The compiler clock alphabet in G₃₁

E8.G31.CLOCK_ALPHABET.01

Original ledger classification: Exact corpus compression + selectivity audit [typed evidence line: 19/19 checks, zero fails; the honesty line kept (degree-only audits cannot separate G31 from the reducible torus Z8 x Z12 x Z20 x Z24 -- frozen scope the irreducible rank-4 Shephard-Todd list); the Molien uniqueness and the full W(D5)-fence structural refutation CITED (v634), not recomputed; X1/X2 controls fire]; a normal form plus an armed kill rule for a deployed object; NO RH claim; no marker move (Python-only per GATE.WOLFRAM.02)

Original claim text

v858 -- E8.G31.CLOCK_ALPHABET.01: the E8 audits, part II -- the compiler clock alphabet is a NORMAL FORM of G31: Deg(G31)/4 = {2, 3, 5, 6} = {|Z2|, N_fam, g_car, |R+(A3)|} with the audit identity (sum d/4)(sum (d-1)/4) = 16 x 15 = 240 REALIZED on the roots, (8, 12, 20, 24) the UNIQUE integer quadruple with product 46080 and sum 64, the 607-group rank-4 kill scan typed (G31 the SOLE full-battery passer; weak-triple and raw-sum-64 impostors named), and the W(D5) x W(A3) fence re-killed with COMPUTED centers 4 vs 1, ONE module from one probe (19/19 checks, zero fails, verdict CLOCK-ALPHABET-SELECTIVE; discovery probe g31_clock_alphabet_probe.py, 2026-08-08, re-run identically at promotion, embedded BYTE-EXACT and executed verbatim, ~4 s). THE REBUILD (warded against v634): 240 doubled-integer E8 roots, J free of order 4 with EXACTLY 60 Gaussian lines, 60 order-2 reflections (one per line, all distinct), BFS closure |G| = 46080 = 8 x 12 x 20 x 24 = prod Deg(G31), every element complex-linear (commutes with J), the base certified by an exact adjugate identity (det = -512), and the EXACT reflection census: 60 elements with rank(M - I) = 2, all order 2, all generators -- ZERO quasi-reflections (v634 TEST2 reproduced; sum (d-1) = 60 anchors the degrees; the Molien uniqueness CITED, not recomputed). THE NORMAL FORM: Deg(G31) = |mu4| x {2, 3, 5, 6} with gcd = 4 = |mu4|, lcm = 120 = |R+(E8)|, lcm/gcd = 30 = h(E8); sum d/4 = 16 (the class size) and sum (d-1)/4 = 15 (the classes) with 16 x 15 = 240 = |R(E8)| -- and the 60/4 = 15 line is REALIZED on the roots (the 15 classes at 4 lines per class, the same 15 x 16 census as the package-A code side); the alphabet {2, 3, 5, 6} = {|Z2|, N_fam, g_car, |R+(A3)|} is EXACTLY the compiler clock alphabet; the Springer footprint is typed (exponents (7, 11, 19, 23) all = 3 mod 4; the d/4-clock realization x^{d/4} in {J, J^3} cited v654). THE ARITHMETIC PIN (measured): the integer-quadruple census with product 46080 and sum 64 returns EXACTLY [(8, 12, 20, 24)]. THE KILL SCAN (the selectivity theorem, 607 groups): the full rank-4 Shephard-Todd catalog (integrity: prod(degrees) = |G| for every entry -- a table second-source check) is run through the seven-gate battery (product, raw sum 64, gcd normalization, alphabet set, lcm, lcm/gcd = 30, audit identity): G31 is the SOLE full-battery passer; the weak-triple (gcd/lcm/30) impostors are typed by name (G(20,10,4), G(20,20,4), G29) and the raw-sum-64 coincidences too (G(8,2,4), G(10,10,4), G30 = W(H4)); the family boundary is verified in range (no G(m,p,4) matches the alphabet for any 2 <= m <= 120, with the typed argument: 3m/g <= 6 forces g >= m/2, then {d/g} contains 1 or 4); THE HONESTY LINE (typed, no kill): degree-only audits cannot separate G31 from the reducible torus Z8 x Z12 x Z20 x Z24 (identical degrees by construction) -- the frozen scope is the irreducible rank-4 list; the controls fire (the perturbed quadruple (8,12,20,28) fails the battery; the wrong normalization gcd 2 breaks the alphabet). THE FENCE (re-killed with a computed one-number invariant): |W(D5)| x |W(A3)| = 1920 x 24 = 46080 = |G31| stays NUMEROLOGY -- the computed |Z(G31)| = 4 with Z = <J> = mu4, versus |Z(W(D5) x W(A3))| = 1 x 1 = 1 (odd-rank D5: -1 not in W(D5); Z(S4) = 1): NOT isomorphic, NOT a decomposition (v634 T1.3/T1.4/T1.5 CITED for the full structural refutation: G = (Z4 o 2^{1+4}).Sp4(2) and W(D5) embeds in NO rank-4 group). The module is an exact-arithmetic corpus compression and a selectivity audit of a DEPLOYED object (v634/v654) -- a normal form plus an armed kill rule, no new physical claim, no marker moves. NO RH claim. Python-only per GATE.WOLFRAM.02. [2026-08-08 round-LXXIX note (v874): THE ALPHABET HAS A DECODER -- E8.WINDING.DECODER.01 proves that the v857 winding quadratic maps the clock alphabet {2,3,5,6} to (-|mu4|,-|mu4|,|Z2|,h(D5)) with the decoder polynomial (y+4)^2(y-2)(y-8) carrying budget coefficients (2, 48, 32, 256) and the exact bridge |mu4| q(|Z2|)/|R(E8)| = -1/15 = the Walsh message eigenvalue; typed AUDIT theorem, not a functor; the 6-vs-7 honesty note of this row carries over verbatim.]

Original classification detail

CLOCK-ALPHABET-SELECTIVE: Deg(G31)/4 = {2,3,5,6} = {|Z2|, N_fam, g_car, |R+(A3)|}; gcd 4 = |mu4|, lcm 120 = |R+(E8)|, lcm/gcd 30 = h(E8); (sum d/4)(sum(d-1)/4) = 16 x 15 = 240 realized on the roots (4 lines/class); (8,12,20,24) the unique product-46080/sum-64 quadruple; 607-group scan: G31 sole full-battery passer (weak-triple impostors G(20,10,4)/G(20,20,4)/G29; sum-64 coincidences G(8,2,4)/G(10,10,4)/W(H4)); family boundary m <= 120 empty; fence killed by computed centers 4 vs 1

Dependencies as recorded

v634; v654; CFIN.UNIQUE.01