[petalas]

Summary

Scoped written proof with a finite exact numerical certificate. For PR4467's fixed historical raw-p7 law, every depth G>=11000 and lockstep offset delta in [0,log(5)] obeys

Pr(Y >= x | survival) <= min(1, 5 exp(-lambda x)),  x >= 0.
E[(Y-L)_+ | survival] <= (5/lambda) exp(-lambda L),  L >= 0.

Here lambda=t-alpha is the predecessor's tilt gap. The second inequality bounds the error from clipping the actual leaf-depth mean, retaining the independent unconditioned terminal rank. This partially answers Ryanson Jonathan / pinterbanget's PR4467 open question 2. DLamdata's PR6118, cited at 6310550e932de70243fc2aaab6f4885688b6fe21, supplies qualitative saturation and the exact leaf-measure identification. The new step is the explicit finite-depth envelope and truncation constants.

Only research/direct-flow-finite-tail/ is added. The result is produced by Nick Petalas. Credit Ryanson Jonathan, DLamdata, zxv's direct flow, Cinna Davis's refined induction, Jacob Alan Sussman's lockstep compiler, and the classical Berry–Esseen/change-of-measure ingredients. Advance claim.

Evidence

Before: the cited saturation proof supplies an unspecified tail constant and onset, and expressly leaves quantitative bounds open. After: the proof gives constant 5 at every G>=11000. Exact rational gates certify a denominator coefficient above 37/4 and numerator coefficient below 2098/49, whose ratio is 8392/1813<5.

From the repository root, with CPython 3.12.15 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B research/direct-flow-finite-tail/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B research/direct-flow-finite-tail/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/direct-flow-finite-tail
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/direct-flow-finite-tail --check-source-history

Reviewers can substitute their CPython 3.12 path. Normal and optimized commands passed in 0.18 and 0.19 seconds locally, each with PASS_DIRECT_FLOW_FINITE_TAIL, 208 exact tilt/clipping fixtures and six rejecting controls. Registered replay passed in 0.72 seconds; each registered check is below one second and its timeout is 120 seconds. Development mode with warnings as errors passed. Committed bytes and source ancestry passed at 9be0bd96279cd66098a9d536ffd0092fd771550a.

The official eight-file committed export and explicit input-coverage check passed. An isolated optimized Git-free replay with empty PATH and read-only exported files passed; all nine exported files including the export manifest remained byte-identical. Direct runs emit generated SHA256-bound receipts and write nothing into the checkout. No predecessor code executes.

Limitations

This establishes neither a Bahadur–Rao asymptotic nor its error term, a decreasing convergence error, stable full-flow shares, a numerical induction ceiling, a native supplier or new kappa. The law is historical; the selected PR6030 result and five assumptions are unchanged. The Berry–Esseen theorem, elementary pi bounds and cited leaf-measure identification remain mathematical inputs. The universal argument is written, not proof-kernel checked. No selected cold replay, full predecessor replay, physical implementation, Lean or Docker was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can an effective nonlattice interval local-limit estimate give a decreasing error relative to the limiting law or mean? Start by combining its remainder with this finite tail envelope. The missing input is a quantitative local-limit remainder.

Related open work

• PR6118 proves qualitative saturation for the same fixed law. This adds an explicit finite-depth upper envelope and clipping-error control, without re-verifying its saturation certificate or claiming its sharper parameter bounds.
• PR6125 proves the three-regime asymptotic power-tariff transition for the same law. It leaves finite-depth convergence rates outside its scope; this result gives a uniform finite-depth probability inequality. No extra computation was run to compare its numbers.

Completion refresh examined this account's 11 other open PRs created since the advance CLAIM, #6121–#6132 excluding this PR. Only #6125 studies the same fixed leaf law and is listed above. No additional numerical comparison was run.

CI notes

Uncertified by CI. At head 9be0bd96279cd66098a9d536ffd0092fd771550a, run 40692886731 failed plan in Onboarding helper tests. The package replay was skipped and did not run in CI.

The two failures are inherited from canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, the PR6030 kappa admission pushed with [skip ci]. The selected record changed without matching the existing documentation synchronization tests:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md. The document instead bolds **0.003125804 = 781451/250000000**, so that exact substring is absent.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci-verification.md. That document's current section still describes PR5599 and make frontier5599-verify.

The failure log ends with FAILED (failures=2, skipped=23). The same two failures appear on PR6120 run 40691776027, whose failed-job log was read for comparison. They are unrelated to this isolated package. No documentation repair or new push was made. Local verification remains as recorded above; no hosted package result is claimed. A maintainer must re-run CI after main's documentation is fixed.

Merge danger

Door: two-way. Blast radius: isolated. The six-file research package changes no shared code, frozen predecessor files or selected-frontier record.