[pinterbanget]

Summary (corrected)

This updates research/direct-flow-boundary-share/, which concerns zxv's #4436 open direct-exponent gate.

Correction. The first version (head 04da76eb) claimed the gate reduces to a uniform boundary-exit share bound. That was wrong. Its lemma assumed leaf exits have rank o(n). In #4436's flow, leaf exits reach 0.397·N and carry 85–92 % of the source t-potential. The lemma stays valid only as a conditional statement whose leaf hypothesis fails here.

Claim type: an exact identity, a rigorous uniform bound and finite measurements. No new κ.

Results

• Mirror identity, checked exactly at 4 ranks. Boundary exits and inner sources are one base measure (μ = cnt(ρ)u(a), z = ρc(a)) seen through mirrored windows: (1/r, 1/k] for exits and (1/k, 1/r] for sources, at rank rkz.
• Rigorous bound. Splitting into elementary windows gives β(N) ≤ max_J E_J/S_J = 0.3208 for every N, against a measured maximum of 0.1545. The boundary share is therefore not the bottleneck.
• The exit coefficient against η (Refine PR4359: conditional κ = 0.00230509 and smaller explicit template ranks #4405's refined-induction test), through N ≤ 5·10⁵: it passes at saving 0.0008 for N ≥ 3·10⁵ (sup κ 0.99445 < η 0.99537) and fails at 0.001. This agrees 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 saving of about 0.0008.

Evidence

python3.12 -B scripts/research_package.py --package research/direct-flow-boundary-share --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/direct-flow-boundary-share
  PASS_REGISTERED_RESEARCH_PACKAGE research/direct-flow-boundary-share outcome=completed   (~35 s, stdlib only)

Head: 488a75a9e95e14448b6301719249f876f3999931. Base: canonical main 2a13eb18.

Not covered

• Asymptotics of κ.
• A leaf-exit comparison, which is now the open question.
• 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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