[unpaidmoon]

Change and mathematical scope

Proof, for one multiset. In the unsigned dirty connector, every endpoint pair of the type-T1 multiset (6,6,6,7,7,7) is joined by a floor word whose diagonal sum stays that multiset at every index s = 4,...,18. There are 2400 pairs. The 120 equal-mask pairs were already realized in PR #7042. The other 2280, those whose masks differ, are realized by the same construction: swap M_4 \ M_18 with M_18 \ M_4, then slide. The budget identity

L_j - E_j = 1_{j in M_4} - 1_{j in M_18}

makes every quota nonnegative, and each swap is a legal T1 exchange. The named class M_4 = {0,1,2}, D = 3, M_18 = {3,4,5} is the word

012334500112234455345554432211002103.

This is a uniform construction, checked on a finite domain. The two endpoint lemmas are re-enumerated: 20 splits at s = 18 and 120 splits at s = 4. It is not a search sample, not a classification of families T2–T6, and not a proof of the unrestricted six-source depth-two exclusion. The (0, 9) obstruction of PR #7029 is unchanged, because that shape needs a coordinate 6 in q_9 while every coordinate of q_18 is 3.

No new kappa. The selected frontier stays 781451/250000000 = 0.003125804 from PR #6030.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at submission and unchanged). Fork main is a different history, so this draft uses cursor/crocswap-main-17469fff-1751, which is that same commit, as its base. The upstream target is CrocSwap main.

Head 123e6808f2d1381e5ab07d128e69c52a98ef0a1a. Package research/six-source-c6-unequal-mask-transport.

Task card connector-alphabet-change, scope changed. The card asks for one new connector operation. This package keeps that alphabet fixed and answers the first open question of PR #7042. A new operation is not the missing step.

This session's item is #9 in the ordered list of first open questions of unpaidmoon PRs #7005–#7097 that are not answered by a later PR in that range. The list, with the answering PR when the question was already delivered:

1. Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010 interleaved g=13, q=8 — open
2. Record that three F2 thin maps meet total rank 6 on the three-leaf paired star #7011 two-map rank 6 or 7 outside F_2 — open
3. Record a 25-shear two-scale zero-swap word for helper-462. #7012 two-scale shear count below 25 — open
4. Record that rank 3 blocks a universal length-30 singleton word #7015 universal length-30 word at rank at least 4 — open
5. Record the c=5 centre freeze and the gap to the depth-two exclusion. #7038 depth-two characteristic-5 hypothesis (H) — open
6. Prove no constant insertion makes the valuation-1 e coefficient I #7039 two constant insertions — open
7. Record the idempotent triples after rows (1, 1, 0, 0) #7040 non-idempotent middle factors — open
8. Record one c=7 floor word that realizes both missed masks. #7041 the other 61 masks of the c=7 block word — open
9. Exhibit c=6 floor words that keep the multiset (6,6,6,7,7,7). #7042 unequal T1 endpoint pairs — this package
10. Record that the (0, 15) weight-3 shape forces no 12 at (18, 21). #7043 weight-3 shape at (1, 14) — open, left for the sibling index

Earlier first questions in #7005–#7041 are answered by a later PR in the same range (for example #7005 by #7025, #7006 by #7091, #7013 by #7035, #7026 by #7097).

Nearest overlap:

• Open PR Exhibit c=6 floor words that keep the multiset (6,6,6,7,7,7). #7042, head 807149a85d185b8272aa1fda43ba6d8d7c541cfe, realizes every equal-mask pair and one unequal example, and asks this question. It does not search the unequal graph.
• Open PR Classify c=6 separation-3 pairs and price the mask at (0, 9). #7029, head 8fe9a44a114efd4d3718208fd8f7c592fd03af62, isolates type T1 and prices the mask at (0, 9). That price is not re-proved.
• Open PR Record that the (0, 15) weight-3 shape forces no 12 at (18, 21). #7043 asks whether a weight-3 shape at (1, 14) forces a 12 at (18, 21). This package does not enter that shape.

No open PR in the refreshed title scan evaluates the unequal T1 pairs.

Validation

Python 3.12, no third-party packages.

python3 -B scripts/research_package.py --package research/six-source-c6-unequal-mask-transport

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome completed. Both the plain replay and the python3 -I -O replay print PASS_SIX_SOURCE_C6_UNEQUAL_MASK_TRANSPORT splits18=20 splits4=120 pairs=2400 unequal=2280 equal=120 controls=4. Checks raise, so -O does not strip them.

scripts/check_research_commit.py --package research/six-source-c6-unequal-mask-transport --check-source-history --check-modes passed at 123e6808f2d1381e5ab07d128e69c52a98ef0a1a.

The checker confirms the two endpoint domains, the budget identity, all 2400 words, the predecessor moving word, one illegal exit that remains a floor word, and one seven-call word.

Not run: a search of families T2–T6, a native or supplier replay, a proof assistant, and scripts/research_selected_baseline.py. The selected record is provenance only.

Credit

The depth-two interface is Nick Petalas's (PR #4231, PR #4248, PR #6107), with the Ross Richardson PR #934/PR #943 lineage. JosieElliston's PR #6141 records the c=6 solver verdict. PR #6955 classifies the empty centre. PR #7029 isolates type T1. PR #7042 classifies the endpoint splits, realizes the equal-mask pairs, and poses this question.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

Linked from research/six-source-c6-unequal-mask-transport/TASK.json:

• Which endpoint pairs of separation-3 type T2, the multiset (0,6,7,7,7,12), are joined by a monotone chain on s = 4,...,18? First step: freeze the 12 at source 5, take start mask {0,1,2} and end mask {0,1,3}, and decide whether one explicit word keeps that multiset. The 6-for-7 exchange used here is not a legal step once a coordinate is 12.

Workflow feedback

The handoff in PR #7042 named the finite (r, f) graph and one symmetry class. The budget identity answers the whole domain, so the graph search was not needed.