Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Odd-prime synthesis and complete scope}\label{app:odd_synthesis}
3 
4 Date: 2026-10-05. Parent: \path{Paper_XI_Base_Realization165.zip}.
5 
6 This is a synthesis of the independently reviewed proofs in this
7 continuation. The result is an ordinary mathematical proof, using the
8 stated classical simple-type theorems. It is not a Lean certificate.
9 Universal claim 66 remains open. Original H5 remains the original
10 conditional premise about the same fixed candidate; no unconditional
11 pair-factor comparison is asserted here.
12 
13 \hypertarget{odd_synthesis-the-arithmetic-construction-now-proved}{%
14 \subsection{1. The arithmetic construction now
15 proved}\label{odd_synthesis-the-arithmetic-construction-now-proved}}
16 
17 Let \(p\) be odd, \(F=\mathbb F_p((t))\), and \[
18 0<h_1<\cdots<h_d,\qquad p\nmid h_i,\qquad
19 q_i=(p+1)h_i,\qquad N=p^d.
20 \] For every such tuple there is an explicit matrix \(\beta\in M_N(F)\)
21 whose field \(K=F[\beta]\) is totally ramified of degree \(N\),
22 normalizes the complete lattice chain, and gives a maximal simple
23 stratum. It has a literal defining sequence of degrees
24 \(p,p^2,\ldots,p^d\), with leading level \(n_1\) and successive critical
25 levels \(n_2,\ldots,n_d\) given by \[
26 n_j=p^{d-j}q_{d-j+1}\quad(1\le j\le d).
27 \] Every native parameter in the construction, every amplified
28 approximation, and every consecutive correction has matrix trace zero
29 over its base field.
30 
31 For this actual stratum there exists a full simple character \(\theta\)
32 with \(\theta|_{U_F^1}=1\). Retaining its canonical beta extension, the
33 residue character and noncompact extension can be selected so that the
34 resulting irreducible admissible supercuspidal \(\pi\) has any
35 prescribed smooth central quasicharacter \(\omega\) trivial on
36 \(U_F^1\). These completion choices must be free or already compatible;
37 incompatible immutably supplied choices are not replaced.
38 
39 The same candidate has \[
40 \begin{aligned}
41 \operatorname{rank}(\pi)&=N, &t(\pi)&=1,\\
42 \operatorname{Sw}(\pi)&=p^{d-1}(p+1)h_d, &
43 \dim_{\mathbb F_p}(J^1/H^1)&=N-1,\\
44 \dim\eta&=p^{(N-1)/2},\\
45 a(\pi^\vee\times\pi)
46  &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.
47 \end{aligned}
48 \] These numerical identities do not identify a Weil representation, an
49 endo-class, or the pair epsilon factors required by H5.
50 
51 \hypertarget{odd_synthesis-construction-and-proof}{%
52 \subsubsection{Construction and proof}\label{odd_synthesis-construction-and-proof}}
53 
54 At a recursive base \((L,t_L)\), introduce \[
55 E=L(\alpha),\qquad \alpha^p-\alpha=t_L^{-q_d}.
56 \] This is a separable totally ramified extension of degree \(p\), with
57 \(v_E(\alpha)=-q_d\) and \(\operatorname{Tr}_{E/L}\alpha=0\). Choose
58 \(1\le b<p\) with \(q_db\equiv-1\pmod p\), set \(a=(q_db+1)/p\), and put
59 \(z=t_L^a\alpha^b\). Then \(z\) is an explicit uniformizer. Write its
60 Eisenstein polynomial as \(f(X)=X^p+\cdots+a_0\).
61 
62 Let \(P_0\) extract the constant coefficient in the basis
63 \(1,z,\ldots,z^{p-1}\). The normalized corestriction
64 \(s=\kappa\operatorname{pr}\), where \(\kappa=-f'(z)/z^{p-1}\),
65 satisfies \[
66 u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times,
67 \qquad
68 \operatorname{Lift}_E(C)=u_0^{-1}CP_0,
69 \qquad s(\operatorname{Lift}_E(C))=C.
70 \] The lift preserves the exact filtration valuation. Its crucial
71 additional identity is \[
72 \operatorname{tr}_L(\operatorname{Lift}_E(C))
73  =\operatorname{Tr}_{E/L}
74    \bigl(\kappa^{-1}\operatorname{tr}_E(C)\bigr).
75 \] Define \[
76 B(L,t_L;h_1,\ldots,h_d)
77  =\alpha I_{p^{d-1}}+
78    \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
79 \] with the degree-\(p\) multiplication matrix as the base case.
80 Induction proves trace zero at every native rank. This is stronger than
81 the automatic zero trace obtained by multiplying any matrix multiplicity
82 by \(p\) in characteristic \(p\).
83 
84 For the inductive field step \(d\ge2\), put \(n=p^{d-1}q_d\) and
85 \(R=p^{d-2}q_{d-1}\). The derived matrix is the recursively constructed
86 maximal-field parameter, with \(k_0=-q_1\). The inequalities \(n>pR>2R\)
87 imply \(-2R>-n-q_1\). Bushnell--Kutzko Remark 2.2.7 therefore applies to
88 the actual displayed matrix. It supplies its maximal field,
89 normalization, and critical exponent. Valuation preservation and literal
90 truncation of the recursion supply the whole defining sequence. The full
91 details, quotient count, and conductor dependencies are in
92 \path{tower/TRACE_ZERO_AS_TOWER166.md} and
93 \path{tower/RECURSIVE_TOWER166.md}.
94 
95 The full-character argument uses a specific order of operations. First
96 construct a full character at native rank \(p^j\), trivial on \(U_F^1\).
97 Transfer its full beta-determinant unit character to multiplicity \(p\),
98 using BK 3.6.13(ii) and 3.6.14. On a scalar \(u\in U_F^1\), this gives
99 \(\theta_j(u^p)=1\). Next add the actual trace-zero correction, using BK
100 3.3.18. It remains trivial on the scalar intersection of its domain.
101 Finally extend at the next \textbf{maximal native parameter field}: the
102 forced deep character and the trivial character of \(U_F^1\) agree on
103 their intersection, so they extend through a finite abelian unit
104 quotient. BK 3.2.5 constructs the full simple character.
105 
106 This procedure never substitutes an unjustified extension through a
107 nonmaximal centralizer. It proves full-character compatibility,
108 including shallow units, rather than merely cancellation of a leading
109 trace. The residue factor fixes \(\omega\) on \(\mathbb F_p^\times\); an
110 \(N\)-th-root choice for the noncompact generator fixes \(\omega(t)\).
111 Every choice is recorded before H5. See
112 \path{central/TRACE_ZERO_CHARACTER_INDUCTION166.md} for the
113 complete induction.
114 
115 \hypertarget{odd_synthesis-the-finite-packet-that-this-same-candidate-realizes}{%
116 \subsection{2. The finite packet that this same candidate
117 realizes}\label{odd_synthesis-the-finite-packet-that-this-same-candidate-realizes}}
118 
119 Assume the one-vertex scalar-pushout contract: there are \(d\) labelled,
120 actually supplied full character extensions with whole symplectic
121 quotients \(V_i\) of dimension two over \(\mathbb F_p\). Their balanced
122 \(\mathbb C^\times\)-central product has quotient
123 \(V_{\mathrm{src}}=\bigoplus_iV_i\) and Heisenberg carrier dimension
124 \(p^d=N\). Initially take the declared stabilizer \(C=\{1\}\).
125 
126 For \(p=3\), this applies directly to the retained cubic family with
127 whole rank-three break-type quotients. For \(p>3\), it applies only when
128 the supplied plane data genuinely satisfy the stated original bridge
129 contract. A native minimal rank-\(p\) type quotient has dimension
130 \(p-1\), so it must not automatically be called a plane.
131 
132 The actual candidate from Section 1 has a nondegenerate leading subspace
133 \[
134 S\subset J^1/H^1,\qquad
135 \dim_{\mathbb F_p}S=(p-1)p^{d-1}\ge2d.
136 \] Here \(S\) is the image of the concrete congruence subgroup at half
137 the leading even level. BK's high-depth equality preserves this subgroup
138 and its commutator pairing through every later approximation; see
139 \path{capacity/LEADING_LAYER_CAPACITY166.md}, Section 1. This part
140 of that note applies to the direct constructor and does not require its
141 separate classical arithmetic-induction application.
142 
143 Choose a labelled symplectic embedding \(V_{\mathrm{src}}\to S\). For
144 each original full character form \[
145 \mathcal P_i=(\mathbb C^\times\times J_i^1)/
146  \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
147 \] The balanced product and the corresponding host restriction have the
148 same alternating commutator. The center-fixing extension isomorphism,
149 obtained with matched Darboux lifts and complex roots, therefore
150 preserves every supplied value \(\theta_i(h)\), including the shallow
151 values.
152 
153 Write \(W=V_{\mathrm{src}}\perp V_{\mathrm{src}}^\perp\). Finite
154 Stone--von Neumann gives
155 \(\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp\). Choose an
156 isotropic eigenspace giving a line in the complementary factor. Its
157 actual Heisenberg image-algebra projector \(e\) has rank \(N\). The
158 retained induction/idempotent theorem recovers the \textbf{same}
159 inducing representation \(\Lambda\) from \(e\Lambda\), and then the same
160 \(\pi\) by compact induction. No candidate is changed at recovery.
161 
162 This attachment preserves the full source characters as scalar-pushout
163 data. It does not identify the old matrix groups with literal subgroups
164 of the new type, retain the old lower summand conjugacy classes, or
165 place each source label at its old numerical host depth. Those stronger
166 requirements have not been proved.
167 
168 When the retained independent extraspecial tensor source has \(d\ge2\),
169 its full determinant is trivial on wild inertia: each factor determinant
170 has order dividing \(p\), and tensor determinants raise it to
171 \(p^{d-1}\). Thus its central character belongs to the class in Section
172 1. For \(d=1\), wild triviality of the source determinant must be
173 checked separately. Independence of the tensor factors is an input, not
174 a consequence of distinct \(h_i\).
175 
176 \hypertarget{odd_synthesis-a-proved-extension-to-genuine-scalar-inertia-phases}{%
177 \subsection{3. A proved extension to genuine scalar inertia
178 phases}\label{odd_synthesis-a-proved-extension-to-genuine-scalar-inertia-phases}}
179 
180 There is also a positive result beyond \(C=\{1\}\). Suppose the finite
181 prime-to-\(p\) stabilizer acts trivially on the source scalar extension,
182 and its prescribed honest operators on the \textbf{entire} rank-\(N\)
183 Weil carrier are \(\chi(c)I_N\), each lying in the actual inertia image.
184 Assume the candidate's central character is the determinant of that same
185 source under the fixed rank-one reciprocity convention.
186 
187 Then a finite cover \(\widetilde C\twoheadrightarrow C\) acts by actual
188 central matrices in the unchanged candidate. Its represented operators
189 descend to exactly \(\chi\) on \(C\). Hence the attachment and recovery
190 above are equivariant for that action. A direct matrix homomorphism from
191 \(C\) itself is not asserted and can fail.
192 
193 Indeed, for residue cardinality \(q\), let
194 \(B=\omega(\operatorname{Teich}(k^\times))\). The actual inertia
195 hypothesis gives \(\chi(C)^N\subset B\). Since \(N\) is a power of \(p\)
196 and \(\chi(C)\) has order prime to \(p\), this implies
197 \(\chi(C)\subset B\). Choose \(k_0\) inverse to \(N\) modulo \(q-1\),
198 and set \[
199 \delta(u)=\omega(\operatorname{Teich}(u))^{k_0},\qquad
200 \widetilde C=\{(c,u):\chi(c)=\delta(u)\}.
201 \] The matrix \(\operatorname{Teich}(u^{k_0})I_N\) acts by \(\chi(c)\).
202 The kernel acts trivially on the entire sector induction and ambient
203 compact induction, because these matrices are central. This preserves
204 the fixed \(\Lambda\), its canonical beta-extension factor, and all
205 completion choices. It does not assert that the canonical beta extension
206 alone has scalar character \(\chi\).
207 
208 The full theorem and the nonsplitting example are in
209 \path{stabilizer/SCALAR_NATIVE_ATTACHMENT166.md}, with an
210 independent review in \path{audit/SCALAR_NATIVE_REVIEW166.md}. An
211 arbitrary abstract phase or a phase on a proper multiplicity space does
212 not satisfy its hypotheses merely because it is scalar.
213 
214 \hypertarget{odd_synthesis-what-was-repaired-and-what-remains}{%
215 \subsection{4. What was repaired, and what
216 remains}\label{odd_synthesis-what-was-repaired-and-what-remains}}
217 
218 The old fixed-field central argument genuinely failed. At \(p=3\),
219 \((h_1,h_2)=(1,5)\), its naturally transferred full character takes the
220 value \(\zeta_3\ne1\) on \(1+t^2\), although the desired wild center is
221 trivial. A shallow character choice cannot repair a forced value. The
222 two-stage changing-field repair is retained in
223 \path{central/CENTRAL_REPAIR166.md}; the trace-zero tower above
224 resolves this central issue uniformly in every length.
225 
226 The construction does not use an inverse established correspondence or
227 the classical arithmetic-induction/descent candidate to select \(\pi\).
228 It does use the named BK construction and type theorems, the retained
229 type/formal-degree conductor formulas, and exact sector recovery. The
230 capacity note separately retains an alternative consequence for a
231 candidate supplied by the substantial classical arithmetic constructor;
232 that is a different route and is not an input to Section 1.
233 
234 The unresolved universal obligations are precise:
235 
236 \begin{enumerate}
237 \def\labelenumi{\arabic{enumi}.}
238 \tightlist
239 \item
240   \textbf{Nontrivial prescribed symplectic stabilizer actions.} For this
241   totally ramified full-degree host, the stronger inner-normalizer
242   interpretation forces prime-to-\(p\) actions on the quotient to be
243   trivial. It therefore cannot implement a genuinely nontrivial supplied
244   symplectic action. The raw original contract does not automatically
245   require that stronger interpretation; it still needs an actual valid
246   arithmetic comparison, not relabelling of external operators.
247 \item
248   \textbf{Arbitrary admissible source packets and transport diagrams.}
249   The proved attachment has the whole-plane, one-vertex, finite
250   scalar-pushout hypotheses above. It does not establish all source
251   classes, arbitrary graphs, or prescribed numerical layer placement.
252 \item
253   \textbf{The dyadic construction with all markings.} A separate
254   field-level recursion and some capacity statements exist at \(p=2\).
255   The odd-prime trace-zero/full-character theorem here does not settle
256   that case.
257 \end{enumerate}
258 
259 Original H5 stays exactly where it was: if supplied for this fixed
260 candidate and the stipulated tests, the retained conditional endpoint
261 applies. No claim that the displayed profiles imply H5, or that one of
262 the noncompact root choices must satisfy H5, is made. The unresolved
263 construction issues above are not renamed epsilon-factor obligations.
264 
265 \hypertarget{odd_synthesis-evidence-and-reproducibility}{%
266 \subsection{5. Evidence and
267 reproducibility}\label{odd_synthesis-evidence-and-reproducibility}}
268 
269 The arbitrary-length theorem is proved symbolically and independently
270 reviewed. The exact \(p=3,d=2,(h_1,h_2)=(1,2)\) computation in
271 \texttt{tower/as\_rank9check/} verifies the displayed matrix, trace
272 identities, corestriction, and levels; it is a finite check, not the
273 universal proof. The separate rank-27 run in \texttt{tower/run\_rank27/}
274 checks the earlier non-trace-zero recursion and must not be described as
275 an AS central completion example. No Lean build was run.
276 
277 \path{STATUS166.json} records the closed and open obligations. The
278 component audits identify the exact reviewed files and hashes. The
279 dependency file pins prior checkpoints and the authorized primary-source
280 copy without redistributing that publication.

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