[pinterbanget]

Summary

This adds research/pr4535-smaller-rank/. hexapode's #4535 (head a6d7237f) freezes its critical-root compiler word at N = 2^(10^12). Its own certificate already gives absorption L(N) < N^z at

N = 2^77,014,944,503, a 13.0× smaller exponent,

with the same κ = 0.003020165 and the same parameters, root bracket and gap. This partly answers the "funded cutoff" open question of #4532 and #4535. No new κ.

Why

Absorption needs only ln A + p·ln(c + ln N) < (z − θ_hi) ln N. The left side is about 24 nats at #4535's rank, against about 261 on the right, so the rank is far larger than needed.

• Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535's own ln A bound. Even with it, (1−θ_lo) ln n0 ≤ 13.62, the least certified rank is 2^90,180,626,561 (11.1×).
• Sharper bound. Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535's own definition A = n0^(1−θ)/(c+ln n0)^p gives ln A ≤ 10.245, and the least rank is 2^77,014,944,503.

Checks (own certified logarithms, exact rationals)

• Minimal rank. K is the least integer certified. The inequality fails at K − 1000 even with the most favourable bounds, so K is minimal for these parameters.
• Larger ranks. It holds at every larger N (z − θ_hi > p/(c+ln N)), and N > n0.
• Envelopes. The frozen word's envelopes shrink accordingly. G_depth falls from 2·10^11 to 1.54·10^10, and log₂ C_N from 3.4·10^12 to 2.62·10^11.
• Consistency. At K = 10^12 the margin is 240.17 nats (236.79 with the coarse ln A), consistent with Improve conditional multiplication kappa to 0.003020165 via critical-root transfer #4535 and my Independently check #4535's arithmetic (κ = 0.003020165): critical root, gap 377e-12, mirror, absorption, binding outer row (no new κ) #4536.

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

Limitations:

• Only the frozen rank changes. Physical constants, the outer cutoff and selection are untouched.
• p, c, n0 and d are not retuned.

Open questions:

• Halving the outer slack d would cut K a further 1.8×, but needs a 54-predicate outer replay.
• A majorant retune needs Prove adaptive gate criterion and critical-root leaf bounds (no κ) #4532's gates rechecked at θ.

Head: 7db8ee8d97d456f120cb534d8cf356a1b8b02ca8. Base: canonical main f626db5c.

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

🤖 Generated with Claude Code