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

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