[unpaidmoon]

Change and mathematical scope

Scoped exchange census. 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. On a chain whose (9, 15) sum is not weight 3, (10, 14) and (11, 13) can be new at the same time. The weight-3 cover is then exactly 3, not 20.

Both pairs are weight 3 only for 720 joined shapes of one common mask M. The prefix s = 4..10 and the tail s = 14..21 are frozen on M, and s = 12 carries either M or the mask of (11, 13). (0, 3) and (33, 33) add nothing. (9, 15) adds the complement only when the endpoints are the complement pair already classified by #7032. Off that pair the cover is 3. On that pair the cover is 4, which sharpens the “at most 6” bound of #7032 for this subfamily and does not rerun its 31,788-walk census.

The word

012220001121343344555345555443343121100022

has diagonal image {(0, 1, 2)}, realizes (0, 3, 4) at (10, 14) and (0, 2, 3) at (11, 13), and does not realize (9, 15). Its weight-3 cover is 3. Its s = 9 usage is (3, 1, 2, 0, 0, 0), not the complement.

This does not decide whether a six-source depth-two block exists at c = 7. Masks of every weight other than 3, shared-mask walks with only one of the two sums 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-bd5c at that same commit. The diff is only research/six-source-c7-noncomplement-pairs/.

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 #7032. A new operation is not the missing step.

Overlap checked 2026-11-10 against open upstream PRs #7005–#7038, fork branches, and discussion #1166. The nearest results are open #7032, which releases this non-complement question after settling the complement shape, and open #7016, which is the one-pair obstruction. This package does not repeat either census. No open PR enumerates these non-complement joint walks.

Validation

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

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

Both replay commands print:

PASS_SIX_SOURCE_C7_NONCOMPLEMENT_PAIRS shapes=180/180 types=720/720 prefix_paths_per_state=210 cone=3,6,8,8,6,3,1 mids_per_type=3 mask_M_routes=2 both_new_windows_per_mask=576 complement_windows_per_mask=72 noncomplement_cover=3 complement_cover=4 witness_cover=3 witness_both_new=1 sharp_word_cover=4 slots_blocked=82 graph_slots=25 shared_mask_paths=7818720 shared_mask_paths_agree=1 controls=16

The registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE research/six-source-c7-noncomplement-pairs outcome=negative-result when replay is requested. The metadata check prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check).

Evidence class: exhaustive finite census of a finite free-type graph. The shape list, the 720 types, the frozen prefix and tail, and the slot partition are enumerated completely; the cover arithmetic is uniform on that census. This is not a SAT certificate, not a search-failure claim, and not an unrestricted nonexistence proof. A forward count and a backward count agree on 7,818,720 shared-mask paths from s = 10 to s = 14; only 2,160 of them make both sums weight 3. 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 is upstream PR #7016, head 3a89abb195ec9a63476c12042df5fbb0760e3534. The complement two-pair census and the question answered here are upstream PR #7032, head 6616162b4c49e84ae67921013d03151297f86879.

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

Open next questions

1. On a centre-only chain where s = 10 and s = 14 share a mask, s = 11 and s = 13 share a different mask, and exactly one of those two sums is weight 3, what is the weight-3 cover? Both sums being weight 3 forces the masks to agree and drops the cover to 3 or 4. A shared-mask walk with only one new pair is the remaining way the 7,818,720 paths could approach a large cover. Separate endpoint equalities still suggest an image of size at most 16 and a cover of at most 19, and that counting is not proved here. First step: fix the weight-3 end to one of the 180 shapes and enumerate the states at the other pair whose common mask is different. Blocked because those paths are outside the 2,160 both-shape paths classified here.

Workflow feedback (optional)

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