KB3i: the 18-point cover remainder for F2^7 - parity shortcut DEMOTED, m=5 obstruction localized (outcome=partial, no new kappa)

[navinor]

KB3i: the 18-point cover remainder for F2^7 - parity shortcut DEMOTED, m=5 obstruction localized (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); PR 2911 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; PR 2933 classified the vanishing configurations and recorded the class floors; PR 2961 decided the three tightest classes and fully excluded (10,8); PR 3023 killed (12,6) A-vanish (5,1) and recorded the remainder this PR takes: the surviving (12,6) A-vanish classes (4,2), (3,3) plus the both-odd splits (11,7), (9,9), which carry no vanishing character. This PR is strictly SET-level: no words are enumerated.

Record chain authors, all verified against current ground truth with gh pr view NNN --json author,title at writing: PR 2542/2634/2773/2870/2911/2933/2961/3023 are Aaron Wang (navinor); PR 2691 and PR 2973 are Nicholas Stone (NicholasStone); PR 3005 is Nick Petalas (petalas); PR 1185 is Mister-Meeseeks.

Concurrent word-level state, recorded honestly: PR 2973 (merged) corrected a false-prune defect in the word-census budgets and records 22 <= f(7) <= 24; PR 3005 independently corroborates it. PR 2973's own scope note records the set-level chain PR 2773-3023 as unaffected. Under 22 <= f(7) <= 24 this PR's question is the deciding necessary condition for f(7) = 22: if no 18-point set covers F2^7, every universal word at L = 22 needs at least 19 distinct prefixes.

Verdict per class (exact integers)

Split	Class	PR 2933 floor	Status here	Deciding evidence
(12,6)	A-vanish (4,2)	44 >= 32	NOT decided (walk-ready)	exact reduction D1 = {0,1} under GL(5,2), budgets 20/20/24; walk too large
(12,6)	A-vanish (3,3)	45 >= 32	NOT decided (walk-ready)	four L3 lines direct, all budgets slack, no forced structure
(11,7)	both-odd, no vanishing char	-	NOT excluded; parity shortcut DEMOTED	both-odd covers exist at m=3 (complete census); frontier E_o = 13, min sum rho^2 = 17
(9,9)	both-odd, no vanishing char	-	NOT excluded; parity shortcut DEMOTED	both-odd covers exist at m=3 (complete census); frontier E_o = 17, min sum rho^2 = 19

No class is decided here. The (4,2)/(3,3) reductions and budgets are necessary conditions and exact finite search domains (PR 3023's classification, reproduced), not exclusions.

Route (b): the parity shortcut is PROVEN false (DEMOTED)

The cleanest way to break (11,7)/(9,9) would be a parity rule: a covering reduced pair (A,B) of F2^m (i.e. A + B = F2^m and (A+A) u (B+B) = F2^m) must have |A| and |B| both even. Then the both-odd splits (11,7)/(9,9) would be excluded and the 18-point question would collapse to (12,6)/(10,8).

This is false. A complete enumeration of the reduced covering problem in F2^3 (all 2^8 x 2^8 subset pairs) finds covering pairs in every both-odd split (3,5), (3,7), (5,3), (5,5), (5,7), (7,3), (7,5), (7,7) - for example (5,5) has 3136 covers and (7,7) has 64. So there is no dimension-free parity obstruction, and none can be used to kill (11,7)/(9,9). (Among small both-odd splits only (3,3) is empty.) This is a PROVEN negative on the shortcut, not on the splits themselves.

The m=5 ground truth and its obstruction (route b, "why at m=5")

The recorded ground truth - F2^5 covers exist only in splits (2,8), (4,6), (6,4), (8,2) - is a statement about the reduced problem in F2^4 with |A| + |B| = 10. It is confirmed by complete enumeration, and the both-odd emptiness is localized to three distinct mechanisms:

m=4 split	covers	why empty (both-odd)
(2,8)	15360 (even)	- (witness A=[0,1] B=[0,2,4,6,8,10,12,15])
(4,6)	143360 (even)	- (witness A=[0,1,2,3] B=[0,1,2,4,8,15])
(1,9)	0	counting: st = 9 < 16, so A+B never covers (self covers for all 183040 pairs)
(3,7)	0	cross never covers: st = 21 >= 16 but A+B != F2^4 for every pair
(5,5)	0	self never covers when cross covers: all 4085760 cross-covering pairs have `

So the "even-splits-only" pattern at m=5 is a genuine fact, but its mechanism is split- and scale-specific (counting / cross / self), not a parity rule. Because both-odd covers exist at m=3 and the (5,5) self-sum bound is scale-dependent, the m=5 obstruction does not lift to a valid exclusion of (11,7)/(9,9) at m=6. No parity or self-sum-size argument is available there. (DEMOTED: the lift.)

The (11,7)/(9,9) identity frontier (reproduced, quantified)

Both splits survive the rho^2/identity layer; the exact remaining conditions (PR 2961's, reproduced here) are:

Split	odd excess E_o	even-half bound	min sum rho^2 (partition)	effective min (post PR 2961)	squeeze (opposite-parity)
(11,7)	13	42	13	17 (all-ones and (2,1^11) excluded)	inapplicable (s+t even)
(9,9)	17	34	17	19 (all-ones excluded)	inapplicable (s+t even)

E_o = s*t - 64; the even-half bound is the max of f_hat^2 + g_hat^2 off the all-ones hyperplane from the self-sum half of coverage (s + t + 2*P - 128, P = C(s,2)+C(t,2)). The PR 2870 Parseval squeeze uses only odd coverage and requires opposite parity; its hypothesis fails for every 18-point split (all have s + t = 18 even), so its deficit argument reaches no contradiction here. Breaking (11,7)/(9,9) requires pushing the even-half (self-sum) coverage multiplicities at the 501/196 surviving rows (PR 2911) - the direction PR 2870 did not use. This is PR 2961's open question 2, untouched by any prior PR and not settled here.

Route (c): bounded construction search (no cover found)

A bounded, seeded local search finds no covering 18-set:

search	best coverage	record ceiling
direct k=18 on F2^7 (3 seeds)	118 / 128	119 / 128 (PR 2634)
reduced (12,6), (11,7), (10,8) on F2^6	118 / 128	-
reduced (9,9) on F2^6	117 / 128	-

Seed-invariant best 118/128; no cover reached. A failed bounded search is a scoped search failure, never an exclusion. The search ceiling is unchanged.

Anchors reproduced before any new number

Anchor	Expected	Status here
PR 2870 OBSTRUCTION	parseval=696 allowed=504 deficit=192	recomputed fresh
PR 2911 ROWS18	808/501/251/196	recomputed fresh
PR 2911 NOVANISH	(12,6),(10,8) infeasible	recomputed fresh
PR 2933 CLASSIFY	2 lines value=32 need=32 TIGHT, cases=6720	recomputed fresh
PR 2961 FOURIER (7,1)	need=175 lower=239	recomputed fresh
PR 2961 CLASS82	checked=28815360 covering=0	read back from PR 3023 receipts
PR 2961 CLASS60	d_checked=169911 d_with_transversals=0	read back from PR 3023 receipts
PR 3023 AV51	u_checked=169911 uv_pairs=3049492320 covering=0	read back from PR 3023 receipts

The light anchors are recomputed live from the byte-vendored engines; the heavy walk anchors (CLASS82/CLASS60/AV51) are read back from PR 3023's per-shard receipts (PR 3023 re-walked them from empty receipts; reproduced values match exactly).

Reproduce

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

python3 -I -B research/f7-18point-remainder-kb3i-a1/verify.py quick
python3 -I -B research/f7-18point-remainder-kb3i-a1/verify.py full
python3 -I -B research/f7-18point-remainder-kb3i-a1/verify.py search

Expected markers: PASS_KB3I_QUICK, PASS_KB3I_FULL, PASS_KB3I_SEARCH. quick reproduces the anchors, the complete m=3 parity-demotion census and the (11,7)/(9,9) frontier in well under a second; full re-runs the complete m=4 sum-10 censuses and the (5,5) self-obstruction localization (271 s); search re-runs the bounded construction search. Registered replay: python3 scripts/research_package.py --package research/f7-18point-remainder-kb3i-a1 --check-only passes PASS_PACKAGE_METADATA_AND_PINS.

Scope and limitations

Set-level only inside PR943's abstract interface. Not checked: anything word-level (PR 2973/PR 3005's corrected word census and L = 22 versus L = 24 are word-level work, untouched here), walk-realizability of any cover, and any kappa, supplier, tariff or selected-frontier implication. (4,2), (3,3), (11,7), (9,9) are all undecided here; nothing in this PR excludes any of them, and the parity-shortcut demotion is a negative result about the method, not about the splits. No construction search found a cover (route c is a scoped search failure, not an exclusion). The recorded set-level search ceilings stay 118/128 at k = 18 (this run) / 119/128 (PR 2634) and 124/128 at k = 19. All arithmetic is exact Python integers (int bitmasks); no floats are used. Frozen predecessor packages are unchanged; the six vendored engines are byte-exact with SHA-256 pins.

Open next questions

1. Push the even-half (self-sum) coverage identity for (11,7)/(9,9) (PR 2961's open question 2): derive the self-sum multiplicities at the 501/196 surviving rows and check them against every cover of F2^5 (PR 2933's census) before lifting to m = 6. This is the one direction the PR 2870 odd-coverage squeeze did not use. Blocked by: no known cross/self coupling identity; the parity shortcut is DEMOTED.
2. Decide (12,6) A-vanish (4,2) with PR 3023's recorded reduced form (W 4-sets, D1 = {0,1}) and budgets |U+W| >= 20, |V+W| >= 20, |U+V| >= 24: a complete sharded walk with generous per-shard budgets and per-shard receipts. Blocked by: nothing structural; a few hours of sharded walk (strictly larger than (5,1)'s 3,049,492,320-pair walk).
3. Decide or identify (12,6) A-vanish (3,3) via the direct normalized walk or a cross-line multiplicity identity. Blocked by: all four line budgets are slack and no forced structure exists.
4. Find a valid lift of the (5,5) self-sum obstruction to (9,9): the m=4 mechanism (|A+A| <= 11 for cross-covering (5,5)) is scale-dependent and fails at m=3; determine the exact m/split threshold, if any, at which a self-sum-size bound forbids a both-odd self cover.

Credit

PR 2542/2634/2773/2870/2911/2933/2961/3023 (Aaron Wang): the 17-point exclusion chain, the KB3b search ceilings, the pair-representation rows, the odd-coverage Fourier/Parseval identity, the (A,D) reduction and vanishing-character restriction, the class floors, the tight-class decisions, and the (5,1) decision whose recorded remainder this PR takes. PR 943 (Ross Richardson): the prefix-walk model. PR 1185 (Mister-Meeseeks): the pinned f(5)=12, f(6)=16 record. PR 2691/2973 (Nicholas Stone) and PR 3005 (Nick Petalas): the word-level census line and its correction (recorded state; PR 2973 leaves the set-level chain intact). The PR 2961/2773 engines (tight.py, cover18.py, vanish.py, agent64_cover17.py, agent64_search17.py, anchors.py) are vendored byte-exact with SHA-256 pins and credited in NOTICE.md/SOURCES.json.