KB3g: the tight classes of the 18-point cover question for F2^7 - (7,1)/(8,2)/(6,0) PROVEN impossible, (11,7)/(9,9) gap pressed (outcome=partial, no new kappa)

[navinor]

KB3g: the tight classes of the 18-point cover question for F2^7 (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. This package executes the recorded remainder of PR 2933 (its open questions 1 and 3) at the SET level, exact integers only: it decides the three tightest surviving classes of the 18-point cover question and presses the identity gap at (11,7)/(9,9). The question itself is not decided (no 18-point cover constructed, no split excluded outright), but its three tightest classes are gone.

Record chain, all authors 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, PR 2933 by Aaron Wang (navinor); PR 2691 by Nicholas Stone (NicholasStone) is the concurrent L=20 WORD census (recorded state only; untouched here).

Verdict per class (exact integers throughout)

Split	Class	PR 2933 status	Verdict here	Deciding evidence
(10,8)	A-vanish (7,1), tight 32=32	pinned to Sidon	PROVEN impossible	Fourier contradiction 175 vs 239; zero Sidon 5-pairs exist
(10,8)	D-vanish (8,2), tight 32=32	pinned to Sidon	PROVEN impossible	all 28,815,360 equality configurations fail all three coverage lines
(12,6)	A-vanish (6,0), full odd half	pinned to full sumset	PROVEN impossible	no 6-set D of F2^5 has any 6-set X with X + D = F2^5 (all 169,911 normalized D walked)
(11,7)	no vanishing possible (L2)	row remainder 501	narrowed: sum rho^2 >= 17 PROVEN	cross-sum multiplicity identity; no contradiction
(9,9)	no vanishing possible (L2)	row remainder 196	narrowed: sum rho^2 >= 19 PROVEN	same identity; no contradiction

Surviving after this package: (12,6) A-vanish (5,1)(4,2)(3,3) and D-vanish (9,3)(8,4)(7,5)(6,6); (10,8) A-vanish (6,2)(5,3)(4,4) and D-vanish (7,3)(6,4)(5,5); (11,7)/(9,9) at their row remainders with the new multiplicity conditions. Mirrors are identical by odd translation.

(10,8) A-vanish (7,1): an exact Fourier contradiction, and empty enumeration

PR 2933's L7 pins every cover of the class to |A0+A1| = 25 with |A0| = |A1| = 5 (all 25 cross sums distinct) inside F2^5. With f = 1_A0, g = 1_A1, h = 1_S on S = A0+A1, Fourier over F2^5 gives h_hat = f_hat * g_hat pointwise, so Parseval forces sum_{v != 0} h_hat^2 = 32*25 - 625 = 175 while sum_{v != 0} f_hat^2 = sum_{v != 0} g_hat^2 = 32*5 - 25 = 135. Both sizes are odd, so both Walsh transforms are odd and f_hat^2 >= 1 pointwise; from (f_hat^2 - 1)(g_hat^2 - 1) >= 0 one gets sum h_hat^2 >= 135 + 135 - 31 = 239 > 175: contradiction. The m=4 analogue computes need = lower = 7 exactly (no contradiction), matching its 512 real covers - the identity is tight exactly where covers exist. Independently, the complete enumeration of normalized Sidon 5-pairs finds zero pairs (validated equal to full brute force, 5040 = 5040 at m=4; 2M random F2^5 pairs top out at |A0+A1| = 23), and coverage is independent of the free parameter d1 (proven; machine-checked).

(10,8) D-vanish (8,2): 28,815,360 equality configurations, all failing

L7 pins the class to |A0+A1| = 16 and |D0+D1| = 16, disjoint with union the whole odd half. Normalizing 0 in D0, 0 in D1 and delta = 1 under GL(5,2) (half-translations preserve all four coverage lines - machine-checked), the A-side is parametrized by w (32 values, coverage independent of w, proven) and one choice element per {x, x+1} pair inside the complement of D0+D1 (256 choices). Complete walk: 4,287,920 normalized Sidon 4-pairs, 112,560 with 1-invariant complement; all 28,815,360 (pair, choice) configurations fail all three translated coverage lines simultaneously (fail_hist = {("I","II","III"): 28815360}), zero covers. The reduction chain is validated end-to-end at m=4: the four L3 lines and a DIRECT S + S = F2^5 cover test agree on all 4,096 analogue configurations, 512 covers accepted by both routes.

(12,6) A-vanish (6,0): no full-sumset pair of 6-sets exists in F2^5

The class forces A0+A1 = F2^5 (36 pairs, excess 4) and A0 + D = A1 + D = F2^5 with |D| = 6. Two exact layers. First, a multiplicity classification (PROVEN): for any 6-set pair X, Y with X + Y = F2^5, r = 1 + rho with sum rho = 4, and per nonzero v the product f_hat(v) * g_hat(v) lies in {0, 4, -4}; exactly two excess patterns survive the Walsh constraints (rho = 2+2 on two points, and rho = 1+1+1+1 on an affine 2-flat {p1, p2, p3, p1+p2+p3}), while 4*delta_p, 3+1 and 2+1+1 are PROVEN dead by exact integer constraints. Second, a complete transversal walk (PROVEN): X + D = F2^5 says X hits all 32 translates of D; all 169,911 normalized 6-sets D were walked with a complete branch-on-uncovered search and every D has zero 6-set transversals, so lines (I'')/(II'') are unsatisfiable. Positive control: the same search recovers real transversals at m=3; a miniature cross-check finds no 3+3 full pair in F2^3 either (all 3,136 pairs).

(11,7)/(9,9) identity gap: pressed, not cut

One bounded pivot on the named next identity (cross-sum multiplicities; the vanishing route is vacuous there by PR 2933's L2). With A an s-set and D a t-set covering A + D = E = F2^6, r = 1 + rho with sum rho = s*t - 64; both sides odd make both Walsh transforms odd, and the same pointwise inequality yields the exact necessary condition 64 * sum rho^2 - (s*t - 64)^2 >= (64*s - s^2) + (64*t - t^2) - 63, i.e. sum rho^2 >= 17 at (11,7) (excess 13) and >= 19 at (9,9) (excess 17). Both bounds are attainable, so the identity does not contradict coverage: DEMOTED as an exclusion mechanism (the excess is too large for the rigidity that killed (7,1)). It does give PROVEN new necessary conditions: the all-ones excess pattern is impossible on both splits, and at (11,7) also the single-double pattern (2, 1^11). Validated positively on every odd-odd-split cover of F2^4. The row remainders 501/196 stand.

Anchors reproduced before any new number

Anchor	Value	Status here
PR 2870 OBSTRUCTION	parseval=696 allowed=504 deficit=192	recomputed fresh
PR 2911 SQUEEZE table	1248/696; 1166/798 vs 21294; 1080/896; 990/990	recomputed fresh
PR 2911 NOVANISH	(12,6), (10,8) infeasible: vanishing forced	recomputed fresh
PR 2911 ROWS18	808/501/251/196	recomputed fresh
PR 2933 CLASSIFY lines	both (7,1)/(8,2) value=32 need=32 TIGHT; L7 m=3 cases=6720	recomputed fresh
PR 2773 k10 budget-3	149,990,400	re-walked in full from empty receipts, 62/62 sub-shards
PR 2773 k11 levels 3-4	0, 0	re-walked in full from empty receipts, 62/62 sub-shards

Receipt trap recorded in ANCHOR-REPRO.json: a resumed walk against a committed receipt silently passes; the two heavy anchors here started from a missing ANCHOR-PROGRESS.json (fresh receipts confirmed printed by the runner).

Reproduce

Standard library only, CPython 3.12, from the repository root:

python3 -I -B research/f7-18point-tight-classes-seed4c1e7b30/test_engine.py
python3 -I -B research/f7-18point-tight-classes-seed4c1e7b30/verify.py quick
python3 -I -B research/f7-18point-tight-classes-seed4c1e7b30/verify.py novanish
python3 -I -B research/f7-18point-tight-classes-seed4c1e7b30/verify.py rows
python3 -I -B research/f7-18point-tight-classes-seed4c1e7b30/verify.py classes
python3 -I -B research/f7-18point-tight-classes-seed4c1e7b30/verify.py anchors

Expected markers: Ran 13 tests ... OK, PASS_TIGHT_QUICK, PASS_TIGHT_NOVANISH, PASS_TIGHT_ROWS, PASS_TIGHT_CLASSES, PASS_TIGHT_ANCHORS. The class walks are resumable with per-shard receipts (TIGHT-PROGRESS.json, one section per mode; TIGHT-C60-*.json per shard of the (6,0) walk; run_tight.py merge60 merges them). Registered replay: python3 scripts/research_package.py --package research/f7-18point-tight-classes-seed4c1e7b30 --check-only passes PASS_PACKAGE_METADATA_AND_PINS.

Scope and limitations

Set-level only inside PR943's abstract interface. Not checked: anything word-level (the repeated-state L=20 word census is Nicholas Stone's concurrent claim, discussion 1166 comment 18954844; untouched here), the 19-point cover question, walk-realizability of any cover, and any kappa, supplier, tariff or selected-frontier implication. The surviving non-tight classes are not decided and no claim is made about them. The (11,7)/(9,9) multiplicity conditions are necessary, not sufficient; no row of the PR 2911 remainder is excluded by them. The m=4 analogue controls validate the reduction chain but are not a proof at m=6; the m=6 results rest on the exhaustive enumerations (13 machine tests, byte-pinned engines) and the exact identities. The gap 20 <= f(7) <= 24 is unchanged. Discussion post: FINDING/DONE comment 18975496.

Open next questions

1. Decide the surviving non-tight classes of (12,6) and (10,8): press L5's multi-vanishing grid (counts (x, y, y, x)) at two characters against the class floors, starting with (5,1) and (6,2). Why it matters: clearing both splits settles two of the four 18-point splits. Blocked by: nothing structural; a bounded joint enumeration.
2. Find the sharper (11,7)/(9,9) identity: combine the proven sum-rho^2 conditions with the even half (A+A) u (D+D) = E multiplicities (PR 2870 used only odd coverage) as a second layer on the 501/196 rows. Why it matters: those splits carry the other half of the question. Blocked by: no known coupling identity between odd-line cross-sum multiplicities and even-line self-sums.
3. Search for an 18-point cover outside the decided classes: seeded search over the (11,7)/(9,9) row remainder profiles with the new multiplicity conditions as admissibility filters. Why it matters: a construction keeps the L=22 route alive; search best remains 118/128 at k=18. Blocked by: nothing structural; a failed bounded search is not an exclusion.