dyadic_capacity.tex
Realization Calculus
dyadic_capacity.tex
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1 \clearpage
2 \section{Exact dyadic capacity and numerical-profile obstruction}\label{app:dyadic_capacity}
3
4 Status: ordinary proof from the supplied actual defining sequence and
5 the retained primary Bushnell--Kutzko results. No Lean or numerical
6 computation is used. This note calculates the quotient of an actual
7 simple type. It does not prove universal H5 or identify an explicitly
8 constructed host with the source-defined arithmetic candidate.
9
10 \hypertarget{dyadic_capacity-exact-hypotheses-and-result}{%
11 \subsection{1. Exact hypotheses and
12 result}\label{dyadic_capacity-exact-hypotheses-and-result}}
13
14 Let F=F\_2((t)). Fix d\textgreater=1 and increasing positive odd
15 integers
16
17 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
18 h_1<...<h_d, q_i=3h_i, n=2^d.
19 \end{Verbatim}
20
21 Suppose the actual arithmetic constructor supplies a full-degree totally
22 ramified parameter beta on the complete period-n chain, with actual
23 approximating fields of degrees
24
25 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
26 2,4,...,2^d
27 \end{Verbatim}
28
29 and ambient critical levels
30
31 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
32 nu_j=2^(d−j) q_(d−j+1), 1<=j<=d.
33 \end{Verbatim}
34
35 These are the supplied dyadic specialization of the recursive tower's
36 field/profile conclusions. The calculation applies equally to the
37 original polynomial tower of Block166 and to an Artin--Schreier variant
38 once that variant's field/profile conclusions are established. No
39 trace-zero assertion is needed here.
40
41 The exact actual Heisenberg quotient is
42
43 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
44 dim_F2(J^1/H^1)=2^d−2,
45 dim eta=2^((2^d−2)/2).
46 \end{Verbatim}
47
48 Consequently its symplectic dimension is at least the 2d dimensions of
49 the whole d-plane source packet exactly when d\textgreater=3. In rank~8,
50 d=3, the quotient has dimension 6 and eta has dimension 8: the source
51 carrier fits exactly, with no complementary Heisenberg factor needed.
52
53 \begin{longtable}[]{@{}
54 >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
55 >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
56 >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
57 >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
58 >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
59 >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}@{}}
60 \toprule\noalign{}
61 \begin{minipage}[b]{\linewidth}\raggedright
62 d
63 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
64 Ambient rank
65 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
66 Actual quotient dimension
67 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
68 eta dimension
69 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
70 Source carrier dimension
71 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
72 Capacity
73 \end{minipage} \\
74 \midrule\noalign{}
75 \endhead
76 \bottomrule\noalign{}
77 \endlastfoot
78 1 & 2 & 0 & 1 & 2 & Fails \\
79 2 & 4 & 2 & 2 & 4 & Fails \\
80 3 & 8 & 6 & 8 & 8 & Exact \\
81 4 & 16 & 14 & 128 & 16 & Sufficient \\
82 general & 2\^{}d & 2\^{}d−2 & \(2^{2^{d-1}-1}\) & 2\^{}d &
83 Sufficient iff d\textgreater=3 \\
84 \end{longtable}
85
86 This improves the sufficient threshold obtained from the leading layer alone
87 d\textgreater=4 to the exact threshold obtained from the whole profile d\textgreater=3. It
88 also exposes a genuine low-rank limitation of this proposed
89 whole-carrier realization.
90
91 \hypertarget{dyadic_capacity-floors-and-ceilings-in-the-actual-bk-orders}{%
92 \subsection{2. Floors and ceilings in the actual BK
93 orders}\label{dyadic_capacity-floors-and-ceilings-in-the-actual-bk-orders}}
94
95 Write P=rad(A), and let B\_j be the A-centralizer of the degree-2\^{}j
96 approximation. Its complete-chain grade has F\_2-dimension
97
98 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
99 b_j=2^(d−j), b_0=n.
100 \end{Verbatim}
101
102 Put
103
104 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
105 a_j=ceil(nu_j/2), t_j=floor(nu_j/2)+1.
106 \end{Verbatim}
107
108 The exact BK definitions, 3.1.7--8, use a\_j as the J cutoff and t\_j as
109 the H cutoff. Thus a\_j=t\_j for an odd critical level, while
110 t\_j=a\_j+1 for an even critical level.
111
112 The increasing h\_i give
113
114 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
115 nu_j>2 nu_(j+1), 1<=j<d.
116 \end{Verbatim}
117
118 This strict separation is useful: the centralizer approximation estimate
119 from BK 3.1.5 puts each newly introduced centralizer into the preceding
120 one modulo a term above the preceding half-level. In the notation of
121 these stages, the relevant error level is
122 nu\_j−nu\_(j+1)\textgreater nu\_j/2. Therefore the successive grade dimensions are b\_j. There is no unproved assumption that the
123 approximation fields are literally nested inside the final field.
124
125 Applying the ring recursion gives the J-grade staircase:
126
127 \begin{itemize}
128 \tightlist
129 \item
130 below a\_d the grade dimension is b\_d=1;
131 \item
132 on a\_(j+1)\textless=k\textless a\_j it is b\_j;
133 \item
134 from a\_1 upward it is b\_0=n.
135 \end{itemize}
136
137 The H staircase uses the corresponding t\_j. Hence the exact
138 contribution of transition j to \(J^1/H^1\) is
139
140 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
141 (b_(j−1)−b_j) · 1_(nu_j even).
142 \end{Verbatim}
143
144 One can also obtain this by subtracting the exact logarithmic indices of
145 H and J in 1+P. It is an equality of the group index
146 \textbar \(J^1/H^1\)\textbar, not a guess from the visible
147 leading matrix support. BK 3.4.1 makes the resulting commutator pairing
148 nondegenerate.
149
150 For j\textless d the level nu\_j has a factor of 2 and is even. The final
151 level nu\_d=q\_1 is odd. Therefore
152
153 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
154 dim_F2 J^1/H^1
155 =sum_(j=1)^(d−1) (2^(d−j+1)−2^(d−j))
156 =2^d−2.
157 \end{Verbatim}
158
159 The omitted final increment would have dimension 1. It cannot be treated
160 as a symplectic plane or silently restored: at that odd level the two
161 defining lattice cutoffs are exactly equal. Its disappearance is also
162 consistent with the even dimension required of a nondegenerate
163 alternating space.
164
165 The source packet requires dimension 2d. At d=1,2, 2\^{}d−2\textless2d.
166 At d=3 equality holds. If the inequality holds for d\textgreater=3, then
167
168 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
169 2^(d+1)−2−2(d+1) = (2^d−2−2d)+(2^d−2)>0,
170 \end{Verbatim}
171
172 so it continues to hold for every larger d.~This proves the exact
173 threshold.
174
175 \hypertarget{dyadic_capacity-conductors-still-match-despite-the-lost-final-contribution}{%
176 \subsection{3. Conductors still match despite the lost final
177 contribution}\label{dyadic_capacity-conductors-still-match-despite-the-lost-final-contribution}}
178
179 This section reuses the same retained formal-degree and self-pair
180 conductor identities already audited in Blocks 161 and 165. It is not
181 needed to prove the dimension formula, but shows why numerical conductor
182 checks do not expose the low-rank capacity failure.
183
184 Put Delta\_j=b\_(j−1)−b\_j. The exact grade count gives
185
186 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
187 I=log_2[(1+P):J^1]
188 =sum_j Delta_j ceil(nu_j/2)−(n−1).
189 \end{Verbatim}
190
191 For this profile only the final level is odd and Delta\_d=1. Therefore
192
193 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
194 2I=sum_j Delta_j nu_j+1−2(n−1).
195 \end{Verbatim}
196
197 The Heisenberg dimension exponent is D/2, where D=n−2. The retained
198 formal-degree identity for a full-degree, totally ramified, torsion-one
199 maximal type gives
200
201 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
202 a(Pi^vee×Pi)=n^2−1+(n−1)+2I+D.
203 \end{Verbatim}
204
205 Substitution yields exact cancellation of the odd-cutoff correction:
206
207 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
208 a(Pi^vee×Pi)=n^2−1+sum_j Delta_j nu_j
209 =2^(2d)−1+sum_(i=1)^d 3·4^(i−1) h_i.
210 \end{Verbatim}
211
212 The ordinary conductor is n+nu\_1, so Swan(Pi)=2\^{}(d−1)·3h\_d, and the
213 torsion number is one. These are the usual three numerical invariants of
214 the stipulated independent dyadic Heisenberg tensor source.
215
216 Thus all three numerical invariants can match while eta is too small for
217 the whole source carrier at ranks 2 and 4. This is a capacity failure, not
218 a conductor mismatch. It does not by itself refute any original H1--H5
219 statement.
220
221 \hypertarget{dyadic_capacity-rank2-failure-is-forced-by-swan-and-torsion-alone}{%
222 \subsection{4. Rank 2 failure is forced by Swan and torsion
223 alone}\label{dyadic_capacity-rank2-failure-is-forced-by-swan-and-torsion-alone}}
224
225 Let Pi be any actual supercuspidal representation of GL2(F) with torsion
226 number one and odd Swan exponent N=3h\_1. BK 6.2.5 forces its maximal
227 parameter field K/F to be totally ramified of degree~2 and its hereditary
228 order to have complete period 2. The ordinary conductor formula then
229 fixes its native level as N.
230
231 Every parameter beta of a maximal simple type in Pi has v\_K(beta)=−N,
232 which is odd. Since the residue extension is trivial, BK 1.4.14--15 say
233 beta is minimal over F. The two order definitions are therefore
234
235 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
236 J_order=O_K+P^ceil(N/2),
237 H_order=O_K+P^(floor(N/2)+1).
238 \end{Verbatim}
239
240 The exponents coincide because N is odd. Hence \(J^1=H^1\),
241 eta is one-dimensional, and no two-dimensional symplectic source plane
242 can embed into this quotient.
243
244 This proves failure for every maximal simple type of every candidate
245 with these two invariants. It is stronger than failure of one explicit
246 matrix formula. It remains a statement about attachment inside the
247 actual maximal-simple-type Heisenberg carrier, not a proof that no other
248 arithmetic realization relation could work.
249
250 \hypertarget{dyadic_capacity-rank4-failure-is-also-forced-by-swan-and-torsion}{%
251 \subsection{5. Rank 4 failure is also forced by Swan and
252 torsion}\label{dyadic_capacity-rank4-failure-is-also-forced-by-swan-and-torsion}}
253
254 Let Pi be any actual supercuspidal representation of GL4(F) with torsion
255 number one and Swan exponent N=6h\_2, where h\_2 is odd. Its maximal
256 parameter field is totally ramified of degree~4, its order has period 4,
257 and its native level is N.
258
259 Here v\_2(N)=1. The defining-sequence/minimality argument of Block166
260 forces the terminal minimal approximation field E/F to have degree~2.
261 There is no possible intermediate degree between 2 and 4. The scalar
262 terminal case is excluded by 4 not dividing N, and a minimal
263 full-degree~4 parameter is excluded because gcd(N,4)=2. Thus the actual
264 degree pattern is forced to be 4→2.
265
266 Write c=−k\_0(beta,A), so 1\textless=c\textless N. Select the critical
267 approximation furnished by the construction in BK 2.4.1 and the remark
268 preceding 2.4.7. Because the original field is maximal, this
269 approximation can be selected with a simple maximal derived stratum in
270 B=End\_E(F\^{}4)≅M2(E). Equivalently one can replace its derived stratum
271 by an equivalent simple representative and use BK 2.2.2--3. The derived
272 field has degree~2 and ramification index 2 over E. A simple stratum
273 {[}B\_order,c,c−1,alpha{]} is minimal. Since the complete B-chain has
274 period 2 and the derived field is totally ramified, BK 1.4.14 forces
275
276 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
277 gcd(c,2)=1.
278 \end{Verbatim}
279
280 Therefore c is odd. It cannot be made even while preserving this degree
281 pattern and a genuine maximal derived field.
282
283 Let j=(c+1)/2. The actual orders are
284
285 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
286 J_order=O_K+Q^j+P^(N/2),
287 H_order=O_K+Q^j+P^(N/2+1).
288 \end{Verbatim}
289
290 The final low transition contributes nothing, while the high transition
291 has dimension 4−2=2. Thus every such Pi has
292
293 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
294 dim_F2 J^1/H^1=2, dim eta=2.
295 \end{Verbatim}
296
297 The desired two-plane source carrier has dimension 4, so it cannot be a
298 sector of this eta. Requiring the self-pair conductor additionally fixes
299 c=3h\_1 through
300
301 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
302 a(Pi^vee×Pi)=15+2N+c,
303 \end{Verbatim}
304
305 but is unnecessary for the capacity failure.
306
307 \hypertarget{dyadic_capacity-rank8-the-three-invariants-do-not-force-the-good-profile}{%
308 \subsection{6. Rank 8: the three invariants do not force the good
309 profile}\label{dyadic_capacity-rank8-the-three-invariants-do-not-force-the-good-profile}}
310
311 The full 2→4→8 tower has quotient dimension 6 and enough capacity.
312 Nevertheless Swan, torsion number and self-pair conductor alone do not
313 force that degree pattern. Here is an actual skipped-degree arithmetic
314 construction with the same three numerical invariants and a smaller
315 quotient.
316
317 Fix increasing odd h\_1\textless h\_2\textless h\_3 and put
318
319 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
320 q=3h_3, N=4q=12h_3, c=h_1+4h_2.
321 \end{Verbatim}
322
323 Then c is odd and 0\textless c\textless N. Indeed c\textless5h\_3, so
324 even 2c\textless N.
325
326 Construct the separable totally ramified quadratic extension
327
328 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
329 E=F(alpha), alpha^2+alpha=t^(−q).
330 \end{Verbatim}
331
332 Put a=(q+1)/2 and z=t\^{}a alpha. Then z is a uniformizer and
333
334 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
335 z^2+t^a z+t=0.
336 \end{Verbatim}
337
338 The element alpha has E-valuation −q and is minimal over F. On E\^{}4,
339 gamma=alpha I\_4 is minimal of ambient level N in the complete
340 period-eight order A.
341
342 Now construct the separable Eisenstein quartic extension over E
343
344 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
345 L=E(e), e^4+z e+z=0,
346 \end{Verbatim}
347
348 and let C be multiplication by e\^{}(−c) on L as a four-dimensional
349 E-space. Since c is odd, e\^{}(−c) is minimal, generates L/E, and
350 defines a simple maximal derived stratum {[}B\_order,c,c−1,C{]}.
351
352 Let P0 extract the coefficient of 1 on each E-coordinate in the basis
353 1,z. The normalized outer corestriction from the Artin--Schreier
354 right-inverse calculation has
355
356 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
357 s(P0)=t/z^2.
358 \end{Verbatim}
359
360 Consequently the explicit F-linear lift
361
362 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
363 b=(z^2/t) C P0
364 \end{Verbatim}
365
366 satisfies s(b)=C and has valuation −c.~Define the actual rank-eight
367 matrix
368
369 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
370 beta=alpha I_4+(z^2/t) C P0.
371 \end{Verbatim}
372
373 BK 2.2.3 applies because gamma is simple at level c\textless N and the
374 derived stratum is simple maximal. It proves that F{[}beta{]}/F is
375 totally ramified of degree~8, normalizes the complete chain, has leading
376 level N, and has critical exponent −c.~Its defining degree pattern is
377 8→2, skipping degree~4.
378
379 Since N is even and c is odd, the actual Heisenberg quotient has
380 dimension
381
382 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
383 (8−4)·1+(4−1)·0=4,
384 \end{Verbatim}
385
386 so eta has dimension 4. Yet its invariants are
387
388 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
389 Swan=N=12h_3,
390 torsion=1,
391 a(Pi^vee×Pi)=63+4N+3c
392 =63+48h_3+12h_2+3h_1,
393 \end{Verbatim}
394
395 exactly the same as the full three-stage profile. Any maximal simple
396 character and its standard type completion give an actual supercuspidal
397 Pi with these values. The required full three-plane source carrier would
398 have dimension 8, larger than this eta.
399
400 This is an arithmetic counterexample to the proposed implication from
401 those three numerical invariants to adequate rank-eight Heisenberg
402 capacity. It is not an H5 counterexample, does not identify the skipped
403 host with the source-defined classical candidate, and does not claim
404 that it preserves the supplied full source characters.
405
406 \hypertarget{dyadic_capacity-which-repairs-have-actually-been-tested}{%
407 \subsection{7. Which repairs have actually been
408 tested}\label{dyadic_capacity-which-repairs-have-actually-been-tested}}
409
410 \hypertarget{dyadic_capacity-replacing-the-displayed-type-or-parameter-at-fixed-swan-and-torsion}{%
411 \subsubsection{Replacing the displayed type or parameter at fixed Swan
412 and
413 torsion}\label{dyadic_capacity-replacing-the-displayed-type-or-parameter-at-fixed-swan-and-torsion}}
414
415 At ranks 2 and 4 this fails by Sections~4--5 for every maximal simple type
416 satisfying those invariants. A new matrix search or a different critical
417 coefficient cannot manufacture the missing quotient dimensions.
418
419 At rank~8 the full tower supplies the good profile, but the three
420 invariants alone do not force it inside a separately selected candidate.
421 Section~6 proves the failure of that proposed inference with an explicit
422 actual host. A finer endo-class or ramification comparison would be
423 needed for that same-candidate assertion.
424
425 \hypertarget{dyadic_capacity-skipping-approximation-degrees-or-changing-critical-parity}{%
426 \subsubsection{Skipping approximation degrees or changing critical
427 parity}\label{dyadic_capacity-skipping-approximation-degrees-or-changing-critical-parity}}
428
429 At rank~2 there is no degree to skip. At rank~4 the level excludes both a
430 scalar minimal approximation and a full-degree minimal approximation.
431 The remaining quadratic derived step has odd critical exponent by
432 minimality. Thus neither skipping nor declaring that exponent even is a
433 valid repair at fixed invariants.
434
435 At rank~8 a degree skip is genuinely possible, but it decreases capacity
436 in the explicit example above.
437
438 \hypertarget{dyadic_capacity-scalar-twists-shallow-or-deep}{%
439 \subsubsection{Scalar twists, shallow or
440 deep}\label{dyadic_capacity-scalar-twists-shallow-or-deep}}
441
442 A scalar twist of the existing actual inducing representation does not
443 increase its dimension. The exact Block164 pushout transport identifies
444 the same quotient J/H and leaves its commutator pairing unchanged. Thus
445 it cannot turn an insufficient eta into a whole source carrier.
446
447 For an actual parameter, replacing beta by beta+aI with a∈F leaves its
448 field, centralizer and commutator map unchanged. A shallow scalar
449 addition does not change the leading level. A dominating scalar addition
450 can change that level, but it contributes no new noncentral Heisenberg
451 directions: in the defining sequence the added scalar stage has equal H
452 and J orders. This agrees with the exact representation-level twist
453 transport. Enlarging a displayed numerical level is therefore not a
454 capacity repair.
455
456 This paragraph does not assert that every arbitrary scalar character
457 twist is represented by one globally specified additive parameter. The
458 already proved group/representation transport suffices for its negative
459 capacity conclusion.
460
461 \hypertarget{dyadic_capacity-adding-residue-multiplicity-or-increasing-ambient-rank}{%
462 \subsubsection{Adding residue multiplicity or increasing ambient
463 rank}\label{dyadic_capacity-adding-residue-multiplicity-or-increasing-ambient-rank}}
464
465 The torsion-one condition forces a full-degree totally ramified
466 parameter field and residue multiplicity one. A host with an unramified
467 or residual multiplicity factor does not preserve that invariant.
468 Increasing the ambient rank changes the required source rank. Neither
469 maneuver closes the current fixed-rank construction.
470
471 \hypertarget{dyadic_capacity-using-a-larger-arithmetic-carrier-inside-the-same-representation}{%
472 \subsubsection{Using a larger arithmetic carrier inside the same
473 representation}\label{dyadic_capacity-using-a-larger-arithmetic-carrier-inside-the-same-representation}}
474
475 A larger compact subgroup representation, a genuinely different
476 arithmetic quotient, or an induction-stage carrier might have enough
477 dimension. Such an enlargement is not ruled out here. It would need an
478 actual finite Heisenberg action carrying the prescribed full source data
479 and a proved recovery in the unchanged candidate. Artificially adding
480 vector-space copies or transporting abstract operators to an induced
481 space does not provide that arithmetic attachment or prove its kernel
482 compatibility. No such repair is constructed in this note.
483
484 \hypertarget{dyadic_capacity-quadratic-refinements-and-marked-centers}{%
485 \subsection{8. Quadratic refinements and marked
486 centers}\label{dyadic_capacity-quadratic-refinements-and-marked-centers}}
487
488 For d\textgreater=3, the dimension result gives a symplectic embedding
489 of the source quotient into the actual host quotient. With C trivial, a
490 C×-central pushout can then be matched by its alternating commutator,
491 and Stone--von Neumann supplies the corresponding source sector. This is
492 the same finite scalar-pushout attachment contract used previously.
493
494 At p=2, that statement must not be promoted to an identification of
495 arbitrary preselected native quadratic refinements. The C× pushout
496 permits rescaling lifts by i and can identify the pushouts of the
497 dihedral and quaternion extraspecial planes, even though their native
498 square functions differ. A chosen source embedding can preserve that
499 source's full finite central extension and its character values. It does
500 not thereby identify a separately marked target mu\_2 extension or its
501 Arf invariant.
502
503 In particular, symplectic-isotropic reduction respects a prescribed
504 quadratic form only when the reduced subspace is totally singular for
505 that form. The retained \path{DYADIC_AND_WEAK_ATTACHMENT159.md} gives the
506 exact quadratic/Arf obstruction. Additional source or target marked
507 centers, chosen lifts, quadratic refinements and nontrivial stabilizer
508 actions therefore require their own comparison. They are neither erased
509 from the task nor supplied by a dimension inequality.
510
511 \hypertarget{dyadic_capacity-scope-and-source-provenance}{%
512 \subsection{9. Scope and source
513 provenance}\label{dyadic_capacity-scope-and-source-provenance}}
514
515 The strongest proved conclusion is exact capacity for the specified
516 actual full defining profile, together with invariant low-rank failures
517 and an explicit rank-eight counterexample to numerical profile forcing.
518 No universal impossibility for original Theorem~16.2 is asserted. In
519 particular this note supplies no complete original H1--H5 counterexample
520 and does not decide whether that endpoint requires this precise
521 whole-maximal-type-carrier realization.
522
523 Primary source directly reread: Bushnell--Kutzko, \emph{The admissible
524 dual of GL(N) via compact open subgroups}, 1.4.14--15; 2.2.2--3;
525 2.4.1--2 and the maximal-field remark immediately preceding 2.4.7;
526 3.1.5, 3.1.7--9; 3.2.1--4; 3.4.1; 6.2.5 and its following paragraph.
527
528 Authorized retained text:
529
530 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
531 closure_recovery/extracted/Paper_XI_Continuation158_General_Sectors_and_Arit
532 hmetic_Constraints/scratch158/information158/input_cache/BK_source.txt
533 SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8
534 \end{Verbatim}
535
536 Internal inputs reused at their stated scope:
537
538 \begin{itemize}
539 \tightlist
540 \item
541 \path{attachment166/tower/RECURSIVE_TOWER166.md}: actual defining degrees,
542 levels, filtered right inverse and separated grade count. Its field constructor permits p=2.
543 \item
544 \path{attachment166/tower/TRACE_ZERO_AS_TOWER166.md}, Sections~1--2:
545 normalized corestriction and explicit right inverse. Their algebraic formulas apply at p=2. The odd-prime trace-zero assertion is not used.
546 \item
547 \path{attachment166/capacity/LEADING_LAYER_CAPACITY166.md}: the leading
548 subspace argument and the minimal terminal degree calculation.
549 \item
550 \path{base_realization165/polarization/PROFILE_RIGIDITY165.md} and
551 \path{closure_recovery/full161/scratch161/placement161/ADJOINT_CONDUCTOR_TEST161.md}:
552 retained formal-degree/self-pair conductor identities and
553 normalization.
554 \item
555 \path{input_pair164/TWIST_TRANSPORT.md}: exact character-twist/pushout
556 transport.
557 \item
558 \path{closure_recovery/extracted/Paper_XI_Continuation159_Type_Completion_and_Sector_Recovery/scratch159/coverage159/DYADIC_AND_WEAK_ATTACHMENT159.md}:
559 distinction between symplectic and marked quadratic reduction.
560 \end{itemize}
561
562 No full copyrighted source is redistributed here. No new external
563 correspondence theorem or H5 premise is introduced.
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