Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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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 completed independent review
8 is retained in the accompanying audit files.
9 
10 \hypertarget{odd_capacity-the-useful-simplification}{%
11 \subsection{The useful simplification}\label{odd_capacity-the-useful-simplification}}
12 
13 Full critical-profile rigidity is unnecessary for the trivial-stabilizer
14 finite attachment contract. An even leading level produces a symplectic
15 subspace that survives every later simple approximation. For the
16 independent rank-p Heisenberg tensor sources, this single subspace
17 already has enough dimension to hold every source plane, however many
18 there are.
19 
20 The argument below concerns the actual type in the source-defined
21 arithmetic candidate. It does not identify that type with a separately
22 guessed binder parameter.
23 
24 \hypertarget{odd_capacity-leading-layer-theorem}{%
25 \subsection{1. Leading-layer theorem}\label{odd_capacity-leading-layer-theorem}}
26 
27 Let F have residue field k of cardinality q and residue characteristic
28 p.~Let pi be an actual irreducible supercuspidal representation of
29 GL\_n(F) with torsion number one, where n=p\^{}r. Suppose
30 
31 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
32 a(pi)=n+N,     N>0 even,     v_p(N)=r−1.
33 \end{Verbatim}
34 
35 Then an actual maximal simple type in pi, with Heisenberg quotient
36 \(W=J^1/H^1\), contains a nondegenerate symplectic
37 F\_p-subspace S of dimension
38 
39 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
40 [k:F_p] (p−1)p^(r−1).
41 \end{Verbatim}
42 
43 For k=F\_p this is exactly (p−1)p\^{}(r−1). The subspace is the image of
44 the concrete congruence group U\^{}(N/2)(A) in \(J^1/H^1\).
45 It is present regardless of the later critical exponents, their
46 parities, or the number of intermediate approximation fields.
47 
48 \hypertarget{odd_capacity-actual-parameter-field-and-leading-level}{%
49 \subsubsection{1.1. Actual parameter field and leading
50 level}\label{odd_capacity-actual-parameter-field-and-leading-level}}
51 
52 By BK maximal-type existence, pi contains a maximal simple type. Its
53 positive level follows from a(pi)\textgreater n.~Write its stratum as
54 {[}A,N\_A,0,beta{]}, with parameter field K=F{[}beta{]}. BK 6.2.5 gives
55 
56 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
57 t(pi)=n/e(K/F).
58 \end{Verbatim}
59 
60 Thus e(K/F)=n.~Since {[}K:F{]} divides n, K/F is totally ramified of
61 degree n and has residue degree one. The principal-order assertion after
62 BK 6.2.5 gives e(A)=n.~Accordingly A is the complete principal
63 hereditary order. The established conductor formula
64 
65 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
66 a(pi)=n+n N_A/e(A)
67 \end{Verbatim}
68 
69 now implies N\_A=N. This is the same source-checked formula used in
70 Block165, originating in BH1999 Section 6.1 Lemma 2 and retained in
71 \path{PRINCIPAL_ORDER_CONDUCTOR161.md}.
72 
73 Put P=rad(A), \(U^a(A)=1+P^a\) for a\textgreater=1, and
74 m=N/2.
75 
76 \hypertarget{odd_capacity-the-terminal-minimal-approximation-has-degree-p}{%
77 \subsubsection{1.2. The terminal minimal approximation has degree
78 p}\label{odd_capacity-the-terminal-minimal-approximation-has-degree-p}}
79 
80 Apply the defining-sequence construction BK 2.4.1--2 to beta. Write
81 gamma for its terminal minimal approximation. Every transition has a
82 critical level c\textless N, and gamma has the same A-level N as beta.
83 All approximation residue degrees divide f(K/F)=1, and all approximation
84 ramification degrees divide n.~Therefore E=F{[}gamma{]} is totally
85 ramified of degree d dividing n.
86 
87 The scalar case is impossible: a scalar of F has A-valuation divisible
88 by n, whereas v\_p(N)=r−1. Thus gamma is noncentral and minimal in the
89 ordinary field sense. Its field valuation is
90 
91 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
92 −v_E(gamma)=Nd/n.
93 \end{Verbatim}
94 
95 Minimality, by BK 1.4.14--15, requires gcd(Nd/n,d)=1. Since d is a
96 p-power and v\_p(N)=r−1, integrality first gives p\textbar d, while
97 coprimality forces d=p.~Hence E/F is totally ramified of degree p.
98 
99 Set B=End\_E(F\^{}n), B\_order=A∩B, and Q=rad(B\_order). The same
100 complete lattice chain has period n/p over E. Each P-grade has
101 k-dimension n and each Q-grade has k-dimension n/p.~In particular,
102 
103 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
104 dim_k P^m/(Q^m+P^(m+1))=n−n/p.
105 \end{Verbatim}
106 
107 No assertion that E is literally a subfield of the final field K is
108 needed. It is the actual minimal approximation field in the defining
109 sequence.
110 
111 \hypertarget{odd_capacity-the-high-subgroup-survives-every-approximation}{%
112 \subsubsection{1.3. The high subgroup survives every
113 approximation}\label{odd_capacity-the-high-subgroup-survives-every-approximation}}
114 
115 The minimal definitions BK 3.1.7--8 give
116 
117 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
118 J_order(gamma)=B_order+P^m,
119 H_order(gamma)=B_order+P^(m+1).
120 \end{Verbatim}
121 
122 At a nonminimal transition beta\_i→beta\_(i+1), with critical level
123 c\_i\textless N, the recursive J-order definition uses the threshold
124 ceil(c\_i/2). Since ceil(c\_i/2)\textless=m, the inclusion
125 P\^{}m⊂J\_order(beta\_(i+1)) implies P\^{}m⊂J\_order(beta\_i). Starting
126 at gamma, induction proves
127 
128 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
129 U^m(A)⊂J^1(beta).
130 \end{Verbatim}
131 
132 For H, BK 3.1.9(ii) states that
133 \(H^a(\beta_i)=H^a(\beta_{i+1})\) as soon as
134 
135 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
136 a>=floor(c_i/2)+1.
137 \end{Verbatim}
138 
139 Every c\_i is at most 2m−1, so this threshold is at most m. Consequently
140 
141 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
142 H^m(beta)=H^m(gamma).
143 \end{Verbatim}
144 
145 Here H\^{}m means \(H^1\cap U^m(A)\). Therefore the image S of
146 U\^{}m(A) in W is canonically
147 
148 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
149 S = U^m(A)/H^m(beta)
150   = U^m(A)/H^m(gamma)
151   ≅ P^m/(Q^m+P^(m+1)).
152 \end{Verbatim}
153 
154 The last group identification uses m\textgreater=1, so multiplication is
155 additive modulo P\^{}(m+1). Thus \textbar S\textbar=q\^{}(n−n/p).
156 
157 \hypertarget{odd_capacity-its-actual-commutator-form-is-nondegenerate}{%
158 \subsubsection{1.4. Its actual commutator form is
159 nondegenerate}\label{odd_capacity-its-actual-commutator-form-is-nondegenerate}}
160 
161 Choose the actual simple character theta contained in the type of pi. By
162 the standard deep-restriction property of simple characters, theta
163 restricts to psi\_beta on U\^{}(m+1)(A). This follows directly from BK
164 3.2.1--4 by induction along the defining sequence.
165 
166 For u,v in U\^{}m(A), their group commutator is in
167 \(U^{2m}(A)=U^N(A)\), which lies in U\^{}(m+1)(A). The
168 terminal approximation satisfies
169 
170 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
171 beta−gamma∈P^(1−N),
172 \end{Verbatim}
173 
174 because every critical level in the sequence is less than N. Hence
175 
176 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
177 (beta−gamma)([u,v]−1)∈P.
178 \end{Verbatim}
179 
180 With the standard level-one additive character psi\_A=psi\_F∘tr\_A/F,
181 psi\_A is trivial on P. We therefore have the exact equality
182 
183 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
184 theta([u,v]) = psi_beta([u,v]) = psi_gamma([u,v]).
185 \end{Verbatim}
186 
187 The right side is the Heisenberg commutator pairing for the minimal
188 simple stratum of gamma. Its quotient
189 \(J^1(\gamma)/H^1(\gamma)\) is precisely
190 \(U^m(A)/H^m(\gamma)\), because
191 \(J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)\), with
192 \(U^1(B_{\mathrm{order}})\subset H^1(\gamma)\). BK's nondegeneracy theorem for
193 simple-character commutators therefore says that the form on S is
194 nondegenerate.
195 
196 For general k, this is a nondegenerate alternating F\_p-form of
197 dimension \([k:\FF_p](n-n/p)\). For \(k=\FF_p\), it has dimension \(n-n/p\). This
198 proves the theorem.
199 
200 The argument uses no self-pair conductor and no determination of the
201 later critical levels. Those invariants remain useful for finer
202 arithmetic identification, but they are not needed for this
203 finite-sector capacity claim.
204 
205 \hypertarget{odd_capacity-arbitrary-length-odd-prime-source-family}{%
206 \subsection{2. Arbitrary-length odd-prime source
207 family}\label{odd_capacity-arbitrary-length-odd-prime-source-family}}
208 
209 Fix an odd prime p and F=F\_p((t)). Let the complete marked Weil source
210 rho be an independent tensor product of r rank-p wild Heisenberg factors
211 at distinct positive levels
212 
213 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
214 0<h_1<...<h_r,     p does not divide h_i.
215 \end{Verbatim}
216 
217 Here independence means the stipulated balanced-central-product
218 description of the wild image is valid, and rho\textbar P\_F is
219 irreducible. We do not infer independence from the inequalities alone.
220 Suppose the retained Tsushima source calculation supplies the usual
221 highest slope
222 
223 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
224 slope(rho)=(p+1)h_r/p.
225 \end{Verbatim}
226 
227 Then rank(rho)=n=p\^{}r and
228 
229 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
230 Swan(rho)=N=(p+1)p^(r−1)h_r.
231 \end{Verbatim}
232 
233 Let Pi\_BH(rho) be the published source-defined arithmetic
234 induction/descent candidate, with its full prescribed central character,
235 as audited in Block165. The construction is applied to the complete
236 actual rho, including its scalar normalization. Its codomain is the
237 torsion-one supercuspidal class, and its standard epsilon-function
238 property gives a(Pi\_BH(rho))=n+N. These are uses of the substantial
239 classical constructor, not consequences of finite Heisenberg algebra
240 alone and not a selection of an inverse LLC image.
241 
242 For odd p, N is even and v\_p(N)=r−1. The leading-layer theorem
243 therefore puts a symplectic space of dimension
244 
245 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
246 D=(p−1)p^(r−1)
247 \end{Verbatim}
248 
249 inside the actual Heisenberg quotient of a type in this same
250 Pi\_BH(rho).
251 
252 The source packet is the orthogonal sum of r planes, so its symplectic
253 dimension is 2r. The elementary inequality
254 
255 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
256 (p−1)p^(r−1)>=2r
257 \end{Verbatim}
258 
259 holds for every odd p and r\textgreater=1: it suffices that
260 3\^{}(r−1)\textgreater=r, which follows by induction. Thus the actual
261 candidate's leading symplectic layer alone can receive the entire source
262 packet.
263 
264 \hypertarget{odd_capacity-full-characters-and-recovery-for-declared-c1}{%
265 \subsubsection{2.1. Full characters and recovery for declared
266 C=\{1\}}\label{odd_capacity-full-characters-and-recovery-for-declared-c1}}
267 
268 Work under the same finite scalar-pushout contract as
269 \path{SAME_CANDIDATE_ATTACHMENT165.md}: declared stabilizer C=\{1\}, one
270 vertex, and full source character extensions retained, with no
271 requirement of literal old matrix-group inclusion or equality of old
272 numerical depths with host depths.
273 
274 Let P\_src be the balanced C×-central product of the r supplied full
275 source character extensions. Its quotient V\_src has dimension 2r over
276 F\_p and its central-character-one Heisenberg representation has
277 dimension p\^{}r=n.~Choose a symplectic embedding V\_src→S⊂W.
278 
279 The C×-pushout of the actual type character, restricted to this
280 subspace, has the same alternating commutator as P\_src. The retained
281 extension theorem supplies a center-fixing isomorphism from P\_src to
282 that restriction. Consequently every source h∈H\_i\^{}1 is carried to
283 the scalar theta\_i(h), preserving the full character value. No shallow
284 source values are discarded.
285 
286 Let V be the image and write W=V⊥V\_perp. Stone--von Neumann gives
287 
288 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
289 eta_W ≅ eta_src⊗eta_perp.
290 \end{Verbatim}
291 
292 Choose a line in eta\_perp and the corresponding rank-n projector e in
293 the actual finite Heisenberg image algebra. Then e eta\_W≅eta\_src.
294 Applying the retained Block159 induction/idempotent theorem to the
295 unchanged actual extended type Lambda in Pi\_BH(rho) recovers Lambda and
296 then this exact Pi\_BH(rho). All source normalization and the actual
297 noncompact scalar remain fixed.
298 
299 This proves same-candidate finite attachment for arbitrary r in this
300 C=\{1\} odd-prime family. Original H5, when supplied for this same
301 candidate, remains exactly the original conditional comparison premise.
302 We do not prove it here.
303 
304 \hypertarget{odd_capacity-first-new-rank-gl_27}{%
305 \subsubsection{2.2. First new rank: GL\_27}\label{odd_capacity-first-new-rank-gl_27}}
306 
307 For p=3,r=3, the source has rank27 and V\_src has dimension6. Put h=h\_3
308 with 3∤h. The actual candidate has
309 
310 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
311 N=36h,       m=18h,
312 dim_F3 S=27−9=18.
313 \end{Verbatim}
314 
315 Thus its high layer contains the entire three-plane source packet. At
316 least a 3\^{}9-dimensional Heisenberg contribution is present, while the
317 required source sector has dimension27=3\^{}3. The six surplus
318 symplectic pairs can be projected onto a line in their representation,
319 together with any additional lower layers. This proves the required
320 capacity and sector realization for the new rank27 family without
321 solving its two possible lower critical levels.
322 
323 In particular, for (h\_1,h\_2,h\_3)=(1,2,4), N=144 and the same high
324 subspace has dimension18. These numbers are a specialization of the
325 universal proof, not a numerical experiment.
326 
327 \hypertarget{odd_capacity-why-trying-to-force-every-critical-level-was-unnecessary}{%
328 \subsection{3. Why trying to force every critical level was
329 unnecessary}\label{odd_capacity-why-trying-to-force-every-critical-level-was-unnecessary}}
330 
331 At rank27, a hypothetical complete degree pattern 27→9→3 has leading
332 level N=36h\_3 and lower critical levels a\textless b\textless N. The
333 standard full-profile conductor count would give
334 
335 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
336 Swan(End rho)=18N+6b+2a.
337 \end{Verbatim}
338 
339 Comparing with the stipulated source formula 8(h\_1+9h\_2+81h\_3) only
340 gives
341 
342 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
343 a+3b=4h_1+36h_2.
344 \end{Verbatim}
345 
346 That single linear relation does not determine both a and b, and by
347 itself does not force each parity. Nor does it establish that all three
348 degrees occur rather than skipping degree9. This observation is only a
349 warning about the insufficient equations; it is not presented as an
350 arithmetic counterexample or as existence of every numerical profile
351 satisfying the equation.
352 
353 The leading-layer theorem bypasses these unresolved profile choices. It
354 supplies a fixed nondegenerate high subspace in every actual candidate
355 with the two required invariants.
356 
357 \hypertarget{odd_capacity-dyadic-capacity-boundary}{%
358 \subsection{4. Dyadic capacity
359 boundary}\label{odd_capacity-dyadic-capacity-boundary}}
360 
361 The leading-layer theorem itself is valid at p=2. If an actual
362 torsion-one candidate of rank2\^{}r has
363 
364 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
365 N=3·2^(r−1)h,     h odd,     r>=2,
366 \end{Verbatim}
367 
368 then its high subspace has dimension 2\^{}(r−1) over F\_2. This alone is
369 at least 2r for r\textgreater=4, with equality at r=4. It does not
370 supply enough capacity by itself for r=2 or3. For r=1, N is odd and this
371 theorem does not apply.
372 
373 This is a capacity statement only. Applying it to a dyadic source
374 requires the actual source family, its arithmetic candidate and whatever
375 quadratic/phase markings the original contract requires. No universal
376 dyadic attachment theorem is claimed from dimension alone.
377 
378 \hypertarget{odd_capacity-boundaries-that-remain}{%
379 \subsection{5. Boundaries that remain}\label{odd_capacity-boundaries-that-remain}}
380 
381 \begin{enumerate}
382 \def\labelenumi{\arabic{enumi}.}
383 \tightlist
384 \item
385   The full character attachment here is through center-fixing
386   scalar-pushout maps. Literal embeddings of prescribed old arithmetic
387   matrix groups are not proved.
388 \item
389   No nontrivial prescribed stabilizer action is attached. Capacity does
390   not imply equivariant placement.
391 \item
392   All planes may be placed in the leading host layer. Preservation of
393   supplied numerical ramification depths, or a specified map into
394   individual host depth layers, would be an additional requirement and
395   is not claimed.
396 \item
397   General packet transport graphs are not handled here. The statement is
398   for C=\{1\} and one vertex.
399 \item
400   The arithmetic induction/descent theorem is an explicit major external
401   input. This is a new capacity/attachment consequence for its actual
402   output, not a new elementary reconstruction of that classical
403   constructor.
404 \item
405   H5 remains conditional and refers to that unchanged actual candidate.
406   No additional pair-factor premise is introduced.
407 \item
408   No Lean verification is claimed.
409 \end{enumerate}
410 
411 \hypertarget{odd_capacity-primary-locators-and-retained-dependencies}{%
412 \subsection{6. Primary locators and retained
413 dependencies}\label{odd_capacity-primary-locators-and-retained-dependencies}}
414 
415 The authorized primary BK text was read at these exact results:
416 1.4.14--15 (minimality), 2.4.1--2 (defining sequence and degree
417 divisibility), 3.1.7--9 (recursive orders and high-depth equality),
418 3.2.1--4 (deep restriction of simple characters), 3.4.1 (nondegenerate
419 Heisenberg commutator), and 6.2.5 with the following paragraph (torsion
420 number and principal period).
421 
422 The standard conductor and published arithmetic-constructor properties
423 are reused from the source-pinned Block165 audit and Block161 conductor
424 proof. The finite scalar-pushout and sector-recovery arguments are
425 exactly those retained and restated in
426 \path{SAME_CANDIDATE_ATTACHMENT165.md}. No entire copyrighted source is
427 copied into this continuation.

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