[pinterbanget]

Summary

This adds research/pr4535-outer-slack-rank/, a follow-up to my #4563. It keeps κ = 0.003020165 and #4535's certificate, and shrinks the outer slack from d = 10^−10 to d = 10^−12. Running the exact outer-ledger source #4535 pins (pr4359-p7-padding/.../endpoint.py, SHA256-checked), all 54 strict predicates, 4 functional guards and 9 margins above κ still pass.

The absorption gap rises from 377/10^12 to 971/10^12. The least certified frozen rank falls to

N = 2^29,328,002,299, a 34.1× smaller exponent than #4535's 2^(10^12), and 2.6× below #4563.

No new κ, and no selection change.

Checks

• Control. At d = 10^−10, the ledger reproduces Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535's recorded minimum margin 29969798347/10^20 and its full outer_parameters exactly.
• New slack. At d = 10^−12, every predicate passes, with minimum margin 2996979834997/10^24 above κ.
• Rank. The rank is certified with my own logarithms in exact rationals: least, monotone beyond, with the same A, p, c and n0. It reproduces Certify #4535 at a 13x smaller finite rank: N = 2^77,014,944,503 instead of 2^10^12 (same κ = 0.003020165) #4563's K at d = 10^−10.
• Envelope. The log₂ class envelope is about 9.97·10^10, against 3.4·10^12 in Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535.

Trade-off, stated plainly. Smaller d shrinks the outer power gaps (about 3d), which raises the eventual input cutoff roughly like 1/d. The frozen word's constants, though, scale at least like 2^K, and K drops by 4.77·10^10. The net effect on that crude estimate is favourable. No explicit cutoff is computed, here or in #4535.

python3.12 -B scripts/research_package.py --package research/pr4535-outer-slack-rank
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4535-outer-slack-rank outcome=completed   (~2 s, stdlib only)

Limitations:

• Only d, the derived outer parameters and the frozen rank change. The root bracket, mirror and majorant are Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535's.
• Physical constants are not computed.

Head: fbcc9d63446cd4fc8eade8fc29b149fb60fa79ce. Base: canonical main f626db5c.

Credit: hexapode (#4535), Dillon (#4532), zxv and Jacob Alan Sussman, Cinna Davis (#4470), and the pr4359-p7-padding outer-ledger lineage. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code