Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{The actual enlarged rank-four inducing carrier}\label{app:rank4}
3 
4 Date: 2026-10-05. Ordinary proof submitted for independent review.
5 
6 \hypertarget{rank4-result-and-boundary}{%
7 \subsection{Result and boundary}\label{rank4-result-and-boundary}}
8 
9 For the actual rank-four Artin--Schreier candidate of Block167, one can
10 enlarge the compact group J1 to an actual compact subgroup P with
11 
12 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
13 J1 normal in P,  [P:J1]=2,
14 E=Ind_(J1)^P(eta) irreducible of dimension 4.
15 \end{Verbatim}
16 
17 The representation E occurs inside the restriction of the same actual
18 supercuspidal candidate pi. There is an explicit normal subgroup H0 of P
19 with {[}P:H0{]}=16 such that P/H0 is elementary abelian and the
20 commutator form supplied by the actual character on H0 is nondegenerate.
21 Thus the carrier has an actual four-dimensional binary symplectic
22 quotient. No abstract transfer of matrix operators is used to create
23 this action.
24 
25 The finite image of P has scalar center theta(H0), which can be larger
26 than \{+1,-1\}. It is literally an extraspecial group if this scalar
27 center has order two. That additional assertion is not assumed or proved
28 here. The construction supplies an actual finite Heisenberg central
29 extension with nondegenerate projective quotient F\_2\^{}4. Matching an
30 independently prescribed quadratic phase is also a separate assertion.
31 
32 \hypertarget{rank4-actual-arithmetic-groups-and-notation}{%
33 \subsection{1. Actual arithmetic groups and
34 notation}\label{rank4-actual-arithmetic-groups-and-notation}}
35 
36 Let F=F\_2((t)), and take the audited rank-four constructor with odd
37 0\textless h\_1\textless h\_2. Write
38 
39 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
40 q_i=3h_i,  n=2q_2,  a=n/2=q_2,
41 r=q_1=2m+1,  m=(r-1)/2>=1.
42 \end{Verbatim}
43 
44 We have a\textgreater r, so n\textgreater2r. Let A be the complete
45 period-four hereditary order, with radical P\_A. Let gamma be the actual
46 degree-two outer minimal approximation, E=F{[}gamma{]},
47 B=End\_E(F\^{}4)=M\_2(E), B0=A cap B, and Q=rad(B0). The actual final
48 parameter is beta=gamma+delta, and
49 
50 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
51 s(delta)=c,
52 \end{Verbatim}
53 
54 where c is the minimal degree-two Artin--Schreier parameter over E, of
55 B0-level r. The final field K=F{[}beta{]} is totally ramified of degree
56 four; no embedding E subset K is asserted.
57 
58 Use \(U_A^j=1+P_A^j\) and \(U_B^j=1+Q^j\).
59 BK3.1.7--9 give the actual groups
60 
61 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
62 H=H1(beta)=U_K^1 U_B^(m+1) U_A^(a+1),
63 J=J1(beta)=U_K^1 U_B^(m+1) U_A^a.
64 \end{Verbatim}
65 
66 In particular J/H has dimension two over F\_2. Let theta be the actual
67 full simple character constructed for beta, and let eta be its unique
68 Heisenberg representation on J, of dimension two. The argument below
69 works with this supplied actual theta; it does not replace its values.
70 
71 Choose the outer minimal simple character theta\_gamma from its defining
72 sequence. On H cap U\_A\^{}(m+1), the recursion says
73 
74 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
75 theta=theta_gamma psi_delta.
76 \end{Verbatim}
77 
78 Moreover H is contained in H1(gamma)=U\_B\^{}1 U\_A\^{}(a+1). The
79 character theta\_gamma is invariant under U\_B\^{}1: on U\_B\^{}1 it
80 factors through the E-determinant, and on U\_A\^{}(a+1) it is the
81 gamma-additive character.
82 
83 \hypertarget{rank4-the-actual-final-field-has-the-needed-graded-line}{%
84 \subsection{2. The actual final field has the needed graded
85 line}\label{rank4-the-actual-final-field-has-the-needed-graded-line}}
86 
87 For every integer j\textgreater=1 and k in K cap P\_A\^{}j, the
88 minimal-approximation commutator estimate gives
89 
90 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
91 k=b+e,  b in Q^j,  e in P_A^(j+n-r).              (1)
92 \end{Verbatim}
93 
94 Indeed {[}gamma,k{]}=-{[}delta,k{]} belongs to P\_A\^{}(j-r), and the
95 minimal corestriction exact sequence for gamma solves its commutator
96 with an error of grade j+n-r. The remaining difference centralizes gamma
97 and therefore belongs to B. This is the usual minimal case of BK1.4.10.
98 
99 Applying s to {[}beta,k{]}=0 and using (1) gives
100 
101 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
102 [c,b] in Q^(j+n-2r).
103 \end{Verbatim}
104 
105 Hence the leading B-grade of b commutes with the leading grade of c. Let
106 R be the complete period-two E-chain shift, with R\^{}2=z I, where z is
107 the chosen uniformizer of E. In the constructor's basis
108 
109 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
110 c congruent to R^(-r) modulo Q^(1-r).
111 \end{Verbatim}
112 
113 Because r is odd, the centralizing subspace on every B-grade j is the
114 one-dimensional line spanned by R\^{}j. On the other hand K is a totally
115 ramified maximal field normalizing the complete A-chain, so (K cap
116 P\_A\^{}j)/(K cap P\_A\^{}(j+1)) has dimension one over F\_2. Estimate
117 (1) is injective on the leading grade, so its image is exactly that
118 centralizing line.
119 
120 This proves the required graded assertion for the actual K, without
121 replacing it by the derived model E{[}c{]} or assuming nested fields.
122 
123 \hypertarget{rank4-normality-and-index-of-the-enlarged-group}{%
124 \subsection{3. Normality and index of the enlarged
125 group}\label{rank4-normality-and-index-of-the-enlarged-group}}
126 
127 Set
128 
129 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
130 P=J U_B^m.
131 \end{Verbatim}
132 
133 We first prove this is a group. The group U\_B\^{}m normalizes
134 U\_B\^{}(m+1) and every U\_A\^{}j. For k in U\_K\^{}1, (1) approximates
135 k by an element of U\_B\^{}1 with error in U\_A\^{}(n-r+1). Therefore
136 
137 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
138 [U_B^m,U_K^1]
139    subset U_B^(m+1) U_A^(m+n-r+1)
140    subset H,
141 \end{Verbatim}
142 
143 where the last inclusion follows from a\textgreater r=2m+1. Thus
144 U\_B\^{}m normalizes both H and J, and P is an actual compact subgroup
145 with J normal in P.
146 
147 The quotient \(U_B^m/U_B^{m+1}\) has dimension two. By
148 Section2 the image of J cap U\_B\^{}m on this grade is exactly its
149 centralizing line, of dimension one. For the converse inclusion needed
150 here, the B-approximation of a K-element of grade m differs from it by
151 grade m+n-r\textgreater a; that error lies in U\_A\^{}a subset J. Thus
152 the whole centralizing line really occurs in the intersection, not only
153 in an abstract associated graded space. Consequently
154 
155 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
156 [P:J]=[U_B^m:J cap U_B^m]=2.                    (2)
157 \end{Verbatim}
158 
159 For a concrete representative choose D=diag(1,0) in B and put
160 
161 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
162 x=R^m D,  u=1+x.
163 \end{Verbatim}
164 
165 Its leading grade is outside the centralizing line, so u represents the
166 nontrivial coset of P/J. Since \(u^2=1+x^2\) in
167 characteristic two, we have u\^{}2 in U\_B\^{}(2m) subset U\_B\^{}(m+1)
168 subset H.
169 
170 \hypertarget{rank4-the-actual-character-has-an-orbit-of-length-two}{%
171 \subsection{4. The actual character has an orbit of length
172 two}\label{rank4-the-actual-character-has-an-orbit-of-length-two}}
173 
174 Define the character of H
175 
176 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
177 lambda(h)=theta(u h u^(-1))/theta(h).
178 \end{Verbatim}
179 
180 For h in H the preceding normality argument gives {[}u,h{]} in H cap
181 U\_A\^{}(m+1). Since theta\_gamma is invariant under u,
182 
183 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
184 lambda(h)=psi_delta([u,h]) in {+1,-1}.          (3)
185 \end{Verbatim}
186 
187 Thus lambda has order dividing two, and its values are actual values of
188 the supplied simple character. It is J-invariant, since both theta and
189 its u-conjugate are J-invariant.
190 
191 It is nontrivial. Take y=R\^{}(m+1)D and h=1+y in U\_B\^{}(m+1). The
192 leading commutator in (3), after corestriction, has value
193 
194 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
195 psi_E(tr_(M_2(E)/E)(c[x,y]))=-1.
196 \end{Verbatim}
197 
198 Here the commutator remainder has B-grade at least 3m+1, which becomes
199 positive after multiplication by c of grade -2m-1. On the leading
200 grades, writing X=R\^{}m diag(A,B) and Y=R\^{}(m+1)diag(C,D), one
201 obtains
202 
203 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
204 tr(R^(-2m-1)[X,Y])=(A+B)(C+D) in F_2.
205 \end{Verbatim}
206 
207 Both sums equal one for the chosen D=diag(1,0). The normalized psi\_E is
208 nontrivial on the residue field and trivial on its maximal ideal. For
209 this AS normalization one can check the sign directly from
210 psi\_E(1)=psi\_F(Tr\_(E/F)(alpha))=psi\_F(1)=-1.
211 
212 Hence theta\^{}u differs from theta, eta\^{}u differs from eta, and
213 Clifford theory applied to (2) gives
214 
215 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
216 E_carrier=Ind_J^P(eta) irreducible, dim(E_carrier)=4.       (4)
217 \end{Verbatim}
218 
219 \hypertarget{rank4-an-actual-nondegenerate-symplectic-quotient-of-dimension-four}{%
220 \subsection{5. An actual nondegenerate symplectic quotient of dimension
221 four}\label{rank4-an-actual-nondegenerate-symplectic-quotient-of-dimension-four}}
222 
223 Put H0=ker(lambda) inside H, and theta0=theta restricted to H0. In the
224 two-coset induction model, h in H acts by
225 
226 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
227 diag(theta(h) I_2, theta^u(h) I_2).
228 \end{Verbatim}
229 
230 Thus it acts scalarly precisely when h belongs to H0. We check that H0
231 is the entire scalar preimage and that its quotient is symplectic.
232 
233 First, lambda is invariant under u. Indeed u\^{}2 belongs to J and theta
234 is J-invariant, so \(\lambda^u=\lambda^{-1}=\lambda\) by (3).
235 Consequently H0 is normal in P, and theta0 is P-invariant.
236 
237 Second, lambda is trivial on U\_A\^{}(a+1). There the correction formula
238 is valid, gamma commutes with u, and the delta contribution to a
239 commutator has grade at least a+1+m-r=a-m\textgreater0. Therefore it
240 lies in the kernel of the additive character.
241 
242 Write J=H U\_A\^{}a. Since lambda is a character and J-invariant, it
243 kills {[}H,H{]} and {[}H,U\_A\^{}a{]}. It also kills
244 \([U_A^a,U_A^a]\), which lies in U\_A\^{}(2a). Hence
245 lambda({[}J,J{]})=1. It kills every square in J as well: squares of H
246 have lambda-value one, squares of U\_A\^{}a lie in U\_A\^{}(2a), and the
247 mixed commutator contributions have just been handled.
248 
249 Finally lambda({[}u,H{]})=1 by u-invariance, and {[}u,U\_A\^{}a{]}
250 belongs to U\_A\^{}(a+m) subset U\_A\^{}(a+1), so lambda({[}u,J{]})=1.
251 Also lambda(u\^{}2)=1 because u commutes with u\^{}2. It follows that
252 
253 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
254 [P,P] subset H0,  g^2 in H0 for every g in P,
255 [P:H0]=[P:J][J:H][H:H0]=2*4*2=16.
256 \end{Verbatim}
257 
258 Therefore P/H0 is an actual elementary abelian group of dimension four.
259 If an element of J acts scalarly on the carrier, its first eta-block is
260 scalar, so it belongs to H by the original nondegenerate Heisenberg
261 quotient J/H. It then belongs to H0. An element outside J exchanges the
262 two coset summands and cannot act scalarly. Thus H0 is precisely the
263 scalar preimage.
264 
265 The pairing
266 
267 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
268 (gH0,kH0) |-> theta0([g,k])
269 \end{Verbatim}
270 
271 is alternating and well defined. Since the projective quotient has
272 exponent two, its values lie in \{+1,-1\}. It is nondegenerate: an
273 element in its radical commutes with the entire irreducible carrier
274 image and therefore acts scalarly by Schur's lemma, forcing it into H0.
275 
276 This is the actual four-dimensional binary Heisenberg quotient sought
277 for capacity. Its operators are supplied by elements of P.
278 
279 \hypertarget{rank4-the-carrier-occurs-in-the-same-supercuspidal-candidate}{%
280 \subsection{6. The carrier occurs in the same supercuspidal
281 candidate}\label{rank4-the-carrier-occurs-in-the-same-supercuspidal-candidate}}
282 
283 Let pi=c-Ind\_N\^{}(GL\_4(F))(Lambda) be the existing actual candidate,
284 with N=K\^{}times J. The residue field is F\_2, so the compact extended
285 type restricts to eta on J. Since P is contained in U\_A\^{}1,
286 
287 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
288 N cap P=J.
289 \end{Verbatim}
290 
291 Indeed an element k j in P with k in K\^{}times has k in U\_A\^{}1,
292 hence k in U\_K\^{}1 subset J. The identity double-coset term in the
293 restriction of compact induction to P is therefore
294 
295 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
296 Ind_J^P(eta)=E_carrier.
297 \end{Verbatim}
298 
299 It is realized concretely by the functions supported on N P. Thus the
300 four-dimensional carrier occurs in the same pi, with its already fixed
301 full type character and noncompact extension scalar. No new arithmetic
302 candidate, beta extension, or inverse correspondence is used.
303 
304 \hypertarget{rank4-exact-limits-finite-image-versus-literal-extraspecial-group}{%
305 \subsection{7. Exact limits: finite image versus literal extraspecial
306 group}\label{rank4-exact-limits-finite-image-versus-literal-extraspecial-group}}
307 
308 The actual finite image is P/ker(theta0), because H0 is the scalar
309 preimage. Its center is the scalar cyclic 2-group theta0(H0), and its
310 commutator subgroup has order two. Its quotient by its center is
311 F\_2\^{}4.
312 
313 If theta0(H0) has order two, this finite image is extraspecial of order
314 32. If its scalar center is larger, quotienting arbitrary central
315 elements need not preserve its commutator: in a cyclic 2-group the
316 order-two subgroup can lie inside the squares. Thus a literal
317 extraspecial quotient is not justified solely by the calculations above.
318 
319 The stronger statement proved is the one with full actual character: an
320 actual compact subgroup, actual induced carrier, actual scalar subgroup
321 and nondegenerate actual commutator pairing. Passing to the usual full
322 scalar pushout retains all theta0-values. It does not prove that an
323 independently supplied quadratic refinement, stabilizer action, or
324 source phase agrees with this carrier. Those requirements must be
325 checked separately. No H5 or universal claim66 closure is asserted.
326 
327 \hypertarget{rank4-provenance}{%
328 \subsection{Provenance}\label{rank4-provenance}}
329 
330 Uses the audited dyadic parameter/character construction, BK1.4.10,
331 BK3.1.7--9, and the defining-sequence character formula. The source
332 normalization and all field/chain data are the actual ones in
333 \path{dyadic167/tower/DYADIC_AS_TOWER167.md}. This note changes neither that
334 proof nor its recorded exact computation. Clifford induction and the
335 identity compact-induction double-coset term are used in their ordinary
336 finite-index forms, with hypotheses proved above.

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