[unpaidmoon]

Change and mathematical scope

Scoped audit of one CrocSwap#4100 precision/conditioning step. No new κ. Selected κ stays 0.001672660197076918.

After the retained dyadic carve, with physical half-bandwidth W and at least W retained coordinates between successive interiors, the Schur complement in boundary order has bandwidth at most 2W-1. That matches the one-interval cap and is the O(W) claim in GAUSSIAN.md. The constant is sharp: one middle interval, and a pair of intervals separated by exactly b survivors, both attain 2b-1. A separator of only b-1 survivors can make the bandwidth grow with the number of interiors.

This is not a supplier admission, a Strassen repricing, or a complete multiplication theorem.

Task and overlap

• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7.
• Card: uniform-transfer-audit, scope refined to this bandwidth proposition. CRT repair, the free-algebra lift, and fixed-tape induction stay open.
• Pinned inputs: research/unconditional-batched-crt/GAUSSIAN.md, research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.md.
• Package: research/gaussian-carve-schur-bandwidth/.

Nearest overlap, and why this is different:

• Cap the Schur bandwidth of a middle interval by 2b-1 CrocSwap/integer-mult-bounds#4285 proves the cap 2b-1 for one middle interval and leaves boundary order of the Gaussian caller open.
• On a width-2 path, one interior has boundary-order half-bandwidth 3 (no new κ) CrocSwap/integer-mult-bounds#4298 and On a width-3 path, one interior has boundary-order half-bandwidth 5 (no new κ) CrocSwap/integer-mult-bounds#4299 certify one interior on one equal-weight path (bandwidth 2 and 3, a few lengths). Both say they do not measure the truncated Gaussian row, and neither bounds a second interior.
• Show that circular Schur bandwidth follows the arc length CrocSwap/integer-mult-bounds#4286 is the circular arc-length case. This proof is linear and uses the retained endpoint strips and period cuts as boundary.
• Map independent audit coverage of #4100's obligations (navigation; no new κ) CrocSwap/integer-mult-bounds#4311 lists the carve comparison as still open. Later open PRs through Classify all seven scalar fit cutoffs of the retuned ordinary selector CrocSwap/integer-mult-bounds#4353 do not take it.

The fork PR is based on cursor/upstream-main-pin-a7d3, which is that canonical main commit. Fork main is not canonical main.

Validation

CPython 3.12, standard library only:

python3 -B research/gaussian-carve-schur-bandwidth/verify.py
python3 scripts/research_package.py --package research/gaussian-carve-schur-bandwidth

Both passed here (PASS gaussian-carve-schur-bandwidth, PASS_REGISTERED_RESEARCH_PACKAGE).

Checked: the published 7×7 witness (far entry -1/432, both inverse norms 488/389, bandwidth 3); separated layouts of bandwidth 1, 2 and 3 stay within 2b-1 and one two-interval host attains it; the bandwidth-2 chain with one survivor between singleton interiors has Schur bandwidth equal to the interior count for 1..8 interiors; an out-of-band entry and a singular pivot fail closed; pinned carve sentences and 7^16 < 2^45.

The all-size support bound is the counting argument in the README. The finite hosts are certificates, not a stand-in for that argument. Not run: tensor-caller replay, narrow-solver window inspection, circular cells, Lean, Docker, selected cold replay.

Credit

Cinna Davis, CrocSwap#4100 GAUSSIAN.md. The capped-interior loop is research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.md. The one-interval cap and witness ratios are jwr, PR CrocSwap#4285, recomputed here. PR CrocSwap#4298 and PR CrocSwap#4299 are the one-interior boundary-order checks. PR CrocSwap#4311 is the coverage map.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review is claimed.

Open next questions

1. Does any narrow boundary solve use triangular windows of width exactly W? That would miss the 2W-1 fill even though the O(W) cost sentence survives. First step: bind one shallow-Toeplitz narrow solve to a concrete window. Blocker: GAUSSIAN.md states O(W) and an RHS batch of width at most 2W, not an executable window width.
2. Does every Propose unconditional κ = 0.00173097911567079247276057 via batched CRT CrocSwap/integer-mult-bounds#4100 phase cell retain a period cut through elimination? A circular cell with no retained cut is the Show that circular Schur bandwidth follows the arc length CrocSwap/integer-mult-bounds#4286 arc-length setting. First step: one full-period cell with its cut marked boundary. Blocker: this package does not replay the tensor caller.

Workflow feedback (optional)

The CrocSwap#4311 coverage map made the remaining CrocSwap#4100 obligations enumerable. Its "still open" cell for this carve was still accurate after the one-interior path checks in CrocSwap#4298 and CrocSwap#4299.