[petalas]

Summary

Scoped proof produced by Nick Petalas: the fixed raw-p7 direct-flow leaf tariff has an exact three-regime asymptotic. With lambda=t-alpha>0, v=lambda+u, and A(u)=121^(-u)Z(t+u)/Z(t):

v > 0: M_G(u,delta) -> A(u) lambda/v
v = 0: M_G(u,delta) ~ A(u) lambda sigma sqrt(pi G/2)
v < 0: M_G(u,delta) ~ A(u) lambda sigma sqrt(2 pi G) exp(-v delta) R(u)^G
       R(u)=Z(t+u)/Z(alpha) exp(-v log5) > 1

The threshold is exactly t+u=alpha, for every fixed real shift u, uniformly across bounded lockstep offsets. The closed upper endpoint is explicitly a fixed-depth extension; actual integer ranks use the original floor depth and a half-open offset interval. A bounded mean log-depth therefore does not justify every power tariff.

This extends DLamdata's #6118 tilted local-limit framework and Ryanson Jonathan / pinterbanget's #4467 leaf model. The mathematical citation is pinned to #6118 at 6310550e932de70243fc2aaab6f4885688b6fe21; its newer head reorganizes documentation without changing PROOF.md. No predecessor certificate is replayed or claimed as this contribution. Credit also zxv's #4436 direct flow, Cinna Davis / 1337isnot1337's #4405 refined induction, Jacob Alan Sussman's compiler and Stone's classical local limit theorem.

Claim type: scoped proof, corroborated by exact finite identities. No new κ or selected-frontier change. Base: 721c03822c421b7f7fcbb2598993b947c4d88419.

Evidence

Before this contribution, #6118 supplied exponential deficit and mean convergence, without the full power tariff or its critical/divergent regimes. PROOF.md derives an exact tilted quotient and the three asymptotics; verify.py checks the new finite algebra with two separately implemented prefix/leaf enumerations.

Local normal and optimized checks pass: 15 fixtures, 225 moment identities, 450 tilted quotients, 210 second-measure-change identities, 10,191 accepted ordered prefixes counted across fixtures, and eight rejecting controls. The checks compute exact rational values at integer exponents; they do not numerically certify the infinite-parameter theorem. Both registered commands ran locally in under one second each.

From the repository root, using the tested CPython 3.12 interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/direct-flow-leaf-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/direct-flow-leaf-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/direct-flow-leaf-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/direct-flow-leaf-tariff --check-source-history --check-modes

Other hosts may substitute their CPython 3.12 executable. Registered replay, metadata/source bindings, committed-byte/source-history/mode checks and the official eight-file export/input-coverage check pass. Direct normal/optimized replay also passes from a read-only export without Git.

Limitations and unrun obligations

Stone's nonlattice interval local-limit theorem, as specialized in #6118, is a cited mathematical dependency. Independent human and formal proof-kernel review are unclaimed. The intended raw-p7 crossing exponent is inherited; this package proves its theorem for every fixed t>alpha and does not recalculate that crossing.

No finite-depth error rate, practical rank threshold, source/boundary/leaf share theorem, complete refined-induction ceiling, native implementation or supplier admission is established. The historical raw-p7 measure is not the selected #6030 invoice. All five selected framework assumptions remain. Full selected replay, Lean and Docker were not run.

Open next questions

• Derive the compatible non-leaf asymptotics and relative masses needed for the complete normalized exit tariff; begin with source and boundary measures at the same tilted barrier.
• Give explicit finite-depth errors for the three regimes; begin with quantitative interval local limits for the fixed log{2,3,5} support, retaining its recurrent phases.

Related open work

• Prove saturation of the fixed #4467 leaf deficit (no new κ) #6118 proves qualitative leaf-deficit saturation for this same law. This contribution builds on its framework and evaluates the full power tariff, including critical and divergent regimes; its saturation certificate is not audited here.

• Portable-replay repair of my packages: 18 VERIFY input lists, 3 at-base receipts, 4 selected-record drifts, 1 timeout (no science change) #4547 repairs replay metadata in predecessor packages, including the historical direct-flow chain. This isolated package adds a mathematical cost calculation and adopts no pending repair.

• Prove an explicit finite-depth leaf tail bound (no new κ) #6128 proves an explicit finite-depth tail envelope and clipping-error bound for the same historical raw-p7 law. This PR derives asymptotic power tariffs and their critical/divergent regimes; it supplies no finite-depth rate. No extra computations were run to compare the results.

CI notes

Uncertified by CI at commit f58e29052a0bbb4e123ea777b5e03321fde6f31f. In run 40692604534, plan failed during Onboarding helper tests and the package replay was skipped. The package replay did not run in CI.

The failure is inherited from current main 721c03822c421b7f7fcbb2598993b947c4d88419, the PR6030 κ-admission commit pushed with [skip ci]. Its selected record and onboarding docs do not satisfy two existing synchronization tests:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the exact marker **0.003125804** in docs/research/current-status.md. That document instead bolds the longer expression **0.003125804 = 781451/250000000**, so the literal marker is absent.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The failed-job log reports 678 onboarding tests, two failures and 23 skips. PR #6120's run 40691776027 fails the same two tests, confirmed from its failed-job log. Neither failing document is changed by this PR. These are baseline failures, not failures caused by the leaf-tariff package.

All listed local checks passed, but this hosted run certified no package replay. This PR needs a maintainer re-run after main is fixed. No push, main-document fix or CI re-run was performed for this blocker.

Merge Danger

Door: two-way. Blast Radius: isolated research package. Existing sources, selected records and frozen evidence are unchanged.