All Research · Luminary Research Archive

79 papers · Page 1 of 4

Linear Algebra and Matrix Inequalities

Edition 1

Zero geometry and sharp compensated permanent derivatives on a complex Gram stratum

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.

October 25, 2026Read paper

Edition 1

Asymptotically optimal bounds for permanental cofactor matrices

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.

October 25, 2026Read paper

Edition 10

Occupation bounds for coupled permanent cofactors

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.

October 24, 2026Read paper

Complex Analysis

Edition 2

All-depth root-Laguerre positivity with nonreal Fourier zeros

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis
Paper type: Research Article
Manuscript stage: Preprint

We give an explicit admissible Fourier kernel for which every continuous root-coordinate Laguerre iterate is positive and uniformly strictly log-concave, although its Fourier transform has nonreal zeros. Every root iterate also defines a strictly totally positive translation kernel of all orders, with explicit signed confluent determinants. The fixed kernel is φ(s) = exp(−exp(s²) − 999s²). Its Fourier transform is entire of exact order one. Two whole-line rational Gaussian-moment enclosures certify the negative sixth cumulant, the sharper bound being κ₆ < −1.8 × 10⁻¹⁸. The real-zero canonical product would require that cumulant to be nonnegative. Exact quadratic-tilt invariance explains why the entire root hierarchy misses this obstruction. We also give a general Gaussian-localization formula for higher cumulants, including an effective finite-jet error bound. Newman’s Gaussian-damping classification already implies existence of a bad tilt for the underlying double-exponential family; the present conclusion is an explicit certified all-depth counterexample. It concerns general kernels and gives no conclusion about the Riemann theta kernel.

October 25, 2026Read paper

Edition 1

Global finite-depth stability and nonreal zeros of admissible Fourier transforms

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis
Paper type: Research Article
Manuscript stage: Preprint

We prove a global finite-jet stability theorem for the differential iteration Tf = (f′)² − ff″. A uniformly positive reference curvature and bounded curvature-normalized higher derivatives give explicit amplitude budgets for positive, uniformly log-concave iterates through any prescribed finite depth. Intensified hyperbolic-cosh mixtures realize the required jets on the whole half-line, with all spatial tails controlled. Combining this realization with a credited finite-atomic Fourier multiplier gives a simultaneous separation theorem. For every finite generalized-Laguerre cutoff, finite root-coordinate iteration depth and 0 < Y < H, one fully admissible kernel has an order-one Fourier transform satisfying all retained global Laguerre inequalities and all polynomial moment orders throughout 0 ≤ y ≤ Y, yet has nonreal zeros inside |Im z| < H and negative physical polynomial squares at a later parameter. Parameters depend on the prescribed finite depths. The construction concerns general admissible kernels; it gives no conclusion about the all-order positivity of the actual Riemann theta kernel.

October 25, 2026Read paper

Edition 1

Admissible kernel lifts with positive endpoint forms and nonreal Fourier zeros

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis
Paper type: Research Article
Manuscript stage: Preprint

We give a finite-atomic construction of admissible Fourier kernels with prescribed Fourier multipliers of positive finite atomic measures. A sufficiently broad hyperbolic-cosh base makes the translated mixture uniformly strictly log-concave while retaining an order-one entire Fourier transform. A global centered-moment and cumulant estimate also preserves strict concavity of G(u) = log φ(√u) and of log(S′(u)² − S(u)S″(u)), where S(u) = φ(√u). The required scales are explicit and conservative. An exact four-atom multiplier produces kernels satisfying these shape conditions whose physical endpoint moment matrices are positive definite at every finite order, while their Fourier transforms have nonreal zeros. All zeros can be confined to any prescribed horizontal strip; an interior physical parameter still has negative polynomial squares in both parities. Classical Turán consequences are stated with their coefficient normalization and do not imply real-zero closure. The construction strengthens a generic nonimplication within the full admissible class. It is not a counterexample to the Riemann theta kernel and does not resolve the Riemann hypothesis.

October 24, 2026Read paper

Edition 1

Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis · Algebra · Linear Algebra and Matrix Theory
Paper type: Research Article
Manuscript stage: Preprint

On the complex zero-sum hyperplane in C^n, consider the symmetric quartics A = n∑ᵢ|vᵢ|⁴ − |∑ᵢvᵢ²|² and Q = ‖v‖⁴ − |∑ᵢvᵢ²|². For every odd n ≥ 3 and every α, β > 0, we prove that a holomorphic squared-norm representation of ‖v‖²ᵐ(αA + βQ) requires m ≥ n − 2. For each fixed odd n, equality holds for all sufficiently small positive β/α, whereas at β = 0 the least exponent is zero. The lower bound uses an explicit Fischer annihilator and a positive Hamming-distance correlation identity on the odd-dimensional sign cube. The upper bound uses the classical quadratic complete intersection associated with the zero-sum hyperplane. For n = 5, we determine the entire first-degree window: the least exponent is three exactly when 0 < β/α ≤ ρ*, where 305ρ*³ + 786ρ*² + 351ρ* − 162 = 0. An explicit four-by-four pencil and harmonic decomposition give this sharp endpoint, including its four-dimensional kernel. We also derive the quartic as a coefficient in a centered permanent deficit, which motivates the family without implying the full permanent inequality.

October 24, 2026Read paper

Edition 1

Compact capture and protected bridges on the Koras–Russell cylinder

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis · Geometry
Paper type: Research Article
Manuscript stage: Preprint

On an adapted model of the Koras-Russell cylinder, we give explicit compression of a closed C-star carrier and a sufficient criterion for ambient compact capture: the planar polynomial hull of the scalar image under q = xy+1 must avoid 1. This yields fixed Runge capture domains, including the connected inverse image of the plane slit along [1,infinity). Joint capture can be corrected to approximate the identity on a globally holomorphically convex parametrized ball inside a fixed Runge Euclidean domain. Prescribed compact bridges of positive thickness can thus be captured while protecting the ball without scalar-range separation. A complementary convex-ball argument treats certain protected bodies through q=1, and an analytic disk explains the obstruction to extending weighted contraction to the entire punctured chart. These are qualitative capture criteria, not a biholomorphic identification of the whole cylinder with C^4 or an improved stabilization bound.

October 24, 2026Read paper

Edition 2

Finite-projection stabilization of contractible affine hypersurfaces

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis · Geometry
Paper type: Research Article
Manuscript stage: Preprint

Let X be a smooth contractible affine hypersurface of complex dimension n >= 2 with the holomorphic density property. We prove that X times C^n is biholomorphic to C^(2n). A finite monic projection replaces the proper-embedding parameter in Huang’s stabilization construction, retaining colliding fibers as curvilinear finite schemes. Stratified Oka interpolation and parametric Andersén-Lempert approximation yield compact capture in a fixed Runge Euclidean basin. Applications include McLean’s varieties and the Koras-Russell cubic. The sufficient ambient dimensions in specified nonlinearizable-action constructions are lowered by one, including disconnected cyclic fixed loci in C^6 and a fixed contractible non-Euclidean surface in C^8.

October 24, 2026Read paper

Analysis and Spectral Theory

Edition 1

Finite-order positivity and growing negative inertia for averaged Riemann theta products

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory · Number Theory and Special Functions · Linear Algebra and Matrix Inequalities
Paper type: Research Article
Manuscript stage: Preprint

We study the contrast between finite-order positivity and all-order failure for averaged Riemann theta products. Our main new theorem proves that, at the exact upper-center cutoff A=23/2, each polynomial parity through degree N has negative inertia at least 3 log(N)/(8 pi)-O(log log N). A fully explicit conservative bound constructs common negative subspaces, with every theta term and approximation error retained. Consequently a correction making the full form positive semidefinite must have rank at least 3 log(N)/(4 pi)-O(log log N), even if the correction changes with N or couples parities. The proof combines classical sinc concentration with gamma-reference cancellation and uniform Meixner-Pollaczek projection. We consolidate, with explicit attribution, previously established positivity through degree eight, degree-dependent terminal positivity, exact rational negative witnesses of degrees 4096 and 4097, and eventual witness degree at most ceil(A^(31/4)) in both parities. A classical lognormal counterexample explains why finite moments and a negative density region alone are insufficient. The physical all-order problem remains open; no zero theorem or optimal inertia asymptotic is claimed.

October 24, 2026Read paper

Number Theory

Edition 6

Monotone center weights and the averaged odd-quintic theta gate

Scott Schubr

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

We prove averaged Riemann theta moment positivity through degree eight for nonzero nonnegative nondecreasing center weights under the respective finite-moment hypotheses. Every fixed degree has a positive terminal center region X>=1000 N^2, yet the all-order monotone cone fails. At A=23/2 we supply exact rational even and odd polynomial witnesses of degrees 4096 and 4097. Their complete normalized integrals lie strictly between -3e-7 and -2e-7, with all theta products, outward rounding, continuous sampling error and infinite tails paid. Positive, strictly increasing bounded real-analytic center weights also fail. For every sufficiently large real cutoff A, both parities admit rational negative witnesses of degree at most ceil(A^(31/4)), while any such witness has degree exceeding sqrt(A/1000). The quantitative proof combines classical Meixner-Pollaczek projection and Kaiser-Bessel concentration with reference cancellation and quadratic sign transfer. No minimal degree, numerical universal onset, physical-weight counterexample or RH theorem is claimed.

October 24, 2026Read paper

Classical and Real Analysis

Edition 1

Three-point positivity for weighted Riemann theta products

Scott Schubr

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

We give an explicit curvature argument for three-point positivity of weighted products of the Riemann theta density. For every alpha at least 5/8 and every nonnegative Borel center measure with a finite positive integral of Phi^(2alpha), the averaged product Phi(s+t/2)^alpha Phi(s-t/2)^alpha has positive definite three-point translation matrices at distinct real positions. Repeated positions give positive semidefinite matrices. A general curvature criterion combines with positive-Laplace estimates for the complete theta-series tail; all infinite-tail bounds and rational interval estimates are explicit. Positive definiteness of arbitrary order is not established.

October 24, 2026Read paper

Edition 1

Sharp Quadratic Smoothing for Truncated Simplex Sums

Scott Schubr

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

For every nonnegative integer radius, we prove joint log-concavity of the raw three-dimensional truncated simplex lattice sum at arbitrary real endpoints when each independent real power is at least two. The coordinatewise threshold two is sharp. Exhaustive cell coverage and exact positive-coefficient polynomial identities certify the quadratic endpoint over unbounded parameter orthants, rather than numerical samples or finite parameter cutoffs. In every dimension, the same endpoint holds with up to two unrestricted occupancies and arbitrarily many binary matroid companions with activities in [0,1]. Degree-elevated homogenization, classical complete-log-concavity and compensated composition supply that extension. The unrestricted quadratic simplex problem in dimensions four and higher remains open.

October 24, 2026Read paper

Edition 1

Dimension Thresholds for Concavity of Normalized Mixed Moments

Scott Schubr

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

For a nonnegative coordinatewise nondecreasing integrable weight on a finite d-dimensional box, joint Gamma-normalized mixed moments are log-concave when every exponent is at least d/4, for d at least two. No input log-concavity, independence or positive association is required. Finite product grids with ordered log-concave coordinate values and index-increasing weights admit the sufficient bound d/4+4. Continuous densities and finite-grid examples show that both thresholds have optimal leading constant one quarter; additive constants remain open. Conditional logarithmic-deficit estimates, a signed-block covariance argument and trigamma bounds also give spatial product-power smoothing for arbitrary decreasing inputs at powers d/4+4 in the continuous case and d/4+7 for arrays. A classical matroid input yields a dimension-free binary endpoint, while unweighted hereditary binary supports require dimension-linear powers.

October 24, 2026Read paper

Edition 1

A Sharp Zero-Sector Criterion for Polynomial Prefix Sums

Scott Schubr

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

Let h be a real polynomial with h(0)>0 whose zeros lie in the closed sector of half-angle pi/4 around the negative real axis. For every nonnegative real step and geometric bias at least one, its geometrically weighted integer prefix sums admit a finite concave logarithmic extension to the nonnegative quadrant. This implies weighted logarithmic Jensen inequalities for arbitrary probability laws with finite first moments and integral mean prefix length. The universal sector angle and bias range are sharp, including integral-height counterexamples. Joint log-concavity of a two-variable truncated-square lattice sum follows from complete geometry coverage and exact rational certificates; classical Prekopa marginalization removes the degree bound. Applications cover split projective-line bundles over projective space in every dimension. The extension agrees with integer-prefix rows rather than asserting ordinary polynomial continuation in the prefix variable.

October 24, 2026Read paper

Algebra

Edition 1

Concavity of Lattice Counts on Toric Threefolds

Scott Schubr

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

The fourth root of the ordinary Riemann-Roch count polynomial is concave on the entire closed nef cone of every smooth toric Fano threefold. This exponent is uniformly optimal for the ordinary real polynomial extensions. Precisely nine of the eighteen threefolds have concave cube root; the other nine have cube-root failure, certified by exact fan and polynomial computations. Integral nonbig nef classes are included. For split projective-line bundles over the projective plane, the ordinary polynomial has optimal power one third at twists zero, one and two, while no positive concave power exists at any integer twist at least three. Geometrically weighted lattice counts with bias at least one admit nonpolynomial concave logarithmic extensions at every twist, whereas every fixed bias strictly between zero and one has integral midpoint counterexamples at large twists. Ordinary-polynomial optimality and lattice-only sharpness are distinguished.

October 24, 2026Read paper

Combinatorics and Graph Theory

Edition 3

Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scott Schubr

Primary subject: Mathematics
Subject subfields: Combinatorics and Graph Theory
Paper type: Research Article
Manuscript stage: Preprint

We prove a global sixth-root Jensen inequality for bounded unimodular fibers whose central affine dimension is at most three. Every integrable integer-valued demand with feasible fibers almost surely and integral mean satisfies the inequality when the central fiber is bounded and has affine dimension at most three, including countable support. A monotone concave extension of the sixth root of every type-A3 alcoved lattice count allows all twelve directed bounds to vary. Classical chamber formulas and Lorentzian volume theory combine with degree-six homogenization, wall gluing, shortest-path closure and a common integral rank-three reduction. Stronger planar powers, K4 extension sharpness and integral cube-root counterexamples are retained. Directed rounding and a triangle obstruction distinguish local methods from global Jensen. The general sixth-root exponent is not claimed optimal; unrestricted higher-dimensional unimodular Jensen remains open.

October 24, 2026Read paper

Edition 1

Degree laws and edge correlations for random forests

Scott Schubr

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

For the unrooted spanning-forest measure on any finite simple graph with arbitrary positive edge weights, the full incident-edge marginal at every vertex is real stable. Each star is strongly Rayleigh, incident edges are nonpositively correlated, and vertex degrees have Poisson-binomial laws. The proof combines a classical optional-component-mark forest polynomial with a stable determinant differential operator and a signed matching polynomial. We also prove strict pairwise negative edge correlation on every complete graph with at least three vertices and every complete bipartite graph with both parts at least two, at every common positive fugacity. Exact Hermite and Laguerre counts, continued fractions, scalar crossing barriers and polynomial certificates treat disjoint edges in these families. Quantitative bipartite critical and supercritical deficit asymptotics are retained. Arbitrary-host disjoint-edge correlation and negative association of the full forest law remain outside the results.

October 24, 2026Read paper

Functional Analysis

Edition 1

Mean ergodic convergence from near-diagonal subadditivity

Scott Schubr

Primary subject: Mathematics
Subject subfields: Functional Analysis · Dynamical Systems
Paper type: Research Article
Manuscript stage: Preprint

We prove a mean ergodic theorem under subadditivity required only for nearly equal block lengths. The operator is positive, unital and integral-preserving, without ergodicity or invertibility. Convergence of invariant projections of uniformly integrable normalized inputs, with both ordered adjacent-two inequalities, forces L1 convergence. An existing scalar shrinking-corridor theorem supplies invariant Lp convergence for every finite p when the nonincreasing relative width has divergent square integral and eventually admits length differences two. Nonnegative inputs require individual Lp membership; for signed inputs, uniform integrability of normalized negative-part pth powers is necessary and sufficient. Under the same scalar-admissible width and norm criterion, signed integrable inputs have almost-sure limsup equal to the invariant limit, uniformly over macroscopic operator powers and lengths. The corresponding lower-limit assertion remains open.

October 24, 2026Read paper

Optimization and Control

Edition 1

Self-concordance and allocation for spatial power sums

Scott Schubr

Primary subject: Mathematics
Subject subfields: Optimization and Control · Classical and Real Analysis
Paper type: Technical Report
Manuscript stage: Preprint

This mathematical supplement studies the negative logarithm of one-sided product-power smoothing of arbitrary coordinatewise decreasing weights. A common real power p >= 16d gives a standard self-concordant barrier with parameter at most 2pd. For decreasing lattice inputs with at most r positive coordinates per state, p >= 32r suffices independently of ambient dimension and occupancy magnitudes. Residual Pareto tails, an exact third-cumulant identity and a sparse L3 bound supply the proof without input independence or log-concavity. Decreasing binary mixtures require power growing linearly with r for uniform convexity. Log-concave binary generating polynomials with deletion bounds admit dimension-free fixed-power statements and squared-logarithmic activity dependence. Classical damped Newton and projected-gradient methods give explicit allocation consequences. Constants are not sharp and no bit-complexity bound is asserted.

October 24, 2026Read paper

Statistical and Nonlinear Physics

Edition 1

Bose gases in weakly twisted annular-sector waveguides

Scott Schubr

Primary subject: Physics and Astronomy
Subject subfields: Statistical and Nonlinear Physics · Mathematical Physics
Paper type: Research Article
Manuscript stage: Preprint

We study an ideal Bose gas in an infinite-volume Dirichlet tube whose annular-sector cross-section rotates with speed c(1+z^2)^(p/2) as c decreases to zero. At particle scale c^(-1/p), critical scaled density is finite exactly for 0 < p < 1, with the excess in the single ground mode. An operator-valued symbol retains every transverse band. For p > 1, the denser scale c^(-2/(p+1)) produces positive limiting occupation fractions in every fixed mode of an anharmonic oscillator. The marginal logarithmic scale is identified by a sector-specific uniform harmonic comparison. Actual fixed-mode spectral limits, intermediate-energy counts, full fixed-time heat traces and singular excited occupations are controlled. All statistical conclusions concern grand-canonical mean occupations; classical power-trap and free-Gibbs laws are credited.

October 24, 2026Read paper

Statistical Theory

Edition 1

A sharp logarithmic inverse bound for Gaussian estimation

Scott Schubr

Primary subject: Statistics and Data Science
Subject subfields: Statistical Theory · Geometry and Optimal Transport
Paper type: Research Article
Manuscript stage: Preprint

At fixed dimension and positive Gaussian noise level, fix a centered isotropic reference whose twice continuously differentiable potential has Hessian between two positive constant multiples of the identity. Uniformly over isotropic log-concave inputs, we prove a sharp inverse-logarithmic Wasserstein modulus of order 3/2 from either noisy-output W1 distance or excess quadratic risk of the reference Bayes estimator. Explicit narrow-Gaussian perturbations with zero mass, mean and covariance corrections give the matching lower order at every permitted reference. The Gaussian-reference case yields a sharp total linear-MMSE gap modulus at every fixed positive SNR. Curvature-uniform constants yield sharp minimax W2 estimation rates of order (log n)^(-3/2) on two-sided curvature classes and fixed-reference testing at the same separation order. Inputs may have compact support or nonsmooth convex potentials.

October 24, 2026Read paper

Differential Equations

Edition 2

Single-time boundary heat-flux rigidity for smooth domains

Scott Schubr

Primary subject: Mathematics
Subject subfields: Differential Equations
Paper type: Research Article
Manuscript stage: Preprint

For a bounded connected smooth Euclidean domain in dimension at least two, consider Dirichlet heat evolution with initial temperature identically one. If the signed outward normal derivative is spatially constant on the entire boundary at one positive time, the domain is a ball; connected boundary is not assumed. The proof uses moving planes, an L2 initial-corner comparison, component reconstruction and a parabolic corner barrier. We also give a local near-ball quantitative stability estimate in strong boundary norms, with volume and centroid normalization, at times in a fixed compact positive interval. In contrast, one fixed time observed on a moving proper boundary cap admits an infinite-dimensional family of nearby normalized nonballs with exactly the ball flux on that cap. Global stability and reconstruction from a fixed physical patch are not asserted.

October 24, 2026Read paper

Probability and Statistics

Edition 1

Local and eventual entropy concavity under Gaussian smoothing

Scott Schubr

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

For independent copies of a symmetric variance-one input, we study entropy of their variance-preserving weighted sum after Gaussian smoothing. For every Q < 1/2, we prove all-weight strict entropy concavity for 0 < q <= Q in an explicit infinite-dimensional Gaussian-relative bounded-density neighborhood. A uniform moment envelope with finite nonzero Hermite discrepancy gives a common eventual threshold; every fixed symmetric non-Gaussian all-moment input has eventual all-weight strict concavity. For the symmetric log-concave class, any counterexample sequence as q decreases to zero must converge weakly and in each fixed absolute moment to the Gaussian, while endpoint curvature is strictly negative for q <= 1/20. An exact moment-matched exponential-tail construction delineates the remaining near-Gaussian gap. A common all-weight threshold for the full symmetric log-concave class remains open.

October 24, 2026Read paper