Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{The actual enlarged rank-two inducing carrier}\label{app:rank2}
3 
4 Status: ordinary proof. The construction below supplies an actual
5 arithmetic finite Heisenberg pair and exact recovery of an already fixed
6 representation. It does not change the previously audited
7 zero-dimensional BK quotient at rank two. No Lean verification or H5
8 comparison is claimed.
9 
10 \hypertarget{rank2-exact-setting-and-result}{%
11 \subsection{1. Exact setting and
12 result}\label{rank2-exact-setting-and-result}}
13 
14 Let F=F\_2((t)) and G=GL2(F). Let {[}A,q,0,beta{]} be a minimal simple
15 stratum with K=F{[}beta{]} totally ramified of degree2 and A its
16 complete period-two hereditary order. Assume
17 
18 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
19 q=2m+1>=3,       m>=1.
20 \end{Verbatim}
21 
22 Write R=rad(A), \(U_A^a=1+R^a\) for a\textgreater=1, and
23 \(U_K^a=1+\mathfrak p_K^a\). Let theta be a full simple character.
24 Since q is odd and the residue field is F2, the actual maximal simple
25 type group and character groups coincide:
26 
27 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
28 J=J^1=H^1=U_K^1 U_A^(m+1).
29 \end{Verbatim}
30 
31 Fix the full actual extension Lambda of theta to
32 
33 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
34 N=K×J,
35 \end{Verbatim}
36 
37 and the already fixed irreducible supercuspidal candidate
38 
39 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
40 Pi=c-Ind_N^G Lambda.
41 \end{Verbatim}
42 
43 Lambda is one-dimensional here. Its value on a uniformizer of K,
44 including the noncompact scalar normalization, is part of the fixed
45 data.
46 
47 Define the larger compact open arithmetic subgroup
48 
49 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
50 P=U_K^1 U_A^m.
51 \end{Verbatim}
52 
53 Then the following statements hold.
54 
55 \begin{enumerate}
56 \def\labelenumi{\arabic{enumi}.}
57 \item
58   J is normal of index2 in P, and P∩N=J.
59 \item
60   E=Ind\_J\^{}P theta is an actual irreducible two-dimensional
61   P-representation.
62 \item
63   There is an explicitly defined normal subgroup H\_new⊂J of index2 such
64   that P/H\_new≅F2² and theta\_new=theta\textbar H\_new has
65   nondegenerate commutator pairing on that quotient. E is its Heisenberg
66   representation.
67 \item
68   E embeds into the restriction of the same Pi, as the subspace of
69   inducing functions supported on NP.
70 \item
71   With X=c-Ind\_J\^{}G theta, a uniformizer varpi\_K and
72   s=Lambda(varpi\_K), there is an exact quotient
73 
74 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
75 Pi ≅ X/(T−s)X,
76 (Tf)(g)=f(varpi_K g).
77 \end{Verbatim}
78 
79   Induction in stages identifies X with c-Ind\_P\^{}G E. This gives an
80   exact same-candidate recovery using the retained noncompact scalar,
81   not merely a nonzero cyclic-generation assertion.
82 \end{enumerate}
83 
84 The new Heisenberg pair is an actual pair of compact subgroups of
85 GL2(F), but is not asserted to be the BK \(H^1/J^1\) pair of
86 a maximal simple type. Whether the original H2 realization relation
87 allows this enlargement is a separate obligation.
88 
89 \hypertarget{rank2-the-compact-enlargement-is-a-group-with-normal-subgroup-of-index-two}{%
90 \subsection{2. The compact enlargement is a group, with normal subgroup
91 of index
92 two}\label{rank2-the-compact-enlargement-is-a-group-with-normal-subgroup-of-index-two}}
93 
94 K× normalizes A and every power of its radical, so U\_K\^{}1 normalizes
95 U\_A\^{}m. Therefore P is a subgroup and is compact open. It is
96 contained in U\_A\^{}1 because m\textgreater=1 and
97 \(U_K^1\subset U_A^1\).
98 
99 The standard radical calculation gives
100 
101 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
102 [U_A^a,U_A^b]⊂U_A^(a+b),       a,b>=1.
103 \end{Verbatim}
104 
105 Indeed, for x∈R\^{}a and y∈R\^{}b, the difference (1+x)(1+y)−(1+y)(1+x)
106 is in R\^{}(a+b), and multiplication by unit inverses preserves that
107 ideal.
108 
109 Consequently
110 
111 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
112 [U_A^m,U_K^1]⊂U_A^(m+1),
113 [U_A^m,U_A^(m+1)]⊂U_A^(2m+1)⊂U_A^(m+1).
114 \end{Verbatim}
115 
116 These facts, and normalization by U\_K\^{}1, show that J is normal in P.
117 They also show
118 
119 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
120 [P,J]⊂U_A^(m+1).                                      (1)
121 \end{Verbatim}
122 
123 To calculate the index, use
124 
125 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
126 U_A^m∩J=U_K^m U_A^(m+1).
127 \end{Verbatim}
128 
129 For if ku∈U\_A\^{}m with k∈U\_K\^{}1 and u∈U\_A\^{}(m+1), then
130 \(k\in K^\times\cap U_A^m=U_K^m\). The reverse inclusion is immediate.
131 The complete period-two A-grade \(R^m/R^{m+1}\) has
132 F2-dimension2, and its field grade \(\mathfrak p_K^m/\mathfrak p_K^{m+1}\) has
133 dimension1. Thus
134 
135 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
136 [P:J]=[U_A^m:U_K^m U_A^(m+1)]=2.                      (2)
137 \end{Verbatim}
138 
139 Finally, if kj∈P with k∈K× and j∈J, then \(k\in Pj^{-1}\subset U_A^1\).
140 Hence k∈U\_K\^{}1 and kj∈J. Therefore
141 
142 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
143 P∩N=J.                                                (3)
144 \end{Verbatim}
145 
146 These arguments use the actual group intersections. They do not replace
147 P or J by a vector-space surrogate.
148 
149 \hypertarget{rank2-irreducibility-and-the-conjugate-character-ratio}{%
150 \subsection{3. Irreducibility and the conjugate-character
151 ratio}\label{rank2-irreducibility-and-the-conjugate-character-ratio}}
152 
153 For the minimal simple character, the primary BK intertwining theorem
154 gives
155 
156 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
157 I_G(theta)=J K×J=N.
158 \end{Verbatim}
159 
160 Here K× normalizes J and theta. In particular, for g∈P\textbackslash J,
161 (3) says g does not intertwine theta. Because J is normal in P, this
162 means that the two characters
163 
164 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
165 theta,
166 theta^g(j)=theta(gjg^−1)
167 \end{Verbatim}
168 
169 are distinct. The index-two induction E=Ind\_J\^{}P theta is therefore
170 irreducible. This also follows directly from the two distinct J-weight
171 lines, which are interchanged by g.
172 
173 Define the actual character of J
174 
175 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
176 lambda(j)=theta^g(j)/theta(j).
177 \end{Verbatim}
178 
179 It is nontrivial. By (1),
180 
181 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
182 lambda(j)=theta(gjg^−1j^−1),
183 \end{Verbatim}
184 
185 and the argument of theta lies in U\_A\^{}(m+1). The full simple
186 character has exact restriction
187 
188 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
189 theta(1+x)=psi_F(tr_F(beta x)),       x∈R^(m+1).
190 \end{Verbatim}
191 
192 Since F has characteristic2, its additive characters take values in
193 \{1,−1\}. Consequently
194 
195 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
196 lambda(J)={1,−1}.                                    (4)
197 \end{Verbatim}
198 
199 No assertion about shallow values of theta is used here. Theta may have
200 a larger finite image on J. Only its ratio with its actual conjugate is
201 proved to have order2.
202 
203 As g²∈J, conjugation by g² fixes the character theta. Thus
204 \(\lambda^g=\lambda^{-1}=\lambda\). It follows that
205 
206 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
207 H_new=ker(lambda)
208 \end{Verbatim}
209 
210 is normal in P. It has index2 in J and index4 in P.
211 
212 \hypertarget{rank2-the-actual-finite-heisenberg-pair}{%
213 \subsection{4. The actual finite Heisenberg
214 pair}\label{rank2-the-actual-finite-heisenberg-pair}}
215 
216 Use the function model of E with f(jp)=theta(j)f(p) and right
217 translation by P. Coordinates are f(1),f(g). Then for j∈J,
218 
219 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
220 E(j)=diag(theta(j),theta^g(j))
221     =theta(j) diag(1,lambda(j)),
222 \end{Verbatim}
223 
224 while
225 
226 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
227 E(g)=[[0,1],[theta(g²),0]].                            (5)
228 \end{Verbatim}
229 
230 Therefore E(g)²=theta(g²)I. Choose j\_0∈J with lambda(j\_0)=−1, and put
231 
232 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
233 Z=theta(j_0)^−1 E(j_0)=diag(1,−1),
234 X=E(g).
235 \end{Verbatim}
236 
237 The exact matrix identity is
238 
239 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
240 XZX^−1=−Z.                                           (6)
241 \end{Verbatim}
242 
243 The projective image of J has order2, generated by Z, while X is
244 off-diagonal and contributes an independent projective involution. Thus
245 the projective image of P is exactly C2×C2, with nontrivial commutator
246 (6). Its projective kernel is precisely H\_new. In particular
247 
248 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
249 P/H_new≅F2².
250 \end{Verbatim}
251 
252 On H\_new, E is the scalar theta\_new=theta\textbar H\_new. This
253 character is P-invariant: conjugation by J fixes theta and conjugation
254 by g changes theta by lambda, which is trivial on H\_new. The resulting
255 alternating form
256 
257 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
258 (pH_new,p'H_new)↦theta_new([p,p'])
259 \end{Verbatim}
260 
261 is nondegenerate, since its two displayed generators have commutator −1.
262 Therefore (H\_new,P,theta\_new) is an actual arithmetic finite
263 Heisenberg pair, and its actual irreducible carrier E has dimension2.
264 
265 All representations here have finite image on the compact group P: theta
266 has open kernel, the induction has finite dimension, and an open normal
267 core of that kernel acts trivially. Thus no infinite projective-image
268 issue is being suppressed.
269 
270 This proof does not require rescaling the actual matrix g inside GL2(F).
271 Its represented square theta(g²) can be an arbitrary retained scalar.
272 Rescaling X is only allowed later in a C× pushout if that is the
273 attachment contract.
274 
275 \hypertarget{rank2-explicit-embedding-into-the-same-fixed-candidate}{%
276 \subsection{5. Explicit embedding into the same fixed
277 candidate}\label{rank2-explicit-embedding-into-the-same-fixed-candidate}}
278 
279 Realize Pi as functions F:G→C with
280 
281 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
282 F(nx)=Lambda(n)F(x),
283 \end{Verbatim}
284 
285 compact support modulo N, and right translation by G. For f∈E define
286 i(f) by
287 
288 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
289 i(f)(np)=Lambda(n)f(p),       n∈N, p∈P,
290 \end{Verbatim}
291 
292 and zero off NP. This is well-defined because N∩P=J and
293 Lambda\textbar J=theta. The set NP is a finite union of N-cosets, by
294 {[}P:J{]}=2, so the support is compact modulo N. The map i is
295 P-equivariant and injective.
296 
297 Its image is exactly the two-dimensional P-stable subspace of Pi
298 consisting of inducing functions supported on NP. Thus the enlarged
299 carrier is already inside the given Pi. No new candidate is chosen, and
300 all noncompact values of Lambda remain fixed.
301 
302 Since Pi is irreducible and i(E) is nonzero, its G-span is Pi. More
303 precisely, there is the canonical G-equivariant quotient
304 
305 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
306 Q_E:c-Ind_P^G E→Pi,
307 Q_E(F)=sum_(Px∈P\G) Pi(x^−1)i(F(x)).                 (7)
308 \end{Verbatim}
309 
310 The sum is finite and independent of representatives. If x is replaced
311 by px, then F(px)=E(p)F(x) and P-equivariance of i cancels the
312 additional Pi(p\^{}−1). Formula (7) is nonzero and hence onto. The next
313 section identifies its kernel exactly.
314 
315 \hypertarget{rank2-exact-recovery-by-the-noncompact-hecke-eigenvalue}{%
316 \subsection{6. Exact recovery by the noncompact Hecke
317 eigenvalue}\label{rank2-exact-recovery-by-the-noncompact-hecke-eigenvalue}}
318 
319 Let X=c-Ind\_J\^{}G theta, in the same left-covariant/right-translation
320 convention. Induction in stages gives an explicit isomorphism
321 
322 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
323 c-Ind_P^G E → X,
324 F ↦ [x↦F(x)(1)].                                     (8)
325 \end{Verbatim}
326 
327 The inverse sends f∈X to the E-valued function F(x)(p)=f(px). Both
328 covariance statements follow directly from the definitions, and the
329 finite index {[}P:J{]} ensures the support conditions agree.
330 
331 Fix a uniformizer varpi=varpi\_K. Because K× normalizes theta, the
332 operator
333 
334 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
335 (Tf)(x)=f(varpi x)                                   (9)
336 \end{Verbatim}
337 
338 is a well-defined invertible G-equivariant operator on X. Put
339 s=Lambda(varpi), which is nonzero. Since k\_K=F2 and J contains
340 U\_K\^{}1,
341 
342 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
343 N/J≅K×/U_K^1≅Z,
344 \end{Verbatim}
345 
346 with generator varpi. Define
347 
348 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
349 Q:X→Pi,
350 (Qf)(x)=sum_(ell∈Z) s^(−ell) f(varpi^ell x).          (10)
351 \end{Verbatim}
352 
353 For each x the sum is finite, since f has compact support modulo the
354 open subgroup J. Equation (10) satisfies
355 
356 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
357 (Qf)(varpi x)=s(Qf)(x),
358 (Qf)(jx)=theta(j)(Qf)(x),       j∈J,
359 \end{Verbatim}
360 
361 and therefore the required N-covariance. Its support is compact modulo
362 N. It commutes with G-right translation and satisfies
363 
364 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
365 QT=sQ.                                               (11)
366 \end{Verbatim}
367 
368 It is onto: a function in Pi has finite support modulo N, and on each
369 occupied N-coset its value can be lifted to a function supported on one
370 J-coset. Extend those finitely many choices using theta.
371 
372 To compute the kernel, decompose X as the algebraic direct sum of its
373 subspaces supported on the left N-cosets in G. Each such summand is the
374 space of finitely supported sequences on N/J≅Z, with the fixed theta
375 covariance. The operator T is a bilateral shift. On that summand, (10)
376 is evaluation of its Laurent polynomial at the nonzero scalar s, up to
377 the harmless convention of replacing the Laurent variable by its
378 inverse. The kernel is therefore the principal ideal generated by T−s.
379 The elementary identity
380 
381 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
382 ker(C[z,z^−1]→C, z↦s)=(z−s)
383 \end{Verbatim}
384 
385 proves, summand by summand, that
386 
387 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
388 ker Q=(T−s)X.
389 \end{Verbatim}
390 
391 All vectors occupy finitely many such summands, so no completion or
392 closure is needed. Thus
393 
394 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
395 Pi ≅ X/(T−s)X ≅ (c-Ind_P^G E)/(T−s)(c-Ind_P^G E),    (12)
396 \end{Verbatim}
397 
398 where T is transported through (8).
399 
400 Under (8), formula (10) agrees with (7): on the standard copy of E
401 supported on P, it gives exactly i, and G-equivariance then gives
402 equality everywhere. Thus (12) identifies the kernel of the canonical
403 recovery map, not merely the kernel of an unrelated surjection.
404 
405 The same description can be phrased as evaluation of the type Hecke
406 algebra at the character determined by Lambda. Its intertwining support
407 is N, each supported intertwining space is one-dimensional, and the
408 retained extension Lambda normalizes the Laurent generator. The explicit
409 formulas above avoid any ambiguity about convolution orientation.
410 
411 The scalar s is essential. E alone, being compact data, does not
412 determine it. Changing s can change the unramified normalization of the
413 candidate. The construction retains the original Lambda and s, so it
414 recovers the exact original Pi rather than an unspecified member of its
415 inertial class.
416 
417 \hypertarget{rank2-full-scalar-pushout-attachment-and-remaining-marking-obligations}{%
418 \subsection{7. Full scalar-pushout attachment and remaining marking
419 obligations}\label{rank2-full-scalar-pushout-attachment-and-remaining-marking-obligations}}
420 
421 Form the C× pushout of the actual pair (H\_new,P,theta\_new). It is a
422 Heisenberg extension of the actual symplectic plane P/H\_new. Any
423 supplied source plane with the same chosen alternating form has a
424 center-preserving pushout isomorphism into it. Every source
425 full-character value is then preserved in the center, as in the retained
426 full-character attachment theorem. Stone--von Neumann identifies its
427 source carrier with the actual two-dimensional E.
428 
429 Together with the embedding i and exact quotient (12), this provides an
430 actual arithmetic carrier and recovery inside the same fixed candidate.
431 It closes the carrier-capacity failure of the old rank-two eta by
432 changing the compact arithmetic pair, not by increasing the old eta or
433 discarding its dimension calculation.
434 
435 At p=2, a C× pushout does not automatically identify a separately marked
436 native mu\_2 extension or quadratic refinement. The source finite
437 central extension can be retained through its chosen embedding, while a
438 required comparison with the target's preselected square function
439 remains an additional question. No arbitrary native Arf invariant is
440 asserted here.
441 
442 Nontrivial stabilizer operators, transport graphs, and original H2's
443 requirement that a target be a particular BK type-side quotient are
444 likewise separate. In particular, the theorem proves that the new pair
445 is an actual arithmetic finite Heisenberg pair, but does not rename it
446 as the unchanged maximal-simple-type pair. No new hypothesis is silently
447 added to original Theorem16.2.
448 
449 \hypertarget{rank2-scope-smallest-boundary-and-sources}{%
450 \subsection{8. Scope, smallest boundary, and
451 sources}\label{rank2-scope-smallest-boundary-and-sources}}
452 
453 The inequality q\textgreater=3 matters. If q=1, then m=0 and the
454 displayed enlargement is not the group used above. Over residue field F2
455 the complete Iwahori unit group equals U\_A\^{}1, and a naive level-zero
456 replacement does not give the asserted index-two enlargement. No q=1
457 claim is made. The source levels q=3h with positive odd h satisfy the
458 required q\textgreater=3.
459 
460 The proof uses the actual one-dimensional Lambda because the maximal
461 parameter field has full degree and residue field F2. It does not apply
462 the same scalar argument automatically to a higher-rank inducing
463 representation.
464 
465 Primary BK passages: 3.1.7--8 (odd minimal H/J equality); 3.2.1--2 (deep
466 restriction and K× normalization); 3.3.2 and 5.1.1/5.1.8 (intertwining);
467 and 6.2.1--5 (the fixed maximal-type extension and compact induction).
468 The subgroup, matrix, Mackey-support, and Laurent-evaluation
469 calculations are proved explicitly above. No external correspondence
470 theorem is used in this enlargement.
471 
472 Retained primary text:
473 
474 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
475 closure_recovery/extracted/Paper_XI_Continuation158_General_Sectors_and_Arit
476 hmetic_Constraints/scratch158/information158/input_cache/BK_source.txt
477 SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8
478 \end{Verbatim}
479 
480 The previously reviewed negative statement concerned eta of the original
481 BK \(J^1/H^1\). That quotient remains zero. The new positive
482 statement concerns a different, explicitly constructed arithmetic pair
483 inside the same ambient representation, with exact
484 noncompact-normalization-aware recovery.

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