[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration: the dimension-two case of David Ganz's atlas B099 has the sharp bound E <= 3^(-1/4) sqrt(delta). The package determines the exact minimum and every minimizing commuting triple, preserves the contraction constraint, and proves that neither the constant nor the one-half exponent can improve near zero commutator. This contribution was produced by Nick Petalas.

A_i = a_i I + v_i dot sigma
S = sum_i v_i v_i^T, eigenvalues lambda_1 >= lambda_2 >= lambda_3
E^2 = lambda_2 + lambda_3
B_i = a_i I + (v_i dot n)(n dot sigma), n any top unit eigenvector

Only research/qubit-commuting-repair/ is added. This changes the formal-interface-lemma card to an explicit fixed-dimensional side question, as recorded in TASK.json. No new kappa or selected-frontier change is claimed. Credit David Ganz / #2849 for the repair question and normalization, and the classical Pauli representation and best rank-one approximation methods. Related qubit rounding and normalized-HS literature is linked in the README. Global literature priority is unclaimed.

Evidence

Before, B099 leaves the all-dimension square-root repair question unproved. After, the written proof settles the complete two-by-two case with the sharp constant, equality classification and explicit contraction-preserving optimum. The all-dimension problem remains open.

Both warning-strict normal and optimized local commands pass PASS_QUBIT_COMMUTING_REPAIR. Each was timed locally at about 1.1 seconds, below its 240-second registered limit. Exact checks cover 729 literal rational matrix pairs, 19,683 covariance triples, 1,000 spectral repair fixtures, 150 noncommuting equality fixtures, 32 scaled Pauli attainers and eleven rejection controls. The written proof supplies the universal real statement; these finite checks corroborate its identities.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -Werror -B research/qubit-commuting-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -Werror -B research/qubit-commuting-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/qubit-commuting-repair
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/qubit-commuting-repair --check-source-history

Elsewhere substitute a CPython 3.12 executable for the environment-specific path. Only the standard library is required. The registered replay passes PASS_REGISTERED_RESEARCH_PACKAGE with outcome=completed; metadata, exact source pins and committed-byte/source-history preflight pass. Official committed export and integrity checks pass. Both direct commands also pass from a read-only Git-free export, with all seven exported files unchanged. Receipts are generated to stdout and bound to package hashes; no receipt is hand-written.

AST syntax/function-size inspection passes, with the longest function at 28 lines. No external static analyzer was installed. The bounded managed-runtime allocation deviation is documented in the package. The selected baseline orientation command passes its metadata-only check; it is not a scientific replay.

Limitations and unrun obligations

This is a two-by-two Hermitian contraction theorem. It does not resolve B099 in arbitrary dimension, give a native supplier or price a physical caller. Compiler, dirty restoration, routing, precision/tape, stock and complete supplier costs remain outside scope. The selected cold replay, Lean, Docker, predecessor scientific suites and repository-wide historical replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Extend the constructive bound to a known common invariant decomposition into blocks of size at most two. Start by combining qubit repairs with exact normalized-HS block weights; distinguish a feasible upper bound from a global optimum that might mix blocks.
2. Find a square-root bound for general three-by-three Hermitian contractions. Start with one spectral clustering rule and the complete summed repair error; Pauli collinearity is specific to dimension two.

Related open work

No open PR studying the same qubit-triple optimal repair distance or sharp normalized-HS constant appeared in the refreshed live lists and completed open-PR/Discussion cache. The atlas predecessor #2849 and its separate qutrit MUB stability successor #5227 are merged. The latter concerns unbiased bases and a different residual and repair orbit; this is not its follow-up. Other concurrent atlas questions concern different algebraic or combinatorial objects. Sparse local notes, unpublished work and a complete historical code/literature audit remain coverage limits. No extra computations were run to validate other contributions.

The completion refresh read the titles and bodies of all seven other open account PRs created since the CLAIM: #5246 studies F4 Hermitian hyperplane intersections, #5247 downset inertia, #5248 symplectic partial-spread completion, #5249 P112 diagonal-completion rank, #5250 matrix-pencil determinant images, #5251 a projective design and #5252 finite etale tensor determinant images. None studies the qubit-triple repair distance or supplies a competing value of its sharp constant. No additional computations were run to validate those results or compare unrelated quantities.

CI notes

Both hosted checks passed at exact head 895e3db4982ad4bba6af3fbfbc6caf7b5f97ae1f in Registered research packages run 39965503591: planning in 5m7s and the focused registered replay in 1m19s. Run metadata confirms completed status, success and the matching head SHA. The registered replay executes both warning-strict normal and optimized VERIFY.json commands.

All local checks and publication/export checks listed above also passed. Hosted replay certifies execution of the bounded checks and rejection controls; it does not supply independent human or formal proof-kernel acceptance of the written theorem. The unrun obligations listed above remain outside this contribution.

Merge Danger

Two-way door. Blast radius: research. The change is confined to one new opt-in package and its declared atlas input.