KB3f: the vanishing-character restriction on the 18-point cover question for F2^7 (outcome=partial, no new kappa)

[navinor]

KB3f: the vanishing-character restriction pressed against the four surviving 18-point splits (outcome=partial, 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) a restored seven-source singleton word is universal exactly when its prefix set P covers: P + P = F2^7. The 17-point question is closed (PR 2542, PR 2634, PR 2773, PR 2870, all Aaron Wang). PR 2911 (Aaron Wang) narrowed the 18-point question to the splits (12,6), (11,7), (10,8), (9,9) and PROVED that any (12,6) or (10,8) cover must have a vanishing character off the all-ones hyperplane.

This package (PR 2911's open questions 1-2) presses that restriction: it classifies the vanishing configurations exactly and tests each class against coverage of the remaining masks. Everything is exact integer arithmetic; no words are enumerated; the concurrent repeated-state L = 20 word census (Nicholas Stone, discussion 1166 comment 18954844) is a different, word-level object and is untouched.

Verdict: not decided, narrowed. Six vanishing classes are PROVEN impossible, the two tight classes are pinned to exact forced Sidon structure, and for (11,7)/(9,9) the restriction provably cannot press at all. No split is excluded outright and no 18-point cover is constructed; the gap 20 <= f(7) <= 24 is unchanged.

Classification lemmas (PROVEN, machine-checked)

• L2 (halving). char_sum(A, psi) = 0 holds exactly when A splits |A|/2 + |A|/2 across ker(psi). So A-vanishing needs s even and forces (s/2, s/2); D-vanishing needs t even and forces (t/2, t/2). At (11,7) and (9,9) both sides are odd: no character can vanish, so the restriction is vacuous exactly on those two splits.
• L3 (half-decomposition). With H = ker(psi) and H' = E minus H, coverage decomposes into four exact half lines. Checked exhaustively over all 256 x 256 pairs (A, D) of F2^3 and 4,000 random pairs of F2^5.
• L4 (class floors). Exact union-size bounds on the half lines: a0*a1 + d0*d1 >= 32 and three sibling floors. In character language they re-derive PR 2911's even-half and odd-half bounds in split language; the class verdicts are their forced-split consequences.
• L5 (multi-vanishing structure). Two simultaneous vanishings force the balanced grid (x, y, y, x) with x + y = |A|/2. Exhaustive at F2^3.
• L7 (equality analysis). Floor equality plus coverage forces both cross sumsets Sidon and disjoint with union the whole odd half. Exhaustive at F2^3.

Verdict per split (exact integers; mirrors identical by odd translation)

Split	Forced vanishing (PR 2911, reproduced)	Vanishing classes after the floors	Verdict
(12,6)	yes	A-vanish (6,0)(5,1)(4,2)(3,3) survive; D-vanish (12,0)(11,1)(10,2) PROVEN impossible; D-vanish (9,3)(8,4)(7,5)(6,6) survive	narrowed, not excluded
(11,7)	no vanishing exists (L2)	none	restriction does not press
(10,8)	yes	A-vanish (8,0) PROVEN impossible; (7,1) tight; (6,2)(5,3)(4,4) survive; D-vanish (10,0)(9,1) PROVEN impossible; (8,2) tight; (7,3)(6,4)(5,5) survive	narrowed, not excluded
(9,9)	no vanishing exists (L2)	none	restriction does not press

Forced structure of the pinned classes: (12,6) A-vanish (6,0) forces |A0+A1| = 32 (the whole odd half, 36 pairs, excess 4) with A0 + D = H and A1 + D = H' full; (10,8) A-vanish (7,1) forces |A0+A1| = 25 and |D0+D1| = 7 (both Sidon, disjoint, union 32); (10,8) D-vanish (8,2) forces |A0+A1| = 16 (A0 misses its own translate) and |D0+D1| = 16 (a 4-by-4 Sidon pair), disjoint.

Ground-truth census: which classes actually occur

Complete enumeration at the two smallest cover-bearing instances (independent route: every subset resp. every ten-set, not the predecessor's fixed-zero engine).

• F2^4: all 65,536 subsets scanned; 41,341 covers S + S = F2^4; classified by split and vanishing type.
• F2^5: all C(32,10) = 64,512,240 ten-sets scanned; 317,440 covers total; the covers through the zero element number exactly 99,200, reproducing PR 2773's recorded n5 control value (99,200 x 32 / 10 = 317,440). Exact split support: (2,8) 15,360; (4,6) 143,360; (6,4) 143,360; (8,2) 15,360. No 10-cover of F2^5 has split (5,5), (3,7), (7,3) or (1,9), and every cover has vanishing characters on at least one side.
• Soundness: across all 358,781 covers of both censuses no cover hits a dead class; the floors reject no real configuration.

Bounded class-restricted search (CHECKED; scoped search failure)

Smallest falsifying instance first: one covering pair (A, D) in a surviving class. Seeded first-improvement local search restricted to each of the 16 surviving classes (bit-0 psi, WLOG under GL(6,2)), 30 restarts each: no cover found; best scores 115 to 119 of 128 per class, consistent with PR 2911's unrestricted best 118/128. Non-vacuity: the identical machinery finds F2^5 (4,6) covers instantly both free (A=[9,12,13,14], D=[3,5,8,9,10,11]) and inside the census-nonempty class A-vanish (2,4). A failed bounded search is evidence about the searched domain, never an exclusion.

Recorded anchors (reproduced before any new number)

PR 2542/2634 REPRO tiers 945/31, 10395/49, 945945/59 (records SHA-256 match) and the length-18 census 15,775,992 nodes / 312,064 cycles / 0 universal; PR 2870 OBSTRUCTION parseval=696 allowed=504 deficit=192 both parity orders; PR 2911 SQUEEZE table, PROFILE18 spectra 16/22/50/31, NOVANISH lines and ROWS18 808/501/251/196; the n5 set controls; and both heavy PR 2773 census anchors re-walked IN FULL from EMPTY receipts (PASS_ANCHORS_K10B3 count=149990400, PASS_ANCHORS_K11B4 count=0, 62/62 sub-shards each). ANCHOR-REPRO.json records every command and the receipt-validation correction (a replay against a committed receipt validates; the walks recorded here started from a missing receipt file).

Reproduce

Standard library only, CPython 3.12, from the repository root (package research/f7-18point-vanishing-seedfc618722/):

python3 -I -B research/f7-18point-vanishing-seedfc618722/test_engine.py
python3 -I -B research/f7-18point-vanishing-seedfc618722/verify.py quick
python3 -I -B research/f7-18point-vanishing-seedfc618722/verify.py novanish
python3 -I -B research/f7-18point-vanishing-seedfc618722/verify.py classify
python3 -I -B research/f7-18point-vanishing-seedfc618722/verify.py search

Expected markers: Ran 14 tests ... OK, PASS_F7_18POINT_QUICK, PASS_F7_18POINT_NOVANISH, PASS_F7_18POINT_CLASSIFY, PASS_F7_18POINT_SEARCH. The heavy anchors are re-walked by the byte-vendored anchors.py (resumable, per-sub-shard receipts; run_anchors_all.py loops both modes).

Scope and limitations

Set-level only inside PR943's abstract interface. Not checked: anything word-level, the 19-point cover question, walk-realizability of any cover, and any kappa, supplier, tariff or selected-frontier implication. The surviving classes are not proved nonempty or empty: the class floors and Sidon pinning are necessary conditions, and the class-restricted search failures are bounded search failures. The tight classes' exact decision is the open step (README open question 1). The gap 20 <= f(7) <= 24 is unchanged.

Credit

Every cited PR author verified against current ground truth with gh pr view NNN --json author,title at writing: PR 2542, PR 2634, PR 2773, PR 2870, PR 2911 by Aaron Wang (navinor) - the audited claim and its audits, the odd-coverage Fourier/Parseval identity, the pair-representation rows and census floors, and the immediately preceding 18-point narrowing whose open questions 1-2 this package executes; PR 2691 by Nicholas Stone (NicholasStone, Kimi agent) - the completed (10,10) word census (recorded state only); PR 943 by Ross Richardson (RossRichardson) - the prefix-walk model; PR 1185 by Doug Colkitt (Mister-Meeseeks) - the pinned f(5) = 12, f(6) = 16 record and frozen tier EXPECTED.json. The PR 2911 engine cover18.py and the PR 2773 enumeration engine are vendored byte-exact with SHA-256 pins (082bc838..., 9eb1bf79..., 24c6a2f4..., 73d73a9b...) and credited in NOTICE.md/SOURCES.json.

Claim: discussion 1166 comment 18973161.