[petalas]

Summary

Unpaidmoon's #7099 asks which T2 endpoint pairs are joined by a weight-3 chain on the six-source c=6 diagonal. The answer on [4,18] is empty. This package proves the sharp window of every separation-3 family and supplies explicit attaining words.

S(s) = q_s + q_(39-s)
weight-3 steps preserve each coordinate's base
L low coordinates require at least 5L letters in a suffix of length s-3
least window start = max(4, 3+5L)

Claim type: scoped written mathematical proof with an exhaustive finite certificate. On T2's longest window [8,18], exactly 720 of 21,600 endpoint pairs are joined. The package also proves that weight-3 at (0,9) coexists with a window [s0,18] exactly when s0 >= 8. This last threshold extends the advance claim's scope, as recorded in the progress update.

Only research/six-source-c6-chain-windows/ is added. The window theorem, threshold and verifier were produced by Nick Petalas. Credit is retained for unpaidmoon #7099/#7029/#7042 and the merged centre-classification package; Nick Petalas #4231/#4248/#6107; Ross Richardson #934/#943; and JosieElliston #6141.

Evidence

• Before: Prove every T1 endpoint pair has a monotone chain #7099 leaves the T2 endpoint relation open on [4,18], and Classify c=6 separation-3 pairs and price the mask at (0, 9). #7029 excludes the joint (0,9) hypothesis using that full window.
• After: a suffix-count proof gives sharp family windows; a graph-to-word argument proves the finite state graph is exact. Exhaustive backward and forward reachability agree on all six joined-pair counts. Seven stored words pass literal prefix-count checks. A separately written reverse graph agrees on all six pair counts, and five verifier mutations were rejected locally.

Reproduce from the repository root with CPython 3.12.15 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/six-source-c6-chain-windows/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/six-source-c6-chain-windows/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/six-source-c6-chain-windows/verify.py --print-witnesses
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/six-source-c6-chain-windows
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/six-source-c6-chain-windows --check-source-history --check-modes

At head af3647d85d6f52956b1423ec8b22646f0deac196, the two registered commands pass locally, each below 30 seconds. The package runner prints PASS_REGISTERED_RESEARCH_PACKAGE. The committed-byte, source-history and mode check prints PASS_RESEARCH_COMMIT_BYTES for all six package files and the one declared input. The scientific marker is:

PASS_SIX_SOURCE_C6_CHAIN_WINDOWS states=77220 steps=644820 first=[(0, 4), (1, 8), (2, 13), (3, 18)] joined=[360, 720, 20, 360, 1080, 2400] forward=6 starts09=[0, 0, 0, 0, 180, 480] witnesses=7 controls=7 inputs=1

SOURCES.json binds the interface README, the original task catalogue and the selected-record context to main e54523517d0f03e58352e647c403a5af3e525a57. The question source is #7099 at 123e6808f2d1381e5ab07d128e69c52a98ef0a1a. The T1 by-product, 2,400 pairs, agrees with #7099; the 700 endpoint states and six family sizes agree with #7029. These arise inside the new window computation, without replaying either predecessor's package.

Limitations and unrun obligations

The result fixes six sources, c=6, length 36 and the diagonal pairs (s,39-s), 4 <= s <= 18, with the additional early pair (0,9). The six-window census fixes the family at (18,21); intermediate multisets may differ. It does not classify constant-multiset windows. T2's impossibility on [4,18] holds for either reading. Other masks, source counts and characteristics are outside scope. The unrestricted depth-two exclusion remains #6141's solver result. No solver, Lean, Docker, native supplier, physical price or multiplication-theorem replay was run. Independent human and formal proof-kernel review are unclaimed. No kappa or selected-frontier change.

Open next questions

• Which additional weight-3 pairs can coexist with (0,9) and [8,18]? Start from the 180 extendible states at s=8 and retain enough prefix-order information for one additional pair.
• Which analogous suffix bounds are sharp for other characteristics and source counts? The c=7 existence question is separate open work and is not taken here.

Related open work

• Prove every T1 endpoint pair has a monotone chain #7099 proves all 2,400 T1 endpoint pairs join on [4,18] and poses the T2 question. This package answers T2 and gives all six sharp windows; its T1 count agrees.
• Classify c=6 separation-3 pairs and price the mask at (0, 9). #7029 supplies the six c=6 families and the conditional (0,9) exclusion. This package gives its exact window threshold. The common 700 count pairs and family sizes agree.
• Exhibit c=6 floor words that keep the multiset (6,6,6,7,7,7). #7042 constructs T1 chains and counts their endpoint domains. The common endpoint counts, 120 at s=4 and 20 at s=18, agree. This package's new windows concern the other five families.
• Record that the (0, 15) weight-3 shape forces no 12 at (18, 21). #7043 treats (0,15) and asks about (1,14). Those early pairs are outside this package; no shared extension count is computed.
• Prove one off-diagonal pair cannot finish the c=7 weight-3 count. #7016 excludes one off-diagonal replacement in the neighboring c=7 model; this package computes no c=7 quantity.
• Prove two off-diagonal pairs do not finish the c=7 weight-3 count. #7032 treats two off-diagonal replacements at c=7; this package keeps c=6 fixed.
• Record the non-weight-3 masks missed by the c=7 witness. #7037 records masks missed by a c=7 witness; no mask coverage of that word is checked here.
• Record one c=7 floor word that realizes both missed masks. #7041 gives a c=7 word realizing two missed masks; no c=7 word is computed here.
• Prove non-complement c=7 pairs (10, 14) and (11, 13) have cover 3. #7045 proves cover three for two non-complement c=7 pairs; this package studies c=6 diagonal reachability instead.

Overlap was refreshed against 466 open PR bodies, this account's recent PRs, the latest 30 handoff comments, a paginated 7,089-record Discussion snapshot, targeted searches of main READMEs and merged PRs, including general-characteristic floor-word results. Sparse local-note coverage is incomplete and snapshots are non-atomic. No duplicate of this result was found within that coverage.

Completion refresh read all twelve other open account PRs created after the CLAIM: #7132, #7133, #7134, #7135, #7167, #7168, #7169, #7170, #7172, #7173, #7174 and #7175. Their objects are stock fit, dyadic guards, rectangle residual ranks, splitting tax, E8 hosting/COPY/Gram geometry, tensor Gram stabilization, commuting-stage ranks and fixed-invoice helper splitting. None studies the six-source c=6 diagonal endpoint windows or computes the same quantity. No new same-object line or numerical comparison is warranted.

Merge Danger

Door: Two-way.

Blast Radius: Package.

One isolated research package; no shared code or selected-construction files change.

CI notes

CI did not certify this package. At head af3647d85d6f52956b1423ec8b22646f0deac196, run 42984461381, plan job 130402225182, failed in Onboarding helper tests. The completed checks were read once and the job log inspected. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found in the CI verification document

The current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify, while certificates/selected-result.json names make frontier7000-verify at base e54523517d0f03e58352e647c403a5af3e525a57. I read those committed main files and confirmed that this PR changes neither file. This is a repository-level navigation mismatch. The package replay job 130404267591 was skipped after plan failed.

The following failed job logs were also inspected. Each shows the same sole test failure, missing command and 760-test summary with 24 skipped:

• Other contributors: 42905094314 and 42903313790.
• Lower P104 frame-0 hosting from 40 to 14 monomials (no κ) #7167: 42983769381.
• Exclude the proposed 96-line E8 complement at inner dimension 8k (no κ) #7174: 42984295238.
• Earlier Close the six-bit fit window of the selected BIT stage's stock prescription at log2 e = 52182660963730.92 (no κ) #7132: 42914704054.
• Earlier Prove the exact odd-rectangle residual-rank floor for any stage size and hub count: 72ρ at u = h = ρ (no κ) #7134: 42914997118.

The scientific evidence remains the local normal/optimized verifiers, registered replay, committed-byte/source-history/mode checks and committed export checks described above. No fix, retrigger commit, job rerun or push was made during closeout. The repository-level repair is left to the maintainers.