M The actual enlarged rank-four inducing carrier
On this page · Visual guide: Low-rank repair enlarges an actual compact group · Visual guide: Rank four gains a four-dimensional carrier · Permalink · Source line 2
Date: 2026-10-05. Ordinary proof submitted for independent review.
M.1 Result and boundary
On this page · Permalink · Source line 7
For the actual rank-four Artin–Schreier candidate of Block167, one can enlarge the compact group J1 to an actual compact subgroup P with
J1 normal in P, [P:J1]=2,
E=Ind_(J1)^P(eta) irreducible of dimension 4.The representation E occurs inside the restriction of the same actual supercuspidal candidate pi. There is an explicit normal subgroup H0 of P with [P:H0]=16 such that P/H0 is elementary abelian and the commutator form supplied by the actual character on H0 is nondegenerate. Thus the carrier has an actual four-dimensional binary symplectic quotient. No abstract transfer of matrix operators is used to create this action.
The finite image of P has scalar center theta(H0), which can be larger than {+1,-1}. It is literally an extraspecial group if this scalar center has order two. That additional assertion is not assumed or proved here. The construction supplies an actual finite Heisenberg central extension with nondegenerate projective quotient F_2^4. Matching an independently prescribed quadratic phase is also a separate assertion.
M.2 1. Actual arithmetic groups and notation
On this page · Permalink · Source line 33
Let F=F_2((t)), and take the audited rank-four constructor with odd 0< h_1< h_2. Write
q_i=3h_i, n=2q_2, a=n/2=q_2,
r=q_1=2m+1, m=(r-1)/2>=1.We have a> r, so n>2r. Let A be the complete period-four hereditary order, with radical P_A. Let gamma be the actual degree-two outer minimal approximation, E=F[gamma], B=End_E(F^4)=M_2(E), B0=A cap B, and Q=rad(B0). The actual final parameter is beta=gamma+delta, and
s(delta)=c,where c is the minimal degree-two Artin–Schreier parameter over E, of B0-level r. The final field K=F[beta] is totally ramified of degree four; no embedding E subset K is asserted.
Use \(U_A^j=1+P_A^j\) and \(U_B^j=1+Q^j\). BK3.1.7–9 give the actual groups
H=H1(beta)=U_K^1 U_B^(m+1) U_A^(a+1),
J=J1(beta)=U_K^1 U_B^(m+1) U_A^a.In particular J/H has dimension two over F_2. Let theta be the actual full simple character constructed for beta, and let eta be its unique Heisenberg representation on J, of dimension two. The argument below works with this supplied actual theta; it does not replace its values.
Choose the outer minimal simple character theta_gamma from its defining sequence. On H cap U_A^(m+1), the recursion says
theta=theta_gamma psi_delta.Moreover H is contained in H1(gamma)=U_B^1 U_A^(a+1). The character theta_gamma is invariant under U_B^1: on U_B^1 it factors through the E-determinant, and on U_A^(a+1) it is the gamma-additive character.
M.3 2. The actual final field has the needed graded line
On this page · Permalink · Source line 84
For every integer j>=1 and k in K cap P_A^j, the minimal-approximation commutator estimate gives
k=b+e, b in Q^j, e in P_A^(j+n-r). (1)Indeed [gamma,k]=-[delta,k] belongs to P_A^(j-r), and the minimal corestriction exact sequence for gamma solves its commutator with an error of grade j+n-r. The remaining difference centralizes gamma and therefore belongs to B. This is the usual minimal case of BK1.4.10.
Applying s to [beta,k]=0 and using (1) gives
[c,b] in Q^(j+n-2r).Hence the leading B-grade of b commutes with the leading grade of c. Let R be the complete period-two E-chain shift, with R^2=z I, where z is the chosen uniformizer of E. In the constructor's basis
c congruent to R^(-r) modulo Q^(1-r).Because r is odd, the centralizing subspace on every B-grade j is the one-dimensional line spanned by R^j. On the other hand K is a totally ramified maximal field normalizing the complete A-chain, so (K cap P_A^j)/(K cap P_A^(j+1)) has dimension one over F_2. Estimate (1) is injective on the leading grade, so its image is exactly that centralizing line.
This proves the required graded assertion for the actual K, without replacing it by the derived model E[c] or assuming nested fields.
M.4 3. Normality and index of the enlarged group
On this page · Permalink · Source line 124
Set
P=J U_B^m.We first prove this is a group. The group U_B^m normalizes U_B^(m+1) and every U_A^j. For k in U_K^1, (1) approximates k by an element of U_B^1 with error in U_A^(n-r+1). Therefore
[U_B^m,U_K^1]
subset U_B^(m+1) U_A^(m+n-r+1)
subset H,where the last inclusion follows from a> r=2m+1. Thus U_B^m normalizes both H and J, and P is an actual compact subgroup with J normal in P.
The quotient \(U_B^m/U_B^{m+1}\) has dimension two. By Section2 the image of J cap U_B^m on this grade is exactly its centralizing line, of dimension one. For the converse inclusion needed here, the B-approximation of a K-element of grade m differs from it by grade m+n-r> a; that error lies in U_A^a subset J. Thus the whole centralizing line really occurs in the intersection, not only in an abstract associated graded space. Consequently
[P:J]=[U_B^m:J cap U_B^m]=2. (2)For a concrete representative choose D=diag(1,0) in B and put
x=R^m D, u=1+x.Its leading grade is outside the centralizing line, so u represents the nontrivial coset of P/J. Since \(u^2=1+x^2\) in characteristic two, we have u^2 in U_B^(2m) subset U_B^(m+1) subset H.
M.5 4. The actual character has an orbit of length two
On this page · Permalink · Source line 171
Define the character of H
lambda(h)=theta(u h u^(-1))/theta(h).For h in H the preceding normality argument gives [u,h] in H cap U_A^(m+1). Since theta_gamma is invariant under u,
lambda(h)=psi_delta([u,h]) in {+1,-1}. (3)Thus lambda has order dividing two, and its values are actual values of the supplied simple character. It is J-invariant, since both theta and its u-conjugate are J-invariant.
It is nontrivial. Take y=R^(m+1)D and h=1+y in U_B^(m+1). The leading commutator in (3), after corestriction, has value
psi_E(tr_(M_2(E)/E)(c[x,y]))=-1.Here the commutator remainder has B-grade at least 3m+1, which becomes positive after multiplication by c of grade -2m-1. On the leading grades, writing X=R^m diag(A,B) and Y=R^(m+1)diag(C,D), one obtains
tr(R^(-2m-1)[X,Y])=(A+B)(C+D) in F_2.Both sums equal one for the chosen D=diag(1,0). The normalized psi_E is nontrivial on the residue field and trivial on its maximal ideal. For this AS normalization one can check the sign directly from psi_E(1)=psi_F(Tr_(E/F)(alpha))=psi_F(1)=-1.
Hence theta^u differs from theta, eta^u differs from eta, and Clifford theory applied to (2) gives
E_carrier=Ind_J^P(eta) irreducible, dim(E_carrier)=4. (4)M.6 5. An actual nondegenerate symplectic quotient of dimension four
On this page · Permalink · Source line 220
Put H0=ker(lambda) inside H, and theta0=theta restricted to H0. In the two-coset induction model, h in H acts by
diag(theta(h) I_2, theta^u(h) I_2).Thus it acts scalarly precisely when h belongs to H0. We check that H0 is the entire scalar preimage and that its quotient is symplectic.
First, lambda is invariant under u. Indeed u^2 belongs to J and theta is J-invariant, so \(\lambda^u=\lambda^{-1}=\lambda\) by (3). Consequently H0 is normal in P, and theta0 is P-invariant.
Second, lambda is trivial on U_A^(a+1). There the correction formula is valid, gamma commutes with u, and the delta contribution to a commutator has grade at least a+1+m-r=a-m>0. Therefore it lies in the kernel of the additive character.
Write J=H U_A^a. Since lambda is a character and J-invariant, it kills [H,H] and [H,U_A^a]. It also kills \([U_A^a,U_A^a]\), which lies in U_A^(2a). Hence lambda([J,J])=1. It kills every square in J as well: squares of H have lambda-value one, squares of U_A^a lie in U_A^(2a), and the mixed commutator contributions have just been handled.
Finally lambda([u,H])=1 by u-invariance, and [u,U_A^a] belongs to U_A^(a+m) subset U_A^(a+1), so lambda([u,J])=1. Also lambda(u^2)=1 because u commutes with u^2. It follows that
[P,P] subset H0, g^2 in H0 for every g in P,
[P:H0]=[P:J][J:H][H:H0]=2*4*2=16.Therefore P/H0 is an actual elementary abelian group of dimension four. If an element of J acts scalarly on the carrier, its first eta-block is scalar, so it belongs to H by the original nondegenerate Heisenberg quotient J/H. It then belongs to H0. An element outside J exchanges the two coset summands and cannot act scalarly. Thus H0 is precisely the scalar preimage.
The pairing
(gH0,kH0) |-> theta0([g,k])is alternating and well defined. Since the projective quotient has exponent two, its values lie in {+1,-1}. It is nondegenerate: an element in its radical commutes with the entire irreducible carrier image and therefore acts scalarly by Schur's lemma, forcing it into H0.
This is the actual four-dimensional binary Heisenberg quotient sought for capacity. Its operators are supplied by elements of P.
M.7 6. The carrier occurs in the same supercuspidal candidate
On this page · Permalink · Source line 280
Let pi=c-Ind_N^(GL_4(F))(Lambda) be the existing actual candidate, with N=K^times J. The residue field is F_2, so the compact extended type restricts to eta on J. Since P is contained in U_A^1,
N cap P=J.Indeed an element k j in P with k in K^times has k in U_A^1, hence k in U_K^1 subset J. The identity double-coset term in the restriction of compact induction to P is therefore
Ind_J^P(eta)=E_carrier.It is realized concretely by the functions supported on N P. Thus the four-dimensional carrier occurs in the same pi, with its already fixed full type character and noncompact extension scalar. No new arithmetic candidate, beta extension, or inverse correspondence is used.
M.8 7. Exact limits: finite image versus literal extraspecial group
On this page · Permalink · Source line 305
The actual finite image is P/ker(theta0), because H0 is the scalar preimage. Its center is the scalar cyclic 2-group theta0(H0), and its commutator subgroup has order two. Its quotient by its center is F_2^4.
If theta0(H0) has order two, this finite image is extraspecial of order 32. If its scalar center is larger, quotienting arbitrary central elements need not preserve its commutator: in a cyclic 2-group the order-two subgroup can lie inside the squares. Thus a literal extraspecial quotient is not justified solely by the calculations above.
The stronger statement proved is the one with full actual character: an actual compact subgroup, actual induced carrier, actual scalar subgroup, and nondegenerate actual commutator pairing. Passing to the usual full scalar pushout retains all theta0-values. It does not prove that an independently supplied quadratic refinement, stabilizer action, or source phase agrees with this carrier. Those requirements must be checked separately. No H5 or universal claim66 closure is asserted.
M.9 Provenance
On this page · Permalink · Source line 328
Uses the audited dyadic parameter/character construction, BK1.4.10, BK3.1.7–9, and the defining-sequence character formula. The source normalization and all field/chain data are the actual ones in dyadic167/tower/DYADIC_AS_TOWER167.md. This note changes neither that proof nor its recorded exact computation. Clifford induction and the identity compact-induction double-coset term are used in their ordinary finite-index forms, with hypotheses proved above.