The G₃₁ reflection structure

E8.ST31.01

Original ledger classification: Exact [E] group/Molien/Springer census + [E, KILL] the numerology kill + honest clock typing (Python-only per GATE.WOLFRAM.02; Lean order identities).

Original claim text

The ST31 structure round (one module, 56 checks, ~10 s; discovery probe st31_structure_probe.py 56/56). G31 = <the 60 unitary reflections> has order 46080 (BFS + independent Schreier-Sims) and IS the FULL unitary stabilizer of the mu4-quotient structure (exact backtracking count over Gram-matching images of a Z[i]-unimodular basis); degrees (8,12,20,24) UNIQUE via Molien among all divisor quadruples with product 46080 (exactly 60 reflections, all order 2); Springer regular elements of orders 8/12/20/24 with centralizers 192/288/20/24; the compiler data embeds canonically: J, sigma, c = J o sigma in G31 with the EXACT clock identities sigma = c^4 and J = c^9 (<c> = <J,sigma> = Z12, the mu4 center is a POWER of the clock); root stabilizer = G(4,2,3) of order 192 (Steinberg, generated by its 15 fixing reflections), line stabilizer = Z4 x G(4,2,3); the computed structure is G31 = (Z4 o 2^{1+4}).Sp4(2). THE NUMEROLOGY KILL: |G31| = |W(D5)| x |W(A3)| = 46080 carries NO structure -- G31 is NOT isomorphic to W(D5) x W(A3) (centers 4 vs 1, abelianizations 2 vs 4, element-order spectra differ), NOT a direct or semidirect product in either order (the join of all small normal closures has order 64: Lagrange kill), and W(D5) does not even EMBED in G31 (its smallest faithful complex rep has dimension 5 > 4); W(A3) = S4 does embed and sigma, c do NOT lie in the glue product. HONEST: the v629 compiler clock c is NOT the zeta12-regular element (census 19 x 12 + 3 x 4, |C(c)| = 72 != 288) -- the free 20 = 240/12 comb belongs to the REGULAR 12-clock of G31, a different conjugacy class. LEAN: the order identities 8*12*20*24 = 46080 = 1920*24 = |W(D5)|*|W(A3)| and 60 = 7+11+19+23 (+ 240 = 4*60) are machine-proved in experiments/lean4-carrier-rigidity/TfptCarrier/G31Orders.lean (lake build green, no sorry). [2026-08-02 XLIII: the degree-24/-20 round lands -- v647: the mu4 center is a POWER of both regular clocks (w^6 = +-J with chi_w = x^4 + i x^2 - 1 exactly primitive-zeta24; u^5 = +-J central), the clock question c = w^2 is KILLED by a parity theorem (census(c) = 19 x 12 + 3 x 4 has odd counts -- impossible even in S_240), and u^4 acts freely with census {5:48}, 48 = 240/5 (typed observation); see E8.ST31DEG.01.] [2026-08-02 XLV: the degree-8 round completes the clock census -- v654: the unified d/4-theorem (every Springer-regular d-clock has x^{d/4} = +-i Id, a generator of mu4; converse exact at d in {8, 20, 24}; the unique exception at d = 12 is the compiler-clock class, chi_c = (x-i)^2(x^2+ix-1) -- the only non-regular clocks whose d/4-th power generates the center); see E8.ST31D8.01.] [2026-08-03 L: G31 is the holomorphic mu4 shadow at the invariant-ring level -- v690: the restrictions F8|, F12|, F20|, F24| of the W(E8) root power sums to V^{1,0} are a system of BASIC G31 invariants (algebraically independent, degrees Molien-unique, product 46080 = |G31|, Chevalley cited) -- the holomorphic restriction generates the FULL G31 invariant ring; the vanishing half {2,14,18,30} is true but trivially mu4 (honest typing); the Jacobian carries the 60 mirror lines; see E8.QUARTICHALF.01.]

Original classification detail

ST31-STRUCTURE-DECIDED: G31 = full unitary stabilizer, degrees (8,12,20,24) Molien-unique, sigma = c^4 / J = c^9 exact, real structure (Z4 o 2^{1+4}).Sp4(2) computed; |G31| = |W(D5)|*|W(A3)| stays NUMEROLOGY (killed three ways); the compiler clock is NOT the regular 12-element; no marker moves.

Dependencies as recorded

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