[petalas]

Summary

Scoped written proof with exact finite corroboration. In the unchanged PR628 balanced dirty word, a protected source and unread target improve the general joint-error interval coefficient bound for h=ceil(log2 n)>=1:

- (h+2)^2
+ h^2+2h+3

The improvement is exactly 2h+1. Height four improves from 36 to 27. Every ordered pair of complete-leaf cuts, both orientations and shared dirty scratch are covered. Contractive factors need no commutation. Commuting factors with a common subset norm budget G give the new bound multiplied by G. The one-factor bound is two. This is not a sharpness theorem, a new kappa or a complete multiplication theorem.

The new step is the source/scratch/target block inequality in research/protected-boundary-interval-bound/PROOF.md. Hanson Sun's PR1085 prefix-entry invariant and Charles Bethin's PR1278 squarefree intervals are cited premises, not audit deliverables. This takes Open next question 2 of Arham Tahir's PR1374. Its lower family remains compatible with the new upper bound. Philippe Van Velzen's PR2177 settles the separate destination-only problem.

Produced by Nick Petalas. Credit Chafik Boukhalfa PR628/905, Hanson Sun PR1085, Charles Bethin PR1278/1416, Arham Tahir PR1374, Philippe Van Velzen PR2177, Philipp/mightyOil, Gareth Davidson, 0xkilgore and the retained precision/restoration lineage. Exact main-tree proof/context blobs are pinned in SOURCES.json. No predecessor executable is imported or run.

Advance claim.

Evidence

Before, the cited general upper bound was the squared prefix coefficient. After, the written block proof gives h^2+2h+3 uniformly. The standalone verifier passes eight generic formal block products, 10,000 exact strict-gain identities, 23,736 nonempty free noncommutative intervals at n=1,2,3,5,8,11 in both orientations, complete arbitrary-dirty endpoints and fourteen controls. Generic block equality is not claimed to be attainable by two prefixes of one word.

Local CPython 3.12.15 normal and optimized registered commands passed in 0.200 and 0.199 seconds. Their generated six-file SHA256 receipts agree. The registered package runner, committed-byte/source-history/mode checks, official export integrity and both warning-strict read-only Git-free empty-PATH replays passed. All seven exported files including the export manifest retained their bytes and modes.

Exact reproduction from the repository root, with Python 3.12 and its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev research/protected-boundary-interval-bound/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev -O research/protected-boundary-interval-bound/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/protected-boundary-interval-bound
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/protected-boundary-interval-bound --check-source-history --check-modes

Any CPython 3.12 installation may replace the local interpreter path. Expected markers are PASS_PROTECTED_BOUNDARY_INTERVAL_BOUND, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Current selected-metadata orientation also passed. Canonical main was refreshed at de5ec2e, priority revision 8, before submission.

Limitations and unrun obligations

The quantified result rests on the written block proof and cited lemmas. Finite checks do not establish those predecessor premises. The verifier uses fixed finite capacities and no external dependencies. Independent human and formal proof-kernel review are unclaimed.

No rounded leaf implementation, negative-leaf preimage proof, internal magnitude/precision budget, native/tape realization, stock/ownership admission or complete supplier invoice is supplied. Lean, Docker and the selected scientific replay were not run. Every registered package command ran locally. The selected frontier and its five framework assumptions remain unchanged.

Open next questions

• Close the remaining gap to PR1374's floor(h^2/2)+h+2 lower family. Begin with the chronological relation between the two prefixes in the six phase-pair formulas; the block count does not control all cancellation sharply.
• Bind the new bound to one exact rounded caller while retaining destination support, negative-leaf preimages, preparation, internal magnitudes and the complete physical bill. No such caller is supplied here.

Related open work

This account's open PRs created since the CLAIM at 2026-10-27T10:04:50Z were refreshed before DONE. The other nine PRs have different objects and quantities:

• Prove one-column completion over composite cyclotomic alphabets #5865: missing-column completion over cyclotomic alphabets.
• Prove exact phase defects for charts fixing any binary subspace #5866: phase defects of binary orthogonal charts fixing a subspace.
• Count every finite-field connector chart repair rank (no new κ) #5867: finite-field connector chart repair-rank distributions.
• Certify the sharp 13-vertex Clebsch query threshold (no new κ) #5868: induced-subset query savings for the Clebsch matrix.
• Prove singular reachability and a Veronese obstruction (no κ) #5869: singular right-vector reachability over commutative domains.
• Prove sharp integer inertia for every sunflower downset (no new κ) #5870: inertia of sunflower-downset fitting matrices.
• Prove exact matrix-tensor determinant images and SL orbit counts #5871: determinant images and special-linear orbits of matrix multiplication tensors.
• Classify binary C5 optima at total payload seven (no new κ) #5873: local-similarity classes of binary C5 fitting matrices.
• Prove the sharp rank cap for three-call A5 conjugate pairs #5874: paid-rank thresholds for three-call A5 conjugate pairs.

None studies the same PR628 joint interval coefficient envelope, and no shared computed quantity was identified. No extra mathematical computations were run to check another contribution. PR1374, PR1416 and PR2177 remain the merged predecessors. Sparse local-note and code-diff coverage is incomplete; absence of a match is not a novelty certificate.

CI notes

At contribution commit 50a0edd82520770b6eefc76e32163746fc6330cb, Registered research packages run 40487121457 failed in plan, specifically the Onboarding helper tests step. All ten failure entries identify tests/test_research_current_assignment.py. The log ends Ran 511 tests and FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failed assertions are in that same test file.

This is the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. The same 8-failure/2-error step is also documented in #5867's run 40486502162. Maintainers will fix it. No tests or files outside this package were changed.

The replay job was skipped. Hosted replay has not certified this package; only local scientific checks have run. Local warning-strict normal/optimized direct verification, both registered commands, package metadata, committed-byte/source-history/mode checks and both read-only Git-free exported replays passed as recorded above. The package and mathematical result are unchanged after publication.

Merge Danger

Door: two-way. Blast radius: isolated. Only six files in the new research package are added.