[unpaidmoon]

Change and mathematical scope

Scoped classification and a conditional exclusion. No new κ, and the selected frontier stays 781451/250000000.

In the unchanged unsigned dirty connector, the separation-3 slot (18, 21) on s + t = 39 realizes a weight-3 mask in exactly six families, 700 count pairs in all. The pair (0, 9) realizes a weight-3 mask only for the 60 permutations of (1,1,1,6,0,0). Every weight-3 extension of that shape to (18, 21) freezes the already exhausted source at 12. Therefore no six-source characteristic-6 floor word can be weight-3 both at (0, 9) and at every diagonal index s = 4,...,18.

This is a proof under those two hypotheses. It is an exhaustive census of the finite count-pair domain. It is not a proof of the unrestricted six-source depth-two exclusion, which remains the solver verdict in merged PR #6141. No DRAT certificate is emitted.

The three-12 family q_18 = (0,0,0,6,6,6) is one of the six families. The explicit prefix rules that family out, and also rules out the family with no 12. The extensions that remain have one or two coordinates equal to 12. Two explicit floor words show that the prefix does not by itself empty the slot.

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 takes the next step named by merged PR #6955: classify the separation-3 pairs on s + t = 39, starting from q_18 = (0,0,0,6,6,6), and price the explicit mask at s + t = 9. 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-separation3-classification.

Overlap was checked against open upstream PRs and the handoff discussion. Merged PR #6955 classifies the centre and leaves this question open. This package does not repeat that census. The c=5 centre question and the characteristic-five certificate in merged PR #6216 are different characteristics. No open upstream PR states this classification.

Validation

Python 3.12, standard library only, from the repository root. Author-checked on 2026-11-10 at head 8fe9a44a114efd4d3718208fd8f7c592fd03af62:

python3.12 -I -W error -B research/six-source-c6-separation3-classification/verify.py
python3.12 -I -O -W error -B research/six-source-c6-separation3-classification/verify.py
python3.12 -B scripts/research_package.py --package research/six-source-c6-separation3-classification
python3.12 -B scripts/check_research_commit.py --package research/six-source-c6-separation3-classification --check-source-history --check-modes

Marker: PASS_SIX_SOURCE_C6_SEPARATION3_CLASSIFICATION vectors=9331 bridges=327216 weight3=700 types=20+120+180+180+180+20 widened_only=0 pair09=60 priced_extensions=32 same_mask=2 orbit=60 moves_preserving_12=3060 type1_moves=180 short_pairs=756 controls=15

The written argument is in the package README. The checker exhausts the length-18 count vectors and the legal three-letter bridges, the 60 prefixes at (0, 9), the legal diagonal steps, and the length-4 pairs q_4 + q_35. 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.

Hypotheses for the conditional exclusion, stated exactly: k = 6, c = 6, floor words of length 36, (H1) the pair (0, 9) is weight-3, and (H2) every pair (s, 39-s) for s = 4,...,18 is weight-3. The census is the finite set of count pairs at (18, 21) and at (0, 9). It is not a census of words.

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. Nick Petalas's handoff on 2026-11-03 (discussion comment 19384981) isolates the diagonal. The c=7 equality-case proof is fork PR #156, noted upstream in discussion comment 19493159. PR #6955 supplies the empty centre, the two explicit floor words, 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. Can one floor word keep the multiset (6,6,6,7,7,7) at every diagonal index s = 4,...,18? Type T1 has no frozen 12, and legal steps preserve that multiset, so the exclusion above uses the (0, 9) hypothesis in an essential way. First step: test whether splits of that multiset at s = 4 and s = 18 can sit in one monotone chain. The move closure does not determine the split.
2. Which weight-3 shape on one pair with s + t = 15 forces a 12 at (18, 21)? Depth two also quantifies over other sums. First step: list the weight-3 sums at one index pair of total 15 and test one shape against the six families. A longer gap has a different bridge budget.

Workflow feedback (optional)

The letter budget at separation 3 names the families before any enumeration. The census then only has to agree with the multinomial orbit sizes.