Record the coset-wide one-point pair ceiling

[unpaidmoon]

Change and mathematical scope

Negative result, as a finite certificate. For every n=7 weight split that survives the basic pair test B0 + a*b >= need, with one side already at its target size and the other side one parity-legal point short, the maximum completed pair product is at least need. The strict test that deletes a branch when every such completion lies below need therefore deletes nothing.

The bound is sharp at (9,9). The sets E = {0, 3, 5, 9, 10, 33, 34, 40, 43} and O = {1, 4, 16, 21, 64, 69, 81, 84} have pair product S = 4 and outside value f(p) = 5 at every remaining odd point, so every one-point completion has product 9 = need. These are parity-legal sets, not a prefix walk.

No scalar saving, paid-profile change, or κ improvement. Selected κ stays 0.001672660197076918. f(7) is not decided. This is not a length-22 census and not a multiplication theorem.

Remaining obligations: whether the hypercube-neighbor restriction of the same ceiling can prune, and whether an exact two-point ceiling prunes (9,9).

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6. Task census-pruning-audit. Scope refined: the card's first experiment, the word 01021323, is already recorded in #3821 and #3699. This package takes the uncovered one-point partial form of the pair identity in research/f7-weight-parity-census-correction/MATH.md section 3, which is the length-22 handoff's next cheap obligation. The live input is that MATH.md file, pinned in SOURCES.json.

Nearest open work, and why this is not a repeat:

• Record the pair-collision identity #3827 records sum C(r_y, 2) = 2 * sum A_x B_x for every subset of F_2^4 and every nonzero linear functional. This package rechecks that identity only for weight parity, as a lemma, and adds the sums of f and the completion increment.
• Strengthen the length-22 seven-label census with a union-support budget; exclude the (9,9) split (no new kappa) #3818 claims a different union-support exclusion of (9,9). The witness here is not a walk and does not revisit that claim.
• Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731 claims census exclusions of (11,7), (10,8), and (11,8) by a different probe. Those rows remain in this table only to show that the one-point pair test is not the test that removes them.
• Derive the complete length-22 parity-split domain at seven labels (no census, no f(7)) #3708, Eleven length-22 splits survive, and nine survive the 17-point import #3712, and Independent check of the f7 functional classification, plus the (5,14) domain (no new kappa) #3733 are the split domain and the functional classification. Not repeated.

Validation

Python 3.12.3, standard library only. From the repository root:

python3.12 -B scripts/research_package.py --package research/f7-one-point-pair-ceiling

Observed: PASS_REGISTERED_RESEARCH_PACKAGE research/f7-one-point-pair-ceiling outcome=negative-result.

verify.py recomputes all 3820 basic-identity survivors with both parts at most 64 and checks the one-point bound, checks the ceiling drop for every residue with M from 2 through 64, rebuilds the (9,9) odd set by the deterministic greedy, recomputes every completion, and checks the f identities on every even/odd subset pair in F_2^3 and F_2^4. It also recomputes the pinned 01021323 fixture (weight split (4,4), bit-0 split (6,2), excess 8). test_verify.py repeats the (9,9) and (7,11) arithmetic with literals. The floor swap drops (7,11) and (11,7) from 7 to 6; replacing witness point 84 by 2 moves the ceiling from 9 to 17.

Not run: a length-22 census, a prefix-walk neighbor search, a two-point lower bound, and any Lean check. The rectangle checksum is a regression lock of the closed form. The independent evidence is the witness, the literal arithmetic, and the subset enumeration.

Credit

The pair identity, the basic row test, and 01021323 come from Nicholas Stone's package research/f7-weight-parity-census-correction (substantial OpenAI Codex assistance; Apache-2.0). Parallel PRs #3821, #3699, #3827, #3818, and #3731 are left in place. Prepared by unpaidmoon with substantial AI assistance (Cursor Grok), including the averaging argument and the verifier. No independent human or formal-kernel review.

Open next questions

From research/f7-one-point-pair-ceiling/README.md:

1. Does restricting the same maximum to the n hypercube neighbors of the current prefix vertex fall below need on any open length-22 split? The coset-wide maximum is at least that walk-legal maximum, so a neighbor restriction can still prune. First step: evaluate S + f(p) only at the neighbors of one explicit partial prefix whose coset-wide ceiling equals need. Blocked here because this package does not search walks.
2. Does an exact two-point completion ceiling strictly prune (9,9)? First step: lower-bound S + f(p) + f(r) + A_{p xor r} over odd pairs outside a 7-point odd set paired with a 9-point even set. A greedy sample reached 17, above need 9, and that sample is not a lower bound.

Workflow feedback (optional)

The useful split was between the coset-wide maximum and the walk-neighbor maximum. A one-point identity that is sharp on sets can still look like a pruning rule for the census engine, and it is not one.