Paper 16 - A Reproducible Walsh–Torus Heat Map and Its Limits for Navier–Stokes Dynamics
Year: 2026
Type: preprint
Abstract: 1. About This Research This paper explores a mathematical connection between Walsh functions and Fourier functions. Walsh functions help us study patterns in binary data, where each value can have two possible states. Fourier functions help us describe waves, periodic patterns, and changes over time. Our goal is to explain how these two mathematical systems can be connected, where the connection works, and where its limits appear. 2. What We Have Demonstrated We present a clear mathematical method for representing a function from a Boolean cube in a periodic mathematical space called a torus. The method has three important properties: It preserves the original values at the selected Boolean points. It preserves the squared mathematical norm. It preserves the corresponding linear heat evolution. These properties are supported by mathematical proofs and reproducible calculations. 3. Understanding the Heat Connection The heat equation describes how a quantity spreads or changes smoothly over time. In this paper, we show how a selected group of Walsh functions can follow the same linear heat evolution as corresponding Fourier functions. We also explain the exact condition under which a finite collection of Fourier waves can share a consistent heat rule when restricted to Boolean points. Simple mathematical examples help readers understand this condition. 4. Connection with Navier–Stokes Equations The Navier–Stokes equations are important in mathematics and fluid mechanics. They describe how fluids such as water and air move. These equations contain both linear and nonlinear mathematical operations. Our research explains why a connection that works for linear heat evolution does not automatically work for nonlinear fluid motion. We provide a three-dimensional mathematical example in which a nonlinear operation creates a new frequency outside the original selected range. This helps identify the limits of the mathematical connection. 5. Important Mathematical Findings The paper presents an explicit real-valued mathematical map, a condition for consistent heat evolution, and examples that explain the map's limitations. The calculations show that preserving values and squared norms does not automatically mean preserving multiplication or nonnegative values. These results help distinguish a useful mathematical representation from a complete physical model. 6. Reproducible Calculations and Data To make the work easier to check, we provide Python code and supporting calculation files. The numerical checks include small Boolean systems and examples of heat evolution. We also include an illustration based on eight counts derived from an earlier QDL analysis of a published IBM quantum-error-correction dataset. This illustration checks mathematical formulas using data-derived numbers. It is not an experiment involving real fluid motion. 7. Why This Research May Be Useful This work brings together mathematical ideas from Boolean Fourier analysis, heat equations, and spectral methods used in fluid dynamics. It provides a clear starting point for readers who want to understand how a finite mathematical representation can be connected to a continuous system. The proofs, examples, and supporting code allow interested readers to examine and reproduce the calculations. The underlying Walsh, Fourier, and Navier–Stokes methods are established mathematics. This paper presents a focused comparison of these methods within the Quantum Diverter Loop research program. 8. Research Scope and Limitations This paper is a mathematical study. It does not claim to solve the three-dimensional Navier–Stokes existence and smoothness problem. It does not establish a new physical law or demonstrate that Quantum Diverter Loop mathematics fully describes fluid dynamics. The originality of the combined mathematical presentation has not yet been independently established. The conclusions are limited to the specific mathematical identities, conditions, and examples presented in the paper. 9. Supporting Files and Reproducibility Two files accompany this publication. Main Research Paper: A PDF containing the mathematical definitions, proofs, examples, conclusions, and references. Supplementary Reproducibility Package: A ZIP file containing the LaTeX source, Python verification code, calculation results, derived numerical data, supporting documentation, and file-integrity checksums. The original third-party experimental data are credited to their source and are not redistributed in this package. 10. About the Research Program This study is part of the independent Quantum Diverter Loop (QDL) Research Program. It builds on earlier QDL mathematical investigations involving Walsh representations, interaction coordinates, and the comparison of mathematical structures. The purpose of this publication is to make the relevant mathematical results easier to understand, review, and reproduce. Author: Roshankumar Chandaliya Research Program: Quantum Diverter Loop (QDL) Publication Type: Research Preprint Version: 1.0.0
Author: chandaliya, Roshankumar
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224085449