[pinterbanget]

Summary

This adds research/pr4554-native-fft-check/, an independent check of the native FFT multiplier in Dugongue's #4554 (head 4f46c6a9): the parameters in PROOF.md and the recovery bound 216N⁶β²/S < 2^(−10ℓ−6) < 1/4 in FFT-ERROR.md. It follows my #4556 on the same PR. All checked claims are confirmed. No new κ.

Both texts are vendored (blobs c7b4777b and 53691b10, SHA256-pinned). Nothing is imported from #4554's code.

Checks

• Error chain, exact rationals. Worst-case recurrences run for N ≤ 2^20 and β ≤ 2^8 (80 cases):

  • forward amplitudes ≤ 2·4^jβ;
  • (1) 36N³β/S, (2) 112N⁵β²/S, (3) 5N⁴β²;
  • inverse amplitudes ≤ 6·4^jN⁴β² and 212N⁷β²/S;
  • (4) 216N⁶β²/S ≤ (5).

  The twiddle recurrence gives a_j ≤ 8jδ.

• Parameters, exact integers, every ℓ = 2…200. N < 2^(ℓ+2), F+1 ≤ 2^(ℓ+7), (5) < 2^(−10ℓ−6), δ, η < 1, and the raw-product word budget is within 16(F+16)+20.

• Separately written fixed-grid multiplier. It follows the PROOF.md construction: my own Machin/Taylor root floored to the grid, floor-rounded twiddle products, conjugated forward twiddles, an unnormalized inverse, floor division by N, nearest-integer recovery and radix carries.

  • Exactness. All 367 products are exact, covering every pair for n ≤ 4, plus random and all-ones operands up to 12,000 bits.
  • Observed error. The worst observed coefficient error is 5.7·10⁻⁷ of (4).
  • Magnitudes and width. Inverse magnitudes stay below 6N⁶β²S, and carries stay at most L(β−1). Every stored value fits the global word width; the FFT uses at most 12% of it, and the root routine sets the rest.

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

Limitations:

• ROOTS-RECOVERY.md's root-precision proof and the O(n) operation ledger are not re-derived.
• The Gaussian adapters are out of scope here; Independently check #4554's native banded inverse: exact elimination bounds, Gaussian tail, exact periodic Woodbury solves (no new κ) #4556 covers the banded inverse.

Head: ebc09ef98f107da49b78b450642fd00ca1a93a74. Base: canonical main f626db5c.

Credit: Dugongue (#4554); Fürer (2014). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code