KB3j: the (4,2) and (3,3) A-vanish walks - budget-honest partial with receipts; even-half self-sum profiles classified (outcome=partial; no new kappa)

[navinor]

KB3j: the (4,2)/(3,3) A-vanish walks and the even-half self-sum profiles - (4,2) walk partial with receipts, self-sum-bound lift DEMOTED (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) 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); PR 2933 classified the vanishing configurations; PR 2961 fully excluded (10,8); PR 3023 killed (12,6) A-vanish (5,1) and reduced (4,2)/(3,3) to exact walkable forms; PR 3034 demoted the parity shortcut and recorded the remainder this PR takes. This PR is strictly SET-level: no words are enumerated.

Record chain authors, all verified against current ground truth at writing with gh pr view NNN --json author,title: PR 2542/2634/2773/2870/2911/2933/2961/3023/3034 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) records 22 <= f(7) <= 24 and PR 3005 corroborates it; PR 2973's own scope note records the set-level chain as unaffected. Under that gap this PR's question is the deciding necessary condition for f(7) = 22.

Verdict per split (exact integers)

Split	Status here	Deciding evidence
(12,6) A-vanish (4,2)	NOT decided (walk partial)	29/2048 shards, 249,244,455 pairs, w_candidates=0 throughout; complete walk sized at 1.722e10 pairs, about 608 core-hours
(12,6) A-vanish (3,3)	NOT decided (no forced structure)	complete k=3 shape census: matched class (3,3) has 296,352 covers of 1,382,976 configs; 10 of 64 classes empty, killing at (I)/(II)/(III) or (IV)-only
(11,7) both-odd	NOT excluded (route c pressed)	m=3 both-odd self-sum profiles completely classified; self-sum-bound lift DEMOTED; frontier unchanged (E_o=13, min sum rho^2=17)
(9,9) both-odd	NOT excluded (route c pressed)	same route-(c) status; frontier unchanged (E_o=17, min sum rho^2=19)
construction	no 18-set found	no new search run; recorded ceilings 118/128 (PR 3034 run) and 119/128 (PR 2634 record) stand

No class is decided here and nothing here excludes any class; the (4,2) zero counts are facts about walked pairs only.

(4,2): exact reduction, validated engine, budget-honest partial walk

PR 3023's reduction extends: D1 = {d1, d1+delta} normalizes to {0, 1} (delta to 1 under GL(5,2), d1 absorbed by global translation). The class is EXACTLY the triples (U, V 6-sets, W 4-set) of F2^5 with lines (a) (U+W) u (V+D1), (b) (U+D1) u (V+W), (c) (U+V) u (W+D1), (d) selfsums of U, V, W, D1 each unioning to all of F2^5. The four lines are invariant under (U+a, V+b, W+a+b) and the mirror U/V swap (co-translation of each line's summands), so 0 in U, 0 in V WLOG - machine-checked on 300 random configurations. Line (c) forces M = F2^5 \ (U+V) into W u (W+1), so |U+V| >= 24; per pair a complete M-cluster W search decides the class exactly. PR 3023's budgets |U+W| >= 20, |V+W| >= 20, |U+V| >= 24 hold.

Sizing (CHECKED): a 12-U probe walked 1,216,422 pairs in 154.6 s (7,866 pairs/s); the complete walk is 1.722e10 normalized pairs, about 608 core-hours at the measured rate. The complete decision is far outside one session, so the walk is recorded as a budget-honest PARTIAL with per-shard receipts (explicit complete/budget_hit flags, generous 3,600 s budgets, actual 1,411-1,583 s per shard):

Quantity	Value
shards complete of 2048	29 (all complete=true, budget_hit=false)
normalized U slices walked	2,407
normalized pairs walked	249,244,455
pairs with sumset 24 / 25 / 26 / 27 / 28 / 29 / 30	89,073,054 / 71,435,214 / 45,702,660 / 20,350,656 / 22,151,421 / 425,160 / 106,290
W candidates passing (a),(b),(c)	0 on every walked pair
covering configurations	0
core-seconds spent	42,667.82

Recomputation proof (receipt-validation trap): shard 1 was re-run from an emptied receipt, its first run preserved; counters identical (83 U slices, 8,592,684 pairs, w_candidates=0, identical histogram; timing fields excluded).

The (5,1) kill shape is present on every walked pair (zero W candidates), but the (5,1) counting shortcut is DEMOTED for (4,2): (4,2)'s requirement window is 20-21 points per side against capacity 24, looser than (5,1)'s 26 against 30 with W forced to equal M. A structural contradiction is open (open question 2).

Small-space ground truth (PROVEN, complete): the k=3 census of the whole (4,2) shape has 576 classes, 484 with covers and 86 pair-killed - e.g. class (2,2,2) pair-killed, (2,2,3) line-(d)-only (32 candidates, all fail (d)), (2,2,4) with 80 covers of 322 candidates; the k=4 census (floor-surviving grid) has 96 classes, 10 with covers, 79 pair-killed. The shape is satisfiable at small scale, so the F2^5 zero counts cannot be a broken-machinery false empty.

(3,3): complete small-space map, no forced structure

No singleton absorption exists; the four PR 2933 L3 lines apply directly with 0 in A0, 0 in A1 WLOG. The complete k=3 census (64 classes, 44,166,408 normalized configurations) maps the shape: matched class (3,3) has 296,352 covers; (2,3) and (3,2) are IV-only kills (2,688 / 6,048 configurations pass (I),(II),(III), none passes (IV)); eight small classes fail (I)/(II)/(III) outright; 10 of 64 classes empty in total. PR 3034's "no forced structure" is confirmed at the smallest scale: there is no small-instance contradiction to lift, and the class remains walk-ready but larger than (4,2).

Route (c): even-half self-sum profiles at (11,7)/(9,9)

For a reduced covering pair (A, B) define the even-half multiplicity mu2 = |(A+A) n (B+B)|. Complete enumerations at the two testable scales (PROVEN):

Scale	Split	Covers	Self-sum profile, all covers	Self-union
m=3	(5,5)	3,136	(8,8,8) unique	8/8 full
m=3	(7,7)	64	(8,8,8) unique	8/8 full
m=3	(3,5) and (5,3)	2,688 each	(4,8,4) and (8,4,4) unique	8/8
m=3	(3,7) and (7,3)	448 each	(4,8,4) and (8,4,4) unique	8/8
m=3	(5,7) and (7,5)	448 each	(8,8,8) unique	8/8
m=4	(2,8)	15,360	(2,15,1) unique	16/16
m=4	(4,6)	143,360	(4,13,4) x 35,840 and (7,16,4) x 107,520, mu2=4 all	16/16

The m=4 (5,5) self-obstruction (PR 3034: 4,085,760 cross-covering pairs, self-when-cross 0, max self-union 15/16) is the quantified contrast: at m=3 every both-odd cover has FULL self-union and full selfsums at (5,5)/(7,7). The scale-dependent self-sum bound therefore does not lift - DEMOTED, exactly the failure PR 3034 predicted and now measured per profile.

What survives scaling (CHECKED at both scales): mu2 is constant per split in every complete census here (m=3 all splits; m=4 (2,8) = 1, (4,6) = 4). This per-split constancy is the narrowed even-half condition to carry to the 501/196 surviving rows (PR 2911) before lifting to m=6. It is CHECKED at two scales, not proven at m=6, and no identity is claimed.

Anchors reproduced before any new number

Anchor	Expected	Status here
PR 2870 OBSTRUCTION	parseval=696 allowed=504 deficit=192	recomputed live
PR 2933 CLASSIFY	2 TIGHT, cases=6720	recomputed live
PR 2961 FOURIER (7,1)	need=175 lower=239	recomputed live
PR 2961 CLASS82	checked=28,815,360 covering=0	read back from PR 3023's pinned receipts
PR 2961 CLASS60	d_checked=169,911 d_with_transversals=0	read back from PR 3023's pinned receipts
PR 3023 AV51	uv_pairs=3,049,492,320 covering=0	read back from PR 3023's pinned receipts
PR 3034 m=5 table	(5,5) cross=4,085,760 self=0 max_self_union=15/16	read back; re-derived live earlier this session via PR 3034's own verifier (PASS_KB3I_FULL seconds=267.8)

Reproduce

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

python3 -I -B research/f7-18point-av42-kb3j-b1/test_engine.py
python3 -I -B research/f7-18point-av42-kb3j-b1/verify.py quick
python3 -I -B research/f7-18point-av42-kb3j-b1/verify.py walks

Expected markers: Ran 9 tests ... OK, PASS_KB3J_QUICK, PASS_KB3J_WALKS. The test suite validates every engine against complete brute force on the smallest instances (all C(32,4) W 4-sets on sampled F2^5 pairs; all W subsets at m=3; pair walk against all normalized V; the reduction against direct coverage on every halved F2^3 configuration). The walk resumes with run42.py av42 --shards 2048 --shard k --budget 3600 and run42.py merge av42; the censuses re-run with run42.py k3 (196 s), run42.py k4 (3,923 s) and selfsum_classify.py. Registered replay: python3 scripts/research_package.py --package research/f7-18point-av42-kb3j-b1 --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 hold 22 <= f(7) <= 24; untouched here), walk-realizability of any cover, the (12,6) D-vanish classes, the surviving (10,8) classes, the 19-point question, and any kappa, supplier or selected-frontier implication. (4,2), (3,3), (11,7), (9,9) are all undecided here. The (4,2) walk covers 2,407 of 169,911 U slices (1.42%) and proves nothing beyond its walked pairs; the route-(c) profile facts are complete at m=3/m=4 only. All arithmetic is exact Python integers (int bitmasks); no floats are used. Frozen predecessor packages are unchanged; the ten pinned files are byte-exact with SHA-256 pins.

Open next questions

1. Complete the (4,2) walk (resume from shard 29; about 605 core-hours remain at the measured rate; receipts make it deterministic and resumable). Blocked by: compute only. Clearing (4,2) forces any (12,6) A-vanish cover into (3,3).
2. Isolate the w_candidates=0 mechanism at |U+V| = 24 (|M| = 8: W must be independent with W u (W+1) = M exactly, at most 16 candidates per pair). A pair-level contradiction would decide the class without the remaining walk. Blocked by: the (5,1) counting shape does not transfer; the mechanism is not isolated.
3. Decide or identify (3,3) via a new reduction (no singleton absorption) or a cross-line multiplicity identity coupling (I)/(II) with (III)/(IV). Blocked by: all four line budgets are slack; no forced structure at any scale tested.
4. Lift the mu2 per-split constancy to the 501/196 surviving rows (PR 2911) and then to m=6 as an exact even-half identity, or find where it breaks. Blocked by: constancy is CHECKED at two scales only; no cross/self coupling identity is known.

Credit

PR 2542/2634/2773/2870/2911/2933/2961/3023/3034 (Aaron Wang): the 17-point exclusion chain, the 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, the (5,1) decision and its walk engine, and the KB3i 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/PR 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 vendored engines are byte-exact with SHA-256 pins and credited in NOTICE.md/SOURCES.json.