Realization Calculus

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1 \clearpage
2 \section{Rank-four normalization and induction in stages}\label{app:rank4_stages}
3 
4 This is a corollary of the enlarged-carrier construction in
5 Appendix~\ref{app:rank4}.
6 
7 Use that note's actual groups and notation:
8 
9 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
10 n=2a,  r=2m+1,  a>r,
11 H subset J=J1(beta) normal in P=J U_B^m,
12 [P:J]=2,  N=K^times J,
13 pi=c-Ind_N^G(Lambda),  Lambda|J=eta.
14 \end{Verbatim}
15 
16 The field K normalizes P. Consequently the following are actual groups
17 and actual smooth representations:
18 
19 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
20 M=K^times P,
21 Sigma=Ind_N^M(Lambda),
22 dim Sigma=4,
23 Sigma|P=Ind_J^P(eta),
24 pi is isomorphic to c-Ind_M^G(Sigma).
25 \end{Verbatim}
26 
27 The representation Sigma is irreducible, and M is compact modulo the
28 center of G. Thus the enlarged carrier has an exact arithmetic inducing
29 datum for the same candidate, not merely an occurrence in its
30 restriction.
31 
32 \hypertarget{rank4_stages-normalization-by-the-actual-parameter-field}{%
33 \subsection{1. Normalization by the actual parameter
34 field}\label{rank4_stages-normalization-by-the-actual-parameter-field}}
35 
36 The units O\_K\^{}times equal U\_K\^{}1 because the residue field is
37 F\_2. They belong to J and hence normalize P. It remains to check a
38 uniformizer k of K. This element has A-grade one and normalizes A.
39 
40 The approximation estimate proved for the actual field in Section~2 of the
41 carrier note gives
42 
43 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
44 k=b+e,  b in Q^1,  e in P_A^(n-r+1).
45 \end{Verbatim}
46 
47 The leading grade of b is the nonzero centralizing line of the primitive
48 derived parameter. Over F\_2 it is exactly the leading grade of R, the
49 period-two E-chain shift. Therefore
50 
51 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
52 b=R v,  v in 1+Q^1.
53 \end{Verbatim}
54 
55 In particular b is invertible and normalizes both the complete E-chain
56 and the complete interleaved F-chain. We can write
57 
58 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
59 k=b(1+epsilon),  epsilon in P_A^(n-r).
60 \end{Verbatim}
61 
62 For u in U\_B\^{}m, the usual filtration commutator bound gives
63 
64 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
65 (1+epsilon)u(1+epsilon)^(-1)u^(-1)
66    in U_A^(n-r+m).
67 \end{Verbatim}
68 
69 This group is contained in U\_A\^{}a subset J, because
70 
71 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
72 n-r+m=2a-m-1>a.
73 \end{Verbatim}
74 
75 Conjugation by b preserves both relevant filtrations. Since b normalizes
76 U\_B\^{}m, it follows that
77 
78 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
79 k U_B^m k^(-1) subset U_B^m J=P.
80 \end{Verbatim}
81 
82 Also kJk\^{}(-1)=J, as K\^{}times normalizes its actual simple-character
83 groups. Thus kPk\^{}(-1) is contained in P and contains J. Its index
84 over J is two, equal to {[}P:J{]}, so it equals P. This proves
85 normalization by k, its inverse, and hence all K\^{}times.
86 
87 No equality or inclusion between the fields E and K was used. The
88 element b belongs to the actual outer centralizer B and only
89 approximates k to the stated precision.
90 
91 \hypertarget{rank4_stages-the-two-inducing-groups-have-index-two}{%
92 \subsection{2. The two inducing groups have index
93 two}\label{rank4_stages-the-two-inducing-groups-have-index-two}}
94 
95 Since K\^{}times normalizes P, the product M=K\^{}times P is a group.
96 Both P/J and the conjugation action of K\^{}times on it are defined. P/J has order two, whose automorphism group is trivial. Therefore
97 
98 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
99 [K^times,P] subset J.
100 \end{Verbatim}
101 
102 Together with J normal in P and K\^{}times normalizing J, this proves
103 N=K\^{}times J normal in M. The intersection K\^{}times cap P is
104 U\_K\^{}1, which is contained in J, since P subset U\_A\^{}1.
105 Consequently
106 
107 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
108 [M:N]=[P:J]=2.
109 \end{Verbatim}
110 
111 The quotient \(K^\times/F^\times\) is compact because K/F is a
112 finite totally ramified extension, and P is compact. Hence M is compact
113 modulo F\^{}times. It is open in G because it contains the open subgroup
114 J.
115 
116 \hypertarget{rank4_stages-the-enlarged-inducing-representation-is-irreducible}{%
117 \subsection{3. The enlarged inducing representation is
118 irreducible}\label{rank4_stages-the-enlarged-inducing-representation-is-irreducible}}
119 
120 Let u in P represent its nontrivial coset modulo J. The carrier note
121 proved that theta\^{}u differs from theta, and therefore eta\^{}u
122 differs from eta. It follows that Lambda\^{}u differs from Lambda, since
123 their restrictions to J already differ. Applying finite-index Clifford
124 theory to N normal in M gives
125 
126 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
127 Sigma=Ind_N^M(Lambda) irreducible,
128 dim Sigma=2 dim Lambda=4.
129 \end{Verbatim}
130 
131 There is one N--P double coset in M, and N cap P=J. Thus restriction to
132 P gives the actual four-dimensional Heisenberg carrier:
133 
134 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
135 Sigma|P is isomorphic to Ind_J^P(eta).
136 \end{Verbatim}
137 
138 Its central character on F\^{}times is unchanged from Lambda, because
139 F\^{}times is central and lies in N.
140 
141 \hypertarget{rank4_stages-exact-recovery-by-induction-in-stages}{%
142 \subsection{4. Exact recovery by induction in
143 stages}\label{rank4_stages-exact-recovery-by-induction-in-stages}}
144 
145 Induction in stages for these open subgroups gives
146 
147 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
148 c-Ind_M^G(Sigma)
149    is isomorphic to c-Ind_M^G(Ind_N^M Lambda)
150    is isomorphic to c-Ind_N^G(Lambda)
151    =pi.
152 \end{Verbatim}
153 
154 The second induction is finite-index, so ordinary and compact induction
155 from N to M agree. This is an exact equality of the resulting
156 isomorphism classes and applies to the same fixed Lambda, including its
157 noncompact scalar choice. It introduces no new candidate and no H5
158 premise.
159 
160 \hypertarget{rank4_stages-scope}{%
161 \subsection{Scope}\label{rank4_stages-scope}}
162 
163 This establishes the arithmetic inducing datum and exact recovery for
164 the enlarged carrier. It leaves unchanged the distinction between the
165 original two-dimensional BK quotient J/H and the new four-dimensional
166 actual quotient P/H0. It also leaves unchanged the source-phase and
167 literal extraspecial-center limitations recorded in the carrier note.
168 
169 Dependency pin: \path{RANK4_ENLARGED_CARRIER167.md} SHA256
170 \path{1517e26138776fe0f32d74a174d309ec5dd6a0f572a979473d8af3d20b7a4155}.

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