[pinterbanget]

Summary

This adds research/pr4554-banded-inverse-check/, an independent check of NATIVE-BANDED-INVERSE.md in Dugongue's #4554 (head 4f46c6a9). The component has three parts: pivot-free band elimination with a rounding error bound, a proved Gaussian tail and bandwidth, and the periodic rank-2h Woodbury reduction. All checked claims are confirmed. No new κ.

The spec is vendored (blob 8be8f992, SHA256-pinned). Nothing is imported from #4554's solvers.

Checks

• Band theorem, exact Gaussian-rational arithmetic. 20 near-identity complex band matrices with ρ = 1/2, including adversarial all-negative E, and h ≤ 4.

  • Exact elimination. Pivots are ≥ 1/2, row sums ≤ 3/2, multipliers ≤ 3, and ‖A⁻¹b‖ ≤ 2‖b‖.
  • 60 rounded solves. u ranges from the limit 1/(1000H²) down to 2⁻³⁰. Pivots are ≥ 1/4, multipliers ≤ 8, ‖L̂Û − A‖ ≤ 6H²u, ‖L̂‖ ≤ 9H, ‖Û‖ ≤ 2, and the ŷ and x̂ growth bounds hold. The final error is at most 1000H⁴u(‖b‖+1), with worst observed ratio 8.8·10⁻⁵, consistent with the stated loose constants.

• Gaussian tail.

  • The per-summand distance ≥ d − 1/2 holds.
  • The tail ≤ 4exp(−πα²h(h+1)) holds in log domain.
  • The stated h gives a tail < 2^(−P)/4 for P ≤ 4096.

  Together these make 5,346 checks.

• Periodic Woodbury. 7 actual instances of N, down to r = 1.1875, all with ‖N−I‖ ≤ 1/8:

  • the row tails meet the bound, and the corner entries lie only in the first or last h columns;
  • ‖A0−I‖ ≤ 1/4, ‖U‖ ≤ 1/4 and ‖S−I‖ ≤ 1/3;
  • x = y − Xz satisfies A x = b exactly.

• Replay. Native Word RAM Gaussian adapters: chirp blocks and reusable banded inverse #4554's own package replays at its head (outcome=completed, 16.6 s).

python3.12 -B scripts/research_package.py --package research/pr4554-banded-inverse-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4554-banded-inverse-check outcome=completed   (~16 s, stdlib only)

Limitations:

• The rest of Native Word RAM Gaussian adapters: chirp blocks and reusable banded inverse #4554 is out of scope.
• ‖N − I‖ ≤ 2^(−α²θ) is a predecessor fact; only instance membership is checked.
• Norm comparisons use floats; matrix arithmetic is exact.

Head: 5193f368a77a1533abcf8e8bafc97486f85fcb02. Base: canonical main f626db5c.

Credit: Dugongue (#4554) and the Harvey–van der Hoeven Gaussian resampling sources. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code