Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- 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 subfield
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study the quartics hλ = 4e₁e₃ + λa²b², where the elementary symmetric polynomials are taken in four variables. We prove that their real-stability interval is exactly [0, 64/27]. Throughout its positive part, the parameter is a genuine complex projective modulus, and the full linear automorphism group of the positive hyperbolicity cone consists of positive scalings and the independent interchanges of the two distinguished variable pairs. These cones are consequently pairwise linearly nonisomorphic and nonhomogeneous, including at the endpoint. The projective surfaces have eight singularities of type A₁ and one of type A₃ in the interior; one additional A₁ point appears at the endpoint. Correspondingly, the dual cone has exactly four Lorentz facets in the interior and exactly five at the endpoint. The arguments give an exact classification for this coefficient deformation of the specialized Vámos quartic, using classical hyperbolicity, singularity, and tangent-cone methods.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- 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 3
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
Let P be a probability law on nonnegative integer count vectors of total size n, with a homogeneous real-stable probability generating polynomial and positive mean vector t. We compare P with M = Mult(n, t/n) using the weighted covariance deficit W = ½(m − 1 − ∑ᵢ Var(Aᵢ)/tᵢ). We prove TV(P, M) ≤ min{1, (147/68)W}, independently of degree, category count and the mean vector. The argument transfers the centered elementary-symmetric-polynomial inequality of Han and Niles-Weed through Banach’s theorem and an orthogonal multinomial density expansion. It retains explicit low orders, handles exceptional discrete root profiles, and combines exact polynomial certificates with positivity of the likelihood ratio. A classical hyperbolic-slice reduction also gives the stronger bound TV(P, M) ≤ min{1, 2W} for every degree from one through fifty. Standalone rational-arithmetic code verifies all finite and continuous-domain certificates. Every all-degree linear coefficient must be at least two; whether two suffices in all degrees remains open. The results include nonfactorizable stable generating polynomials and make no algorithmic or statistical-estimation claim.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- 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 subfield
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
A fixed connected polar section can hide every odd logarithmic harmonic of inverse twisting speed from the leading Weyl profile. We show that its original pure cosine perturbation is nevertheless audible in the next heat-trace term. The second coefficient contains the relative heat trace of a connected step strip; replacing the exposed radial face by an independent angular channel gives the wrong model. We derive a uniform momentum-integrated radial expansion and use an exact angular clock, a small real shear and weighted Dirichlet–Neumann bracketing to obtain an actual two-term heat expansion for the Euclidean tube. A positive corner-loop remainder is nonincreasing in heat time. Its Stieltjes representation controls the complex Mellin transform sharply enough to prove noncancellation at the original fundamental frequency. Thus a parameter invisible in the leading Weyl law is determined by the lower-order heat profile, without adding any lower-homogeneity perturbation. Higher odd frequencies and a two-term unsmoothed counting law are left open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- 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 subfield
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We determine the leading Dirichlet eigenvalue count for rapidly twisted Euclidean tubes whose inverse twisting speed is a power times a smooth positive logarithmically periodic function. The actual operator count is an explicit periodic profile, without a regular-variation hypothesis. The Fourier multiplier factors into a nonvanishing universal factor and a complex angular-width moment. This gives a necessary and sufficient criterion for recovery of the inverse-speed modulation from the complete leading profile. A fixed connected two-width cross-section hides every odd logarithmic harmonic. In particular, smooth strictly increasing non-regularly-varying speeds on this same geometry have the pure-power law N(Λ) = Λ²/(6π) + o(Λ²) for two identical ends. The ambiguity is infinite dimensional, whereas visible modes admit conditional Sobolev recovery. For constant-step sections and inverse speeds with derivative bounded above and away from zero, we also prove an O(Λ⁷⁄⁴) counting remainder. A mixed-homogeneity inverse speed then has a genuine second periodic spectral profile: every mode of its lower-order modulation is visible on the example geometry. Whether lower terms reveal the original pure blind modulation remains open, as does equality of complete spectra.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
We estimate fixed smooth pullback entries of the inverse weighted-Laplacian metric of an unknown smooth strictly positive density on a flat torus. The estimator uses symmetric heat-kernel Sinkhorn normalization of the sample and prescribed score functions, without density evaluations. For every fixed dimension we establish a joint, population-centered root-n central limit theorem as the heat bandwidth tends to zero. The proof gives a signed-measure Taylor expansion on one Sobolev space, an explicit influence function including normalization and inverse response, and a consistent empirical covariance estimator using exact solves. A scalar Hessian identity with the moving integration measure improves the sufficient remainder budget. The population tensor has even finite asymptotic jets; fixed multiband extrapolation removes enough bias to retain the same limiting covariance. The influence is the canonical gradient, and the estimator attains the pointwise information bound under smooth local density submodels. Finally, an observed scaling, spectral-gap and inverse-residual certificate produces an actual finite first-hit output with negligible root-n metric error. Its certified-arithmetic realization requires effective convergent input enclosures. Bandwidth conditions are conservative; no practical runtime, optimal smoothing rate or numerically certified covariance algorithm is asserted.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- 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 subfield
- Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
For each fixed degree n, we prove that among homogeneous real-stable polynomials in n variables with nonnegative coefficients, value one at the all-ones vector, and a prescribed gradient sufficiently close to the all-ones vector, the power of the corresponding linear form uniquely minimizes the squarefree coefficient. This answers Question 1.8 of Gurvits–Leake, arXiv:2609.21083v2. We strengthen the minimum statement to a uniform lower bound for the coefficient deficit by the coefficient ℓ¹ distance to that power. The distance order is optimal. The proof uses centered real-root dispersion, an exact coefficient identity and the classical balanced uniqueness theorem. The neighborhood and constants are dimension-dependent and qualitative. We also give a total-variation interpretation for stable count distributions and an illustrative nondeterminantal quartic family.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We determine exact eigenvalue-counting asymptotics for infinite-volume Dirichlet tubes confined by rotation. Continuous regularly varying reference speeds of positive index require neither monotonicity nor derivative bounds; a second class includes smooth rapidly varying speeds such as exponential and double-exponential growth. The two axial ends may have different types. Finite-band cross-sections allow continuous angular endpoints, pinching widths, multiple components and finitely many topology transitions. The law remains unchanged under angle perturbations with finite limits, without control of their derivatives, and therefore permits infinitely many actual speed reversals and zeros. In the positive-index regime the coefficient is a nonlinear angular-width moment; in the rapid regime it depends only on area. We give full box and localization proofs, logarithmic trace thresholds, width-distribution recovery from a family of coefficients and a finite-data obstruction.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfield
- 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 subfield
- 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 subfield
- 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.