H The explicit dyadic arithmetic tower
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Date: 2026-10-05. Ordinary proof. No character, quotient-dimension, source attachment, pair-factor, or universal Paper XI closure is asserted here.
H.1 1. Statement
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Let F=F_2((t)), let 0< h_1<…< h_d be odd positive integers, and set
q_i=3h_i, N=2^d.The explicit recursion below constructs B_d in M_N(F) such that
F[B_d] is a totally ramified maximal field normalizing
the complete period-N lattice chain,
v_A(B_d)=-2^(d-1)q_d,
k_0(B_d,A)=-q_1,
tr_F(B_d)=1.There are explicit native constructors of ranks 2^j for the largest j exponents. Their identity-amplifications give the actual defining sequence with field degrees 2,4,…,2^d and critical levels
n_j=2^(d-j)q_(d-j+1), 1<=j<=d.Every parameter in this ambient defining sequence has the same leading level n_1. The quantities n_2,…,n_d are its successive critical approximation levels, not new leading levels of the ambient matrices.
Every native constructor has trace 1. Every correction from an identity-2 amplification of the preceding native constructor to the next native constructor also has trace 1. In full ambient rank 2^d, the earlier amplified approximations instead have trace 0, because their multiplicities are even. This distinction is essential.
H.2 2. The minimal degree-two building block
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Let L be a complete discretely valued field of characteristic 2 with residue field F_2, and choose a uniformizer t_L. For odd q>0 put a=(q+1)/2 and define
E=L(alpha), alpha^2-alpha=t_L^(-q),
z=t_L^a alpha.The Artin–Schreier polynomial is irreducible: an element x^2-x of negative L-valuation has even negative valuation. Its derivative is 1, and the valuation equation forces ramification index 2. Thus E/L is separable and totally ramified, v_E(alpha)=-q, and alpha is minimal.
The element z has E-valuation 2a-q=1 and satisfies
f(z)=z^2+t_L^a z+t_L=0.Thus 1,z is a uniformizer power basis, and
alpha=z/t_L^a, Tr_(E/L)(alpha)=1.H.3 3. An explicit normalized corestriction and lift
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On E use the coefficient basis 1,z. Its trace-dual elements are
z_0^*=(z+t_L^a)/t_L^a,
z_1^*=1/t_L^a.Let pr(T_ij)=z^i z_j^* be the trace corestriction. Put
kappa=t_L^a/z=alpha^(-1),
s=kappa pr,
u_0=s(P0)=t_L/z^2.Here P0 is constant-coefficient projection, applied separately to each coordinate when acting on E^m. The identity for u_0 follows from f(z)=0. It is a unit: v_E(t_L)=v_E(z^2)=2. More explicitly,
s(T_i0)=z^i(1+t_L^a/z),
s(T_i1)=z^(i-1).Since v_E(t_L^a/z)=2a-1=q>0, these preserve the expected leading grades. On End_L(E^m), take the complete interleaved chain in the basis w_(mi+b)=z^i v_b. Its centralizer is M_m(E) with its complete E-chain. The formulas give s(P^r) contained in Q^r for all integers r.
For C in M_m(E), define the explicit L-linear operator
Lift_E(C)=u_0^(-1) C P0=(z^2/t_L) C P0.B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0^(-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular \(s(P^r)=Q^r\).
H.4 4. Exact trace transport
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For every C in M_m(E), ordinary matrix traces satisfy
tr_L(Lift_E(C))
=Tr_(E/L)(z_0^* u_0^(-1) tr_E(C))
=Tr_(E/L)(kappa^(-1) tr_E(C))
=Tr_(E/L)(alpha tr_E(C)). (T2)The first equality follows by taking the L-trace of each rank-one operator yP0, which is the constant coefficient of y. This coefficient is Tr_(E/L)(z_0^* y).
Thus the trace of a lift is 1 whenever the inner ordinary E-matrix trace is 1. It is not asserted that arbitrary trace values are unchanged.
H.5 5. Recursive actual matrices and fields
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Define B(L,t_L;h_1,…,h_d) as follows. For d=1 use the multiplication matrix of alpha from Section2 with q=q_1. For d>=2 apply Section2 with q=q_d, set m=2^(d-1), recursively construct
C=B(E,z;h_1,...,h_(d-1)),and put
B(L,t_L;h_1,...,h_d)=alpha I_m+(z^2/t_L) C P0. (D2)This is a finite algebraic recipe. It uses specified separable quadratic extensions and no unknown support, genericity choice, or correspondence inverse.
The degree and simplicity proof is by induction. The base alpha is minimal and generates E. At a later step let gamma=alpha I_m and b=Lift_E(C). The outer minimal stratum has
n=2^(d-1)q_d, k_0(gamma,A)=-n.The inner induction hypothesis gives an actual maximal field E[C] normalizing the complete E-chain with
R=2^(d-2)q_(d-1), v(C)=-R, k_0(C)=-q_1.Because h_d> h_(d-1), we have n>2R. Therefore
-2R > -n-q_1 = k_0(gamma,A)+k_0(C).The derived stratum with parameter s(b)=C is pure, and its field is maximal. Bushnell–Kutzko Remark2.2.7 consequently gives the field generated by the actual gamma+b, not merely an equivalent replacement. It has ramification and degree 2^d, residue degree one, normalizes the complete chain, and has k_0=max(-n,-q_1)=-q_1. Since v(b)> v(gamma), its level is n and [A,n,0,gamma+b] is simple.
The derived model E[C] is not identified with L[gamma+b]. In particular no literal embedding of E into the latter field is claimed.
H.6 6. Trace 1 at every native stage
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The base matrix has trace Tr_(E/L)(alpha)=1. Inductively tr_E(C)=1, so (T2) gives tr_L(Lift_E(C))=1. For d>=2 the L-trace of alpha I_m is
m Tr_(E/L)(alpha)=0,because m is even. Hence tr_L(B)=1 at every native rank.
For any j>=1, let A_j denote the native rank-2^j constructor using a fixed list of the largest j exponents. The next native rank-2^(j+1) constructor A_(j+1) uses the same largest exponents and adds the next smaller one. With their compatible interleaved bases, put
Delta_(j+1)=A_(j+1)-A_j tensor I_2.Then
tr(Delta_(j+1))=1-2*1=1 in characteristic 2.This native trace identity is stronger and more relevant than a claim obtained merely by amplifying to a rank divisible by two.
H.7 7. Full arithmetic defining sequence
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For the final rank 2^d define
B^[j]=B(F,t;h_(d-j+1),...,h_d) tensor I_(2^(d-j)).All truncations use the same top q_d to select the first extension and uniformizer, the same next q_(d-1) to select the next extension, and so on. The coefficient projection and lift commute with amplification. Thus these are literal truncations of (D2).
Their parameter fields have degree 2^j, and their critical levels in the ambient complete chain are n_j=2^(d-j)q_(d-j+1). Valuation-preserving lifts give
v_A(B^[j+1]-B^[j])=-n_(j+1).Later differences have strictly greater valuation, since n_j>2n_(j+1). This proves that the displayed parameters form an actual defining sequence of simple approximations.
Their ambient traces are
tr_F(B^[j])=0 for j<d, tr_F(B^[d])=1.Consequently ambient corrections before the last step have trace 0, and the last correction has trace 1. Every one of these trace values is integral. On a scalar principal unit u, the additive correction character has value psi_F((u-1)tr(delta)); this is 1 for an additive character trivial on the maximal ideal. No assertion about full simple character extension follows here without its separate intersection and transfer argument.
H.8 8. Concrete rank-eight instance and limits
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For h=(1,3,5), the construction gives actual approximation-field degrees 2,4,8 and critical levels 60,18,3. The supplementary exact program constructs its Laurent-polynomial matrices, verifies the normalized lift identities, and checks the native/ambient trace distinctions and matrix grades. It does not prove field existence or simplicity by bounded computation: those are consequences of Section5.
The odd-prime Heisenberg quotient and conductor counting from Block166 cannot simply be copied: q_1 is odd here. This note makes no dyadic quotient, full-character, stabilizer, phase, source-attachment, or H5 claim. It also changes lower parameter conjugacy classes, as in the corresponding odd-prime Artin–Schreier construction.
H.9 Dependencies and provenance
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Primary theorem: Bushnell–Kutzko, The admissible dual of GL(N) via compact open subgroups, Remark2.2.7 (printed p55), together with the amplification normalization and defining-sequence conventions already checked in Block166.
Retained primary source text SHA256: d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8.
Block166 inputs are pinned separately in INPUT_PINS167.json and are not modified by this continuation. The Artin–Schreier, trace-dual, and trace calculations needed specifically for p=2 are proved above.