[unpaidmoon]

Change and mathematical scope

Scoped negative result. No new κ, and the selected frontier stays 781451/250000000.

In the unchanged unsigned dirty connector, the only weight-3 value of q_15 is a permutation of (6,6,1,1,1,0). There are 60 such vectors. That shape meets none of the six separation-3 families recorded in PR #7029. The set of weight-3 extensions to (18, 21) is empty: 23820 legal three-letter bridge pairs, and none is a weight-3 mask. The shape therefore does not force a 12 onto that slot in the sense used for the mask at (0, 9), where the extension set is nonempty and every extension carries a 12.

This is a proof of the shape list by case analysis, and an exhaustive finite census of the length-15 count vectors and of the bridges from this one shape. It is not a proof of the unrestricted six-source depth-two exclusion, which remains the solver verdict in PR #6141. No DRAT certificate is emitted. The empty-set reading of "every extension has a 12" is not reported as a price.

Hypothesis (H15), that (0, 15) is weight-3, is incompatible with a weight-3 sum at (18, 21). It is therefore incompatible with hypothesis (H2) of PR #7029, and the reason is the single index s = 18. The indices s = 4,...,17 are not used. Depth two does not require (H15).

Task and overlap

Base: CrocSwap/integer-mult-bounds 17469fff78c22941280f04d39dc5a1faee09c840, re-fetched at session start and again before submission; both reads were that commit. This draft uses cursor/upstream-17469fff-pin-4e14 as its base because that branch is exactly that commit. The fork's main is an older snapshot.

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 2 of PR #7029 for the one pair (0, 15), whose gap to index 18 is three letters. A new operation is not the missing step. This is a side result on the recursive-interface schedule. It does not change a supplier, a consumer, or κ.

Package: research/six-source-c6-sum15-pair015.

Overlap was checked against open upstream PRs, including #7005–#7038, and against discussion #1166. PR #7029 poses this question and stops before a census of s + t = 15. PR #6955 classifies the centre and is not repeated. The separation-3 classification and the (0, 9) price are not redone. The (6,6,6,7,7,7) chain and characteristic 5 are different questions. No open upstream PR states this emptiness result.

Validation

Python 3.12, standard library only, from the repository root. Author-checked on 2026-11-10 at head 4628769582285b5fd8a9d59352e229337b53733c:

python3.12 -I -W error -B research/six-source-c6-sum15-pair015/verify.py
python3.12 -I -O -W error -B research/six-source-c6-sum15-pair015/verify.py
python3.12 -B scripts/research_package.py --package research/six-source-c6-sum15-pair015
python3.12 -B scripts/check_research_commit.py --package research/six-source-c6-sum15-pair015 --check-source-history --check-modes

Marker: PASS_SIX_SOURCE_C6_SUM15_PAIR015 vectors=7872 weight3=60 shape=(0, 1, 1, 1, 6, 6) families=6 family_hits=0 shape_trials=23820 shape_extensions=0 live_prefix_extensions=13 live_prefix_twelves=13 controls=19

The written argument is in the package README. The checker exhausts the length-15 count vectors and the legal three-letter bridges from the 60 prefixes. A non-shape prefix has 13 weight-3 extensions, all carrying a 12, so the empty count is not a dead enumerator. It does not search depth-two words and does not rerun the SAT encoding. Both command modes, the registered package replay, and the commit-byte check passed. No Lean, Docker, or historical suite was run.

Evidence class: (1) a proof of the shape list and of the family incompatibility under the six recorded q_18 multisets; (2) an exhaustive finite census of the 7872 length-15 vectors and of the 23820 bridges, which proves the extension set is empty without assuming the family list is complete. This is not a heuristic or incomplete search, and it is not a proof of unrestricted nonexistence.

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 records the c=6 solver verdict. PR #6955 supplies the empty centre and the two explicit floor words used as controls. PR #7029 supplies the six families, the nonempty 12-price of the mask at (0, 9), and the question answered here. queelius's PR #6216 is the characteristic-five certificate. zxv's PR #6057 and JosieElliston's PR #6061 supply the c ≥ 8 words.

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

Open next questions

1. Does one weight-3 shape at the pair (1, 14) force a 12 at (18, 21) on a nonempty extension set? The gap there is four letters, so the bridge budget is different. First step: list the weight-3 sums q_1 + q_14 and test one shape against the same six q_18 multisets.
2. Can the residue shape (12,1,1,1,0,0), which is not a length-15 prefix, occur as q_6 + q_9 and still force a coordinate 12 at (18, 21) on a nonempty extension set? A coordinate 12 in a sum of total 15 requires both indices to be at least 6. The gap from 9 to 18 is nine letters.

Workflow feedback (optional)

The three-letter gap makes the first bridge a stars-and-bars count, and the family multisets then fail by pigeonhole before any second bridge is chosen.

Link once: #1200