[petalas]

Summary

Classify every finite call budget at which PR6139's hidden-window lower bound is attainable, answering its Open next question 1. Claim type: scoped mathematical proof with exact finite corroboration. No new kappa.

S=6^h/2, L=floor(N/S), Z=floor((L-1)/2)
bound: w >= log_2((Z+1)/(Z-1))
attainable iff L in {4n^2-1,4n^2:n>=3} union {2n(n+1):n>=4}

The proof gives every attaining residual and permitted common padding, including budgets with two distinct residuals. It reduces equality to Delta=1, z=Z and k=R(1) in {1,2}. Attainability is existential; other profiles at the same budget can have wider windows. The actual optimum at other budgets remains open.

Karnaugh33's PR6139 at 3b9de03623e8bf05af49f9f9c5fc7a426cbde9e5 supplies the bound, first family at L=4n^2-1 and second family's n=4 example. This contribution adds the exhaustive classification, adjacent budgets and general second family. Credit zxv's PR6049 for the minimal-support framework, Nick Petalas's PR5174, Ali Romman's rank-budget lineage and the classical quadratic identities. This contribution was produced by Nick Petalas.

Evidence

Before: PR6139 establishes sharpness at an infinite sequence and a further example, leaving the complete attainment set open. After: PROOF.md supplies both directions for all positive integers h,N, with a standalone exact constructor and two finite oracles.

Local CPython 3.12.15, standard library only, normal and optimized warning-strict checks pass:

• 4097 normalized budgets, with 100 attaining budgets and 101 residuals; 260610 divisor trials.
• All 142880 positive coefficient triples through normalized budget 96; 134 eligible residuals and 2760 residual/budget comparisons.
• 606 signed profile constructions with exact counts, rank mass, endpoint multiplicities and window signs; 768 common-padding cases.
• 19 rejected inputs, five semantic controls and nine large-integer cases. Checks remain active under optimization.
• Registered package replay, metadata/pins, whitespace and committed-byte/source-history checks pass for 0ef36537f64f6d11e7574268266ef7aeffc4df1e. The two registered scientific commands took 0.118 and 0.120 seconds locally, each with a 120-second limit. Generated stdout receipts bind all six package files by SHA256.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/hidden-window-attainment-budgets/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/hidden-window-attainment-budgets/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/hidden-window-attainment-budgets
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/hidden-window-attainment-budgets --check-source-history

Any CPython 3.12 installation may replace the displayed interpreter. Expected markers are PASS_HIDDEN_WINDOW_ATTAINMENT_BUDGETS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The theorem concerns the minimal power-of-two alphabet and the stated endpoint/interior-root pattern. Finite checks corroborate the written proof. The reference functions accept 1<=h<=16 and 1<=N<=2^256; the theorem is uniform. Source results are cited at exact public heads, with no predecessor executable or foreign Git commit required at runtime.

These arithmetic profiles do not establish a native word, independent targets, source compatibility, dirty restoration, compiler admission, stock, precision/tape or a complete consumer. All five framework assumptions remain. Selected cold replay, predecessor scientific replays, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed. Selected records and existing packages are unchanged.

Open next questions

Determine the exact minimum width at normalized budgets outside the classified set. Begin with a discriminant-sensitive minimization of Delta/z^2 and an integer optimality certificate. That optimization remains unresolved; this PR stops at the first verified equality classification.

Related open work

• PR6139 proves the discrete bound and poses the equality question answered here. Its N=1000 census and larger-support counterexample are cited, not replayed.
• PR6049 constructs hidden losing windows with minimal rank support and unbounded multiplicities. This result fixes the call budget and classifies equality in the subsequent bound; it supplies no physical realization.

The completion refresh inspected all 13 other open account PRs created since the advance CLAIM at 2026-10-28T16:11:14Z. None studies this hidden-window equality question or computes a conflicting same-instance value. The related work above remains the applicable list.

CI notes

Uncertified by CI at 0ef36537f64f6d11e7574268266ef7aeffc4df1e. In run 40725333120, plan was cancelled at the workflow's timeout-minutes: 10 limit. It ran 16:22:34–16:32:37 UTC. The available job log ends during checkout with The operation was canceled.; it does not show onboarding tests executing in this run. The package replay was skipped and did not run in CI. The local scientific and publication checks above passed, but this hosted run certified none of them.

Independently of that timeout, the unchanged main baseline has two onboarding documentation failures. Thus allowing planning to finish would still encounter these failures:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the exact standalone **0.003125804** text in docs/research/current-status.md; its bold span instead includes the rational value.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md; that document still describes PR5599.

Canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, the PR6030 admission pushed with [skip ci], changed the selected record without synchronizing those documents. Both exact failures and FAILED (failures=2, skipped=23) were confirmed in PR6150 run 40724809789 and another contributor's PR6120 run 40691776027. Open PR6050 targets the shared documentation synchronization.

A maintainer re-run is needed after main is fixed. No files were changed, no commit or push was made, and no CI rerun was requested for these unrelated blockers.

Merge Danger

Door: two-way. Blast radius: isolated research package. Six files are added; shared code, frozen evidence and the selected frontier are unchanged.