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1 \clearpage
2 \section{The surviving leading symplectic layer}\label{app:odd_capacity}
3 
4 Status: ordinary proof from the retained Bushnell--Kutzko theorems and,
5 for the source-defined candidate, the substantial published
6 Bushnell--Henniart arithmetic induction/descent constructor. No Lean
7 build or bounded computation is used. The accompanying audit files identify the scope of the internal review.
8 
9 \hypertarget{odd_capacity-the-useful-simplification}{%
10 \subsection{The useful simplification}\label{odd_capacity-the-useful-simplification}}
11 
12 Full critical-profile rigidity is unnecessary for the trivial-stabilizer
13 finite attachment contract. An even leading level produces a symplectic
14 subspace that survives every later simple approximation. For the
15 independent rank-p Heisenberg tensor sources, this single subspace
16 already has enough dimension to hold every source plane, however many
17 there are.
18 
19 The argument below concerns the actual type in the source-defined
20 arithmetic candidate. It does not identify that type with a separately
21 guessed binder parameter.
22 
23 \hypertarget{odd_capacity-leading-layer-theorem}{%
24 \subsection{1. Leading-layer theorem}\label{odd_capacity-leading-layer-theorem}}
25 
26 Let F have residue field k of cardinality q and residue characteristic
27 p.~Let pi be an actual irreducible supercuspidal representation of
28 GL\_n(F) with torsion number one, where n=p\^{}r. Suppose
29 
30 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
31 a(pi)=n+N,     N>0 even,     v_p(N)=r−1.
32 \end{Verbatim}
33 
34 Then an actual maximal simple type in pi, with Heisenberg quotient
35 \(W=J^1/H^1\), contains a nondegenerate symplectic
36 F\_p-subspace S of dimension
37 
38 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
39 [k:F_p] (p−1)p^(r−1).
40 \end{Verbatim}
41 
42 For k=F\_p this is exactly (p−1)p\^{}(r−1). The subspace is the image of
43 the concrete congruence group U\^{}(N/2)(A) in \(J^1/H^1\).
44 It is present regardless of the later critical exponents, their
45 parities, or the number of intermediate approximation fields.
46 
47 \hypertarget{odd_capacity-actual-parameter-field-and-leading-level}{%
48 \subsubsection{1.1. Actual parameter field and leading
49 level}\label{odd_capacity-actual-parameter-field-and-leading-level}}
50 
51 By BK maximal-type existence, pi contains a maximal simple type. Its
52 positive level follows from a(pi)\textgreater n.~Write its stratum as
53 {[}A,N\_A,0,beta{]}, with parameter field K=F{[}beta{]}. BK 6.2.5 gives
54 
55 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
56 t(pi)=n/e(K/F).
57 \end{Verbatim}
58 
59 Thus e(K/F)=n.~Since {[}K:F{]} divides n, K/F is totally ramified of
60 degree n and has residue degree one. The principal-order assertion after
61 BK 6.2.5 gives e(A)=n.~Accordingly A is the complete principal
62 hereditary order. The established conductor formula
63 
64 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
65 a(pi)=n+n N_A/e(A)
66 \end{Verbatim}
67 
68 now implies N\_A=N. This is the same source-checked formula used in
69 Block165, originating in BH1999 Section 6.1 Lemma 2 and retained in
70 \path{PRINCIPAL_ORDER_CONDUCTOR161.md}.
71 
72 Put P=rad(A), \(U^a(A)=1+P^a\) for a\textgreater=1, and
73 m=N/2.
74 
75 \hypertarget{odd_capacity-the-terminal-minimal-approximation-has-degree-p}{%
76 \subsubsection{1.2. The terminal minimal approximation has degree
77 p}\label{odd_capacity-the-terminal-minimal-approximation-has-degree-p}}
78 
79 Apply the defining-sequence construction BK 2.4.1--2 to beta. Write
80 gamma for its terminal minimal approximation. Every transition has a
81 critical level c\textless N, and gamma has the same A-level N as beta.
82 All approximation residue degrees divide f(K/F)=1, and all approximation
83 ramification degrees divide n.~Therefore E=F{[}gamma{]} is totally
84 ramified of degree d dividing n.
85 
86 The scalar case is impossible: a scalar of F has A-valuation divisible
87 by n, whereas v\_p(N)=r−1. Thus gamma is noncentral and minimal in the
88 ordinary field sense. Its field valuation is
89 
90 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
91 −v_E(gamma)=Nd/n.
92 \end{Verbatim}
93 
94 Minimality, by BK 1.4.14--15, requires gcd(Nd/n,d)=1. Since d is a
95 p-power and v\_p(N)=r−1, integrality first gives p\textbar d, while
96 coprimality forces d=p.~Hence E/F is totally ramified of degree p.
97 
98 Set B=End\_E(F\^{}n), B\_order=A∩B, and Q=rad(B\_order). The same
99 complete lattice chain has period n/p over E. Each P-grade has
100 k-dimension n and each Q-grade has k-dimension n/p.~In particular,
101 
102 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
103 dim_k P^m/(Q^m+P^(m+1))=n−n/p.
104 \end{Verbatim}
105 
106 No assertion that E is literally a subfield of the final field K is
107 needed. It is the actual minimal approximation field in the defining
108 sequence.
109 
110 \hypertarget{odd_capacity-the-high-subgroup-survives-every-approximation}{%
111 \subsubsection{1.3. The high subgroup survives every
112 approximation}\label{odd_capacity-the-high-subgroup-survives-every-approximation}}
113 
114 The minimal definitions BK 3.1.7--8 give
115 
116 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
117 J_order(gamma)=B_order+P^m,
118 H_order(gamma)=B_order+P^(m+1).
119 \end{Verbatim}
120 
121 At a nonminimal transition beta\_i→beta\_(i+1), with critical level
122 c\_i\textless N, the recursive J-order definition uses the threshold
123 ceil(c\_i/2). Since ceil(c\_i/2)\textless=m, the inclusion
124 P\^{}m⊂J\_order(beta\_(i+1)) implies P\^{}m⊂J\_order(beta\_i). Starting
125 at gamma, induction proves
126 
127 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
128 U^m(A)⊂J^1(beta).
129 \end{Verbatim}
130 
131 For H, BK 3.1.9(ii) states that
132 \(H^a(\beta_i)=H^a(\beta_{i+1})\) as soon as
133 
134 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
135 a>=floor(c_i/2)+1.
136 \end{Verbatim}
137 
138 Every c\_i is at most 2m−1, so this threshold is at most m. Consequently
139 
140 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
141 H^m(beta)=H^m(gamma).
142 \end{Verbatim}
143 
144 Here H\^{}m means \(H^1\cap U^m(A)\). Therefore the image S of
145 U\^{}m(A) in W is canonically
146 
147 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
148 S = U^m(A)/H^m(beta)
149   = U^m(A)/H^m(gamma)
150   ≅ P^m/(Q^m+P^(m+1)).
151 \end{Verbatim}
152 
153 The last group identification uses m\textgreater=1, so multiplication is
154 additive modulo P\^{}(m+1). Thus \textbar S\textbar=q\^{}(n−n/p).
155 
156 \hypertarget{odd_capacity-its-actual-commutator-form-is-nondegenerate}{%
157 \subsubsection{1.4. Its actual commutator form is
158 nondegenerate}\label{odd_capacity-its-actual-commutator-form-is-nondegenerate}}
159 
160 Choose the actual simple character theta contained in the type of pi. By
161 the standard deep-restriction property of simple characters, theta
162 restricts to psi\_beta on U\^{}(m+1)(A). This follows directly from BK
163 3.2.1--4 by induction along the defining sequence.
164 
165 For u,v in U\^{}m(A), their group commutator is in
166 \(U^{2m}(A)=U^N(A)\), which lies in U\^{}(m+1)(A). The
167 terminal approximation satisfies
168 
169 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
170 beta−gamma∈P^(1−N),
171 \end{Verbatim}
172 
173 because every critical level in the sequence is less than N. Hence
174 
175 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
176 (beta−gamma)([u,v]−1)∈P.
177 \end{Verbatim}
178 
179 With the standard level-one additive character psi\_A=psi\_F∘tr\_A/F,
180 psi\_A is trivial on P. We therefore have the exact equality
181 
182 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
183 theta([u,v]) = psi_beta([u,v]) = psi_gamma([u,v]).
184 \end{Verbatim}
185 
186 The right side is the Heisenberg commutator pairing for the minimal
187 simple stratum of gamma. Its quotient
188 \(J^1(\gamma)/H^1(\gamma)\) is precisely
189 \(U^m(A)/H^m(\gamma)\), because
190 \(J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)\), with
191 \(U^1(B_{\mathrm{order}})\subset H^1(\gamma)\). BK's nondegeneracy theorem for
192 simple-character commutators therefore says that the form on S is
193 nondegenerate.
194 
195 For general k, this is a nondegenerate alternating F\_p-form of
196 dimension \([k:\FF_p](n-n/p)\). For \(k=\FF_p\), it has dimension \(n-n/p\). This
197 proves the theorem.
198 
199 The argument uses no self-pair conductor and no determination of the
200 later critical levels. Those invariants remain useful for finer
201 arithmetic identification, but they are not needed for this
202 finite-sector capacity claim.
203 
204 \hypertarget{odd_capacity-arbitrary-length-odd-prime-source-family}{%
205 \subsection{2. Arbitrary-length odd-prime source
206 family}\label{odd_capacity-arbitrary-length-odd-prime-source-family}}
207 
208 Fix an odd prime p and F=F\_p((t)). Let the complete marked Weil source
209 rho be an independent tensor product of r rank-p wild Heisenberg factors
210 at distinct positive levels
211 
212 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
213 0<h_1<...<h_r,     p does not divide h_i.
214 \end{Verbatim}
215 
216 Here independence means the stipulated balanced-central-product
217 description of the wild image is valid, and rho\textbar P\_F is
218 irreducible. We do not infer independence from the inequalities alone.
219 Suppose the retained Tsushima source calculation supplies the usual
220 highest slope
221 
222 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
223 slope(rho)=(p+1)h_r/p.
224 \end{Verbatim}
225 
226 Then rank(rho)=n=p\^{}r and
227 
228 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
229 Swan(rho)=N=(p+1)p^(r−1)h_r.
230 \end{Verbatim}
231 
232 Let Pi\_BH(rho) be the published source-defined arithmetic
233 induction/descent candidate, with its full prescribed central character,
234 as audited in Block165. The construction is applied to the complete
235 actual rho, including its scalar normalization. Its codomain is the
236 torsion-one supercuspidal class, and its standard epsilon-function
237 property gives a(Pi\_BH(rho))=n+N. These are uses of the substantial
238 classical constructor, not consequences of finite Heisenberg algebra
239 alone and not a selection of an inverse LLC image.
240 
241 For odd p, N is even and v\_p(N)=r−1. The leading-layer theorem
242 therefore puts a symplectic space of dimension
243 
244 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
245 D=(p−1)p^(r−1)
246 \end{Verbatim}
247 
248 inside the actual Heisenberg quotient of a type in this same
249 Pi\_BH(rho).
250 
251 The source packet is the orthogonal sum of r planes, so its symplectic
252 dimension is 2r. The elementary inequality
253 
254 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
255 (p−1)p^(r−1)>=2r
256 \end{Verbatim}
257 
258 holds for every odd p and r\textgreater=1: it suffices that
259 3\^{}(r−1)\textgreater=r, which follows by induction. Thus the actual
260 candidate's leading symplectic layer alone can receive the entire source
261 packet.
262 
263 \hypertarget{odd_capacity-full-characters-and-recovery-for-declared-c1}{%
264 \subsubsection{2.1. Full characters and recovery for declared
265 C=\{1\}}\label{odd_capacity-full-characters-and-recovery-for-declared-c1}}
266 
267 Work under the same finite scalar-pushout contract as
268 \path{SAME_CANDIDATE_ATTACHMENT165.md}: declared stabilizer C=\{1\}, one
269 vertex, and full source character extensions retained, with no
270 requirement of literal old matrix-group inclusion or equality of old
271 numerical depths with host depths.
272 
273 Let P\_src be the balanced C×-central product of the r supplied full
274 source character extensions. Its quotient V\_src has dimension 2r over
275 F\_p and its central-character-one Heisenberg representation has
276 dimension p\^{}r=n.~Choose a symplectic embedding V\_src→S⊂W.
277 
278 The C×-pushout of the actual type character, restricted to this
279 subspace, has the same alternating commutator as P\_src. The retained
280 extension theorem supplies a center-fixing isomorphism from P\_src to
281 that restriction. Consequently every source h∈H\_i\^{}1 is carried to
282 the scalar theta\_i(h), preserving the full character value. No shallow
283 source values are discarded.
284 
285 Let V be the image and write W=V⊥V\_perp. Stone--von Neumann gives
286 
287 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
288 eta_W ≅ eta_src⊗eta_perp.
289 \end{Verbatim}
290 
291 Choose a line in eta\_perp and the corresponding rank-n projector e in
292 the actual finite Heisenberg image algebra. Then e eta\_W≅eta\_src.
293 Applying the retained Block159 induction/idempotent theorem to the
294 unchanged actual extended type Lambda in Pi\_BH(rho) recovers Lambda and
295 then this exact Pi\_BH(rho). All source normalization and the actual
296 noncompact scalar remain fixed.
297 
298 This proves same-candidate finite attachment for arbitrary r in this
299 C=\{1\} odd-prime family. Original H5, when supplied for this same
300 candidate, remains exactly the original conditional comparison premise.
301 We do not prove it here.
302 
303 \hypertarget{odd_capacity-first-new-rank-gl_27}{%
304 \subsubsection{2.2. First new rank: GL\_27}\label{odd_capacity-first-new-rank-gl_27}}
305 
306 For p=3,r=3, the source has rank~27 and V\_src has dimension 6. Put h=h\_3
307 with 3∤h. The actual candidate has
308 
309 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
310 N=36h,       m=18h,
311 dim_F3 S=27−9=18.
312 \end{Verbatim}
313 
314 Thus its high layer contains the entire three-plane source packet. At
315 least a 3\^{}9-dimensional Heisenberg contribution is present, while the
316 required source sector has dimension 27=3\^{}3. The six surplus
317 symplectic pairs can be projected onto a line in their representation,
318 together with any additional lower layers. This proves the required
319 capacity and sector realization for the new rank~27 family without
320 solving its two possible lower critical levels.
321 
322 In particular, for (h\_1,h\_2,h\_3)=(1,2,4), N=144 and the same high
323 subspace has dimension 18. These numbers are a specialization of the
324 universal proof, not a numerical experiment.
325 
326 \hypertarget{odd_capacity-why-trying-to-force-every-critical-level-was-unnecessary}{%
327 \subsection{3. Why trying to force every critical level was
328 unnecessary}\label{odd_capacity-why-trying-to-force-every-critical-level-was-unnecessary}}
329 
330 At rank~27, a hypothetical complete degree pattern 27→9→3 has leading
331 level N=36h\_3 and lower critical levels a\textless b\textless N. The
332 standard full-profile conductor count would give
333 
334 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
335 Swan(End rho)=18N+6b+2a.
336 \end{Verbatim}
337 
338 Comparing with the stipulated source formula 8(h\_1+9h\_2+81h\_3) only
339 gives
340 
341 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
342 a+3b=4h_1+36h_2.
343 \end{Verbatim}
344 
345 That single linear relation does not determine both a and b, and by
346 itself does not force each parity. Nor does it establish that all three
347 degrees occur rather than skipping degree~9. This observation is only a
348 warning about the insufficient equations. It is not presented as an
349 arithmetic counterexample or as existence of every numerical profile
350 satisfying the equation.
351 
352 The leading-layer theorem bypasses these unresolved profile choices. It
353 supplies a fixed nondegenerate high subspace in every actual candidate
354 with the two required invariants.
355 
356 \hypertarget{odd_capacity-dyadic-capacity-boundary}{%
357 \subsection{4. Dyadic capacity
358 boundary}\label{odd_capacity-dyadic-capacity-boundary}}
359 
360 The leading-layer theorem itself is valid at p=2. If an actual
361 torsion-one candidate of rank~2\^{}r has
362 
363 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
364 N=3·2^(r−1)h,     h odd,     r>=2,
365 \end{Verbatim}
366 
367 then its high subspace has dimension 2\^{}(r−1) over F\_2. This alone is
368 at least 2r for r\textgreater=4, with equality at r=4. It does not
369 supply enough capacity by itself for r=2 or 3. For r=1, N is odd and this
370 theorem does not apply.
371 
372 This is a capacity statement only. Applying it to a dyadic source
373 requires the actual source family, its arithmetic candidate and whatever
374 quadratic/phase markings the original contract requires. No universal
375 dyadic attachment theorem is claimed from dimension alone.
376 
377 \hypertarget{odd_capacity-boundaries-that-remain}{%
378 \subsection{5. Boundaries that remain}\label{odd_capacity-boundaries-that-remain}}
379 
380 \begin{enumerate}
381 \def\labelenumi{\arabic{enumi}.}
382 \tightlist
383 \item
384   The full character attachment here is through center-fixing
385   scalar-pushout maps. Literal embeddings of prescribed old arithmetic
386   matrix groups are not proved.
387 \item
388   No nontrivial prescribed stabilizer action is attached. Capacity does
389   not imply equivariant placement.
390 \item
391   All planes may be placed in the leading host layer. Preservation of
392   supplied numerical ramification depths, or a specified map into
393   individual host depth layers, would be an additional requirement and
394   is not claimed.
395 \item
396   General packet transport graphs are not handled here. The statement is
397   for C=\{1\} and one vertex.
398 \item
399   The arithmetic induction/descent theorem is an explicit major external
400   input. This is a new capacity/attachment consequence for its actual
401   output, not a new elementary reconstruction of that classical
402   constructor.
403 \item
404   H5 remains conditional and refers to that unchanged actual candidate.
405   No additional pair-factor premise is introduced.
406 \item
407   No Lean verification is claimed.
408 \end{enumerate}
409 
410 \hypertarget{odd_capacity-primary-locators-and-retained-dependencies}{%
411 \subsection{6. Primary locators and retained
412 dependencies}\label{odd_capacity-primary-locators-and-retained-dependencies}}
413 
414 The authorized primary BK text was read at these exact results:
415 1.4.14--15 (minimality), 2.4.1--2 (defining sequence and degree
416 divisibility), 3.1.7--9 (recursive orders and high-depth equality),
417 3.2.1--4 (deep restriction of simple characters), 3.4.1 (nondegenerate
418 Heisenberg commutator), and 6.2.5 with the following paragraph (torsion
419 number and principal period).
420 
421 The standard conductor and published arithmetic-constructor properties
422 are reused from the source-pinned Block165 audit and Block161 conductor
423 proof. The finite scalar-pushout and sector-recovery arguments are
424 exactly those retained and restated in
425 \path{SAME_CANDIDATE_ATTACHMENT165.md}. No entire copyrighted source is
426 copied into this continuation.

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