FREQUENCY INTERACTIONS AND CURVATURE IN FOURIER DENSITY MODELS
Scott Schubr

This archive contains the paper, its complete LaTeX sources and mathematical
reproducibility material. Curvature assertions concern the intrinsic pullback
metric of the specified aligned-cosine density family at uniform density.
They are not assertions about negative ambient Wasserstein curvature.

Read Frequency_Interactions_Curvature_Fourier_Density_Models.pdf first. Its analytic proofs establish the infinite
families of claims; the finite checks below supplement those proofs.

SOURCE COMPILATION

The four LaTeX files are complete. Run tectonic main.tex, or run xelatex
main.tex twice with the standard AMS, newtx, microtype, geometry, booktabs,
needspace and hyperref packages available. A local rebuild is named main.pdf.

REPRODUCIBILITY

The geometry and certificate postprocessors use Python 3.10 or newer and only
the Python standard library. From the reproducibility directory, run:

    python check_geometry.py
    python check_certificates.py --out regenerated_comparisons.json

The first command checks 9,880 triples with frequencies at most 40, compares
223 operators against an independent coordinate metric-jet calculation,
and checks the sharp-range identities, sparse examples, threshold witnesses
and two four-mode counterexamples. Finite enumeration is not the proof of
the general classification or of the sharp spacing constant.

The second command reads the supplied 10 continuum certificates (540 scalar
intervals) and independent limiting-metric enclosures. It regenerates the
centered whole-box comparisons and exact rational generalized relative-error
certificates. data/comparisons.json is the corresponding saved result. Decimal
endpoints are outward bounds; fractions are exact. Fields marked display are
ordinary floating-point approximations and must not be read as rigorous bounds.

For full interval recomputation install the pinned dependency:

    python -m pip install -r requirements.txt

Example (a center certificate at the smallest studied bandwidth):

    python interval_metric.py --h .000625 --point --N 16 --L 44 --M 20 --J 48 --Q 512 --out regenerated_point.json

Omit --point for the whole parameter box. The five h values are .01, .005,
.0025, .00125 and .000625. All supplied whole-box certificates use these same
controls. The center certificates at .01 and .005 use Q=640; the other centers
use Q=512. The portable producer retains heat-kernel cancellation before
interval evaluation. Its algebraically equivalent interval expression can
produce different, overlapping enclosures from earlier center evaluations;
bit-for-bit equality with every saved certificate is not promised.

The independent limiting-metric calculation uses exact Fourier integration
of a truncated reciprocal-density series and an explicit infinite tail:

    python poisson_metric.py --L 12 --out regenerated_poisson_box.json
    python poisson_metric.py --L 12 --point --out regenerated_poisson_point.json

The interval computation can take several minutes per certificate. Its
continuum justification is in the paper and cited predecessor; executing
postprocessing alone does not independently establish those analytic bounds.
The error-budget checks bound finite construction error, not the entire
finite-bandwidth bias relative to the limiting metric and not interval width.

DATA AND SCOPE

The density is 1+theta1*cos(2*pi*x)+theta2*cos(4*pi*x) on the unit circle,
with theta1 in [.019,.021] and theta2 in [.00475,.00525]. The center is
(.02,.005). Each metric certificate contains three symmetric matrix entries,
six parameter derivatives per entry and three tensors: corrected, uncorrected
and limiting. Derivatives are actual derivatives, not Taylor coefficients.
The certified comparisons apply to these five bandwidths only. Path-length
consequences concern curves staying in the specified family and box; no global
ambient transport-distance or industry performance claim is made.

The code and data are supplementary mathematical material. The general
proofs, not the number of computed examples, support the classification.
