Row 579 (DEG2 = C6): the last open row of the one-sided ledger, 603/603 KILLED (conditional on one rerun slot); follow-up to #3154

[DreamingOfClouds]

Follow-up to #3154 (research/p112-reduction-to-conjectures). No kappa claim, no window, no supplier word. Pure finite combinatorics on the E8 root lines.

What this adds

• research/p112-row579-deg2/ closes conjecture C6 = DEG2 of C(P112) >= 64: complete reduction to stated conjectures #3154 (ledger row 579 [7^8,4,4,4,3], d = 8). Chain: Theorem K5 (no five sevens with an A4 pair pattern, size-only) and Lemmas K5^- and K5^-2 give at least 6 non-root pair directions among the 28 pairs; the only possible non-root types are D7, E6A1, E6, A7, D6 (minimum leftovers 14, 23, 24, 25, 26; all other supports >= 31); DEG2 (no seven carries two non-root pair directions) makes the non-root graph a matching, e_N <= 4 < 6, a contradiction.
• DEG2 is closed in a mask model (necessary conditions only: finite mask set, parallel exclusion, Lemma F triangle condition, |union| <= 28, 3F4+F3 cover); MODEL.md proves that this is a sound relaxation of Lemma F. Implementation A (DEG2A): 4-seven survivors only for La-orbit reps 1 and 8, five-sets exist, six-sets 0 (all 10 reps; rerunnable in chunks). Implementation C (DEG2C/d2fix.c, an independently written search with the FIXED cover oracle; a bug in the first version of its oracle was found and fixed, see MODEL.md).
• Ledger: data/ledger4_consolidated.csv = C(P112) >= 64: complete reduction to stated conjectures #3154's ledger3 with row 579 KILLED: 603/603 KILLED, recomputed by verify.sh.
• Every claim has a rerunnable check with two implementations where claimed; PROOF_MAP.md maps claim to check; verify.sh default run reproduces K579T (impls A and B), R579b K5^-/K5^-2 (impls A and B), the type list, and the DEG2 impl A five-/six-set stages; FULL=1 adds pairmin and every quad-stage chunk.

Open slot (not a claim)

SLOT-C6-IMPL-C: impl C fixed rerun: six-set count = ?, controls = ? (to be filled in by the author before this leaves draft). Until it is filled, agreement of the second implementation is not claimed and the row-579 kill is conditional on it.

Credit

• conorholds' Prove the E8 d=0 kernel floor and scoped rigid C1 bounds #3166 closed C7 (the d = 0 sliver). With this PR (C6) the whole one-sided case K2 (L120 suffices, via One-sided padding lemma: P112 to P120 at cost exactly +8 (L120 suffices for Delta >= n_nf) #3074) is complete.
• petalas's C5 work (test(research): certify the P112 deletion quantifier separation #3158 to Research: prove the sharp fifteen-line P112 two-six-flat cover floor #3185) is progress on C5 (two-sided FINITE beta R = 7, p = 4), which this PR does not touch; the two-sided case K1 still depends on C1-C5.

How to check

PYTHON=/path/to/python3 bash verify.sh [outdir] (Python >= 3.11 with numpy, cc; <= 3 processes, each under timeout 1200), FULL=1 bash verify.sh for the heavy reruns.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.

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