A registered result in its source context

E8.SIMPLEX.FOURIER.01

Original ledger classification: Exact corpus compression [typed evidence line: 26/26 + 20/20 checks, zero fails, exact integer/Fraction/sympy arithmetic throughout; the identification typed SPECTRAL-only (the packet and code gradings do not refine each other); the spinor halving stays numerical fiat; q_wind(6) = 8 = det R typed a measured coincidence; the integer-window 2xsquare census measured, not narrated]; audits of DEPLOYED claims (v817/v833/v844, tfpt_2 I.5 keyboxes) -- normal forms and identities, no new physical claim; NO RH claim; no marker move (Python-only per GATE.WOLFRAM.02)

Original claim text

v857 -- E8.SIMPLEX.FOURIER.01 + FLAVOR.WINDING.QUADRATIC.01: the E8 audits, part I -- the Gaussian census IS a Fourier law and v817's per-prime measurement is the prime restriction of a CHARACTER THEOREM (mh2(n) = -1/15 as an integer identity at EVERY odd n <= 16500), and the tfpt_2 winding line collapses to ONE rank-one determinant quadratic q_wind = t^2 - g_car t + |Z2| with the triple lock s = 6 and the reality threshold typed, ONE module from two probes (26/26 + 20/20 checks, zero fails, verdicts SIMPLEX-FOURIER-EXACT + INTERTWINER-IDENTIFIED and WINDING-QUADRATIC-EXACT; discovery probes simplex_fourier_probe.py and flavor_winding_quadratic_probe.py, 2026-08-08, re-run identically at promotion, embedded BYTE-EXACT and executed verbatim, exact integer/Fraction/sympy arithmetic, ~2 s). PART A, THE FOURIER LAW (audit package A): the Construction-A rebuild is warded (30 placements, 2 invariant under pi_J and pi_sigma; 240 roots, all norm 4, all in L; the quotient V = L/(1+i)L = F_2^4 with class addition well-defined on all 256 pairs and every class 2-torsion), the census r = (0; 16^15) (zero class EMPTY, v833/v844) has the EXACT Walsh transform rhat = (240; -16 x15) with the inversion sum rule, i.e. the normalized spectrum {1, (-1/15)^15}: the uniform-nonzero channel P = (J - I)/15 entrywise in Fractions (doubly stochastic; eigenaction exact on all 16 characters; two-step P^2 = (I + 14J)/225 with contraction 1/225); THE FIVE-LINE THEOREM measured at every level: (i) RAMIFICATION -- Theta_0(n) = 0 for EVERY odd n <= 16 with the one-line proof (x = (1+i)y forces even level) and Theta_0(2m) = 240 sigma_3(m); (ii) EQUIDISTRIBUTION -- Theta_v(n) is ONE value over the 15 nonzero classes at every level n <= 16 (the Sp(4,2) footprint); (iii) hence Thetahat_u(odd n) = -Theta_L(n)/15; (iv) THE INTERTWINER IDENTITY -- the v817 packet numerator (Th0-Th1+Th2-Th3)(n) EQUALS the code Walsh transform Thetahat_u(n) for every nonzero u at EVERY level n <= 16, odd AND even (both ladders read [-16, 112, -448, 1136, -2016, 3136, -5504, 9328] at n = 1..8; the Eisenstein normal form 15 Theta_D8 = 7 Theta_L + 8 Theta_0 across machineries); (v) THE VALUE THEOREM -- 15 (Th0-Th1+Th2-Th3)(n) = -240 sigma_3(n) for EVERY odd n <= 16500, hence mh2(odd n) = -1/15 IDENTICALLY, covering all 1911 odd primes <= 16500: v817's per-prime measurement is a character theorem of the code, one Walsh transform away from the deployed census with NO new input. THE HONEST TYPE (measured, kept): the identification is SPECTRAL (numerator q-series equal) -- the packet 5|3-parity grading (52, 60, 128) and the code grading (15 x 16) do NOT refine each other (the S6.3 cross table: 4 + 3 + 8 mixed classes); the spinor halving Th1 = Th3 stays numerical fiat (typed); 52 = 40 + 12 = |R(D5)| + |R(A3)| is the seam reading. Controls fire (occupied zero class, one class at 15, wrong quotient dim F_2^3 -- each breaks the law at a stated integer), invariances hold (24 explicit isometries of (L, J) preserve census and spectrum; the OTHER invariant placement C** gives the identical law). PART B, THE WINDING QUADRATIC (audit package for tfpt_2, section I.5): against the deployed residue matrix (det R = 8 = h(D5), tr R = 9, PrinMin2 = (5,3,2), SNF (1,1,8), anchor R e1 = (1,1,2), cofactor seam normal adj(R) row 1 = (5,-9,6) with n.1 = 2 = |Z2|), THE MINI-THEOREM: chi_{R_s} - chi_R = -s q_wind with q_wind = e1^T adj(tI - R) 1 = t^2 - 5t + 2 = t^2 - g_car t + |Z2| EXACT in sympy -- the whole winding line is one rank-one matrix-determinant lemma; the keybox coefficients (9+s, 10+5s, 8+2s), the per-winding det increment d(det R_s)/ds = q_wind(0) = 2 = |Z2| and the minor law PrinMin2(R_s) = (5, 3(s+1), 2(s+1)) (carrier minor 5 s-INVARIANT) are all readouts of the single quadratic; q_wind is irreducible over Q (disc 17 nonsquare) and the two base points are s-invariant (rem(chi_{R_s}, q_wind) = -12t, s-free); THE TRIPLE LOCK: trace (tr R_s = 15), determinant (det R_s = 20) and Coxeter lift ((R_s^T a)_1 = 30) each solve to s = 6 ALONE, intersection {6} = {|R+(A3)|}, and at s = 6: chi_L = t^3 - 15t^2 + 40t - 20 with (15, 40, 20) = (dim A3, |R(D5)|, |R(A4)|), 3 real positive roots; THE LOCUS (exact Sturm/resultant): Delta(s) = disc_t chi_{R_s} = 17s^4 - 18s^3 + 709s^2 + 588s - 7996 (irreducible), Delta(6) = 39200 = 2^5 5^2 7^2 = 2 x 140^2 > 0 -- s = 6 is NOT a collision point; the reality threshold is the unique Delta-root in (2, 3), s* ~ 2.8250 (matching the deployed ~2.83; Sturm real-eigenvalue counts jump 1 -> 3 between s = 2 and s = 3); the integer-window census types s = 6 as the ONLY 2xsquare hit in s = -2..12 (measured, typed -- and q_wind(6) = 8 = det R reported as a measured coincidence). Controls fire: the sibling directions 1 e2^T and 1 e3^T change the quadratic AND break the lock (intersections EMPTY; 1 e3^T even reducible), the sibling ROW transport e1 1^T keeps -g_car t but LOSES |Z2| (locks pairwise disjoint 6/12/24), and the sibling minor branch {1,3,4} is unreachable on the whole winding line (integer scan s = -100..100 empty; the carrier minor 5 invariant) -- the triple (1, -5, 2) is direction-specific. Both parts are exact-arithmetic corpus compressions and audits of DEPLOYED claims (v817, v833, v844, v4/v94 keyboxes) -- normal forms and identities, no new physical claim, no marker moves. NO RH claim. Python-only per GATE.WOLFRAM.02. [2026-08-08 round-LXXIX note (v875): THE CENSUS IS NOW A THEOREM -- E8.SHELL.GLOBAL.01 proves Thetahat_u/Theta_L = -1/15 for ALL odd shells (Sp(4,2) transitivity of the G31 class action + (1+i) doubling + Theta_L = 240 sigma_3), replacing the n <= 16500 finite-census argument behind the value theorem; the census range is retained as the exact ward and the character-theorem reading of this row is unchanged; the consequence chain is kernel-checked in FORM.E8.SHELL.GLOBAL.01 (TfptCarrier/GaussianShells.lean).]

Original classification detail

SIMPLEX-FOURIER-EXACT + INTERTWINER-IDENTIFIED + WINDING-QUADRATIC-EXACT: rhat = (240; -16 x15), P = (J-I)/15 with spec {1, (-1/15)^15}, two-step 1/225 exact; ramification + equidistribution at every level <= 16; packet numerator == code Walsh at every level (both [-16, 112, -448, 1136, ...]); mh2(odd n) = -1/15 for every odd n <= 16500 (integer identity, 1911 primes); q_wind = t^2 - 5t + 2 exact by the adjugate lemma; triple lock s = 6; Delta(6) = 39200 = 2 x 140^2, sole 2xsquare hit; s* ~ 2.8250 the unique Delta-root in (2,3); 7 controls fire, 2 invariances hold

Dependencies as recorded

CFIN.UNIQUE.01; MESSAGE.LADDER.01; ARF.BENT.CSS.01; v4; v94; v817; v833; v844