[petalas]

Summary

Scoped all-size proof for every completed atomic downset of contiguous block elimination, under ||T||₂<=Λ, ||T_k^-1||₂<=Q for every original leading section, and Λ,Q>=1.

- common envelope: 32 N^(3/2) (ΛQ)^8   [Brian, #4788]
+ common envelope:  2 sqrt(N) (ΛQ)^2

The new argument partitions the execution into maximal finished and untouched intervals. Original leading solves bound each forward block row and each inverse block column directly. It also proves ||P||₂<=sqrt(N)(Λ+Λ²Q) and ||P^-1||₂<=sqrt(N(1+(ΛQ)²)), with a finer frontier-count form. Arbitrary complex matrices, unbalanced trees and legal child interleavings are allowed. README contains the complete written proof.

Produced by Nick Petalas. Credit Brian/JoyboyBrian #4788 for the global LDU restrictions and frontier viewpoint; Jose Manuel/jmbarrancoml #4205 for the original full-recursion proof; Vlado Vojdanovski #1673 for the compiler. Their results are cited predecessors, not audit deliverables. No predecessor verifier is imported or replayed.

Evidence

The self-contained exact checker passes 46 trees and all 2,380 downsets of ten real/nonreal rational fixtures through five logical coordinates. It checks 9,520 matrix identities, 9,372 partition blocks and 23,504 downset spectral certificates using exact rational PSD elimination. Eleven control groups include wrong factor order, omitted finished endpoints, wrong inverse signs and the insufficient whole-inverse-only premise. A generated stdout receipt binds these counts to package-file hashes.

Both warning-strict normal and optimized direct checks passed, timed at 17.034s and 17.690s respectively. These are check durations, not comparative performance claims. Registered package replay, metadata/provenance pins, committed-byte/source-history preflight and official committed export integrity pass locally. Both final committed registered checks pass with hashes matching all five publication files. Official export integrity and its Git-free optimized replay pass with all six exported files unchanged, on Python 3.12.15 / macOS-27.0.1-arm64-arm-64bit. Export SHA-256: f6432f3939d92408679fe803532ac8db40df844bbb92fbb9287f381d6295e1d6.

Reproduce from the repository root with CPython 3.12 and the standard library:

~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/downset-block-solve-bound/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/downset-block-solve-bound/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/downset-block-solve-bound
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/downset-block-solve-bound --check-source-history

Baseline: canonical main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. The selected-baseline source-binding control passed; no selected scientific cold replay was run. The overlap helper covered 367 open PRs and 3,821 Discussion records; its sparse local inventory was incomplete. Separate main README and merged searches complement it. Live relevant PRs and recent claims were refreshed before committing. This is bounded coverage, not a novelty certificate.

Limitations

This is a scoped mathematical inequality with finite corroboration, not formal-kernel acceptance or a complete multiplication theorem. No rounded Full-factory substitution, unfinished FFT/generator-fragment bound, native cost, supplier admission, asymptotic sharpness, new κ or selected-frontier change is claimed. Lean, Docker and full historical replays were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Substitute the envelope into a complete rounded Full allocation after proving each downstream coefficient, fragment and data-grid inequality.
• Determine whether the square-root dependence on frontier-block count is necessary when original leading conditioning stays bounded.

Related open work

• Sharpen leading-section bounds for all completed downsets #4788 studies exactly the same completed-downset norm. Its global-factor/frontier result is the credited predecessor; this adds the sharper direct block-solve estimate.
• Prove sharp dirty Toeplitz shear gate-cut norm bounds (no new κ) #4829 bounds cuts inside a dirty Toeplitz FFT shear. Its fragment boundary complements this completed-call theorem; no constant from that different boundary is recomputed here.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T21:47:11Z were refreshed. The nine other submissions in that snapshot concern different objects: exterior-face tariffs, a fresh-row quotient, shortest S3 words, weighted-reversal rescaling, spectator workspace, UT3 couplings/tariffs, a CRT batch bill and center-output compression. None studies this completed-elimination-downset norm or reports a competing value for it. No additional numerical comparison was run. Unpublished work and arbitrary historical diffs remain outside coverage.

CI notes

Both hosted checks passed on publication head 8acf04b4c42fe75894a87a3b2d37ddb68e96a798 in run 39861025848: plan in 3m25s and the registered research/downset-block-solve-bound replay in 3m17s. Run metadata confirms the exact head, completed status and success. The package replay runs both registered warning-strict normal and optimized commands. These automated checks certify the bounded replay; independent human and formal proof-kernel review and the unrun rounded/native/compiler obligations remain unclaimed.

Merge Danger

Door: two-way. Blast Radius: package.

Only five new files in research/downset-block-solve-bound/; no shared implementation, frozen source, selected record or predecessor credit is changed.