Realization Calculus

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1 \clearpage
2 \section{The actual enlarged rank-four inducing carrier}\label{app:rank4}
3 
4 
5 \hypertarget{rank4-result-and-boundary}{%
6 \subsection{Result and boundary}\label{rank4-result-and-boundary}}
7 
8 For the actual rank-four Artin--Schreier candidate of Block167, one can
9 enlarge the compact group J1 to an actual compact subgroup P with
10 
11 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
12 J1 normal in P,  [P:J1]=2,
13 E=Ind_(J1)^P(eta) irreducible of dimension 4.
14 \end{Verbatim}
15 
16 The representation E occurs inside the restriction of the same actual
17 supercuspidal candidate pi. There is an explicit normal subgroup H0 of P
18 with {[}P:H0{]}=16 such that P/H0 is elementary abelian and the
19 commutator form supplied by the actual character on H0 is nondegenerate.
20 Thus the carrier has an actual four-dimensional binary symplectic
21 quotient. No abstract transfer of matrix operators is used to create
22 this action.
23 
24 The finite image of P has scalar center theta(H0), which can be larger
25 than \{+1,-1\}. It is literally an extraspecial group if this scalar
26 center has order two. That additional assertion is not assumed or proved
27 here. The construction supplies an actual finite Heisenberg central
28 extension with nondegenerate projective quotient F\_2\^{}4. Matching an
29 independently prescribed quadratic phase is also a separate assertion.
30 
31 \hypertarget{rank4-actual-arithmetic-groups-and-notation}{%
32 \subsection{1. Actual arithmetic groups and
33 notation}\label{rank4-actual-arithmetic-groups-and-notation}}
34 
35 Let F=F\_2((t)), and take the audited rank-four constructor with odd
36 0\textless h\_1\textless h\_2. Write
37 
38 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
39 q_i=3h_i,  n=2q_2,  a=n/2=q_2,
40 r=q_1=2m+1,  m=(r-1)/2>=1.
41 \end{Verbatim}
42 
43 We have a\textgreater r, so n\textgreater2r. Let A be the complete
44 period-four hereditary order, with radical P\_A. Let gamma be the actual
45 degree-two outer minimal approximation, E=F{[}gamma{]},
46 B=End\_E(F\^{}4)=M\_2(E), B0=A cap B, and Q=rad(B0). The actual final
47 parameter is beta=gamma+delta, and
48 
49 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
50 s(delta)=c,
51 \end{Verbatim}
52 
53 where c is the minimal degree-two Artin--Schreier parameter over E, of
54 B0-level r. The final field K=F{[}beta{]} is totally ramified of degree four. No embedding E subset K is asserted.
55 
56 Use \(U_A^j=1+P_A^j\) and \(U_B^j=1+Q^j\).
57 BK 3.1.7--9 give the actual groups
58 
59 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
60 H=H1(beta)=U_K^1 U_B^(m+1) U_A^(a+1),
61 J=J1(beta)=U_K^1 U_B^(m+1) U_A^a.
62 \end{Verbatim}
63 
64 In particular J/H has dimension two over F\_2. Let theta be the actual
65 full simple character constructed for beta, and let eta be its unique
66 Heisenberg representation on J, of dimension two. The argument below
67 works with this supplied actual theta. It does not replace its values.
68 
69 Choose the outer minimal simple character theta\_gamma from its defining
70 sequence. On H cap U\_A\^{}(m+1), the recursion says
71 
72 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
73 theta=theta_gamma psi_delta.
74 \end{Verbatim}
75 
76 Moreover H is contained in H1(gamma)=U\_B\^{}1 U\_A\^{}(a+1). The
77 character theta\_gamma is invariant under U\_B\^{}1: on U\_B\^{}1 it
78 factors through the E-determinant, and on U\_A\^{}(a+1) it is the
79 gamma-additive character.
80 
81 \hypertarget{rank4-the-actual-final-field-has-the-needed-graded-line}{%
82 \subsection{2. The actual final field has the needed graded
83 line}\label{rank4-the-actual-final-field-has-the-needed-graded-line}}
84 
85 For every integer j\textgreater=1 and k in K cap P\_A\^{}j, the
86 minimal-approximation commutator estimate gives
87 
88 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
89 k=b+e,  b in Q^j,  e in P_A^(j+n-r).              (1)
90 \end{Verbatim}
91 
92 Indeed {[}gamma,k{]}=-{[}delta,k{]} belongs to P\_A\^{}(j-r), and the
93 minimal corestriction exact sequence for gamma solves its commutator
94 with an error of grade j+n-r. The remaining difference centralizes gamma
95 and therefore belongs to B. This is the usual minimal case of BK 1.4.10.
96 
97 Applying s to {[}beta,k{]}=0 and using (1) gives
98 
99 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
100 [c,b] in Q^(j+n-2r).
101 \end{Verbatim}
102 
103 Hence the leading B-grade of b commutes with the leading grade of c. Let
104 R be the complete period-two E-chain shift, with R\^{}2=z I, where z is
105 the chosen uniformizer of E. In the constructor's basis
106 
107 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
108 c congruent to R^(-r) modulo Q^(1-r).
109 \end{Verbatim}
110 
111 Because r is odd, the centralizing subspace on every B-grade j is the
112 one-dimensional line spanned by R\^{}j. On the other hand K is a totally
113 ramified maximal field normalizing the complete A-chain, so (K cap
114 P\_A\^{}j)/(K cap P\_A\^{}(j+1)) has dimension one over F\_2. Estimate
115 (1) is injective on the leading grade, so its image is exactly that
116 centralizing line.
117 
118 This proves the required graded assertion for the actual K, without
119 replacing it by the derived model E{[}c{]} or assuming nested fields.
120 
121 \hypertarget{rank4-normality-and-index-of-the-enlarged-group}{%
122 \subsection{3. Normality and index of the enlarged
123 group}\label{rank4-normality-and-index-of-the-enlarged-group}}
124 
125 Set
126 
127 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
128 P=J U_B^m.
129 \end{Verbatim}
130 
131 We first prove this is a group. The group U\_B\^{}m normalizes
132 U\_B\^{}(m+1) and every U\_A\^{}j. For k in U\_K\^{}1, (1) approximates
133 k by an element of U\_B\^{}1 with error in U\_A\^{}(n-r+1). Therefore
134 
135 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
136 [U_B^m,U_K^1]
137    subset U_B^(m+1) U_A^(m+n-r+1)
138    subset H,
139 \end{Verbatim}
140 
141 where the last inclusion follows from a\textgreater r=2m+1. Thus
142 U\_B\^{}m normalizes both H and J, and P is an actual compact subgroup
143 with J normal in P.
144 
145 The quotient \(U_B^m/U_B^{m+1}\) has dimension two. By
146 Section~2 the image of J cap U\_B\^{}m on this grade is exactly its
147 centralizing line, of dimension one. For the converse inclusion needed
148 here, the B-approximation of a K-element of grade m differs from it by
149 grade m+n-r\textgreater a. That error lies in U\_A\^{}a subset J. Thus
150 the whole centralizing line really occurs in the intersection, not only
151 in an abstract associated graded space. Consequently
152 
153 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
154 [P:J]=[U_B^m:J cap U_B^m]=2.                    (2)
155 \end{Verbatim}
156 
157 For a concrete representative choose D=diag(1,0) in B and put
158 
159 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
160 x=R^m D,  u=1+x.
161 \end{Verbatim}
162 
163 Its leading grade is outside the centralizing line, so u represents the
164 nontrivial coset of P/J. Since \(u^2=1+x^2\) in
165 characteristic two, we have u\^{}2 in U\_B\^{}(2m) subset U\_B\^{}(m+1)
166 subset H.
167 
168 \hypertarget{rank4-the-actual-character-has-an-orbit-of-length-two}{%
169 \subsection{4. The actual character has an orbit of length
170 two}\label{rank4-the-actual-character-has-an-orbit-of-length-two}}
171 
172 Define the character of H
173 
174 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
175 lambda(h)=theta(u h u^(-1))/theta(h).
176 \end{Verbatim}
177 
178 For h in H the preceding normality argument gives {[}u,h{]} in H cap
179 U\_A\^{}(m+1). Since theta\_gamma is invariant under u,
180 
181 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
182 lambda(h)=psi_delta([u,h]) in {+1,-1}.          (3)
183 \end{Verbatim}
184 
185 Thus lambda has order dividing two, and its values are actual values of
186 the supplied simple character. It is J-invariant, since both theta and
187 its u-conjugate are J-invariant.
188 
189 It is nontrivial. Take y=R\^{}(m+1)D and h=1+y in U\_B\^{}(m+1). The
190 leading commutator in (3), after corestriction, has value
191 
192 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
193 psi_E(tr_(M_2(E)/E)(c[x,y]))=-1.
194 \end{Verbatim}
195 
196 Here the commutator remainder has B-grade at least 3m+1, which becomes
197 positive after multiplication by c of grade -2m-1. On the leading
198 grades, writing X=R\^{}m diag(A,B) and Y=R\^{}(m+1)diag(C,D), one
199 obtains
200 
201 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
202 tr(R^(-2m-1)[X,Y])=(A+B)(C+D) in F_2.
203 \end{Verbatim}
204 
205 Both sums equal one for the chosen D=diag(1,0). The normalized psi\_E is
206 nontrivial on the residue field and trivial on its maximal ideal. For
207 this AS normalization one can check the sign directly from
208 psi\_E(1)=psi\_F(Tr\_(E/F)(alpha))=psi\_F(1)=-1.
209 
210 Hence theta\^{}u differs from theta, eta\^{}u differs from eta, and
211 Clifford theory applied to (2) gives
212 
213 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
214 E_carrier=Ind_J^P(eta) irreducible, dim(E_carrier)=4.       (4)
215 \end{Verbatim}
216 
217 \hypertarget{rank4-an-actual-nondegenerate-symplectic-quotient-of-dimension-four}{%
218 \subsection{5. An actual nondegenerate symplectic quotient of dimension
219 four}\label{rank4-an-actual-nondegenerate-symplectic-quotient-of-dimension-four}}
220 
221 Put H0=ker(lambda) inside H, and theta0=theta restricted to H0. In the
222 two-coset induction model, h in H acts by
223 
224 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
225 diag(theta(h) I_2, theta^u(h) I_2).
226 \end{Verbatim}
227 
228 Thus it acts scalarly precisely when h belongs to H0. We check that H0
229 is the entire scalar preimage and that its quotient is symplectic.
230 
231 First, lambda is invariant under u. Indeed u\^{}2 belongs to J and theta
232 is J-invariant, so \(\lambda^u=\lambda^{-1}=\lambda\) by (3).
233 Consequently H0 is normal in P, and theta0 is P-invariant.
234 
235 Second, lambda is trivial on U\_A\^{}(a+1). There the correction formula
236 is valid, gamma commutes with u, and the delta contribution to a
237 commutator has grade at least a+1+m-r=a-m\textgreater0. Therefore it
238 lies in the kernel of the additive character.
239 
240 Write J=H U\_A\^{}a. Since lambda is a character and J-invariant, it
241 kills {[}H,H{]} and {[}H,U\_A\^{}a{]}. It also kills
242 \([U_A^a,U_A^a]\), which lies in U\_A\^{}(2a). Hence
243 lambda({[}J,J{]})=1. It kills every square in J as well: squares of H
244 have lambda-value one, squares of U\_A\^{}a lie in U\_A\^{}(2a), and the
245 mixed commutator contributions have just been handled.
246 
247 Finally lambda({[}u,H{]})=1 by u-invariance, and {[}u,U\_A\^{}a{]}
248 belongs to U\_A\^{}(a+m) subset U\_A\^{}(a+1), so lambda({[}u,J{]})=1.
249 Also lambda(u\^{}2)=1 because u commutes with u\^{}2. It follows that
250 
251 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
252 [P,P] subset H0,  g^2 in H0 for every g in P,
253 [P:H0]=[P:J][J:H][H:H0]=2*4*2=16.
254 \end{Verbatim}
255 
256 Therefore P/H0 is an actual elementary abelian group of dimension four.
257 If an element of J acts scalarly on the carrier, its first eta-block is
258 scalar, so it belongs to H by the original nondegenerate Heisenberg
259 quotient J/H. It then belongs to H0. An element outside J exchanges the
260 two coset summands and cannot act scalarly. Thus H0 is precisely the
261 scalar preimage.
262 
263 The pairing
264 
265 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
266 (gH0,kH0) |-> theta0([g,k])
267 \end{Verbatim}
268 
269 is alternating and well defined. Since the projective quotient has
270 exponent two, its values lie in \{+1,-1\}. It is nondegenerate: an
271 element in its radical commutes with the entire irreducible carrier
272 image and therefore acts scalarly by Schur's lemma, forcing it into H0.
273 
274 This is the actual four-dimensional binary Heisenberg quotient sought
275 for capacity. Its operators are supplied by elements of P.
276 
277 \hypertarget{rank4-the-carrier-occurs-in-the-same-supercuspidal-candidate}{%
278 \subsection{6. The carrier occurs in the same supercuspidal
279 candidate}\label{rank4-the-carrier-occurs-in-the-same-supercuspidal-candidate}}
280 
281 Let pi=c-Ind\_N\^{}(GL\_4(F))(Lambda) be the existing actual candidate,
282 with N=K\^{}times J. The residue field is F\_2, so the compact extended
283 type restricts to eta on J. Since P is contained in U\_A\^{}1,
284 
285 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
286 N cap P=J.
287 \end{Verbatim}
288 
289 Indeed an element k j in P with k in K\^{}times has k in U\_A\^{}1,
290 hence k in U\_K\^{}1 subset J. The identity double-coset term in the
291 restriction of compact induction to P is therefore
292 
293 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
294 Ind_J^P(eta)=E_carrier.
295 \end{Verbatim}
296 
297 It is realized concretely by the functions supported on N P. Thus the
298 four-dimensional carrier occurs in the same pi, with its already fixed
299 full type character and noncompact extension scalar. No new arithmetic
300 candidate, beta extension, or inverse correspondence is used.
301 
302 \hypertarget{rank4-exact-limits-finite-image-versus-literal-extraspecial-group}{%
303 \subsection{7. Exact limits: finite image versus literal extraspecial
304 group}\label{rank4-exact-limits-finite-image-versus-literal-extraspecial-group}}
305 
306 The actual finite image is P/ker(theta0), because H0 is the scalar
307 preimage. Its center is the scalar cyclic 2-group theta0(H0), and its
308 commutator subgroup has order two. Its quotient by its center is
309 F\_2\^{}4.
310 
311 If theta0(H0) has order two, this finite image is extraspecial of order
312 32. If its scalar center is larger, quotienting arbitrary central
313 elements need not preserve its commutator: in a cyclic 2-group the
314 order-two subgroup can lie inside the squares. Thus a literal
315 extraspecial quotient is not justified solely by the calculations above.
316 
317 The stronger statement proved is the one with full actual character: an
318 actual compact subgroup, actual induced carrier, actual scalar subgroup
319 and nondegenerate actual commutator pairing. Passing to the usual full
320 scalar pushout retains all theta0-values. It does not prove that an
321 independently supplied quadratic refinement, stabilizer action, or
322 source phase agrees with this carrier. Those requirements must be
323 checked separately. No H5 or universal claim~66 closure is asserted.
324 
325 \hypertarget{rank4-provenance}{%
326 \subsection{Provenance}\label{rank4-provenance}}
327 
328 Uses the audited dyadic parameter/character construction, BK 1.4.10,
329 BK 3.1.7--9, and the defining-sequence character formula. The source
330 normalization and all field/chain data are the actual ones in
331 \path{dyadic167/tower/DYADIC_AS_TOWER167.md}. This note changes neither that
332 proof nor its recorded exact computation. Clifford induction and the
333 identity compact-induction double-coset term are used in their ordinary
334 finite-index forms, with hypotheses proved above.

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