KB3c: 17-point cover audit - PR 2542's 'no 17-point set covers F2^7' against its own lemma table (no new kappa)

[navinor]

KB3c: 17-point cover audit — PR 2542's "no 17-point set covers F2^7" against its own lemma table (no new kappa)

Contributor: Aaron Wang, with help from opencode (substantial AI assistance in scoping, implementation, verification and writing; no model named; no independent human review claimed). No kappa, supplier, certificate or selected-frontier change.

Change and mathematical scope

In the PR943 prefix-walk model (PR 943 by Ross Richardson, research/universal-connector-boundary), universality of a restored seven-source singleton word is the covering condition P + P = F2^7 on its prefix set. PR 2542 by Aaron Wang (research/universal-f7-singleton-seedb859517b) excludes length 18 and asserts in its README: "No 17-point set covers F2^7". Its machine-checked lemma_a_table contradicts only the parity split rows (9,8)/(8,9), and its certificate records size17_splits = {"9_8": false, "8_9": false} only. The counting-feasible 17-point splits (|E|,|O|) are exactly (9,8),(8,9),(10,7),(7,10),(11,6),(6,11); the four non-(9,8) rows survive the identity as written (lemma_a_table(7, 10, 7) is feasible; PR 2634 by Aaron Wang already pins that in test_lemma_rows_survive_at_L20).

This package audits the claim at the set level and decides the surviving rows exactly:

1. The pair-representation identity is extended to all six rows with the sharp collision-multiplicity constraint: cross-sum collisions satisfy sum_o C(r_o, 2) = 2 sum_x A_x B_x with odd excess sum_o (r_o - 1) = s*t - 64, and C(r, 2) >= r - 1 per mask turns each excess into an integer collision profile. Rows survive iff a profile's c is exactly attainable over the shared support values. (9,8)/(8,9) re-killed (best 1, need 4); (10,7) forces c in {3,4,5,6,8} (six rows); (11,6) collapses to the single row class (e_ex, b_ex, c) = (e, 6-e, 1) with shared = 1.
2. An exact Sidon/near-Sidon census of the even group E = F2^6 shows the minimum pair-sum excess of any 10-point set is at least 3 (in particular F2^6 has no Sidon 10-set), and likewise at least 3 for 11-point sets. Complete shards through {0, 3}: 9-sets Sidon 17,776,640; 10-sets excess <= 2: 0; 11-sets excess <= 2: 0; 7-sets in the odd coset Sidon 34,122,816. The shard is existence-complete by GL(6,2) transitivity and invariance (the extension to the odd coset preserves pair sums, cross sums and coverage).
3. Result: the (10,7)/(7,10) splits are excluded exactly — after the census floor e_ex >= 3 the only identity row left attains c = 0, which is not a profile value. (11,6)/(6,11) are reduced to four rows (3,3),(4,2),(5,1),(6,0), each with one shared sum carrying A_x = B_x = 1. PR 2542's literal claim is now established for every split its length-18 exclusion needed and for the length-20-relevant splits; the length-22-relevant (11,6)/(6,11) rows remain open.

Exact controls validate the machinery: the enumeration agrees with naive brute force on a 16-element subgroup at budgets 0/1/2; independent.py re-derives every row by a different code path (direct count-vector loops); the exact n = 5 set censuses reproduce the community facts (zero nine-point covers, max 29 of 32; 99,200 ten-point covers of all C(31,9) = 20,160,075 sets); the odd-translation mirror is machine-checked.

Reproduce

From the repository root, standard library only, Python 3.12:

python3 -I -B research/f7-17point-cover-audit-seedcfa215c2/verify.py
python3 -I -B research/f7-17point-cover-audit-seedcfa215c2/independent.py
python3 -I -B research/f7-17point-cover-audit-seedcfa215c2/test_engine.py
python3 scripts/research_package.py --package research/f7-17point-cover-audit-seedcfa215c2 --check-only

Observed markers: PASS_17POINT_VERIFY seconds=148.5, PASS_17POINT_INDEPENDENT {...}, Ran 13 tests ... OK, PASS_PACKAGE_METADATA_AND_PINS. verify.py --full re-runs the four load-bearing zero-count census shards (~17 min) and compares them with SIDON-RECEIPT.json.

Before any new number the recorded predecessor state is reproduced byte-exactly (vendored engine SHA-256 pinned 82e4c407b58864c7…, enforced by test): the pinned tiers 945/31, 10395/49, 945945/59 with records SHA-256 against research/universal-connector-six-source/EXPECTED.json, and PR 2542's decisive length-18 census re-run in full: 15,775,992 nodes, 312,064 simple 18-cycles, zero universal words. In-session runs of PR 2634's own verifier at head e2f30d93 additionally reproduced its search campaign 120/126/128 at lengths 20/22/24 and its then-partial 20-cycle census (11/15 shards, 798,901,058 nodes, 67,674,177 cycles). The completed (10,10) census is recorded state from PR 2691 by Nicholas Stone (6,305,209,079 nodes, 597,148,314 simple 20-cycles, zero universal, 15/15 shards) and is not re-run here.

Pair-collision disclosure and object change

This session was assigned KB3c (finish the L=20 census and attack its recorded remainder: the repeated-state splits). The entry pair-collision check found both assigned options claimed or completed: option (a) is completed by PR 2691 (Nicholas Stone), and options (b)/(c) are the active claim of Nicholas Stone (Kimi agent) on the repeated-state L=20 word census (discussion 1166 comment 18954844). The object was therefore changed, per the pair-collision rule, to the set-level question the assignment itself names as the recorded remainder: PR 2542's overreaching "no 17-point set" claim. This package enumerates no words. Handoff to the concurrent claim: its (10,7)/(7,10) L=20 word splits are empty a fortiori, since a universal word there would yield a 17-point cover of split (10,7)/(7,10), excluded here.

Scope and limits

A finite set-level audit inside PR943's abstract interface. Not checked: realizability of the four surviving (11,6) rows (the joint (A, B) tree needs stabilizer canonical forms; the GL(6,2)-invariance lemma reduces the domain to 0, 3 in A but does not canonicalize pairs — this is the open next question of the package README), the 18-/19-point cover question r((C2)^7), and every word-level question. No kappa, supplier, tariff or selected-frontier change; the recorded gap 20 <= f(7) <= 24 is unchanged. Canonical main moved to 1696cf14e403b3eab35cd9eab27a4c2eb4e4ae38 during the session (priority revision 5); the base commit 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 is retained per the assignment pin, and certificates/selected-result.json is byte-identical between the two.

Credit

• PR 2542 by Aaron Wang (navinor): the audited claim, the pair-representation identity (lemma_a_table), the length-18 exclusion; its engine.py is vendored byte-exact with a hash pin.
• PR 2634 by Aaron Wang (navinor): the recorded KB3b state, the L=20 campaign and census, and the pinned (10,7) identity row.
• PR 2691 by Nicholas Stone (NicholasStone, Kimi agent, substantial AI assistance): the completed (10,10) simple-20-cycle census and its released independent audit.
• PR 943 by Ross Richardson (RossRichardson): the prefix-walk model.
• PR 1185 by Doug Colkitt (Mister-Meeseeks): the pinned record f(5) = 12, f(6) = 16, 18 <= f(7) <= 24.

[navinor]

k=11 follow-up census complete (commit 6739c799) — the (3,3) and (4,2) rows of the (11,6)/(6,11) family are excluded; the row table is now 2 rows.

The resumable raw-domain walk of 11-point sets of E = F2^6 through {0, 3} (existence-complete by GL(6,2) transitivity; sub-sharded by the smallest free element, 62/62 sub-shards with per-sub-shard checkpoints in RAW11-PROGRESS.json) finds zero 11-sets of pair-sum excess <= 4:

• level 3 (excess exactly 3): 0
• level 4 (excess exactly 4): 0

So the minimum pair-sum excess of an 11-point subset of F2^6 is at least 5 (for comparison, 10-point sets attain exactly 3, with 149,990,400 excess-3 sets in the same sharding). Applying that floor to the sharp identity rows of this PR: rows (3,3) and (4,2) die (e_ex too small), and the surviving (11,6)/(6,11) rows are exactly two: (5,1) and (6,0), each with one shared sum carrying A_x = B_x = 1. Final row table for the audited claim "No 17-point set covers F2^7":

• (9,8)/(8,9) — dead in the identity (PR 2542's own rows, re-derived)
• (10,7)/(7,10) — excluded exactly (zero surviving rows at the census floor e_ex >= 3)
• (11,6)/(6,11) — two rows open: (5,1), (6,0)

The literal claim is therefore established for four of the six counting-feasible splits; the two (11,6)/(6,11) rows are the whole remaining content, and they are parity-relevant only at L = 22.

Verification (fresh, CPython 3.12, registered replay): PASS_17POINT_VERIFY seconds=115.3 (pinned tiers 945/31, 10395/49, 945945/59; PR 2542's L18 census 15,775,992 / 312,064 / 0; K11 census complete: 62/62 sub-shards, level3=0 level4=0), PASS_17POINT_INDEPENDENT (receipt consistency with progress_agrees: true, rows re-derived by a second code path), Ran 13 tests ... OK, PASS_REGISTERED_RESEARCH_PACKAGE research/f7-17point-cover-audit-seedcfa215c2 outcome=partial, PASS_PACKAGE_METADATA_AND_PINS.

Credit (authors verified against current ground truth at writing): PR 2542 by Aaron Wang (navinor) — the audited claim, the identity, the vendored engine; PR 2634 by Aaron Wang (navinor) — the recorded KB3b state and the pinned (10,7) row; PR 2691 by Nicholas Stone (NicholasStone, Kimi agent) — the completed (10,10) census whose recorded remainder named this follow-up.

Contributor: Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed). No kappa, supplier or selected-frontier change; the gap 20 <= f(7) <= 24 is unchanged.