Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Dyadic full characters and the odd critical endpoint}\label{app:dyadic_character}
3 
4 Checkpoint: Paper\_XI\_Dyadic167. Parent: final
5 Paper\_XI\_Attachment166. Author scope: this file only; no Block166 file
6 is modified.
7 
8 Evidence: ordinary mathematical proof using the actual Artin--Schreier
9 matrix tower and the retained Bushnell--Kutzko character and transfer
10 theorems. No Lean verification is claimed. This note proves no dyadic
11 quotient-capacity, quadratic-phase, packet-attachment, or H5 result.
12 
13 \hypertarget{dyadic_character-exact-theorem}{%
14 \subsection{1. Exact theorem}\label{dyadic_character-exact-theorem}}
15 
16 Let F=F\_2((t)), let psi\_F be an additive character of conductor t O\_F
17 (trivial there and nontrivial on O\_F), and let
18 
19 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
20 0<h_1<...<h_d,       each h_i odd,
21 q_i=3h_i,           N=2^d.
22 \end{Verbatim}
23 
24 Use the dyadic Artin--Schreier recursion specified in Section2. It gives
25 the actual maximal simple parameter beta\_d with its native arithmetic
26 defining sequence. There is an actual full simple character theta of its
27 maximal simple stratum such that
28 
29 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
30 theta restricted to U_F^1 is trivial.                         (1)
31 \end{Verbatim}
32 
33 The construction retains the actual canonical beta extension and can be
34 completed with any prescribed central quasicharacter omega\_F trivial on
35 U\_F\^{}1. Because \(k_F^\times=\FF_2^\times\) is trivial,
36 such an omega\_F is unramified. Thus this covers every tamely ramified
37 central quasicharacter in this particular residue-field setting.
38 
39 The corresponding compact induction is an actual irreducible admissible
40 supercuspidal of rank N, by the retained maximal-simple-type theorem. No
41 claim is made here about its Heisenberg quotient size or about
42 realization of a supplied dyadic finite packet.
43 
44 The trace-zero argument from166 does not apply verbatim: at p=2 the
45 primitive Artin--Schreier parameter has trace one. The correct invariant
46 is \textbf{integral trace}, not necessarily zero trace. Since psi\_F
47 kills t O\_F, integral correction trace is sufficient on scalar
48 principal units. The recursion below preserves the stronger exact value
49 one.
50 
51 \hypertarget{dyadic_character-actual-dyadic-recursion-and-exact-native-traces}{%
52 \subsection{2. Actual dyadic recursion and exact native
53 traces}\label{dyadic_character-actual-dyadic-recursion-and-exact-native-traces}}
54 
55 At a recursive step let L be a characteristic-two local field with
56 residue field F\_2, uniformizer t\_L, and a positive odd q. Define
57 
58 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
59 E=L(alpha),       alpha^2-alpha=t_L^(-q).
60 \end{Verbatim}
61 
62 The right side is not an Artin--Schreier coboundary: a negative
63 valuation of x\^{}2-x is even, whereas -q is odd. The degree-two
64 polynomial is therefore irreducible and separable. Its valuation
65 equation gives total ramification and v\_E(alpha)=-q. Put
66 
67 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
68 a=(q+1)/2,       z=t_L^a alpha.
69 \end{Verbatim}
70 
71 Then z is an actual uniformizer and its Eisenstein polynomial is
72 
73 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
74 f(Z)=Z^2+t_L^a Z+t_L.                                      (2)
75 \end{Verbatim}
76 
77 Its two conjugate alpha values are alpha and alpha+1, so
78 
79 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
80 Tr_(E/L)(alpha)=1.                                         (3)
81 \end{Verbatim}
82 
83 Use the coefficient basis 1,z, the trace corestriction pr, and the same
84 normalized corestriction convention as166:
85 
86 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
87 kappa=-f'(z)/z = t_L^a/z = alpha^(-1),
88 s=kappa pr.
89 \end{Verbatim}
90 
91 Minus and plus agree in characteristic two. If P0 extracts the constant
92 coefficient on every E-coordinate, then
93 
94 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
95 u_0=s(P0)=t_L/z^2 is a unit,
96 Lift_E(C)=u_0^(-1) C P0,
97 s(Lift_E(C))=C.
98 \end{Verbatim}
99 
100 The trace-dual calculation in166 gives, as an identity of L-matrix
101 traces for every matrix C over E,
102 
103 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
104 tr_L Lift_E(C)
105    =Tr_(E/L)(kappa^(-1) tr_E(C))
106    =Tr_(E/L)(alpha tr_E(C)).                               (4)
107 \end{Verbatim}
108 
109 The lift preserves the complete-chain filtration valuation, by expansion
110 in 1,z. Thus it is the actual filtered right inverse used in the field
111 constructor, not an arbitrary inverse on vector spaces.
112 
113 Define B(L,t\_L;h\_1,\ldots,h\_s) recursively. At s=1 use the L-matrix
114 of multiplication by alpha with q=q\_1. At s\textgreater=2 use (2) with
115 q=q\_s, m=2\^{}(s-1), and
116 
117 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
118 C=B(E,z;h_1,...,h_(s-1)),
119 B(L,t_L;h_1,...,h_s)=alpha I_m+Lift_E(C).                    (5)
120 \end{Verbatim}
121 
122 The base trace is one by (3). If tr\_E(C)=1, then (4) gives lift trace
123 one. Since m is even, the trace of alpha I\_m as an L-matrix is m
124 Tr\_(E/L)(alpha)=0. Consequently
125 
126 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
127 tr_L B(L,t_L;h_1,...,h_s)=1                                (6)
128 \end{Verbatim}
129 
130 for every native recursion, by induction. There is no division by the
131 rank and no confusion between a native trace and its amplified trace.
132 
133 The actual field proof from166 applies with this degree-two extension
134 and normalized lift. At a nontrivial step its hypotheses are
135 n=2\^{}(s-1)q\_s, R=2\^{}(s-2)q\_(s-1), s(Lift(C))=C, and
136 
137 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
138 n>2R,
139 -2R>-n-q_1.
140 \end{Verbatim}
141 
142 The first strict inequality follows from h\_s\textgreater h\_(s-1).
143 These are the conditions used in BK Remark2.2.7. The displayed beta
144 itself generates the totally ramified degree-2\^{}s field normalizing
145 the complete chain, and its final critical exponent is -q\_1. This
146 statement does not identify the final field with its auxiliary derived
147 field.
148 
149 For the final length d write the native shorter matrices as
150 
151 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
152 b_j=B(F,t;h_(d-j+1),...,h_d),       1<=j<=d.
153 \end{Verbatim}
154 
155 They are the compatible native truncations of (5), because the outer
156 extension, uniformizer, and lift depend on the same retained largest h
157 at each stage. For \(1\le j<d\), in rank 2\^{}(j+1), put
158 
159 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
160 gamma_j=b_j tensor I_2,
161 delta_j=b_(j+1)-gamma_j.
162 \end{Verbatim}
163 
164 The defining-sequence calculation of166 gives gamma\_j as a simple
165 approximation at the critical level r\_j=q\_(d-j) of b\_(j+1), and
166 delta\_j has grade -r\_j. Equation (6) gives the exact scalar identity
167 
168 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
169 tr_F(gamma_j)=0,
170 tr_F(delta_j)=1.                                          (7)
171 \end{Verbatim}
172 
173 Thus for every scalar u in U\_F\^{}1 on which its additive correction is
174 defined,
175 
176 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
177 psi_(delta_j)(u I)=psi_F((u-1)tr_F(delta_j))
178                  =psi_F(u-1)=1.                           (8)
179 \end{Verbatim}
180 
181 It is (8), not a false trace-zero assertion, that drives the full
182 character construction.
183 
184 \hypertarget{dyadic_character-the-odd-critical-level-rounding}{%
185 \subsection{3. The odd-critical-level
186 rounding}\label{dyadic_character-the-odd-critical-level-rounding}}
187 
188 The critical levels r=q\_i are odd. Write r=2s+1, and set
189 
190 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
191 m=floor(r/2)=s,        m+1=(r+1)/2.
192 \end{Verbatim}
193 
194 The relevant BK statements use the floor, with no assumption that r or
195 the residue characteristic is odd or even:
196 
197 \begin{itemize}
198 \tightlist
199 \item
200   BK3.1.9(ii) identifies H\^{}k(beta) and H\^{}k(gamma) for
201   k\textgreater=floor(r/2)+1.
202 \item
203   BK3.3.18 gives the correction bijection at
204   floor(r/2)\textless=m\textless=r-1.
205 \item
206   BK3.2.5 extends the resulting character at floor(r/2) to a full
207   character, its fibers being the indicated determinant characters.
208 \end{itemize}
209 
210 At our exact endpoint m=s, the correction character is legitimate. For
211 x,y in the radical power P\^{}(s+1),
212 
213 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
214 delta*x*y belongs to P^(2(s+1)-r)=P^1.
215 \end{Verbatim}
216 
217 The level-one additive matrix character is trivial on P\^{}1, so the
218 multiplicative cross term contributes one. Therefore psi\_delta is a
219 character on U\^{}(s+1), and hence on the recursive subgroup used below.
220 This checks the borderline odd-level rounding directly. Replacing m by
221 r/2 as if it were an integer would be incorrect.
222 
223 The same check applies at the native minimal base level q, which is odd:
224 its forced additive character begins at floor(q/2)+1=(q+1)/2, and
225 2*(q+1)/2-q=1.
226 
227 \hypertarget{dyadic_character-finite-character-extension-at-maximal-rank}{%
228 \subsection{4. Finite character extension at maximal
229 rank}\label{dyadic_character-finite-character-extension-at-maximal-rank}}
230 
231 We reuse the elementary extension procedure with its compatibility
232 condition explicit. If K/F is a finite totally ramified field, and chi
233 is a smooth character of U\_K\^{}a trivial on U\_K\^{}a intersect
234 U\_F\^{}1, define a character on U\_K\^{}a U\_F\^{}1 by
235 
236 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
237 chi_0(ku)=chi(k).
238 \end{Verbatim}
239 
240 It is well defined by the intersection condition. Choose M\textgreater=a
241 with chi\textbar U\_K\^{}M=1. Then chi\_0 is a character of a subgroup
242 of the finite abelian group \(U_K^1/U_K^M\). Extend it by
243 adjoining generators and choosing the required complex roots, and pull
244 it back. The resulting chi\_K extends chi and is trivial on U\_F\^{}1.
245 
246 All freely chosen character extensions below take place at maximal
247 native rank, where the centralizer is K itself. We do not apply this
248 scalar-intersection argument directly inside a nonmaximal matrix
249 centralizer, which would require a separate determinant-image check.
250 
251 \hypertarget{dyadic_character-base-full-character-with-trace-one}{%
252 \subsection{5. Base full character with trace
253 one}\label{dyadic_character-base-full-character-with-trace-one}}
254 
255 At native rank two, let E=F{[}alpha{]}, with q=q\_d and a=floor(q/2)+1.
256 The forced character on U\_E\^{}a is
257 
258 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
259 phi(v)=psi_F(Tr_(E/F)(alpha(v-1))).
260 \end{Verbatim}
261 
262 For scalar u in U\_E\^{}a intersect U\_F\^{}1, equation (3) gives
263 
264 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
265 phi(u)=psi_F((u-1)Tr_(E/F)(alpha))=psi_F(u-1)=1.
266 \end{Verbatim}
267 
268 Section4 constructs chi\_E on U\_E\^{}1 extending phi and trivial on
269 U\_F\^{}1. Define theta\_1 on \(H^1=U_E^1\) U\_(A\_1)\^{}a
270 by chi\_E on the first factor and psi\_alpha on the second. The
271 restrictions agree on the intersection. BK3.2.1 gives the actual full
272 simple character, with theta\_1\textbar U\_F\^{}1=1.
273 
274 This uses the true trace value one and only the stated conductor
275 condition on psi\_F. It does not require an artificial zero-trace
276 primitive parameter in a quadratic extension.
277 
278 \hypertarget{dyadic_character-native-rank-induction-via-full-beta-determinant-transfer}{%
279 \subsection{6. Native-rank induction via full beta-determinant
280 transfer}\label{dyadic_character-native-rank-induction-via-full-beta-determinant-transfer}}
281 
282 Assume theta\_j is a full simple character for native b\_j in rank
283 2\^{}j, with theta\_j\textbar U\_F\^{}1=1. Let K\_j=F{[}b\_j{]}. Since
284 K\_j is maximal, the character chi\_j=theta\_j\textbar U\_(K\_j)\^{}1 is
285 its full beta-determinant character.
286 
287 Transfer theta\_j to the multiplicity-two realization gamma\_j=b\_j
288 tensor I\_2 in rank 2\^{}(j+1), using BK3.6.14. Denote the full
289 transferred character by Theta\_j. BK3.6.13(ii) preserves the entire
290 beta-determinant character on U\_(K\_j)\^{}1 at level m0=0. Hence for
291 every literal scalar u in U\_F\^{}1,
292 
293 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
294 Theta_j(u I)=chi_j(det_(K_j)(u I_2))
295             =chi_j(u^2)=theta_j(u^2)=1.                    (9)
296 \end{Verbatim}
297 
298 This statement includes nonminimal b\_j. The determinant exponent two is
299 essential and has not been suppressed.
300 
301 Now set beta=b\_(j+1), gamma=gamma\_j, delta=delta\_j, r=r\_j,
302 m=floor(r/2), and
303 
304 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
305 D=H^(m+1)(beta)=H^(m+1)(gamma).
306 \end{Verbatim}
307 
308 The equality and endpoint m were verified in Section3. BK3.3.18
309 constructs the actual recursive simple character
310 
311 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
312 vartheta=(Theta_j restricted to D)*psi_delta.
313 \end{Verbatim}
314 
315 For u in D intersect U\_F\^{}1, equations (8) and (9) prove
316 
317 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
318 vartheta(u)=1.                                             (10)
319 \end{Verbatim}
320 
321 The new field K=F{[}beta{]} is maximal and totally ramified in native
322 rank 2\^{}(j+1). Its chain has K-period one, so
323 
324 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
325 D intersect U_K^1=U_K^(m+1),
326 D intersect U_F^1=U_F^ceil((m+1)/2^(j+1)).
327 \end{Verbatim}
328 
329 Thus (10) is exactly the compatibility required by Section4. Extend
330 vartheta\textbar U\_K\^{}(m+1) to chi\_K on U\_K\^{}1 trivial on
331 U\_F\^{}1. BK3.2.5 constructs the full simple character by
332 
333 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
334 theta_(j+1)(kd)=chi_K(k)*vartheta(d),
335 k in U_K^1, d in D.
336 \end{Verbatim}
337 
338 That proposition supplies normalization and well-definedness; its proof
339 is a character-extension argument and imposes no exclusion of residue
340 characteristic two. The new theta\_(j+1) is trivial on U\_F\^{}1.
341 Induction proves (1) in every native rank 2\^{}d.
342 
343 More generally the character argument needs only integral base field
344 trace and integral traces for the native corrections. The dyadic tower
345 establishes their exact value one, while the odd-prime tower of166
346 establishes zero.
347 
348 \hypertarget{dyadic_character-canonical-beta-extension-and-prescribed-tame-center}{%
349 \subsection{7. Canonical beta extension and prescribed tame
350 center}\label{dyadic_character-canonical-beta-extension-and-prescribed-tame-center}}
351 
352 Let K/F be the final totally ramified degree-N parameter field and let
353 eta be the Heisenberg representation for the constructed theta. The
354 compact type group is
355 
356 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
357 J=O_K^times J^1.
358 \end{Verbatim}
359 
360 Here \(\OO_K^\times=U_K^1\) because the residue field is
361 F\_2. Since U\_K\^{}1 is contained in J\^{}1, we have
362 
363 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
364 J=J^1,       J/J^1=F_2^times={1}.                           (11)
365 \end{Verbatim}
366 
367 Thus the actual canonical beta extension on J equals eta; there is no
368 residue-character choice and no change of that extension. It acts
369 trivially on the central U\_F\^{}1 by (1). Equation (11) does not
370 identify any externally prescribed dyadic quadratic phases.
371 
372 Let omega\_F be any smooth quasicharacter trivial on U\_F\^{}1. Since
373 \(\OO_F^\times=U_F^1\), it is determined by its value
374 omega\_F(t). The actual maximal type extends to K\^{}times J.
375 Concretely, (K\^{}times J)/J is infinite cyclic, generated by a
376 K-uniformizer. An intertwiner for its conjugation action on eta gives an
377 extension Lambda\_0. Denote its central character by omega\_0.
378 
379 Choose a complex scalar s with
380 
381 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
382 s^N=omega_F(t)/omega_0(t).
383 \end{Verbatim}
384 
385 Multiply Lambda\_0 by the character of K\^{}times J trivial on J and
386 taking the K-uniformizer to s. Because t has valuation N in K, this
387 changes its scalar by s\^{}N and gives the requested central
388 quasicharacter on all F\^{}times. The value at every element of J,
389 including the canonical beta extension, is unchanged. This does not
390 assume that a chosen K-uniformizer raised to N equals t.
391 
392 The retained maximal-simple-type theorem now gives the actual
393 irreducible admissible supercuspidal pi=c-Ind\_(K\^{}times
394 J)\^{}GL\_N(F) Lambda. The noncompact scalar is fixed before any H5
395 comparison. An incompatible immutably prescribed extension is not
396 overwritten; as in166, its value must be free or already compatible.
397 
398 \hypertarget{dyadic_character-what-this-closes-and-what-it-does-not}{%
399 \subsection{8. What this closes and what it does
400 not}\label{dyadic_character-what-this-closes-and-what-it-does-not}}
401 
402 This closes the full-character and actual tame-central-character
403 construction for the explicit dyadic arithmetic tower. The failure of
404 trace zero at p=2 is repaired by the correct integral trace invariant.
405 Both the odd-level floor and the multiplicity-two determinant transfer
406 are accounted for explicitly.
407 
408 This note does not compute \(J^1/H^1\) or prove that it
409 contains a particular dyadic source packet. It does not establish a
410 quadratic refinement, canonical phase matching, nontrivial stabilizer
411 action, general transport graph, source admission, or H5. In particular,
412 an actual supercuspidal host with the required central character is not
413 asserted to be a completed realization of the original dyadic packet
414 without those separate proofs.
415 
416 \hypertarget{dyadic_character-exact-source-checks-and-input-pins}{%
417 \subsection{9. Exact source checks and input
418 pins}\label{dyadic_character-exact-source-checks-and-input-pins}}
419 
420 The following final166 files were read and reused without editing:
421 
422 \begin{itemize}
423 \tightlist
424 \item
425   \path{attachment166/central/TRACE_ZERO_CHARACTER_INDUCTION166.md}
426   SHA256
427   \path{99cb6e80c419e0bf7325e2b2b3fbaba1391903448a3b6a0c4cd9f3944ed406a1}.
428 \item
429   \path{attachment166/tower/TRACE_ZERO_AS_TOWER166.md} SHA256
430   \path{49a18fb733f0209b3e38118f5132839a060961edd0bdc0211e1f7b441a4e5653}.
431 \item
432   \path{attachment166/tower/RECURSIVE_TOWER166.md} SHA256
433   \path{6f39a077df18eb03f9fab2b6e426529dbe14cf4f7e1cfce95ddb0fcd187d7b33}.
434 \item
435   \path{attachment166/audit/TOWER_CENTRAL_REVIEW166.md} SHA256
436   \path{4bc71dd050ddb321a253105c8a4ee7a86185af255532d5605caa7a9de383e26b}.
437 \end{itemize}
438 
439 Primary retained source: Bushnell--Kutzko, \emph{The admissible dual of
440 GL(N) via compact open subgroups}, text SHA256
441 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8}.
442 The current dyadic check reread definitions3.1.7--8 and
443 Proposition3.1.9(ii) (printed90--91), Definition3.2.1 and
444 Proposition3.2.5 (printed98--100), and Corollary3.3.18
445 (printed112--113). Their level thresholds are floor(r/2)+1 or
446 floor(r/2), as used above, and their statements do not exclude p=2. The
447 transfer's preservation of the full beta-determinant character is
448 BK3.6.13(ii), with existence Theorem3.6.14 (printed136--137), already
449 directly checked in166 and reread for this application.
450 
451 The character-extension and correction proofs use neither division by
452 two nor an odd-order averaging argument. The exact dyadic trace
453 calculation and the borderline correction-character check are given in
454 Sections2--3. No new publication lookup, external
455 arithmetic-induction/descent constructor, or inverse LLC is used.

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