| # Theorem ledger - v1.0.0 |
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| Every matrix pair below consists of same-order complex Hermitian positive semidefinite matrices. Rank bounds apply separately to both factors, not to their Hadamard product. |
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| | ID | Statement | Hypotheses | Dependency and status | |
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| | Q7 | per(A circle B) <= (24/25) per(A) per(B) | Order 7; each rank <=3 | Full written analytic proof; 49 finite exact checks include endpoint and tail checks | |
| | Q8 | Same inequality with coefficient 3/5 | Order 8; each rank <=3 | Same analytic estimates; 21 closed intervals cover entire spectral domain | |
| | Q9 | Same inequality with coefficient 2/5 | Order 9; each rank <=3 | Same analytic estimates; 12 closed intervals cover entire spectral domain | |
| | QT | Same inequality with coefficient 4/5 | Every order >=10; each rank <=3 | Analytic bound T_n; rational base case and decreasing-tail proof | |
| | QU | Same inequality with coefficient 24/25 | Every order >=7; each rank <=3 | Consequence of Q7, Q8, Q9, QT | |
| | QC | Ordinary Chollet inequality, coefficient 1 | Every order; each rank <=3 | Conditional on accepting cited, unaudited through-order-six preprint in addition to supplied high-order arguments | |
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| The bounds include zero-diagonal cases, where the relevant permanents vanish. Correlation matrices have positive permanent under the supplied Gaussian-sphere estimate. Diagonal normalization preserves the self-conjugate ratio. The permutation-monomial Cauchy-Schwarz argument transfers that estimate to matrix pairs. |
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| ## Trust boundaries |
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| The finite verifiers check rational inequalities and complete interval coverage. They are not a proof assistant and do not establish all analytic lemmas automatically. The additional AI review found no fatal error; it is not external human peer review. |
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| No unrestricted arbitrary-rank theorem, optimal coefficient, exhaustive novelty audit, new claim about neural learning, or proof of the small-order external paper is included. The Qingxiao-inspired label records the origin of the research direction; the proof itself uses combinatorics, probability, spectral geometry and scalar convexity. |
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