[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. For positive integers d0,a,b, let D=d0*b. A sufficient threshold for (m+1)^D < 2^a*m^D is

T = 1                             if a>D,
    max(2,ceil((3D-a)/(2a)))       otherwise.

It never exceeds the cited ceil(27D/(17a)) threshold. It answers Open next question 1 of unpaidmoon #7450 with a bound that depends on m:

 (d0,a,b) = (1,1,5)
- sufficient threshold 8
+ sufficient threshold 7, proved sharp

The sharp four at (1,2,5) is preserved. The new step applies the classical decreasing function (2x+1)*log(1+1/x) to the auxiliary persistence comparison. No novelty is claimed for the analytic inequality itself. The self-contained proof is in research/shifted-persistence-threshold/PROOF.md.

Source question: unpaidmoon #7450 at f9b9057f9befe744df7f6b58b61b53e77001af1f; its census and persistence construction are cited, not audited or replayed. Credit its retained persistence-note lineage, Mister-Meeseeks's #7000 application review, and classical convexity and induction. Canonical base e54523517d0f03e58352e647c403a5af3e525a57; card uniform-transfer-audit, changed to this auxiliary threshold lemma. Advance CLAIM.

Evidence

Before: the uniform power-pair method in #7450 leaves the specified m=7 step outside its sufficient region. After: the written proof includes it for every later integer, and the exact failure at m=6 proves sharpness.

Local CPython 3.12.15, standard library only. Normal and optimized commands pass, timed separately at 0.033 and 0.039 seconds. The registered workload checks 255 shifted-power inequalities, 255 consecutive decreases, all 9,216 parameter pairs a,D=1..96 and 27,648 one-step evaluations. In that rectangle, 4,753 threshold pairs strictly improve. Nine semantic controls and eight exact-reason invalid-input controls pass. Output marker PASS_SHIFTED_PERSISTENCE_THRESHOLD comes with generated SHA256 hashes for all six package files. Package metadata/pins and committed bytes/modes/source history pass at 844ea8372f3e1605bed1368f0a0135a177d6aa2f.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/shifted-persistence-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/shifted-persistence-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/shifted-persistence-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/shifted-persistence-threshold --check-source-history --check-modes

Other hosts may substitute their CPython 3.12 path. Registered commands use {python}. No predecessor scientific verifier was run.

Limitations and unrun obligations

This is a sufficient threshold, not the general exact minimum. At D=10,a=1 it gives fifteen while fourteen already works. It proves persistence once an initial comparison holds; it does not construct the initial successful index or a complete selected cutoff. No physical supplier, new kappa, assumption discharge or complete multiplication theorem is claimed. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

The selected cold replay, predecessor suites, physical/compiler/native/ownership/precision/tape/recovery proofs, Lean and Docker were not run. Ruff and Mypy are unavailable on the local PATH; in-memory compilation and warning-strict normal/optimized replay pass. The isolated verifier documents its bounded managed-runtime allocation deviation.

Open next questions

Can another explicit rational term in the logarithm bound reduce the remaining sufficient-versus-necessary gap, starting with D=10,a=1? First derive the corrected integer test and prove it is never worse. Complete cutoffs and physical costs remain separate obligations. No follow-up is reserved.

Related open work

• Prove the section-6 one-step ratio 27/17. #7450, unpaidmoon: supplies the prior 27/17 threshold and asks about m=7. This proof includes that case and retains the same sharp four; no predecessor census is replayed.
• Prove the sharp n-axis monomial elimination record count. #7286, unpaidmoon: improves a monomial elimination record count elsewhere in the same application review. Its object and count are independent of this one-step inequality.

The completion refresh inspected this account's five other open PRs created since the CLAIM, #7470 through #7474. They study different objects and do not compute a competing threshold for this one-step comparison. No extra mathematical computation was run to compare their results.

CI notes

CI did not certify this package. At unchanged head 844ea8372f3e1605bed1368f0a0135a177d6aa2f, run 43289715434, plan job 131432409153, failed in Onboarding helper tests. I read the completed checks once and independently read the failed job log.

The log reports exactly one failing test among 760 tests, with 24 skipped: test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. At line 58, its AssertionError says that `make frontier7000-verify` was not found in the CI verification document. The replay job was skipped because plan failed; no hosted package replay ran.

I inspected the base files at canonical main e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The selected record, CI guide and failing test are unchanged by this PR. This is a repository-level mismatch, separate from this package's passing local verification.

I also independently read the failed-job logs for run 42905094314, run 42903313790, and run 43280441536. Each has the same sole failing test, the same missing-command assertion, and the same 760-test total with 24 skipped. No fix, retrigger commit, push or workflow rerun was made; maintainers own the repair of main.

Merge danger

Door: two-way. Blast radius: isolated research package. The selected frontier, shared code and pinned evidence are unchanged.