References and Literature Context

This bibliography supports the curated research report and its companion pages. Sources were checked on 18 October 2026. The descriptions identify why a reference is relevant; they do not claim an exhaustive priority search or an independent verification of every result in the cited literature. Attribution is provided through authors, titles, years, and publication identifiers, without outgoing website links. The cited external papers are not redistributed in this archive. Historical source notes retain their original form and are supplied as research records.

Chowla's problem and recent progress

JMTZ2025

Zhihan Jin, Aleksa Milojević, István Tomon, and Shengtong Zhang. From small eigenvalues to large cuts, and Chowla's cosine problem. arXiv:2509.03490, version 2, 27 October 2025.

The paper connects spectral graph methods to polynomial lower bounds for the negative minimum of cosine sums. It provides context for the gap between available polynomial bounds and the square-root target. The present report does not reprove its graph-theoretic results.

Bedert2026

Benjamin Bedert. Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260, version 3, 24 July 2026; first submitted in 2025.

The version checked reports a lower bound of order $N^{1/5-o(1)}$. Older cached summaries of the first version state a different exponent. The version number matters when describing the mathematical landscape.

Shankar2026

Abhishek Shankar. A Log-Free $n^{1/5}$ Bound for Chowla's Cosine Problem. arXiv:2609.05338, 4 September 2026.

This preprint reports removal of a subpolynomial loss from a related lower bound. It is cited as recent literature, with its preprint status retained. Its proof is not used as a verification certificate for any computation in this package.

BuEtAl2026

Bryan Bu, Fayzan Khan, Hyunjun Yi, Atharv Jha, Vihaan Vajpeyi, Ishaan Chandupatla, and Madhav Nagpal. A Half-Constant Upper Bound for Chowla's Cosine Problem. arXiv:2610.17160, 12 October 2026.

This very recent preprint concerns constructions giving upper bounds on the extremal function. It is included in the literature watch because construction upper bounds and universal lower bounds play different roles. No comparison in this report is presented as an independent validation of that preprint or a claim of priority over it.

Established optimization methods

Lasserre2021

Jean B. Lasserre. Connecting optimization with spectral analysis of tri-diagonal matrices. Mathematical Programming 190, 795–809 (2021), published online 8 August 2020. DOI: 10.1007/s10107-020-01549-3. Author preprint, arXiv:1907.09784.

This is the direct prior art for the univariate pushforward measure hierarchy used in the report. The general approximation mechanism, its eigenvalue interpretation, and basic convergence are established methods. The archive's polynomial quotient is a specialization of this framework, not a new hierarchy.

SlotLaurent2021

Lucas Slot and Monique Laurent. Near-optimal analysis of Lasserre's univariate measure-based bounds for multivariate polynomial optimization. Mathematical Programming 188, 443–460 (2021), published online 30 October 2020. DOI: 10.1007/s10107-020-01586-y. Author preprint, arXiv:2001.11289.

The paper studies convergence of the same pushforward approach using polynomial concentration and estimates for mass near a minimum. The present report's explicit degree bound is compared with an earlier bound from this project, not claimed to improve the best general result in this literature. A separate priority analysis would be needed for the specialization that is uniform in the largest frequency.

SlotLaurent2022

Lucas Slot and Monique Laurent. Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets. Mathematical Programming 193, 831–871 (2022), published online 25 January 2020. DOI: 10.1007/s10107-020-01468-3. Author preprint, arXiv:1905.08142.

Its Chebyshev and needle-polynomial estimates are relevant prior art for the filtering step. Journal issue dates, online publication dates, and preprint dates are different and should not be conflated.

NISTDLMF

NIST Digital Library of Mathematical Functions. Section 18.5, Explicit Representations; equation 18.5.1.

Used for the standard representation of Chebyshev polynomials, including $T_d(\cos\theta)=\cos(d\theta)$ and its continuation outside the unit interval.

PerlazaBisson2025

Samir M. Perlaza and Gaetan Bisson. Variations on the Expectation Due to Changes in the Probability Measure. Entropy 27(8), 865 (2025), published 14 August 2025. DOI: 10.3390/e27080865.

The paper connects changes of expectation to Gibbs measures, information projections, and relative-entropy identities. It supplies context for the observation-measure interpretation in Appendix L. The cosine-specific estimates are proved in that appendix; the underlying Gibbs framework is established prior work.

Physical interpretation

BakerAcoustics

Richard Baker, University of Manchester. Basic Acoustics, Chapter 3: Sound Intensity and Pressure. University teaching resource.

Supports the distinction between acoustic pressure and intensity, with the impedance relation under the plane progressive-wave assumptions. The report's ideal cosine model does not substitute for a measured acoustic system.

TongAQM

David Tong, University of Cambridge. Lectures on Applications of Quantum Mechanics, Chapter 3: Electrons in Solids.

Context for tight-binding operators, dispersion relations, and spectral bands on a lattice. The Fourier identities in the appendix are proved directly.

Bardyn2014

Charles-Edouard Bardyn, Sebastian D. Huber, and Oded Zilberberg. Seeing bulk topological properties of band insulators in small photonic lattices. New Journal of Physics 16, 123013 (2014). DOI: 10.1088/1367-2630/16/12/123013. Author preprint, arXiv:1312.6894.

An example of established use of twisted boundary conditions. The report does not claim to invent this spectral technique or to obtain a topological result about audio.

Cappellaro2012

Paola Cappellaro. 22.51 Quantum Theory of Radiation Interactions, Complete Course Notes. MIT OpenCourseWare, Fall 2012; especially the harmonic-oscillator, coherent-state, and phonon sections. Official course notes.

Provides the physical vocabulary for number states, coherent states, and phonons. The report keeps mean displacement, variance, and number-operator eigenvalues distinct. Number states with nonzero occupation should not all be described as minimum-uncertainty states.

Shapiro6453

Jeffrey H. Shapiro. 6.453 Quantum Optical Communication. MIT OpenCourseWare, Fall 2016. Selected official readings:

• Reading 4: quantum harmonic oscillator and number-state formalism.
• Reading 5: coherent-state properties.
• Reading 6: quadrature noise and squeezed states.

These readings support the physical distinctions used in the companion essay. A classical Fourier coefficient is not forced to be an integer by the number spectrum of a quantum oscillator.

OConnell2010

A. D. O'Connell and coauthors. Quantum ground state and single-phonon control of a mechanical resonator. Nature 464, 697–703 (2010). DOI: 10.1038/nature08967. Research-group-hosted paper and supplementary material.

An experimental reference establishing the physical context of single mechanical quanta. It supplies no direct theorem about the unit-amplitude cosine sums studied here.

Formula rendering for the website

KaTeXDocs

KaTeX official documentation: Node.js rendering and rendering options. The implementation handoff recommends build-time rendering with output: "htmlAndMathml", local CSS and fonts, and a failed build for unsupported formulas rather than silent raw-TeX output.

MathJaxDocs

MathJax official documentation: What is MathJax?. It is an alternative if the existing site already uses MathJax or an expression cannot be handled by the selected KaTeX pipeline. Running both renderers on the same expressions is unnecessary.

Scope of the literature comparison

The principal attribution correction is explicit: the moment hierarchy is Lasserre's, and the Chebyshev concentration mechanism has established precedents. The appendix's frequency-independent sufficient degree is supplied with a full proof, but no claim of scientific priority. This package reports its own verification status separately from the publication status of external sources. A future journal submission would require an expert assessment of novelty, positioning, and the proofs.