[petalas]

Summary

Scoped written algebraic proof with exact finite corroboration, produced by Nick Petalas. For any field, nominal matrix B and independently uncertain coordinate set E, the exact common identity-pairing capacity is

max { r : X (B + sum_E t_ij E_ij) Y = I_r for every t }
  = max { rank B[A,C] : (A times C) intersect E is empty }.

X and Y may be dense but must be fixed across alternatives. A nonsingular clean minor supplies attaining maps. The nominal map plus each single-entry alternative already forces the full bound.

For B=I, the exact capacity is the independence number of the error-support graph, with loops forbidden. A directed n-cycle has capacity floor(n/2), even though every nominal/single-entry map is invertible. The five-cycle therefore has exactly two common channels despite each map individually having rank five. Correlated-error and nonidentity-nominal witnesses delimit the theorem.

Only six files under research/robust-coordinate-pairing/ are added. This is an original common-contract obstruction for an optimistic algebraic relaxation. It does not assert that a selected supplier has this uncertainty law. No new kappa, supplier admission or selected-frontier change is claimed.

The proof uses classical rank factorization, nonsingular minors and outer products. aliromman #5615 supplies surrounding dyadic helper-pairing context. Nick Petalas #7269 concerns prescribed denominators for one fixed pairing; #6585 concerns common-chart isolation with both cross blocks zero. Those packages are background, not premises or replay deliverables.

Evidence

Before this result, pointwise invertibility alone does not provide a common-pairing capacity test. The new theorem gives the exact clean-submatrix criterion and constructs its attainments.

• Exhaustive comparison on 10,266 nominal/support cases from 231 matrices: 943,861 left/right sandwiches, 7,869 nominal identity sandwiches, and constructed attainments for all 8,400 positive-capacity cases. Domains are all 1-by-1 and 2-by-2 matrices over F2/F3, all 2-by-3 and 3-by-2 matrices over F2, and I3 over F2, with every uncertainty support in each case.
• All 530 directed graphs with loops on one through three vertices; fifteen cycle instances of lengths three through five over Q/F2/F3/F5/F7; 112 rational nominal/support cases; eight boundary equalities and four rejecting omitted-error controls.
• Warning-strict normal and optimized direct checks pass with equal generated source-hash receipts. Local direct times were 0.52 and 0.54 seconds, below separate registered 120-second limits. Registered metadata/pins and both commands pass in 1.30 seconds total. These timings establish timeout headroom only.
• In-memory compilation, all 22 function sizes, whitespace, exact committed bytes/modes/source history, official export/input integrity and read-only Git-free normal/optimized exported replays pass. All functions are at most 25 physical lines. No external linter/type checker was installed or claimed.
• CPython 3.12.15 on macOS 27.0.1 arm64, standard library only. The selected source-binding orientation passed at canonical main e545235. The selected scientific construction and predecessor scientific suites were not replayed.

From the worktree root:

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

Another absolute CPython 3.12 path may replace the local interpreter. Expected markers are PASS_ROBUST_COORDINATE_PAIRING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier writes no files, uses no network or Git, and generates rather than hand-writes its receipt.

Advance CLAIM. The delivered scope matches that claim. Current live claims, open PRs and distinctive main/merged formulas were checked before claiming and committing; text searches do not establish novelty or cover unpublished work.

Limitations and unrun obligations

The universal all-field statement rests on the written proof. Finite enumeration is not formal proof-kernel acceptance. Independent human and formal proof-kernel review are unclaimed.

Only the retained identity block is constrained. Spectator cross blocks, arbitrary dirty restoration, chronology, extra workspace, adaptive charts, correlated errors, approximation and rings with zero divisors require additional arguments. Field-valued attainments need not be dyadic. No actual cut, producer, consumer, scalar schedule, native realization, precision/storage policy, physical hosting or full paid/target invoice is supplied. Unknown costs are not zero. All five selected assumptions remain. Lean, Docker and the selected scientific cold replay are unrun.

Open next questions

• Bind one actual helper cut to independently varying coordinate alternatives and permitted common charts. Its missing producer/consumer contract currently prevents a physical application or new supplier bill.
• Determine capacity for two specified correlated error generators. The skew-symmetric witness shows why the union of entry supports cannot replace their correlation law. Physical delivery and full pricing remain separate.

Related open work

• Prove exact helper capacity under odd-denominator budgets (no κ) #7269 proves exact helper-channel capacity under prescribed odd-denominator budgets for a fixed dyadic pairing. This result changes the contract to common exact pairings across independent coordinate alternatives over a field; it computes no Smith-budget capacity or shared numerical value from Prove exact helper capacity under odd-denominator budgets (no κ) #7269.

• Exclude common-hyperplane public helper recharts #7459 excludes common-hyperplane recharts in an actual P104 word. This theorem concerns an abstract independently uncertain pairing and neither computes that word's rechart saving nor audits its exclusion.

• Reduce mixed line/hyperplane recharts to two-bit closure (no κ) #7465 reduces mixed common line/hyperplane helper recharts to maximum closure for dimension-linear prices. This result instead computes common identity-pairing capacity across independent coordinate alternatives; it computes none of that package's rechart optima.

The closeout account-open-PR check inspected the six other PRs created since the CLAIM: #7461, #7462, #7463, #7464, #7465 and #7467. They concern quadratic embeddings, a binary map, a spectral projector, compiler width divisors, mixed recharts and a multiplicity operator. The adjacent #7465 is distinguished above; none supplies a competing value for this package's common-pairing capacity. No additional computations were run to audit those results.

CI notes

CI did not certify this package. At unchanged head 0a49cfa436fc2ad9cd8fffa84ebb35fb376b06a1, run 43280276547 failed in plan, Onboarding helper tests. I read the completed checks once and inspected the actual job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError says that `make frontier7000-verify` was not found in the CI verification document. The package replay job was skipped because plan failed. No hosted package execution or certification follows from this run.

This mismatch is already present at canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md describes #6030 and make frontier6030-verify. I inspected those base files and confirmed that both files and the failing test are byte-identical at this PR head. The package does not cause the failure.

I also inspected the actual plan-job logs in all five comparison runs below. Each has the same sole failing test, missing-command assertion, 760 tests, one failure and 24 skips:

• Run 42905094314, job 130140894651.
• Run 42903313790, job 130134720799.
• Run 43279348945, job 131396171336.
• Run 43279460328, job 131396549910.
• Run 43279626746, job 131397128979.

The local checks listed above remain the package's execution evidence. No source fix, retrigger commit, push or workflow rerun was made for this repository-level failure.

Merge Danger

Door: two-way.

Blast Radius: package.

Only an opt-in proof/verifier package is added. Shared code, frozen predecessor evidence and the selected frontier retain their bytes.