Phase transitions and effective curvature bounds in Fourier density models
Scott Schubr, October 22, 2026

CONTENTS

The PDF contains the complete paper and proofs. LaTeX sources are in manuscript/.
The reproducibility directory contains exact-rational coordinate curvature,
four-frequency maximum root isolation, and full nonuniform density-box certificates.
This is a follow-up to Frequency interactions and curvature in Fourier density
models (Luminary Research Archive, edition 1, 2026), cited in the paper.

REPRODUCTION

Python 3.10 or later is sufficient; all Python code uses the standard library.
From the package root run:

  python -B reproducibility/check_results.py --output reproduced_results.json

This independently reconstructs 75 harmonic phase matrices, 1170 nonharmonic
phase matrices and 72 four-frequency matrices from metric two-jets. It also
checks exact rational maximum enclosures and freshly reproduces every one of
the 120 saved metric-derivative enclosures and both complete-box conclusions.
These finite checks supplement the general analytic proofs.

To regenerate the saved box certificate:

  python -B reproducibility/density_box.py --radius 1/100 --degree 12 --output regenerated_box.json

The results array equals reproducibility/data/density_box.json exactly. The
certificate contains full reciprocal-series infinite tails and outward decimal
enclosures computed by exact integer/fraction arithmetic. The manuscript supplies the analytic box proof.
The source_sha256 field identifies density_box.py. Curvature conventions and
the conversion from the scaled metric to physical curvature are in the paper.

To enclose the maximum at a chosen rational ratio n/m > 5:

  python -B reproducibility/four_mode.py --ratio 10 --bits 100

The two fractions bound K_max/(4*pi^2*m^2). Root-equation signs are evaluated
with exact rational comparisons, without floating square roots. Ratios at or
above the saturation threshold return the exact value -9/2. The model itself
uses distinct positive integer frequencies; a rational ratio is realized by
choosing positive integers n,m with n>5m.

BUILDING THE PAPER

With a standard LaTeX installation, run pdflatex twice on manuscript/main.tex
from that directory. Alternatively use Tectonic on manuscript/main.tex.
The PDF uses standard AMS mathematics, New TX fonts, geometry, microtype,
booktabs and hyperref packages. No Python package is needed to build the paper.

SCOPE

The curvature is intrinsic to the finite-dimensional density model with the
periodic Wasserstein metric. The phase classification is at uniform density,
with one fixed phase per distinct frequency. The four-frequency maximum is
for aligned phases at uniform density. Nonuniform certificates concern the
complete closed coefficient cube [-1/100,1/100]^3 for the aligned spectra
{m,2m,3m} and {m,2m,4m}, with curvature scaling by m^2.


The paper credits the established Gauss, four-form and interval methods.

SHA256SUMS.txt lists hashes of all other package members.
