TFPT scientific source excerpt
Original repository path: verification/v690_quartic_half.py
Original file SHA-256: 489c6c034e8a79cbe281dc09cb17d3d84ffe7e3eb46df4b7b92c46d6ccacb324
Inclusive original lines: 6-40
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000006 | HYPOTHESIS (external review, theorem candidate): restricting the
000007 | W(E8)-invariant root power sums f_d(x) = sum_roots <alpha, x>^d
000008 | (d in Deg W(E8) = {2,8,12,14,18,20,24,30}) to the holomorphic
000009 | eigenspace V^{1,0} = ker(J - i) (dim 4) kills exactly the half
000010 | {2,14,18,30} (degrees = 2 mod 4) and keeps {8,12,20,24} = Deg(G31)
000011 | alive -- 'G31 is the holomorphic mu4 shadow of the E8 compiler'.
000012 | 
000013 | RADICAL HONESTY -- the review's own Fallstrick, resolved in layers:
000014 | 
000015 |   (L1) the mu4-orbit mechanism: for x in V^{1,0} and any root alpha,
000016 |        <J alpha, x> = -i <alpha, x> EXACTLY (verified for all 240
000017 |        roots as a weight identity), so summing over each J-orbit
000018 |        {a, Ja, -a, -Ja} gives the factor 1 + (-i)^d + (-1)^d + i^d
000019 |        = 0 for ALL d != 0 mod 4.  The claimed vanishing {2,14,18,30}
000020 |        is therefore TRIVIAL and not degree-list-specific: 6, 10, 22,
000021 |        26 vanish identically too (checked).  This is a symbolic
000022 |        PROOF of the vanishing half, but it carries no G31 content.
000023 |   (L2) the isotropy layer: V^{1,0} of ANY orthogonal complex
000024 |        structure is q-isotropic (q = <x,x>), and W(E8) has NO basic
000025 |        invariant in degrees 4, 6, 10 (f_4 ~ q^2, f_6 ~ q^3, f_10 in
000026 |        q-multiples -- f_4, f_6 proportionality verified SYMBOLICALLY
000027 |        as 8-variable polynomial identities here).  So F_4| = F_6| =
000028 |        F_10| = 0 persists even on non-compiler complex structures.
000029 |   (L3) the SUBSTANCE: F_8|, F_12|, F_20|, F_24| are nonzero (exact
000030 |        points; a nonzero value at one exact point is a rigorous
000031 |        nonvanishing proof), G31-invariant (all 60 reflections + the
000032 |        clock generators sigma, J, exact Gaussian arithmetic), and
000033 |        ALGEBRAICALLY INDEPENDENT (exact Jacobian != 0).  With
000034 |        degrees (8,12,20,24) = the Molien-unique degree quadruple of
000035 |        G31 (v634) and 8*12*20*24 = 46080 = |G31|, Chevalley's
000036 |        theorem [maths-lit fact, cited not proved] makes them a
000037 |        system of BASIC INVARIANTS: the restriction of the W(E8)
000038 |        invariant ring to V^{1,0} generates the FULL G31 invariant
000039 |        ring.  That -- not the vanishing pattern -- is the theorem
000040 |        candidate's load-bearing content.