[unpaidmoon]

Change and mathematical scope

Exhaustive finite census, with an explicit floor word for every pair. No new κ. The selected record stays κ = 781451/250000000. This package does not edit that record.

In the unsigned six-source characteristic-6 floor model, all 2280 endpoint pairs whose masks differ are realized by a word that keeps the multiset (6,6,6,7,7,7) at every diagonal index s = 4,...,18. The equal-mask pairs are the 120 pure slides already constructed in PR CrocSwap#7042. They are not the new claim.

The pairs fall into ten orbits under permutation of the six sources. One 14-step path is found in each orbit. Transport along the region bijection produces a word for every pair, and each of those 2280 words is checked. The domain is finite, the enumeration is every unequal pair of 3-subsets together with every final letter, and the completeness argument is in research/c6-t1-unequal-mask-chains/PROOF.md.

Evidence class: exhaustive finite census. This is not a failed search and not a proof about any other weight-3 shape. It does not say that an arbitrary six-source word exists or does not exist outside this multiset and these endpoint lemmas.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at session start and before submission; both reads were this SHA). This fork draft uses cursor/crocswap-main-17469fff-1751 as its base because that branch is the same commit. The fork's own main is older. The upstream target is CrocSwap main.

Head 62e5bce7e8ca888fed8809d76b8049abece2ae7d. Package research/c6-t1-unequal-mask-chains.

Task card connector-alphabet-change, priority revision 8. Scope status: changed. The card asks for one new connector operation. This package keeps the alphabet fixed and answers open question 1 of PR CrocSwap#7042.

Session r235 took item 8 of the residual open-next list from unpaidmoon PRs CrocSwap#7005–CrocSwap#7097, ordered by PR number and question index. A question was removed when a later PR in that range cites the source PR and shares at least five of the question's first eight significant words. The first eight remaining items were:

1. Split the c=1 sign characters that g and t move CrocSwap/integer-mult-bounds#7009 Q3, residual signed permutations that split the c = 3 characters.
2. Exhibit a 104 by 13 factorisation of the P104 window over F_5. CrocSwap/integer-mult-bounds#7014 Q1, hosting the 13 frame-indicator rows as a legal P104 word.
3. Prove no constant insertion makes the valuation-1 e coefficient I CrocSwap/integer-mult-bounds#7039 Q1, two constant matrices in two gaps of the diagonal word.
4. Prove no constant insertion makes the valuation-1 e coefficient I CrocSwap/integer-mult-bounds#7039 Q2, one constant matrix beside the non-idempotent factor N.
5. Record the idempotent triples after rows (1, 1, 0, 0) CrocSwap/integer-mult-bounds#7040 Q1, which of the 65,536 middle matrices satisfy the three-gate identity.
6. Record one c=7 floor word that realizes both missed masks. CrocSwap/integer-mult-bounds#7041 Q1, which of the other 61 nonzero masks the block word misses.
7. Record one c=7 floor word that realizes both missed masks. CrocSwap/integer-mult-bounds#7041 Q2, adjacent transpositions of the Record the non-weight-3 masks missed by the c=7 witness. CrocSwap/integer-mult-bounds#7037 witness.
8. Exhibit c=6 floor words that keep the multiset (6,6,6,7,7,7). CrocSwap/integer-mult-bounds#7042 Q1, which unequal endpoint masks are joined by a monotone chain of (6,6,6,7,7,7). This package.

Nearest overlap:

• PR Exhibit c=6 floor words that keep the multiset (6,6,6,7,7,7). CrocSwap/integer-mult-bounds#7042 constructs the 120 equal-mask words, one moving word, and the two endpoint lemmas. Those lemmas are rechecked. The equal-mask family is not rebuilt.
• PR Classify c=6 separation-3 pairs and price the mask at (0, 9). CrocSwap/integer-mult-bounds#7029 isolates type T1 and prices the mask at (0, 9). That price is not re-proved. Lemma 1 still forces q_18 to be identically 3, so these words stay outside that shape.
• PR Classify the c=6 centre on s+t=39 as empty of weight-3 masks. CrocSwap/integer-mult-bounds#6955 is the empty-centre classification. It is not rerun.
• A title search on 2026-11-10 found no other pull request stating this 2280-pair transport. The hit for the numeral 2280 is an unrelated R439 note.

Validation

Python 3.12, standard library only.

python3 -B scripts/research_package.py --package research/c6-t1-unequal-mask-chains

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome completed. Both the plain replay and the python3 -O -B replay print

PASS_C6_T1_UNEQUAL_MASK_CHAINS pairs=2280 orbits=10 witnesses=10 checked=2280 controls=7

Checks raise, so -O does not strip them.

scripts/check_research_commit.py --package research/c6-t1-unequal-mask-chains --check-source-history --check-modes passed at 62e5bce7e8ca888fed8809d76b8049abece2ae7d.

Not run: a proof assistant, a SAT solver, a supplier or native replay, and scripts/research_selected_baseline.py. The selected record is read only to pin κ.

Credit

The depth-two interface is Nick Petalas's (PR CrocSwap#4231, PR CrocSwap#4248, PR CrocSwap#6107), with the Ross Richardson PR CrocSwap#934/PR CrocSwap#943 lineage. JosieElliston's PR CrocSwap#6141 records the c=6 solver verdict. PR CrocSwap#6955 classifies the empty centre. PR CrocSwap#7029 isolates type T1. PR CrocSwap#7042, head 807149a85d185b8272aa1fda43ba6d8d7c541cfe, supplies the equal-mask words, the moving word, and the endpoint lemmas.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed.

Open next questions

• For one separation-3 family other than type T1, which unequal endpoint masks are joined by a monotone chain of that family's multiset? T1 is the only family with no coordinate 12, and its transport is now complete. A family that uses a 12 has a different legal step. First step: fix that family's multiset, write the legal exchange, and search one symmetry class of unequal masks. The step rule in this package only exchanges a 6 with a 7.

Workflow feedback (optional)

The useful split was the ten region types. One path per orbit, then a source bijection, checks every pair without a separate search.

Feedback discussion: CrocSwap#1200