All Research · Luminary Research Archive

79 papers · Page 2 of 4

Algorithms and Complexity

Edition 4

A linear query lower bound for graph-isomorphism testing at maximum degree three

Scott Schubr

Primary subject: Computer Science & Software Systems
Subject subfields: Algorithms and Complexity · Combinatorics and Graph Theory
Paper type: Research Article
Manuscript stage: Preprint

For every sufficiently large even N, we exhibit existentially a known connected simple 3-regular N-vertex target whose randomized two-sided isomorphism testing requires Ω(N) incidence queries at literal proximity 0.01, even with connected cubic inputs promised. This gives a negative answer to Goldreich’s Open Problem 4.1 in its stated distance normalization. We also retain maximum-degree-three fixed targets at every sufficiently large integer order, reaching the minimum maximum-degree threshold since degree two admits size-independent query bounds. We use Dror’s sparse-code and adaptive indistinguishability framework. A no-rung 16n-vertex parity encoding has the arbitrary-bijection estimate dist_H(b, Im A) ≤ (2/3) dist_iso(G_b, G_0). Adaptive coloring of imported target middle vertices, a joint defective-row budget, and median assignment rounding handle arbitrary mixing of graph roles. Bounded equivariant surgeries cover every even order with additive distance loss at most 322. An upper bound 4/429 < 0.01 for the older capped family explains why that completion method alone cannot reach the numerical endpoint.

October 24, 2026Read paper

Classical and Real Analysis

Edition 1

Oscillating sequences with subadditivity on shrinking corridors

Scott Schubr

Primary subject: Mathematics
Subject subfields: Classical and Real Analysis
Paper type: Research Article
Manuscript stage: Preprint

For measurable nonincreasing relative widths w, this paper identifies a square-integral threshold for exact eventual subadditivity on the integer corridor n ≤ m ≤ n + nw(log n). Divergence of ∫w² forces normalized-slope convergence for nondecreasing nonnegative linearly bounded sequences, without continuity or an adjacent-pair assumption. If adjacent integer pairs are eventually admissible, a prefix-envelope and dyadic-record argument removes the monotonicity, sign and growth hypotheses: every real sequence obeying the corridor inequality has a normalized limit in ℝ ∪ {−∞}. In particular, n/√log n is a sufficient sublinear corridor for arbitrary real sequences. For a finite square integral, a shifted-tail correction produces strictly increasing integer submultiplicative and supermultiplicative counterexamples with logarithmic growth endpoints 1.99 and 2.01. The resulting dichotomy is exact in the monotone counting class and, with the adjacent-pair condition, for arbitrary real sequences. Divergent widths with missing adjacent pairs remain unresolved in the unrestricted real-sequence setting.

October 23, 2026Read paper

Combinatorics and Graph Theory

Edition 2

Prescribed-gradient minimizers and coefficient stability for homogeneous real-stable polynomials

Scott Schubr

Primary subject: Mathematics
Subject subfields: 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. We also classify strict local minimality at every positive gradient and give a cubic example separating local and global minimization. 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.

October 23, 2026Read paper

Algebra

Edition 1

Exponential growth of sparse Euler products and normal graded domains

Scott Schubr

Primary subject: Mathematics
Subject subfields: Algebra · Combinatorics
Paper type: Research Article
Manuscript stage: Preprint

Let dᵢ be strictly increasing positive integer degrees and let Mᵢ be positive integers with log(Mᵢ)/dᵢ → δ ∈ (0, ∞ ). For the Euler product with Mᵢ independent polynomial generators of degree dᵢ and one degree-one generator, we prove that the upper logarithmic coefficient rate is δ and the lower rate is max{δ − θ, 0}, where θ = lim supᵢ log(dᵢ₊₁)/dᵢ. This gives an exact convergence criterion with no regularity assumption on the support gaps. We contrast it with normal connected locally finite graded domains of Krull dimension two, obtained by adapting an infinite-pole construction of Maclean. Their lower rate is δ/R, where R = lim supᵢ dᵢ₊₁/dᵢ, and their upper rate is δ. In particular, normality, finite dimension and a degree-one nonzerodivisor do not jointly force convergence of the exponential Hilbert rate.

October 23, 2026Read paper

Probability and Statistics

Edition 6

Moment certificates for homogeneous stable count laws

Scott Schubr

Primary subject: Mathematics
Subject subfields: Probability and Statistics
Paper type: Research Article
Manuscript stage: Preprint

This paper compares a homogeneous real-stable count law with its mean-matched multinomial law through the normalized covariance deficit D. Finite chi-square bounds retain Tr(D) and Tr(D²), with an exact covariance-determined quadratic likelihood component and a uniform remainder. Matching mean and covariance gives a sharp total-variation rate of order Tr(D)^{3/2}. The optimal centered real elementary-symmetric coefficient inequality is proved in every order with its equality classification. A sharp centered complex degree-five coefficient bound in six variables yields the optimal factor 3/4 for the fifth likelihood component of every degree-six stable count law, at arbitrary covariance rank. Together with the other orders, this establishes the complete-homogeneous factor-one target throughout degree six. General degrees retain a factor-24 envelope, with stronger order-five bounds and structured spectral regions. Higher-interior factor one in degrees at least seven remains open. A fixed-law Gaussian mutual-total-variation limit is retained. Classical root-extremum, determinantal, centered-coefficient and orthogonal-expansion methods are credited; standalone code reconstructs finite exact certificates.

October 23, 2026Read paper

Edition 1

Covariance calibration and parity certificates for stable count laws

Scott Schubr

Primary subject: Mathematics
Subject subfields: Probability and Statistics
Paper type: Methods Paper
Manuscript stage: Preprint

We study the normalized covariance deficit of a homogeneous real-stable count law on m categories with total count k. This deficit is a rescaling of an established canonical dependence matrix for two balls drawn without replacement. Its positive semidefiniteness gives a diagonal domination property that controls rare categories and normalization error. We use this structure to construct an upper confidence bound for its operator norm without a minimum category-probability assumption and without knowing the mean. Independent batches of full count observations suffice: the mean stage uses O(m(k − 1)ε^{−2} log(m/δ)/k) observations and the second stage uses O(m(k − 1)²ε^{−2} log(m/δ)). A stable degree-two subclass gives a matching linear category and inverse-square accuracy lower bound up to logarithms. Real-rooted one-dimensional projections then turn a covariance bound and a reference Fourier margin into simultaneous positive-parity certificates. We give a sharp positivity boundary on a binary subclass and stable laws with identical means and covariance but opposite parity. The constants in the confidence bound are conservative, and the observation model is classical labeled counts.

October 23, 2026Read paper

Edition 1

Statistical inference for empirical heat-kernel Sinkhorn metrics

Scott Schubr

Primary subject: Mathematics
Subject subfields: 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.

October 22, 2026Read paper

Quantum Information Theory

Edition 1

Single-copy learning of stabilizers with correlated diagonal noise

Scott Schubr

Primary subject: Mathematics
Subject subfields: Quantum Information Theory
Paper type: Research Article
Manuscript stage: Preprint

This paper studies exact identification of an unknown qubit stabilizer from a state diagonal in its stabilizer eigenbasis. A positive lower bound on every target-signed stabilizer expectation unifies global white noise, strict-majority diagonal mixtures and further correlated mixtures. For a supplied floor μ₀ ≥ 2^{−(log₂ n)^γ}, with fixed γ < 1, fixed-confidence recovery uses O(n^{1+ε}) original copies, polynomial ideal quantum work and 2^{O(n/log log n)} classical time. A uniformly spread successful component in a learned, verified Pauli direction contracts surviving error odds without independent syndrome errors. An unknown target-containing stabilizer-code background admits polynomial recovery under a relative diagonal-visibility promise. Dense CSS targets with sufficiently small background-code rank and a bounded-weight local-Pauli regime are included. Known noisy-parity solvers, polynomial subspace lifts and high-fidelity learning are explicit ingredients. General polynomial-time recovery and arbitrary coherent noise remain unresolved.

October 23, 2026Read paper

Analysis and Spectral Theory

Edition 2

Positive Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory · Differential Equations
Paper type: Research Article
Manuscript stage: Preprint

We construct bounded connected nonconvex planar pentagons for which the affine 2-Laplacian has a strictly positive Dirichlet eigenfunction at an eigenvalue strictly above its first variational eigenvalue. The same domain therefore supports positive affine eigenfunctions at two distinct values. The construction gives a negative answer, for n = p = 2, to the general-domain positive-eigenfunction question of Haddad, Jiménez and Montenegro. The mechanism is an affine-isotropic saddle formed by pinching a pentagon into two right isosceles triangles. A symmetry-preserving diagonal normalization has positive curvature, while a shear couples the ground state to a nearly coincident odd mode. We prove the sharp limit (λ₂ − λ₁)∂²λ₁/∂q² → −32768/81 at the isotropic positions. An area-preserving apex slide has the corresponding limit −50π⁴. The saddles persist under small vertex perturbations, including examples with nonparallel convex-hull sides. Consequences include distinct isotropic positions on one linear orbit and failure of positive-definite-deformation log-convexity outside convexity. We further preserve the saddle under smooth inner exhaustion and use a symmetry-balanced product construction to obtain positive affine 2-eigenfunctions at two distinct eigenvalues on smooth connected nonconvex domains in every dimension n ≥ 2.

October 23, 2026Read paper

Edition 1

Finite-spectral recovery of Weyl-blind twisting modes

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory
Paper type: Research Article
Manuscript stage: Preprint

Odd logarithmic twisting modes in a fixed polar-strip tube are invisible to the leading Weyl profile but detectable in the next heat profile. This paper gives a quantitative continuation of that complete-spectrum result. An estimate on the actual curved normal cap yields a heat remainder O(T¹⁷⁄⁶) and normalized bias O(T⁻¹⁄⁶). A complete finite spectral prefix, certified uncertainties, a counting envelope and a certified complex multiplier enclosure give a deterministic recovery disk. With exact inputs, the fixed-mode cutoff rate is O((log Λ/Λ)¹⁄¹²); the relative-noise balance is O(σ¹⁄⁷) before heat-multiplier inversion. A long-straight-core construction shows why a uniform geometric prior is necessary, and two-sided multiplier bounds quantify exponential conditioning. The geometric remainder constants are not fully evaluated numerically in this manuscript. The concrete fundamental-mode interval execution is valid but noninformative; practical numerical reconstruction is not claimed.

October 23, 2026Read paper

Edition 1

A Positive Corner Limit for Inverse Half-Powers of Vámos Quartics

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory
Paper type: Research Article
Manuscript stage: Preprint

For each fixed parameter 0 < λ ≤ 64/27, this paper determines an anisotropic corner limit of the inverse Laplace distribution of a stable Vámos quartic raised to the power −1/2. The limit is an explicit positive density with arcsine slope sections. Convergence holds jointly in all scaled variables, including the variable where the exact partial inverse has a logarithmic singularity. The proof uses a modified Bessel partial inverse, differentiation to remove logarithmic counterterms and quantitative estimates for inversion on growing frequency regions. It also determines the first correction in the same joint distribution topology. Positivity of the full inverse distribution is not assumed. The result does not establish positivity at a positive scale or justify restriction of the converging distributions to a graph.

October 23, 2026Read paper

Edition 1

An Unrestricted Half-Power Obstruction for the Specialized Vámos Quartic

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory
Paper type: Research Article
Manuscript stage: Preprint

The inverse square root of the specialized four-variable Vámos quartic is shown not to be completely monotone. More generally, the obstruction holds for every interior stable member hλ = 4e₁e₃ + λa²b² with 0 < λ < 64/27. The proof computes a regular correction to a singular positive leading inverse, establishes distributional Laplace-inversion convergence through uniform complex estimates, and identifies a negative density outside the leading support. A smooth-section argument transfers this sign to the unrestricted inverse. For the original quartic an explicit interval and uniform negative bound are obtained. The result gives a qualitative strict exponent gap above 1/2, uniform on compact interior parameter sets. At λ = 64/27 the first correction is nonnegative outside the leading support; complete monotonicity at that endpoint remains open.

October 23, 2026Read paper

Edition 1

Audibility of all odd blind modes in rapidly twisted polar-strip tubes

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory
Paper type: Research Article
Manuscript stage: Preprint

A fixed rapidly twisted Euclidean tube has an infinite lattice of logarithmic inverse-speed harmonics invisible to its leading Weyl profile. This paper proves that every odd harmonic on that lattice is detected by the next heat profile. The connected-versus-cut relative heat trace of the step aperture is represented as a Laplace transform of a positive measure through a finite positive-root matrix lemma, threshold atoms and controlled aperture limits. A noncommuting trace estimate gives an analytic mass bound, and Euler-product domination yields uniform noncancellation at every original odd frequency. Combined with the preceding actual-tube heat expansion, this recovers each admissible pure amplitude and gives injectivity on admissible finite odd Fourier profiles. The argument requires no sampled aperture calculation or floating-point PDE enclosure. Frequency-dependent lower bounds do not establish recovery stable under finite spectral data.

October 23, 2026Read paper

Edition 1

Left-Sector Mellin Certification of a Weyl-Blind Twisting Mode

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory
Paper type: Research Article
Manuscript stage: Preprint

A complex Mellin invariant of a fixed connected step junction is certified on a ray with negative real part. Positive-axis exponential flatness is propagated through an exterior sector using a classical Phragmén–Lindelöf multiplier with explicit constants. A left-sector contour suppresses reciprocal-Gamma conditioning and detects the original fifth inverse-speed harmonic, which is invisible to the leading Weyl profile of the same Euclidean tube. Complete residual bounds retain the noncommuting aperture inverse and infinite input tails. The certificate includes 67 complex nodes, the endpoint calculation, infinite quadrature tails and stated arithmetic errors. The junction Mellin modulus is below 3.198 × 10⁻⁹, while cancellation requires a modulus above 1.18 × 10⁻⁸; the fifth heat multiplier therefore has modulus greater than 2.16 × 10⁻⁷. Higher modes and finite-data stability are outside this certificate.

October 23, 2026Read paper

Edition 1

Certified Resolvent Detection of a Third Weyl-Blind Twisting Mode

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory
Paper type: Research Article
Manuscript stage: Preprint

The leading Weyl profile of a rapidly twisted tube with a fixed connected step section annihilates every odd logarithmic harmonic of inverse twist speed. A preceding heat-trace result detects the fundamental harmonic. This paper gives a computer-assisted resolution for the original third harmonic without changing the section or adding a lower-order perturbation. A half-strip resolvent formula and complete residual estimates enclose a complex Mellin invariant of the connected junction. Sectorial trace bounds reduce the certified quadrature to 27 complex nodes while retaining infinite tails, arithmetic error, node conversion and aliasing. The resulting Mellin modulus is below 5.653 × 10⁻⁶, whereas cancellation requires a modulus above 1.13 × 10⁻⁵; the third heat coefficient therefore has modulus greater than 5.08 × 10⁻⁵. The source certificate states its arithmetic assumptions and regenerates its inputs. Higher odd modes and finite-spectrum stability are outside this paper’s scope.

October 23, 2026Read paper

Edition 1

Heat-Trace Recovery of a Weyl-Blind Twisting Mode

Scott Schubr

Primary subject: Mathematics
Subject subfields: 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.

October 22, 2026Read paper

Edition 1

Geometric Blind Modes in the Weyl Asymptotics of Twisted Tubes

Scott Schubr

Primary subject: Mathematics
Subject subfields: 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.

October 22, 2026Read paper

Linear Algebra and Matrix Inequalities

Edition 1

Strong and energy-bounded permanent inequalities on bipartite supports

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.

October 23, 2026Read paper

Edition 1

Row-dependent bounds for strong permanent contraction on four-by-eight supports

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.

October 22, 2026Read paper

Edition 1

Strong permanent contraction on four-by-seven bipartite supports

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.

October 22, 2026Read paper

Edition 1

Strong permanent contraction on four-row bipartite supports through six columns

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.

October 22, 2026Read paper

Edition 1

Strong permanent inequalities for three-row bipartite and triangle-free supports

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.

October 22, 2026Read paper

Edition 1

Strong permanent inequalities for complex two- and three-chord seven-cycle supports

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.

October 22, 2026Read paper

Geometry and Optimal Transport

Edition 1

Sharp Stability and Geometric Classification of a Vámos Quartic Deformation

Scott Schubr

Primary subject: Mathematics
Subject subfields: 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.

October 22, 2026Read paper