From Hypothesis to Evidence: An Open Research Journey in Computational Electrostatics, Discrete Exterior Calculus, and Spectral Geometry

This repository documents an ongoing independent research project at the intersection of Computational Electrostatics, Discrete Exterior Calculus (DEC), Spectral Geometry, Discrete Morse Theory, and Computational Topology. Unlike conventional research repositories that present only final results, this archive intentionally preserves the complete research process, including the original hypotheses, mathematical developments, software implementations, numerical experiments, literature reviews, critical revisions, corrected mistakes, abandoned conjectures, and the evolution of the research questions themselves. The project began with the objective of exploring whether new relationships could be established between the spectrum of the Hodge Laplacian and the critical-point structure of electrostatic potential functions on simplicial complexes. As the investigation progressed, extensive implementation, numerical validation, and comparison with the scientific literature revealed that several initially promising ideas had already been established in different mathematical settings, while others required substantial revision or rejection. Rather than concealing these findings, the repository documents them openly as part of a transparent scientific process. The repository includes, among other components: Discrete Exterior Calculus (DEC) implementations. Hodge Laplacian operators (L₀ and L₁). Simplicial mesh generators. Spectral analysis tools. Banchoff discrete Morse critical-point classification. Whitney reconstruction experiments. Numerical validation and reproducibility tools. Statistical analyses. Literature reviews and comparative studies. Experimental reports. Source code from multiple development stages. Documentation of corrections, redesigns, and methodological decisions. Artificial intelligence systems were extensively used throughout the project as computational assistants for programming, debugging, mathematical discussion, literature exploration, documentation, and critical review. Nevertheless, all research decisions, interpretations, validation procedures, and responsibility for the published material remain entirely those of the author. The repository is published in the spirit of Open Science to facilitate: independent verification, reproducibility, constructive criticism, collaboration, and future research. This work has not undergone peer review. Nothing contained in this repository should be interpreted as an established scientific result, new mathematical theorem, or validated physical theory unless independently verified. Many documents reflect intermediate stages of the research and are intentionally preserved to provide a complete record of the scientific process. The current outcome of the project is best understood as the development of a reproducible computational research framework upon which future mathematical, computational, and physical investigations may be built. The research remains active, and feedback from mathematicians, physicists, computational scientists, engineers, and the broader scientific community is sincerely welcomed.