A registered result in its source context

E8.TORSOR.FOURIER.01

Original ledger classification: Exact [E] integer/cyclotomic character theory (canonical rays, exact Clebsch spectrum, exact overlap census) with the predeclared P3 failure typed -- the weight and character gradings mutually unbiased; ROOTCLASS-MIXED (v775) intact and re-confirmed; no marker move (Python-only per GATE.WOLFRAM.02).

Original claim text

the projective Fourier modes of the 15 Gaussian root torsors -- carrier-block purity at CHARACTER level with the mutually-unbiased honest cap (30/30 checks, ~60 s; discovery probe e8_torsor_fourier_probe.py, 2026-08-06, verdict TORSOR-FOURIER-PARTIAL; FROZEN_SPEC SHA-hashed before any root data). THE CONSTRUCTION: because the v782 class fibers are torsors WITHOUT origin, their character data is canonical where their point data is not -- 120 = 15 x 8 projective Fourier rays m(v, u) with coefficients (-1)^{hbar(u, x(r))} on the 16-root fiber; PHASE LAW EXACT on all 960 origin shifts (a shift by t multiplies by the GLOBAL sign (-1)^{hbar(u,t)}: the rays are canonical); orthogonal ray basis of the (-1)-symmetric 120-dim pair space (integer Gram = 16 I; the antisymmetric half receives NO canonical mode -- the v782 sign theorem). EQUIVARIANCE: deck J diagonal with eigenvalue (-1)^{hbar(u,v)} (60 even + 60 odd, 4 + 4 per fiber); ALL 16 Pauli translations diagonal by characters (15 x 15 x 8 exact); G31 MONOMIAL (7200 generator matches; label action closes to Sp(4,2), order 720, kernel 64 = mu4 x Pauli16; label orbits {15 uniform, 45 deck-even, 60 deck-odd}); Burnside exact (transitive on pairs and on lines); sigma (Z3) multiplicities integral. THE CARRIER BLOCK CLEAN AT CHARACTER LEVEL (P1a/P1b/P2): the spinor sector = EXACTLY the 8 R-fibers (split-independent), so M_S = the span of their 64 whole Fourier rays -- a Fourier-ALIGNED invariant block (impossible at point level per v775); M_S factorizes as a W(D5) x W(A3)-set as 16 x 4 (W(D5)/W(A3) rebuilt by BFS from the certified v775 split, 1920/24), W(D5) transitive of permutation rank 3 on the 16 half-spinor weight pairs with the folded-5-cube/CLEBSCH charpoly (x-5)(x-1)^10(x+3)^5 EXACT => isotypic dimensions 1 + 5 + 10 -- the carrier Pascal AT CHARACTER LEVEL -- and the tetrad is S4-NATURAL (faithful, transitive, stabilizer S3, permutation rep 1 + standard 3) with sigma = the family 3-CYCLE + fixed anchor; W(A3)-tables exact (the 20 D5-adjoint pairs family-invariant; M_S = (1+5+10) x (1+3)); W(D5) Burnside on the spinor pairs: 4 orbits (the tetrad), sum m_i^2 = 48. P3 FAILS AS PREDECLARED (the honest cap): every mode has full 16-root support and |<pair|mode>|^2 = 1/8 EXACTLY on every cell -- the D5-weight basis and the character basis are MUTUALLY UNBIASED on each carrier fiber; no weight axis is a Fourier ray; the pointwise code -> matter reading stays dead (ROOTCLASS-MIXED v775 intact, re-confirmed). OVERLAP CENSUS: every certified stabilizer ray expands exactly in the 4 deck-even modes of its fiber (squared overlap 1/4 each; deck-odd and foreign fibers orthogonal -- the deck-odd half is a pure phase register invisible to stabilizer rays); the 45 nontrivial deck-even modes ARE, under the certified chart U, the 45 (context, Pauli) incidences (each nonzero Pauli covered exactly 3x; uniform modes -> identity, Sum z z^dag = 4I) -- a canonical G31-equivariant operator dictionary, noted (report only) as exactly the SHAPE the v791 carrier-intertwiner demand asks for on the finite register. CONTROLS: fiber scramble destroys deck equivariance; the wrong character group F2^4 = V collides 2:1 (rank 8 < 16); seeded random sign systems fail the certificates. Everything exact integer/Fraction/Z[i]/F2 (sympy gated to two charpolys). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Original classification detail

TORSOR-FOURIER-PARTIAL: 120 canonical projective Fourier rays (phase law exact on 960 shifts, Gram = 16 I); deck/Pauli diagonal, G31 monomial through Sp(4,2); the carrier block = the 8 R-fibers' 64 rays = (1+5+10) x (1+3) under W(D5) x W(A3) with sigma = family 3-cycle + anchor -- the carrier Pascal at character level, impossible at point level (v775); weight/character bases exactly mutually unbiased (the honest cap); the 45 deck-even modes = the 45 context Pauli observables under the certified chart; no RH statement.

Dependencies as recorded

E8.TRANSITIONBUS.01; E8.CLIFFORD2Q.01; ARF.ROOTCLASS.01; PRIME.POSITIVE_DESCENT.01; v782; v783; v775; v752; v791