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52 papers · Page 3 of 3

Probability and Statistics

Edition 1

Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores

Scott Schubr

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

We study how numerical errors in symmetric kernel scaling affect the spatial score obtained from its smooth kernel extension. A full square-root-mass residual gives a sharp kernel-energy bound for every positive candidate scaling, including singular positive semidefinite Gram matrices. For Gaussian kernels this yields computable, scale-invariant pointwise score certificates. For iid samples from a strictly positive C³ density on the d-dimensional torus, we prove a simultaneous nonlinear score bound and determine the exact local modulus in a shrinking-bandwidth regime, with leading constant C_d/√f(x). Under the stated normalization and sampling assumptions, residual o(1/√n) suffices for negligible normalized score error; positive sample-dependent candidates with residual asymptotic to c/√n attain nonzero normalized error. The sufficient sampling regime is not claimed optimal. A reproducible solver experiment illustrates the certificate on actual iterates. These numerical-accuracy results do not assert a new statistical central limit theorem or an iteration lower bound.

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    Number Theory and Special Functions

    Edition 2

    Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem

    Scott Schubr

    Primary subject
    Mathematics
    Subject subfields
    Number Theory and Special Functions
    Paper type
    Research Article
    Manuscript stage
    Preprint

    We give a reproducible method for locating and certifying quintic obstructions for the normalized fixed-center Riemann theta product, and prove a unique transition in its odd quintic moment block on [3.64, 3.66]. The transition parameter lies strictly between 3.64888758188562 and 3.64888758188563. Within this interval, below the transition, degree five is the minimum degree of a polynomial whose square has negative signed-moment value; at the transition the degree-five form is positive semidefinite with a one-dimensional nullspace; above it the form is positive definite. The proof combines a Schur-complement reduction with uniform directed interval bounds, including the complete infinite theta tails. An exact rational arithmetic replay verifies the bounds on the entire interval. An explicit integer-coefficient quintic illustrates why testing one fixed polynomial does not locate the transition of the full quintic block.

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      Edition 1

      An Explicit Quintic Obstruction for the Riemann Theta Kernel: An Independent Rational Reconstruction

      Scott Schubr

      Primary subject
      Mathematics
      Subject subfields
      Number Theory and Special Functions
      Paper type
      Replication and Reproducibility Study
      Manuscript stage
      Preprint

      We give a complete rational reconstruction of an explicit obstruction to positive definiteness for a fixed-center product of the Riemann theta density. At s₀ = ¼ log(13/(4π)), the signed derivative functional evaluated on the square of P(X) = X⁵ − 264X³ + 10952X lies in the exact rational interval [−1412370791.837791749, −1412370791.837791748]. The proof constructs derivatives by formal power series, retains twelve theta summands, bounds the entire remaining series, and encloses exponentials by range-reduced alternating series. A printed table of rational moment intervals proves the negative contraction. The odd derivative Hankel block is indefinite, and the six-point translation Gram has a negative direction at every sufficiently small positive spacing.

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        Edition 1

        An Explicit Quintic Obstruction for the Riemann Theta Kernel

        Scott Schubr

        Primary subject
        Mathematics
        Subject subfields
        Number Theory and Special Functions
        Paper type
        Research Article
        Manuscript stage
        Preprint

        We exhibit an explicit polynomial square on which the signed derivative functional of a fixed-center Riemann theta product is negative. At the center s₀ = ¼ log(13/(4π)), the polynomial P(X) = X⁵ − 264X³ + 10952X satisfies ℒₛ₀(P²) ≤ −1,351,625,671. The certificate uses rational interval arithmetic for derivatives through order ten and an explicit bound on the complete infinite theta tail. It follows that the normalized product at this center is not positive definite and that its odd derivative Hankel block is indefinite.

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