The holomorphic restriction of E₈ invariants

E8.QUARTICHALF.01

Original ledger classification: Symbolic [PROOF] mu4-orbit vanishing + [E] G31 invariance/independence/mirror census + [E, kill] complex-structure controls + [cited] Chevalley basic-invariant theorem (exact arithmetic; Python-only per GATE.WOLFRAM.02).

Original claim text

The quartic invariant restriction (review contract E8.QUARTIC.HALF.01) -- is G31 the holomorphic mu4 shadow of the E8 Coxeter compiler? ALIVE, with the honest split (one module, 22 checks, ~1 s, exact Gaussian-integer/Fraction arithmetic, no floats; discovery probe quartic_half_probe.py 22/22, verdict QUARTIC-HALF-ALIVE). THE VANISHING HALF [PROOF, typed honest]: {2,14,18,30} is TRUE but TRIVIAL -- W(J alpha) = -i W(alpha) for all 240 roots => every J-orbit contributes t^d (1 + (-i)^d + (-1)^d + i^d) = 0 for every d != 0 mod 4 (holds for 6, 10, 22, 26 too: mu4 orbit factor, NOT degree-list-specific); degrees 4/6 vanish on ANY q-isotropic subspace (f_4 = 576 q^2, f_6 = 1920 q^3 symbolically); F_4| has all 35 monomial coefficients = 0 exactly (the first G31-consistency point: mod-4 alone would allow F_4 != 0). THE LOAD-BEARING CONTENT [E, central]: F_8|, F_12|, F_20|, F_24| are G31-invariant (60/60 hermitian reflections + clock generators, exact), ALGEBRAICALLY INDEPENDENT (exact Jacobian != 0 off the mirrors), degrees (8,12,20,24) Molien-unique with product 46080 = |G31| => by Chevalley [cited] a system of BASIC INVARIANTS: the holomorphic restriction of the W(E8) invariant ring generates the FULL G31 invariant ring; the Jacobian carries the 60 mirror lines (deg Jac = 60 = sum(d_i - 1), on-mirror vanishing point-verified 60/60; the full factorization = cited standard theorem, NAMED residual). MUST-FAIL CONTROLS [E, kill]: J' = RJR^T (orthogonal, NOT root-preserving) breaks the informative half {14,18,22,26,30} while the trivial isotropy layers persist; the root-preserving J'' keeps the full pattern (the theorem is about the W(E8)-conjugacy class of compiler-compatible J); a random non-isotropic 4-space breaks even F_2; honest negative: signed-permutation J-conjugates canNOT serve as controls (they stay in W(D8)). No marker move. [2026-08-03 LI: formally verified -- the mu4-orbit vanishing mechanism (orbit factor 1 + (-i)^d + (-1)^d + i^d = 0 iff d != 0 mod 4, per-pair weight law, the full vanishing half incl. the trivial extras) and the quartic power sum Sum <alpha,x>^4 = 576 (Sum x_i^2)^2 over the explicit 240 roots (+ holomorphic isotropy) are kernel-checked in Lean 4 (experiments/lean4-carrier-rigidity/TfptCarrier/QuarticHalf.lean, lake build green, no sorry / native_decide); the Chevalley basic-invariant step remains CITED, not formalized.]

Original classification detail

QUARTIC-HALF-ALIVE: the vanishing half {2,14,18,30} proven symbolically but TRIVIAL (mu4 orbit factor, honest typing); the substance: F8/F12/F20/F24 algebraically independent G31 basic invariants (holomorphic restriction generates the FULL G31 ring, Chevalley cited); Jacobian carries the 60 mirrors (60/60 point-verified); controls fire exactly as demanded.

Dependencies as recorded

E8.ST31.01; E8.ST31DEG.01; v634; v647; v654