Realization Calculus

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1 \clearpage
2 \section{Exact enlarged-carrier recovery and comparison with the original contract}\label{app:lowrank_contract}
3 
4 Evidence: a bounded contract audit and elementary conditional recovery
5 lemmas. This note does not prove that the proposed compact enlargement
6 is a group, that its induced representation is Heisenberg, or that the
7 source markings attach to it. Those are the arithmetic inputs to be
8 verified by the separate rank-two and rank-four constructions.
9 
10 \textbf{Verdict:} an enlarged compact carrier is not, merely by being
11 Heisenberg or an equivalent type, a new instance of the original H2
12 quotient. It can serve as a common assembly carrier for the already
13 supplied H2 targets if its actual occurrence, marked attachment, and
14 same-candidate recovery are proved and retained. The recovery mechanism
15 differs from the finite sector idempotent used when the packet lies
16 inside the original inducing carrier.
17 
18 \hypertarget{lowrank_contract-what-original-h2-literally-requires}{%
19 \subsection{1. What original H2 literally
20 requires}\label{lowrank_contract-what-original-h2-literally-requires}}
21 
22 Original Section~10 fixes an actual maximal simple character theta, its
23 actual groups H\_theta¹ subset J\_theta¹ subset J\_theta, and
24 
25 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
26 V_theta=J_theta¹/H_theta¹.
27 \end{Verbatim}
28 
29 Its Heisenberg representation is the particular eta\_theta containing
30 theta. Definition~11.1 supplies an equivariant symplectic isometry from a
31 source envelope to that type-side quotient, together with the central
32 character, and at p=2 the compatible quadratic refinements and
33 central-extension lift. Standard Fact 10.2 separately supplies the
34 canonical beta extension of this simple-character datum.
35 
36 Consequently the following implications are invalid without more proof:
37 
38 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
39 Xi is a finite Heisenberg representation
40    therefore Xi=eta_theta' for a maximal simple character theta';
41 
42 (P,Xi) is equivalent to a known arithmetic type
43    therefore it has that type's J¹/H¹ or canonical beta extension.
44 \end{Verbatim}
45 
46 An equivalence of generated representation categories does not identify
47 the particular finite group quotient in Definition~11.1. If a fixed
48 conductor and torsion profile forces every maximal simple-character
49 carrier in the candidate to have dimension one or two, constructing a
50 larger Xi in that candidate does not refute the dimension theorem. Xi is
51 another arithmetic carrier.
52 
53 There is nevertheless an important distinction between \textbf{replacing
54 an H2 target} and \textbf{assembling already supplied H2 targets}. The
55 accepted sector construction itself uses a finite sector that need not
56 be a whole maximal simple-character carrier. Its old actual break
57 targets remain unchanged and are attached to that sector. An enlarged
58 P-carrier can play the analogous assembly role, provided the complete
59 dictionary and recovery below are proved. Calling it merely an arbitrary
60 vector space would understate that result. Calling it a new canonical maximal simple type would overstate it.
61 
62 \hypertarget{lowrank_contract-precise-conditional-setup}{%
63 \subsection{2. Precise conditional
64 setup}\label{lowrank_contract-precise-conditional-setup}}
65 
66 Let G=GL\_n(F), Z=F×I\_n, and let the already constructed candidate be
67 
68 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
69 pi=c-Ind_N^G Lambda,
70 \end{Verbatim}
71 
72 with its fixed actual extended type and fixed central character omega.
73 Let J be a compact open normal subgroup of N such that
74 
75 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
76 eta=Lambda|J is irreducible,       N/J is infinite cyclic.
77 \end{Verbatim}
78 
79 In the proposed F2-residue, full-degree totally ramified setting, J is
80 the actual compact type group J¹=J, not a new group called J merely to
81 fit this notation. Let P be an actual compact open subgroup of G
82 satisfying
83 
84 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
85 J subset P,          N intersect P=J.
86 \end{Verbatim}
87 
88 Define the genuine finite-dimensional compact induction
89 
90 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
91 Xi=Ind_J^P eta.                                       (1)
92 \end{Verbatim}
93 
94 Neither irreducibility nor Heisenberg structure of Xi is assumed to
95 follow from its dimension. They must be proved separately when needed.
96 
97 If the known eta intertwining satisfies I\_G(eta)=N, then Xi is
98 irreducible: the finite Mackey endomorphism formula gives
99 
100 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
101 End_P(Ind_J^P eta)=End_J(eta)=C,
102 \end{Verbatim}
103 
104 because no nontrivial double coset in P lies in I\_G(eta).
105 Finite-dimensional smooth representations of compact P factor through a
106 finite group and are semisimple, so scalar endomorphisms imply
107 irreducibility. This criterion requires its actual group and intertwining hypotheses. Dimension alone is not a substitute.
108 
109 \hypertarget{lowrank_contract-xi-occurs-in-the-same-pi-and-can-be-an-equivalent-compact-type}{%
110 \subsection{3. Xi occurs in the same pi, and can be an equivalent
111 compact
112 type}\label{lowrank_contract-xi-occurs-in-the-same-pi-and-can-be-an-equivalent-compact-type}}
113 
114 The subspace of the actual compact-induction model of pi consisting of
115 functions supported on NP is stable under P. Mackey restriction
116 identifies that subspace with
117 
118 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
119 Ind_(N intersect P)^P(Lambda|_(N intersect P))=Xi.     (2)
120 \end{Verbatim}
121 
122 Thus Xi is an actual compact P-subrepresentation of the same pi, with no
123 change of arithmetic candidate or noncompact scalar.
124 
125 Induction in stages also gives the exact G-isomorphism
126 
127 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
128 c-Ind_P^G Xi isomorphic to c-Ind_J^G eta.             (3)
129 \end{Verbatim}
130 
131 If (J,eta) is already a type for a stated Bernstein block and Xi is
132 irreducible, then (P,Xi) detects the same irreducible representations:
133 
134 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
135 Hom_P(Xi,V) isomorphic to Hom_J(eta,V).
136 \end{Verbatim}
137 
138 This is the precise statement about equivalent types that follows from (1). It
139 does not assert that Xi is the eta of another maximal simple character,
140 or attach a canonical beta extension to Xi. The known canonical
141 extension in Lambda remains the one originally supplied.
142 
143 \hypertarget{lowrank_contract-exact-recovery-before-fixing-the-center-a-quotient}{%
144 \subsection{4. Exact recovery before fixing the center: a
145 quotient}\label{lowrank_contract-exact-recovery-before-fixing-the-center-a-quotient}}
146 
147 Choose u in N mapping to a generator of N/J, and retain the actual
148 matrix Lambda(u). Use the model
149 
150 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
151 X=c-Ind_J^G eta,
152 f(jg)=eta(j)f(g),
153 \end{Verbatim}
154 
155 with G acting by right translation. Define the invertible G-equivariant
156 shift operator
157 
158 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
159 (Tf)(g)=Lambda(u)^(-1)f(ug).                          (4)
160 \end{Verbatim}
161 
162 Normality of J and the actual intertwining relation for Lambda(u) ensure
163 that Tf has the same J-covariance. Its normalization explicitly retains the fixed noncompact extension. Changing Lambda(u) changes the relation
164 below.
165 
166 Define
167 
168 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
169 (Qf)(g)=sum_(k in Z) Lambda(u)^(-k)f(u^k g).         (5)
170 \end{Verbatim}
171 
172 The sum is finite on each argument and has the required compact support
173 modulo N. It is N-covariant and therefore lies in pi. It is surjective
174 and satisfies Q(Tf)=Q(f). Decomposing support into N/J-orbits reduces
175 its kernel to the elementary augmentation kernel on finitely supported
176 sequences on Z. That kernel consists exactly of consecutive differences.
177 Hence
178 
179 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
180 X/(T-1)X isomorphic to pi.                           (6)
181 \end{Verbatim}
182 
183 Through (3), this is a recovery of pi from the enlarged compact carrier
184 plus the original noncompact relation. It does not select the correct
185 relation from the finite Heisenberg packet alone.
186 
187 In general this quotient cannot be replaced by a G-equivariant split
188 inclusion before central normalization. A nonzero vector of X has finite
189 support modulo compact J. The valuation of its determinant support is a
190 finite subset of Z. A scalar uniformizer z=tI\_n shifts these valuations by n. Consequently X has no nonzero eigenvector for z. Every vector of
191 pi is such an eigenvector, with eigenvalue omega(t). Therefore no
192 nonzero G-map pi -\textgreater{} X exists. In particular (6) does not
193 split.
194 
195 This is why the Block159 sector summand projector cannot simply be
196 quoted: there the finite sector was inside Lambda and its stabilizer
197 contained the noncompact center. Here Xi is larger than Lambda and lies
198 in pi\textbar P, with P compact. These are different induction problems.
199 
200 \hypertarget{lowrank_contract-after-retaining-the-center-a-finite-hecke-idempotent}{%
201 \subsection{5. After retaining the center: a finite Hecke
202 idempotent}\label{lowrank_contract-after-retaining-the-center-a-finite-hecke-idempotent}}
203 
204 There is also an exact summand version once the already fixed central
205 character is imposed. Put J\_Z=ZJ and P\_Z=ZP. Assume, as in the
206 proposed totally ramified full-degree setting, that
207 
208 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
209 N/J_Z is finite cyclic of order ell
210 \end{Verbatim}
211 
212 (here ell=n). Let Lambda\_0=Lambda\textbar J\_Z. Because Xi occurs in
213 the same pi, its action on Z intersect P agrees with omega. It therefore
214 extends to P\_Z by letting Z act through omega. Call this Xi\_omega.
215 Induction in stages gives
216 
217 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
218 Y=c-Ind_(P_Z)^G Xi_omega
219   isomorphic to c-Ind_(J_Z)^G Lambda_0.              (7)
220 \end{Verbatim}
221 
222 Using the same normalized shift (4), now on Y, one has T\^{}ell=1.
223 Indeed u\^{}ell belongs to J\_Z and the J\_Z covariance cancels
224 Lambda(u)\^{}ell. The finite operator
225 
226 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
227 e_Lambda=(1/ell) sum_(j=0)^(ell-1) T^j               (8)
228 \end{Verbatim}
229 
230 is a G-equivariant idempotent whose image is exactly the fixed pi. To
231 see this directly, N-covariant functions embed in J\_Z-covariant
232 functions because N/J\_Z is finite. On that embedded pi the finite
233 version of (5) equals ell times the identity. Its average is precisely
234 (8), and its image consists exactly of the N-covariant functions.
235 
236 Equivalently, Ind\_(J\_Z)\^{}N Lambda\_0 is Lambda tensored with the
237 regular representation of N/J\_Z. Formula (8) selects the Lambda summand.
238 It uses the already fixed Lambda(u), including its root choice, so it
239 preserves the same candidate. Other root choices select the other
240 extensions in this finite induction. They are not interchangeable input
241 data.
242 
243 This is a finite Hecke/induction projector after normalization of the central character. It is not the original finite projector in the image algebra onto a sector of eta. A final artifact must retain (7), (8), and the
244 normalization Lambda(u), rather than describing recovery merely as ``the
245 carrier is an equivalent type.''
246 
247 \hypertarget{lowrank_contract-the-exact-remaining-admissibility-checks}{%
248 \subsection{6. The exact remaining admissibility
249 checks}\label{lowrank_contract-the-exact-remaining-admissibility-checks}}
250 
251 For either proposed low-rank enlargement, the new proof must supply all
252 of the following concrete inputs:
253 
254 \begin{enumerate}
255 \def\labelenumi{\arabic{enumi}.}
256 \tightlist
257 \item
258   \textbf{Actual compact group:} prove that the proposed P is a
259   subgroup, is compact open, contains the actual J, and satisfies P
260   intersect N=J. Compute {[}P:J{]} and the resulting Xi dimension.
261 \item
262   \textbf{Actual carrier:} verify eta=Lambda\textbar J, its actual
263   intertwining if the irreducibility criterion is used, and the Mackey
264   realization (2).
265 \item
266   \textbf{Actual finite Heisenberg structure:} exhibit the scalar
267   preimage in P and the resulting elementary abelian projective
268   quotient. Compute its represented commutator and prove nondegeneracy.
269   An irreducible carrier of dimension two or four need not automatically
270   have this structure.
271 \item
272   \textbf{Original attachments:} attach the unchanged supplied actual H2
273   break targets, retaining their full source character values and dyadic
274   finite extension/lift/square markings. The precise distinction between transported and native markings remains the one proved in
275   \path{DYADIC_MARKINGS167.md}.
276 \item
277   \textbf{Recovery data:} retain the actual Lambda, central character
278   and noncompact generator normalization, and prove the isomorphisms and
279   quotient/idempotent above for those exact groups.
280 \end{enumerate}
281 
282 When these are supplied, the result is stronger than occurrence of an
283 unrelated finite vector space: it is a marked arithmetic carrier in the
284 same candidate, equivalent through explicit induction and recovering
285 that candidate exactly. For C=\{1\} and one vertex, it can repair the
286 failed low-rank \textbf{assembly-carrier} realization under the retained
287 scalar-pushout contract. The original H2 targets remain their original simple-character quotients. They are not renamed Xi.
288 
289 If the intended target instead requires the source packet itself to
290 equal J\_theta'¹/H\_theta'¹ for a maximal simple character in this same
291 candidate, the enlargement does not solve it. The retained low-rank
292 dimension obstruction still applies. A separate theorem identifying Xi
293 with such an eta would be needed, and under the fixed obstructed profile
294 it cannot be inferred from type equivalence.
295 
296 No H5 condition is added or discharged. If H5 is subsequently supplied,
297 it must concern this fixed recovered candidate. Nontrivial stabilizer
298 actions, canonical native quadratic comparison, and arbitrary transport
299 graphs remain outside this bounded argument.
300 
301 \hypertarget{lowrank_contract-evidence-and-self-review}{%
302 \subsection{Evidence and self-review}\label{lowrank_contract-evidence-and-self-review}}
303 
304 Read directly: original Section~10 and Definition~11.1 in the raw
305 manuscript; the Block159 \path{NONINVARIANT_SECTOR_RECONSTRUCTION159.md};
306 current \path{PROOF167.md} and \path{TARGET167.md}; the retained dyadic marking
307 theorem. The Mackey, induction, shift-quotient, and central-normalized
308 projector arguments above are proved conditionally for the displayed
309 groups, rather than assumed for the proposed congruence formula. No
310 actual low-rank P computation or new Lean execution is claimed in this
311 contract note.

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