Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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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{Revised edition 1.1.0, October 18, 2026}
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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173 \special{pdf:trailerid [<0e9d864536f570d7480471e0f99695cb><0e9d864536f570d7480471e0f99695cb>]}
174 \begin{abstract}
175 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.
176 \end{abstract}
177 \makeatletter\@setabstract\makeatother
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187 
188 \section{Introduction}\label{sec:introduction}
189 
190 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}. The present article 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?
191 
192 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}.
193 
194 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.
195 
196 We therefore separate three levels of assertion. First, the article 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.
197 
198 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}.
199 
200 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 the paper.
201 
202 \begin{figure}[t]
203 \centering
204 \includegraphics[width=.98\textwidth]{01_architecture.pdf}
205 \caption{The finite theorem layers and the remaining wild boundary.}
206 \label{fig:architecture}
207 \end{figure}
208 
209 This reading edition separates the missing arithmetic realization from the
210 conditional identification of one actual candidate. The endpoint formerly
211 listed as proof obligation 66 (historical Theorem 16.2, row 95 of the
212 all-records register) is now \cref{conj:arithmetic_realization}.
213 The finite theorem statements and proofs include the geometric-Frobenius
214 convention correction and the additional local corrections recorded in this
215 edition's reading guide. The later
216 restricted constructions are developed in \cref{sec:arithmetic_progress} and
217 the detailed construction appendices. Their evidence is ordinary proof with
218 retained internal reviews, rather than new Lean certificates. These reviews
219 are not external peer review.
220 
221 The accompanying release includes Markdown (\texttt{.md}) files containing
222 detailed proofs, construction notes, proof audits, and research records.
223 These documents distinguish established results, conditional results, and
224 the remaining obligations associated with Conjecture 66.
225 
226 The article 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 local-factor boundary appears in \cref{sec:boundary}. Examples and scope
227 follow in \cref{sec:examples,sec:scope}. Section~\ref{sec:arithmetic_progress}
228 and the appendices develop the restricted arithmetic constructions. The
229 separate progress paper reconciles subsequent routes, corrections and
230 remaining obligations without replacing these source records.
231 
232 \section{Weil Deligne chains}\label{sec:wd}
233 
234 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
235 \[
236 r(w)Nr(w)^{-1}=\abs{w}_K N.
237 \]
238 
Statement: Indecomposable monodromy blocks239 \begin{standardfact}[Indecomposable monodromy blocks]\label{fact:spdecomp}
240 Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks
241 \[
242 \operatorname{Sp}_m(\tau),
243 \]
244 where \(\tau\) is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length \(m\).
245 \end{standardfact}
246 
247 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.
248 
Statement: Monodromy packet chain249 \begin{definition}[Monodromy packet chain]\label{def:monodromy_chain}
250 Let \(V_0,\ldots,V_{m-1}\) be copies of the space of \(\tau\). Put
251 \[
252 V=\bigoplus_{j=0}^{m-1}V_j.
253 \]
254 A monodromy packet chain is a nilpotent operator \(N\) with
255 \[
256 N(V_j)\subseteq V_{j-1}
257 \]
258 for \(j>0\), with \(N(V_0)=0\), together with a Frobenius operator whose conjugation scales \(N\) by \(q^{-1}\).
259 \end{definition}
260 
Statement: Exact block reconstruction · Go to proof · Read statement and proof side by side261 \begin{proposition}[Exact block reconstruction]\label{prop:monodromy_reconstruction}
262 Let \(P_j\) be the projector onto \(V_j\). Then
263 \[
264 N=\sum_{j=1}^{m-1}P_{j-1}NP_j,
265 \qquad
266 N^m=0.
267 \]
268 Every power \(N^r\) is the sum of the packet paths of length \(r\) along the chain.
269 \end{proposition}
270 
Proof block: Exact block reconstruction · Go to statement · Read statement and proof side by side271 \begin{proof}
272 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 \(N^m=0\).
273 \end{proof}
274 
Statement: Scalar covariance model · Go to proof · Read statement and proof side by side275 \begin{proposition}[Scalar covariance model]\label{prop:scalar_covariance}
276 Let
277 \[
278 F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
279 \]
280 and let \(N_{j-1,j}=1\). Then
281 \[
282 FNF^{-1}=q^{-1}N.
283 \]
284 \end{proposition}
285 
Proof block: Scalar covariance model · Go to statement · Read statement and proof side by side286 \begin{proof}
287 The only nonzero entry in column \(j\) and row \(j-1\) is multiplied by \(q^{j-1}q^{-j}=q^{-1}\) under conjugation.
288 \end{proof}
289 
290 \begin{figure}[t]
291 \centering
292 \includegraphics[width=.9\textwidth]{02_weil_deligne_chain.pdf}
293 \caption{A Weil Deligne chain is reconstructed from one atom and finitely many nearest neighbor transfers.}
294 \label{fig:wdchain}
295 \end{figure}
296 
Statement: Formal atomic reconstruction · Go to proof · Read statement and proof side by side297 \begin{theorem}[Formal atomic reconstruction]\label{thm:formal_atomic}
298 Assume that every irreducible Weil representation \(\tau\) has been assigned a supercuspidal representation \(\sigma_\tau\), compatibly with isomorphism and unramified twist. Then the rules
299 \[
300 \operatorname{Sp}_m(\tau)
301 \longmapsto
302 \operatorname{St}_m(\sigma_\tau)
303 \]
304 and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
305 \end{theorem}
306 
Proof block: Formal atomic reconstruction · Go to statement · Read statement and proof side by side307 \begin{proof}
308 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.
309 \end{proof}
310 
311 \begin{remark}\label{rem:atomic_boundary}
312 \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.
313 \end{remark}
314 
315 \section{Tame Frobenius packets}\label{sec:tame}
316 
317 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
318 \[
319 P_j
320 =
321 \frac1M
322 \sum_{a=0}^{M-1}
323 \zeta_M^{-ja}r(t)^a.
324 \]
325 
Statement: Tame character projectors · Go to proof · Read statement and proof side by side326 \begin{proposition}[Tame character projectors]\label{prop:tame_projectors}
327 The operators \(P_j\) are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by
328 \[
329 \Phi P_j\Phi^{-1}=P_{qj}.
330 \]
331 \end{proposition}
332 
Proof block: Tame character projectors · Go to statement · Read statement and proof side by side333 \begin{proof}
334 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.
335 \end{proof}
336 
Statement: Tame orbit induction · Go to proof · Read statement and proof side by side337 \begin{theorem}[Tame orbit induction]\label{thm:tame_induction}
338 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
339 \[
340 \boxed{
341 \tau\cong\Ind_{W_{K_f}}^{W_K}\xi.
342 }
343 \]
344 \end{theorem}
345 
Proof block: Tame orbit induction · Go to statement · Read statement and proof side by side346 \begin{proof}
347 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.
348 \end{proof}
349 
Statement: Dimension and regularity · Go to proof · Read statement and proof side by side350 \begin{corollary}[Dimension and regularity]\label{cor:tame_dimension}
351 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.
352 \end{corollary}
353 
Proof block: Dimension and regularity · Go to statement · Read statement and proof side by side354 \begin{proof}
355 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 \(f\).
356 \end{proof}
357 
358 
Statement: Dual and self dual tame packets · Go to proof · Read statement and proof side by side359 \begin{proposition}[Dual and self dual tame packets]\label{prop:tame_dual}
360 For a tame atom
361 \[
362 \tau_\xi=\Ind_{W_{K_f}}^{W_K}\xi,
363 \]
364 we have
365 \[
366 \tau_\xi^\vee
367 \cong
368 \Ind_{W_{K_f}}^{W_K}\xi^{-1}.
369 \]
370 The atom is self dual exactly when
371 \[
372 \xi^{-1}=\xi^{\Phi^j}
373 \]
374 for some \(j\in\ZZ/f\ZZ\).
375 \end{proposition}
376 
Proof block: Dual and self dual tame packets · Go to statement · Read statement and proof side by side377 \begin{proof}
378 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.
379 \end{proof}
380 
Statement: Unramified twist · Go to proof · Read statement and proof side by side381 \begin{proposition}[Unramified twist]\label{prop:tame_twist}
382 Let \(\nu\) be an unramified character of \(W_K\). Then
383 \[
384 \tau_\xi\otimes\nu
385 \cong
386 \Ind_{W_{K_f}}^{W_K}
387 \left(
388 \xi\,\nu|_{W_{K_f}}
389 \right).
390 \]
391 The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar \(a\), the twisted cycle has closing scalar
392 \[
393 a\nu(\Phi)^f.
394 \]
395 \end{proposition}
396 
Proof block: Unramified twist · Go to statement · Read statement and proof side by side397 \begin{proof}
398 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 \(\nu(\Phi)^f\).
399 \end{proof}
400 
401 \begin{figure}[t]
402 \centering
403 \includegraphics[width=.62\textwidth]{03_tame_frobenius_orbit.pdf}
404 \caption{A tame irreducible atom is one transitive Frobenius orbit.}
405 \label{fig:tameorbit}
406 \end{figure}
407 
408 \subsection{Regular orbit counts}
409 
410 Characters of \(\FF_{q^n}^{\times}\) are indexed by exponents modulo \(q^n-1\), and Frobenius acts by multiplication by \(q\).
411 
Statement: Möbius orbit count · Go to proof · Read statement and proof side by side412 \begin{proposition}[Möbius orbit count]\label{prop:mobius_count}
413 The number of regular Frobenius orbits of length \(n\) is
414 \[
415 \boxed{
416 c_n(q)
417 =
418 \frac1n
419 \sum_{d\mid n}
420 \mu(d)
421 \bigl(q^{n/d}-1\bigr).
422 }
423 \]
424 \end{proposition}
425 
Proof block: Möbius orbit count · Go to statement · Read statement and proof side by side426 \begin{proof}
427 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.
428 \end{proof}
429 
430 \begin{example}[Degree four over \(\FF_2\)]\label{ex:degree_four}
431 The regular exponent orbits modulo fifteen are
432 \[
433 (1,2,4,8),
434 \qquad
435 (3,6,12,9),
436 \qquad
437 (7,14,13,11).
438 \]
439 The formula gives
440 \[
441 \frac14\bigl((2^4-1)-(2^2-1)\bigr)=3.
442 \]
443 \end{example}
444 
445 \begin{figure}[t]
446 \centering
447 \includegraphics[width=.76\textwidth]{04_regular_orbit_counts.pdf}
448 \caption{Regular Frobenius orbit counts for three residue field sizes.}
449 \label{fig:orbitcounts}
450 \end{figure}
451 
452 \section{Finite field cuspidal projectors}\label{sec:finitefield}
453 
454 Let
455 \[
456 G_n=\GL_n(\FF_q),
457 \qquad
458 T_n\cong\FF_{q^n}^{\times}
459 \]
460 with \(T_n\) an elliptic maximal torus.
461 
Statement: Regular elliptic torus characters462 \begin{standardfact}[Regular elliptic torus characters]\label{fact:deligne_lusztig}
463 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.
464 \end{standardfact}
465 
466 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.
467 
Statement: Primitive cuspidal idempotent · Go to proof · Read statement and proof side by side468 \begin{proposition}[Primitive cuspidal idempotent]\label{prop:cuspidal_idempotent}
469 Let \(\chi_\theta\) be the irreducible character of \(\kappa_\theta\), and let \(d_\theta=\chi_\theta(1)\). Then
470 \[
471 e_\theta
472 =
473 \frac{d_\theta}{\abs{G_n}}
474 \sum_{g\in G_n}
475 \chi_\theta(g^{-1})g
476 \]
477 is the primitive central idempotent selecting the \(\kappa_\theta\) block in \(\CC[G_n]\).
478 \end{proposition}
479 
Proof block: Primitive cuspidal idempotent · Go to statement · Read statement and proof side by side480 \begin{proof}
481 On an irreducible representation of character \(\psi\), the central element acts by a scalar. Taking traces shows that the scalar is
482 \[
483 \frac{d_\theta}{\abs{G_n}\psi(1)}
484 \sum_{g\in G_n}
485 \chi_\theta(g^{-1})\psi(g).
486 \]
487 Character orthogonality makes this scalar one for \(\psi=\chi_\theta\) and zero otherwise. The element is therefore central, idempotent, and primitive.
488 \end{proof}
489 
Statement: Parabolic annihilation · Go to proof · Read statement and proof side by side490 \begin{proposition}[Parabolic annihilation]\label{prop:parabolic_annihilation}
491 Let \(P=LU\) be a proper parabolic subgroup of \(G_n\), and put
492 \[
493 e_U=\frac1{\abs U}\sum_{u\in U}u.
494 \]
495 If \(\kappa_\theta\) is cuspidal, then
496 \[
497 \boxed{
498 e_Ue_\theta=e_\theta e_U=0.
499 }
500 \]
501 \end{proposition}
502 
Proof block: Parabolic annihilation · Go to statement · Read statement and proof side by side503 \begin{proof}
504 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.
505 \end{proof}
506 
507 \begin{figure}[t]
508 \centering
509 \includegraphics[width=.92\textwidth]{05_cuspidal_projector.pdf}
510 \caption{A regular torus character determines a cuspidal block and its primitive central projector.}
511 \label{fig:cuspidal}
512 \end{figure}
513 
514 \subsection{Depth zero reconstruction and cycle orientation}
515 
516 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}.
517 
518 The induced tame Weil atom has an elementary cycle invariant. In an orbit basis \(e_0,\ldots,e_{n-1}\), write
519 \[
520 \tau(\Phi)e_j=e_{j+1}
521 \]
522 for \(j<n-1\), and
523 \[
524 \tau(\Phi)e_{n-1}=ae_0.
525 \]
526 
Statement: Frobenius cycle polynomial · Go to proof · Read statement and proof side by side527 \begin{proposition}[Frobenius cycle polynomial]\label{prop:frobenius_cycle}
528 The Frobenius transfer matrix has characteristic polynomial
529 \[
530 X^n-a
531 \]
532 and determinant
533 \[
534 \boxed{
535 \det\tau(\Phi)=(-1)^{n-1}a.
536 }
537 \]
538 \end{proposition}
539 
Proof block: Frobenius cycle polynomial · Go to statement · Read statement and proof side by side540 \begin{proof}
541 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 \(a\).
542 \end{proof}
543 
Statement: Unramified orientation character · Go to proof · Read statement and proof side by side544 \begin{corollary}[Unramified orientation character]\label{cor:orientation}
545 The sign \((-1)^{n-1}\) is the orientation of the closed Frobenius packet cycle. It is unchanged by rescaling the individual orbit basis vectors.
546 \end{corollary}
547 
Proof block: Unramified orientation character · Go to statement · Read statement and proof side by side548 \begin{proof}
549 A basis rescaling conjugates the transfer matrix by a diagonal matrix, so the determinant and characteristic polynomial remain unchanged.
550 \end{proof}
551 
552 \section{Tensor refinement of tame atoms}\label{sec:tensor}
553 
554 Let
555 \[
556 \tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi,
557 \qquad
558 \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
559 \]
560 where \(K_n/K\) and \(K_m/K\) are unramified. Put
561 \[
562 d=\gcd(n,m),
563 \qquad
564 L=\lcm(n,m).
565 \]
566 
Statement: Greatest common divisor and least common multiple packets · Go to proof · Read statement and proof side by side567 \begin{theorem}[Greatest common divisor and least common multiple packets]\label{thm:gcd_lcm}
568 There is a decomposition
569 \[
570 \boxed{
571 \tau_\xi\otimes\tau_\eta
572 \cong
573 \bigoplus_{i=0}^{d-1}
574 \Ind_{W_{K_L}}^{W_K}
575 \left(
576 \xi_{L,n}\eta_{L,m}^{\Phi^i}
577 \right),
578 }
579 \]
580 where \(\xi_{L,n}\) and \(\eta_{L,m}\) are the norm lifts to \(K_L\).
581 \end{theorem}
582 
Proof block: Greatest common divisor and least common multiple packets · Go to statement · Read statement and proof side by side583 \begin{proof}
584 Apply Mackey decomposition \cite{Mackey1952}. The double cosets
585 \[
586 W_{K_n}\backslash W_K/W_{K_m}
587 \]
588 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.
589 \end{proof}
590 
Statement: Packet count and dimension · Go to proof · Read statement and proof side by side591 \begin{corollary}[Packet count and dimension]\label{cor:gcd_lcm_count}
592 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
593 \[
594 dL=nm.
595 \]
596 \end{corollary}
Proof block: Packet count and dimension · Go to statement · Read statement and proof side by side597 \begin{proof}
598 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.
599 \end{proof}
600 
601 
602 \begin{figure}[t]
603 \centering
604 \includegraphics[width=.82\textwidth]{06_gcd_lcm_refinement.pdf}
605 \caption{Tensor refinement separates into greatest common divisor many packets, each with least common multiple carrier dimension.}
606 \label{fig:gcdlcm}
607 \end{figure}
608 
609 \subsection{Exponent coordinates}
610 
611 Represent the residue characters by exponents
612 \[
613 a\pmod{q^n-1},
614 \qquad
615 b\pmod{q^m-1}.
616 \]
617 
Statement: Refined exponent formula · Go to proof · Read statement and proof side by side618 \begin{proposition}[Refined exponent formula]\label{prop:refined_exponent}
619 The \(i\)th interaction packet on \(\FF_{q^L}^{\times}\) has exponent
620 \[
621 \boxed{
622 c_i
623 =
624 a\frac{q^L-1}{q^n-1}
625 +bq^i\frac{q^L-1}{q^m-1}
626 \pmod{q^L-1}.
627 }
628 \]
629 It is trivial on tame inertia exactly when \(c_i=0\).
630 \end{proposition}
631 
Proof block: Refined exponent formula · Go to statement · Read statement and proof side by side632 \begin{proof}
633 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 \(q^L-1\).
634 \end{proof}
635 
636 \subsection{Tame local polynomials}
637 
638 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.
639 
Statement: Local polynomial of one tame packet · Go to proof · Read statement and proof side by side640 \begin{proposition}[Local polynomial of one tame packet]\label{prop:tame_local_polynomial}
641 Suppose an interaction packet has orbit length \(r\) and closing scalar \(a\). Its local polynomial is
642 \[
643 \det\left(I-u\Phi\mid V^{I_K}\right)
644 =
645 \begin{cases}
646 1-a u^r,&\text{if the inertia character is trivial},\\
647 1,&\text{otherwise}.
648 \end{cases}
649 \]
650 \end{proposition}
651 
Proof block: Local polynomial of one tame packet · Go to statement · Read statement and proof side by side652 \begin{proof}
653 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 \(1-a u^r\).
654 \end{proof}
655 
Statement: Tame tensor local factor · Go to proof · Read statement and proof side by side656 \begin{corollary}[Tame tensor local factor]\label{cor:tame_tensor_factor}
657 The local factor of the tame tensor product is the product of
658 \[
659 (1-a_i u^{r_i})^{-1}
660 \]
661 over precisely those refined packets whose inertia exponent vanishes.
662 \end{corollary}
663 
Proof block: Tame tensor local factor · Go to statement · Read statement and proof side by side664 \begin{proof}
665 The invariant space is the direct sum of the invariant refined packets, and determinant is multiplicative on direct sums.
666 \end{proof}
667 
Statement: Trivial tensor packet criterion · Go to proof · Read statement and proof side by side668 \begin{proposition}[Trivial tensor packet criterion]\label{prop:trivial_tensor_packet}
669 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
670 \[
671 c_i=0
672 \qquad\text{and}\qquad
673 a_i=1.
674 \]
675 \end{proposition}
676 
Proof block: Trivial tensor packet criterion · Go to statement · Read statement and proof side by side677 \begin{proof}
678 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.
679 \end{proof}
680 
Statement: Tame Hom test · Go to proof · Read statement and proof side by side681 \begin{corollary}[Tame Hom test]\label{cor:tame_hom}
682 For irreducible tame atoms,
683 \[
684 \dim\Hom_{W_K}(\tau_\xi^\vee,\tau_\eta)
685 \]
686 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.
687 \end{corollary}
688 
Proof block: Tame Hom test · Go to statement · Read statement and proof side by side689 \begin{proof}
690 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.
691 \end{proof}
692 
693 \subsection{Tame conductor}
694 
Statement: Conductor of a nontrivial tame atom · Go to proof · Read statement and proof side by side695 \begin{proposition}[Conductor of a nontrivial tame atom]\label{prop:tame_conductor}
696 Let \(\tau\) be an irreducible tame Weil representation of dimension \(n>1\). Then
697 \[
698 \Swan(\tau)=0,
699 \qquad
700 \dim\tau^{I_K}=0,
701 \qquad
702 \Art(\tau)=n.
703 \]
704 \end{proposition}
705 
Proof block: Conductor of a nontrivial tame atom · Go to statement · Read statement and proof side by side706 \begin{proof}
707 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
708 \[
709 \dim\tau-\dim\tau^{I_K}+\Swan(\tau)=n.
710 \]
711 \end{proof}
712 
713 \section{Finite Gauss phases}\label{sec:gauss}
714 
715 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
716 \[
717 G(\chi,\psi)
718 =
719 \sum_{x\in k^{\times}}\chi(x)\psi(x).
720 \]
721 The standard theory of Gauss and Jacobi sums will be used in the form recorded in \cite{DavenportHasse1935,BerndtEvansWilliams1998}.
722 
Statement: Gauss magnitude · Go to proof · Read statement and proof side by side723 \begin{proposition}[Gauss magnitude]\label{prop:gauss_magnitude}
724 If \(\chi\) is nontrivial, then
725 \[
726 \boxed{
727 \abs{G(\chi,\psi)}=q^{1/2}.
728 }
729 \]
730 \end{proposition}
731 
Proof block: Gauss magnitude · Go to statement · Read statement and proof side by side732 \begin{proof}
733 Compute
734 \[
735 \abs{G(\chi,\psi)}^2
736 =
737 \sum_{x,y\in k^{\times}}
738 \chi(xy^{-1})\psi(x-y).
739 \]
740 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 \(q\).
741 \end{proof}
742 
Statement: Normalized Gauss phase743 \begin{definition}[Normalized Gauss phase]\label{def:gauss_phase}
744 For nontrivial \(\chi\), define
745 \[
746 \gamma(\chi,\psi)
747 =
748 q^{-1/2}G(\chi,\psi).
749 \]
750 Then \(\abs{\gamma(\chi,\psi)}=1\).
751 \end{definition}
752 
Statement: Duality of Gauss phases · Go to proof · Read statement and proof side by side753 \begin{proposition}[Duality of Gauss phases]\label{prop:gauss_duality}
754 For nontrivial \(\chi\),
755 \[
756 \boxed{
757 \gamma(\chi,\psi)\gamma(\chi^{-1},\psi)
758 =
759 \chi(-1).
760 }
761 \]
762 \end{proposition}
763 
Proof block: Duality of Gauss phases · Go to statement · Read statement and proof side by side764 \begin{proof}
765 The standard Gauss identity gives
766 \[
767 G(\chi,\psi)G(\chi^{-1},\psi)=\chi(-1)q.
768 \]
769 Divide by \(q\).
770 \end{proof}
771 
Statement: Hasse and Davenport lifting772 \begin{standardfact}[Hasse and Davenport lifting]\label{fact:hasse_davenport}
773 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.
774 \end{standardfact}
775 
776 \begin{figure}[t]
777 \centering
778 \includegraphics[width=.53\textwidth]{07_gauss_phase_circle.pdf}
779 \caption{Normalized finite Gauss sums lie on the unit circle.}
780 \label{fig:gausscircle}
781 \end{figure}
782 
783 \begin{remark}\label{rem:gauss_boundary}
784 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.
785 \end{remark}
786 
787 \section{Finite Heisenberg packets}\label{sec:heisenberg}
788 
789 Let \(p\) be odd. Let \(X\) be an \(r\) dimensional vector space over \(\FF_p\), and put
790 \[
791 V=X\oplus X^*.
792 \]
793 Define the alternating form
794 \[
795 b\bigl((x,\xi),(y,\eta)\bigr)
796 =
797 \eta(x)-\xi(y).
798 \]
799 Let \(Z=\FF_p\), and define the Heisenberg group
800 \[
801 \frakH(V)=V\times Z
802 \]
803 with multiplication
804 \[
805 (v,z)(w,t)
806 =
807 \left(v+w,z+t+\frac12 b(v,w)\right).
808 \]
809 Fix the additive character
810 \[
811 \psi(z)=\e^{2\pi iz/p}.
812 \]
813 
Statement: Schrödinger operators814 \begin{definition}[Schrödinger operators]\label{def:schrodinger}
815 On \(\calH=\CC^X\), define
816 \[
817 \bigl(\rho(x,\xi,z)f\bigr)(u)
818 =
819 \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
820 \]
821 \end{definition}
822 
Statement: Finite Schrödinger representation · Go to proof · Read statement and proof side by side823 \begin{theorem}[Finite Schrödinger representation]\label{thm:schrodinger_rep}
824 The map \(\rho:\frakH(V)\to\GL(\calH)\) is a representation. It is irreducible, its center acts by \(\psi\), and
825 \[
826 \dim\calH=p^r=\sqrt{\abs V}.
827 \]
828 \end{theorem}
829 
Proof block: Finite Schrödinger representation · Go to statement · Read statement and proof side by side830 \begin{proof}
831 A direct calculation gives
832 \[
833 \rho(v,z)\rho(w,t)
834 =
835 \rho\left(v+w,z+t+\frac12b(v,w)\right).
836 \]
837 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 \(\abs X=p^r\).
838 \end{proof}
839 
Statement: Character support · Go to proof · Read statement and proof side by side840 \begin{proposition}[Character support]\label{prop:heisenberg_trace}
841 The character of the Schrödinger representation satisfies
842 \[
843 \boxed{
844 \Tr\rho(v,z)
845 =
846 \begin{cases}
847 p^r\psi(z),&v=0,\\
848 0,&v\ne0.
849 \end{cases}
850 }
851 \]
852 \end{proposition}
853 
Proof block: Character support · Go to statement · Read statement and proof side by side854 \begin{proof}
855 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
856 \[
857 \psi(z)\sum_{u\in X}\psi(\xi(u))=0.
858 \]
859 When \(x=\xi=0\), the operator is the scalar \(\psi(z)\).
860 \end{proof}
861 
Statement: Finite Stone von Neumann theorem862 \begin{standardfact}[Finite Stone von Neumann theorem]\label{fact:stone_von_neumann}
863 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.
864 \end{standardfact}
865 
866 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}.
867 
868 \subsection{Maximal isotropic selectors}
869 
870 Let \(L\subseteq V\) be maximal isotropic. Choose a subgroup lift \(\widetilde L\subseteq\frakH(V)\) on which the central coordinate is zero.
871 
Statement: Isotropic averaging projector · Go to proof · Read statement and proof side by side872 \begin{proposition}[Isotropic averaging projector]\label{prop:isotropic_selector}
873 The operator
874 \[
875 P_L
876 =
877 \frac1{\abs L}
878 \sum_{\ell\in\widetilde L}
879 \rho(\ell)
880 \]
881 is an orthogonal projector. In the Schrödinger model with \(L=X\), it has rank one and selects the constant functions.
882 \end{proposition}
883 
Proof block: Isotropic averaging projector · Go to statement · Read statement and proof side by side884 \begin{proof}
885 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
886 \[
887 (P_Lf)(u)
888 =
889 \frac1{\abs X}
890 \sum_{x\in X}f(u+x),
891 \]
892 which is the constant function with value equal to the mean of \(f\). Its range is one dimensional.
893 \end{proof}
894 
Statement: Primitive line selector · Go to proof · Read statement and proof side by side895 \begin{corollary}[Primitive line selector]\label{cor:primitive_selector}
896 Every maximal isotropic subgroup selects a one dimensional fixed line in the irreducible Heisenberg representation.
897 \end{corollary}
898 
Proof block: Primitive line selector · Go to statement · Read statement and proof side by side899 \begin{proof}
900 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
901 \[
902 \Tr P_L
903 =
904 \frac{p^r}{p^r}=1.
905 \]
906 Thus the projector has rank one.
907 \end{proof}
908 
909 \begin{figure}[t]
910 \centering
911 \includegraphics[width=.9\textwidth]{08_heisenberg_geometry.pdf}
912 \caption{The finite Heisenberg packet and its maximal isotropic selector.}
913 \label{fig:heisenberg}
914 \end{figure}
915 
916 \section{Twisted group algebras and commutator envelopes}\label{sec:twisted}
917 
918 Let \(A\) be a finite abelian group, and let
919 \[
920 c:A\times A\to\CC^{\times}
921 \]
922 be a normalized two cocycle. The twisted group algebra \(\CC^c[A]\) has basis \((u_a)_{a\in A}\) and multiplication
923 \[
924 u_au_b=c(a,b)u_{a+b}.
925 \]
926 Define the commutator bicharacter
927 \[
928 b_c(a,b)=c(a,b)c(b,a)^{-1}.
929 \]
930 
Statement: Scalar choice invariance · Go to proof · Read statement and proof side by side931 \begin{proposition}[Scalar choice invariance]\label{prop:cocycle_choice}
932 Replacing \(u_a\) by \(\lambda_au_a\), with \(\lambda_a\in\CC^{\times}\), changes \(c\) by a coboundary and leaves \(b_c\) unchanged.
933 \end{proposition}
934 
Proof block: Scalar choice invariance · Go to statement · Read statement and proof side by side935 \begin{proof}
936 The new cocycle is
937 \[
938 c'(a,b)
939 =
940 \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
941 \]
942 The scalar coboundary factor is symmetric in \(a\) and \(b\), so it cancels from \(c'(a,b)c'(b,a)^{-1}\).
943 \end{proof}
944 
Statement: Alternating commutator form · Go to proof · Read statement and proof side by side945 \begin{proposition}[Alternating commutator form]\label{prop:commutator_form}
946 The map \(b_c\) is an alternating bicharacter. Its radical
947 \[
948 R=\{a\in A:b_c(a,x)=1\text{ for every }x\in A\}
949 \]
950 is a subgroup, and \(b_c\) descends to a nondegenerate alternating bicharacter on \(A/R\).
951 \end{proposition}
952 
Proof block: Alternating commutator form · Go to statement · Read statement and proof side by side953 \begin{proof}
954 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.
955 \end{proof}
956 
Statement: Nondegenerate twisted group algebra · Go to proof · Read statement and proof side by side957 \begin{theorem}[Nondegenerate twisted group algebra]\label{thm:twisted_matrix}
958 Assume that \(b_c\) is nondegenerate. Then \(\abs A\) is a square, and
959 \[
960 \boxed{
961 \CC^c[A]
962 \cong
963 M_{\sqrt{\abs A}}(\CC).
964 }
965 \]
966 Consequently, \(\CC^c[A]\) has one irreducible module up to isomorphism, of dimension \(\sqrt{\abs A}\).
967 \end{theorem}
968 
Proof block: Nondegenerate twisted group algebra · Go to statement · Read statement and proof side by side969 \begin{proof}
970 Let
971 \[
972 z=\sum_{a\in A}z_au_a
973 \]
974 be central. Commuting with \(u_x\) gives
975 \[
976 z_ab_c(x,a)=z_a
977 \]
978 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
979 \[
980 \overline P=\frac{1}{\abs A}\sum_{a\in A}u_a P u_a^{-1}.
981 \]
982 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 \(\sqrt{\abs A}\).
983 \end{proof}
984 
Statement: Radical free envelope · Go to proof · Read statement and proof side by side985 \begin{corollary}[Radical free envelope]\label{cor:radical_free}
986 For an arbitrary scalar projective action of \(A\), the quotient
987 \[
988 \boxed{
989 A^{\mathrm{symp}}=A/\rad b_c
990 }
991 \]
992 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.
993 \end{corollary}
994 
Proof block: Radical free envelope · Go to statement · Read statement and proof side by side995 \begin{proof}
996 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 \(\FF_p\).
997 \end{proof}
998 
999 \begin{figure}[t]
1000 \centering
1001 \includegraphics[width=.94\textwidth]{09_commutator_envelope.pdf}
1002 \caption{Projective intertwiners determine a radical free symplectic envelope.}
1003 \label{fig:commutator}
1004 \end{figure}
1005 
1006 \begin{remark}\label{rem:projective_scope}
1007 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.
1008 \end{remark}
1009 
1010 \section{A restricted one break Galois envelope}\label{sec:onebreak}
1011 
1012 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 \(G/N\).
1013 
Statement: One break Clifford datum1014 \begin{definition}[One break Clifford datum]\label{def:one_break_datum}
1015 A one break Clifford datum consists of finite groups and representations
1016 \[
1017 N\triangleleft G,
1018 \qquad
1019 A=G/N,
1020 \qquad
1021 \Pi:G\to\GL(\calX),
1022 \qquad
1023 \rho:N\to\GL(W),
1024 \]
1025 with the following properties.
1026 
1027 \begin{enumerate}
1028 \item The quotient \(A\) is a finite elementary abelian \(p\) group.
1029 \item The representations \(\Pi\) and \(\rho\) are irreducible.
1030 \item The isomorphism class of \(\rho\) is fixed by \(G\).
1031 \item The restriction of \(\Pi\) to \(N\) is \(\rho\) isotypic.
1032 \item The quotient records one declared ramification break.
1033 \end{enumerate}
1034 
1035 The multiplicity space is
1036 \[
1037 M_\rho=\Hom_N(W,\calX).
1038 \]
1039 \end{definition}
1040 
1041 The isotypic condition gives an \(N\) equivariant evaluation isomorphism
1042 \[
1043 W\otimes M_\rho
1044 \longrightarrow
1045 \calX.
1046 \]
1047 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
1048 \[
1049 \Pi(s(a))=I_a\otimes T_a.
1050 \]
1051 The operators \(T_a\) form a projective representation of \(A\).
1052 
Statement: Clifford multiplicity action · Go to proof · Read statement and proof side by side1053 \begin{proposition}[Clifford multiplicity action]\label{prop:clifford_action}
1054 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.
1055 \end{proposition}
1056 
Proof block: Clifford multiplicity action · Go to statement · Read statement and proof side by side1057 \begin{proof}
1058 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.
1059 
1060 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 \(\Pi\).
1061 \end{proof}
1062 
Statement: Clifford cocycle class1063 \begin{definition}[Clifford cocycle class]\label{def:clifford_class}
1064 Choose scalars \(c(a,b)\) such that
1065 \[
1066 T_aT_b=c(a,b)T_{a+b}.
1067 \]
1068 The cohomology class
1069 \[
1070 [c_\rho]\in H^2(A,\CC^{\times})
1071 \]
1072 is the Clifford cocycle class of the datum.
1073 \end{definition}
1074 
Statement: Intrinsic commutator pairing · Go to proof · Read statement and proof side by side1075 \begin{proposition}[Intrinsic commutator pairing]\label{prop:one_break_form}
1076 The pairing
1077 \[
1078 b_\rho(a,b)
1079 =
1080 c(a,b)c(b,a)^{-1}
1081 \]
1082 is independent of every choice in \cref{def:one_break_datum}. It is an alternating bicharacter
1083 \[
1084 b_\rho:A\times A\to\mu_p.
1085 \]
1086 \end{proposition}
1087 
Proof block: Intrinsic commutator pairing · Go to statement · Read statement and proof side by side1088 \begin{proof}
1089 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 \(p\).
1090 \end{proof}
1091 
Statement: One break symplectic envelope1092 \begin{definition}[One break symplectic envelope]\label{def:one_break_envelope}
1093 The one break symplectic envelope is
1094 \[
1095 \boxed{
1096 V_\rho=A/\rad b_\rho.
1097 }
1098 \]
1099 \end{definition}
1100 
Statement: Nondegenerate one break packet · Go to proof · Read statement and proof side by side1101 \begin{theorem}[Nondegenerate one break packet]\label{thm:one_break_matrix}
1102 If \(b_\rho\) is nondegenerate, then
1103 \[
1104 \CC^{c_\rho}[A]
1105 \cong
1106 \End(M_\rho),
1107 \qquad
1108 \dim M_\rho=\sqrt{\abs A}.
1109 \]
1110 \end{theorem}
1111 
Proof block: Nondegenerate one break packet · Go to statement · Read statement and proof side by side1112 \begin{proof}
1113 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.
1114 \end{proof}
1115 
Statement: Isomorphism invariance · Go to proof · Read statement and proof side by side1116 \begin{proposition}[Isomorphism invariance]\label{prop:one_break_invariance}
1117 The cocycle class and the equivariant isometry class of \(V_\rho\) depend only on the isomorphism class of the one break Clifford datum.
1118 \end{proposition}
1119 
Proof block: Isomorphism invariance · Go to statement · Read statement and proof side by side1120 \begin{proof}
1121 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.
1122 \end{proof}
1123 
Statement: Tame stabilizer action · Go to proof · Read statement and proof side by side1124 \begin{proposition}[Tame stabilizer action]\label{prop:tame_stabilizer}
1125 Suppose a finite group \(C\) acts by automorphisms of the one break Clifford datum. Then \(C\) acts on \(V_\rho\) by symplectic automorphisms.
1126 \end{proposition}
1127 
Proof block: Tame stabilizer action · Go to statement · Read statement and proof side by side1128 \begin{proof}
1129 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.
1130 \end{proof}
1131 
1132 \begin{figure}[t]
1133 \centering
1134 \includegraphics[width=.96\textwidth]{10_one_break_envelope.pdf}
1135 \caption{The restricted one break construction is made on the Clifford multiplicity space.}
1136 \label{fig:onebreak}
1137 \end{figure}
1138 
Statement: Heisenberg representation attached to one break · Go to proof · Read statement and proof side by side1139 \begin{corollary}[Heisenberg representation attached to one break]\label{cor:one_break_heisenberg}
1140 Fix a nontrivial central character. The envelope \(V_\rho\) determines one irreducible finite Heisenberg representation up to isomorphism.
1141 \end{corollary}
1142 
Proof block: Heisenberg representation attached to one break · Go to statement · Read statement and proof side by side1143 \begin{proof}
1144 Apply the finite Stone von Neumann theorem to the nondegenerate symplectic quotient.
1145 \end{proof}
1146 
1147 \begin{remark}\label{rem:clifford_scope}
1148 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.
1149 \end{remark}
1150 
1151 \section{The type side symplectic quotient}\label{sec:typeside}
1152 
1153 Let \(\theta\) be a maximal simple character for \(\GL_n(K)\), with compact open subgroups
1154 \[
1155 H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
1156 \]
1157 The quotient
1158 \[
1159 V_\theta=J_\theta^1/H_\theta^1
1160 \]
1161 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
1162 \[
1163 \psi_0\bigl(h_\theta(\bar x,\bar y)\bigr)
1164 =
1165 \theta([x,y])
1166 \]
1167 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.
1168 
Statement: Type side Heisenberg quotient1169 \begin{standardfact}[Type side Heisenberg quotient]\label{fact:type_heisenberg}
1170 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
1171 \[
1172 \sqrt{\abs{J_\theta^1/H_\theta^1}}.
1173 \]
1174 \end{standardfact}
1175 
1176 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.
1177 
Statement: Canonical beta extension1178 \begin{standardfact}[Canonical beta extension]\label{fact:canonical_beta}
1179 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.
1180 \end{standardfact}
1181 
1182 The normalization and its relation to finite Weil representations are developed in \cite{Blondel2012,Dotto2022}. The present article uses this theorem as type side input. It does not rederive simple character theory.
1183 
Statement: Type side projective commutator · Go to proof · Read statement and proof side by side1184 \begin{proposition}[Type side projective commutator]\label{prop:type_commutator}
1185 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.
1186 \end{proposition}
1187 
Proof block: Type side projective commutator · Go to statement · Read statement and proof side by side1188 \begin{proof}
1189 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}.
1190 \end{proof}
1191 
1192 \section{The restricted Galois to type bridge}\label{sec:bridge}
1193 
1194 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.
1195 
Statement: Bridge datum1196 \begin{definition}[Bridge datum]\label{def:bridge_datum}
1197 A restricted bridge datum consists of
1198 
1199 \begin{enumerate}
1200 \item a one break Galois envelope \((V_\rho,b_\rho)\)
1201 \item a type side quotient \((V_\theta,h_\theta)\)
1202 \item an equivariant symplectic isometry
1203 \[
1204 \iota:V_\rho\longrightarrow V_\theta
1205 \]
1206 for a finite cyclic stabilizer \(C=\langle c\rangle\)
1207 \item the same nontrivial central character on the two Heisenberg groups
1208 \item when \(p=2\), compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting \(\iota\).
1209 \end{enumerate}
1210 \end{definition}
1211 
Statement: Restricted Heisenberg bridge · Go to proof · Read statement and proof side by side1212 \begin{theorem}[Restricted Heisenberg bridge]\label{thm:restricted_bridge}
1213 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
1214 \[
1215 U:\calH_\rho\longrightarrow\calH_\theta
1216 \]
1217 such that
1218 \[
1219 U\omega_\rho(h)
1220 =
1221 \omega_\theta(\iota_H(h))U
1222 \]
1223 for every Heisenberg element \(h\), and every other such map is a scalar multiple of \(U\).
1224 \end{theorem}
1225 
Proof block: Restricted Heisenberg bridge · Go to statement · Read statement and proof side by side1226 \begin{proof}
1227 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.
1228 \end{proof}
1229 
1230 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.
1231 
Statement: Cyclic discrepancy character · Go to proof · Read statement and proof side by side1232 \begin{proposition}[Cyclic discrepancy character]\label{prop:cyclic_discrepancy}
1233 Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group \(C\). Then there is a character
1234 \[
1235 \delta:C\to\CC^{\times}
1236 \]
1237 such that
1238 \[
1239 U\Omega_\rho(g)
1240 =
1241 \delta(g)\Omega_\theta(g)U
1242 \]
1243 for every \(g\in C\).
1244 \end{proposition}
1245 
Proof block: Cyclic discrepancy character · Go to statement · Read statement and proof side by side1246 \begin{proof}
1247 For each \(g\), the operator
1248 \[
1249 \Omega_\theta(g)^{-1}U\Omega_\rho(g)U^{-1}
1250 \]
1251 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.
1252 \end{proof}
1253 
Statement: Linearization over a cyclic stabilizer · Go to proof · Read statement and proof side by side1254 \begin{lemma}[Linearization over a cyclic stabilizer]\label{lem:cyclic_linearization}
1255 Every finite dimensional projective representation of a finite cyclic group over \(\CC\) is projectively equivalent to an honest representation.
1256 \end{lemma}
1257 
Proof block: Linearization over a cyclic stabilizer · Go to statement · Read statement and proof side by side1258 \begin{proof}
1259 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.
1260 \end{proof}
1261 
Statement: One phase closes the cyclic bridge · Go to proof · Read statement and proof side by side1262 \begin{theorem}[One phase closes the cyclic bridge]\label{thm:one_phase_bridge}
1263 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.
1264 \end{theorem}
1265 
Proof block: One phase closes the cyclic bridge · Go to statement · Read statement and proof side by side1266 \begin{proof}
1267 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.
1268 \end{proof}
1269 
1270 \begin{figure}[t]
1271 \centering
1272 \includegraphics[width=.94\textwidth]{11_restricted_bridge.pdf}
1273 \caption{An equivariant symplectic isometry reduces the finite comparison to one scalar cyclic phase.}
1274 \label{fig:restrictedbridge}
1275 \end{figure}
1276 
Statement: Bridge uniqueness · Go to proof · Read statement and proof side by side1277 \begin{corollary}[Bridge uniqueness]\label{cor:bridge_uniqueness}
1278 When the semidirect product representation is irreducible, the completed bridge is unique up to scalar.
1279 \end{corollary}
1280 
Proof block: Bridge uniqueness · Go to statement · Read statement and proof side by side1281 \begin{proof}
1282 Any two completed bridges differ by an endomorphism commuting with the semidirect product representation. Schur's lemma applies.
1283 \end{proof}
1284 
1285 \begin{remark}\label{rem:bridge_scope}
1286 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.
1287 \end{remark}
1288 
1289 \section{Odd residue characteristic phases}\label{sec:oddphase}
1290 
1291 Let \(p\) be odd, let \((V,b)\) be a finite symplectic \(\FF_p\) space of dimension \(2r\), and let
1292 \[
1293 g\in\Sp(V)
1294 \]
1295 have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature \cite{Gerardin1977,Neuhauser2002,GurevichHadani2007}.
1296 
Statement: Fixed point free Weil trace1297 \begin{standardfact}[Fixed point free Weil trace]\label{fact:weil_trace}
1298 Under the declared normalization,
1299 \[
1300 \boxed{
1301 \Tr\omega(g)
1302 =
1303 \left(\frac{-1}{p}\right)^r
1304 \left(\frac{\det(g-I)}{p}\right).
1305 }
1306 \]
1307 Here \((\frac{\cdot}{p})\) denotes the quadratic character of \(\FF_p^{\times}\).
1308 \end{standardfact}
1309 
Statement: Odd finite Weil phase1310 \begin{definition}[Odd finite Weil phase]\label{def:odd_phase}
1311 For fixed point free \(g\), define
1312 \[
1313 \Phi_p(g,V)
1314 =
1315 \left(\frac{-1}{p}\right)^r
1316 \left(\frac{\det(g-I)}{p}\right).
1317 \]
1318 \end{definition}
1319 
Statement: Symplectic invariance · Go to proof · Read statement and proof side by side1320 \begin{proposition}[Symplectic invariance]\label{prop:phase_invariance}
1321 If
1322 \[
1323 \iota:(V,b)\longrightarrow(V',b')
1324 \]
1325 is a symplectic isometry and \(g'=\iota g\iota^{-1}\), then
1326 \[
1327 \Phi_p(g,V)=\Phi_p(g',V').
1328 \]
1329 \end{proposition}
1330 
Proof block: Symplectic invariance · Go to statement · Read statement and proof side by side1331 \begin{proof}
1332 The dimensions agree, and conjugate matrices have the same determinant of \(g-I\).
1333 \end{proof}
1334 
Statement: Orthogonal multiplicativity · Go to proof · Read statement and proof side by side1335 \begin{proposition}[Orthogonal multiplicativity]\label{prop:odd_multiplicativity}
1336 Suppose
1337 \[
1338 V=V_1\perpbig\cdots\perpbig V_s
1339 \]
1340 and \(g\) preserves every summand. If no \(g|_{V_j}\) has a fixed vector, then
1341 \[
1342 \boxed{
1343 \Phi_p(g,V)
1344 =
1345 \prod_{j=1}^{s}
1346 \Phi_p(g|_{V_j},V_j).
1347 }
1348 \]
1349 \end{proposition}
1350 
Proof block: Orthogonal multiplicativity · Go to statement · Read statement and proof side by side1351 \begin{proof}
1352 The half dimensions add, and
1353 \[
1354 \det(g-I\mid V)
1355 =
1356 \prod_j\det(g-I\mid V_j).
1357 \]
1358 The quadratic character is multiplicative.
1359 \end{proof}
1360 
Statement: Characteristic polynomial certificate · Go to proof · Read statement and proof side by side1361 \begin{corollary}[Characteristic polynomial certificate]\label{cor:characteristic_phase}
1362 Let
1363 \[
1364 P_g(X)=\det(XI-g\mid V).
1365 \]
1366 Then
1367 \[
1368 \boxed{
1369 \Phi_p(g,V)
1370 =
1371 \left(\frac{-1}{p}\right)^r
1372 \left(\frac{P_g(1)}{p}\right).
1373 }
1374 \]
1375 \end{corollary}
1376 
Proof block: Characteristic polynomial certificate · Go to statement · Read statement and proof side by side1377 \begin{proof}
1378 Since \(\dim V\) is even,
1379 \[
1380 \det(g-I)=\det(I-g)=P_g(1).
1381 \]
1382 \end{proof}
1383 
1384 
Statement: Fixed and moving decomposition · Go to proof · Read statement and proof side by side1385 \begin{proposition}[Fixed and moving decomposition]\label{prop:fixed_moving}
1386 Assume that \(g\in\Sp(V)\) has order prime to \(p\). Then
1387 \[
1388 V=\ker(g-I)\perpbig\im(g-I).
1389 \]
1390 The restriction of the symplectic form to \(\im(g-I)\) is nondegenerate, and \(g-I\) is invertible on that moving summand.
1391 \end{proposition}
1392 
Proof block: Fixed and moving decomposition · Go to statement · Read statement and proof side by side1393 \begin{proof}
1394 The order hypothesis makes \(g\) semisimple, so
1395 \[
1396 V=\ker(g-I)\oplus\im(g-I).
1397 \]
1398 If \(x\in\ker(g-I)\) and \(y=(g-I)z\), then
1399 \[
1400 b(x,y)=b(x,gz)-b(x,z)=b(g^{-1}x,z)-b(x,z)=0.
1401 \]
1402 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.
1403 \end{proof}
1404 
Statement: Moving phase1405 \begin{definition}[Moving phase]\label{def:moving_phase}
1406 Under the hypotheses of \cref{prop:fixed_moving}, define
1407 \[
1408 \Phi_p^{\mathrm{mov}}(g,V)
1409 =
1410 \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
1411 \]
1412 \end{definition}
1413 
Statement: Hyperbolic phase formula · Go to proof · Read statement and proof side by side1414 \begin{proposition}[Hyperbolic phase formula]\label{prop:hyperbolic_phase}
1415 Let \(U\) be a finite dimensional \(\FF_p\) space, let \(a\in\GL(U)\) have no eigenvalue one, and let \(g\) act on
1416 \[
1417 H(U)=U\oplus U^*
1418 \]
1419 by
1420 \[
1421 g=(a,(a^{-1})^*).
1422 \]
1423 Then
1424 \[
1425 \boxed{
1426 \Phi_p(g,H(U))
1427 =
1428 \left(\frac{\det a}{p}\right).
1429 }
1430 \]
1431 \end{proposition}
1432 
Proof block: Hyperbolic phase formula · Go to statement · Read statement and proof side by side1433 \begin{proof}
1434 Put \(d=\dim U\). We have
1435 \[
1436 \det\bigl((a^{-1})^*-I\bigr)
1437 =
1438 (-1)^d(\det a)^{-1}\det(a-I).
1439 \]
1440 Hence
1441 \[
1442 \det(g-I)
1443 =
1444 (-1)^d(\det a)^{-1}\det(a-I)^2.
1445 \]
1446 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 \((\frac{\det a}{p})\).
1447 \end{proof}
1448 
Statement: Phase only bridge test · Go to proof · Read statement and proof side by side1449 \begin{corollary}[Phase only bridge test]\label{cor:phase_only_test}
1450 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.
1451 \end{corollary}
1452 
Proof block: Phase only bridge test · Go to statement · Read statement and proof side by side1453 \begin{proof}
1454 Apply the definition of \(\Phi_p\).
1455 \end{proof}
1456 
1457 \begin{figure}[t]
1458 \centering
1459 \includegraphics[width=.88\textwidth]{12_odd_phase_blocks.pdf}
1460 \caption{Odd finite Weil phases multiply across orthogonal atomic blocks.}
1461 \label{fig:oddphase}
1462 \end{figure}
1463 
1464 \begin{remark}\label{rem:fixed_space}
1465 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.
1466 \end{remark}
1467 
1468 \section{Dyadic quadratic refinements}\label{sec:dyadic}
1469 
1470 Let \((V,b)\) be a symplectic vector space over \(\FF_2\). The alternating form alone is not enough to determine the finite phase.
1471 
Statement: Quadratic refinement1472 \begin{definition}[Quadratic refinement]\label{def:quadratic_refinement}
1473 A quadratic refinement of \(b\) is a map
1474 \[
1475 q:V\to\FF_2
1476 \]
1477 satisfying
1478 \[
1479 q(x+y)-q(x)-q(y)=b(x,y).
1480 \]
1481 \end{definition}
1482 
Statement: Affine space of refinements · Go to proof · Read statement and proof side by side1483 \begin{proposition}[Affine space of refinements]\label{prop:refinement_affine}
1484 If one quadratic refinement exists, then the set of all quadratic refinements is an affine space over \(V^*\).
1485 \end{proposition}
1486 
Proof block: Affine space of refinements · Go to statement · Read statement and proof side by side1487 \begin{proof}
1488 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.
1489 \end{proof}
1490 
Statement: Arf invariant1491 \begin{definition}[Arf invariant]\label{def:arf}
1492 Choose a symplectic basis \((e_i,f_i)_{i=1}^{r}\). Define
1493 \[
1494 \Arf(q)
1495 =
1496 \sum_{i=1}^{r}q(e_i)q(f_i)
1497 \in\FF_2.
1498 \]
1499 \end{definition}
1500 
Statement: Arf invariance and classification1501 \begin{standardfact}[Arf invariance and classification]\label{fact:arf_classification}
1502 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.
1503 \end{standardfact}
1504 
1505 This is the classical classification of nonsingular quadratic forms over \(\FF_2\) \cite{MilnorHusemoller1973}.
1506 
Statement: Dyadic Gauss sign · Go to proof · Read statement and proof side by side1507 \begin{theorem}[Dyadic Gauss sign]\label{thm:arf_gauss}
1508 Let \(\dim V=2r\), and let \(q\) be nondegenerate. Then
1509 \[
1510 \boxed{
1511 \sum_{x\in V}(-1)^{q(x)}
1512 =
1513 (-1)^{\Arf(q)}2^r.
1514 }
1515 \]
1516 \end{theorem}
1517 
Proof block: Dyadic Gauss sign · Go to statement · Read statement and proof side by side1518 \begin{proof}
1519 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.
1520 \end{proof}
1521 
Statement: Orthogonal dyadic multiplicativity · Go to proof · Read statement and proof side by side1522 \begin{corollary}[Orthogonal dyadic multiplicativity]\label{cor:arf_multiplicative}
1523 For an orthogonal sum
1524 \[
1525 (V,q)=(V_1,q_1)\perpbig(V_2,q_2),
1526 \]
1527 we have
1528 \[
1529 \Arf(q)=\Arf(q_1)+\Arf(q_2)
1530 \]
1531 and the normalized Gauss sign is the product of the two normalized signs.
1532 \end{corollary}
1533 
Proof block: Orthogonal dyadic multiplicativity · Go to statement · Read statement and proof side by side1534 \begin{proof}
1535 Take symplectic bases of the two summands and concatenate them. The Gauss sum factors as a product over the Cartesian decomposition.
1536 \end{proof}
1537 
Statement: The form alone does not determine the phase · Go to proof · Read statement and proof side by side1538 \begin{proposition}[The form alone does not determine the phase]\label{prop:dyadic_need_q}
1539 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.
1540 \end{proposition}
1541 
Proof block: The form alone does not determine the phase · Go to statement · Read statement and proof side by side1542 \begin{proof}
1543 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}.
1544 \end{proof}
1545 
1546 
Statement: Translation of quadratic refinements · Go to proof · Read statement and proof side by side1547 \begin{proposition}[Translation of quadratic refinements]\label{prop:quadratic_translation}
1548 For \(v\in V\), put
1549 \[
1550 q_v(x)=q(x)+b(v,x).
1551 \]
1552 Then
1553 \[
1554 \boxed{
1555 \Arf(q_v)=\Arf(q)+q(v).
1556 }
1557 \]
1558 \end{proposition}
1559 
Proof block: Translation of quadratic refinements · Go to statement · Read statement and proof side by side1560 \begin{proof}
1561 Using the quadratic identity,
1562 \[
1563 q_v(x)=q(x+v)+q(v).
1564 \]
1565 Therefore
1566 \[
1567 \sum_x(-1)^{q_v(x)}
1568 =
1569 (-1)^{q(v)}
1570 \sum_x(-1)^{q(x)}.
1571 \]
1572 Compare the signs in \cref{thm:arf_gauss}.
1573 \end{proof}
1574 
Statement: Number of dyadic refinements · Go to proof · Read statement and proof side by side1575 \begin{theorem}[Number of dyadic refinements]\label{thm:quadratic_counts}
1576 Let \(\dim V=2r\). The number of quadratic refinements with Arf invariant zero is
1577 \[
1578 \boxed{
1579 2^{r-1}(2^r+1),
1580 }
1581 \]
1582 and the number with Arf invariant one is
1583 \[
1584 \boxed{
1585 2^{r-1}(2^r-1).
1586 }
1587 \]
1588 \end{theorem}
1589 
Proof block: Number of dyadic refinements · Go to statement · Read statement and proof side by side1590 \begin{proof}
1591 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
1592 \[
1593 Z+O=2^{2r}
1594 \]
1595 and \cref{thm:arf_gauss} gives
1596 \[
1597 Z-O=(-1)^{\Arf(q)}2^r.
1598 \]
1599 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.
1600 \end{proof}
1601 
1602 \begin{figure}[t]
1603 \centering
1604 \includegraphics[width=.92\textwidth]{13_dyadic_quadratic_refinement.pdf}
1605 \caption{At residue characteristic two, the quadratic refinement and its Arf invariant carry phase information not contained in the alternating form.}
1606 \label{fig:dyadic}
1607 \end{figure}
1608 
1609 \section{Matrix holonomy and finite gluing}\label{sec:holonomy}
1610 
1611 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
1612 \[
1613 A_e:V_u\longrightarrow V_v,
1614 \qquad
1615 B_e:W_u\longrightarrow W_v
1616 \]
1617 be invertible. The reversed edge carries the inverse map.
1618 
Statement: Path transport1619 \begin{definition}[Path transport]\label{def:path_transport}
1620 For an oriented path
1621 \[
1622 \gamma=(v_0\to v_1\to\cdots\to v_m),
1623 \]
1624 define
1625 \[
1626 A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0},
1627 \qquad
1628 B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
1629 \]
1630 For a loop based at \(r\), these are the based holonomies.
1631 \end{definition}
1632 
Statement: Matrix holonomy criterion · Go to proof · Read statement and proof side by side1633 \begin{theorem}[Matrix holonomy criterion]\label{thm:matrix_holonomy}
1634 Fix a root \(r\) and an isomorphism
1635 \[
1636 U_r:V_r\longrightarrow W_r.
1637 \]
1638 The following conditions are equivalent.
1639 
1640 \begin{enumerate}
1641 \item There are isomorphisms \(U_v:V_v\to W_v\), with the declared \(U_r\), such that
1642 \[
1643 U_vA_e=B_eU_u
1644 \]
1645 for every edge \(e:u\to v\).
1646 \item For one fundamental cycle basis based at \(r\),
1647 \[
1648 U_rA_\gamma=B_\gamma U_r.
1649 \]
1650 \item The same identity holds for every based loop \(\gamma\).
1651 \end{enumerate}
1652 \end{theorem}
1653 
Proof block: Matrix holonomy criterion · Go to statement · Read statement and proof side by side1654 \begin{proof}
1655 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.
1656 
1657 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.
1658 \end{proof}
1659 
Statement: Unbased gauge classification · Go to proof · Read statement and proof side by side1660 \begin{corollary}[Unbased gauge classification]\label{cor:unbased_holonomy}
1661 There exists some family of vertex isomorphisms if and only if the two holonomy representations
1662 \[
1663 \pi_1(\Gamma,r)\longrightarrow\GL(V_r),
1664 \qquad
1665 \pi_1(\Gamma,r)\longrightarrow\GL(W_r)
1666 \]
1667 are conjugate.
1668 \end{corollary}
1669 
Proof block: Unbased gauge classification · Go to statement · Read statement and proof side by side1670 \begin{proof}
1671 Apply \cref{thm:matrix_holonomy} and allow \(U_r\) to vary.
1672 \end{proof}
1673 
Statement: Scalar and projective gluing · Go to proof · Read statement and proof side by side1674 \begin{corollary}[Scalar and projective gluing]\label{cor:scalar_holonomy}
1675 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
1676 \[
1677 \boxed{
1678 \abs E-\abs V+1.
1679 }
1680 \]
1681 \end{corollary}
1682 
Proof block: Scalar and projective gluing · Go to statement · Read statement and proof side by side1683 \begin{proof}
1684 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.
1685 \end{proof}
1686 
Statement: Projective cocycle form · Go to proof · Read statement and proof side by side1687 \begin{proposition}[Projective cocycle form]\label{prop:projective_holonomy}
1688 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.
1689 \end{proposition}
1690 
Proof block: Projective cocycle form · Go to statement · Read statement and proof side by side1691 \begin{proof}
1692 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.
1693 \end{proof}
1694 
1695 
Statement: Exact edge residual from cycle defect · Go to proof · Read statement and proof side by side1696 \begin{proposition}[Exact edge residual from cycle defect]\label{prop:holonomy_residual}
1697 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
1698 \[
1699 D_e
1700 =
1701 U_rA_{p_v}^{-1}A_eA_{p_u}
1702 -
1703 B_{p_v}^{-1}B_eB_{p_u}U_r.
1704 \]
1705 Then the edge residual is exactly
1706 \[
1707 \boxed{
1708 U_vA_e-B_eU_u
1709 =
1710 B_{p_v}D_eA_{p_u}^{-1}.
1711 }
1712 \]
1713 \end{proposition}
1714 
Proof block: Exact edge residual from cycle defect · Go to statement · Read statement and proof side by side1715 \begin{proof}
1716 Substitute
1717 \[
1718 U_u=B_{p_u}U_rA_{p_u}^{-1},
1719 \qquad
1720 U_v=B_{p_v}U_rA_{p_v}^{-1}
1721 \]
1722 into the left side and factor \(B_{p_v}\) on the left and \(A_{p_u}^{-1}\) on the right.
1723 \end{proof}
1724 
Statement: Quantitative finite gluing · Go to proof · Read statement and proof side by side1725 \begin{corollary}[Quantitative finite gluing]\label{cor:quantitative_holonomy}
1726 With the notation of \cref{prop:holonomy_residual},
1727 \[
1728 \norm{U_vA_e-B_eU_u}
1729 \leq
1730 \norm{B_{p_v}}\norm{D_e}\norm{A_{p_u}^{-1}}.
1731 \]
1732 Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.
1733 \end{corollary}
1734 
Proof block: Quantitative finite gluing · Go to statement · Read statement and proof side by side1735 \begin{proof}
1736 Take operator norms in the exact identity.
1737 \end{proof}
1738 
1739 \begin{figure}[t]
1740 \centering
1741 \includegraphics[width=.72\textwidth]{14_matrix_holonomy.pdf}
1742 \caption{A spanning tree fixes the vertex gauges. The non tree edges test the remaining cycle holonomies.}
1743 \label{fig:holonomy}
1744 \end{figure}
1745 
1746 \begin{remark}\label{rem:holonomy_boundary}
1747 The theorem is an exact finite gluing result. It assumes that the packet spaces and edge transports have already been constructed on both sides.
1748 \end{remark}
1749 
1750 \section{Breakwise assembly}\label{sec:breakwise}
1751 
1752 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.
1753 
Statement: Orthogonal break decomposition1754 \begin{definition}[Orthogonal break decomposition]\label{def:break_decomposition}
1755 An orthogonal break decomposition is a symplectic direct sum
1756 \[
1757 V=\perpbig_{s\in S}V_s
1758 \]
1759 that is preserved by the declared tame stabilizer. Each \(V_s\) is the finite symplectic packet assigned to one break.
1760 \end{definition}
1761 
Statement: Heisenberg tensor factorization · Go to proof · Read statement and proof side by side1762 \begin{proposition}[Heisenberg tensor factorization]\label{prop:heisenberg_tensor}
1763 Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands:
1764 \[
1765 \boxed{
1766 \omega_V
1767 \cong
1768 \bigotimes_{s\in S}\omega_{V_s}.
1769 }
1770 \]
1771 \end{proposition}
1772 
Proof block: Heisenberg tensor factorization · Go to statement · Read statement and proof side by side1773 \begin{proof}
1774 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
1775 \[
1776 \prod_s\sqrt{\abs{V_s}}
1777 =
1778 \sqrt{\abs V}.
1779 \]
1780 It is irreducible by the finite Stone von Neumann theorem, so it is the required representation.
1781 \end{proof}
1782 
Statement: Breakwise phase product · Go to proof · Read statement and proof side by side1783 \begin{corollary}[Breakwise phase product]\label{cor:break_phase}
1784 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
1785 \[
1786 \boxed{
1787 \Phi_p(g|_M,M)
1788 =
1789 \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s).
1790 }
1791 \]
1792 In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
1793 \end{corollary}
1794 
Proof block: Breakwise phase product · Go to statement · Read statement and proof side by side1795 \begin{proof}
1796 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}.
1797 \end{proof}
1798 
Statement: Breakwise restricted bridge · Go to proof · Read statement and proof side by side1799 \begin{theorem}[Breakwise restricted bridge]\label{thm:breakwise_bridge}
1800 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}.
1801 \end{theorem}
1802 
Proof block: Breakwise restricted bridge · Go to statement · Read statement and proof side by side1803 \begin{proof}
1804 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.
1805 \end{proof}
1806 
1807 \begin{figure}[t]
1808 \centering
1809 \includegraphics[width=.9\textwidth]{15_breakwise_decomposition.pdf}
1810 \caption{Orthogonal break packets produce tensor factors of the finite Heisenberg representation.}
1811 \label{fig:breakwise}
1812 \end{figure}
1813 
1814 \section{Local factor tests and the remaining bridge}\label{sec:boundary}
1815 
1816 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.
1817 
1818 \Needspace{13\baselineskip}
Statement: Finite tame comparison package · Go to proof · Read statement and proof side by side1819 \begin{proposition}[Finite tame comparison package]\label{prop:tame_package}
1820 For a tame induced atom, the following data are determined by its packet coordinates:
1821 
1822 \begin{enumerate}
1823 \item the dimension and inertia orbit
1824 \item the Frobenius cycle polynomial
1825 \item the Artin conductor
1826 \item the local polynomial
1827 \item the tensor refinement into interaction packets
1828 \item the finite Gauss sum of each declared residue character, with unit-modulus phase normalization when that character is nontrivial.
1829 \end{enumerate}
1830 \end{proposition}
1831 
Proof block: Finite tame comparison package · Go to statement · Read statement and proof side by side1832 \begin{proof}
1833 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.
1834 \end{proof}
1835 
1836 \subsection{Arithmetic realization and the conditional comparison}
1837 
1838 Fix the paper's reciprocity, additive-character, local-factor, and phase
1839 normalizations. Let \(\mathcal A\) be the declared class of irreducible wild
1840 Weil atoms under consideration. For \(\rho\in\mathcal A\), the supplied
1841 assembled finite datum \(d_\rho\) consists of the following inputs,
1842 historically denoted H1--H4:
1843 \begin{enumerate}
1844 \item an intrinsic finite break decomposition on the Galois side;
1845 \item the declared equivariant symplectic bridges to the supplied breakwise
1846 type-side quotients;
1847 \item compatible odd or dyadic central-extension and phase data on every
1848 break, including every prescribed finite marking;
1849 \item a compatible transport system with vanishing matrix holonomy
1850 obstruction.
1851 \end{enumerate}
1852 These are finite comparison inputs. Supplying separate breakwise target
1853 types does not itself supply one common arithmetic candidate. If a common
1854 candidate is already supplied, that is a stronger restricted hypothesis.
1855 
1856 An \emph{arithmetic realization witness}
1857 \(\mathcal R(d_\rho,\pi)\) consists of actual arithmetic inducing data
1858 for an admissible supercuspidal \(\pi\), together with explicit
1859 comparison and recovery maps carrying the prescribed assembled finite
1860 packet. The maps preserve its declared stabilizer action, all character values
1861 explicitly included in the supplied datum, central-extension and phase
1862 data, and compatible transports.
1863 Recovery returns that same \(\pi\), with its fixed central character and
1864 noncompact completion. An enlarged actual inducing carrier is permitted;
1865 the whole packet is not required to be the original maximal-simple-type
1866 quotient. Any stronger native-marking or numerical-layer requirement is
1867 part of the declared datum and must be verified separately.
1868 
1869 \begingroup
1870 \renewcommand{\theconjecture}{66}
Statement: Arithmetic packet realization1871 \begin{conjecture}[Arithmetic packet realization]
1872 \label{conj:arithmetic_realization}\label{thm:conditional_atomic}
1873 For every \(\rho\in\mathcal A\) with its supplied H1--H4 assembled
1874 finite datum \(d_\rho\), there exists an actual admissible supercuspidal
1875 representation \(\pi\) of \(\GL_{\dim\rho}(K)\) and an arithmetic
1876 realization witness \(\mathcal R(d_\rho,\pi)\).
1877 \end{conjecture}
1878 \endgroup
1879 
1880 \noindent This is the realization problem isolated from historical
1881 Theorem 16.2 and original proof obligation 66 (row 95 of the historical
1882 all-records register). It asks for one witness for each
1883 admitted source, rather than for success of every proposed host or every
1884 noncompact completion. No proof of the unrestricted statement is claimed.
1885 
Statement: Comparison of one realized candidate · Go to proof · Read statement and proof side by side1886 \begin{proposition}[Comparison of one realized candidate]
1887 \label{prop:realized_candidate_comparison}
1888 Let \(\rho\in\mathcal A\), and let \(\pi\) be one actual admissible
1889 supercuspidal representation of \(\GL_{\dim\rho}(K)\) with
1890 \(\mathcal R(d_\rho,\pi)\). Suppose H5 holds for this same \(\pi\):
1891 its required pair local factors equal those prescribed for \(\rho\)
1892 on the stipulated characterizing family of twists, with the fixed
1893 normalizations. Assume the stated local-factor characterization applies
1894 to that family and class. Then \(\pi\) is isomorphic to the representation
1895 assigned to \(\rho\) by the established local Langlands correspondence.
1896 \end{proposition}
1897 
Proof block: Comparison of one realized candidate · Go to statement · Read statement and proof side by side1898 \begin{proof}
1899 H5 identifies the tested local factors for this actual candidate. The
1900 applicable local-factor characterization identifies its isomorphism
1901 class with the established correspondence \cite{Henniart1993}. The
1902 realization witness ensures that it is this same candidate which carries
1903 the prescribed assembled packet. No existence conclusion is inferred
1904 from the finite gluing conditions alone.
1905 \end{proof}
1906 
1907 \begin{remark}\label{rem:conditional_not_proof}
1908 H5 remains a conditional premise about the same fixed candidate. It is
1909 not a formal consequence of the finite symplectic bridge, conductor
1910 profiles, or a choice of noncompact root. This proposition adds no
1911 unconditional pair-factor obligation to the construction conjecture.
1912 \end{remark}
1913 
1914 \subsection{Exact open boundary}
1915 
1916 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.
1917 
1918 \begin{enumerate}
1919 \item Construct a choice free finite symplectic envelope from an arbitrary irreducible wild inertia atom, with its ramification filtration and tame stabilizer action.
1920 \item Prove a canonical equivariant symplectic isometry from that envelope to the type side quotient \(J_\theta^1/H_\theta^1\).
1921 \item Recover the canonical beta extension phase in every break, including the required dyadic quadratic refinement.
1922 \item Prove matrix holonomy compatibility for the resulting packet transport system.
1923 \item Compare the full family of pair \(L\), epsilon, and gamma factors without invoking the established correspondence.
1924 \end{enumerate}
1925 
1926 \begin{figure}[t]
1927 \centering
1928 \includegraphics[width=.78\textwidth]{16_boundary_ladder.pdf}
1929 \caption{The proved finite layers and the remaining independent local correspondence steps.}
1930 \label{fig:boundary}
1931 \end{figure}
1932 
1933 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.
1934 
1935 \section{Examples and finite certificates}\label{sec:examples}
1936 
1937 The examples in this section illustrate the exact finite formulas. They are not used as proofs of the general theorems.
1938 
1939 \begin{example}[A degree four tame orbit]\label{ex:tame_four_again}
1940 Over \(\FF_2\), the exponent orbit
1941 \[
1942 1\longmapsto2\longmapsto4\longmapsto8\longmapsto1
1943 \]
1944 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}.
1945 \end{example}
1946 
1947 \begin{example}[Tensor packets of lengths four and six]\label{ex:tensor_46}
1948 For \(n=4\) and \(m=6\),
1949 \[
1950 \gcd(4,6)=2,
1951 \qquad
1952 \lcm(4,6)=12.
1953 \]
1954 The tensor product splits into two interaction packets of depth twelve. Their total dimension is \(2\cdot12=24\), equal to the product \(4\cdot6\).
1955 \end{example}
1956 
1957 \begin{example}[Rank one Heisenberg group]\label{ex:heisenberg_rank_one}
1958 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.
1959 \end{example}
1960 
1961 \begin{example}[Nondegenerate twisted algebra]\label{ex:twisted_plane}
1962 Let \(A=\FF_p^2\) with its standard alternating form. A cocycle with that commutator form gives
1963 \[
1964 \CC^c[A]\cong M_p(\CC).
1965 \]
1966 The unique irreducible projective representation has dimension \(p\).
1967 \end{example}
1968 
1969 \begin{example}[Dyadic phase ambiguity]\label{ex:dyadic_plane}
1970 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
1971 \[
1972 \sum_x(-1)^{q_0(x)}=2,
1973 \qquad
1974 \sum_x(-1)^{q_1(x)}=-2.
1975 \]
1976 Thus the alternating packet alone cannot select the dyadic sign.
1977 \end{example}
1978 
1979 \begin{example}[One cycle gluing obstruction]\label{ex:one_cycle}
1980 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.
1981 \end{example}
1982 
1983 \section{Scope of the finite reconstruction}\label{sec:scope}
1984 
1985 The article proves the following statements under their displayed hypotheses.
1986 
1987 \begin{enumerate}
1988 \item Tame irreducible Weil atoms are transitive Frobenius packets induced from unramified extensions.
1989 \item Regular tame packets have the stated Möbius count.
1990 \item Finite cuspidal blocks have exact primitive central projectors.
1991 \item Tame tensor products satisfy the greatest common divisor and least common multiple packet formula.
1992 \item Tame local polynomials, conductors, and finite Gauss phases are explicit.
1993 \item Nondegenerate scalar projective commutators give full matrix twisted group algebras and unique finite Heisenberg representations.
1994 \item A declared one break Clifford datum has a choice independent radical free symplectic envelope.
1995 \item An equivariant symplectic isometry gives a Heisenberg bridge unique up to scalar.
1996 \item One generator phase removes the scalar discrepancy for a cyclic stabilizer.
1997 \item Odd finite Weil phases and dyadic Arf signs are multiplicative across orthogonal break packets.
1998 \item Matrix holonomy on a fundamental cycle basis is the exact finite gluing obstruction.
1999 \end{enumerate}
2000 
2001 The article 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.
2002 
2003 \section{Restricted arithmetic realization theorems}
2004 \label{sec:arithmetic_progress}
2005 
2006 The later constructions provide actual arithmetic candidates and exact
2007 finite packet recovery in substantial restricted families. They do not
2008 establish \cref{conj:arithmetic_realization} for every admitted source.
2009 Their detailed ordinary proofs are typeset in the appendices, and the
2010 original reviewed Markdown files accompany this edition. In these
2011 appendices \(F\) denotes the base local field, while \(K=F[\beta]\)
2012 denotes a parameter field; these are local notation conventions.
2013 All references to ``claim 66'' in the retained notes mean the general
2014 endpoint now isolated as Conjecture 66.
2015 
2016 \subsection{The odd-prime trace-zero construction}
2017 
Statement: Restricted odd-prime arithmetic realization · Go to proof · Read statement and proof side by side2018 \begin{theorem}[Restricted odd-prime arithmetic realization]
2019 \label{thm:odd_arithmetic}
2020 Let \(p\) be an odd prime, \(F=\FF_p((t))\), fix \(d\ge1\), and let
2021 \[
2022 0<h_1<\cdots<h_d,\qquad p\nmid h_i,
2023 \qquad q_i=(p+1)h_i,\qquad N=p^d.
2024 \]
2025 There is an explicit maximal simple stratum in \(M_N(F)\) with totally
2026 ramified full-degree parameter field, a literal defining sequence of
2027 degrees \(p,p^2,\ldots,p^d\), and levels
2028 \(n_j=p^{d-j}q_{d-j+1}\). Every native parameter and consecutive
2029 correction has trace zero over its recursive base field.
2030 The stratum admits a full simple character trivial on \(U_F^1\).
2031 Its fixed canonical beta extension can be completed to an actual
2032 irreducible admissible supercuspidal \(\pi\) with any prescribed smooth
2033 central quasicharacter trivial on \(U_F^1\), when its completion choices
2034 are free or already compatible.
2035 
2036 Assume actually supplied labelled full-character extensions with whole
2037 two-dimensional symplectic quotients whose balanced complex-scalar
2038 central product has \(d\) planes, one vertex, and trivial stabilizer.
2039 Then the same \(\pi\) admits a full scalar-pushout realization of that
2040 packet and exact recovery of its unchanged inducing representation.
2041 The prescribed central character must lie in the covered tame class.
2042 No literal embedding of the old matrix groups, canonical native marking,
2043 or prescribed numerical host-depth placement is inferred.
2044 \end{theorem}
2045 
Proof block: Restricted odd-prime arithmetic realization · Go to statement · Read statement and proof side by side2046 \begin{proof}
2047 The explicit Artin--Schreier recursion and its normalized filtered lift
2048 construct the maximal parameter and literal defining sequence; see
2049 \cref{app:odd_tower}. Its trace transport identity makes every native
2050 trace zero, rather than relying on amplification by \(p\). Native-rank
2051 full-character construction, full beta-determinant transfer, trace-zero
2052 correction, and maximal-field character extension give the scalar
2053 triviality and central completion; see \cref{app:odd_character}.
2054 The actual leading symplectic layer has dimension
2055 \((p-1)p^{d-1}\ge2d\) and survives every later approximation
2056 (\cref{app:odd_capacity}). A labelled symplectic embedding gives a
2057 center-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann
2058 decomposition separates the source factor from its orthogonal
2059 complement. Maximal-isotropic averaging on the complementary Heisenberg
2060 factor selects a one-dimensional representation line and hence a
2061 source-sized sector. The retained induction/idempotent recovery returns the same
2062 \(\Lambda\) and \(\pi\), as proved in \cref{app:odd_synthesis}.
2063 \end{proof}
2064 
2065 For \(p>3\), a native minimal rank-\(p\) type quotient has dimension
2066 \(p-1\), so it is not automatically a plane. The theorem requires
2067 genuinely supplied whole-plane data. Independence of source tensor
2068 factors is also an input. An arbitrary single-factor wild determinant
2069 need not be tame.
2070 
2071 Under additional actual-inertia hypotheses, genuine scalar phases on
2072 the entire source carrier are realized by actual central matrices of a
2073 finite cover and descend to the stipulated operators. A direct matrix
2074 homomorphism from the stabilizer itself need not exist; the exact
2075 statement and descent proof are in \cref{app:odd_scalar}.
2076 
2077 \subsection{The weighted dyadic construction and enlarged carriers}
2078 
Statement: Restricted dyadic marked assembly realization · Go to proof · Read statement and proof side by side2079 \begin{theorem}[Restricted dyadic marked assembly realization]
2080 \label{thm:dyadic_arithmetic}
2081 Let \(F=\FF_2((t))\), fix an additive character of conductor
2082 \(t\OO_F\), fix \(d\ge1\), and let \(h_1<\cdots<h_d\) be positive odd integers.
2083 Put \(q_i=3h_i\), \(N=2^d\), and
2084 \(n_j=2^{d-j}q_{d-j+1}\). The explicit recursive matrix generates a
2085 separable totally ramified degree-\(N\) maximal parameter field,
2086 normalizes the complete lattice chain, and has the literal defining
2087 sequence of degrees \(2,4,\ldots,2^d\). It admits a full simple character
2088 trivial on \(U_F^1\) and an actual supercuspidal completion with any
2089 prescribed central quasicharacter trivial on \(U_F^1\), when the
2090 noncompact completion is free or already compatible.
2091 
2092 For actually supplied full marked whole-plane scalar-pushout data with
2093 one vertex, trivial stabilizer, and prescribed source central character
2094 trivial on \(U_F^1\), an arithmetic assembly carrier exists
2095 at every length and recovers the same candidate: inside its original
2096 type for \(d\ge3\), and in explicit enlarged inducing carriers for
2097 \(d=1,2\). Every supplied full character value and transported whole
2098 finite subgroup, lift, square, and phase marking is retained at the
2099 represented marking level. Any central-character kernel already factored
2100 out remains recorded separately; scalar pushout does not recover its
2101 invisible elements. This does
2102 not identify the transported marking with an independently prescribed
2103 canonical native marking.
2104 \end{theorem}
2105 
Proof block: Restricted dyadic marked assembly realization · Go to statement · Read statement and proof side by side2106 \begin{proof}
2107 The Artin--Schreier filtered lift has native trace one. Consequently
2108 each native correction \(\delta\) satisfies
2109 \[
2110 v_F(\operatorname{tr}\delta)+s\ge1 \quad (s=1),
2111 \]
2112 which makes its additive scalar correction trivial on \(U_F^1\).
2113 The actual field and defining-sequence proof is
2114 \cref{app:dyadic_tower}. Full beta-determinant transfer and character
2115 extension at each new maximal native field prove the full-character
2116 claim, including the odd-critical endpoint
2117 \(2(\lfloor r/2\rfloor+1)-r=1\); see \cref{app:dyadic_character}.
2118 The exact original quotient has dimension \(2^d-2\), which accommodates
2119 the required \(2d\) source dimensions precisely for \(d\ge3\)
2120 (\cref{app:dyadic_capacity}).
2121 
2122 At rank two, with \(q=2m+1\), enlarge
2123 \(J=U_K^1U_A^{m+1}\) to \(P=U_K^1U_A^m\). The actual character
2124 ratio \(\lambda=\theta^g/\theta\) for \(g\notin J\) is nontrivial
2125 and sign-valued. With \(H_0=\ker\lambda\), the quotient is
2126 \(P/H_0\simeq\FF_2^2\), and \(\Ind_J^P\theta\) is an
2127 irreducible two-dimensional Heisenberg carrier
2128 (\cref{app:rank2}). At rank four, the actual outer quadratic
2129 centralizer gives \(P=J U_B^m\), \([P:J]=2\), and
2130 \(P/H_0\simeq\FF_2^4\), with carrier \(\Ind_J^P\eta\) of
2131 dimension four (\cref{app:rank4}). The finite scalar center can be
2132 larger than \(\mu_2\); no literal extraspecial quotient is claimed.
2133 
2134 At both ranks \(K^\times\) normalizes \(P\). Thus
2135 \(\mathcal N=K^\times J\) is normal of index two in
2136 \(\mathcal M=K^\times P\), and
2137 \[
2138 \Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda,
2139 \qquad
2140 \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
2141 \]
2142 Induction in stages preserves the full original \(\Lambda\), its
2143 central character, and its noncompact root; see
2144 \cref{app:rank4_stages,app:lowrank_contract}. The full marked
2145 scalar-pushout comparison applies to these actual enlarged pairs as
2146 well as to the original higher-rank types (\cref{app:dyadic_markings}).
2147 The full synthesis and scope are in \cref{app:dyadic_synthesis}.
2148 \end{proof}
2149 
2150 The original low-rank maximal-type capacity obstruction remains true.
2151 The new pair is a common arithmetic assembly carrier; it is not renamed
2152 as the original maximal-simple-character pair. For an independently
2153 fixed native quadratic marking, a preassigned embedding must satisfy
2154 \(q_W(f(v))=q_{\rm src}(v)\). At equal capacity, a free embedding
2155 requires equal Arf invariant. Spare capacity for \(d\ge4\) admits
2156 either source Arf type when the embedding is free. Retaining only the
2157 central \(\{\pm1\}\) would erase the distinction between whole marked
2158 finite extensions such as \(D_8\) and \(Q_8\).
2159 
2160 \subsection{Scope, numerical invariants, and evidence}
2161 
2162 For the same candidates the constructions give
2163 \[
2164 \begin{array}{c|c|c}
2165  & p\text{ odd} & p=2\\ \hline
2166 \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\
2167 \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d
2168 \end{array}
2169 \]
2170 and torsion number one. Here \(\Swan\) denotes the Swan conductor,
2171 and \(a\) below denotes the ordinary pair conductor. An epsilon exponent
2172 for a specified additive character is a different quantity. The separate
2173 progress paper, Sections 22--24, reconciles these conventions and the
2174 actual-order hypotheses. Their self-pair conductor formulas are
2175 \[
2176 \begin{aligned}
2177 a(\pi^\vee\times\pi)
2178  &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i
2179  &&(p\text{ odd}),\\
2180 a(\pi^\vee\times\pi)
2181  &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i
2182  &&(p=2).
2183 \end{aligned}
2184 \]
2185 These identities do not prove H5, identify an endo-class, or construct
2186 a canonical source-to-host marking. At rank eight, the retained
2187 skipped-degree host has the same displayed numerical profiles as the
2188 full tower and a smaller Heisenberg quotient. The explicit construction,
2189 rather than numerical matching alone, supplies the arithmetic object.
2190 
2191 The proofs use the stated Bushnell--Kutzko simple-type theorems
2192 \cite{BushnellKutzko1993} and the retained conductor inputs. Exact source
2193 locations and hypothesis checks are recorded in the component proofs
2194 and accompanying audits. The finite rank-nine and rank-eight runs check
2195 particular matrices, traces, normalized lifts, and filtration grades.
2196 They are bounded checks, not arbitrary-length proofs. No new Lean build
2197 of these arithmetic constructions is claimed.
2198 
2199 Nontrivial prescribed symplectic stabilizer actions, independently fixed
2200 canonical native markings, arbitrary source/transport diagrams, and any
2201 required numerical layer placement remain outside the restricted results.
2202 The current totally ramified host cannot implement nontrivial
2203 prime-to-\(p\) symplectic actions by the stronger character-preserving
2204 inner-normalizer mechanism; this obstruction is stated precisely in
2205 \cref{app:stabilizer_contract}. That stronger mechanism is not silently
2206 imposed on the original conjecture. H5 remains conditional for the same
2207 fixed candidate.
2208 
2209 The accompanying archive retains the mathematical reviews, computational evidence, and source records underlying the displayed scope.
2210 
2211 \clearpage
2212 \appendix
2213 \section*{Detailed ordinary proofs and construction records}
2214 \phantomsection
2215 \addcontentsline{toc}{section}{Detailed ordinary proofs and construction records}
2216 
2217 The following appendices typeset the complete selected current construction
2218 proofs from Blocks 166 and 167. Their source notation, numbered source
2219 subsections, exact hypotheses, and mathematical scope are retained.
2220 The synthesis appendices state the final combined results; component notes
2221 also document narrower lemmas, alternative routes, and failed interpretations.
2222 In particular, the separate classical arithmetic-induction application in
2223 the leading-capacity note is not an input to the direct constructor.
2224 
2225 Historical research identifiers, filenames, and references to earlier
2226 blocks are preserved to make the arguments traceable to the accompanying
2227 Markdown files. Earlier ``pending'' review language is superseded by the
2228 retained internal review records included with those source files, at their
2229 recorded dates and scopes. No external peer review is claimed.
2230 Negative capacity conclusions concern the original maximal-type carrier;
2231 the later low-rank enlargement does not repeal them.
2232 These appendices are ordinary mathematical proofs. Neither their typesetting
2233 nor the inherited formalization certificates constitute a fresh Lean proof
2234 of these arithmetic results or of Conjecture 66.
2235 
2236 \input{appendices/odd_synthesis}
2237 \input{appendices/odd_tower}
2238 \input{appendices/odd_character}
2239 \input{appendices/odd_capacity}
2240 \input{appendices/odd_scalar}
2241 \input{appendices/stabilizer_contract}
2242 \input{appendices/dyadic_synthesis}
2243 \input{appendices/dyadic_tower}
2244 \input{appendices/dyadic_character}
2245 \input{appendices/dyadic_capacity}
2246 \input{appendices/dyadic_markings}
2247 \input{appendices/rank2}
2248 \input{appendices/rank4}
2249 \input{appendices/rank4_stages}
2250 \input{appendices/lowrank_contract}
2251 
2252 \clearpage
2253 \bibliographystyle{amsplain}
2254 \bibliography{Paper_XI_Traditional_Final}
2255 
2256 \end{document}

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