[pinterbanget]

Summary

This adds research/pr4405-refill-check/. It independently checks 1337isnot1337's #4405, which refines zxv's #4359 in two ways: smaller explicit uniform ranks (TEMPLATE.md), and an endpoint family admitting every rational κ below the complex mixed root (ENDPOINT.md). It follows on from my #4404 audit of #4359's original rank chain.

Claim type: an independent exact check plus a lever analysis. No new κ. Nothing is imported from #4405 or #4359.

Results

• Refined ranks reproduce. The induction needs only an exit coefficient ≤ η, and it is valid. These match Refine PR4359: conditional κ = 0.00230509 and smaller explicit template ranks #4405:

  • complex H0 = 335,449 and H1 = 1,546,084,442;
  • BIT H0 = 69,740 and H1 = 42,843,607;
  • the endpoint's H1 = 2,099,384,703,303,215,709.

  The H0 values are certified minimal over the whole log enclosure.

• Refine PR4359: conditional κ = 0.00230509 and smaller explicit template ranks #4405's suggested levers.

  • z0: the midpoint is already optimal. A 199-point grid finds nothing better, because the leading factor 1/((z0−t)(z−z0)) is minimized there.
  • The exit constant 16: its share of H0 is 1.2·10⁻⁴ (complex) and 9.2·10⁻⁴ (BIT), so sharpening it is negligible.
  • The uniform saving: this is the real lever, since H1 ∝ 1/δ². At a uniform saving of 0.0023, H1 falls to 61,765,945 (complex) and 13,943 (BIT). Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359 admits BIT at 0.0027 only so that BIT does not cap the complex supplier, so a much smaller BIT rank may be available. That depends on the cover fallback, which is not checked here.

• The endpoint.

  • The mixed moment at C_test = 0.00230509019 is certified positive (about 4.7·10⁻⁷), so C_test < C*. Also C* < 0.002306.
  • ENDPOINT.md's self-contained gap identities hold exactly, with positive values, at κ = 0.00230509 and 24 more rational κ. These cover ξ, εq, the stopped-supplier gaps d/2, 3d/2 and d/2, the stop gap, ε−ξ, the precision-guard gap, and the parameter and geometric order.

Evidence

python3.12 -B scripts/research_package.py --package research/pr4405-refill-check --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/pr4405-refill-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4405-refill-check outcome=completed   (~1 s, stdlib only)

Head: 269b9cca40e88fb6616806ec39a7464c6c17351e. Base: canonical main 12fa8c5c, priority revision 7. The audited files' SHA256 prefixes are in the README.

Not covered

• The outer cost-row formulas.
• Native closure, adapters and the delayed cover or its fallback.
• Whether a smaller uniform saving is admissible.

Task and overlap

• Card: independent-moment, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: no review of Refine PR4359: conditional κ = 0.00230509 and smaller explicit template ranks #4405 had been posted.

Credit

@1337isnot1337 (#4405), @zxv (#4359), and Jacob Alan Sussman (Theorem 1G, #4268).

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