Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Rank-four normalization and induction in stages}\label{app:rank4_stages}
3 
4 Date: 2026-10-05. Corollary to \path{RANK4_ENLARGED_CARRIER167.md}, whose
5 reviewed bytes are kept unchanged. Ordinary proof submitted for
6 independent audit.
7 
8 Use that note's actual groups and notation:
9 
10 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
11 n=2a,  r=2m+1,  a>r,
12 H subset J=J1(beta) normal in P=J U_B^m,
13 [P:J]=2,  N=K^times J,
14 pi=c-Ind_N^G(Lambda),  Lambda|J=eta.
15 \end{Verbatim}
16 
17 The field K normalizes P. Consequently the following are actual groups
18 and actual smooth representations:
19 
20 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
21 M=K^times P,
22 Sigma=Ind_N^M(Lambda),
23 dim Sigma=4,
24 Sigma|P=Ind_J^P(eta),
25 pi is isomorphic to c-Ind_M^G(Sigma).
26 \end{Verbatim}
27 
28 The representation Sigma is irreducible, and M is compact modulo the
29 center of G. Thus the enlarged carrier has an exact arithmetic inducing
30 datum for the same candidate, not merely an occurrence in its
31 restriction.
32 
33 \hypertarget{rank4_stages-normalization-by-the-actual-parameter-field}{%
34 \subsection{1. Normalization by the actual parameter
35 field}\label{rank4_stages-normalization-by-the-actual-parameter-field}}
36 
37 The units O\_K\^{}times equal U\_K\^{}1 because the residue field is
38 F\_2. They belong to J and hence normalize P. It remains to check a
39 uniformizer k of K. This element has A-grade one and normalizes A.
40 
41 The actual-field approximation estimate proved in Section2 of the
42 carrier note gives
43 
44 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
45 k=b+e,  b in Q^1,  e in P_A^(n-r+1).
46 \end{Verbatim}
47 
48 The leading grade of b is the nonzero centralizing line of the primitive
49 derived parameter. Over F\_2 it is exactly the leading grade of R, the
50 period-two E-chain shift. Therefore
51 
52 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
53 b=R v,  v in 1+Q^1.
54 \end{Verbatim}
55 
56 In particular b is invertible and normalizes both the complete E-chain
57 and the complete interleaved F-chain. We can write
58 
59 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
60 k=b(1+epsilon),  epsilon in P_A^(n-r).
61 \end{Verbatim}
62 
63 For u in U\_B\^{}m, the usual filtration commutator bound gives
64 
65 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
66 (1+epsilon)u(1+epsilon)^(-1)u^(-1)
67    in U_A^(n-r+m).
68 \end{Verbatim}
69 
70 This group is contained in U\_A\^{}a subset J, because
71 
72 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
73 n-r+m=2a-m-1>a.
74 \end{Verbatim}
75 
76 Conjugation by b preserves both relevant filtrations. Since b normalizes
77 U\_B\^{}m, it follows that
78 
79 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
80 k U_B^m k^(-1) subset U_B^m J=P.
81 \end{Verbatim}
82 
83 Also kJk\^{}(-1)=J, as K\^{}times normalizes its actual simple-character
84 groups. Thus kPk\^{}(-1) is contained in P and contains J. Its index
85 over J is two, equal to {[}P:J{]}, so it equals P. This proves
86 normalization by k, its inverse, and hence all K\^{}times.
87 
88 No equality or inclusion between the fields E and K was used. The
89 element b belongs to the actual outer centralizer B and only
90 approximates k to the stated precision.
91 
92 \hypertarget{rank4_stages-the-two-inducing-groups-have-index-two}{%
93 \subsection{2. The two inducing groups have index
94 two}\label{rank4_stages-the-two-inducing-groups-have-index-two}}
95 
96 Since K\^{}times normalizes P, the product M=K\^{}times P is a group.
97 Both P/J and the conjugation action of K\^{}times on it are defined; P/J
98 has order two, whose automorphism group is trivial. Therefore
99 
100 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
101 [K^times,P] subset J.
102 \end{Verbatim}
103 
104 Together with J normal in P and K\^{}times normalizing J, this proves
105 N=K\^{}times J normal in M. The intersection K\^{}times cap P is
106 U\_K\^{}1, which is contained in J, since P subset U\_A\^{}1.
107 Consequently
108 
109 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
110 [M:N]=[P:J]=2.
111 \end{Verbatim}
112 
113 The quotient \(K^\times/F^\times\) is compact because K/F is a
114 finite totally ramified extension, and P is compact. Hence M is compact
115 modulo F\^{}times. It is open in G because it contains the open subgroup
116 J.
117 
118 \hypertarget{rank4_stages-the-enlarged-inducing-representation-is-irreducible}{%
119 \subsection{3. The enlarged inducing representation is
120 irreducible}\label{rank4_stages-the-enlarged-inducing-representation-is-irreducible}}
121 
122 Let u in P represent its nontrivial coset modulo J. The carrier note
123 proved that theta\^{}u differs from theta, and therefore eta\^{}u
124 differs from eta. It follows that Lambda\^{}u differs from Lambda, since
125 their restrictions to J already differ. Applying finite-index Clifford
126 theory to N normal in M gives
127 
128 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
129 Sigma=Ind_N^M(Lambda) irreducible,
130 dim Sigma=2 dim Lambda=4.
131 \end{Verbatim}
132 
133 There is one N--P double coset in M, and N cap P=J. Thus restriction to
134 P gives the actual four-dimensional Heisenberg carrier:
135 
136 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
137 Sigma|P is isomorphic to Ind_J^P(eta).
138 \end{Verbatim}
139 
140 Its central character on F\^{}times is unchanged from Lambda, because
141 F\^{}times is central and lies in N.
142 
143 \hypertarget{rank4_stages-exact-recovery-by-induction-in-stages}{%
144 \subsection{4. Exact recovery by induction in
145 stages}\label{rank4_stages-exact-recovery-by-induction-in-stages}}
146 
147 Induction in stages for these open subgroups gives
148 
149 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
150 c-Ind_M^G(Sigma)
151    is isomorphic to c-Ind_M^G(Ind_N^M Lambda)
152    is isomorphic to c-Ind_N^G(Lambda)
153    =pi.
154 \end{Verbatim}
155 
156 The second induction is finite-index, so ordinary and compact induction
157 from N to M agree. This is an exact equality of the resulting
158 isomorphism classes and applies to the same fixed Lambda, including its
159 noncompact scalar choice. It introduces no new candidate and no H5
160 premise.
161 
162 \hypertarget{rank4_stages-scope}{%
163 \subsection{Scope}\label{rank4_stages-scope}}
164 
165 This establishes the arithmetic inducing datum and exact recovery for
166 the enlarged carrier. It leaves unchanged the distinction between the
167 original two-dimensional BK quotient J/H and the new four-dimensional
168 actual quotient P/H0. It also leaves unchanged the source-phase and
169 literal extraspecial-center limitations recorded in the carrier note.
170 
171 Dependency pin: \path{RANK4_ENLARGED_CARRIER167.md} SHA256
172 \path{1517e26138776fe0f32d74a174d309ec5dd6a0f572a979473d8af3d20b7a4155}.

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