Realization Calculus

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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 degree~2 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 index 2 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 index 2 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-dimension 2, and its field grade \(\mathfrak p_K^m/\mathfrak p_K^{m+1}\) has
133 dimension 1. 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 characteristic 2, 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 order 2.
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 index 2 in J and index 4 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 order 2, 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 dimension 2.
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 issue with an infinite projective image is being suppressed.
268 
269 This proof does not require rescaling the actual matrix g inside GL2(F).
270 Its represented square theta(g²) can be an arbitrary retained scalar.
271 Rescaling X is only allowed later in a C× pushout if that is the
272 attachment contract.
273 
274 \hypertarget{rank2-explicit-embedding-into-the-same-fixed-candidate}{%
275 \subsection{5. Explicit embedding into the same fixed
276 candidate}\label{rank2-explicit-embedding-into-the-same-fixed-candidate}}
277 
278 Realize Pi as functions F:G→C with
279 
280 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
281 F(nx)=Lambda(n)F(x),
282 \end{Verbatim}
283 
284 compact support modulo N, and right translation by G. For f∈E define
285 i(f) by
286 
287 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
288 i(f)(np)=Lambda(n)f(p),       n∈N, p∈P,
289 \end{Verbatim}
290 
291 and zero off NP. This is well-defined because N∩P=J and
292 Lambda\textbar J=theta. The set NP is a finite union of N-cosets, by
293 {[}P:J{]}=2, so the support is compact modulo N. The map i is
294 P-equivariant and injective.
295 
296 Its image is exactly the two-dimensional P-stable subspace of Pi
297 consisting of inducing functions supported on NP. Thus the enlarged
298 carrier is already inside the given Pi. No new candidate is chosen, and
299 all noncompact values of Lambda remain fixed.
300 
301 Since Pi is irreducible and i(E) is nonzero, its G-span is Pi. More
302 precisely, there is the canonical G-equivariant quotient
303 
304 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
305 Q_E:c-Ind_P^G E→Pi,
306 Q_E(F)=sum_(Px∈P\G) Pi(x^−1)i(F(x)).                 (7)
307 \end{Verbatim}
308 
309 The sum is finite and independent of representatives. If x is replaced
310 by px, then F(px)=E(p)F(x) and P-equivariance of i cancels the
311 additional Pi(p\^{}−1). Formula (7) is nonzero and hence onto. The next
312 section identifies its kernel exactly.
313 
314 \hypertarget{rank2-exact-recovery-by-the-noncompact-hecke-eigenvalue}{%
315 \subsection{6. Exact recovery by the noncompact Hecke
316 eigenvalue}\label{rank2-exact-recovery-by-the-noncompact-hecke-eigenvalue}}
317 
318 Let X=c-Ind\_J\^{}G theta, in the same left-covariant/right-translation
319 convention. Induction in stages gives an explicit isomorphism
320 
321 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
322 c-Ind_P^G E → X,
323 F ↦ [x↦F(x)(1)].                                     (8)
324 \end{Verbatim}
325 
326 The inverse sends f∈X to the E-valued function F(x)(p)=f(px). Both
327 covariance statements follow directly from the definitions, and the
328 finite index {[}P:J{]} ensures the support conditions agree.
329 
330 Fix a uniformizer varpi=varpi\_K. Because K× normalizes theta, the
331 operator
332 
333 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
334 (Tf)(x)=f(varpi x)                                   (9)
335 \end{Verbatim}
336 
337 is a well-defined invertible G-equivariant operator on X. Put
338 s=Lambda(varpi), which is nonzero. Since k\_K=F2 and J contains
339 U\_K\^{}1,
340 
341 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
342 N/J≅K×/U_K^1≅Z,
343 \end{Verbatim}
344 
345 with generator varpi. Define
346 
347 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
348 Q:X→Pi,
349 (Qf)(x)=sum_(ell∈Z) s^(−ell) f(varpi^ell x).          (10)
350 \end{Verbatim}
351 
352 For each x the sum is finite, since f has compact support modulo the
353 open subgroup J. Equation (10) satisfies
354 
355 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
356 (Qf)(varpi x)=s(Qf)(x),
357 (Qf)(jx)=theta(j)(Qf)(x),       j∈J,
358 \end{Verbatim}
359 
360 and therefore the required N-covariance. Its support is compact modulo
361 N. It commutes with G-right translation and satisfies
362 
363 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
364 QT=sQ.                                               (11)
365 \end{Verbatim}
366 
367 It is onto: a function in Pi has finite support modulo N, and on each
368 occupied N-coset its value can be lifted to a function supported on one
369 J-coset. Extend those finitely many choices using theta.
370 
371 To compute the kernel, decompose X as the algebraic direct sum of its
372 subspaces supported on the left N-cosets in G. Each such summand is the
373 space of finitely supported sequences on N/J≅Z, with the fixed theta
374 covariance. The operator T is a bilateral shift. On that summand, (10)
375 is evaluation of its Laurent polynomial at the nonzero scalar s, up to
376 the harmless convention of replacing the Laurent variable by its
377 inverse. The kernel is therefore the principal ideal generated by T−s.
378 The elementary identity
379 
380 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
381 ker(C[z,z^−1]→C, z↦s)=(z−s)
382 \end{Verbatim}
383 
384 proves, summand by summand, that
385 
386 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
387 ker Q=(T−s)X.
388 \end{Verbatim}
389 
390 All vectors occupy finitely many such summands, so no completion or
391 closure is needed. Thus
392 
393 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
394 Pi ≅ X/(T−s)X ≅ (c-Ind_P^G E)/(T−s)(c-Ind_P^G E),    (12)
395 \end{Verbatim}
396 
397 where T is transported through (8).
398 
399 Under (8), formula (10) agrees with (7): on the standard copy of E
400 supported on P, it gives exactly i, and G-equivariance then gives
401 equality everywhere. Thus (12) identifies the kernel of the canonical
402 recovery map, not merely the kernel of an unrelated surjection.
403 
404 The same description can be phrased as evaluation of the type Hecke
405 algebra at the character determined by Lambda. Its intertwining support
406 is N, each supported intertwining space is one-dimensional, and the
407 retained extension Lambda normalizes the Laurent generator. The explicit
408 formulas above avoid any ambiguity about convolution orientation.
409 
410 The scalar s is essential. E alone, being compact data, does not
411 determine it. Changing s can change the unramified normalization of the
412 candidate. The construction retains the original Lambda and s, so it
413 recovers the exact original Pi rather than an unspecified member of its
414 inertial class.
415 
416 \hypertarget{rank2-full-scalar-pushout-attachment-and-remaining-marking-obligations}{%
417 \subsection{7. Full scalar-pushout attachment and remaining marking
418 obligations}\label{rank2-full-scalar-pushout-attachment-and-remaining-marking-obligations}}
419 
420 Form the C× pushout of the actual pair (H\_new,P,theta\_new). It is a
421 Heisenberg extension of the actual symplectic plane P/H\_new. Any
422 supplied source plane with the same chosen alternating form has a
423 center-preserving pushout isomorphism into it. Every source
424 full-character value is then preserved in the center, as in the retained
425 full-character attachment theorem. Stone--von Neumann identifies its
426 source carrier with the actual two-dimensional E.
427 
428 Together with the embedding i and exact quotient (12), this provides an
429 actual arithmetic carrier and recovery inside the same fixed candidate.
430 It closes the carrier-capacity failure of the old rank-two eta by
431 changing the compact arithmetic pair, not by increasing the old eta or
432 discarding its dimension calculation.
433 
434 At p=2, a C× pushout does not automatically identify a separately marked
435 native mu\_2 extension or quadratic refinement. The source finite
436 central extension can be retained through its chosen embedding, while a
437 required comparison with the target's preselected square function
438 remains an additional question. No arbitrary native Arf invariant is
439 asserted here.
440 
441 Nontrivial stabilizer operators, transport graphs, and original H2's
442 requirement that a target be a particular BK type-side quotient are
443 likewise separate. In particular, the theorem proves that the new pair
444 is an actual arithmetic finite Heisenberg pair, but does not rename it
445 as the unchanged maximal-simple-type pair. No new hypothesis is silently
446 added to original Theorem~16.2.
447 
448 \hypertarget{rank2-scope-smallest-boundary-and-sources}{%
449 \subsection{8. Scope, smallest boundary, and
450 sources}\label{rank2-scope-smallest-boundary-and-sources}}
451 
452 The inequality q\textgreater=3 matters. If q=1, then m=0 and the
453 displayed enlargement is not the group used above. Over residue field F2
454 the complete Iwahori unit group equals U\_A\^{}1, and a naive level-zero
455 replacement does not give the asserted index-two enlargement. No q=1
456 claim is made. The source levels q=3h with positive odd h satisfy the
457 required q\textgreater=3.
458 
459 The proof uses the actual one-dimensional Lambda because the maximal
460 parameter field has full degree and residue field F2. It does not apply
461 the same scalar argument automatically to a higher-rank inducing
462 representation.
463 
464 Primary BK passages: 3.1.7--8 (odd minimal H/J equality); 3.2.1--2 (deep
465 restriction and K× normalization); 3.3.2 and 5.1.1/5.1.8 (intertwining);
466 and 6.2.1--5 (the fixed maximal-type extension and compact induction).
467 The subgroup, matrix, Mackey-support, and Laurent-evaluation
468 calculations are proved explicitly above. No external correspondence
469 theorem is used in this enlargement.
470 
471 Retained primary text:
472 
473 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
474 closure_recovery/extracted/Paper_XI_Continuation158_General_Sectors_and_Arit
475 hmetic_Constraints/scratch158/information158/input_cache/BK_source.txt
476 SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8
477 \end{Verbatim}
478 
479 The previously reviewed negative statement concerned eta of the original
480 BK \(J^1/H^1\). That quotient remains zero. The new positive
481 statement concerns a different, explicitly constructed arithmetic pair
482 inside the same ambient representation, with exact recovery that preserves the noncompact normalization.

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