A registered result in its source context

COX.STAGEA.TENSOR.01

Original ledger classification: Identity [E] (exact order/charpoly/trace arithmetic + honest kill-scope + coprimality controls -- sympy exact, Wolfram-mirrored) + [C] (the identification with the E8 Coxeter clock, as v223/v314) + [O] (the flip-atlas intertwiner COX.FLIP.INTERTWINER; Stage B metrization)

Original claim text

COXETER STAGE A CLOSED ON THE PRIMITIVE-CHARACTER SPACE (2026-07-23; 15 checks, <1 s, sympy exact): the tfpt_1 two-stage Coxeter audit asked for a natural order-30 operator with characteristic polynomial Phi_30 on the even-flip atlas OR its 8-dimensional primitive-character space. The explicit integer operator T30 = Comp(Phi_5) (x) Comp(Phi_6) (the v418 carrier clock tensor the family/CP hexagon, 8x8) delivers the character-space alternative: [E] T30^30 = I with T30^k != I for 1 <= k < 30 (exact order 30) and chi_T30 = Phi_30 = x^8+x^7-x^5-x^4-x^3+x+1 exactly; CRT mechanism gcd(5,6) = 1: the eigenvalues are (primitive 5th) x (primitive 6th) = exactly the eight primitive 30th roots -- the v315/COX.COUPLE.01 group split Z/30 = Z/5 x Z/6 realised as ONE integer operator (carrier clock x family clock = Coxeter clock). [E] RAMANUJAN TRACE TOMOGRAPHY: tr(T30^k) = c_30(k) for ALL k = 1..30 with the multiplicative factorisation c_30 = c_5 c_6; signed divisor table (-1,1,2,4,-2,-4,-8,8) at k = (1,2,3,5,6,10,15,30); atom readout (|tr T^3|,|tr T^5|,|tr T^15|) = (2,4,8) = (|Z2|,|mu4|,rank E8) -- eight integer traces type the Coxeter clock without diagonalisation or metrization (a cheap kill test for flip-operator candidates). [E] HONEST KILL SCOPE: Comp(Phi_15) PASSES the unsigned triple (Phi_30(x) = Phi_15(-x)) -- the discriminator is the SIGNED table (c_15(1) = +1 vs c_30(1) = -1); the other degree-8 candidates Phi_16/Phi_20/Phi_24 die already on the triple (traces 0,0,0). [E] COPRIMALITY CONTROLS: C6 (x) C6 has order 3 (not 36) with eigenvalue 1, C5 (x) C5 order 5 (not 25) -- CRT is load-bearing. [O] WHAT STAYS OPEN (no upgrade claimed): a canonical intertwiner from the concrete even-flip atlas (240 = 160+80) onto this primitive space (COX.FLIP.INTERTWINER), and Stage B (norm-2, E8-profile, Weyl-closure metrization); the tfpt_1 Stage A prose is sharpened accordingly, the atlas route itself stays [O]. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23).

Original classification detail

[E] Stage A answered on the 8-dim primitive-character alternative: T30 = C_Phi5 (x) C_Phi6, exact order 30, chi = Phi_30, traces = c_30(k) = c_5 c_6; [O] flip-atlas intertwiner + Stage B stay open; no marker moves beyond the Stage A prose sharpening

Dependencies as recorded

COX.CLOCK.01; COX.COUPLE.01; v223; v418; v55