KB3h: (12,6) A-vanish (5,1) PROVEN impossible; (4,2)/(3,3) reduced to exact walkable forms (outcome=partial; no new kappa)

[navinor]

KB3h: (12,6) A-vanish (5,1) PROVEN impossible; (4,2)/(3,3) reduced to exact walkable forms (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 recorded the remainder this PR takes - its open question 1: the surviving non-tight (12,6) A-vanish classes (5,1), (4,2), (3,3) (D-half splits; floors 36 + x*y = 41/44/45 against need 32). This PR is strictly SET-level: no words are enumerated.

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

Concurrent word-level state, recorded honestly: PR 2973 (merged during this session) corrected a false-prune defect in the PR 2542/2634/2691 word-census budgets (bit-0 groups versus Hamming-weight parity; the n = 4, L = 8 word 01021323 is a counterexample to the old prune) and its corrected census excludes L = 20, recording 22 <= f(7) <= 24; PR 3005 independently corroborates it. PR 2973's own scope note records bit 0 as valid in set-level searches using actual group sizes and the set-level chain PR 2773-2961 as unaffected. Under 22 <= f(7) <= 24 this PR's question is the deciding necessary condition for f(7) = 22: at L = 22 every universal word needs at least 18 distinct prefixes and their set must cover.

Verdict per class (exact integers)

Split	Class	PR 2933 floor	Verdict here	Deciding evidence
(12,6)	A-vanish (5,1)	41 >= 32 survives	PROVEN impossible	complete walk: 0 W candidates over 3,049,492,320 normalized pairs
(12,6)	A-vanish (4,2)	44 >= 32 survives	classified, NOT decided	exact reduction to W 4-sets with D1 = 0,1, budgets 20/20/24
(12,6)	A-vanish (3,3)	45 >= 32 survives	classified, NOT decided	four L3 lines direct, all budgets slack, no forced structure

Mirrors are identical by odd translation. The (12,6) D-vanish classes (9,3)/(8,4)/(7,5)/(6,6) and every surviving (10,8) class are out of scope here.

(5,1): exact reduction, complete walk, exact obstruction

D1 = {d1} is a singleton, so the two half-translations U = A0+d1, V = A1+d1, W = D0+d1 absorb it (each preserves all four PR 2933 L3 lines being all of F2^5; machine-checked on random configurations, with the backward round trip). The class is EXACTLY the triples (U, V 6-sets, W 5-set) of F2^5 satisfying V u (U+W) = F2^5, U u (V+W) = F2^5, (U+V) u W = F2^5 and the selfsum line (U+U) u (V+V) u (W+W) u 0 = F2^5; residual symmetries U -> U+a, W -> W+a, V -> V+b give WLOG 0 in U, 0 in V (every solution maps bijectively to a normalized one).

Line (U+V) u W = F2^5 forces M = F2^5 \ (U+V) into W, so |M| <= 5 and |U+V| >= 27. The walk enumerates every normalized U (all 169,911) and per U every normalized V with |U+V| >= 27 by a monotone excess-budget backtracking, then derives W candidates per pair (W must contain M and hit the requirement translates of both remaining lines) and tests survivors exactly.

Quantity	Value
normalized U walked (0 in U)	169,911
normalized pairs (U, V) walked (0 in V, sumset size at least 27)	3,049,492,320
pairs with sumset size 27	1,439,907,840
pairs with sumset size 28	1,572,086,880
pairs with sumset size 29	29,998,080
pairs with sumset size 30	7,499,520
W candidates passing all three forced lines	0
covering configurations	0

The exact obstruction is recorded where it occurs: for EVERY normalized pair, no 5-set W containing M satisfies the V u (U+W) and U u (V+W) lines simultaneously (w_candidates = 0 over the whole walk), so the class is unsatisfiable before the selfsum line is ever relevant. By the reduction's bijectivity: PROVEN - no covering (A, D) exists in class (12,6) A-vanish (5,1).

The walk is 16 shards with per-shard receipts, all complete=true / budget_hit=false. The first launch hit its 3,000 s per-shard budget with explicit complete=false on all 16 shards (partial-as-complete avoided) and was re-walked from scratch at budget 6,000 s (4,365-4,691 s per shard).

Why a false empty cannot hide here

• Both search engines equal complete brute force on the smallest instances: the per-pair W search equals brute force over all 201,376 five-sets on sampled pairs; the pair backtrack equals brute force over all 169,911 normalized V at needs 24 and 27 on sampled U.
• The optimized decision engine equals the independent reference engine on full U slices in the test suite and on 30 randomized slices in a recorded spot recheck (all equal).
• m=3/m=4 analogue control censuses (22 classes with the singleton-D1 shape) agree exactly between the reduction machinery and direct A + D = E coverage tests, including cover-bearing positive controls (m=4 A=(1,1) D=(7,1): 512 covers by both routes; m=4 A=(2,2) D=(6,1): 11,648) next to floor-surviving empty controls (m=4 A=(2,2) D=(5,1): 0 covers); semantic failure controls: dropping either forced line changes the census counts.

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=28,815,360 covering=0	re-walked from empty receipts
PR 2961 CLASS60	d_checked=169,911 d_with_transversals=0	re-walked from empty receipts

Receipt-validation trap handled: the anchor walks started from missing receipt files (fresh receipts confirmed printed), and two shards (CLASS60 shard 0, CLASS82 shard 2) were re-walked from deleted receipts and compared to their originals semantically with timing fields excluded - identical (ANCHOR-REPRO.json).

(4,2) and (3,3): exact classification, not decided

• (4,2): D1 = {d1, d1+delta} normalizes to {0, 1} (delta = 1 under GL(5,2), d1 absorbed). The class is EXACTLY triples (U, V 6-sets, W 4-set) with (U+W) u (V+0,1) = F2^5, (U+0,1) u (V+W) = F2^5, (U+V) u (W+0,1) = F2^5 and the selfsum line. Necessary budgets |U+W| >= 20, |V+W| >= 20, |U+V| >= 24. The complete walk is strictly larger than (5,1)'s and was not run in budget.
• (3,3): no singleton absorption exists; the four L3 lines apply directly to (A0, A1 6-sets, D0, D1 3-sets) with 0 in A0, 0 in A1 (or 0 in D0, 0 in D1) WLOG; all budgets slack (line IV: 36+9 against 32) and no forced Sidon structure - the (6,0) rigidity A0+A1 = F2^5 does not extend.

Reproduce

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

python3 -I -B research/f7-18point-avtail-seed010d26fa/test_engine.py
python3 -I -B research/f7-18point-avtail-seed010d26fa/verify.py quick
python3 -I -B research/f7-18point-avtail-seed010d26fa/verify.py novanish
python3 -I -B research/f7-18point-avtail-seed010d26fa/verify.py rows
python3 -I -B research/f7-18point-avtail-seed010d26fa/verify.py anchors
python3 -I -B research/f7-18point-avtail-seed010d26fa/verify.py m4
python3 -I -B research/f7-18point-avtail-seed010d26fa/verify.py av51

Expected markers: Ran 10 tests ... OK, PASS_AVTAIL_QUICK, PASS_AVTAIL_NOVANISH, PASS_AVTAIL_ROWS, PASS_AVTAIL_ANCHORS, PASS_AVTAIL_M4, PASS_AVTAIL_AV51. The walks are resumable with per-shard receipts (run_avtail.py class71|class82|class60|av51, then run_avtail.py merge <mode>; explicit complete/budget_hit flags, one progress section per mode). Registered replay: python3 scripts/research_package.py --package research/f7-18point-avtail-seed010d26fa --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's corrected word census and the L = 22 versus L = 24 question are word-level work, untouched here), the 19-point cover question, walk-realizability of any cover, and any kappa, supplier, tariff or selected-frontier implication. (4,2)/(3,3) are not decided; their recorded reductions and budgets are necessary conditions and exact search domains, not exclusions. No construction search was run; the recorded set-level search ceilings stay 118/128 at k = 18 and 124/128 at k = 19 (PR 2634's record as quoted by PR 2961). Frozen predecessor packages are unchanged.

Open next questions

1. Decide (12,6) A-vanish (4,2) with the recorded reduced form and budgets; clearing it forces any (12,6) A-vanish cover into (3,3). Blocked by: nothing structural; a few hours of sharded walk.
2. 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.
3. Find the (11,7)/(9,9) coupling identity (PR 2961's open question 2, untouched here): the even-half coverage multiplicities checked first against every odd-odd-split cover of F2^5 (PR 2933's census) before lifting to m = 6.

Credit

PR 2542/2634/2773/2870/2911/2933/2961 (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 41/44/45 for the classes decided/classified here, and the tight-class decisions whose open question 1 this PR executes. PR 943 (Ross Richardson): the prefix-walk model. PR 1185 (Doug Colkitt): 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 explicitly leaves the set-level chain intact). The PR 2961 engines (tight.py and its vendor tree) are vendored byte-exact with SHA-256 pins and credited in NOTICE.md/SOURCES.json.

Claim: discussion 1166 comment 18976193.