Realization Calculus

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

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