Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{The precise stabilizer-action contract and normalizer obstruction}\label{app:stabilizer_contract}
3 
4 Evidence: direct audit of the original statement, a new elementary
5 reduction, and an explicit actual-source/actual-type H1--H2 witness.
6 Earlier artifacts are read-only. No Lean closure, class-wide arithmetic
7 realization, or H5 witness is claimed.
8 
9 \textbf{Primary audit verdict:} incomplete, with the smallest missing
10 implication being the identification of the declared finite action with
11 the action intended on a newly constructed common arithmetic candidate.
12 The original text does not define that identification by a common matrix
13 normalizer.
14 
15 \hypertarget{stabilizer_contract-exact-contract-and-what-is-already-an-input}{%
16 \subsection{1. Exact contract, and what is already an
17 input}\label{stabilizer_contract-exact-contract-and-what-is-already-an-input}}
18 
19 The controlling text is
20 \path{closure_attempt162/inputs/original_reading.txt}:
21 
22 \begin{itemize}
23 \tightlist
24 \item
25   Proposition9.8, starting line4834: a finite group acting by
26   automorphisms of the one-break Clifford datum acts symplectically on
27   its intrinsic envelope.
28 \item
29   Definition11.1, lines4911--4923: a bridge supplies an equivariant
30   isometry to an actual type quotient for a finite cyclic stabilizer C,
31   with common central character and additional dyadic lift data.
32 \item
33   Proposition11.3 and Theorem11.5: once the two honest cyclic actions
34   are chosen, their discrepancy is a character, killed by
35   generator-phase equality.
36 \item
37   Theorem15.4 and Theorem16.2: breakwise comparisons and transports are
38   supplied and tensored. They do not construct a common arithmetic host
39   from separate actual type targets.
40 \end{itemize}
41 
42 Thus H2 and H3 already settle the finite action comparison \textbf{on
43 each supplied break target}. They do not leave a second free stabilizer
44 action to invent on that same target. Their transport into a new common
45 arithmetic type is another assertion. Neither a map C into its matrix
46 normalizer nor arbitrary transported operators are identified with that
47 assertion by the original text.
48 
49 The cyclic group in Definition11.1 is not explicitly required there to
50 have order prime to p.~The reduction below assumes that order condition;
51 it applies to the declared tame prime-to-p case, and cannot silently be
52 applied to an arbitrary finite-order Frobenius operator.
53 
54 \hypertarget{stabilizer_contract-full-scalar-extension-rigidity-for-a-native-tame-action}{%
55 \subsection{2. Full scalar-extension rigidity for a native tame
56 action}\label{stabilizer_contract-full-scalar-extension-rigidity-for-a-native-tame-action}}
57 
58 Let an actual maximal simple type in GL\_n(F) have a totally ramified
59 parameter field K/F of degree n=p\^{}r and residue matrix rank one. Let
60 
61 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
62 W=J¹/H¹,  P=(C××J¹)/{(theta(h)^(-1),h):h in H¹},
63 \end{Verbatim}
64 
65 and let eta be its irreducible Heisenberg representation. Let C be
66 finite of order prime to p.~Suppose the C action on P fixes the scalar
67 center and is genuinely induced by character-preserving matrix
68 normalizers of this type. It is enough that it is an honest action whose
69 action on W comes from those normalizers. Then:
70 
71 \begin{enumerate}
72 \def\labelenumi{\arabic{enumi}.}
73 \tightlist
74 \item
75   C acts trivially on W.
76 \item
77   C acts trivially on the full scalar extension P.
78 \item
79   Every honest C action Omega on eta implementing this action is
80   Omega(c)=chi(c)I for one character chi of C.
81 \end{enumerate}
82 
83 \textbf{Proof.} The retained normalizer theorem in Block162 gives
84 
85 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
86 N=K×J¹,          N/(F×J¹) = Z/nZ.
87 \end{Verbatim}
88 
89 The action on W factors through this cyclic p-group, so a homomorphic
90 image of the prime-to-p group C is trivial.
91 
92 For any central extension P of the elementary abelian p-group W by C×,
93 the kernel of
94 
95 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
96 Aut_(C×)(P) -> Aut(W)
97 \end{Verbatim}
98 
99 is naturally Hom(W,C×). Indeed, an automorphism phi fixing the quotient
100 and the center has phi(x)=a(x\_bar)x. The scalar a is independent of the
101 lift x and multiplication shows that a is a character. Conversely every
102 such character defines an automorphism. Since W has exponent p, this
103 kernel is a finite elementary abelian p-group. The action of C lies in
104 this kernel and therefore is trivial. Finally an implementing operator
105 commutes with the irreducible P action, so is scalar by Schur. Honesty
106 makes its scalar a character. QED.
107 
108 This strengthens the earlier W-level obstruction: passing to the full
109 scalar pushout cannot hide a nontrivial native prime-to-p action above
110 an identity quotient action.
111 
112 \hypertarget{stabilizer_contract-exact-attachment-criterion-in-this-native-setting}{%
113 \subsection{3. Exact attachment criterion in this native
114 setting}\label{stabilizer_contract-exact-attachment-criterion-in-this-native-setting}}
115 
116 Suppose a marked source scalar extension P\_src has a symplectic
117 embedding into W, and an actual finite sector of eta realizes that
118 extension. Fix its center-preserving attachment and a Heisenberg
119 intertwiner T. If the host has the native tame action of Section2, the
120 attachment intertwines the full honest C actions if and only if
121 
122 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
123 the source action on P_src is trivial, and
124 chi_src = chi_host.
125 \end{Verbatim}
126 
127 Necessity follows from equivariance and Section2. For sufficiency,
128 trivial extension actions make both honest carrier actions scalar by
129 Schur, and equal characters commute through every Heisenberg
130 intertwiner. Any sector is stable under the scalar host action.
131 
132 If the source C action on its symplectic quotient is trivial, its action
133 on the full scalar extension is already trivial by the Hom-kernel
134 argument. For several break factors, the source character is the product
135 of their characters. Thus the criterion is a single scalar-character
136 equality, rather than a further system of symplectic placement
137 equations.
138 
139 This applies to the original C=\{1\} fixture immediately. For nontrivial
140 C, H3 equates source and \textbf{supplied break-target} phases. It does
141 not equate their product with a separately built common host's phase
142 without a dictionary. We do not change a fixed canonical beta extension
143 to enforce this equality.
144 
145 \hypertarget{stabilizer_contract-h2-and-matching-carrier-dimensions-do-not-force-trivial-c}{%
146 \subsection{4. H2 and matching carrier dimensions do not force trivial
147 C}\label{stabilizer_contract-h2-and-matching-carrier-dimensions-do-not-force-trivial-c}}
148 
149 Here is an actual H1--H2 witness under the literal original allowance of
150 separate type targets. It is not a claimed H5-admitted input, and does
151 not assert equal ambient matrix ranks for source and break target.
152 
153 Take F=Fp((t)), p odd. The retained primary source by Tsushima,
154 Definition3.1, Lemmas2.6,2.8,3.9,3.12,3.14 and Corollary3.15, with
155 R(X)=X\^{}p and m=1, constructs a genuine rank-p Weil representation
156 irreducible on wild inertia. Its wild image is extraspecial of order p³.
157 The source envelope is a symplectic plane, and its actual tame image
158 C=mu\_(p+1) acts by multiplication on the roots of X\^{}(p²)+X. This
159 action is nontrivial and has order p+1. The source's finite Heisenberg
160 carrier has dimension p.
161 
162 For the actual type target, take the unramified quadratic extension E/F
163 and the maximal order A=End\_OF(OE). Let beta=t\^{}(-2)a, where a is a
164 unit generating the residue extension. This is a minimal simple stratum
165 of even level two. The standard minimal simple-character formulas give
166 
167 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
168 J¹=U_A¹,  H¹=U_E¹ U_A²,
169 W_theta=End_Fp(Fp²)/Fp²,     dim_Fp W_theta=2,
170 dim eta_theta=p.
171 \end{Verbatim}
172 
173 These are actual groups of a simple character, not an abstract
174 replacement. The type residue units act by conjugation. Write
175 sigma(x)=x\^{}p and decompose
176 
177 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
178 End_Fp(Fp²)=Fp² direct-sum Fp² sigma.
179 \end{Verbatim}
180 
181 On the quotient represented by b sigma, conjugation by u in Fp²× is
182 
183 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
184 b -> u^(1-p)b.
185 \end{Verbatim}
186 
187 Its image is the norm-one torus, of order p+1. Consequently the actual
188 projective residue action of Fp²×/Fp× on the two-dimensional quotient is
189 nontrivial. It has the same symplectic C-module as the source: identify
190 the cyclic groups using the displayed ratio map. The two invariant
191 alternating forms differ by a nonzero Fp scalar, and multiplication by
192 lambda in Fp²× rescales an alternating form by Norm(lambda).
193 Surjectivity of the norm gives an equivariant symplectic isometry.
194 
195 This supplies the actual source, actual type quotient, matching degree-p
196 Heisenberg carriers, and nontrivial equivariant H2 action. For p=3, C
197 has order4 and the action is an irreducible rotation of the F3 plane.
198 
199 The target here is in GL\_2(F), while the source has rank p.~This is
200 permitted by the literal breakwise definitions, which contain no
201 relation between the individual type targets' ambient ranks. Block161
202 already recorded that exact omission. If a stricter intended contract
203 requires every break target in a specified GL\_(p\^{}r)(F), this example
204 does not settle that stricter question. We have not supplied H5 for a
205 common candidate or certified fixed canonical beta-extension phases for
206 this example. It refutes only the proposed inference that actual H2 plus
207 equality of finite carrier dimensions forces C trivial.
208 
209 \hypertarget{stabilizer_contract-result-for-the-next-construction-attempt}{%
210 \subsection{5. Result for the next construction
211 attempt}\label{stabilizer_contract-result-for-the-next-construction-attempt}}
212 
213 There are three distinct cases:
214 
215 \begin{longtable}[]{@{}
216   >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
217   >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
218 \toprule\noalign{}
219 \begin{minipage}[b]{\linewidth}\raggedright
220 Given contract
221 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
222 Exact remaining issue
223 \end{minipage} \\
224 \midrule\noalign{}
225 \endhead
226 \bottomrule\noalign{}
227 \endlastfoot
228 Original C=\{1\} fixture & No stabilizer issue remains. \\
229 Native prime-to-p C action in a degree-p\^{}r totally ramified host &
230 Source extension action must be trivial; then compare one scalar
231 character. \\
232 Nontrivial source symplectic C action under the original finite contract
233 & A common-host arithmetic action dictionary is needed. Requiring an
234 inner normalizer makes this impossible for the preceding host;
235 transporting external operators alone was already done in161. \\
236 \end{longtable}
237 
238 Thus ``solve arbitrary nontrivial stabilizer attachment by adjusting the
239 current totally ramified binder'' is not a viable native-inner target.
240 That is a precise failure of that stronger construction route, not a
241 refutation of16.2.
242 
243 \hypertarget{stabilizer_contract-evidence-and-self-review}{%
244 \subsection{Evidence and self-review}\label{stabilizer_contract-evidence-and-self-review}}
245 
246 Dependencies: original Sections9--11,15--16; Block162's source-backed
247 normalizer theorem; Block165's base realization contract and
248 same-candidate attachment; the retained published Tsushima text; the
249 retained minimal-stratum group formulas. The new Hom-kernel lemma and
250 attachment criterion are ordinary elementary proofs. The witness uses
251 the displayed arithmetic constructor inputs, not a numerical experiment.
252 No independent review of this new note is claimed in this file, and no
253 new Lean compilation was attempted.
254 
255 Self-review explicitly checked: the prime-to-p qualification, honest
256 action versus projective implementers, center-fixed scalar extension
257 automorphisms, separate target rank versus original rank, and the
258 absence of an H5 witness.

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