[petalas]

Summary

Scoped all-size proof with exact finite corroboration: separated multiple-arc circular Schur complements admit linear half-bandwidth at most min(M-1,4w-2). Every surviving run must have at least w vertices. A full negative band with t>=3 surviving runs of exactly w vertices has exact optimum 3w-1, for arbitrary positive eliminated arc lengths.

separated radius-w circle
  compress surviving vertices: circular radius <= 2w-1
  fold the survivor circle:    linear half-bandwidth <= 4w-2
  equal w-vertex runs:         exact optimum = 3w-1

This answers a bounded part of Open next question 2 in PR4879. In particular, at n=9,w=2, deleting {2,5,8} from the specified negative band leaves complete support on six survivors. Every order has half-bandwidth five, so the one-arc constant four cannot extend to separated multiple arcs. No new kappa or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit James Reed / jwr404 PR4286/4289 for the circular sign/support argument, unpaidmoon PR4373 and Ryanson Jonathan PR4407 for linear separated-carve/component context, and Cinna Davis PR4100 and its retained Gaussian/framework lineage. The predecessor executables are neither imported nor replayed. Folding, fill paths, clique replacement and minimum-degree bounds are classical; no literature-wide priority is claimed for them.

Evidence

• Before: PR4879 supplies the one-arc theorem and explicitly leaves multiple-arc separation and ordering open.
• After: the written proof supplies an order for all separated layouts and a matching upper/lower bound for the periodic family. The sharp general constant remains open.
• Literal graph elimination checks all 9,451 separated layouts with 4<=n<=12 and w<=min(3,floor(n/2)); independent exact rational Schur elimination checks all 936 of those with n<=9.
• 144 periodic layouts pass literal support, degree and attaining-order checks. All 87,270 orders across eight periodic cases with at most eight survivors are checked. Ten invalid-input rejections and five semantic controls pass.
• Direct warning-strict normal and optimized commands took 0.89 and 0.90 seconds locally on CPython 3.12.15. Their complete generated receipts are identical and bind all five package files by SHA256. Both registered commands pass. The package metadata/provenance check and exact committed-byte/source-history preflight pass at 04c07ef3d99f08bae8d6b66d242cbc2351e753d7.

From the repository root, using the exact local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/separated-cycle-schur/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -I -W error -B research/separated-cycle-schur/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/separated-cycle-schur
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/separated-cycle-schur --check-source-history

Any installed CPython 3.12 path can replace it. Standard library only. Expected scientific marker: PASS_SEPARATED_CYCLE_SCHUR. The verifier writes no files. All mathematical checks remain active under optimization.

Limitations

The upper bound permits arbitrary coefficients over a field with invertible eliminated principal block. Exact periodic support uses a full negative band, real coefficients and the stated strict row-sum hypothesis. These are structural side results about boundary ordering. Actual Gaussian caller installation, both physical permutations, movement of all right-hand sides, restoration, inverse norms, rounded arithmetic, native/supplier admission and all five selected framework assumptions remain separate. Selected cold replay, Lean and Docker are unrun. Independent human and formal proof-kernel review are unclaimed.

The canonical base remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. Live claims/open PRs and merged/main distinctive-term searches were refreshed before committing. The advance CLAIM preceded the verifier. Cached coverage included all 369 open PRs and 6,286 Discussion records, with sparse local-note omissions; direct main README searches supplemented it. Unpublished work and arbitrary diffs remain outside coverage.

Open next questions

1. Close the gap between the periodic lower bound 3w-1 and the general upper bound 4w-2. Start with small width-two support graphs and orders that split surviving runs. A matching all-layout construction or a larger lower family is missing.
2. Bind both permutations to one actual circular Gaussian boundary solve, retaining every right-hand side, storage and restoration. A complete physical producer/consumer and movement invoice are missing.

Related open work

• PR6404 proves a cross-rim determinant/rank formula for a truncated Gaussian and one contiguous interior; this package studies linear bandwidth after separated circular arcs, with no shared numerical quantity computed.

• PR6405 determines the ordered-minor/sign range of a real truncated Gaussian; this package uses arbitrary band coefficients for its upper bound and a separately stated negative host for exact support.

• PR6413 reduces periodic-Gaussian coefficient setup work; it supplies no multiple-arc ordering or competing bandwidth value. No backend replay was run here.

• PR6426 proves degree-dependent cyclic-inverse row-norm growth. It studies a different invariant and supplies no competing multiple-arc Schur bandwidth value; no extra computation was run for this comparison.

The completion refresh inspected this account's fourteen other open PRs created since the CLAIM: #6418, #6419, #6421, #6422, #6423, #6424, #6425, #6426, #6427, #6428, #6429, #6430, #6431 and #6432. None computes a conflicting value of the separated multiple-arc bandwidth quantity.

CI notes

Both hosted checks passed for head 04c07ef3d99f08bae8d6b66d242cbc2351e753d7 in run 41536191914: plan in 4m36s and registered package replay in 1m37s. Run metadata confirms the exact contribution head, completed status and success. Registered replay executes both warning-strict normal and optimized commands. Checks were read once at completion.

All registered local commands and the committed-byte/source-history preflight also passed. Hosted CI certifies execution of the bounded checks; independent human and formal proof-kernel review remain unclaimed. Selected cold replay, Lean and Docker remain unrun.

Merge danger

Reversible. The change adds only the five files in research/separated-cycle-schur/. Shared code, predecessor packages, frozen evidence and selected-frontier files are unchanged.