[petalas]

Result

Scoped written proof with exact finite corroboration. No κ change. On the six-source centre of the diagonal s + t = 6c + 3, the residue-legal centre pair is classified for every characteristic c >= 2:

• even c >= 4: no residue-legal centre at all. Off the centre every legal count is 0, c/2 or c; at the centre it is 0 or c/2. All counts are multiples of c/2, while the centre length 3c+1 is not, and c/2 >= 2.
• odd c >= 5: exactly 300 residue-legal centres, all of weight 3, in exactly two count shapes. 180 have count vector {0,0,h,h,c,c} with the centre on a zero coordinate; 120 have {0,a,h,h,h,c} with the centre forced at a, where a=(c-1)/2 and h=(c+1)/2. Both shapes contain the value 2c ((2,2,0) has two such coordinates, (1,3,a) one).
• c = 3: 366 residue-legal centres, 300 of weight 3, plus 60 of weight 0 and 6 of weight 6.
• c = 2: 450 residue-legal centres, none of weight 3.

So the c=6 emptiness of merged PR #6955 is the instance c = 6 of an even-characteristic exclusion. Every odd characteristic at least three admits weight-3 centres; the frozen-coordinate induction is not obstructed by this residue count there. Conversely, at even characteristic the centre slot is unavailable for every c >= 4, and the induction must start elsewhere or change mechanism.

This package studies the residue-legal centre pair only. It does not search words, does not rerun the solver verdict of PR #6141, and does not claim a depth-two exclusion or a complete multiplication theorem.

Scope note. The advance claim predicted a centre exclusion for every c >= 3. That prediction was wrong for odd c: the odd case admits centres, and the delivered result classifies the exact split instead of asserting emptiness. Only the written result and census are claimed.

Evidence and reproduction

The package is research/six-source-all-c-centre-emptiness/. verify.py checks the even count restriction and the odd count tables for every c = 3..999, solves the two linear equations for every odd c >= 5, checks the 180 + 120 shape count, and enumerates the complete count-vector census for c = 2..10. The c=6 row reproduces the predecessor's 9156 / 46956 / 0 totals.

From the repository root with CPython 3.12.15:

~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/six-source-all-c-centre-emptiness/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B -O -W error research/six-source-all-c-centre-emptiness/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/six-source-all-c-centre-emptiness
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/six-source-all-c-centre-emptiness --check-source-history

Output: PASS_ALL_C_CENTRE_CLASSIFICATION, then PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, then PASS_RESEARCH_COMMIT_BYTES. Each registered check runs in about 1.5 s locally, well under the CI limit.

Limitations and unrun obligations

• Only the centre pair (3c+1, 3c+2) is classified. Off-centre residue-legal pairs are not.
• The c=3 families of weight 0 and weight 6 are listed by census only; they are not used by any induction here.
• The general proof is a congruence argument on counts. The census corroborates c = 2..10, and the algebra check covers c = 3..999, but no word, depth-two existence or exclusion is decided.
• PR No six-source depth-two connector block in characteristics five and six (local SAT; c = 7 open; no new κ) #6141's characteristic-six exclusion remains a solver verdict and is cited, not replayed. No DRAT certificate is emitted.
• The five framework assumptions and the selected conditional κ are untouched.

Open next questions

1. At odd c >= 5, does a residue-legal centre extend to a frozen-coordinate induction from s = 3c+1 down to s = 4? The shapes supply the 2c coordinate, not the intermediate weight-3 indices.
2. Which off-centre indices s != 3c+1 carry residue-legal weight-3 masks at even c? With the centre slot empty, this is the natural replacement start.
3. Do the c=3 weight-0 and weight-6 families interact with any mask identity, or are they excluded by a different hypothesis?

Related open work

Open PRs studying the same connector centre:

• Record the c=5 centre freeze and the gap to the depth-two exclusion. #7038 (c=5 centre freeze on s + t = 33): its premise that every legal c=5 centre is a weight-3 mask agrees with this census, which gives exactly 300 such centres and the 10 = 2c coordinate.
• Prove the c=5 adjacent centre on sum 13 freezes no coordinate. #7044 (c=5 adjacent centre on sum 13): a different diagonal; this package classifies only s + t = 6c + 3.
• Classify the c=5 adjacent centre on sum 43. #7217 (c=5 adjacent centre on sum 43): same characteristic, different diagonal; no number is compared.
• Record separate c=5 weight-2-family and all-ones-window centre counterexamples #7058 (c=5 window and weight-2 counterexamples): off-centre window statements; this package supplies no window classification.
• Prove one off-diagonal pair cannot finish the c=7 weight-3 count. #7016 (c=7 weight-3 count): it uses that a c=7 centre freezes a coordinate 14. This census gives 300 c=7 centres, all of weight 3, and their shapes contain 14 = 2c.
• Classify c=6 separation-3 pairs and price the mask at (0, 9). #7029 (c=6 separation-3 pairs at (18, 21)): an off-centre even-characteristic classification, complementary to the empty c=6 centre slot proved here.
• Bound distinct adjacent pairs of a length-28 universal word (no kappa change) #7321 (adjacent pairs of a length-28 universal word): the same connector-alphabet card family, but a different quantity; no number is shared with this centre classification.

Merged predecessor: #6955 supplies the six-source model, the residue set, the c=6 result and the c=7 frozen-14 description. The weight identity is #6107 and the interface is #4231 / #4248.

CI notes

Head c3c76667137a1c5e90894304ea6dcbe48aebc1e7. Hosted CI did not certify this package. In run 43105506939 (job 130807785393) the plan job failed in "Onboarding helper tests" on exactly one test of 760 (24 skipped): test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError reporting that make frontier7000-verify was not found in the #6030 CI verification text. On main e54523517 the selected record names make frontier7000-verify while docs/ci-verification.md still describes #6030, so the test fails on main's own files. The replay job was skipped because plan failed.

The same single test fails on other contributors' PRs (runs 42905094314, 42903313790) and on this account's #7247-#7256, #7261-#7270 and #7320 (runs 43029932349, 43030419612, 43105433200). I confirmed the failed plan job on all five listed runs and the same failing test name on this PR's run and on run 43105433200. This is a repository-level failure, not a package failure; no fix or retrigger commit is pushed from this branch. Locally on the same head, research_package.py --package research/six-source-all-c-centre-emptiness returned PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, check_research_commit.py --check-source-history returned PASS_RESEARCH_COMMIT_BYTES, and both registered verify.py commands returned PASS_ALL_C_CENTRE_CLASSIFICATION on CPython 3.12.15 (Darwin).