Prove codimension-one L/2 substitution soundness on double transversals.

[unpaidmoon]

Change and mathematical scope

Proof, plus two finite ledgers. No new κ. Selected conditional κ stays 0.001672660197076918.

On simple restored universal words with n ∈ {3, 4} and L = 2n, every label-basis linear functional of Hamming weight at least n − 1 is sound for the fixed L/2 substitution: it does not falsely prune. The argument is in research/f7-codim-one-substitution-soundness/PROOF.md.

The all-ones functional alternates, so the class defect d is 0. A weight-(n − 1) functional misses one label. Length 2n and positive even counts force that label to occur twice, so there are exactly two preserving steps and |s − t| ≤ 2. The unbalanced classes have sizes n + 1 and n − 1, and their collision excess is exactly 3. That equals base(3, 6) = 3 and sits strictly below base(4, 8) = 5.

The weight pattern is not a general law. At (4, 12), an independent enumeration of 314 restricted-growth universal words finds four false prunes for each of the masks 1011, 1101 and 1110, and none for 0111.

Two further rows are enumerations, not structural theorems. Mask 1010 has excess at most 3 on all 22 words at (4, 8) (base = 5). Mask 11010 has excess at most 14 on all 828 words at (5, 12) (base = 15).

This does not run survey(7, 18) or survey(7, 20), does not decide f(7), and does not repair a retained engine. The word 01021323 remains a bit-0 false prune, with split (6, 2) and excess 8 against base 5.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main). Priority revision 6. Upstream main was unchanged at submission.
• Task: census-pruning-audit, scope refined. The card asks whether the parity-pruning error invalidates a retained exclusion. This package proves one soundness boundary of that substitution and leaves the seven-label census open.
• Handoff: Handoffs and requests for independent checks #1166 (comment) question 2. Question 1 of that comment, the (7, 18) / (7, 20) census, is not done here.
• Session item 14. The ordered eligible list below has 12 entries, so the slot is the refinement at 0-based index (14 − 1) mod 12 = 1, list item 2. Item 2 is the census. The piece taken is the unproved weight statement in the same comment.

Ordered list, chronological on #1166 from comment 19016211, one remaining next task per explicit marker, after removing this round’s already delivered items and tasks already carried by a later DONE note or an open PR at or below #4121. In-flight sibling targets stay numbered.

1. Single-hub two-bearing factorization of PR Classify all thirty two-port support transfers #3573 (comment 19021149).
2. (7, 18) / (7, 20) functional census (comment 19021170, question 1).
3. T488 composed ownership/output invoice (comment 19022532, question 1).
4. Physical rank-8/u2 contract against the 16 extra rank-one children (comment 19029639).
5. Integer-shear supplier/recurrence inequality (comment 19031016).
6. Independent review of the research: certify local-first two-ADD obstruction on staged544 #3942 interaction/new-gap reduction (comment 19032637).
7. Adjacent completed-call pair (comment 19034066).
8. Fiber-dependent chronological row/column pair (comment 19034676).
9. Independent challenge of the A1858 contraction (comment 19035159).
10. Coupled source-normalization caller (comment 19035513).
11. Physical K/adjoint/frame boundary price (comment 19036235).
12. Encoded-boundary review of the many-axis prime producer (comment 19037887).

Sibling lists published in #4124, #4125 and #4126 number some of these differently and several still count comment 19016699, which this round already delivered as #4123. None of those pulls, and none of #4127–#4136, proves this substitution boundary. A search for “falsely prune” returns #3699, #3733, #3708 and merged #2973.

Nearest predecessors:

• research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699 measured the weight table, named the two exceptions, and wrote that the regularity was not a theorem.
• Independent check of the f7 functional classification, plus the (5,14) domain (no new kappa) #3733 reproduced that table and showed that (4, 12) breaks a weight threshold. The counts here are recomputed. The predecessor enumerator is not imported.
• Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 is the corrected parity-budget source. The bit-0 witness is retained as a failure control.

Validation

Environment: CPython 3.12.3, standard library only. From the repository root:

python3 scripts/research_package.py --package research/f7-codim-one-substitution-soundness --check-only
python3.12 scripts/research_package.py --package research/f7-codim-one-substitution-soundness

Observed:

• PASS_PACKAGE_METADATA_AND_PINS
• PASS_F7_CODIM_ONE_SUBSTITUTION_SOUNDNESS with receipt counts 245 two-stay pairs, 560 triples, 56 five-subsets, 4 universal words at (3, 6), 22 at (4, 8), 314 at (4, 12), and exception maxima 3 and 14
• PASS_F7_CODIM_ONE_SUBSTITUTION_TESTS groups=4
• PASS_REGISTERED_RESEARCH_PACKAGE research/f7-codim-one-substitution-soundness outcome=completed

The inequality for n ≤ 4 and L = 2n is a written proof for every such word. The restricted-growth enumeration is a cross-check in label-introduction normal form. The (4, 12) firing table and the two weight-(n − 2) rows are finite certificates. No proof assistant was run. No live inputs. Provenance pins for the selected record and the task catalogue are in SOURCES.json.

Credit

The singleton prefix walk is the PR943 model used by #2542, #2634 and #2691. Nicholas Stone wrote the word-census correction (#2973). The false-prune predicate and the measured table are from #3699. #3733 first recorded the (4, 12) break. James Reed’s bit-0 audit is #3096. Douglas Colkitt supplied the selected frontier this package does not change.

Prepared by unpaidmoon. Substantial Cursor Grok assistance for selection, proof writing, implementation and execution. No independent human or formal-kernel review is claimed. New work is Apache-2.0.

Open next questions

1. Which nonzero functionals falsely prune at (7, 18) and (7, 20)? Comment 19021170 names that census as the domain that can change a maintained exclusion. survey(7, 18) is wired in research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699 and is not run here. Blocker: seven-label state count and a separate resource budget.
2. Why are masks 1010 at (4, 8) and 11010 at (5, 12) the only weight-(n − 2) exceptions? The second ledger meets excess 14 against base 15. A first step is to list those tight words and compare them with a firing neighbor such as mask 10101. Blocker: no structural classification.
3. For which lengths above 2n does weight at least n − 1 stay sound? (4, 12) is already a counterexample to a general law. (4, 10) and the weight-at-least-4 masks at (5, 12) were measured sound by research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699 and Independent check of the f7 functional classification, plus the (5,14) domain (no new kappa) #3733 and are unproved. Blocker: four preserving steps can change the class sizes by 4, and n + 1 > 2^{n−2} only for n ≤ 4.

Workflow feedback (optional)

#1200 — sibling sessions published incompatible #1166 orderings because explicit-next-task markers and later DONE notes are easy to count differently. One maintained eligible-handoff snapshot, with comment ids, would make item numbers comparable across parallel sessions.