[alephnan]

PR #6329 proves the limiting exponent for the historical raw-p7 fixed-height scalar recurrence, but leaves its upper-bound onset and majorant constant ineffective. This package answers its first open question with an explicit uniform local-limit estimate and a proof covering every mixed layer, boundary exit, complement and terminal leaf.

For s = 99739/100000, G₀ = 2·2^(10^12)+2 and N₀ = 121·5^G₀, the exact original recurrence satisfies

L(n) ≤ N₀^(261/100000) n^(99739/100000) for every integer n ≥ 1.

The new Fourier argument combines classical Beurling–Selberg interval approximation with elementary integer separation of powers of 2 and 3. It gives a uniform O(1/log n) interval error with an explicit onset, including atomic endpoints. Full-flow propagation costs less than 1600 against an independently certified limiting margin greater than 40000; conservation supplies the initial-range constant.

The exponent was already proved in #6329. The enormous threshold establishes effectivity, not practical finite-size savings. This historical scalar theorem does not improve the selected conditional κ=0.003125804 (#6030), provide a physical supplier, or discharge any selected framework assumption. TASK.json records the changed scope from uniform-transfer-audit. Frozen predecessors and shared code are unchanged.

Validation on CPython 3.12.2:

• Registered normal and optimized exact replays pass.
• Independent rational arithmetic checks cover 3969 interval pairs and 630 signed ceilings.
• Minimal transport replay rejects all 10 actual source corruptions.
• All four registered commands pass from committed Git-archive bytes with no Git metadata and an empty PATH.
• Committed-byte/source-history preflight passes at cd666f0.
• A separate adversarial AI review found no unresolved objection; its earlier contribution is disclosed. Human review and proof-kernel verification are not claimed.

Base and synchronized fork main: 7c49fe3. Latest bounded overlap refresh covered 663 open PRs and 6789 discussion records; sparse local coverage remains incomplete. Exact sources, review, commands, limitations and open next questions are in research/effective-direct-flow/.

Coordination claim.

Prepared for @alephnan with substantial OpenAI Codex assistance. Predecessor and classical analytic attribution are retained.