[pinterbanget]

Summary

This adds research/pr4554-roots-recovery-check/, an independent check of ROOTS-RECOVERY.md in Dugongue's #4554 (head 4f46c6a9): the root-seed construction and the CRT and carry recovery. #4557 assumed the seed premise (coordinate error < 2^−p), so this completes my coverage of #4554's native multiplier. All checked claims are confirmed. No new κ.

The spec is vendored (blob 5791526c, SHA256-pinned). Nothing is imported from #4554's root_seed.py or recover.py.

Checks

• Root seed, my own implementation from the spec.
  • Per-term rounding. Each term is compared with the exact rational Taylor term. Errors stay within the recurrence E_k ≤ (8/k)E_(k−1) + 1/S; the worst is 6.9/S, against the e^8/S cap. The sum is at most 4096K/S.
  • Machin and angle. |A/S − π| ≤ 20(P+2)/S and the angle bound hold against a separate AGM π.
  • Final error < 2^−p. 112 checks against exact algebraic values (D ∈ {1, 2, 3, 4, 6, 8, 12}, decided by integer square comparisons) and 36 checks against a decimal reference, with D up to 2^40+87 and p ≤ 128.
  • Word budget. All values fit in 2P+20 bits. The observed peak is about 283 bits for p = 1, inside the stated 289-bit 2^(2P+17) product bound.
• Lemmas, exactly.
  • The Machin tangent identity, and e < 11/4 with (11/4)^8 < 4096.
  • The Taylor-tail chain and numerator·2^(p+1) < S for every p ≤ 4096.
  • The factorial tail at K = 256, 384 and 512.
• Recovery. Garner CRT recovers all 23,800 coefficients exactly, unsigned and signed. The carry invariant and bounds hold, and every product is exact on random and extremal arrays.

python3.12 -B scripts/research_package.py --package research/pr4554-roots-recovery-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4554-roots-recovery-check outcome=completed   (~26 s, stdlib only)

Limitations:

• The operation envelopes and CRT setup costs are not re-counted.
• Exact per-term checks cover p ≤ 4 only.
• The phase and exp tables are left open.

Head: 870b86cbd3840dfd0fcd1c8bdab96b699be54159. Base: canonical main f626db5c.

Credit: Dugongue (#4554); the method precedents the PR cites. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code