[petalas]

Summary

A scoped comparison-barrier proof improves the retained #3851/#3872 setup fixture's all-principal-window inverse-error certificate:

- error <= (4/11) * residual infinity norm
+ error <= (83/234) * residual infinity norm

The exact decrease is 23/2574. The sufficient known-integer recovery threshold increases from <11/8 to <117/83. At residual 7/5, the new bound is 581/1170 < 1/2; the old bound is 28/55 > 1/2 and is inconclusive.

For comparison matrix M_ii=|H_ii|, M_ij=-|H_ij|, assume a positive w satisfies Mw>0. The proof shows that h=M^-1 r is the least full-source supersolution and that its restriction bounds error on every ordered principal window. For the fixture, h=(49/156,10/39,83/234). Optimality is confined to comparison supersolutions and the same-magnitude comparison class, rather than the actual signed matrix's inverse norm.

This takes Dugongue's #3872 Open next question 3. Its inverse cache and slide census are cited predecessors, not replay deliverables. The general comparison reasoning is classical; this contribution is the reusable least-barrier contract and new exact fixture/recovery improvement. Produced by Nick Petalas, crediting David Ganz / Dugongue #3872, Jose Manuel #3851/#3787, Chafik Boukhalfa, RaD and the retained Gaussian/FFT lineage.

Evidence

• Before: the retained fixture has minimum row gap 11/4, giving certificate 4/11.
• After: literal rational multiplication gives Mh=1 and bound 83/234.
• Both registered normal/optimized commands pass on CPython 3.12.15/macOS. Each took below 0.1 seconds locally, within its 120-second registered limit. The exact verifier checks 15 ordered windows, 16 off-diagonal sign variants, 1,248 residual-box cases, 78 literal known-integer recoveries, the comparison-class half-unit boundary and 12 rejected certificates. A separate permutation-sum determinant oracle corroborates the elimination-produced barrier.
• Committed-byte/source-history preflight passes for all nine package/input files at d53754b5c7211442aa9df762396c00fe1d84f587. Official export and integrity check pass. Normal and optimized direct replays also pass from a read-only Git-free export with PATH=/nonexistent; all ten exported files, including the export receipt, remain byte-identical.
• The selected source-binding orientation passes. No selected scientific cold replay or predecessor scientific suite was run.

From the repository root:

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

The direct marker is PASS_WINDOW_COMPARISON_BARRIER. The package runner reports PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. Sources, unchanged original card, scope change, proof and bounded commands are in the six-file package.

Limitations

The numerical improvement concerns the retained rational setup fixture, not a production Gaussian window. The dimension-uniform complex theorem is a written proof; finite evidence covers real rational sign fixtures. Rational preparation costs apply when the comparison matrix and envelope are rational. Preparing the barrier adds one separately paid dense solve and retained coefficients. Known-integer output remains an explicit recovery premise.

Production factor-page integration, residual formation, coefficient creation, scaling, physical storage/movement and cleanup remain unpriced obligations. No physical precision guard, supplier admission, selected frontier, new kappa or complete multiplication theorem changes. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Bind this barrier to one actual rounded production factor page, including its global matrix, format, incoming-grid bound, residual envelope and complete preparation/emission/cleanup bill.
• For a specified change to the global source, determine whether its comparison barrier can be updated without a fresh dense solve while preserving a positive dominance witness.

Related open work

• Prove a rigorous broad Gaussian phase-cell inverse bound #5103 studies a broad Gaussian phase-cell inverse and a two-norm bound. This package studies a different retained rational principal-window fixture and its infinity-norm residual certificate; no numerical comparison or verification of that PR is claimed.

The main-tree formula/object search, merged-PR searches, open PR bodies, recent Discussion claims and local reservations found no matching barrier contribution within the stated coverage. Sparse notes, arbitrary diffs and unpublished work are not exhaustively covered.

CI notes

CI blocked upstream for head d53754b5c7211442aa9df762396c00fe1d84f587. In run 39934170394, plan failed in Onboarding helper tests and replay was skipped. The shared command was python -B -m unittest discover -s tests -p 'test_research_*.py' -v; its log reports 408 tests, eight failures and 17 skips. The package's hosted replay did not execute.

The eight failure headers and their causes were inspected directly:

• Seven reported failures/subtests in tests/test_research_descendant_source_roles.py stop at working source drift: scripts/check_research_commit.py. The copied historical research/isolated-descendant-bytecode/SOURCES.json still declares that checker as an input with SHA256 ecf206a82bf1f4e748efb6cc639919582050d173b3537ec12f7265a4f7b55ead. The tested merge 2e3e601f6c946de70f406a7aecb9c9939f0d6b10 contains checker SHA256 28fed1e156fe8601633ed453f682a158cb15e94981e6c89ed96ae72ffc9d0c6f. Its checker bytes are identical to upstream base 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, so this mismatch comes from the upstream shared-file version, not this package.
• test_research_onboarding.OverlapTests.test_report_survives_a_cp1252_stdout fails in the copied temporary scripts/find_overlap.py at line 24, from research_package import git_source_env, with ModuleNotFoundError: No module named 'research_package'. The copied onboarding fixture lacks that import dependency.

The failed job logs for #5116 (run 39933826517), #5117 (run 39933925464) and #5118 were downloaded and compared. Each has the identical eight failure headers, seven checker-pin failures and one missing-import failure, followed by FAILED (failures=8, skipped=17). Their changed-file lists contain only their own research packages and no shared scripts, tests or workflows.

This PR likewise adds only its six package files. Shared scripts, tests and historical pins are unchanged here. The local validation above remains successful; hosted replay and CI certification remain unclaimed. No DONE has been posted while this upstream planning failure blocks replay.

Merge Danger

Door: two-way. Blast radius: isolated. Only research/window-comparison-barrier/ is added; no shared implementation, frozen input or selected-frontier file changes.