[pinterbanget]

Summary

This adds research/direct-flow-log-gap/. It follows my #4459 correction, where leaf exits turned out to dominate, and asks what the total exit coefficient of zxv's #4436 direct flow can certify under #4405's refined induction (κ(N,z0) ≤ η = z0 m^{z0−1}).

Claim type: a first-order argument plus finite floating-point evidence. No new κ.

Results

• Balance. At the unit root t* (Σ W_k k^t = Σ P_r r^t, s* = 0.0026865), exit and source potentials balance to 10⁻⁹.
• First order. κ ≈ exp(−(z0−t*)g), where g = E_S[ln s] − E_X[ln e] is the t*-weighted log-gap. This matches the exact κ to 3·10⁻⁵. Since η ≈ exp(−(1−z0)(1+ln m)), the test certifies savings up to cap = s*·g/(g + 1 + ln m).
• Measured. g is a sawtooth in N, but its per-level minima rise by about 0.23 per lockstep level: 1.98, 2.20, 2.54, 2.75, 2.95 (in the verifier), then 3.21, 3.37 (local run to N = 1.73·10⁷). The cap at those minima goes 0.00068 → 0.00099. Saving 0.0008 passes for N ≥ 3·10⁵, and 0.001 does not yet. This is consistent with Independently check #4436's direct overlapping flow: all 9 recorded flows reproduced, exact potential identity to rank 1000, L(1000) matches (no new κ) #4456's effective savings.
• Driver (update). Only the leaf exits sink: their log-depth goes 5.02 → 6.21, while boundary exits (≈1.84) and sources (≈2.45) stay put. A leaf exit is a tilted P-walk below the barrier G ln 5, with relative drift +0.0165. That suggests a meander law, g ~ a + b√G.
• Reading. The ceiling is not bounded away from s* if g → ∞, but convergence is extremely slow. Saving 0.0023 needs g ≈ 34.5: about N ~ 10^{100} under linear growth, or N ~ 10^{900} under √G. Both are heuristic. The direct flow is not a practical substitute for the tensor/padding route near the selected κ unless the induction itself improves.

Evidence

python3.12 -B scripts/research_package.py --package research/direct-flow-log-gap --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/direct-flow-log-gap
  PASS_REGISTERED_RESEARCH_PACKAGE research/direct-flow-log-gap outcome=completed   (~7 s, stdlib only)

The local large run is evidence/gap_large.py 2e7, output evidence/LOGGAP-LARGE.txt (sha256 855a9e8a… (committed LF bytes)). Head: 36d9c475d4bbc99e93952e0ad86a802e1bc581e8. Base: canonical main 2a13eb18.

Not covered

• A proof that g → ∞.
• Multi-step exit 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.

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