[unpaidmoon]

Change and mathematical scope

Scoped exchange obstruction. No new κ, and the selected frontier is unchanged.

In the unchanged unsigned dirty connector, keep the centre as the only failed index in 4..22. Every free-type chain on which (9, 15) is weight 3 realizes the complement of one mask M, and the diagonal image has size at most 6. The early pair (0, 3) and the block (33, 33) add no mask outside that image. The pairs (9, 15), (10, 14), and (11, 13) add at most one mask each, so the weight-3 cover is at most 9. An exhaustive walk census tightens this to 7. The value 7 is attained, but only when the complement is the single new mask. Two or three of those masks can be new at the same time, and different from the mask of (0, 3); the cover is then at most 6. Neither case reaches C(6, 3) = 20.

The word

012001122012345334455345554433543210221100

has diagonal image {(0, 1, 2)}, realizes (0, 3) as that same mask, and realizes three further masks: (3, 4, 5) at (9, 15), (0, 3, 4) at (10, 14), and (0, 1, 3) at (11, 13). Its weight-3 cover is 4. The word

012001122301452554433345334455543210221100

has image size 6 plus the complement, cover 7, and does not realize (10, 14) or (11, 13).

This does not decide whether a six-source depth-two block exists at c = 7. Masks of every weight other than 3, chains on which (9, 15) is not weight 3, and two or more failed diagonal indices stay open. No supplier, consumer, or selected conditional κ changes. The selected record remains 781451/250000000 from PR #6030.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (upstream main at session start; re-fetched, unchanged). This fork's main is older, so the draft base is the pin branch cursor/upstream-17469fff-pin-20e5 at that same commit. The diff is only research/six-source-c7-two-pair-count/.

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 upstream PR #7016. A new operation is not the missing step.

Overlap checked 2026-11-10 against open upstream PRs and fork branches. The nearest result is open #7016, which proves one off-diagonal pair cannot finish the count and releases this two-pair question. This package starts there: it does not repeat the one-index reduction or the cover-19 witness. Merged #6941 proved the diagonal equality case empty. Merged #6141 excluded c = 5 and c = 6 by local SAT and left c = 7 open. No open PR enumerates these joint realizations.

Validation

Python 3.12, standard library only, from the repository root:

python3.12 -I -W error -B research/six-source-c7-two-pair-count/verify.py
python3.12 -I -O -W error -B research/six-source-c7-two-pair-count/verify.py
python3.12 -B scripts/research_package.py --package research/six-source-c7-two-pair-count --check-only

Both replay commands print:

PASS_SIX_SOURCE_C7_TWO_PAIR_COUNT endpoint_pairs=20/20 walks_per_mask=31788 image_at_most=6 structural_cover_at_most=9 census_cover_at_most=7 several_walks=1044 several_cover_at_most=6 three_new_walks=72 cover7_walks=3276 three_word_cover=4 sharp_word_cover=7 block33_outside=0 unclassified=0 controls=12

The registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE research/six-source-c7-two-pair-count outcome=negative-result. The metadata check prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check).

The shape scan is an exhaustive finite census of the free-type states at s = 9 and s = 15: 700 sum-24 weight-3 vectors, 20 hits, all complements. The walk census enumerates every in-type walk between those endpoints; a forward count and a classifying recursion both give 31,788 walks per mask. The two words are checked by reading every pair whose index sum is congruent to 3 modulo 7. These are not a SAT certificate and not a search-failure claim. No Lean, Docker, or unrestricted word search was run.

Credit

The depth-two interface and the weight identity are Nick Petalas's (PR #4231, PR #4248, PR #6107), with the Ross Richardson PR #934 / PR #943 lineage. JosieElliston's PR #6141 leaves c = 7 open. The empty equality case is upstream PR #6941. The one-index exchange and the question answered here are upstream PR #7016, head 3a89abb195ec9a63476c12042df5fbb0760e3534.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. On a centre-only chain whose (9, 15) sum is not weight 3, can (10, 14) and (11, 13) be new at the same time and push the weight-3 cover to 20? The collapse here uses the complement shape. Separate endpoint equalities suggest an image of size at most 16 and a cover of at most 19, but that counting is not proved in this package. First step: enumerate the joint walks on which s = 10 and s = 14 share a mask, s = 11 and s = 13 share a mask, and the states at s = 9 and s = 15 are not a complement pair. Blocked because those walks are outside the 31,788 complement windows classified here.

Workflow feedback (optional)

Linking this once to the feedback discussion is enough: #1200