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1 \clearpage
2 \section{Native-rank full characters and tame central completion}\label{app:odd_character}
3 
4 Source record: Paper\_XI\_Base\_Realization165. 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 Section~6 of the tower note.
100 
101 \hypertarget{odd_character-a-finite-character-extension-fact}{%
102 \subsection{3. A finite character extension fact}\label{odd_character-a-finite-character-extension-fact}}
103 
104 Let K/F be a finite totally ramified local extension. Suppose chi is a
105 smooth character of U\_K\^{}a, and chi is trivial on U\_K\^{}a intersect
106 U\_F\^{}1. Then chi extends to a smooth character of U\_K\^{}1 trivial
107 on U\_F\^{}1.
108 
109 Indeed, on U\_K\^{}a U\_F\^{}1 define chi\_0(ku)=chi(k). Agreement on
110 the intersection makes this a character. Choose M\textgreater=a such
111 that chi is trivial on U\_K\^{}M. The character chi\_0 descends to a
112 subgroup of the finite abelian group \(U_K^1/U_K^M\). It
113 extends to that whole group by successively adjoining generators and
114 selecting complex roots. Pulling it back proves the assertion. This also
115 describes an actual finite procedure for making the extension choices.
116 
117 This fact will be used only when the parameter field is maximal in the
118 current native matrix rank. Direct use in a nonmaximal centralizer would
119 require checking its determinant image and would not follow from the
120 scalar intersection alone.
121 
122 \hypertarget{odd_character-base-character-at-native-rank-p}{%
123 \subsection{4. Base character at native rank
124 p}\label{odd_character-base-character-at-native-rank-p}}
125 
126 Let E=K\_1, and let n\_1=q\_d be the native level of alpha. Since n\_1
127 is even, set a=n\_1/2+1. On U\_E\^{}a the forced character of field units is
128 
129 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
130 phi(v)=psi_F(Tr_(E/F)(alpha(v-1))).
131 \end{Verbatim}
132 
133 It is trivial on U\_E\^{}a intersect U\_F\^{}1 because
134 Tr\_(E/F)(alpha)=0. Section~3 gives a full character chi\_E of U\_E\^{}1
135 extending phi and trivial on U\_F\^{}1. No normalization of the additive character is needed for this implication: the trace is exactly zero.
136 
137 For the native minimal stratum, \(H^1=U_E^1\)
138 U\_(A\_1)\^{}(a). Define theta\_1 by chi\_E on U\_E\^{}1 and by
139 psi\_alpha on U\_(A\_1)\^{}a. They agree on the intersection, so BK 3.2.1
140 supplies the actual full simple character. It satisfies
141 theta\_1\textbar U\_F\^{}1=1.
142 
143 \hypertarget{odd_character-transfer-the-full-field-character-before-adding-the-next-layer}{%
144 \subsection{5. Transfer the full field character before adding the next
145 layer}\label{odd_character-transfer-the-full-field-character-before-adding-the-next-layer}}
146 
147 Suppose theta\_j has been constructed on the native rank-p\^{}j stratum
148 of b\_j, with theta\_j\textbar U\_F\^{}1=1. Its restriction to
149 U\_(K\_j)\^{}1 is the full beta-determinant character chi\_j, since its
150 native centralizer is K\_j itself.
151 
152 Use the standard simple-character transfer to the multiplicity-p
153 realization gamma\_j=b\_j tensor I\_p in rank p\^{}(j+1). Denote the
154 transferred full character by Theta\_j. The exact primary result is
155 BK 3.6.13(ii), with existence in Theorem~3.6.14: transfer preserves the
156 full beta-determinant character on U\_(K\_j)\^{}1. Consequently its
157 restriction to the centralizer order is chi\_j composed with
158 det\_(K\_j).
159 
160 For every literal scalar u in U\_F\^{}1 this gives
161 
162 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
163 Theta_j(u I)=chi_j(u^p)=theta_j(u^p)=1.            (2)
164 \end{Verbatim}
165 
166 The determinant exponent p is essential. We do not replace (2) by an
167 unsupported pullback along a scalar matrix embedding. This transfer also
168 deals with nonminimal b\_j: beta-determinant preservation is the general
169 theorem, not just the minimal-character formula.
170 
171 \hypertarget{odd_character-add-the-actual-correction-and-extend-at-maximal-rank}{%
172 \subsection{6. Add the actual correction and extend at maximal
173 rank}\label{odd_character-add-the-actual-correction-and-extend-at-maximal-rank}}
174 
175 Let r=r\_j=q\_(d-j), m=floor(r/2)=r/2, beta=b\_(j+1), gamma=gamma\_j,
176 and delta=beta-gamma. The actual defining sequence gives
177 
178 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
179 D=H^(m+1)(beta)=H^(m+1)(gamma).
180 \end{Verbatim}
181 
182 By BK 3.3.18, the character
183 
184 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
185 vartheta=Theta_j restricted to D times psi_delta
186 \end{Verbatim}
187 
188 is an actual simple character at level m for beta. For every scalar u in
189 D intersect U\_F\^{}1, equation (2) and the exact trace identity give
190 
191 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
192 vartheta(u)=Theta_j(u)
193                 psi_F((u-1)tr_F(delta))=1.        (3)
194 \end{Verbatim}
195 
196 The new parameter field K=K\_(j+1) is maximal and totally ramified, and
197 the native chain has K-period one. Thus
198 
199 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
200 U_K^1 intersect D=U_K^(m+1),
201 U_F^1 intersect D=U_F^ceil((m+1)/p^(j+1)).
202 \end{Verbatim}
203 
204 In particular (3) supplies the exact compatibility condition for
205 extending vartheta\textbar U\_K\^{}(m+1) while imposing triviality on
206 all U\_F\^{}1. Apply Section~3 to obtain such a character chi\_K of
207 U\_K\^{}1.
208 
209 BK 3.2.5, including its explicit extension formula, now constructs
210 
211 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
212 theta_(j+1)(kd)=chi_K(k) vartheta(d),
213 k in U_K^1, d in D.
214 \end{Verbatim}
215 
216 The proposition supplies normalization of vartheta by the field units,
217 well-definedness, and the full simple-character condition. This
218 theta\_(j+1) has the claimed trivial wild center. Induction proves (1)
219 at the actual final rank p\^{}d.
220 
221 The proof deliberately performs every arbitrary character extension at
222 maximal native rank. Extending directly along amplified intermediate
223 centralizers M\_(p\^{}(d-j))(K\_j) would introduce determinant-image
224 compatibility conditions. Merely knowing that their ambient matrix trace
225 is zero would not discharge those conditions. Transfer first, followed
226 by maximal-rank extension, removes that ambiguity rigorously.
227 
228 \hypertarget{odd_character-complete-this-same-type-with-the-prescribed-tame-center}{%
229 \subsection{7. Complete this same type with the prescribed tame
230 center}\label{odd_character-complete-this-same-type-with-the-prescribed-tame-center}}
231 
232 Keep the actual canonical beta extension kappa of the resulting theta.
233 The final K/F is totally ramified of degree N, with residue rank one.
234 The compact type group satisfies
235 
236 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
237 J/J^1 = k_K^times = F_p^times.
238 \end{Verbatim}
239 
240 On U\_F\^{}1 the beta extension acts by theta, hence trivially. For a
241 prescribed omega\_F trivial on U\_F\^{}1, choose the residue character
242 
243 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
244 sigma=omega_F|F_p^times divided by omega_kappa|F_p^times.
245 \end{Verbatim}
246 
247 Its inflation gives lambda=kappa tensor sigma with the right center on
248 O\_F\^{}times. Complete lambda to an actual extension Lambda\_0 on the
249 inducing group K\^{}times J using the retained maximal-type completion
250 theorem. If its center is omega\_0, choose s in C\^{}times with
251 
252 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
253 s^N=omega_F(t)/omega_0(t).
254 \end{Verbatim}
255 
256 Multiply Lambda\_0 by the character of K\^{}times J trivial on J and
257 taking a K-uniformizer to s. The quotient (K\^{}times J)/J is Z, and t has image N. This proves exact agreement of the resulting central
258 character with omega\_F on all F\^{}times. Canonical kappa is unchanged.
259 
260 The resulting pi=c-Ind Lambda is irreducible, admissible and
261 supercuspidal. All these choices are made before H5. If a residue
262 character or noncompact extension has been immutably supplied, it must already satisfy this compatibility. An incompatible datum is not silently replaced.
263 
264 By the exact same parameter and defining-sequence calculations,
265 
266 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
267 rank(pi)=N=p^d,
268 torsion(pi)=1,
269 Swan(pi)=p^(d-1)(p+1)h_d,
270 a(pi^vee x pi)=p^(2d)-1
271                  + sum_i (p^(2i)-p^(2i-2))h_i.
272 \end{Verbatim}
273 
274 No determinant twist changes the depth. These are the proved arithmetic
275 tower profile formulas, not a proof of H5.
276 
277 \hypertarget{odd_character-full-packet-attachment-in-arbitrary-cubic-length}{%
278 \subsection{8. Full packet attachment in arbitrary cubic
279 length}\label{odd_character-full-packet-attachment-in-arbitrary-cubic-length}}
280 
281 For p=3, retain d original supplied full characters theta\_i and their
282 actual character groups H\_i\^{}1 normal in J\_i\^{}1, with whole
283 symplectic quotients V\_i that are the d labeled planes. Assume the
284 declared stabilizer is C=\{1\} and the diagram is one vertex. The source
285 balanced scalar extension has quotient V\_src of dimension 2d and
286 Heisenberg carrier dimension 3\^{}d, the original rank.
287 
288 The actual final host has total quotient dimension N-1. Its leading
289 nondegenerate symplectic layer has dimension
290 
291 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
292 (p-1)p^(d-1)=2*3^(d-1) >= 2d.
293 \end{Verbatim}
294 
295 This follows either from the explicit defining-sequence quotient
296 calculation or the independently proved leading-layer lemma. Choose a
297 labeled symplectic embedding of the supplied V\_src in that layer.
298 Write W=V\_src perpendicular-sum V\_perp. Choose a maximal isotropic
299 subspace L in V\_perp. Its associated eigenspace in the actual
300 Heisenberg carrier has dimension 3\^{}d, and L-perp/L is V\_src.
301 
302 Form the actual reduced scalar extension using the full host character
303 and its extension to the inverse image of L. Form each original full
304 pushout
305 
306 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
307 P_i=(C^times x J_i^1)/
308            {(theta_i(h)^(-1),h): h in H_i^1}.
309 \end{Verbatim}
310 
311 The center-fixing symplectic-extension isomorphism, constructed by
312 matched Darboux lifts and complex roots, maps their balanced product to
313 that actual reduced scalar extension. In particular each original h in
314 H\_i\^{}1 maps to the scalar theta\_i(h). Every prescribed shallow and
315 deep source character value is retained. Source character attachment
316 does not claim that the original field units are the same literal scalar
317 matrices in the new host.
318 
319 Finite Stone--von Neumann supplies the full carrier intertwiner. The
320 retained sector induction and recovery idempotent then recover this same
321 Lambda and this same pi from the source-sized sector. The canonical beta
322 extension, residue factor, central character, and noncompact scalar are
323 those actually selected in Section~7. No source-independent alternate
324 candidate is substituted in recovery.
325 
326 For the retained independent tensor family with d\textgreater=2, each
327 factor's wild determinant has order dividing p, because the
328 abelianization of its extraspecial order-p\^{}3 image is elementary
329 abelian. The tensor determinant raises each factor determinant to
330 p\^{}(d-1), so the full source determinant is trivial on wild inertia.
331 Thus the source's prescribed central quasicharacter lies in the class
332 covered by Section~7. At d=1, matching this central conclusion requires
333 the source determinant itself to be trivial on wild inertia. It is not inferred merely by taking a tensor power of length one.
334 
335 For p\textgreater3, the same arithmetic host and character theorem hold,
336 and the same attachment works for any actually supplied valid packet
337 whose whole quotient and markings meet the stated symplectic contract. A
338 proposed source plane is not automatically a whole rank-p type quotient:
339 that latter quotient has dimension p-1. No admission of all such plane
340 sources into original H2 is inferred.
341 
342 \hypertarget{odd_character-boundary-and-primary-source-check}{%
343 \subsection{9. Boundary and primary-source
344 check}\label{odd_character-boundary-and-primary-source-check}}
345 
346 This provides the previously missing full-character and actual
347 wild-center construction for the explicit odd-prime arithmetic tower in
348 every prime-power rank. The proof uses no classical
349 automorphic-induction/descent candidate. It retains the supplied finite
350 characters through the declared scalar-pushout attachments.
351 
352 It does not preserve old summand matrix conjugacy classes, produce a
353 prescribed nontrivial stabilizer action, preserve arbitrary numerical
354 depth labels, handle arbitrary transport graphs, settle the dyadic case,
355 or prove original same-candidate H5. It therefore does not close
356 unrestricted claim~66.
357 
358 The retained primary BK text, SHA256
359 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8},
360 was directly checked at:
361 
362 \begin{itemize}
363 \tightlist
364 \item
365   Definition~3.2.1 and Proposition 3.2.5, printed pp. 98--100: minimal
366   character condition and explicit full-character extension.
367 \item
368   Corollary 3.3.18, printed pp. 112--113: the actual additive correction takes
369   the approximation character to the next simple character.
370 \item
371   Condition 3.6.13(ii) and Theorem~3.6.14, printed pp. 136--137: transfer
372   preserves the full beta-determinant character, including at m0=0.
373 \end{itemize}
374 
375 The actual field and defining-sequence assertions are the separately
376 proved and independently reviewed trace-zero tower inputs, with BK 2.2.7
377 and the retained simple-stratum theorems. No newly downloaded source or
378 new novelty claim is involved.

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