Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Dyadic synthesis and all-length assembly scope}\label{app:dyadic_synthesis}
3 
4 Date: 2026-10-05. Base: the final reviewed Block166, pinned in
5 \path{TARGET167.md}. Evidence: ordinary mathematical proof and
6 independent component reviews. No Lean verification or universal claim66
7 closure is asserted.
8 
9 \hypertarget{dyadic_synthesis-combined-arithmetic-theorem}{%
10 \subsection{1. Combined arithmetic
11 theorem}\label{dyadic_synthesis-combined-arithmetic-theorem}}
12 
13 Let \(F=\mathbb F_2((t))\), fix an additive character \(\psi_F\) of
14 conductor \(t\mathcal O_F\), and take increasing odd positive integers
15 \(h_1<\cdots<h_d\). Put \[
16 q_i=3h_i,\qquad N=2^d,\qquad n_j=2^{d-j}q_{d-j+1}.
17 \] There is an explicit \(\beta\in M_N(F)\) generating a separable
18 totally ramified degree-\(N\) field that normalizes the complete lattice
19 chain. It gives a maximal simple stratum with leading level \(n_1\),
20 final critical exponent \(-q_1\), and successive approximation-field
21 degrees \(2,4,\ldots,2^d\), with critical levels \(n_2,\ldots,n_d\).
22 
23 The actual stratum admits a full simple character \(\theta\) trivial on
24 \(U_F^1\). Its canonical beta extension can be completed to an actual
25 irreducible admissible supercuspidal \(\pi\) of \(\mathrm{GL}_N(F)\)
26 with any prescribed smooth central quasicharacter \(\omega\) trivial on
27 \(U_F^1\), provided the noncompact completion is free or already
28 compatible. Over this residue field, such \(\omega\) is unramified.
29 
30 For this same constructed candidate, \[
31 \begin{aligned}
32 t(\pi)&=1,& \operatorname{Sw}(\pi)&=3\,2^{d-1}h_d,\\
33 \dim_{\mathbb F_2}W&=N-2,&
34 \dim\eta&=2^{N/2-1},\qquad W=J^1/H^1,\\
35 a(\pi^\vee\times\pi)
36  &=N^2-1+\sum_{i=1}^d3\,4^{i-1}h_i.
37 \end{aligned}
38 \] Matching these profiles does not identify the source, its endo-class,
39 or the pair factors in original H5.
40 
41 \hypertarget{dyadic_synthesis-the-explicit-construction-and-its-depth-invariant}{%
42 \subsection{2. The explicit construction and its depth
43 invariant}\label{dyadic_synthesis-the-explicit-construction-and-its-depth-invariant}}
44 
45 At a recursive coefficient field \(L\) of characteristic two, with
46 residue field \(\mathbb F_2\) and uniformizer \(t_L\), put \[
47 E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad
48 a=(q+1)/2,\qquad z=t_L^a\alpha.
49 \] The extension is separable and totally ramified of degree two; \(z\)
50 is a uniformizer and \[
51 z^2+t_L^az+t_L=0,\qquad \operatorname{Tr}_{E/L}(\alpha)=1.
52 \] For constant-coefficient projection \(P_0\) in the basis \(1,z\), the
53 normalized trace corestriction and exact filtered right inverse are \[
54 \kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad
55 s(P_0)=t_L/z^2,\qquad
56 \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
57 \] They satisfy \(s(\operatorname{Lift}_E(C))=C\), preserve the exact
58 filtration valuation, and give the identity \[
59 \operatorname{tr}_L\operatorname{Lift}_E(C)
60  =\operatorname{Tr}_{E/L}
61    \bigl(\alpha\operatorname{tr}_E C\bigr).
62 \] The actual recursive matrix is \[
63 B(L,t_L;h_1,\ldots,h_d)
64  =\alpha I_{2^{d-1}}+
65    \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
66 \] with multiplication by \(\alpha\) as the rank-two base case.
67 
68 Every native constructor has trace one. At every next native rank, the
69 preceding parameter amplified by two has trace zero, so the native
70 correction also has trace one. At full ambient rank, the earlier
71 amplified approximations instead have trace zero and the final one has
72 trace one. These are different statements; no amplified trace is used as
73 a substitute for the native trace needed in character construction.
74 
75 At an inductive field step \(d\ge2\), set \(n=2^{d-1}q_d\),
76 \(R=2^{d-2}q_{d-1}\). Then \(n>2R\) and \(-2R>-n-q_1\). BK Remark 2.2.7
77 gives the maximal field generated by the actual displayed matrix.
78 Literal truncation and the filtered lift give the defining sequence.
79 Nonzero native trace also proves separability: the irreducible
80 characteristic polynomial has a nonzero coefficient of \(X^{N-1}\), so
81 its derivative is nonzero. See \path{tower/DYADIC_AS_TOWER167.md}
82 for all field and filtration details.
83 
84 The replacement for trace zero is the depth inequality \[
85 \boxed{v_F(\operatorname{tr}\delta)+s\ge1.}
86 \] It makes the additive correction trivial on scalar \(U_F^s\), because
87 \((u-1)\operatorname{tr}\delta\in t\mathcal O_F\). Our native
88 corrections have trace one, so the inequality holds at \(s=1\). This is
89 the exact source-to-target transfer of the weighted-visibility
90 principle; the Euler coefficient theorem is motivation, not an input to
91 the arithmetic proof. In characteristic two, the weights are uniformizer
92 depths, rather than integer powers of two in the coefficient field.
93 
94 \hypertarget{dyadic_synthesis-full-characters-including-the-odd-critical-endpoint}{%
95 \subsection{3. Full characters, including the odd critical
96 endpoint}\label{dyadic_synthesis-full-characters-including-the-odd-critical-endpoint}}
97 
98 At the minimal rank-two base, extend the forced character on
99 \(U_E^{(q+1)/2}\) to \(U_E^1\), imposing triviality on \(U_F^1\). The
100 prescriptions agree on their intersection because the parameter trace is
101 one and \(\psi_F(u-1)=1\). This is an actual finite abelian character
102 extension and gives the full minimal simple character.
103 
104 For a later native step, first transfer the full beta-determinant unit
105 character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar
106 units, \[
107 \Theta_j(uI)=\theta_j(u^2)=1.
108 \] For odd critical level \(r\), use \(m=\lfloor r/2\rfloor\) and
109 \(D=H^{m+1}(\beta)=H^{m+1}(\gamma)\). The endpoint is valid because
110 \(2(m+1)-r=1\): the additive-character cross term lies in the radical on
111 which the matrix additive character is trivial. BK 3.3.18 supplies
112 \(\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}\). Integral trace makes it
113 trivial on \(D\cap U_F^1\).
114 
115 The next parameter field is maximal. Extend the forced character on its
116 deep units together with the trivial character of \(U_F^1\), then apply
117 BK 3.2.5 to obtain the next full character. All new extensions occur at
118 maximal native rank. No unverified determinant-image extension in a
119 nonmaximal centralizer is used.
120 
121 For the final totally ramified field, \(J/J^1=\mathbb F_2^\times=1\), so
122 \(J=J^1\) and the canonical beta extension is \(\eta\). Extend to
123 \(K^\times J\), and choose the noncompact scalar to match \(\omega(t)\).
124 The needed \(N\)-th root exists in \(\mathbb C^\times\). The compact
125 type stays fixed. All choices precede H5. See
126 \path{central/DYADIC_FULL_CHARACTER167.md} for the complete proof.
127 
128 \hypertarget{dyadic_synthesis-exact-packet-capacity-and-marked-attachment}{%
129 \subsection{4. Exact packet capacity and marked
130 attachment}\label{dyadic_synthesis-exact-packet-capacity-and-marked-attachment}}
131 
132 The dyadic floors cannot be replaced by the odd-prime dimension formula.
133 At stage \(j\), the centralizer grade has dimension \(2^{d-j}\). Its
134 contribution to \(J^1/H^1\) is \[
135 \bigl(2^{d-j+1}-2^{d-j}\bigr)\,\mathbf1_{n_j\text{ even}}.
136 \] Every \(n_j\) for \(j<d\) is even; \(n_d=q_1\) is odd and contributes
137 zero. The sum is exactly \(N-2\). BK nondegeneracy makes this a
138 symplectic space. A whole source packet of \(d\) planes fits if and only
139 if \(N-2\ge2d\), equivalently \(d\ge3\).
140 
141 \begin{longtable}[]{@{}
142   >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
143   >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
144   >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
145   >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
146   >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}@{}}
147 \toprule\noalign{}
148 \begin{minipage}[b]{\linewidth}\raggedright
149 Ambient rank
150 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
151 Host quotient dimension
152 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
153 Host Heisenberg carrier
154 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
155 Required source carrier
156 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
157 Outcome
158 \end{minipage} \\
159 \midrule\noalign{}
160 \endhead
161 \bottomrule\noalign{}
162 \endlastfoot
163 2 & 0 & 1 & 2 & Insufficient \\
164 4 & 2 & 2 & 4 & Insufficient \\
165 8 & 6 & 8 & 8 & Exact capacity \\
166 16 & 14 & 128 & 16 & Spare capacity \\
167 \end{longtable}
168 
169 Now assume \(d\ge3\), the one-vertex contract, declared \(C=\{1\}\), and
170 actually supplied full character extensions with labelled whole-plane
171 quotients. Retain their complete dyadic finite central-extension
172 markings, including their specified central embeddings, lifts, square
173 values, refinements and phases. Any central-character kernel already
174 factored out must be recorded; a scalar pushout cannot recover invisible
175 kernel elements.
176 
177 Under the scalar-pushout realization contract, with no independently
178 fixed conflicting host marking, choose a labelled symplectic embedding
179 into \(W\). Matching commutators gives a center-fixing isomorphism
180 between the source \(\mathbb C^\times\) extension and the corresponding
181 host restriction. \textbf{Retain the image of the whole marked finite
182 subgroup and every lift.} For this isomorphism \(F_0\), \[
183 F_0(a)^2=F_0(a^2).
184 \] Thus its square bits, refinement and Gauss sign survive, along with
185 all supplied full character values. They are not inferred from the
186 unmarked pushout. Stone--von Neumann gives the source factor inside
187 \(\eta\); a complementary line gives an actual rank-\(N\) image-algebra
188 projector. The retained induction/recovery idempotent recovers the same
189 \(\Lambda\) and the same \(\pi\).
190 
191 This gives an actual represented marked finite subgroup in the scalar
192 extension of the actual type. It permits complex scalars times type
193 operators, as the declared contract does. It does not assert literal
194 inclusion of old matrix groups, numerical depth placement, or
195 identification with a separately fixed canonical native finite subgroup.
196 
197 If a genuine host-native quadratic refinement and its finite
198 \(\mu_2\)-central extension are independently supplied, quadratic
199 embedding has a sharper condition. With otherwise free embeddings, at
200 rank eight the equal-dimensional source and host must have equal Arf
201 invariants. For \(d\ge4\), the host has at least one spare symplectic
202 plane, so either source Arf type embeds. If the original embedding is
203 preassigned, it must instead satisfy the pointwise equation
204 \(q_W(f(v))=q_{\rm src}(v)\). These conclusions require the actual host
205 marking as input; dimension does not manufacture it.
206 
207 The full finite proofs are in \path{markings/DYADIC_MARKINGS167.md}.
208 In particular, the D8 and Q8 unmarked complex pushouts coincide while
209 their whole marked finite \(\mu_2\)-central extensions differ. Merely
210 recording the central \(\{\pm1\}\) subgroup would lose that distinction.
211 
212 \hypertarget{dyadic_synthesis-failed-repairs-and-boundaries}{%
213 \subsection{5. Failed repairs and
214 boundaries}\label{dyadic_synthesis-failed-repairs-and-boundaries}}
215 
216 The rank-two and rank-four carrier failures cannot be repaired by
217 changing the maximal simple parameter while keeping the specified Swan
218 exponent and torsion number. At rank two, odd Swan forces a minimal
219 odd-level stratum and \(J^1=H^1\). At rank four, Swan \(6h_2\) with
220 \(h_2\) odd forces the degree pattern \(4\to2\); the maximal quadratic
221 derived field forces an odd lower critical level. Every such
222 maximal-type carrier has dimension two, below the required four. Scalar
223 twists preserve this capacity.
224 
225 This is an obstruction to the whole-source sector inside that
226 maximal-type carrier. It is not a counterexample to the original theorem
227 or a proof that a different arithmetic carrier in the same
228 representation cannot work. Section 7 constructs such carriers and
229 proves their exact recovery.
230 
231 At rank eight, numerical profiles alone also fail to determine capacity.
232 \path{capacity/DYADIC_CAPACITY167.md}, Section 6, constructs an
233 actual skipped-degree \(8\to2\) host with the same Swan, torsion and
234 self-pair conductor but quotient dimension four rather than six. Our
235 successful result uses the explicit full tower, rather than inferring
236 its profile from invariants.
237 
238 Thus the completed dyadic scope is:
239 
240 \begin{itemize}
241 \tightlist
242 \item
243   Actual fields, full characters and central completion in every rank
244   \(2^d\).
245 \item
246   Full marked scalar-pushout packet realization for the stated family
247   when \(d\ge3\), with \(C=1\), one vertex, and transported markings
248   retained.
249 \item
250   A conditional quadratic embedding into an independently supplied
251   native marked host, with the exact Arf or pointwise condition above.
252 \end{itemize}
253 
254 Section 7 constructs different actual arithmetic carriers at the two low
255 ranks and recovers the same candidate from them. The maximal-type
256 capacity failure above remains true. Original H5 stays the original
257 same-candidate conditional premise; these conductor identities and
258 noncompact root choices do not prove it.
259 
260 \hypertarget{dyadic_synthesis-evidence}{%
261 \subsection{6. Evidence}\label{dyadic_synthesis-evidence}}
262 
263 The arbitrary-length field and full-character arguments are ordinary
264 proofs. The exact rank-eight run for \(h=(1,3,5)\) checks actual matrix
265 identities, normalized lifts, leading and adjacent-correction grades
266 \(-60,-18,-3\), native traces one and ambient traces \(0,0,1\). It
267 outputs a matrix with 25 nonzero Laurent-polynomial entries. The
268 corresponding critical exponents are proved symbolically. The run does
269 not prove simplicity, full-character existence, or universal marked
270 realization. Those conclusions use the separate proofs and their stated
271 scope. No Lean build was run.
272 
273 The base synthesis before the low-rank continuation is preserved in
274 \path{history/PROOF167_BEFORE_ENLARGEMENT.md}, together with its
275 review. Its previously open enlarged-carrier route is executed next.
276 
277 \hypertarget{dyadic_synthesis-the-low-rank-repair-enlarge-the-actual-inducing-carrier}{%
278 \subsection{7. The low-rank repair: enlarge the actual inducing
279 carrier}\label{dyadic_synthesis-the-low-rank-repair-enlarge-the-actual-inducing-carrier}}
280 
281 The maximal-type carrier is not the only arithmetic carrier available in
282 the constructed representation. The following constructions keep the
283 original \(\beta,\theta,\Lambda\) and \(\pi\) fixed.
284 
285 \hypertarget{dyadic_synthesis-rank-two}{%
286 \subsubsection{Rank two}\label{dyadic_synthesis-rank-two}}
287 
288 Write the odd minimal level as \(q=2m+1\), with \(m\ge1\), and let
289 \(J=H^1=J^1=U_K^1U_A^{m+1}\). Define \[
290 P=U_K^1U_A^m.
291 \] This is an actual compact open group, with \(J\triangleleft P\),
292 \([P:J]=2\) and \(P\cap(K^\times J)=J\). For \(g\in P\setminus J\), the
293 actual intertwining theorem makes \(\lambda=\theta^g/\theta\)
294 nontrivial. The commutator \([P,J]\subset U_A^{m+1}\) lies in the forced
295 additive-character domain, so \(\lambda\) has values \(\{\pm1\}\). Put
296 \(H_0=\ker\lambda\). Then \[
297 P/H_0\simeq\mathbb F_2^2,\qquad
298 E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
299 \] The actual represented generators are a diagonal sign and an
300 off-diagonal matrix that anticommute. Thus their commutator form is
301 nondegenerate, and \(\theta|_{H_0}\) is the full character of an actual
302 new arithmetic Heisenberg pair. No abstract operators create that group
303 action.
304 
305 The identity compact-induction double coset realizes \(E\) inside the
306 same \(\pi\), as functions supported on \((K^\times J)P\). The complete
307 group and recovery proof is
308 \path{lowrank/RANK2_ENLARGED_CARRIER167.md}.
309 
310 \hypertarget{dyadic_synthesis-rank-four}{%
311 \subsubsection{Rank four}\label{dyadic_synthesis-rank-four}}
312 
313 Use the actual rank-four outer approximation field \(E_0/F\) of degree
314 two and its centralizer \(B=\operatorname{End}_{E_0}(F^4)\). The odd
315 lower critical level is \(r=3h_1=2m+1\); the half leading level is
316 \(a=3h_2>r\). Define \[
317 P=J\,U_B^m,
318 \] where \(J=J^1\) is the actual compact type group. The filtered
319 centralizer estimate for the actual final field, without assuming
320 \(E_0\subset K\), proves that \(P\) is a compact group with \(J\) normal
321 of index two. For a concrete new coset representative \(u\),
322 \(\lambda=\theta^u/\theta\) on the actual \(H=H^1\) is nontrivial with
323 values \(\{\pm1\}\). Put \(H_0=\ker\lambda\subset H\). The calculation
324 proves \[
325 P/H_0\simeq\mathbb F_2^4,\qquad
326 E=\operatorname{Ind}_J^P\eta\text{ is irreducible of dimension }4.
327 \] The new represented commutators are scalar, its quotient has exponent
328 two, and nondegeneracy follows from its exact scalar preimage and
329 irreducibility. The finite scalar center may be larger than \(\mu_2\); a
330 literal extraspecial quotient is not asserted. The full character values
331 are retained by the actual finite Heisenberg extension. Again \(E\)
332 occurs inside the same \(\pi\) through the identity double coset. See
333 \path{lowrank/RANK4_ENLARGED_CARRIER167.md}.
334 
335 \hypertarget{dyadic_synthesis-exact-recovery-with-the-noncompact-normalization-retained}{%
336 \subsubsection{Exact recovery, with the noncompact normalization
337 retained}\label{dyadic_synthesis-exact-recovery-with-the-noncompact-normalization-retained}}
338 
339 At both low ranks, \(K^\times\) normalizes the new \(P\). In rank two
340 this follows directly from normalization of \(A\) and its radical. In
341 rank four, a \(K\)-uniformizer is approximated by the derived chain
342 shift \(R\) times a unit of \(B\), with relative error in \(P_A^{n-r}\).
343 Its conjugation error on \(U_B^m\) lies in \(U_A^{n-r+m}\subset J\),
344 which proves normalization. The exact argument is
345 \path{lowrank/RANK4_INDUCTION_BY_STAGES167.md}.
346 
347 Consequently \(\mathcal M=K^\times P\) is an actual open group compact
348 modulo the center, \(\mathcal N=K^\times J\) is normal of index two in
349 \(\mathcal M\), and \[
350 \Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda,
351 \qquad \Sigma|_P\simeq E,
352 \qquad
353 \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
354 \] Restriction to \(P\) is irreducible, so \(\Sigma\) is irreducible.
355 Induction in stages proves the last equality for the original
356 \(\Lambda\), including its fixed central character and noncompact root.
357 This is a genuine enlarged inducing representation; it changes neither
358 the candidate nor its original canonical beta-extension data.
359 
360 There is also a recovery from compact data explicitly recording that
361 root. With \(A_0=\Lambda(\varpi_K)\) and
362 \(X=\operatorname{c\!-\!Ind}_J^G\eta\), define
363 \((Tf)(g)=A_0^{-1}f(\varpi_Kg)\). Then \(X/(T-1)X\simeq\pi\). After
364 imposing the fixed central character, \(\mathcal N/(F^\times J)\) is
365 cyclic of order \(N\), and the finite average
366 \(N^{-1}\sum_{i=0}^{N-1}T^i\) selects this same extension. These
367 operators act on complex representation spaces, so their scalar division
368 by \(N\) is valid. This is a different recovery from the Block159
369 projector onto a sector of \(\eta\). The exact kernel and finite
370 projector are proved in
371 \path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.
372 
373 \hypertarget{dyadic_synthesis-what-the-low-rank-repair-admits}{%
374 \subsubsection{What the low-rank repair
375 admits}\label{dyadic_synthesis-what-the-low-rank-repair-admits}}
376 
377 Apply the marked scalar-pushout theorem of Section 4 to the
378 \textbf{actual new} pair \((H_0,P)\) and its carrier \(E\). It has
379 exactly the capacity of the whole source at each low rank. All full
380 source character values and the transported whole finite subgroup,
381 lifts, squares and phase markings can be attached and retained. The
382 displayed \(\Sigma\) and induction in stages recover the same candidate.
383 
384 The old H2 break targets remain their supplied actual simple-character
385 quotients. The enlarged pair is their common arithmetic assembly
386 carrier; it is not renamed as the original maximal-simple \(H^1,J^1\)
387 pair. If a stronger reading requires the whole packet itself to be the
388 quotient of a maximal simple character in the same candidate, the
389 earlier dimension obstruction still applies. Likewise, agreement with an
390 independently fixed native quadratic marking must be checked separately.
391 The contract note proves this distinction explicitly.
392 
393 \hypertarget{dyadic_synthesis-final-scope-and-next-construction-question}{%
394 \subsection{8. Final scope and next construction
395 question}\label{dyadic_synthesis-final-scope-and-next-construction-question}}
396 
397 The weighted dyadic mechanism now gives actual arithmetic fields, full
398 characters and tame central completion in every rank \(2^d\) of the
399 stated increasing odd-break family. Under the one-vertex, \(C=1\), full
400 marked scalar-pushout \textbf{assembly} contract, the packet carrier can
401 be realized at every length: inside the original type for \(d\ge3\), and
402 in the explicit enlarged inducing carrier at \(d=1,2\). The source's
403 prescribed central character must belong to the covered tame class. In
404 particular no arbitrary single-factor wild determinant is assumed to be
405 trivial.
406 
407 Remaining universal obligations are construction/comparison of
408 independently prescribed canonical native markings, nontrivial
409 symplectic stabilizer actions, general source/transport diagrams and any
410 required numerical layer placement. Original H5 remains conditional for
411 this same fixed candidate. No equality of profiles or choice among
412 noncompact roots is promoted to H5.
413 
414 The next concrete target is equivariance of the arithmetic attachment
415 for a nontrivial prescribed symplectic action. Another matrix search
416 inside the old low-rank maximal-type carrier would repeat its already
417 proved obstruction. \path{STATUS167.json} and \path{ROUTES167.md}
418 record the achieved scopes and open routes.

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