[petalas]

Summary

Scoped all-size proof with finite corroboration: after one nonempty proper arc is eliminated from a full negative circular band of radius w, the exact minimum Schur half-bandwidth over simultaneous row/column permutations is min(M-1,2w), where M is the survivor count. Arbitrary band-supported coefficients with an invertible arc pivot retain the same upper bound.

surviving path: 0, 1, ..., M-1
new order:     0, M-1, 1, M-2, ...
upper bound = minimum-degree lower bound = min(M-1, 2w)

At M=9, w=2, the inherited linear survivor order has bandwidth eight; the new order has four, and no order has three. This answers the ordering question in James Reed / jwr404's merged #4289, following #4286. Their no-cancellation argument is a credited premise. This contribution is produced by Nick Petalas. No predecessor executable is imported or replayed.

Evidence

• Before: Prove that equal-weight cycle arcs meet the touching diameter #4289 determines circular-distance fill and explicitly leaves survivor reordering open.
• After: PROOF.md identifies the path-power-plus-end-clique support, constructs an order and proves a matching degree lower bound. The all-size result rests on this argument.
• Exact finite corroboration passes on 4,656 graph/radius pairs through M=96, all 362,879 vertex permutations for 36 pairs through M=8, and 651 circular graph-elimination layouts through n=16. Nine invalid-input rejections and four semantic controls pass.
• Warning-strict normal and optimized registered checks pass locally on CPython 3.12.15 / macOS 27.0.1 arm64, taking 1.65 and 1.66 seconds in direct timed runs. Generated receipts agree and bind all six package files by SHA256.
• Committed-byte/source-history preflight, official export integrity and a read-only Git-free optimized replay pass. The exported files remain unchanged. A read-only proof/code review found no mathematical blocker; independent human and formal proof-kernel review are unclaimed.

From the repository root, use these exact local reproduction commands; substitute an installed Python 3.12 path on another machine:

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

Expected scientific marker: PASS_CYCLE_SCHUR_OPTIMAL_ORDER. Standard library only. Direct verification is self-contained; generic provenance checks require the pinned Git history. Verification writes no files unless an external new receipt path is explicitly supplied.

Limitations

Exactness uses H=I-E with every in-band off-diagonal E entry positive, all others zero, and row sums below one. Signed or missing couplings retain the upper bound but can have a smaller optimum. Multiple eliminated arcs, the physical cost of permutations and all right-hand-side movement/restoration, inverse norms, rounded arithmetic, tape layout, compiler/hosting and a complete paid caller remain outside the claim. No physical supplier, new kappa or selected-frontier change is claimed. The selected 0.00302015 is unchanged. Full selected cold replay, proof-assistant checking and native Linux execution were not run.

Open next questions

1. Bind the folded order and its inverse to one literal circular Gaussian boundary solve, retaining all right-hand sides and restoration, then price the complete movement. No physical conversion is provided here.
2. Specify a multiple-arc separation rule, derive its support graph, and decide whether its optimal bandwidth remains O(w). The one-arc theorem does not decide merged components.

Related open work

No other open PR studying this one-arc optimal-order quantity was found in current targeted bandwidth, Schur bandwidth and circular arc title/body searches. Cached overlap coverage also included 395 open PRs and 3,889 Discussion records, with sparse local-note omissions. This is search coverage, not proof of novelty.

Merged #4373 by unpaidmoon and #4407 by Ryanson Jonathan concern linear separated carves and the different 2w-1 constant. Their results are cited without a new replay. Cinna Davis's #4100 supplies the Gaussian-boundary motivation.

The completion refresh read this account's nine other open PRs created since the CLAIM at 2026-10-22 22:40:40 UTC: #4873, #4874, #4875, #4876, #4878, #4880, #4881, #4882 and #4883. None studies the one-arc Schur support graph, its optimal linear half-bandwidth or a competing value of min(M-1,2w). No extra predecessor computations were run. #4880 studies Euclidean conditioning of unitary observation cycles, a different matrix family and quantity, so there is no shared numerical result to compare.

CI notes

Both hosted checks passed for head dbd82ad59d8fa0f346bb7f34b5baa731dc994880 in run 39867280886: planning in 3m4s and registered package replay in 1m30s. Run metadata confirms the exact contribution head, completed status and success. The checks were read once at completion. Registered replay executes both warning-strict normal and optimized commands.

All registered local commands and committed-byte/source-history, export integrity and Git-free replay checks also passed. Hosted CI certifies execution of these bounded checks, not independent human or formal proof-kernel acceptance of the universal theorem. No full selected cold replay or extra native-platform campaign was run.

Merge danger

Reversible. The change adds only research/cycle-schur-optimal-order/; shared code and selected-frontier files are unchanged.