Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Native-rank full characters and tame central completion}\label{app:odd_character}
3 
4 Parent checkpoint: Paper\_XI\_Base\_Realization165. New arithmetic
5 input: \path{attachment166/tower/TRACE_ZERO_AS_TOWER166.md},
6 together with its \path{RECURSIVE_TOWER166.md} defining-sequence
7 proof. Input hashes are recorded in
8 \path{TRACE_ZERO_CHARACTER_PINS166.json}.
9 
10 Evidence: ordinary proof using the explicitly constructed arithmetic
11 parameters and the named Bushnell--Kutzko character and transfer
12 theorems. No arithmetic-induction/descent constructor or inverse
13 correspondence is used. No Lean verification is claimed.
14 
15 \hypertarget{odd_character-statement}{%
16 \subsection{1. Statement}\label{odd_character-statement}}
17 
18 Let p be odd, F=F\_p((t)), and
19 
20 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
21 0<h_1<...<h_d,       p not dividing any h_i,
22 q_i=(p+1)h_i,        N=p^d.
23 \end{Verbatim}
24 
25 Let beta\_d be the actual trace-zero Artin--Schreier matrix constructor
26 in the named tower note. There exists an actual full simple character
27 theta for its maximal simple stratum such that
28 
29 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
30 theta restricted to U_F^1 is the trivial character.             (1)
31 \end{Verbatim}
32 
33 Every step constructing theta is specified below: a finite abelian
34 character extension, the standard simple-character transfer, or the
35 explicit additive correction from the displayed defining sequence. The
36 field, order, and parameter are the actual tower ones, not an
37 unspecified arithmetic realization.
38 
39 The actual canonical beta extension can then be completed to a maximal
40 extended type and supercuspidal pi with any prescribed smooth central
41 quasicharacter omega\_F trivial on U\_F\^{}1. Its rank, Swan conductor,
42 torsion number and self-pair conductor remain those proved for the same
43 tower. The residue and noncompact extension choices must be free or
44 already compatible with omega\_F.
45 
46 For p=3, this gives full scalar-pushout attachment and same-candidate
47 sector recovery for any number d of the supplied cubic source planes,
48 with declared stabilizer C=\{1\} and the one-vertex contract. All
49 original full source character values are retained in the attachment. A
50 general nontrivial stabilizer, transport diagram, or H5 conclusion is
51 not part of this statement.
52 
53 \hypertarget{odd_character-the-exact-arithmetic-inputs}{%
54 \subsection{2. The exact arithmetic
55 inputs}\label{odd_character-the-exact-arithmetic-inputs}}
56 
57 For 1\textless=j\textless=d, put
58 
59 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
60 b_j=B(F,t;h_(d-j+1),...,h_d) in M_(p^j)(F),
61 \end{Verbatim}
62 
63 where B is the trace-zero constructor. The tower proof gives:
64 
65 \begin{enumerate}
66 \def\labelenumi{\arabic{enumi}.}
67 \item
68   K\_j=F{[}b\_j{]} is a totally ramified maximal field of degree p\^{}j,
69   normalizing the native complete period-p\^{}j chain A\_j.
70 \item
71   For \(1\le j<d\), in rank p\^{}(j+1), gamma\_j=b\_j tensor I\_p is a simple
72   approximation to b\_(j+1), at the critical level
73   r\_j=q\_(d-j)\textgreater0 of b\_(j+1). They are equivalent as strata
74   at that level.
75 \item
76   For \(1\le j<d\), delta\_j=b\_(j+1)-gamma\_j has grade -r\_j and F-matrix trace zero.
77 \item
78   At native rank p, b\_1 is multiplication by alpha in the explicit
79   Artin--Schreier field
80 
81 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
82 E=F(alpha),       alpha^p-alpha=t^(-q_d).
83 \end{Verbatim}
84 
85   Its actual field trace is zero, since p is odd.
86 \end{enumerate}
87 
88 For completeness, the tower's exact trace identity is
89 
90 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
91 tr_L Lift_E(C)=Tr_(E/L)(kappa^(-1) tr_E(C)),
92 \end{Verbatim}
93 
94 where s=kappa*pr is its normalized trace corestriction and
95 Lift\_E(C)=s(P0)\^{}(-1) C P0. The recursive inner matrix has trace
96 zero, so this proves zero trace at every native rank. It is not merely
97 the automatic zero obtained by amplifying any matrix p times in
98 characteristic p.~The literal compatibility of the truncations is proved
99 in Section6 of the tower note.
100 
101 \hypertarget{odd_character-a-finite-character-extension-fact}{%
102 \subsection{3. A finite character-extension
103 fact}\label{odd_character-a-finite-character-extension-fact}}
104 
105 Let K/F be a finite totally ramified local extension. Suppose chi is a
106 smooth character of U\_K\^{}a, and chi is trivial on U\_K\^{}a intersect
107 U\_F\^{}1. Then chi extends to a smooth character of U\_K\^{}1 trivial
108 on U\_F\^{}1.
109 
110 Indeed, on U\_K\^{}a U\_F\^{}1 define chi\_0(ku)=chi(k). Agreement on
111 the intersection makes this a character. Choose M\textgreater=a such
112 that chi is trivial on U\_K\^{}M. The character chi\_0 descends to a
113 subgroup of the finite abelian group \(U_K^1/U_K^M\). It
114 extends to that whole group by successively adjoining generators and
115 selecting complex roots. Pulling it back proves the assertion. This also
116 describes an actual finite procedure for making the extension choices.
117 
118 This fact will be used only when the parameter field is maximal in the
119 current native matrix rank. Direct use in a nonmaximal centralizer would
120 require checking its determinant image and would not follow from the
121 scalar intersection alone.
122 
123 \hypertarget{odd_character-base-character-at-native-rank-p}{%
124 \subsection{4. Base character at native rank
125 p}\label{odd_character-base-character-at-native-rank-p}}
126 
127 Let E=K\_1, and let n\_1=q\_d be the native level of alpha. Since n\_1
128 is even, set a=n\_1/2+1. On U\_E\^{}a the forced field-unit character is
129 
130 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
131 phi(v)=psi_F(Tr_(E/F)(alpha(v-1))).
132 \end{Verbatim}
133 
134 It is trivial on U\_E\^{}a intersect U\_F\^{}1 because
135 Tr\_(E/F)(alpha)=0. Section3 gives a full character chi\_E of U\_E\^{}1
136 extending phi and trivial on U\_F\^{}1. No additive-character
137 normalization is needed for this implication: the trace is exactly zero.
138 
139 For the native minimal stratum, \(H^1=U_E^1\)
140 U\_(A\_1)\^{}(a). Define theta\_1 by chi\_E on U\_E\^{}1 and by
141 psi\_alpha on U\_(A\_1)\^{}a. They agree on the intersection, so BK3.2.1
142 supplies the actual full simple character. It satisfies
143 theta\_1\textbar U\_F\^{}1=1.
144 
145 \hypertarget{odd_character-transfer-the-full-field-character-before-adding-the-next-layer}{%
146 \subsection{5. Transfer the full field character before adding the next
147 layer}\label{odd_character-transfer-the-full-field-character-before-adding-the-next-layer}}
148 
149 Suppose theta\_j has been constructed on the native rank-p\^{}j stratum
150 of b\_j, with theta\_j\textbar U\_F\^{}1=1. Its restriction to
151 U\_(K\_j)\^{}1 is the full beta-determinant character chi\_j, since its
152 native centralizer is K\_j itself.
153 
154 Use the standard simple-character transfer to the multiplicity-p
155 realization gamma\_j=b\_j tensor I\_p in rank p\^{}(j+1). Denote the
156 transferred full character by Theta\_j. The exact primary result is
157 BK3.6.13(ii), with existence in Theorem3.6.14: transfer preserves the
158 full beta-determinant character on U\_(K\_j)\^{}1. Consequently its
159 restriction to the centralizer order is chi\_j composed with
160 det\_(K\_j).
161 
162 For every literal scalar u in U\_F\^{}1 this gives
163 
164 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
165 Theta_j(u I)=chi_j(u^p)=theta_j(u^p)=1.            (2)
166 \end{Verbatim}
167 
168 The determinant exponent p is essential. We do not replace (2) by an
169 unsupported pullback along a scalar matrix embedding. This transfer also
170 deals with nonminimal b\_j: beta-determinant preservation is the general
171 theorem, not just the minimal-character formula.
172 
173 \hypertarget{odd_character-add-the-actual-correction-and-extend-at-maximal-rank}{%
174 \subsection{6. Add the actual correction and extend at maximal
175 rank}\label{odd_character-add-the-actual-correction-and-extend-at-maximal-rank}}
176 
177 Let r=r\_j=q\_(d-j), m=floor(r/2)=r/2, beta=b\_(j+1), gamma=gamma\_j,
178 and delta=beta-gamma. The actual defining sequence gives
179 
180 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
181 D=H^(m+1)(beta)=H^(m+1)(gamma).
182 \end{Verbatim}
183 
184 By BK3.3.18, the character
185 
186 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
187 vartheta=Theta_j restricted to D times psi_delta
188 \end{Verbatim}
189 
190 is an actual simple character at level m for beta. For every scalar u in
191 D intersect U\_F\^{}1, equation (2) and the exact trace identity give
192 
193 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
194 vartheta(u)=Theta_j(u)
195                 psi_F((u-1)tr_F(delta))=1.        (3)
196 \end{Verbatim}
197 
198 The new parameter field K=K\_(j+1) is maximal and totally ramified, and
199 the native chain has K-period one. Thus
200 
201 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
202 U_K^1 intersect D=U_K^(m+1),
203 U_F^1 intersect D=U_F^ceil((m+1)/p^(j+1)).
204 \end{Verbatim}
205 
206 In particular (3) supplies the exact compatibility condition for
207 extending vartheta\textbar U\_K\^{}(m+1) while imposing triviality on
208 all U\_F\^{}1. Apply Section3 to obtain such a character chi\_K of
209 U\_K\^{}1.
210 
211 BK3.2.5, including its explicit extension formula, now constructs
212 
213 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
214 theta_(j+1)(kd)=chi_K(k) vartheta(d),
215 k in U_K^1, d in D.
216 \end{Verbatim}
217 
218 The proposition supplies normalization of vartheta by the field units,
219 well-definedness, and the full simple-character condition. This
220 theta\_(j+1) has the claimed trivial wild center. Induction proves (1)
221 at the actual final rank p\^{}d.
222 
223 The proof deliberately performs every arbitrary character extension at
224 maximal native rank. Extending directly along amplified intermediate
225 centralizers M\_(p\^{}(d-j))(K\_j) would introduce determinant-image
226 compatibility conditions. Merely knowing that their ambient matrix trace
227 is zero would not discharge those conditions. Transfer first, followed
228 by maximal-rank extension, removes that ambiguity rigorously.
229 
230 \hypertarget{odd_character-complete-this-same-type-with-the-prescribed-tame-center}{%
231 \subsection{7. Complete this same type with the prescribed tame
232 center}\label{odd_character-complete-this-same-type-with-the-prescribed-tame-center}}
233 
234 Keep the actual canonical beta extension kappa of the resulting theta.
235 The final K/F is totally ramified of degree N, with residue rank one.
236 The compact type group satisfies
237 
238 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
239 J/J^1 = k_K^times = F_p^times.
240 \end{Verbatim}
241 
242 On U\_F\^{}1 the beta extension acts by theta, hence trivially. For a
243 prescribed omega\_F trivial on U\_F\^{}1, choose the residue character
244 
245 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
246 sigma=omega_F|F_p^times divided by omega_kappa|F_p^times.
247 \end{Verbatim}
248 
249 Its inflation gives lambda=kappa tensor sigma with the right center on
250 O\_F\^{}times. Complete lambda to an actual extension Lambda\_0 on the
251 inducing group K\^{}times J using the retained maximal-type completion
252 theorem. If its center is omega\_0, choose s in C\^{}times with
253 
254 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
255 s^N=omega_F(t)/omega_0(t).
256 \end{Verbatim}
257 
258 Multiply Lambda\_0 by the character of K\^{}times J trivial on J and
259 taking a K-uniformizer to s. The quotient (K\^{}times J)/J is Z, and t
260 has image N; this proves exact agreement of the resulting central
261 character with omega\_F on all F\^{}times. Canonical kappa is unchanged.
262 
263 The resulting pi=c-Ind Lambda is irreducible, admissible and
264 supercuspidal. All these choices are made before H5. If a residue
265 character or noncompact extension has been immutably supplied, it must
266 already satisfy this compatibility; an incompatible datum is not
267 silently replaced.
268 
269 By the exact same parameter and defining-sequence calculations,
270 
271 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
272 rank(pi)=N=p^d,
273 torsion(pi)=1,
274 Swan(pi)=p^(d-1)(p+1)h_d,
275 a(pi^vee x pi)=p^(2d)-1
276                  + sum_i (p^(2i)-p^(2i-2))h_i.
277 \end{Verbatim}
278 
279 No determinant twist changes the depth. These are the proved arithmetic
280 tower profile formulas, not a proof of H5.
281 
282 \hypertarget{odd_character-full-packet-attachment-in-arbitrary-cubic-length}{%
283 \subsection{8. Full packet attachment in arbitrary cubic
284 length}\label{odd_character-full-packet-attachment-in-arbitrary-cubic-length}}
285 
286 For p=3, retain d original supplied full characters theta\_i and their
287 actual character groups H\_i\^{}1 normal in J\_i\^{}1, with whole
288 symplectic quotients V\_i that are the d labelled planes. Assume the
289 declared stabilizer is C=\{1\} and the diagram is one vertex. The source
290 balanced scalar extension has quotient V\_src of dimension 2d and
291 Heisenberg carrier dimension 3\^{}d, the original rank.
292 
293 The actual final host has total quotient dimension N-1. Its leading
294 nondegenerate symplectic layer has dimension
295 
296 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
297 (p-1)p^(d-1)=2*3^(d-1) >= 2d.
298 \end{Verbatim}
299 
300 This follows either from the explicit defining-sequence quotient
301 calculation or the independently proved leading-layer lemma. Choose a
302 labelled symplectic embedding of the supplied V\_src in that layer.
303 Write W=V\_src perpendicular-sum V\_perp. Choose a maximal isotropic
304 subspace L in V\_perp. Its associated eigenspace in the actual
305 Heisenberg carrier has dimension 3\^{}d, and L-perp/L is V\_src.
306 
307 Form the actual reduced scalar extension using the full host character
308 and its extension to the inverse image of L. Form each original full
309 pushout
310 
311 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
312 P_i=(C^times x J_i^1)/
313            {(theta_i(h)^(-1),h): h in H_i^1}.
314 \end{Verbatim}
315 
316 The center-fixing symplectic-extension isomorphism, constructed by
317 matched Darboux lifts and complex roots, maps their balanced product to
318 that actual reduced scalar extension. In particular each original h in
319 H\_i\^{}1 maps to the scalar theta\_i(h). Every prescribed shallow and
320 deep source character value is retained. Source character attachment
321 does not claim that the original field units are the same literal scalar
322 matrices in the new host.
323 
324 Finite Stone--von Neumann supplies the full carrier intertwiner. The
325 retained sector induction and recovery idempotent then recover this same
326 Lambda and this same pi from the source-sized sector. The canonical beta
327 extension, residue factor, central character, and noncompact scalar are
328 those actually selected in Section7. No source-independent alternate
329 candidate is substituted in recovery.
330 
331 For the retained independent tensor family with d\textgreater=2, each
332 factor's wild determinant has order dividing p, because the
333 abelianization of its extraspecial order-p\^{}3 image is elementary
334 abelian. The tensor determinant raises each factor determinant to
335 p\^{}(d-1), so the full source determinant is trivial on wild inertia.
336 Thus the source's prescribed central quasicharacter lies in the class
337 covered by Section7. At d=1, matching this central conclusion requires
338 the source determinant itself to be trivial on wild inertia; it is not
339 inferred merely by taking a tensor power of length one.
340 
341 For p\textgreater3, the same arithmetic host and character theorem hold,
342 and the same attachment works for any actually supplied valid packet
343 whose whole quotient and markings meet the stated symplectic contract. A
344 proposed source plane is not automatically a whole rank-p type quotient:
345 that latter quotient has dimension p-1. No admission of all such plane
346 sources into original H2 is inferred.
347 
348 \hypertarget{odd_character-boundary-and-primary-source-check}{%
349 \subsection{9. Boundary and primary-source
350 check}\label{odd_character-boundary-and-primary-source-check}}
351 
352 This provides the previously missing full-character and actual
353 wild-center construction for the explicit odd-prime arithmetic tower in
354 every prime-power rank. The proof uses no classical
355 automorphic-induction/descent candidate. It retains the supplied finite
356 characters through the declared scalar-pushout attachments.
357 
358 It does not preserve old summand matrix conjugacy classes, produce a
359 prescribed nontrivial stabilizer action, preserve arbitrary numerical
360 depth labels, handle arbitrary transport graphs, settle the dyadic case,
361 or prove original same-candidate H5. It therefore does not close
362 unrestricted claim66.
363 
364 The retained primary BK text, SHA256
365 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8},
366 was directly checked at:
367 
368 \begin{itemize}
369 \tightlist
370 \item
371   Definition3.2.1 and Proposition3.2.5, printed98--100: minimal
372   character condition and explicit full-character extension.
373 \item
374   Corollary3.3.18, printed112--113: the actual additive correction takes
375   the approximation character to the next simple character.
376 \item
377   Condition3.6.13(ii) and Theorem3.6.14, printed136--137: transfer
378   preserves the full beta-determinant character, including at m0=0.
379 \end{itemize}
380 
381 The actual field and defining-sequence assertions are the separately
382 proved and independently reviewed trace-zero tower inputs, with BK2.2.7
383 and the retained simple-stratum theorems. No newly downloaded source or
384 new novelty claim is involved.

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