[pinterbanget]

Summary

This adds research/direct-flow-leaf-walk/. It explains the log-gap growth from my #4465: only the leaf exits sink. It also predicts where that growth stops, which bounds what zxv's #4436 direct flow can certify under #4405's single-step refined induction.

Claim type: exact finite computation to lockstep depth 60, plus a standard large-deviation heuristic. No new κ.

Results

• The walk. A leaf exit is rρ, with ρ a product of G P-letters. Steps ln r ≥ 0, so only the final barrier S_G ≤ ln(N/m) ≈ G ln 5 binds. The t*-tilted step mean is μ = 1.62592, above ln 5 by 0.01648, with σ = 1.484.
• Exact data (G ≤ 40 in the verifier, G ≤ 60 local). The leaf log-depth is D(G) = 3.26 + 1.125√G, with max residual 0.054, against 0.38 for linear and 0.32 for logarithmic. The intercept is ln m − μ, and the slope is near the CLT value σ√(2/π).
• Saturation (heuristic). For G ≫ σ²/ε² ≈ 8,100, the conditioned overshoot tends to Exp(λ*) with λ* = 0.0075, so D → ≈136.5. g then stays bounded, and the refined-induction cap stays strictly below s*: about 0.0025–0.00256, against s* = 0.0026865. Saving 0.0023 is reached only near N ≈ 10^{700}.

Evidence

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

Head: 2ce79a81ed6be7a19a4b5c0cff3ae81868aa246a. Base: canonical main 2a13eb18.

Not covered

• A proof of the saturation.
• Stability of the shares at large G.
• Multi-step tests.
• Native premises.

Credit: zxv (#4436), 1337isnot1337 (#4405) and Jacob Alan Sussman (Theorem 1G). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code