KB3e: the 18-point cover question for F2^7 narrowed to an exact finite remainder (outcome=partial, no new kappa)

[navinor]

KB3e: the 18-point cover question for F2^7 — narrowed to an exact finite remainder (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 (RossRichardson)), a restored seven-source singleton word is universal exactly when its prefix set P covers: P + P = F2^7 — every mask of F2^7 is a pair-sum a + b of two elements of the set (self-pairs cover mask 0; the sumset is translation-invariant, so WLOG 0 is in S). PR 2542 by Aaron Wang (navinor) excludes length 18 and claims "No 17-point set covers F2^7"; PR 2773 by Aaron Wang audits it split-by-split; PR 2870 by Aaron Wang closes the remainder rows (5,1),(6,0) with an exact odd-coverage Fourier/Parseval identity (parseval 696 versus allowed 504, deficit 192), completing the 17-point question at the set level.

This package (kappa-blocker KB3e, an authorized explicit-open-request scope change recorded in TASK.json) asks the immediate successor, PR 2870's open question 1: does any 18-point set cover F2^7? It is the deciding set-level object for the f(7) L = 22 route: a cover keeps L = 22 alive and supplies its needed prefix set; excluding covers kills L = 22 outright. A parity split S = A u B has A in the even coset (s points) and B in the odd coset (t points); the counting-feasible 18-point splits are (12,6), (11,7), (10,8), (9,9) and their mirrors. Every 18-point split has s and t of equal character parity — exactly the case the PR 2870 squeeze does not reach.

Verdict: outcome (c) of the assignment, labelled outcome=partial in TASK.json. No 18-point cover was constructed and no split was excluded. Each surviving split's exact obstruction is recorded below; the finite remainder is exact and enumerated.

Verdict per split (exact integers)

Split	Odd excess	PR 2870 squeeze	Parseval remainders (A, B)	Allowed per side	Deficit	Profile system	Rows after floors	Search best of 128
(12, 6)	8	none (equal parity)	1248, 696	none (vanishing possible)	non-positive	feasible (16 spectra)	808	118
(11, 7)	13	none (equal parity)	1166, 798	21294	-20128, -20496	feasible (22 spectra)	501	118
(10, 8)	16	none (equal parity)	1080, 896	none (vanishing possible)	non-positive	feasible (50 spectra)	251	118
(9, 9)	17	none (equal parity)	990, 990	36414	-35424, -35424	feasible (31 spectra, partly undecided)	196	117

Controls: the same profile system reports (11,6) infeasible (reproducing PR 2870's exclusion exactly, including the 696/504/192 squeeze numbers) and (10,7) infeasible (consistent with PR 2773's census exclusion), and accepts the split (4,6) of a real 10-point cover of F2^5 with no false contradiction.

The layers and their exact obstructions

1. Reduction lemma (PROVEN). With D = p + B for any odd mask p (so D lies in the even group E), S = A u B covers F2^n if and only if A + D = E and (A + A) u (D + D) = E. Checked exhaustively over all 256 subsets of F2^3 and on the real F2^5 cover. Each split becomes a two-set covering problem in E, isomorphic to F2^6.
2. Fourier/Parseval re-derivation with tracked deficit (PROVEN bookkeeping; no exclusion). The PR 2870 squeeze needs opposite character parity; at 18 points it never holds and all tracked deficits are non-positive (table). The squeeze as an 18-point exclusion mechanism is DEMOTED; PR 2870 predicted this.
3. Per-character profile system (CHECKED; two PROVEN consequences). Necessary conditions: character parities; the odd-half identity f_hat(u) g_hat(u) = rho_hat(u) with absolute value of rho_hat(u) at most E_o and rho_hat(u) congruent to E_o mod 2; the even-half bound f_hat(u)^2 + g_hat(u)^2 at most s + t + 2P - 2^n with P = C(s,2) + C(t,2); Parseval on both sides; and the exact attainable sums of squares of rho. The constraints hold on every real cover of F2^3 and F2^4 (exhaustive witness battery) and F2^5. The system excludes no 18-point split, but the no-vanishing sub-case is infeasible for (12,6) and (10,8) (PROVEN): e.g. for (12,6) a no-vanishing spectrum is forced to 24, 8 and 31 characters of shapes (2,2), (2,4) and (4,2), giving sum rho^2 = 46, which no nonnegative integer rho of total 8 attains (attainable values are 8, 10, 12, through 40, then 50 and 64). So any (12,6) or (10,8) cover must have at least one character outside the span of the all-ones vector where one side's character vanishes.
4. Pair-representation rows with census floors (CHECKED). PR 2773's rows (e_ex, b_ex, c) extended to the four splits (its byte-vendored shape/draw primitives with a contingency-table pairing-sums replacement for its O(m!) pairing loop, validated equal to it wherever it terminates). Floors: 0 for 6 to 9 points (Sidon k-sets exist), 3 for 10 points (fresh-proven here), 5 for 11 points (reproduced walk), 3 for 12 points (counting). Remainders: 808, 501, 251 and 196 surviving rows over 120, 36, 28 and 45 distinct (e_ex, b_ex) pairs. The "small remainder" expectation of the 17-point case is DEMOTED at 18 points; these row lists are the finite remainder.
5. Bounded construction search (CHECKED; scoped search failure). The smallest falsifying instance — one covering set — was searched first: direct seeded local search at k = 18 (200 restarts) tops out at 118 of 128 covered masks; the structured (A, D) searches per split (60 restarts each; the one bounded pivot after the direct search failed) top out at 118, 118, 118 and 117; a k = 19 probe reaches 124 of 128, matching the maximum PR 2634 recorded for k = 19. Non-vacuity: the identical machinery finds 20-point covers in 5 of 5 short runs, matching PR 2634's recorded abundance of 20-point covers and its own failure to find 19-point covers in 200 restarts. A failed bounded search is evidence about the searched domain only and is never an exclusion.

Recorded anchors (reproduced before any new number)

Anchor	Value	Status here
PR 2542/2634 REPRO tiers	945/31, 10395/49, 945945/59 (records SHA-256 match)	recomputed
PR 2542 length-18 census	15,775,992 nodes, 312,064 cycles, 0 universal	recomputed
PR 2870 OBSTRUCTION lines	parseval 696, allowed 504, deficit 192, both parity orders	recomputed
PR 2773 row table	(11,6)/(6,11) remainder (5,1),(6,0)	recomputed
n=5 set controls	0 nine-point covers (max 29 of 32), 99,200 ten-point covers	recomputed
PR 2773 k10 budget-3	149,990,400 (every leaf at exact excess 3)	recomputed in full (fresh empty-receipt walk, 62 of 62 sub-shards)
PR 2773 k11 levels 3 and 4	0, 0	recomputed in full (fresh empty-receipt walk, 62 of 62 sub-shards)

ANCHOR-REPRO.json records commands and observed outputs, including one correction: an earlier PASS_ANCHORS_* print from the extracted PR 2870 package was receipt validation (its committed receipt pre-filled all sub-shards), not a re-walk; the walks recorded here started from an empty receipt. The four zero shards (k10 budgets 0/1/2, k11 budget 2) are receipt-validated, not re-walked; fresh k10-budget-2 and k11-budget-2 walks independently confirm the two load-bearing zero facts.

Reproduce

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

python3 -I -B research/f7-18point-cover-question-seed6d34356c/test_engine.py
python3 -I -B research/f7-18point-cover-question-seed6d34356c/verify.py quick
python3 -I -B research/f7-18point-cover-question-seed6d34356c/verify.py fourier
python3 -I -B research/f7-18point-cover-question-seed6d34356c/verify.py rows
python3 -I -B research/f7-18point-cover-question-seed6d34356c/verify.py search

Observed markers: Ran 21 tests ... OK, PASS_F7_18POINT_QUICK, PASS_F7_18POINT_FOURIER (378.5 s), PASS_F7_18POINT_ROWS, PASS_F7_18POINT_SEARCH. The heavy census anchors recompute via the byte-vendored PR 2870 driver (anchors.py k10b3, anchors.py k11b4; resumable, per-sub-shard receipts in ANCHOR-PROGRESS.json, one progress section per mode): observed PASS_ANCHORS_K10B3 count=149990400 total_hist={'3': 149990400} shards=62 and PASS_ANCHORS_K11B4 count=0 total_hist={} shards=62. Registered replay metadata passes (PASS_PACKAGE_METADATA_AND_PINS).

Labels

• PROVEN: the (A, D) reduction lemma; the vanishing-character necessity for the (12,6) and (10,8) splits; the deficit bookkeeping showing the PR 2870 squeeze cannot reach any 18-point split.
• CHECKED: every recorded anchor reproduced; the exact row tables after floors; the profile-system status per split (with the (9,9) step-cap region labelled undecided, never a proof); the bounded-search record with its non-vacuity control.
• DEMOTED: the direct extension of the PR 2870 squeeze to 18-point splits as an exclusion mechanism; the expectation of a small row remainder at 18 points.

Scope, limitations and concurrent work

Set-level only, inside PR943's abstract interface. Not checked: everything word-level; the 19-point question; walk-realizability of any cover; any kappa, supplier, tariff or selected-frontier implication. The profile system supplies necessary conditions only — feasibility is never a construction. The gap 20 <= f(7) <= 24 is unchanged. The repeated-state L = 20 word census (splits (10,9)/(10,8),(9,9) at the word level) is the concurrent claim of Nicholas Stone (NicholasStone, Kimi agent), discussion 1166 comment 18954844; this package stays strictly on the SET-level object, its row tables cover the (10,8)/(9,9) set-level part of those splits, and (10,9) is a 19-point object untouched here. Pair-collision checks (discussions 1317 and 1166 with nested replies, before computation and again pre-submission) found no claim on the 18-point set object.

Credit

Every cited PR author was verified against current ground truth with gh pr view NNN --json author,title at writing time: PR 2542, PR 2634, PR 2773 and PR 2870 by Aaron Wang (navinor) — the audited claim, the split-by-split audit with the Sidon/pair-sum budget enumeration (cover17.py/search17.py, vendored byte-exact with SHA-256 pins 082bc838.../9eb1bf79..., extracted with the toolkit's pr-tree-extract from PR 2870's hash-pinned vendor/), and the odd-coverage Fourier/Parseval identity plus the resumable anchor driver (anchors.py, vendored byte-exact, pin 73d73a9b...); 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 the frozen tier EXPECTED.json reproduced as an anchor.

CLAIM thread: discussion 1166 comment 18968355.