Realization Calculus

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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 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   Proposition 9.8, starting line 4834: a finite group acting by
26   automorphisms of the one-break Clifford datum acts symplectically on
27   its intrinsic envelope.
28 \item
29   Definition~11.1, lines 4911--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   Proposition 11.3 and Theorem~11.5: once the two honest cyclic actions
34   are chosen, their discrepancy is a character, killed by equality of phases at the generator.
35 \item
36   Theorem~15.4 and Theorem~16.2: breakwise comparisons and transports are
37   supplied and tensored. They do not construct a common arithmetic host
38   from separate actual type targets.
39 \end{itemize}
40 
41 Thus H2 and H3 already settle the finite action comparison \textbf{on
42 each supplied break target}. They do not leave a second free stabilizer
43 action to invent on that same target. Their transport into a new common
44 arithmetic type is another assertion. Neither a map C into its matrix
45 normalizer nor arbitrary transported operators are identified with that
46 assertion by the original text.
47 
48 The cyclic group in Definition~11.1 is not explicitly required there to
49 have order prime to p.~The reduction below assumes that order condition. It applies to the declared tame prime-to-p case, and cannot silently be
50 applied to an arbitrary finite-order Frobenius operator.
51 
52 \hypertarget{stabilizer_contract-full-scalar-extension-rigidity-for-a-native-tame-action}{%
53 \subsection{2. Full scalar-extension rigidity for a native tame
54 action}\label{stabilizer_contract-full-scalar-extension-rigidity-for-a-native-tame-action}}
55 
56 Let an actual maximal simple type in GL\_n(F) have a totally ramified
57 parameter field K/F of degree n=p\^{}r and residue matrix rank one. Let
58 
59 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
60 W=J¹/H¹,  P=(C××J¹)/{(theta(h)^(-1),h):h in H¹},
61 \end{Verbatim}
62 
63 and let eta be its irreducible Heisenberg representation. Let C be
64 finite of order prime to p.~Suppose the C action on P fixes the scalar
65 center and is genuinely induced by character-preserving matrix
66 normalizers of this type. It is enough that it is an honest action whose
67 action on W comes from those normalizers. Then:
68 
69 \begin{enumerate}
70 \def\labelenumi{\arabic{enumi}.}
71 \tightlist
72 \item
73   C acts trivially on W.
74 \item
75   C acts trivially on the full scalar extension P.
76 \item
77   Every honest C action Omega on eta implementing this action is
78   Omega(c)=chi(c)I for one character chi of C.
79 \end{enumerate}
80 
81 \textbf{Proof.} The retained normalizer theorem in Block162 gives
82 
83 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
84 N=K×J¹,          N/(F×J¹) = Z/nZ.
85 \end{Verbatim}
86 
87 The action on W factors through this cyclic p-group, so a homomorphic
88 image of the prime-to-p group C is trivial.
89 
90 For any central extension P of the elementary abelian p-group W by C×,
91 the kernel of
92 
93 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
94 Aut_(C×)(P) -> Aut(W)
95 \end{Verbatim}
96 
97 is naturally Hom(W,C×). Indeed, an automorphism phi fixing the quotient
98 and the center has phi(x)=a(x\_bar)x. The scalar a is independent of the
99 lift x and multiplication shows that a is a character. Conversely every
100 such character defines an automorphism. Since W has exponent p, this
101 kernel is a finite elementary abelian p-group. The action of C lies in
102 this kernel and therefore is trivial. Finally an implementing operator
103 commutes with the irreducible P action, so is scalar by Schur. Honesty
104 makes its scalar a character. QED.
105 
106 This strengthens the earlier W-level obstruction: passing to the full
107 scalar pushout cannot hide a nontrivial native prime-to-p action above
108 an identity quotient action.
109 
110 \hypertarget{stabilizer_contract-exact-attachment-criterion-in-this-native-setting}{%
111 \subsection{3. Exact attachment criterion in this native
112 setting}\label{stabilizer_contract-exact-attachment-criterion-in-this-native-setting}}
113 
114 Suppose a marked source scalar extension P\_src has a symplectic
115 embedding into W, and an actual finite sector of eta realizes that
116 extension. Fix its center-preserving attachment and a Heisenberg
117 intertwiner T. If the host has the native tame action of Section~2, the
118 attachment intertwines the full honest C actions if and only if
119 
120 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
121 the source action on P_src is trivial, and
122 chi_src = chi_host.
123 \end{Verbatim}
124 
125 Necessity follows from equivariance and Section~2. For sufficiency,
126 trivial extension actions make both honest carrier actions scalar by
127 Schur, and equal characters commute through every Heisenberg
128 intertwiner. Any sector is stable under the scalar host action.
129 
130 If the source C action on its symplectic quotient is trivial, its action
131 on the full scalar extension is already trivial by the Hom-kernel
132 argument. For several break factors, the source character is the product
133 of their characters. Thus the criterion is a single equality of scalar characters, rather than a further system of symplectic placement
134 equations.
135 
136 This applies to the original C=\{1\} fixture immediately. For nontrivial
137 C, H3 equates source and \textbf{supplied break-target} phases. It does
138 not equate their product with a separately built common host's phase
139 without a dictionary. We do not change a fixed canonical beta extension
140 to enforce this equality.
141 
142 \hypertarget{stabilizer_contract-h2-and-matching-carrier-dimensions-do-not-force-trivial-c}{%
143 \subsection{4. H2 and matching carrier dimensions do not force trivial
144 C}\label{stabilizer_contract-h2-and-matching-carrier-dimensions-do-not-force-trivial-c}}
145 
146 Here is an actual H1--H2 witness under the literal original allowance of
147 separate type targets. It is not a claimed H5-admitted input, and does
148 not assert equal ambient matrix ranks for source and break target.
149 
150 Take F=Fp((t)), p odd. The retained primary source by Tsushima,
151 Definition~3.1, Lemmas 2.6, 2.8, 3.9, 3.12, 3.14 and Corollary 3.15, with
152 R(X)=X\^{}p and m=1, constructs a genuine rank-p Weil representation
153 irreducible on wild inertia. Its wild image is extraspecial of order p³.
154 The source envelope is a symplectic plane, and its actual tame image
155 C=mu\_(p+1) acts by multiplication on the roots of X\^{}(p²)+X. This
156 action is nontrivial and has order p+1. The source's finite Heisenberg
157 carrier has dimension p.
158 
159 For the actual type target, take the unramified quadratic extension E/F
160 and the maximal order A=End\_OF(OE). Let beta=t\^{}(-2)a, where a is a
161 unit generating the residue extension. This is a minimal simple stratum
162 of even level two. The standard minimal simple-character formulas give
163 
164 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
165 J¹=U_A¹,  H¹=U_E¹ U_A²,
166 W_theta=End_Fp(Fp²)/Fp²,     dim_Fp W_theta=2,
167 dim eta_theta=p.
168 \end{Verbatim}
169 
170 These are actual groups of a simple character, not an abstract
171 replacement. The type residue units act by conjugation. Write
172 sigma(x)=x\^{}p and decompose
173 
174 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
175 End_Fp(Fp²)=Fp² direct-sum Fp² sigma.
176 \end{Verbatim}
177 
178 On the quotient represented by b sigma, conjugation by u in Fp²× is
179 
180 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
181 b -> u^(1-p)b.
182 \end{Verbatim}
183 
184 Its image is the norm-one torus, of order p+1. Consequently the actual
185 projective residue action of Fp²×/Fp× on the two-dimensional quotient is
186 nontrivial. It has the same symplectic C-module as the source: identify
187 the cyclic groups using the displayed ratio map. The two invariant
188 alternating forms differ by a nonzero Fp scalar, and multiplication by
189 lambda in Fp²× rescales an alternating form by Norm(lambda).
190 Surjectivity of the norm gives an equivariant symplectic isometry.
191 
192 This supplies the actual source, actual type quotient, matching degree-p
193 Heisenberg carriers, and nontrivial equivariant H2 action. For p=3, C
194 has order 4 and the action is an irreducible rotation of the F3 plane.
195 
196 The target here is in GL\_2(F), while the source has rank p.~This is
197 permitted by the literal breakwise definitions, which contain no
198 relation between the individual type targets' ambient ranks. Block161
199 already recorded that exact omission. If a stricter intended contract
200 requires every break target in a specified GL\_(p\^{}r)(F), this example
201 does not settle that stricter question. We have not supplied H5 for a
202 common candidate or certified fixed canonical beta-extension phases for
203 this example. It refutes only the proposed inference that actual H2 plus
204 equality of finite carrier dimensions forces C trivial.
205 
206 \hypertarget{stabilizer_contract-result-for-the-next-construction-attempt}{%
207 \subsection{5. Result for the next construction
208 attempt}\label{stabilizer_contract-result-for-the-next-construction-attempt}}
209 
210 There are three distinct cases:
211 
212 \begin{longtable}[]{@{}
213   >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
214   >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
215 \toprule\noalign{}
216 \begin{minipage}[b]{\linewidth}\raggedright
217 Given contract
218 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
219 Exact remaining issue
220 \end{minipage} \\
221 \midrule\noalign{}
222 \endhead
223 \bottomrule\noalign{}
224 \endlastfoot
225 Original C=\{1\} fixture & No stabilizer issue remains. \\
226 Native prime-to-p C action in a degree-p\^{}r totally ramified host &
227 Source extension action must be trivial. Then compare one scalar character. \\
228 Nontrivial source symplectic C action under the original finite contract
229 & A dictionary for the arithmetic action on a common host is needed. Requiring an
230 inner normalizer makes this impossible for the preceding host. Transporting external operators alone was already done in Block 161. \\
231 \end{longtable}
232 
233 Thus ``solve arbitrary nontrivial stabilizer attachment by adjusting the
234 current totally ramified binder'' is not a viable native-inner target.
235 That is a precise failure of that stronger construction route, not a
236 refutation of 16.2.
237 
238 \hypertarget{stabilizer_contract-evidence-and-self-review}{%
239 \subsection{Evidence and self-review}\label{stabilizer_contract-evidence-and-self-review}}
240 
241 Dependencies: original Sections~9--11,15--16; Block162's source-backed
242 normalizer theorem; Block165's base realization contract and
243 same-candidate attachment; the retained published Tsushima text; the
244 retained minimal-stratum group formulas. The new Hom-kernel lemma and
245 attachment criterion are ordinary elementary proofs. The witness uses
246 the displayed arithmetic constructor inputs, not a numerical experiment.
247 No independent review of this new note is claimed in this file, and no
248 new Lean compilation was attempted.
249 
250 Self-review explicitly checked: the prime-to-p qualification, honest
251 action versus projective implementers, center-fixed scalar extension
252 automorphisms, separate target rank versus original rank, and the
253 absence of an H5 witness.

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