Leading bias and positive correction of heat-kernel Sinkhorn metrics
Scott Schubr, October 21, 2026

Contents
  Leading_Bias_Positive_Correction_Sinkhorn_Metrics.pdf - complete 22-page paper.
  main.tex - self-contained LaTeX source with embedded bibliography.
  validated_metric_example.py - reproducible interval continuum calculation.
  validation_results.json - five bandwidth certificates and rational trials.
  fixed_normalization_example.py - prescribed-density interval normalization.
  fixed_normalization_results.json - three normalization certificates and trials.
  README.txt - scope, predecessor and reproduction instructions.
  SHA256SUMS.txt - checksums for the seven public content files.

Relationship
  Follow-up to Sharp convergence of heat-kernel Sinkhorn metrics:
  https://luminaryoddities.com/researcharchive/sharp-convergence-of-heat-kernel-sinkhorn-metrics/v/1
  It also transfers the cubic population-scale mechanism from First-Hit
  Stopping and Bias-Corrected Score Inference in Dimension Three, edition 2,
  with the fixed-dimension requirements proved in the present paper.

Scope
  Exact population operators for smooth positive densities on a fixed-dimensional
  periodic torus. Explicit leading tensor bias, an exact sign obstruction and
  a compact positive corrected tensor uniformly comparable in quadratic form
  to the original tensor, with sharp Sobolev convergence rates. Fourth-order
  accuracy is H6 to L2; full L2 convergence remains first order. The positive
  correction preserves compactness and the original completed tangent form
  space; a noncompact Simpson comparison is also analyzed.  Richardson and Simpson quadrature are credited classical methods.
  A certified nonconstant two-parameter example includes actual continuum
  residual, derivative-form and analytic quadrature error bounds. Its density
  depends on bandwidth; comparisons use the limiting metric at the same density.
  The computed certificates concern metric values with an exact manufactured normalization.
  Section 9 also proves a constructive approximation theorem for prescribed,
  bandwidth-independent analytic densities close to uniform in the stated
  weighted Fourier norm, on every fixed-dimensional flat torus. It includes
  normalization, uniform scaled inverses, all fixed parameter-derivative
  orders and explicit error budgets. Per-coordinate cutoffs and iteration
  counts are logarithmic in inverse bandwidth; total modes and quadrature
  nodes grow as its dimension-th power.  A second supplement certifies normalization at three
  bandwidths for one fixed prescribed density.  

Reproduction
  The core theorems are analytic and independent of the example computation.
  Compile main.tex using a current LaTeX distribution: pdflatex main.tex twice,
  or tectonic main.tex. Packages are declared in the source. No external
  bibliography file, private research document or online data is needed.
  The supplied PDF was built with Tectonic 0.17.0. Compilers may name the
  rebuilt file main.pdf. Creation metadata can change its byte hash; compare
  mathematical content, page geometry and rendering for reproduction.

  To reproduce the certified example, install Python 3 with mpmath==1.3.0.
  No NumPy, external data, private reference file or network access is needed
  after that dependency is installed. Run each of these (all use --J 192):
    python validated_metric_example.py --h 0.01 --N 20 --J 192 --out h001.json
    python validated_metric_example.py --h 0.005 --N 20 --J 192 --out h0005.json
    python validated_metric_example.py --h 0.0025 --N 20 --J 192 --out h00025.json
    python validated_metric_example.py --h 0.00125 --N 28 --J 192 --out h000125.json
    python validated_metric_example.py --h 0.000625 --N 28 --J 192 --out h0000625.json
  Compare each output with the matching row in validation_results.json.
  The trial coefficients are proposed at 75 digits, frozen as decimal
  rationals and certified with 60-digit interval operations; the proposal's
  accuracy is not assumed. The continuum proof and error budgets are in
  Section 8 of the paper. Interval software supplies the stated computational assumptions. The results file records the specified finite calculations.

  To reproduce the fixed prescribed-density normalization certificates:
    python fixed_normalization_example.py --h 0.01 --N 32 --L 28 --out fixed_h001.json
    python fixed_normalization_example.py --h 0.00125 --N 32 --L 28 --out fixed_h000125.json
    python fixed_normalization_example.py --h 0.00015625 --N 32 --L 28 --out fixed_h000015625.json
  Compare with the matching rows in fixed_normalization_results.json. These
  use 80-digit interval operations and freeze 75-digit decimal-rational trial
  coefficients. The full continuum residual, including modes beyond the trial
  cutoff, encloses the unknown infinite-dimensional normalization. The proof
  and scope are in Section 9. The same mpmath dependency suffices.

Integrity
  Verify the seven content files against SHA256SUMS.txt. The manifest does not
  checksum itself. File hashes establish identity, not mathematical validity.
