Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Exact enlarged-carrier recovery and original-contract comparison}\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 Section10 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. Definition11.1 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 Fact10.2 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 Definition11.1. 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
61 maximal simple type would overstate it.
62 
63 \hypertarget{lowrank_contract-precise-conditional-setup}{%
64 \subsection{2. Precise conditional
65 setup}\label{lowrank_contract-precise-conditional-setup}}
66 
67 Let G=GL\_n(F), Z=F×I\_n, and let the already constructed candidate be
68 
69 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
70 pi=c-Ind_N^G Lambda,
71 \end{Verbatim}
72 
73 with its fixed actual extended type and fixed central character omega.
74 Let J be a compact open normal subgroup of N such that
75 
76 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
77 eta=Lambda|J is irreducible,       N/J is infinite cyclic.
78 \end{Verbatim}
79 
80 In the proposed F2-residue, full-degree totally ramified setting, J is
81 the actual compact type group J¹=J, not a new group called J merely to
82 fit this notation. Let P be an actual compact open subgroup of G
83 satisfying
84 
85 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
86 J subset P,          N intersect P=J.
87 \end{Verbatim}
88 
89 Define the genuine finite-dimensional compact induction
90 
91 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
92 Xi=Ind_J^P eta.                                       (1)
93 \end{Verbatim}
94 
95 Neither irreducibility nor Heisenberg structure of Xi is assumed to
96 follow from its dimension. They must be proved separately when needed.
97 
98 If the known eta intertwining satisfies I\_G(eta)=N, then Xi is
99 irreducible: the finite Mackey endomorphism formula gives
100 
101 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
102 End_P(Ind_J^P eta)=End_J(eta)=C,
103 \end{Verbatim}
104 
105 because no nontrivial double coset in P lies in I\_G(eta).
106 Finite-dimensional smooth representations of compact P factor through a
107 finite group and are semisimple, so scalar endomorphisms imply
108 irreducibility. This criterion requires its actual group and
109 intertwining hypotheses; dimension alone is not a substitute.
110 
111 \hypertarget{lowrank_contract-xi-occurs-in-the-same-pi-and-can-be-an-equivalent-compact-type}{%
112 \subsection{3. Xi occurs in the same pi, and can be an equivalent
113 compact
114 type}\label{lowrank_contract-xi-occurs-in-the-same-pi-and-can-be-an-equivalent-compact-type}}
115 
116 The subspace of the actual compact-induction model of pi consisting of
117 functions supported on NP is stable under P. Mackey restriction
118 identifies that subspace with
119 
120 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
121 Ind_(N intersect P)^P(Lambda|_(N intersect P))=Xi.     (2)
122 \end{Verbatim}
123 
124 Thus Xi is an actual compact P-subrepresentation of the same pi, with no
125 change of arithmetic candidate or noncompact scalar.
126 
127 Induction in stages also gives the exact G-isomorphism
128 
129 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
130 c-Ind_P^G Xi isomorphic to c-Ind_J^G eta.             (3)
131 \end{Verbatim}
132 
133 If (J,eta) is already a type for a stated Bernstein block and Xi is
134 irreducible, then (P,Xi) detects the same irreducible representations:
135 
136 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
137 Hom_P(Xi,V) isomorphic to Hom_J(eta,V).
138 \end{Verbatim}
139 
140 This is the precise equivalent-type statement that follows from (1). It
141 does not assert that Xi is the eta of another maximal simple character,
142 or attach a canonical beta extension to Xi. The known canonical
143 extension in Lambda remains the one originally supplied.
144 
145 \hypertarget{lowrank_contract-exact-recovery-before-fixing-the-center-a-quotient}{%
146 \subsection{4. Exact recovery before fixing the center: a
147 quotient}\label{lowrank_contract-exact-recovery-before-fixing-the-center-a-quotient}}
148 
149 Choose u in N mapping to a generator of N/J, and retain the actual
150 matrix Lambda(u). Use the model
151 
152 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
153 X=c-Ind_J^G eta,
154 f(jg)=eta(j)f(g),
155 \end{Verbatim}
156 
157 with G acting by right translation. Define the invertible G-equivariant
158 shift operator
159 
160 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
161 (Tf)(g)=Lambda(u)^(-1)f(ug).                          (4)
162 \end{Verbatim}
163 
164 Normality of J and the actual intertwining relation for Lambda(u) ensure
165 that Tf has the same J-covariance. Its normalization explicitly retains
166 the fixed noncompact extension; changing Lambda(u) changes the relation
167 below.
168 
169 Define
170 
171 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
172 (Qf)(g)=sum_(k in Z) Lambda(u)^(-k)f(u^k g).         (5)
173 \end{Verbatim}
174 
175 The sum is finite on each argument and has the required compact support
176 modulo N. It is N-covariant and therefore lies in pi. It is surjective
177 and satisfies Q(Tf)=Q(f). Decomposing support into N/J-orbits reduces
178 its kernel to the elementary augmentation kernel on finitely supported
179 sequences on Z. That kernel consists exactly of consecutive differences.
180 Hence
181 
182 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
183 X/(T-1)X isomorphic to pi.                           (6)
184 \end{Verbatim}
185 
186 Through (3), this is a recovery of pi from the enlarged compact carrier
187 plus the original noncompact relation. It does not select the correct
188 relation from the finite Heisenberg packet alone.
189 
190 In general this quotient cannot be replaced by a G-equivariant split
191 inclusion before central normalization. A nonzero vector of X has finite
192 support modulo compact J. The valuation of its determinant support is a
193 finite subset of Z. A scalar uniformizer z=tI\_n shifts these valuations
194 by n; consequently X has no nonzero eigenvector for z. Every vector of
195 pi is such an eigenvector, with eigenvalue omega(t). Therefore no
196 nonzero G-map pi -\textgreater{} X exists. In particular (6) does not
197 split.
198 
199 This is why the Block159 sector summand projector cannot simply be
200 quoted: there the finite sector was inside Lambda and its stabilizer
201 contained the noncompact center. Here Xi is larger than Lambda and lies
202 in pi\textbar P, with P compact. These are different induction problems.
203 
204 \hypertarget{lowrank_contract-after-retaining-the-center-a-finite-hecke-idempotent}{%
205 \subsection{5. After retaining the center: a finite Hecke
206 idempotent}\label{lowrank_contract-after-retaining-the-center-a-finite-hecke-idempotent}}
207 
208 There is also an exact summand version once the already fixed central
209 character is imposed. Put J\_Z=ZJ and P\_Z=ZP. Assume, as in the
210 proposed totally ramified full-degree setting, that
211 
212 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
213 N/J_Z is finite cyclic of order ell
214 \end{Verbatim}
215 
216 (here ell=n). Let Lambda\_0=Lambda\textbar J\_Z. Because Xi occurs in
217 the same pi, its action on Z intersect P agrees with omega. It therefore
218 extends to P\_Z by letting Z act through omega; call this Xi\_omega.
219 Induction in stages gives
220 
221 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
222 Y=c-Ind_(P_Z)^G Xi_omega
223   isomorphic to c-Ind_(J_Z)^G Lambda_0.              (7)
224 \end{Verbatim}
225 
226 Using the same normalized shift (4), now on Y, one has T\^{}ell=1.
227 Indeed u\^{}ell belongs to J\_Z and the J\_Z covariance cancels
228 Lambda(u)\^{}ell. The finite operator
229 
230 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
231 e_Lambda=(1/ell) sum_(j=0)^(ell-1) T^j               (8)
232 \end{Verbatim}
233 
234 is a G-equivariant idempotent whose image is exactly the fixed pi. To
235 see this directly, N-covariant functions embed in J\_Z-covariant
236 functions because N/J\_Z is finite. On that embedded pi the finite
237 version of (5) equals ell times the identity. Its average is precisely
238 (8), and its image consists exactly of the N-covariant functions.
239 
240 Equivalently, Ind\_(J\_Z)\^{}N Lambda\_0 is Lambda tensored with the
241 regular representation of N/J\_Z. Formula(8) selects the Lambda summand.
242 It uses the already fixed Lambda(u), including its root choice, so it
243 preserves the same candidate. Other root choices select the other
244 extensions in this finite induction. They are not interchangeable input
245 data.
246 
247 This is a finite Hecke/induction projector after central-character
248 normalization. It is not the original finite image-algebra projector
249 onto a sector of eta. A final artifact must retain (7), (8), and the
250 normalization Lambda(u), rather than describing recovery merely as ``the
251 carrier is an equivalent type.''
252 
253 \hypertarget{lowrank_contract-the-exact-remaining-admissibility-checks}{%
254 \subsection{6. The exact remaining admissibility
255 checks}\label{lowrank_contract-the-exact-remaining-admissibility-checks}}
256 
257 For either proposed low-rank enlargement, the new proof must supply all
258 of the following concrete inputs:
259 
260 \begin{enumerate}
261 \def\labelenumi{\arabic{enumi}.}
262 \tightlist
263 \item
264   \textbf{Actual compact group:} prove that the proposed P is a
265   subgroup, is compact open, contains the actual J, and satisfies P
266   intersect N=J. Compute {[}P:J{]} and the resulting Xi dimension.
267 \item
268   \textbf{Actual carrier:} verify eta=Lambda\textbar J, its actual
269   intertwining if the irreducibility criterion is used, and the Mackey
270   realization (2).
271 \item
272   \textbf{Actual finite Heisenberg structure:} exhibit the scalar
273   preimage in P and the resulting elementary abelian projective
274   quotient. Compute its represented commutator and prove nondegeneracy.
275   An irreducible carrier of dimension two or four need not automatically
276   have this structure.
277 \item
278   \textbf{Original attachments:} attach the unchanged supplied actual H2
279   break targets, retaining their full source character values and dyadic
280   finite extension/lift/square markings. The precise
281   transported-versus-native distinction remains the one proved in
282   \path{DYADIC_MARKINGS167.md}.
283 \item
284   \textbf{Recovery data:} retain the actual Lambda, central character
285   and noncompact generator normalization, and prove the isomorphisms and
286   quotient/idempotent above for those exact groups.
287 \end{enumerate}
288 
289 When these are supplied, the result is stronger than occurrence of an
290 unrelated finite vector space: it is a marked arithmetic carrier in the
291 same candidate, equivalent through explicit induction and recovering
292 that candidate exactly. For C=\{1\} and one vertex, it can repair the
293 failed low-rank \textbf{assembly-carrier} realization under the retained
294 scalar-pushout contract. The original H2 targets remain their original
295 simple-character quotients; they are not renamed Xi.
296 
297 If the intended target instead requires the source packet itself to
298 equal J\_theta'¹/H\_theta'¹ for a maximal simple character in this same
299 candidate, the enlargement does not solve it. The retained low-rank
300 dimension obstruction still applies. A separate theorem identifying Xi
301 with such an eta would be needed, and under the fixed obstructed profile
302 it cannot be inferred from type equivalence.
303 
304 No H5 condition is added or discharged. If H5 is subsequently supplied,
305 it must concern this fixed recovered candidate. Nontrivial stabilizer
306 actions, canonical native quadratic comparison, and arbitrary transport
307 graphs remain outside this bounded argument.
308 
309 \hypertarget{lowrank_contract-evidence-and-self-review}{%
310 \subsection{Evidence and self-review}\label{lowrank_contract-evidence-and-self-review}}
311 
312 Read directly: original Section10 and Definition11.1 in the raw
313 manuscript; the Block159 \path{NONINVARIANT_SECTOR_RECONSTRUCTION159.md};
314 current \path{PROOF167.md} and \path{TARGET167.md}; the retained dyadic marking
315 theorem. The Mackey, induction, shift-quotient, and central-normalized
316 projector arguments above are proved conditionally for the displayed
317 groups, rather than assumed for the proposed congruence formula. No
318 actual low-rank P computation or new Lean execution is claimed in this
319 contract note.

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