry reduces the finite comparison to one scalar cyclic phase.}
1270 \label{fig:restrictedbridge}
1271 \end{figure}
1272 
1273 \begin{corollary}[Bridge uniqueness]\label{cor:bridge_uniqueness}
1274 When the semidirect product representation is irreducible, the completed bridge is unique up to scalar.
1275 \end{corollary}
1276 
1277 \begin{proof}
1278 Any two completed bridges differ by an endomorphism commuting with the semidirect product representation. Schur's lemma applies.
1279 \end{proof}
1280 
1281 \begin{remark}\label{rem:bridge_scope}
1282 The theorem does not construct \(\iota\). It proves that, once an equivariant symplectic isometry exists, the remaining finite comparison is one dimensional for a cyclic stabilizer.
1283 \end{remark}
1284 
1285 \section{Odd residue characteristic phases}\label{sec:oddphase}
1286 
1287 Let \(p\) be odd, let \((V,b)\) be a finite symplectic \(\FF_p\) space of dimension \(2r\), and let
1288 \[
1289 g\in\Sp(V)
1290 \]
1291 have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature \cite{Gerardin1977,Neuhauser2002,GurevichHadani2007}.
1292 
1293 \begin{standardfact}[Fixed point free Weil trace]\label{fact:weil_trace}
1294 Under the declared normalization,
1295 \[
1296 \boxed{
1297 \Tr\omega(g)
1298 =
1299 \left(\frac{-1}{p}\right)^r
1300 \left(\frac{\det(g-I)}{p}\right).
1301 }
1302 \]
1303 Here \((\frac{\cdot}{p})\) denotes the quadratic character of \(\FF_p^{\times}\).
1304 \end{standardfact}
1305 
1306 \begin{definition}[Odd finite Weil phase]\label{def:odd_phase}
1307 For fixed point free \(g\), define
1308 \[
1309 \Phi_p(g,V)
1310 =
1311 \left(\frac{-1}{p}\right)^r
1312 \left(\frac{\det(g-I)}{p}\right).
1313 \]
1314 \end{definition}
1315 
1316 \begin{proposition}[Symplectic invariance]\label{prop:phase_invariance}
1317 If
1318 \[
1319 \iota:(V,b)\longrightarrow(V',b')
1320 \]
1321 is a symplectic isometry and \(g'=\iota g\iota^{-1}\), then
1322 \[
1323 \Phi_p(g,V)=\Phi_p(g',V').
1324 \]
1325 \end{proposition}
1326 
1327 \begin{proof}
1328 The dimensions agree, and conjugate matrices have the same determinant of \(g-I\).
1329 \end{proof}
1330 
1331 \begin{proposition}[Orthogonal multiplicativity]\label{prop:odd_multiplicativity}
1332 Suppose
1333 \[
1334 V=V_1\perpbig\cdots\perpbig V_s
1335 \]
1336 and \(g\) preserves every summand. If no \(g|_{V_j}\) has a fixed vector, then
1337 \[
1338 \boxed{
1339 \Phi_p(g,V)
1340 =
1341 \prod_{j=1}^{s}
1342 \Phi_p(g|_{V_j},V_j).
1343 }
1344 \]
1345 \end{proposition}
1346 
1347 \begin{proof}
1348 The half dimensions add, and
1349 \[
1350 \det(g-I\mid V)
1351 =
1352 \prod_j\det(g-I\mid V_j).
1353 \]
1354 The quadratic character is multiplicative.
1355 \end{proof}
1356 
1357 \begin{corollary}[Characteristic polynomial certificate]\label{cor:characteristic_phase}
1358 Let
1359 \[
1360 P_g(X)=\det(XI-g\mid V).
1361 \]
1362 Then
1363 \[
1364 \boxed{
1365 \Phi_p(g,V)
1366 =
1367 \left(\frac{-1}{p}\right)^r
1368 \left(\frac{P_g(1)}{p}\right).
1369 }
1370 \]
1371 \end{corollary}
1372 
1373 \begin{proof}
1374 Since \(\dim V\) is even,
1375 \[
1376 \det(g-I)=\det(I-g)=P_g(1).
1377 \]
1378 \end{proof}
1379 
1380 
1381 \begin{proposition}[Fixed and moving decomposition]\label{prop:fixed_moving}
1382 Assume that \(g\in\Sp(V)\) has order prime to \(p\). Then
1383 \[
1384 V=\ker(g-I)\perpbig\im(g-I).
1385 \]
1386 The restriction of the symplectic form to \(\im(g-I)\) is nondegenerate, and \(g-I\) is invertible on that moving summand.
1387 \end{proposition}
1388 
1389 \begin{proof}
1390 The order hypothesis makes \(g\) semisimple, so
1391 \[
1392 V=\ker(g-I)\oplus\im(g-I).
1393 \]
1394 If \(x\in\ker(g-I)\) and \(y=(g-I)z\), then
1395 \[
1396 b(x,y)=b(x,gz)-b(x,z)=b(g^{-1}x,z)-b(x,z)=0.
1397 \]
1398 Thus the two summands are orthogonal. Nondegeneracy on each summand follows from nondegeneracy on \(V\). The last statement is immediate from the direct sum decomposition.
1399 \end{proof}
1400 
1401 \begin{definition}[Moving phase]\label{def:moving_phase}
1402 Under the hypotheses of \cref{prop:fixed_moving}, define
1403 \[
1404 \Phi_p^{\mathrm{mov}}(g,V)
1405 =
1406 \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
1407 \]
1408 \end{definition}
1409 
1410 \begin{proposition}[Hyperbolic phase formula]\label{prop:hyperbolic_phase}
1411 Let \(U\) be a finite dimensional \(\FF_p\) space, let \(a\in\GL(U)\) have no eigenvalue one, and let \(g\) act on
1412 \[
1413 H(U)=U\oplus U^*
1414 \]
1415 by
1416 \[
1417 g=(a,(a^{-1})^*).
1418 \]
1419 Then
1420 \[
1421 \boxed{
1422 \Phi_p(g,H(U))
1423 =
1424 \left(\frac{\det a}{p}\right).
1425 }
1426 \]
1427 \end{proposition}
1428 
1429 \begin{proof}
1430 Put \(d=\dim U\). We have
1431 \[
1432 \det\bigl((a^{-1})^*-I\bigr)
1433 =
1434 (-1)^d(\det a)^{-1}\det(a-I).
1435 \]
1436 Hence
1437 \[
1438 \det(g-I)
1439 =
1440 (-1)^d(\det a)^{-1}\det(a-I)^2.
1441 \]
1442 Multiplying its quadratic character by the normalization factor \((\frac{-1}{p})^d\) cancels the sign and the square. Since a quadratic character equals its inverse, the result is \((\frac{\det a}{p})\).
1443 \end{proof}
1444 
1445 \begin{corollary}[Phase only bridge test]\label{cor:phase_only_test}
1446 Two odd characteristic moving packets have the same normalized phase whenever their moving half dimensions agree and the determinants of \(g-I\) lie in the same square class.
1447 \end{corollary}
1448 
1449 \begin{proof}
1450 Apply the definition of \(\Phi_p\).
1451 \end{proof}
1452 
1453 \begin{figure}[t]
1454 \centering
1455 \includegraphics[width=.88\textwidth]{12_odd_phase_blocks.pdf}
1456 \caption{Odd finite Weil phases multiply across orthogonal atomic blocks.}
1457 \label{fig:oddphase}
1458 \end{figure}
1459 
1460 \begin{remark}\label{rem:fixed_space}
1461 When \(g\) has fixed vectors, one first separates the fixed part and applies the formula to the moving quotient. Additional normalization data may enter the full trace on the fixed part.
1462 \end{remark}
1463 
1464 \section{Dyadic quadratic refinements}\label{sec:dyadic}
1465 
1466 Let \((V,b)\) be a symplectic vector space over \(\FF_2\). The alternating form alone is not enough to determine the finite phase.
1467 
1468 \begin{definition}[Quadratic refinement]\label{def:quadratic_refinement}
1469 A quadratic refinement of \(b\) is a map
1470 \[
1471 q:V\to\FF_2
1472 \]
1473 satisfying
1474 \[
1475 q(x+y)-q(x)-q(y)=b(x,y).
1476 \]
1477 \end{definition}
1478 
1479 \begin{proposition}[Affine space of refinements]\label{prop:refinement_affine}
1480 If one quadratic refinement exists, then the set of all quadratic refinements is an affine space over \(V^*\).
1481 \end{proposition}
1482 
1483 \begin{proof}
1484 The difference of two refinements has zero polar form, so it is additive and hence linear over \(\FF_2\). Conversely, adding a linear form does not change the polar form.
1485 \end{proof}
1486 
1487 \begin{definition}[Arf invariant]\label{def:arf}
1488 Choose a symplectic basis \((e_i,f_i)_{i=1}^{r}\). Define
1489 \[
1490 \Arf(q)
1491 =
1492 \sum_{i=1}^{r}q(e_i)q(f_i)
1493 \in\FF_2.
1494 \]
1495 \end{definition}
1496 
1497 \begin{standardfact}[Arf invariance and classification]\label{fact:arf_classification}
1498 The value in \cref{def:arf} is independent of the symplectic basis. Two nondegenerate quadratic refinements of one finite symplectic space are isometric exactly when their Arf invariants agree.
1499 \end{standardfact}
1500 
1501 This is the classical classification of nonsingular quadratic forms over \(\FF_2\) \cite{MilnorHusemoller1973}.
1502 
1503 \begin{theorem}[Dyadic Gauss sign]\label{thm:arf_gauss}
1504 Let \(\dim V=2r\), and let \(q\) be nondegenerate. Then
1505 \[
1506 \boxed{
1507 \sum_{x\in V}(-1)^{q(x)}
1508 =
1509 (-1)^{\Arf(q)}2^r.
1510 }
1511 \]
1512 \end{theorem}
1513 
1514 \begin{proof}
1515 Orthogonally decompose \((V,q)\) into two dimensional nonsingular planes. On a plane with Arf invariant zero, the four values of \((-1)^q\) sum to \(2\). On a plane with Arf invariant one, they sum to \(-2\). The total sum is the product of the plane sums. The parity of the number of anisotropic planes is the Arf invariant.
1516 \end{proof}
1517 
1518 \begin{corollary}[Orthogonal dyadic multiplicativity]\label{cor:arf_multiplicative}
1519 For an orthogonal sum
1520 \[
1521 (V,q)=(V_1,q_1)\perpbig(V_2,q_2),
1522 \]
1523 we have
1524 \[
1525 \Arf(q)=\Arf(q_1)+\Arf(q_2)
1526 \]
1527 and the normalized Gauss sign is the product of the two normalized signs.
1528 \end{corollary}
1529 
1530 \begin{proof}
1531 Take symplectic bases of the two summands and concatenate them. The Gauss sum factors as a product over the Cartesian decomposition.
1532 \end{proof}
1533 
1534 \begin{proposition}[The form alone does not determine the phase]\label{prop:dyadic_need_q}
1535 On every nonzero symplectic \(\FF_2\) space, there are quadratic refinements of both Arf invariants. Their polar forms agree, but their normalized Gauss signs are opposite.
1536 \end{proposition}
1537 
1538 \begin{proof}
1539 On a symplectic plane, the standard hyperbolic refinement has Arf invariant zero. Adding a suitable nonzero linear form produces a refinement with Arf invariant one. Orthogonally adjoining a fixed refinement on the complement gives the two choices in every positive even dimension. The signs differ by \cref{thm:arf_gauss}.
1540 \end{proof}
1541 
1542 
1543 \begin{proposition}[Translation of quadratic refinements]\label{prop:quadratic_translation}
1544 For \(v\in V\), put
1545 \[
1546 q_v(x)=q(x)+b(v,x).
1547 \]
1548 Then
1549 \[
1550 \boxed{
1551 \Arf(q_v)=\Arf(q)+q(v).
1552 }
1553 \]
1554 \end{proposition}
1555 
1556 \begin{proof}
1557 Using the quadratic identity,
1558 \[
1559 q_v(x)=q(x+v)+q(v).
1560 \]
1561 Therefore
1562 \[
1563 \sum_x(-1)^{q_v(x)}
1564 =
1565 (-1)^{q(v)}
1566 \sum_x(-1)^{q(x)}.
1567 \]
1568 Compare the signs in \cref{thm:arf_gauss}.
1569 \end{proof}
1570 
1571 \begin{theorem}[Number of dyadic refinements]\label{thm:quadratic_counts}
1572 Let \(\dim V=2r\). The number of quadratic refinements with Arf invariant zero is
1573 \[
1574 \boxed{
1575 2^{r-1}(2^r+1),
1576 }
1577 \]
1578 and the number with Arf invariant one is
1579 \[
1580 \boxed{
1581 2^{r-1}(2^r-1).
1582 }
1583 \]
1584 \end{theorem}
1585 
1586 \begin{proof}
1587 Every refinement is \(q_v\) for a unique \(v\in V\), because \(b\) identifies \(V\) with \(V^*\). By \cref{prop:quadratic_translation}, the Arf invariant changes according to \(q(v)\). Let \(Z\) and \(O\) be the numbers of zeros and ones of \(q\). Then
1588 \[
1589 Z+O=2^{2r}
1590 \]
1591 and \cref{thm:arf_gauss} gives
1592 \[
1593 Z-O=(-1)^{\Arf(q)}2^r.
1594 \]
1595 Solving gives the two displayed numbers, with their roles exchanged when the starting refinement has Arf invariant one. Since translation runs through every refinement, the final counts are independent of the starting choice.
1596 \end{proof}
1597 
1598 \begin{figure}[t]
1599 \centering
1600 \includegraphics[width=.92\textwidth]{13_dyadic_quadratic_refinement.pdf}
1601 \caption{At residue characteristic two, the quadratic refinement and its Arf invariant carry phase information not contained in the alternating form.}
1602 \label{fig:dyadic}
1603 \end{figure}
1604 
1605 \section{Matrix holonomy and finite gluing}\label{sec:holonomy}
1606 
1607 Let \(\Gamma\) be a connected finite graph. At every vertex \(v\), place finite dimensional complex vector spaces \(V_v\) and \(W_v\) of the same dimension. For every oriented edge \(e:u\to v\), let
1608 \[
1609 A_e:V_u\longrightarrow V_v,
1610 \qquad
1611 B_e:W_u\longrightarrow W_v
1612 \]
1613 be invertible. The reversed edge carries the inverse map.
1614 
1615 \begin{definition}[Path transport]\label{def:path_transport}
1616 For an oriented path
1617 \[
1618 \gamma=(v_0\to v_1\to\cdots\to v_m),
1619 \]
1620 define
1621 \[
1622 A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0},
1623 \qquad
1624 B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
1625 \]
1626 For a loop based at \(r\), these are the based holonomies.
1627 \end{definition}
1628 
1629 \begin{theorem}[Matrix holonomy criterion]\label{thm:matrix_holonomy}
1630 Fix a root \(r\) and an isomorphism
1631 \[
1632 U_r:V_r\longrightarrow W_r.
1633 \]
1634 The following conditions are equivalent.
1635 
1636 \begin{enumerate}
1637 \item There are isomorphisms \(U_v:V_v\to W_v\), with the declared \(U_r\), such that
1638 \[
1639 U_vA_e=B_eU_u
1640 \]
1641 for every edge \(e:u\to v\).
1642 \item For one fundamental cycle basis based at \(r\),
1643 \[
1644 U_rA_\gamma=B_\gamma U_r.
1645 \]
1646 \item The same identity holds for every based loop \(\gamma\).
1647 \end{enumerate}
1648 \end{theorem}
1649 
1650 \begin{proof}
1651 If the vertex maps exist, multiply the edge equations around a loop. Every intermediate vertex map cancels, giving condition three. Condition three implies condition two.
1652 
1653 Conversely, choose a spanning tree. Starting from \(U_r\), define \(U_v\) by transport along the unique tree path from \(r\) to \(v\). The edge equations hold on every tree edge by construction. Every non tree edge closes one fundamental cycle. The edge equation on that edge is equivalent to the root holonomy identity for its fundamental cycle. Condition two therefore gives all remaining edge equations.
1654 \end{proof}
1655 
1656 \begin{corollary}[Unbased gauge classification]\label{cor:unbased_holonomy}
1657 There exists some family of vertex isomorphisms if and only if the two holonomy representations
1658 \[
1659 \pi_1(\Gamma,r)\longrightarrow\GL(V_r),
1660 \qquad
1661 \pi_1(\Gamma,r)\longrightarrow\GL(W_r)
1662 \]
1663 are conjugate.
1664 \end{corollary}
1665 
1666 \begin{proof}
1667 Apply \cref{thm:matrix_holonomy} and allow \(U_r\) to vary.
1668 \end{proof}
1669 
1670 \begin{corollary}[Scalar and projective gluing]\label{cor:scalar_holonomy}
1671 If every based loop acts by a scalar, finite gluing is equivalent to equality of the scalar holonomies on a fundamental cycle basis. The number of independent tests is
1672 \[
1673 \boxed{
1674 \abs E-\abs V+1.
1675 }
1676 \]
1677 \end{corollary}
1678 
1679 \begin{proof}
1680 A connected graph has a spanning tree with \(\abs V-1\) edges. Every remaining edge gives one fundamental cycle. In the scalar case, conjugacy has no effect.
1681 \end{proof}
1682 
1683 \begin{proposition}[Projective cocycle form]\label{prop:projective_holonomy}
1684 Suppose the edge maps are defined only up to scalar, in the setting where all based holonomies of both systems are scalar. Choose representatives. The ratio of the two scalar holonomies on each loop is unchanged by vertex rescaling. It defines a character of the first homology group when the scalar transport is abelian.
1685 \end{proposition}
1686 
1687 \begin{proof}
1688 A vertex rescaling appears once with positive exponent and once with negative exponent around a closed path, so it cancels. Multiplication of loops multiplies the scalar ratios. In the abelian case, the ratio factors through the abelianization of the fundamental group, which is the first homology group.
1689 \end{proof}
1690 
1691 
1692 \begin{proposition}[Exact edge residual from cycle defect]\label{prop:holonomy_residual}
1693 Fix a spanning tree and construct the vertex maps by tree transport from \(U_r\). Let \(e:u\to v\) be a non tree edge. Let \(p_u\) and \(p_v\) be the tree paths from the root to \(u\) and \(v\). Define the fundamental cycle defect
1694 \[
1695 D_e
1696 =
1697 U_rA_{p_v}^{-1}A_eA_{p_u}
1698 -
1699 B_{p_v}^{-1}B_eB_{p_u}U_r.
1700 \]
1701 Then the edge residual is exactly
1702 \[
1703 \boxed{
1704 U_vA_e-B_eU_u
1705 =
1706 B_{p_v}D_eA_{p_u}^{-1}.
1707 }
1708 \]
1709 \end{proposition}
1710 
1711 \begin{proof}
1712 Substitute
1713 \[
1714 U_u=B_{p_u}U_rA_{p_u}^{-1},
1715 \qquad
1716 U_v=B_{p_v}U_rA_{p_v}^{-1}
1717 \]
1718 into the left side and factor \(B_{p_v}\) on the left and \(A_{p_u}^{-1}\) on the right.
1719 \end{proof}
1720 
1721 \begin{corollary}[Quantitative finite gluing]\label{cor:quantitative_holonomy}
1722 With the notation of \cref{prop:holonomy_residual},
1723 \[
1724 \norm{U_vA_e-B_eU_u}
1725 \leq
1726 \norm{B_{p_v}}\norm{D_e}\norm{A_{p_u}^{-1}}.
1727 \]
1728 Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.
1729 \end{corollary}
1730 
1731 \begin{proof}
1732 Take operator norms in the exact identity.
1733 \end{proof}
1734 
1735 \begin{figure}[t]
1736 \centering
1737 \includegraphics[width=.72\textwidth]{14_matrix_holonomy.pdf}
1738 \caption{A spanning tree fixes the vertex gauges. The non tree edges test the remaining cycle holonomies.}
1739 \label{fig:holonomy}
1740 \end{figure}
1741 
1742 \begin{remark}\label{rem:holonomy_boundary}
1743 The theorem is an exact finite gluing result. It assumes that the packet spaces and edge transports have already been constructed on both sides.
1744 \end{remark}
1745 
1746 \section{Breakwise assembly}\label{sec:breakwise}
1747 
1748 Wild inertia carries an upper ramification filtration. In favorable settings, a wild atom separates into finitely many break pieces. We state the finite algebra needed once such pieces are intrinsic.
1749 
1750 \begin{definition}[Orthogonal break decomposition]\label{def:break_decomposition}
1751 An orthogonal break decomposition is a symplectic direct sum
1752 \[
1753 V=\perpbig_{s\in S}V_s
1754 \]
1755 that is preserved by the declared tame stabilizer. Each \(V_s\) is the finite symplectic packet assigned to one break.
1756 \end{definition}
1757 
1758 \begin{proposition}[Heisenberg tensor factorization]\label{prop:heisenberg_tensor}
1759 Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands:
1760 \[
1761 \boxed{
1762 \omega_V
1763 \cong
1764 \bigotimes_{s\in S}\omega_{V_s}.
1765 }
1766 \]
1767 \end{proposition}
1768 
1769 \begin{proof}
1770 The Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension
1771 \[
1772 \prod_s\sqrt{\abs{V_s}}
1773 =
1774 \sqrt{\abs V}.
1775 \]
1776 It is irreducible by the finite Stone von Neumann theorem, so it is the required representation.
1777 \end{proof}
1778 
1779 \begin{corollary}[Breakwise phase product]\label{cor:break_phase}
1780 In odd residue characteristic, let \(g\in\Sp(V)\) preserve every break packet and suppose its restriction to \(M=\im(g-I)\) is fixed point free. Put \(g_s=g|_{V_s}\) and \(M_s=\im(g_s-I)\). The moving restrictions are symplectic, and
1781 \[
1782 \boxed{
1783 \Phi_p(g|_M,M)
1784 =
1785 \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s).
1786 }
1787 \]
1788 In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
1789 \end{corollary}
1790 
1791 \begin{proof}
1792 The invariant orthogonal decomposition gives \(M=\mathop{\perp}\limits_{s\in S}M_s\). Symplectic invariance gives \(M^\perp=\ker(g-I)\). Fixed point freeness on \(M\) therefore makes the restricted form nondegenerate on \(M\) and on each \(M_s\). Each restriction of \(g_s-I\) to \(M_s\) is invertible, so \cref{prop:odd_multiplicativity} applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply \cref{cor:arf_multiplicative}.
1793 \end{proof}
1794 
1795 \begin{theorem}[Breakwise restricted bridge]\label{thm:breakwise_bridge}
1796 Suppose every break has a restricted bridge datum, the symplectic isometries are compatible with the stabilizer action, and the scalar phases agree on a generating set of the stabilizer. Then the tensor product of the breakwise Heisenberg intertwiners gives a bridge for the full finite envelope. Any remaining gluing obstruction is exactly the matrix holonomy obstruction of \cref{thm:matrix_holonomy}.
1797 \end{theorem}
1798 
1799 \begin{proof}
1800 Tensor the breakwise Heisenberg intertwiners from \cref{thm:restricted_bridge}. Phase agreement makes the tensor map equivariant on the chosen stabilizer generators. The breakwise direct sum and tensor factorization make the local comparisons compatible. If the packets are arranged over a transport graph, the only remaining condition is consistency of the local maps around cycles, which is exactly the holonomy criterion.
1801 \end{proof}
1802 
1803 \begin{figure}[t]
1804 \centering
1805 \includegraphics[width=.9\textwidth]{15_breakwise_decomposition.pdf}
1806 \caption{Orthogonal break packets produce tensor factors of the finite Heisenberg representation.}
1807 \label{fig:breakwise}
1808 \end{figure}
1809 
1810 \section{Local factor tests and the remaining bridge}\label{sec:boundary}
1811 
1812 Local class field theory identifies one dimensional Weil characters with characters of \(K^{\times}\) \cite{Serre1967,Tate1979}. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in \cref{prop:tame_local_polynomial,prop:tame_conductor} therefore give explicit finite invariants for the declared tame packets. Finite Gauss sums give the residue field phase. The full local factor, however, also includes ramification depth, Haar measure normalization, the additive character conductor, and wild interaction data.
1813 
1814 \Needspace{13\baselineskip}
1815 \begin{proposition}[Finite tame comparison package]\label{prop:tame_package}
1816 For a tame induced atom, the following data are determined by its packet coordinates:
1817 
1818 \begin{enumerate}
1819 \item the dimension and inertia orbit
1820 \item the Frobenius cycle polynomial
1821 \item the Artin conductor
1822 \item the local polynomial
1823 \item the tensor refinement into interaction packets
1824 \item the finite Gauss sum of each declared residue character, with unit-modulus phase normalization when that character is nontrivial.
1825 \end{enumerate}
1826 \end{proposition}
1827 
1828 \begin{proof}
1829 The dimension and orbit follow from \cref{thm:tame_induction}. The cycle polynomial is \cref{prop:frobenius_cycle}. For rank greater than one, the conductor is \cref{prop:tame_conductor}. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula \(\operatorname{Art}=\dim V-\dim V^{I_K}+\operatorname{Sw}\). The local polynomial is \cref{prop:tame_local_polynomial}. Tensor refinement is \cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase is \cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \(\psi\), the sum is \(G(1,\psi)=-1\), because the sum of \(\psi\) over the whole field is zero.
1830 \end{proof}
1831 
1832 \subsection{Arithmetic realization and the conditional comparison}
1833 
1834 Fix this chapter's reciprocity, additive-character, local-factor, and phase
1835 normalizations. Let \(\mathcal A\) be the declared class of irreducible wild
1836 Weil atoms under consideration. For \(\rho\in\mathcal A\), the supplied
1837 assembled finite datum \(d_\rho\) consists of the following inputs,
1838 historically denoted H1--H4:
1839 \begin{enumerate}
1840 \item an intrinsic finite break decomposition on the Galois side;
1841 \item the declared equivariant symplectic bridges to the supplied breakwise
1842 type-side quotients;
1843 \item compatible odd or dyadic central-extension and phase data on every
1844 break, including every prescribed finite marking;
1845 \item a compatible transport system with vanishing matrix holonomy
1846 obstruction.
1847 \end{enumerate}
1848 These are finite comparison inputs. Supplying separate breakwise target
1849 types does not itself supply one common arithmetic candidate. If a common
1850 candidate is already supplied, that is a stronger restricted hypothesis.
1851 
1852 An \emph{arithmetic realization witness}
1853 \(\mathcal R(d_\rho,\pi)\) consists of actual arithmetic inducing data
1854 for an admissible supercuspidal \(\pi\), together with explicit
1855 comparison and recovery maps carrying the prescribed assembled finite
1856 packet. The maps preserve its declared stabilizer action, all character values
1857 explicitly included in the supplied datum, central-extension and phase
1858 data, and compatible transports.
1859 Recovery returns that same \(\pi\), with its fixed central character and
1860 noncompact completion. An enlarged actual inducing carrier is permitted.
1861 The whole packet is not required to be the original maximal-simple-type
1862 quotient. Any stronger native-marking or numerical-layer requirement is
1863 part of the declared datum and must be verified separately.
1864 
1865 \begingroup
1866 \renewcommand{\theconjecture}{66}
1867 \begin{conjecture}[Arithmetic packet realization]
1868 \label{conj:arithmetic_realization}\label{thm:conditional_atomic}
1869 For every \(\rho\in\mathcal A\) with its supplied H1--H4 assembled
1870 finite datum \(d_\rho\), there exists an actual admissible supercuspidal
1871 representation \(\pi\) of \(\GL_{\dim\rho}(K)\) and an arithmetic
1872 realization witness \(\mathcal R(d_\rho,\pi)\).
1873 \end{conjecture}
1874 \endgroup
1875 
1876 \noindent This is the realization problem isolated from historical
1877 Theorem 16.2 and original proof obligation 66 (row 95 of the historical
1878 all-records register). It asks for one witness for each
1879 admitted source, rather than for success of every proposed host or every
1880 noncompact completion. No proof of the unrestricted statement is claimed.
1881 
1882 \begin{proposition}[Comparison of one realized candidate]
1883 \label{prop:realized_candidate_comparison}
1884 Let \(\rho\in\mathcal A\), and let \(\pi\) be one actual admissible
1885 supercuspidal representation of \(\GL_{\dim\rho}(K)\) with
1886 \(\mathcal R(d_\rho,\pi)\). Suppose H5 holds for this same \(\pi\):
1887 its required pair local factors equal those prescribed for \(\rho\)
1888 on the stipulated characterizing family of twists, with the fixed
1889 normalizations. Assume the stated local-factor characterization applies
1890 to that family and class. Then \(\pi\) is isomorphic to the representation
1891 assigned to \(\rho\) by the established local Langlands correspondence.
1892 \end{proposition}
1893 
1894 \begin{proof}
1895 H5 identifies the tested local factors for this actual candidate. The
1896 applicable local-factor characterization identifies its isomorphism
1897 class with the established correspondence \cite{Henniart1993}. The
1898 realization witness ensures that it is this same candidate which carries
1899 the prescribed assembled packet. No existence conclusion is inferred
1900 from the finite gluing conditions alone.
1901 \end{proof}
1902 
1903 \begin{remark}\label{rem:conditional_not_proof}
1904 H5 remains a conditional premise about the same fixed candidate. It is
1905 not a formal consequence of the finite symplectic bridge, conductor
1906 profiles, or a choice of noncompact root. This proposition adds no
1907 unconditional pair-factor obligation to the construction conjecture.
1908 \end{remark}
1909 
1910 \subsection{Exact open boundary}
1911 
1912 The following issues delimit the unrestricted program. The first four construct and normalize the finite inputs for broader source classes. Conjecture 66 asks for arithmetic realization once H1--H4 are supplied. The fifth is the independent route to replacing the retained conditional H5 input.
1913 
1914 \begin{enumerate}
1915 \item Construct a choice free finite symplectic envelope from an arbitrary irreducible wild inertia atom, with its ramification filtration and tame stabilizer action.
1916 \item Prove a canonical equivariant symplectic isometry from that envelope to the type side quotient \(J_\theta^1/H_\theta^1\).
1917 \item Recover the canonical beta extension phase in every break, including the required dyadic quadratic refinement.
1918 \item Prove matrix holonomy compatibility for the resulting packet transport system.
1919 \item Compare the full family of pair \(L\), epsilon, and gamma factors without invoking the established correspondence.
1920 \end{enumerate}
1921 
1922 \begin{figure}[t]
1923 \centering
1924 \includegraphics[width=.78\textwidth]{16_boundary_ladder.pdf}
1925 \caption{The proved finite layers and the remaining independent local correspondence steps.}
1926 \label{fig:boundary}
1927 \end{figure}
1928 
1929 The first item is a construction problem. The second is a comparison problem. The third is a normalization problem. The fourth is a descent problem. The fifth concerns proving the local factor premise needed for identification. Conjecture 66 does not assert this fifth step. None is replaced by a larger finite computation.
1930 
1931 \section{Examples and finite certificates}\label{sec:examples}
1932 
1933 The examples in this section illustrate the exact finite formulas. They are not used as proofs of the general theorems.
1934 
1935 \begin{example}[A degree four tame orbit]\label{ex:tame_four_again}
1936 Over \(\FF_2\), the exponent orbit
1937 \[
1938 1\longmapsto2\longmapsto4\longmapsto8\longmapsto1
1939 \]
1940 has length four. It therefore gives a tame atom induced from the unramified extension of degree four. The Frobenius transfer has one four cycle, and the number of regular degree four orbits is three by \cref{prop:mobius_count}.
1941 \end{example}
1942 
1943 \begin{example}[Tensor packets of lengths four and six]\label{ex:tensor_46}
1944 For \(n=4\) and \(m=6\),
1945 \[
1946 \gcd(4,6)=2,
1947 \qquad
1948 \lcm(4,6)=12.
1949 \]
1950 The tensor product splits into two interaction packets of length twelve. Their total dimension is \(2\cdot12=24\), equal to the product \(4\cdot6\).
1951 \end{example}
1952 
1953 \begin{example}[Rank one Heisenberg group]\label{ex:heisenberg_rank_one}
1954 Take \(V=\FF_3^2\). The Heisenberg group has order \(3^3=27\), and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.
1955 \end{example}
1956 
1957 \begin{example}[Nondegenerate twisted algebra]\label{ex:twisted_plane}
1958 Let \(A=\FF_p^2\) with its standard alternating form. A cocycle with that commutator form gives
1959 \[
1960 \CC^c[A]\cong M_p(\CC).
1961 \]
1962 The unique irreducible projective representation has dimension \(p\).
1963 \end{example}
1964 
1965 \begin{example}[Dyadic phase ambiguity]\label{ex:dyadic_plane}
1966 On a symplectic plane over \(\FF_2\), choose refinements \(q_0\) and \(q_1\) with Arf invariants zero and one. Their polar forms are the same, while
1967 \[
1968 \sum_x(-1)^{q_0(x)}=2,
1969 \qquad
1970 \sum_x(-1)^{q_1(x)}=-2.
1971 \]
1972 Thus the alternating packet alone cannot select the dyadic sign.
1973 \end{example}
1974 
1975 \begin{example}[One cycle gluing obstruction]\label{ex:one_cycle}
1976 Let a connected graph have one non tree edge. Tree transport determines every vertex map from one root map. The final edge equation holds exactly when the two holonomies around the unique fundamental cycle intertwine. Changing the scalar on that edge changes the cycle holonomy and produces a genuine obstruction.
1977 \end{example}
1978 
1979 \section{Scope of the finite reconstruction}\label{sec:scope}
1980 
1981 This chapter proves the following statements under their displayed hypotheses.
1982 
1983 \begin{enumerate}
1984 \item Tame irreducible Weil atoms are transitive Frobenius packets induced from unramified extensions.
1985 \item Regular tame packets have the stated Möbius count.
1986 \item Finite cuspidal blocks have exact primitive central projectors.
1987 \item Tame tensor products satisfy the greatest common divisor and least common multiple packet formula.
1988 \item Tame local polynomials, conductors, and finite Gauss phases are explicit.
1989 \item Nondegenerate scalar projective commutators give full matrix twisted group algebras and unique finite Heisenberg representations.
1990 \item A declared one break Clifford datum has a choice independent radical free symplectic envelope.
1991 \item An equivariant symplectic isometry gives a Heisenberg bridge unique up to scalar.
1992 \item One generator phase removes the scalar discrepancy for a cyclic stabilizer.
1993 \item Odd finite Weil phases and dyadic Arf signs are multiplicative across orthogonal break packets.
1994 \item Matrix holonomy on a fundamental cycle basis is the exact finite gluing obstruction.
1995 \end{enumerate}
1996 
1997 This chapter does not independently prove the local Langlands correspondence, simple character theory, the canonical beta extension, or universal wild companion existence. It does not construct the Galois to type symplectic isometry for an arbitrary wild atom. It does not compare all pair local factors in the wild case.
1998 
1999 \section{Restricted arithmetic realization theorems}
2000 \label{sec:arithmetic_progress}
2001 
2002 The later constructions provide actual arithmetic candidates and exact
2003 finite packet recovery in substantial restricted families. They do not
2004 establish \cref{conj:arithmetic_realization} for every admitted source.
2005 Their detailed ordinary proofs are typeset in the appendices, and the
2006 original reviewed Markdown files accompany this edition. In these
2007 appendices \(F\) denotes the base local field, while \(K=F[\beta]\)
2008 denotes a parameter field. These are local notation conventions.
2009 All references to ``claim 66'' in the retained notes mean the general
2010 endpoint now isolated as Conjecture 66.
2011 
2012 \subsection{The odd-prime trace-zero construction}
2013 
2014 \begin{theorem}[Restricted odd-prime arithmetic realization]
2015 \label{thm:odd_arithmetic}
2016 Let \(p\) be an odd prime, \(F=\FF_p((t))\), fix \(d\ge1\), and let
2017 \[
2018 0<h_1<\cdots<h_d,\qquad p\nmid h_i,
2019 \qquad q_i=(p+1)h_i,\qquad N=p^d.
2020 \]
2021 There is an explicit maximal simple stratum in \(M_N(F)\) with totally
2022 ramified full-degree parameter field, a literal defining sequence of
2023 degrees \(p,p^2,\ldots,p^d\), and levels
2024 \(n_j=p^{d-j}q_{d-j+1}\). Every native parameter and consecutive
2025 correction has trace zero over its recursive base field.
2026 The stratum admits a full simple character trivial on \(U_F^1\).
2027 Its fixed canonical beta extension can be completed to an actual
2028 irreducible admissible supercuspidal \(\pi\) with any prescribed smooth
2029 central quasicharacter trivial on \(U_F^1\), when its completion choices
2030 are free or already compatible.
2031 
2032 Assume actually supplied labeled full-character extensions with whole
2033 two-dimensional symplectic quotients whose balanced complex-scalar
2034 central product has \(d\) planes, one vertex, and trivial stabilizer.
2035 Then the same \(\pi\) admits a full scalar-pushout realization of that
2036 packet and exact recovery of its unchanged inducing representation.
2037 The prescribed central character must lie in the covered tame class.
2038 No literal embedding of the old matrix groups, canonical native marking,
2039 or prescribed numerical host-depth placement is inferred.
2040 \end{theorem}
2041 
2042 \begin{proof}
2043 The explicit Artin--Schreier recursion and its normalized filtered lift
2044 construct the maximal parameter and literal defining sequence. See
2045 \cref{app:odd_tower}. Its trace transport identity makes every native
2046 trace zero, rather than relying on amplification by \(p\). Native-rank
2047 full-character construction, full beta-determinant transfer, trace-zero
2048 correction, and maximal-field character extension give the scalar
2049 triviality and central completion. See \cref{app:odd_character}.
2050 The actual leading symplectic layer has dimension
2051 \((p-1)p^{d-1}\ge2d\) and survives every later approximation
2052 (\cref{app:odd_capacity}). A labeled symplectic embedding gives a
2053 center-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann
2054 decomposition separates the source factor from its orthogonal
2055 complement. Maximal-isotropic averaging on the complementary Heisenberg
2056 factor selects a one-dimensional representation line and hence a
2057 source-sized sector. The retained induction/idempotent recovery returns the same
2058 \(\Lambda\) and \(\pi\), as proved in \cref{app:odd_synthesis}.
2059 \end{proof}
2060 
2061 For \(p>3\), a native minimal rank-\(p\) type quotient has dimension
2062 \(p-1\), so it is not automatically a plane. The theorem requires
2063 genuinely supplied whole-plane data. Independence of source tensor
2064 factors is also an input. An arbitrary single-factor wild determinant
2065 need not be tame.
2066 
2067 Under additional actual-inertia hypotheses, genuine scalar phases on
2068 the entire source carrier are realized by actual central matrices of a
2069 finite cover and descend to the stipulated operators. A direct matrix
2070 homomorphism from the stabilizer itself need not exist. The exact
2071 statement and descent proof are in \cref{app:odd_scalar}.
2072 
2073 \subsection{The weighted dyadic construction and enlarged carriers}
2074 
2075 \begin{theorem}[Restricted dyadic marked assembly realization]
2076 \label{thm:dyadic_arithmetic}
2077 Let \(F=\FF_2((t))\), fix an additive character of conductor
2078 \(t\OO_F\), fix \(d\ge1\), and let \(h_1<\cdots<h_d\) be positive odd integers.
2079 Put \(q_i=3h_i\), \(N=2^d\), and
2080 \(n_j=2^{d-j}q_{d-j+1}\). The explicit recursive matrix generates a
2081 separable totally ramified degree-\(N\) maximal parameter field,
2082 normalizes the complete lattice chain, and has the literal defining
2083 sequence of degrees \(2,4,\ldots,2^d\). It admits a full simple character
2084 trivial on \(U_F^1\) and an actual supercuspidal completion with any
2085 prescribed central quasicharacter trivial on \(U_F^1\), when the
2086 noncompact completion is free or already compatible.
2087 
2088 For actually supplied full marked whole-plane scalar-pushout data with
2089 one vertex, trivial stabilizer, and prescribed source central character
2090 trivial on \(U_F^1\), an arithmetic assembly carrier exists
2091 at every length and recovers the same candidate: inside its original
2092 type for \(d\ge3\), and in explicit enlarged inducing carriers for
2093 \(d=1,2\). Every supplied full character value and transported whole
2094 finite subgroup, lift, square, and phase marking is retained at the
2095 represented marking level. Any central-character kernel already factored
2096 out remains recorded separately. Scalar pushout does not recover its
2097 invisible elements. This does
2098 not identify the transported marking with an independently prescribed
2099 canonical native marking.
2100 \end{theorem}
2101 
2102 \begin{proof}
2103 The Artin--Schreier filtered lift has native trace one. Consequently
2104 each native correction \(\delta\) satisfies
2105 \[
2106 v_F(\operatorname{tr}\delta)+s\ge1 \quad (s=1),
2107 \]
2108 which makes its additive scalar correction trivial on \(U_F^1\).
2109 The actual field and defining-sequence proof is
2110 \cref{app:dyadic_tower}. Full beta-determinant transfer and character
2111 extension at each new maximal native field prove the full-character
2112 claim, including the odd-critical endpoint
2113 \(2(\lfloor r/2\rfloor+1)-r=1\). See \cref{app:dyadic_character}.
2114 The exact original quotient has dimension \(2^d-2\), which accommodates
2115 the required \(2d\) source dimensions precisely for \(d\ge3\)
2116 (\cref{app:dyadic_capacity}).
2117 
2118 At rank two, with \(q=2m+1\), enlarge
2119 \(J=U_K^1U_A^{m+1}\) to \(P=U_K^1U_A^m\). The actual character
2120 ratio \(\lambda=\theta^g/\theta\) for \(g\notin J\) is nontrivial
2121 and sign-valued. With \(H_0=\ker\lambda\), the quotient is
2122 \(P/H_0\simeq\FF_2^2\), and \(\Ind_J^P\theta\) is an
2123 irreducible two-dimensional Heisenberg carrier
2124 (\cref{app:rank2}). At rank four, the actual outer quadratic
2125 centralizer gives \(P=J U_B^m\), \([P:J]=2\), and
2126 \(P/H_0\simeq\FF_2^4\), with carrier \(\Ind_J^P\eta\) of
2127 dimension four (\cref{app:rank4}). The finite scalar center can be
2128 larger than \(\mu_2\). No literal extraspecial quotient is claimed.
2129 
2130 At both ranks \(K^\times\) normalizes \(P\). Thus
2131 \(\mathcal N=K^\times J\) is normal of index two in
2132 \(\mathcal M=K^\times P\), and
2133 \[
2134 \Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda,
2135 \qquad
2136 \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
2137 \]
2138 Induction in stages preserves the full original \(\Lambda\), its
2139 central character, and its noncompact root. See
2140 \cref{app:rank4_stages,app:lowrank_contract}. The full marked
2141 scalar-pushout comparison applies to these actual enlarged pairs as
2142 well as to the original higher-rank types (\cref{app:dyadic_markings}).
2143 The full synthesis and scope are in \cref{app:dyadic_synthesis}.
2144 \end{proof}
2145 
2146 The original low-rank maximal-type capacity obstruction remains true.
2147 The new pair is a common arithmetic assembly carrier. It is not renamed
2148 as the original maximal-simple-character pair. For an independently
2149 fixed native quadratic marking, a preassigned embedding must satisfy
2150 \(q_W(f(v))=q_{\rm src}(v)\). At equal capacity, a free embedding
2151 requires equal Arf invariant. Spare capacity for \(d\ge4\) admits
2152 either source Arf type when the embedding is free. Retaining only the
2153 central \(\{\pm1\}\) would erase the distinction between whole marked
2154 finite extensions such as \(D_8\) and \(Q_8\).
2155 
2156 \subsection{Scope, numerical invariants, and evidence}
2157 
2158 For the same candidates the constructions give
2159 \[
2160 \begin{array}{c|c|c}
2161  & p\text{ odd} & p=2\\ \hline
2162 \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\
2163 \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d
2164 \end{array}
2165 \]
2166 and torsion number one. Here \(\Swan\) denotes the Swan conductor,
2167 and \(a\) below denotes the ordinary pair conductor. An epsilon exponent
2168 for a specified additive character is a different quantity. The separate
2169 progress paper, Sections 22--24, reconciles these conventions and the
2170 actual-order hypotheses. Their self-pair conductor formulas are
2171 \[
2172 \begin{aligned}
2173 a(\pi^\vee\times\pi)
2174  &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i
2175  &&(p\text{ odd}),\\
2176 a(\pi^\vee\times\pi)
2177  &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i
2178  &&(p=2).
2179 \end{aligned}
2180 \]
2181 These identities do not prove H5, identify an endo-class, or construct
2182 a canonical source-to-host marking. At rank eight, the retained
2183 skipped-degree host has the same displayed numerical profiles as the
2184 full tower and a smaller Heisenberg quotient. The explicit construction,
2185 rather than numerical matching alone, supplies the arithmetic object.
2186 
2187 The proofs use the stated Bushnell--Kutzko simple-type theorems
2188 \cite{BushnellKutzko1993} and the retained conductor inputs. Exact source
2189 locations and hypothesis checks are recorded in the component proofs
2190 and accompanying audits. The finite rank-nine and rank-eight runs check
2191 particular matrices, traces, normalized lifts, and filtration grades.
2192 They are bounded checks, not arbitrary-length proofs. No new Lean build
2193 of these arithmetic constructions is claimed.
2194 
2195 Nontrivial prescribed symplectic stabilizer actions, independently fixed
2196 canonical native markings, arbitrary source/transport diagrams, and any
2197 required numerical layer placement remain outside the restricted results.
2198 The current totally ramified host cannot implement nontrivial
2199 prime-to-\(p\) symplectic actions by the stronger character-preserving
2200 inner-normalizer mechanism. This obstruction is stated precisely in
2201 \cref{app:stabilizer_contract}. That stronger mechanism is not silently
2202 imposed on the original conjecture. H5 remains conditional for the same
2203 fixed candidate.
2204 
2205 The accompanying archive retains the mathematical reviews, computational evidence, and source records underlying the displayed scope.
2206 
2207 \clearpage
2208 \appendix
2209 \section*{Detailed ordinary proofs and construction records}
2210 \phantomsection
2211 \addcontentsline{toc}{section}{Detailed ordinary proofs and construction records}
2212 
2213 The following appendices typeset the complete selected current construction
2214 proofs from Blocks 166 and 167. Their source notation, numbered source
2215 subsections, exact hypotheses, and mathematical scope are retained.
2216 The synthesis appendices state the final combined results. Component notes
2217 also document narrower lemmas, alternative routes, and failed interpretations.
2218 In particular, the separate classical application of arithmetic induction in
2219 the leading-capacity note is not an input to the direct constructor.
2220 
2221 Historical research identifiers, filenames, and references to earlier
2222 blocks are preserved to make the arguments traceable to the accompanying
2223 Markdown files. Earlier ``pending'' review language is superseded by the
2224 retained internal review records included with those source files, at their
2225 recorded dates and scopes. No external peer review is claimed.
2226 Negative capacity conclusions concern the original maximal-type carrier.
2227 The later low-rank enlargement does not repeal them.
2228 These appendices are ordinary mathematical proofs. Neither their typesetting
2229 nor the inherited formalization certificates constitute a fresh Lean proof
2230 of these arithmetic results or of Conjecture 66.
2231 
2232 \input{appendices/odd_synthesis}
2233 \input{appendices/odd_tower}
2234 \input{appendices/odd_character}
2235 \input{appendices/odd_capacity}
2236 \input{appendices/odd_scalar}
2237 \input{appendices/stabilizer_contract}
2238 \input{appendices/dyadic_synthesis}
2239 \input{appendices/dyadic_tower}
2240 \input{appendices/dyadic_character}
2241 \input{appendices/dyadic_capacity}
2242 \input{appendices/dyadic_markings}
2243 \input{appendices/rank2}
2244 \input{appendices/rank4}
2245 \input{appendices/rank4_stages}
2246 \input{appendices/lowrank_contract}
2247 
2248 \clearpage
2249 \bibliographystyle{amsplain}
2250 \bibliography{Paper_XI_Traditional_Final}
2251 
2252 \end{document}

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