Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study a homogeneous compensated derivative arising from permanent inequalities for centered complex positive semidefinite matrices of order six. On the conjugate-disjoint locus K K̄ = 0, the ordinary omission variance has exactly forty trace-normalized zeros, all sparse rank-one cube-root configurations. On the entire flat, equal-spectrum rank-two stratum, including all compatible complex phases, we determine the compensated derivative explicitly and prove its sharp lower bound 18912 − 140√2. The proof retains the common Gram structure: a real balanced-frame complement gives a complete complex-structure chart, and threefold permutation symmetry eliminates the remaining phase from the paired permanent derivative. An explicit Frobenius tube of radius 10⁻⁶ has a positive uniform margin and contains full-dimensional families of centered rank-five matrices, whose positive rank-one anchors have rank six. A conditioning-free positive-part construction also reduces small conjugate overlap to the exact locus. Its nonflat, unequal-spectrum part and the unrestricted compensation inequality remain open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
For a complex correlation matrix B of order n, let C(B)ij = per B(i|j) be its raw permanental cofactor matrix. We bound the optimal constant γn in C(B) ≼ γn(per B)I. An elementary isolated-row induction and an explicit positive-definite equicorrelation family give, for every n ≥ 2, (n+1)²/(4n) < γn ≤ (n²+2n+3)/(4n). Consequently γn = n/4 + 1/2 + Θ(1/n): both the leading coefficient and the constant term are sharp. The upper estimate extends, with the necessary diagonal weights, to all Hermitian positive semidefinite matrices, including zero diagonals. We also obtain an explicit nonnegative reserve for the standard natural-representation block in order six, and give an exact order-six obstruction to the raw cofactor bound with constant two. The proof is analytic and does not require finite polynomial certificates. The exact finite-order constants and the coefficient of 1/n remain open.
Edition 10
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
For a complex positive semidefinite Gram matrix B=VV*, we couple its permanent-cofactor contractions with those of |B|² and prove the sharp all-order operator bound 0≼U−aT/n!≼(n−1)(c−a²/n!)I, where a=per B and c=per|B|². For the actual bordered permanent defect, we identify a nonnegative cofactor-weighted variance E and characterize its zero set. Ordinary occupations of the same Gram factors give the sharp all-order bound E≥(αₙ/n!)(na−t), with an occupation-resolved refinement. Joint trace and frame-variance bounds retain singular factors, arbitrary complex phases and ambient kernel mixtures. At order six the variance, inherited base defect and forward curvature yield explicit sufficient compensation domains, with the sharp curvature floor R_(√2−1)≥(19/240)α₆². A common-factor argument proves compensation when one selected occupation is at most 1/31, giving the corresponding self-conjugate order-seven domain.
For the actual six-factor core F(t)=per(K+tJ), set P=F(1)+F′(1) and L=F(√2)−P. We prove the sharp constraint P+3L≥7200 and classify equality by a fourth-root orbit plus two zero residual factors. It yields actual compensation for 0≤L≤4 without a diagonal cutoff. A separate 1/32 entrywise tube around the equality family has a strict optimized-capacity margin. The intrinsic criterion L≥0, P+3L−7200≤2⁻¹⁶⁰ derives bounded diagonals and two small residual rows, rather than assuming them, and supplies another explicit capacity margin. The tube and intrinsic regions also satisfy the strict self-bound per|K+√2J|²<(3/4)F(√2)².
Weighted coherent projections, occupation geometry and classical permanent bounds provide further phase-dependent certificates, including full-amplitude two-mode and unequal two-third-root-orbit families with explicit higher-rank neighborhoods. Exact obstructions show why several scalar, fixed-dual and coarse-envelope extensions fail. The attributed small-order induction is reconstructed self-containedly, and the original residual occupation and thinning identities are retained. The generic rank-six compensation inequality, the generic three-quarter self-bound and unrestricted Chollet remain open. A uniform six-row creation-operator reserve further gives strict diagonal monotonicity and convexity of the transformed deficit. Two upper coefficients of a two-coordinate residual pencil are controlled, while an exact complex positive-semidefinite example rules out unrestricted directional monotonicity. The first and middle two-coordinate payments remain open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
Strong permanent contraction is proved for every complex positive semidefinite matrix supported in K₄,q, for any finite q, and in K₅,₅. The four-row proof is analytic; the five-by-five proof combines a retained third-minor budget, finite-column moment bounds and an exact tensor Bernstein certificate with 3,145,728 rational coefficients. Independently, a dimension-free energy criterion proves the ordinary self-Chollet inequality under squared row and column bounds of one and total squared energy at most 12 + (201 + 16√537)/988, including all positive semidefinite K₁₂,q supports. Strong five-row estimates with arbitrary finite column count are also obtained near row-isometries and under dispersed fourth moments. Structural proofs, large-entry transfer and analytic reductions are included, with classical inputs credited. Complex phases, missing edges and singular endpoints are covered; full strong contraction for arbitrary K₅,q is not claimed.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove that every complex positive semidefinite matrix A supported in K₄,₈ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. The proof extends an established four-row framework by retaining a necessary lower bound on squared row moments and choosing a convex permanent bound as a polynomial of the row norms. An exact two-point obstruction shows why one fixed mixture in the inherited scalar framework is insufficient. A small-row reduction and a rational tensor Bernstein certificate cover the capped-entry region; a published all-column large-entry theorem covers its complement. The result is computer-assisted, includes arbitrary phases and singular endpoints, and does not establish all four-row supports or unrestricted Chollet.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove that every complex positive semidefinite matrix A supported in K₄,₇ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. The normalized off-diagonal block is a four-by-seven operator contraction. The new argument retains third-order minor contributions and controls their coupled phase cancellations through the row Gram matrix. A split according to the smallest squared row norm separates an analytic corner reduction from a balanced-row polynomial inequality. The latter is verified by an exact rational tensor Bernstein certificate on 256 boxes. An established all-column large-entry theorem covers the complementary region. The computer-assisted result includes arbitrary phases, missing edges and singular endpoints. It does not prove the all-column four-row case, the K₅,₅ case, or unrestricted Chollet.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove that every complex positive semidefinite matrix A supported in K₄,₆ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. The normalized off-diagonal block is a four-by-six operator contraction. For squared entries at most 7/10, the proof combines a four-vertex overlap estimate with a fixed mixture of rectangular permanent bounds. Separate affinity, deletion of inactive rows and piecewise separate concavity reduce the remaining inequality to fifteen rational corner values. A convex-border argument covers larger entries for every number of columns, using Lieb’s block inequality and the published three-row theorem. Arbitrary phases, missing edges and singular endpoints are included. The four-by-four and four-by-five conclusions follow by padding. No all-column four-row or unrestricted Chollet theorem is asserted.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
For every complex 3 × q contraction X, we prove that the permanent of [I₃ X; X* Iq] is at least the permanent of its entrywise squared modulus. The proof retains all square-minor orders and controls the coupled third-order cancellations through the row Gram matrix. It gives an explicit sum of nonnegative lower bounds and includes arbitrary phases, zero entries and singular endpoints. Combining this theorem with three pentagon blow-up cases and an exact finite graph classification proves the same strong inequality for every positive semidefinite matrix with triangle-free support on at most seven vertices, after diagonal scaling. Two previously unpublished blow-up cases are proved here, one analytically and one by an explicit positive integer polynomial certificate. Reproducibility supplements contain the certificate and finite classification verifier. The unrestricted Chollet conjecture is not settled.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove per(|A|²) ≤ per(A) for complex positive semidefinite correlation matrices supported on a seven-cycle with chords 03,15,26. More precisely, the difference dominates one fiftieth of the matching-partition loss. The proof combines elementary cycle estimates with an explicit nonnegative integer polynomial identity, supplied with a standard-library verification program. The two-chord subgraph with chords 03,15 inherits the same bound. For the crossing placement 03,14 we give a separate analytic proof with constant 1/12. All results include missing edges and singular matrices. The normalized arguments require only vertex-star magnitude bounds, and we describe equality and diagonal scaling. These structural results do not establish the unrestricted Chollet conjecture.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study two auxiliary cofactor comparisons motivated by permanent Hadamard inequalities. For a complex positive definite correlation matrix B, let R be its transposed permanental cofactor matrix and L that of its entrywise squared modulus. We compare zᵀLz and zᵀ|R|◦²z, where zᵢ = |xᵢ|², with (per B)(x*B⁻¹x)(x*Rx). Both comparisons have sharp universal constant one in orders one and two. At every fixed order at least three, neither admits any finite universal constant, even among dense irreducible positive definite correlation matrices with nonreal triangle phases. An explicit roots-of-unity family gives quantitative divergence. The cancellation mechanism is classical; the result concerns these auxiliary comparisons and does not disprove Chollet’s inequality.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
For a complex Hermitian positive semidefinite matrix A, write Q(A) = per(A ◦ Ā) and d(A) = ∏ᵢ aᵢᵢ. We prove Q(A) ≤ d(A) per(A) whenever the off-diagonal support of A is contained in a cycle of length at least four, or in a theta graph with endpoint-path lengths 1, b, c, where b, c ≥ 3 and b + c ≥ 7. The entries may have arbitrary phases and unequal magnitudes, and singular matrices are included. For a single cycle, the loss between two weighted matching partition functions pays the worst possible directed-cycle contribution. For a single-chord cycle, a simultaneous matching-loss allocation pays all three overlapping cycles. We give the complex form of the known one-point strong-closure argument and obtain hereditary inequalities for graphs assembled from these blocks. These are structural results; the unrestricted permanent inequalities are not proved.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove quantitative self-conjugate Chollet inequalities for all complex Hermitian positive semidefinite matrices of rank at most four in orders eight and nine, with respective factors 243/320 and 1215/1352. A short extension of the occupation-border comparison of Li, Zhao and Luo balances its two multiplicative losses. The input is the previously established singular order-five factor 3/5; no new certificate is needed. Together with Rodtes’ order-four theorem and Ma’s theorem for orders at least ten, the new estimates imply Chollet’s inequality in every order when both factors have rank at most four. The coefficient-one inequality is strict for positive diagonal in orders at least five. No claim of optimal constants or an unrestricted-rank conclusion is made.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
For every complex Hermitian positive semidefinite matrix A of order seven and rank at most four, we prove per(A ∘ conjugate(A)) ≤ (192/245)(per A)². The inequality is strict when all diagonal entries are positive. The main input is the bound per(A ∘ conjugate(A)) ≤ (3/5)(per A)² for every singular positive semidefinite matrix of order five. An exact rational principal-minor sum-of-squares certificate proves this singular-boundary inequality. A rank-preserving border comparison then gives the same constant in order six and the stated constant in order seven. The certificate and a standalone exact-integer verifier accompany the paper. Cauchy–Schwarz gives the corresponding two-factor inequalities when both factors have rank at most four.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
For each fixed matrix size, we prove that the scalar tuple uniquely minimizes the mixed discriminant of positive semidefinite Hermitian matrices whose sum is the identity and whose individual traces are sufficiently close to one. This answers Question 1.7 of Gurvits-Leake, arXiv:2609.21083v2. The argument combines Gurvits's doubly stochastic uniqueness theorem with a positive constrained Hessian and compactness. We also record the exact region of positive definiteness of that Hessian at a scalar tuple. The global neighborhood in the theorem is qualitative and may depend on the size.