A Structure–Randomness Transfer Theorem for Sparse Data — Preprint & Repro

This repository hosts the preprint and a minimal, reproducible code+data bundle that validates the core functions of the theorem. The structure-randomness transfer theorem unifies concepts from higher-order Fourier analysis, relative Szemerédi-type transference, and compressed sensing. The core idea of the theorem is that a single "pseudorandom majorant" can be used for two purposes: to count structured patterns within sparse data and to stably recover structured signals from a small number of random measurements. The theorem relies on two main components: · A pseudorandom majorant (v), which is a weighting function that acts like a uniform measure on low-complexity linear patterns. · An arithmetic regularity decomposition, which splits the data's indicator function (1A​) into a structured component (gstr​) and a uniform, or pseudorandom, component (gunf​). The theorem has two key parts: · Part (i): Polynomial Pattern Counts — It guarantees that the number of polynomial configurations of a certain degree within a set A matches what you would expect in a random model, with a small error. This extends the transference paradigm, which shows that theorems about dense sets can be applied to sparse, pseudorandom ones. · Part (ii): Stable Structured Recovery — It demonstrates that the same pseudorandom majorant can generate a small number of random measurements that approximately satisfy the Restricted Isometry Property (RIP). This property allows for the stable recovery of a structured signal (one that lives in the span of the structured component) using a convex program.

Quick reproduce

python3 -m venv srtt-env
source srtt-env/bin/activate
pip install --upgrade pip setuptools wheel
pip install -r env/requirements.txt
bash repro/run_repro.sh
# -> outputs/ with recovered.npy, residual.npy, coefficients.csv, srtt_report.html

Expected outcomes (tolerances in repro/params.json):

• Residual U² ≪ Recovered U²

• Residual spectral entropy ≫ recovered spectral entropy

• SNR-like improves when using custom_D.npy vs Fourier

• in your outputs file created by this repro you will find srtt_report.html It will have 3 graphs, the observed data, the structured signal captured, and the residuals in addition to metrics.

Contents

• paper/preprint.pdf — The unreviewed preprint (frozen artifact).
• docs/ — GitHub Pages landing with MathJax + Download PDF button.
• src/ — Minimal code to reproduce (repro_batch.py) and (optionally) run the app code.
• data-min/ — Minimal non-sensitive data to reproduce primary claims.
• repro/ — One-command runner + default params.
• env/ — Python requirements (pip/conda) and an optional Dockerfile.
• .github/workflows/ — CI for reproducibility and Pages deployment.
• CITATION.cff — How to cite this work.
• LICENSE — Code license (MIT). See paper/LICENSE for preprint (CC BY 4.0).

How to cite

See CITATION.cff. A DOI will be minted upon first GitHub Release via Zenodo (see RELEASE_CHECKLIST.md).

Security & data notes

• The sample datasets are small and non-sensitive. Do not add production logs directly to this repo.
• For larger/sensitive datasets, publish separately as Zenodo datasets and link from this README.

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Author: Kyle E. Litz, PhD
Version: v0.1-preprint (2025-09-20)

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Build status

Site: https://THEForgiven.github.io/srtt-preprint/
Repo: https://github.com/THEForgiven/srtt-preprint