[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. A dense bypass around one flat Gaussian child must retain almost the full target matrix rank when rectangular adapter norms are bounded.

T = B K A + H + E
T: d-dimensional unitary; K: N-dimensional flat unitary
rank_C(H)=h; ||E||_(2->2)<=eta<1
M=||A||_1 ||B||_infinity

(d-h)(1-eta)^2 <= ||BKA||_F^2 <= d^2 M^2/N

The bypass may touch every entry and have unbounded coefficients. An orthonormal basis of its kernel proves the lower energy bound. An explicit Gaussian family attains both inequalities, proving the constant sharp.

For two independent width-f targets and one rank-two child, exact output with M<=1 requires h >= 2*2^f-4. If the dense bypass has rank at most 2^f, then M >= (1-eta)*2^((r-1)f/2)/sqrt(2). Here h is complex coordinate-matrix rank, distinct from recursive axis rank. No new kappa or selected-frontier change.

Evidence

• Before: ToikY #7166, exact model head 6efb5d028b3212fba68edfb725608413622d6ab6, supplies the rectangular Gaussian normalization and column-sparse bypass test. A dense rank-one bypass need not leave a missing entry. Its scientific verifier and invoice were not replayed.
• After: PROOF.md supplies the kernel/energy argument and sharp Gaussian construction. The exact Q(i) verifier checks four flat unitary children, 28 equality witnesses, 48 additional adapter fixtures, two full-child controls and 14 rejecting controls.
• CPython 3.12.15 warning-strict normal and optimized direct runs passed in 0.94 seconds each. Generated six-file hash-bound receipts agree byte for byte. Both registered commands passed locally, individually below their 120-second limits.
• Source compilation, metadata/provenance pins, exact committed bytes, executable modes and declared source ancestry passed. The official six-file export passed integrity checking; read-only Git-free normal/optimized exported replays passed with matching receipts.

Exact commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gaussian-low-rank-bypass/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gaussian-low-rank-bypass/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gaussian-low-rank-bypass
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gaussian-low-rank-bypass --check-source-history --check-modes

Other hosts may substitute their CPython 3.12 executable. Expected direct marker: PASS_GAUSSIAN_LOW_RANK_BYPASS.

Limitations and credit

The universal statement rests on the written proof. Finite error fixtures use E=eta*T; the theorem allows arbitrary operator-norm error. The constant is attained at rank deficit one, without claiming attainability for every prescribed h and M. Coordinate rank is not an implementation bill. The witness has dense, almost full-rank bypass, with no physical cost advantage claimed.

Physical allocation, dirty restoration, routing, native adaptation, precision/tape costs, a complete supplier invoice and multiplication theorem remain unproved. No selected cold replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed.

Credit ToikY #7166; Clear1nsight #4932/#4045/#3989; Rohan Arun #7113; and the retained Gaussian native, controller and compiler lineage. The general matrix inequalities are elementary classical facts. The new dense-bypass application, attaining family and verifier were produced by Nick Petalas. No predecessor code or external scientific data is imported.

Canonical base e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000. The coverage-law-change card changes to a local dense-bypass obstruction. Advance CLAIM. Only the six-file package is added.

Open next questions

• Bind an actual eligible bypass to a coordinate-rank cap. Start with its literal input/output spaces and charge every bypass, adapter and restoration operation before claiming a complete obstruction or gain.
• Determine the constrained optimum at a prescribed M and rank deficit greater than one. The sharp constant alone does not settle every such point.

Related open work

• #7166 treats bounded column support and exact endpoints; this permits dense bypasses and restricts matrix rank.
• #7249 proves a sharp column-sparse approximation floor in maximum-column and normalized-Frobenius error. This independently derived parallel result uses dense low-rank bypasses and operator error. It does not settle that PR's open spectral optimum under its different bypass constraint.
• #7245 studies fixed whole-output consolidation; this bounds one changed higher-rank child and constructs no paid multi-output supplier.
• #7246 bounds fixed finite monomial decoder stock; this permits arbitrary rectangular coefficients under norm bounds.
• #7171 bounds timed input support for serial MOVE fusion; this grants arbitrary local adapters and does not analyze chronology.
• #7165 bounds unchanged donor-chart borrowing; no donor census is computed here.
• #7243 bounds existing-paid cross-chart stock; this supplies no full-construction capacity or signed moment.
• #7003 constructs exact fusions by moving contacts; the local dense-bypass theorem does not exclude that changed chronology.
• #7144 bounds fixed-caller diagonal body batching; this does not test or change that scalar word.
• #7161 proposes a complete paid-word improvement; this computes no competing invoice or kappa.
• #7244 proposes a different complete paid word; this theorem is a standalone local contract.

The completion refresh checked this account's nine other open PRs created since the CLAIM, #7247 through #7255. Only #7249 studies this Gaussian-fusion interface; its line above records the different bypass constraint and error metric. No shared numerical optimum is computed by both results.

These lines compare published contracts only. No extra computation checked another contribution's numbers.

CI notes

CI did not certify this package. At unchanged commit 032216617c70a30e785d809795f16af2f34d60fe, run 43024693032, plan job 130536304269, failed in Onboarding helper tests. I read the completed checks once and inspected the job log directly. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

Direct inspection of canonical base e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json requires make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. This PR changes neither file nor the failing test. The mismatch is in the unchanged repository baseline.

The dependent replay job 130539316679 was skipped because plan failed. CI supplies no hosted scientific replay evidence for this package. The local normal/optimized registered checks, committed-byte/mode/source-history checks and read-only Git-free exported replays above remain the execution evidence.

I directly inspected all five comparison logs and verified the same sole failing test, missing-command assertion and 760-test/one-failure/24-skipped totals:

• Run 42905094314.
• Run 42903313790.
• Run 43023514635.
• Run 43024464841.
• Run 43024631324.

No fix, retrigger commit, push or workflow rerun was made. The repository-level repair remains with maintainers. Independent human and formal proof-kernel review are unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

One new research package. Archival intake, scientific acceptance and selected-frontier admission remain separate decisions.