[pinterbanget]

Summary

This adds research/carve-schur-bandwidth-census/. It is an independent, exhaustive check of unpaidmoon's #4373: with at least b survivors between interiors, the boundary-order Schur complement of the carve has bandwidth at most 2b−1. That bound is GAUSSIAN.md's O(W). The check also generalizes the theorem to an exact formula for any eliminated set.

Claim type: a structural lemma plus an exhaustive finite census. No new κ. The code is separately written and imports nothing from #4373.

Lemma

Group the eliminated set into components, joining consecutive eliminated indices at distance at most b. Then the boundary-order bandwidth is

max( max_C #{survivors in [min C − b, max C + b]} − 1 ,  max boundary distance of survivor pairs within b ).

The proof is the fill-path lemma (Rose–Tarjan). It is exact for generic entries and an upper bound for any entries.

• Prove the separated Gaussian carve has Schur bandwidth at most 2W-1. #4373's theorem. Under its separation hypothesis, each interior is its own component and touches at most 2b survivors. So the bandwidth is at most 2b−1, attained exactly by a two-sided interior.
• Necessity. Separators of b−1 survivors give chains with bandwidth 2b−1+(k−1)(b−1). Prove the separated Gaussian carve has Schur bandwidth at most 2W-1. #4373's b = 2 chain gives k, as stated.

Evidence

• 196,512 layouts: every eliminated subset of every path n ≤ 15, for b = 1, 2, 3. On every layout the formula equals the bandwidth of an actual Schur complement with random entries over GF(2^61−1).
• Separated layouts: all 84,491 satisfy the 2b−1 cap, and the two-sided criterion decides sharpness exactly.
• Cap the Schur bandwidth of a middle interval by 2b-1 #4285's witness: far entry −1/432, bandwidth 3 and both inverse norms 488/389, recomputed exactly.

python3.12 -B scripts/research_package.py --package research/carve-schur-bandwidth-census --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/carve-schur-bandwidth-census
  PASS_REGISTERED_RESEARCH_PACKAGE research/carve-schur-bandwidth-census outcome=completed   (~50 s, stdlib only)

Head: 4bd1d57ebee0a6f2270cae117f3dff3a551f6667. Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7.

Not covered

• Inverse norms in general.
• Circular cells (Show that circular Schur bandwidth follows the arc length #4286).
• The narrow solver's window width (Prove the separated Gaussian carve has Schur bandwidth at most 2W-1. #4373's open question 1).
• Whether real Gaussian entries produce all possible fill. Only the upper bound matters.

Task and overlap

• Card: uniform-transfer-audit, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: no review of Prove the separated Gaussian carve has Schur bandwidth at most 2W-1. #4373 had been posted.

Credit

@unpaidmoon for the theorem and its application (#4373). jwr for the one-interval cap and witness (#4285). #4100's GAUSSIAN.md for the carve.

Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review. Apache-2.0. This PR does not merge anything or change the frontier.

🤖 Generated with Claude Code