[pinterbanget]

Summary

This adds research/pr4568-arithmetic-check/, an independent arithmetic check of hexapode's #4568: conditional κ = 0.00302016597 (+9.7·10⁻¹⁰ over #4535), head b269a729. #4568 uses my #4564's outer slack d = 10^−12. All checked claims are confirmed. I also find that combining #4568 with my #4565 retuned majorant shrinks its frozen rank 6.87× at the same κ. No new κ of my own.

Checks

• Outer ledger. I replayed the exact pinned source (pr4359-p7-padding/.../endpoint.py, SHA256-checked) at Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's parameters, which are confirmed to be exactly κ + jd with d = 10^−12. All 54 strict predicates, the 4 guards and the 9 margins pass. The minimum margin 2996979834027/10^24 matches Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's record.
• Saving order. κ < s_C < tested = 1 − θ_hi < 1 − θ holds. The tested saving sits exactly at the bracket endpoint, and stays strictly below the root's saving because the certified root lies strictly inside Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535's bracket. The gap z − θ_hi = 10^−12 is exact.
• Absorption. At N = 2^(10^14) my certified margin is 42.94 nats, above Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's 418/15. The least certified ranks at Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's κ are:
  • 3.75·10^13 with Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's coarse ln A;
  • 3.25·10^13 with A's own definition;
  • 1.456·10^13 with my Retuned p16 critical-root majorant (M=2^24, ell0=64, p=3/10): #4535's frozen rank 2^11,825,387,808, 85x below 2^10^12 (same κ) #4565 retuned witness, 6.87× below Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's 10^14.
• Replay. Conditional κ = 0.00302016597 (+9.7e-10 over #4535) #4568's own package replays (outcome=completed, 8.4 s).

python3.12 -B scripts/research_package.py --package research/pr4568-arithmetic-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4568-arithmetic-check outcome=completed   (~3 s, stdlib only)

Limitations:

• The critical root, mirror and gates are Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535's; my Independently check #4535's arithmetic (κ = 0.003020165): critical root, gap 377e-12, mirror, absorption, binding outer row (no new κ) #4536 and Independent proof check of #4532 §1.1, §2.2-2.3, §3 (critical-root majorant under #4535's κ): every step verified, no gap (no new κ) #4569 cover them.
• Physical premises are out of scope.
• Whether a slack-retuned κ is admitted is for the maintainers; this package takes no position.

Head: b5e119c1760203f10a2c2245bd8a50fa47b7013a. Base: canonical main f626db5c.

Credit: hexapode (#4568, #4535), Dillon (#4532), Cinna Davis (#4470), zxv and Jacob Alan Sussman. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code