N Rank-four normalization and induction in stages
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Date: 2026-10-05. Corollary to RANK4_ENLARGED_CARRIER167.md, whose reviewed bytes are kept unchanged. Ordinary proof submitted for independent audit.
Use that note's actual groups and notation:
n=2a, r=2m+1, a>r,
H subset J=J1(beta) normal in P=J U_B^m,
[P:J]=2, N=K^times J,
pi=c-Ind_N^G(Lambda), Lambda|J=eta.The field K normalizes P. Consequently the following are actual groups and actual smooth representations:
M=K^times P,
Sigma=Ind_N^M(Lambda),
dim Sigma=4,
Sigma|P=Ind_J^P(eta),
pi is isomorphic to c-Ind_M^G(Sigma).The representation Sigma is irreducible, and M is compact modulo the center of G. Thus the enlarged carrier has an exact arithmetic inducing datum for the same candidate, not merely an occurrence in its restriction.
N.1 1. Normalization by the actual parameter field
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The units O_K^times equal U_K^1 because the residue field is F_2. They belong to J and hence normalize P. It remains to check a uniformizer k of K. This element has A-grade one and normalizes A.
The actual-field approximation estimate proved in Section2 of the carrier note gives
k=b+e, b in Q^1, e in P_A^(n-r+1).The leading grade of b is the nonzero centralizing line of the primitive derived parameter. Over F_2 it is exactly the leading grade of R, the period-two E-chain shift. Therefore
b=R v, v in 1+Q^1.In particular b is invertible and normalizes both the complete E-chain and the complete interleaved F-chain. We can write
k=b(1+epsilon), epsilon in P_A^(n-r).For u in U_B^m, the usual filtration commutator bound gives
(1+epsilon)u(1+epsilon)^(-1)u^(-1)
in U_A^(n-r+m).This group is contained in U_A^a subset J, because
n-r+m=2a-m-1>a.Conjugation by b preserves both relevant filtrations. Since b normalizes U_B^m, it follows that
k U_B^m k^(-1) subset U_B^m J=P.Also kJk^(-1)=J, as K^times normalizes its actual simple-character groups. Thus kPk^(-1) is contained in P and contains J. Its index over J is two, equal to [P:J], so it equals P. This proves normalization by k, its inverse, and hence all K^times.
No equality or inclusion between the fields E and K was used. The element b belongs to the actual outer centralizer B and only approximates k to the stated precision.
N.2 2. The two inducing groups have index two
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Since K^times normalizes P, the product M=K^times P is a group. 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
[K^times,P] subset J.Together with J normal in P and K^times normalizing J, this proves N=K^times J normal in M. The intersection K^times cap P is U_K^1, which is contained in J, since P subset U_A^1. Consequently
[M:N]=[P:J]=2.The quotient \(K^\times/F^\times\) is compact because K/F is a finite totally ramified extension, and P is compact. Hence M is compact modulo F^times. It is open in G because it contains the open subgroup J.
N.3 3. The enlarged inducing representation is irreducible
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Let u in P represent its nontrivial coset modulo J. The carrier note proved that theta^u differs from theta, and therefore eta^u differs from eta. It follows that Lambda^u differs from Lambda, since their restrictions to J already differ. Applying finite-index Clifford theory to N normal in M gives
Sigma=Ind_N^M(Lambda) irreducible,
dim Sigma=2 dim Lambda=4.There is one N–P double coset in M, and N cap P=J. Thus restriction to P gives the actual four-dimensional Heisenberg carrier:
Sigma|P is isomorphic to Ind_J^P(eta).Its central character on F^times is unchanged from Lambda, because F^times is central and lies in N.
N.4 4. Exact recovery by induction in stages
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Induction in stages for these open subgroups gives
c-Ind_M^G(Sigma)
is isomorphic to c-Ind_M^G(Ind_N^M Lambda)
is isomorphic to c-Ind_N^G(Lambda)
=pi.The second induction is finite-index, so ordinary and compact induction from N to M agree. This is an exact equality of the resulting isomorphism classes and applies to the same fixed Lambda, including its noncompact scalar choice. It introduces no new candidate and no H5 premise.
N.5 Scope
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This establishes the arithmetic inducing datum and exact recovery for the enlarged carrier. It leaves unchanged the distinction between the original two-dimensional BK quotient J/H and the new four-dimensional actual quotient P/H0. It also leaves unchanged the source-phase and literal extraspecial-center limitations recorded in the carrier note.
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