Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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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 rank8,
50 d=3, the quotient has dimension6 and eta has dimension8: 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 leading-layer-only sufficient threshold
87 d\textgreater=4 to the exact whole-profile threshold 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
123 dimensions really are b\_j; there is no unproved assumption that the
124 approximation fields are literally nested inside the final field.
125 
126 Applying the ring recursion gives the J-grade staircase:
127 
128 \begin{itemize}
129 \tightlist
130 \item
131   below a\_d the grade dimension is b\_d=1;
132 \item
133   on a\_(j+1)\textless=k\textless a\_j it is b\_j;
134 \item
135   from a\_1 upward it is b\_0=n.
136 \end{itemize}
137 
138 The H staircase uses the corresponding t\_j. Hence the exact
139 contribution of transition j to \(J^1/H^1\) is
140 
141 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
142 (b_(j−1)−b_j) · 1_(nu_j even).
143 \end{Verbatim}
144 
145 One can also obtain this by subtracting the exact logarithmic indices of
146 H and J in 1+P. It is an equality of the group index
147 \textbar \(J^1/H^1\)\textbar, not a guess from the visible
148 leading matrix support. BK 3.4.1 makes the resulting commutator pairing
149 nondegenerate.
150 
151 For j\textless d the level nu\_j has a factor of2 and is even. The final
152 level nu\_d=q\_1 is odd. Therefore
153 
154 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
155 dim_F2 J^1/H^1
156   =sum_(j=1)^(d−1) (2^(d−j+1)−2^(d−j))
157   =2^d−2.
158 \end{Verbatim}
159 
160 The omitted final increment would have dimension1. It cannot be treated
161 as a symplectic plane or silently restored: at that odd level the two
162 defining lattice cutoffs are exactly equal. Its disappearance is also
163 consistent with the even dimension required of a nondegenerate
164 alternating space.
165 
166 The source packet requires dimension2d. At d=1,2, 2\^{}d−2\textless2d.
167 At d=3 equality holds. If the inequality holds for d\textgreater=3, then
168 
169 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
170 2^(d+1)−2−2(d+1) = (2^d−2−2d)+(2^d−2)>0,
171 \end{Verbatim}
172 
173 so it continues to hold for every larger d.~This proves the exact
174 threshold.
175 
176 \hypertarget{dyadic_capacity-conductors-still-match-despite-the-lost-final-contribution}{%
177 \subsection{3. Conductors still match despite the lost final
178 contribution}\label{dyadic_capacity-conductors-still-match-despite-the-lost-final-contribution}}
179 
180 This section reuses the same retained formal-degree and self-pair
181 conductor identities already audited in Blocks161 and165. It is not
182 needed to prove the dimension formula, but shows why numerical conductor
183 checks do not expose the low-rank capacity failure.
184 
185 Put Delta\_j=b\_(j−1)−b\_j. The exact grade count gives
186 
187 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
188 I=log_2[(1+P):J^1]
189   =sum_j Delta_j ceil(nu_j/2)−(n−1).
190 \end{Verbatim}
191 
192 For this profile only the final level is odd and Delta\_d=1. Therefore
193 
194 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
195 2I=sum_j Delta_j nu_j+1−2(n−1).
196 \end{Verbatim}
197 
198 The Heisenberg dimension exponent is D/2, where D=n−2. The retained
199 formal-degree identity for a full-degree, totally ramified, torsion-one
200 maximal type gives
201 
202 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
203 a(Pi^vee×Pi)=n^2−1+(n−1)+2I+D.
204 \end{Verbatim}
205 
206 Substitution yields exact cancellation of the odd-cutoff correction:
207 
208 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
209 a(Pi^vee×Pi)=n^2−1+sum_j Delta_j nu_j
210            =2^(2d)−1+sum_(i=1)^d 3·4^(i−1) h_i.
211 \end{Verbatim}
212 
213 The ordinary conductor is n+nu\_1, so Swan(Pi)=2\^{}(d−1)·3h\_d, and the
214 torsion number is one. These are the usual three numerical invariants of
215 the stipulated independent dyadic Heisenberg tensor source.
216 
217 Thus all three numerical invariants can match while eta is too small for
218 the whole source carrier at ranks2 and4. This is a capacity failure, not
219 a conductor mismatch. It does not by itself refute any original H1--H5
220 statement.
221 
222 \hypertarget{dyadic_capacity-rank2-failure-is-forced-by-swan-and-torsion-alone}{%
223 \subsection{4. Rank2 failure is forced by Swan and torsion
224 alone}\label{dyadic_capacity-rank2-failure-is-forced-by-swan-and-torsion-alone}}
225 
226 Let Pi be any actual supercuspidal representation of GL2(F) with torsion
227 number one and odd Swan exponent N=3h\_1. BK 6.2.5 forces its maximal
228 parameter field K/F to be totally ramified of degree2 and its hereditary
229 order to have complete period2. The ordinary conductor formula then
230 fixes its native level as N.
231 
232 Every parameter beta of a maximal simple type in Pi has v\_K(beta)=−N,
233 which is odd. Since the residue extension is trivial, BK 1.4.14--15 say
234 beta is minimal over F. The two order definitions are therefore
235 
236 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
237 J_order=O_K+P^ceil(N/2),
238 H_order=O_K+P^(floor(N/2)+1).
239 \end{Verbatim}
240 
241 The exponents coincide because N is odd. Hence \(J^1=H^1\),
242 eta is one-dimensional, and no two-dimensional symplectic source plane
243 can embed into this quotient.
244 
245 This proves failure for every maximal simple type of every candidate
246 with these two invariants. It is stronger than failure of one explicit
247 matrix formula. It remains a statement about attachment inside the
248 actual maximal-simple-type Heisenberg carrier, not a proof that no other
249 arithmetic realization relation could work.
250 
251 \hypertarget{dyadic_capacity-rank4-failure-is-also-forced-by-swan-and-torsion}{%
252 \subsection{5. Rank4 failure is also forced by Swan and
253 torsion}\label{dyadic_capacity-rank4-failure-is-also-forced-by-swan-and-torsion}}
254 
255 Let Pi be any actual supercuspidal representation of GL4(F) with torsion
256 number one and Swan exponent N=6h\_2, where h\_2 is odd. Its maximal
257 parameter field is totally ramified of degree4, its order has period4,
258 and its native level is N.
259 
260 Here v\_2(N)=1. The defining-sequence/minimality argument of Block166
261 forces the terminal minimal approximation field E/F to have degree2.
262 There is no possible intermediate degree between2 and4. The scalar
263 terminal case is excluded by 4 not dividing N, and a minimal
264 full-degree4 parameter is excluded because gcd(N,4)=2. Thus the actual
265 degree pattern is forced to be 4→2.
266 
267 Write c=−k\_0(beta,A), so 1\textless=c\textless N. Select the critical
268 approximation furnished by the construction in BK 2.4.1 and the remark
269 preceding 2.4.7. Because the original field is maximal, this
270 approximation can be selected with a simple maximal derived stratum in
271 B=End\_E(F\^{}4)≅M2(E). Equivalently one can replace its derived stratum
272 by an equivalent simple representative and use BK 2.2.2--3. The derived
273 field has degree2 and ramification index2 over E. A simple stratum
274 {[}B\_order,c,c−1,alpha{]} is minimal. Since the complete B-chain has
275 period2 and the derived field is totally ramified, BK 1.4.14 forces
276 
277 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
278 gcd(c,2)=1.
279 \end{Verbatim}
280 
281 Therefore c is odd. It cannot be made even while preserving this degree
282 pattern and a genuine maximal derived field.
283 
284 Let j=(c+1)/2. The actual orders are
285 
286 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
287 J_order=O_K+Q^j+P^(N/2),
288 H_order=O_K+Q^j+P^(N/2+1).
289 \end{Verbatim}
290 
291 The final low transition contributes nothing, while the high transition
292 has dimension4−2=2. Thus every such Pi has
293 
294 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
295 dim_F2 J^1/H^1=2,       dim eta=2.
296 \end{Verbatim}
297 
298 The desired two-plane source carrier has dimension4, so it cannot be a
299 sector of this eta. Requiring the self-pair conductor additionally fixes
300 c=3h\_1 through
301 
302 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
303 a(Pi^vee×Pi)=15+2N+c,
304 \end{Verbatim}
305 
306 but is unnecessary for the capacity failure.
307 
308 \hypertarget{dyadic_capacity-rank8-the-three-invariants-do-not-force-the-good-profile}{%
309 \subsection{6. Rank8: the three invariants do not force the good
310 profile}\label{dyadic_capacity-rank8-the-three-invariants-do-not-force-the-good-profile}}
311 
312 The full 2→4→8 tower has quotient dimension6 and enough capacity.
313 Nevertheless Swan, torsion number and self-pair conductor alone do not
314 force that degree pattern. Here is an actual skipped-degree arithmetic
315 construction with the same three numerical invariants and a smaller
316 quotient.
317 
318 Fix increasing odd h\_1\textless h\_2\textless h\_3 and put
319 
320 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
321 q=3h_3,       N=4q=12h_3,       c=h_1+4h_2.
322 \end{Verbatim}
323 
324 Then c is odd and 0\textless c\textless N; indeed c\textless5h\_3, so
325 even 2c\textless N.
326 
327 Construct the separable totally ramified quadratic extension
328 
329 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
330 E=F(alpha),       alpha^2+alpha=t^(−q).
331 \end{Verbatim}
332 
333 Put a=(q+1)/2 and z=t\^{}a alpha. Then z is a uniformizer and
334 
335 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
336 z^2+t^a z+t=0.
337 \end{Verbatim}
338 
339 The element alpha has E-valuation −q and is minimal over F. On E\^{}4,
340 gamma=alpha I\_4 is minimal of ambient level N in the complete
341 period-eight order A.
342 
343 Now construct the separable Eisenstein quartic extension over E
344 
345 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
346 L=E(e),       e^4+z e+z=0,
347 \end{Verbatim}
348 
349 and let C be multiplication by e\^{}(−c) on L as a four-dimensional
350 E-space. Since c is odd, e\^{}(−c) is minimal, generates L/E, and
351 defines a simple maximal derived stratum {[}B\_order,c,c−1,C{]}.
352 
353 Let P0 extract the coefficient of1 on each E-coordinate in the basis
354 1,z. The normalized outer corestriction from the Artin--Schreier
355 right-inverse calculation has
356 
357 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
358 s(P0)=t/z^2.
359 \end{Verbatim}
360 
361 Consequently the explicit F-linear lift
362 
363 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
364 b=(z^2/t) C P0
365 \end{Verbatim}
366 
367 satisfies s(b)=C and has valuation −c.~Define the actual rank-eight
368 matrix
369 
370 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
371 beta=alpha I_4+(z^2/t) C P0.
372 \end{Verbatim}
373 
374 BK 2.2.3 applies because gamma is simple at level c\textless N and the
375 derived stratum is simple maximal. It proves that F{[}beta{]}/F is
376 totally ramified of degree8, normalizes the complete chain, has leading
377 level N, and has critical exponent −c.~Its defining degree pattern is
378 8→2, skipping degree4.
379 
380 Since N is even and c is odd, the actual Heisenberg quotient has
381 dimension
382 
383 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
384 (8−4)·1+(4−1)·0=4,
385 \end{Verbatim}
386 
387 so eta has dimension4. Yet its invariants are
388 
389 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
390 Swan=N=12h_3,
391 torsion=1,
392 a(Pi^vee×Pi)=63+4N+3c
393              =63+48h_3+12h_2+3h_1,
394 \end{Verbatim}
395 
396 exactly the same as the full three-stage profile. Any maximal simple
397 character and its standard type completion give an actual supercuspidal
398 Pi with these values. The required full three-plane source carrier would
399 have dimension8, larger than this eta.
400 
401 This is an arithmetic counterexample to the proposed implication from
402 those three numerical invariants to adequate rank-eight Heisenberg
403 capacity. It is not an H5 counterexample, does not identify the skipped
404 host with the source-defined classical candidate, and does not claim
405 that it preserves the supplied full source characters.
406 
407 \hypertarget{dyadic_capacity-which-repairs-have-actually-been-tested}{%
408 \subsection{7. Which repairs have actually been
409 tested}\label{dyadic_capacity-which-repairs-have-actually-been-tested}}
410 
411 \hypertarget{dyadic_capacity-replacing-the-displayed-type-or-parameter-at-fixed-swan-and-torsion}{%
412 \subsubsection{Replacing the displayed type or parameter at fixed Swan
413 and
414 torsion}\label{dyadic_capacity-replacing-the-displayed-type-or-parameter-at-fixed-swan-and-torsion}}
415 
416 At ranks2 and4 this fails by Sections4--5 for every maximal simple type
417 satisfying those invariants. A new matrix search or a different critical
418 coefficient cannot manufacture the missing quotient dimensions.
419 
420 At rank8 the full tower supplies the good profile, but the three
421 invariants alone do not force it inside a separately selected candidate.
422 Section6 proves the failure of that proposed inference with an explicit
423 actual host. A finer endo-class or ramification comparison would be
424 needed for that same-candidate assertion.
425 
426 \hypertarget{dyadic_capacity-skipping-approximation-degrees-or-changing-critical-parity}{%
427 \subsubsection{Skipping approximation degrees or changing critical
428 parity}\label{dyadic_capacity-skipping-approximation-degrees-or-changing-critical-parity}}
429 
430 At rank2 there is no degree to skip. At rank4 the level excludes both a
431 scalar minimal approximation and a full-degree minimal approximation.
432 The remaining quadratic derived step has odd critical exponent by
433 minimality. Thus neither skipping nor declaring that exponent even is a
434 valid repair at fixed invariants.
435 
436 At rank8 a degree skip is genuinely possible, but it decreases capacity
437 in the explicit example above.
438 
439 \hypertarget{dyadic_capacity-scalar-twists-shallow-or-deep}{%
440 \subsubsection{Scalar twists, shallow or
441 deep}\label{dyadic_capacity-scalar-twists-shallow-or-deep}}
442 
443 A scalar twist of the existing actual inducing representation does not
444 increase its dimension. The exact Block164 pushout transport identifies
445 the same quotient J/H and leaves its commutator pairing unchanged. Thus
446 it cannot turn an insufficient eta into a whole source carrier.
447 
448 For an actual parameter, replacing beta by beta+aI with a∈F leaves its
449 field, centralizer and commutator map unchanged. A shallow scalar
450 addition does not change the leading level. A dominating scalar addition
451 can change that level, but it contributes no new noncentral Heisenberg
452 directions: in the defining sequence the added scalar stage has equal H
453 and J orders. This agrees with the exact representation-level twist
454 transport. Enlarging a displayed numerical level is therefore not a
455 capacity repair.
456 
457 This paragraph does not assert that every arbitrary scalar character
458 twist is represented by one globally specified additive parameter. The
459 already proved group/representation transport suffices for its negative
460 capacity conclusion.
461 
462 \hypertarget{dyadic_capacity-adding-residue-multiplicity-or-increasing-ambient-rank}{%
463 \subsubsection{Adding residue multiplicity or increasing ambient
464 rank}\label{dyadic_capacity-adding-residue-multiplicity-or-increasing-ambient-rank}}
465 
466 The torsion-one condition forces a full-degree totally ramified
467 parameter field and residue multiplicity one. A host with an unramified
468 or residual multiplicity factor does not preserve that invariant.
469 Increasing the ambient rank changes the required source rank. Neither
470 maneuver closes the current fixed-rank construction.
471 
472 \hypertarget{dyadic_capacity-using-a-larger-arithmetic-carrier-inside-the-same-representation}{%
473 \subsubsection{Using a larger arithmetic carrier inside the same
474 representation}\label{dyadic_capacity-using-a-larger-arithmetic-carrier-inside-the-same-representation}}
475 
476 A larger compact subgroup representation, a genuinely different
477 arithmetic quotient, or an induction-stage carrier might have enough
478 dimension. Such an enlargement is not ruled out here. It would need an
479 actual finite Heisenberg action carrying the prescribed full source data
480 and a proved recovery in the unchanged candidate. Artificially adding
481 vector-space copies or transporting abstract operators to an induced
482 space does not provide that arithmetic attachment or prove its kernel
483 compatibility. No such repair is constructed in this note.
484 
485 \hypertarget{dyadic_capacity-quadratic-refinements-and-marked-centers}{%
486 \subsection{8. Quadratic refinements and marked
487 centers}\label{dyadic_capacity-quadratic-refinements-and-marked-centers}}
488 
489 For d\textgreater=3, the dimension result gives a symplectic embedding
490 of the source quotient into the actual host quotient. With C trivial, a
491 C×-central pushout can then be matched by its alternating commutator,
492 and Stone--von Neumann supplies the corresponding source sector. This is
493 the same finite scalar-pushout attachment contract used previously.
494 
495 At p=2, that statement must not be promoted to an identification of
496 arbitrary preselected native quadratic refinements. The C× pushout
497 permits rescaling lifts by i and can identify the pushouts of the
498 dihedral and quaternion extraspecial planes, even though their native
499 square functions differ. A chosen source embedding can preserve that
500 source's full finite central extension and its character values. It does
501 not thereby identify a separately marked target mu\_2 extension or its
502 Arf invariant.
503 
504 In particular, symplectic-isotropic reduction respects a prescribed
505 quadratic form only when the reduced subspace is totally singular for
506 that form. The retained \path{DYADIC_AND_WEAK_ATTACHMENT159.md} gives the
507 exact quadratic/Arf obstruction. Additional source or target marked
508 centers, chosen lifts, quadratic refinements and nontrivial stabilizer
509 actions therefore require their own comparison. They are neither erased
510 from the task nor supplied by a dimension inequality.
511 
512 \hypertarget{dyadic_capacity-scope-and-source-provenance}{%
513 \subsection{9. Scope and source
514 provenance}\label{dyadic_capacity-scope-and-source-provenance}}
515 
516 The strongest proved conclusion is exact capacity for the specified
517 actual full defining profile, together with invariant low-rank failures
518 and an explicit rank-eight counterexample to numerical profile forcing.
519 No universal impossibility for original Theorem16.2 is asserted. In
520 particular this note supplies no complete original H1--H5 counterexample
521 and does not decide whether that endpoint requires this precise
522 whole-maximal-type-carrier realization.
523 
524 Primary source directly reread: Bushnell--Kutzko, \emph{The admissible
525 dual of GL(N) via compact open subgroups}, 1.4.14--15; 2.2.2--3;
526 2.4.1--2 and the maximal-field remark immediately preceding 2.4.7;
527 3.1.5, 3.1.7--9; 3.2.1--4; 3.4.1; 6.2.5 and its following paragraph.
528 
529 Authorized retained text:
530 
531 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
532 closure_recovery/extracted/Paper_XI_Continuation158_General_Sectors_and_Arit
533 hmetic_Constraints/scratch158/information158/input_cache/BK_source.txt
534 SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8
535 \end{Verbatim}
536 
537 Internal inputs reused at their stated scope:
538 
539 \begin{itemize}
540 \tightlist
541 \item
542   \path{attachment166/tower/RECURSIVE_TOWER166.md}: actual defining degrees,
543   levels, filtered right inverse and separated grade count; its field
544   constructor permits p=2.
545 \item
546   \path{attachment166/tower/TRACE_ZERO_AS_TOWER166.md}, Sections1--2:
547   normalized corestriction and explicit right inverse. Their algebraic
548   formulas apply at p=2; the odd-prime trace-zero assertion is not used.
549 \item
550   \path{attachment166/capacity/LEADING_LAYER_CAPACITY166.md}: the leading
551   subspace argument and the minimal terminal degree calculation.
552 \item
553   \path{base_realization165/polarization/PROFILE_RIGIDITY165.md} and
554   \path{closure_recovery/full161/scratch161/placement161/ADJOINT_CONDUCTOR_TEST161.md}:
555   retained formal-degree/self-pair conductor identities and
556   normalization.
557 \item
558   \path{input_pair164/TWIST_TRANSPORT.md}: exact character-twist/pushout
559   transport.
560 \item
561   \path{closure_recovery/extracted/Paper_XI_Continuation159_Type_Completion_and_Sector_Recovery/scratch159/coverage159/DYADIC_AND_WEAK_ATTACHMENT159.md}:
562   distinction between symplectic and marked quadratic reduction.
563 \end{itemize}
564 
565 No full copyrighted source is redistributed here. No new external
566 correspondence theorem or H5 premise is introduced.

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