A registered result in its source context

E8.ST31DEG.01

Original ledger classification: Exact [E] structural clock relations + [E] parity-theorem kill + [C] typed observations (Python-only per GATE.WOLFRAM.02).

Original claim text

The ST31 degree-24/-20 round (one module, 25 checks, ~4 s; discovery probe st31_degree24_probe.py 25/25; exact Fraction/Z[i] arithmetic). THE REGULAR 24-CLOCK [E, structural]: an explicit w with free census {24:10} and |C_G31(w)| = 24 (Springer); on the 60 lines it reads as the free 6-clock {6:10}; EXACT center relations w^12 = -1 and w^6 = J^3 -- the mu4 center is a POWER of the regular 24-clock (mu4 = <w^6>, <w>/mu4 = Z6 = the line 6-clock); chi_w = x^4 + i x^2 - 1 EXACT (Gaussian-integral quartic factor of Phi_24; divides x^24 - 1 and no proper x^d - 1: all four eigenvalues PRIMITIVE zeta24); the half-clock w^2 is the REGULAR 12-element ({12:20}, |C| = 288), NOT the compiler clock. THE CLOCK KILL [E, parity theorem]: c = w^2 is IMPOSSIBLE for any order-24 w, even in S_240 -- the 12-cycles of any square come in PAIRS from 24-cycles (4-cycles in pairs from 8-cycles) and census(c) = 19 x 12 + 3 x 4 has ODD counts; the full power-census scan over all 46080 elements confirms: the only (census, k) with census(x^k) = census(c) are order-12 elements with census(c) itself and k coprime to 12. THE 20-DEGREE [E]: regular u with census {20:12}, |C(u)| = 20; u^5 = J^3 CENTRAL (stronger than J-class); u^4 of order 5 acts FREELY with census {5:48} = 240/5 EXACTLY, lines {5:12}, |C(u^4)| = 20; chi_u primitive-zeta20 exact. HONEST TYPING [C]: the equality 48 = seam sites stays an OBSERVATION (number coincidence, no structural bridge in this module); g_car = 5 has only the census side; no marker moves. [2026-08-02 XLV: v654 completes the degree census (order-8 landscape 10560/4800/480, the unified d/4-theorem, 'free => regular' killed at d = 8 and already at order 4); LEAN ROUND 3: the order statements of the regular words are machine-proved (G31WordOrders.lean: chi_w | X^24 - 1 and chi_u | X^20 - 1 via the conjugate-quartic witnesses chi x conj = Phi_24 / Phi_20, and NO proper-divisor X^d - 1 by companion-matrix kernel decide -- all eigenvalues primitive; W^24 = U^20 = 1), and the combinatorial kernel of the c = w^2 parity kill is certified at the multiset level (SquareParity.lean: the exact count law #{L-cycles of rho^2} = 2 #{2L-cycles of rho} for even L + census_not_square; the permutation-level bridge honestly typed as not formalized); lake build green, 3383 jobs, no sorry.]

Original classification detail

ST31-DEGREE-ANCHORED: the mu4 center is a power of both regular clocks (w^6 = +-J, chi_w = x^4 + i x^2 - 1 primitive-zeta24; u^5 = +-J central); c = w^2 killed by the parity theorem (19/3 odd -- proof-grade); u^4 free with census {5:48}, 48 = 240/5; 48 = seam sites stays a typed observation; no marker moves.

Dependencies as recorded

E8.ST31.01; E8.ZIQUOTIENT.01; E8.INCIDENCE.01; v629; v633; v634