Interpretability and Representability of Commutative Algebra, Algebraic Topology, and Topological Spectral Theory for Real‐World Data

While recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis via persistent homology (PH) that provides explainable AI by extracting multiscale structural features from complex datasets. Interpretability is crucial for world models, the new frontier in AI that can understand and simulate reality. This article investigates the interpretability and representability of three foundational mathematical AI methods, PH, persistent Laplacians (PL) derived from topological spectral theory, and persistent commutative algebra (PCA) rooted in Stanley-Reisner theory. We apply these methods to a set of data, including geometric shapes, synthetic complexes, fullerene structures, and biomolecular systems to examine their geometric, topological, and algebraic properties. PH captures topological invariants such as connected components, loops, and voids through persistence barcodes. PL extends PH by incorporating spectral information, quantifying topological invariants, geometric stiffness, and connectivity via harmonic and nonharmonic spectra. PCA introduces algebraic invariants such as graded Betti numbers, facet persistence, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi> <mml:mo>/</mml:mo> <mml:mi>h</mml:mi></mml:math> -vectors, offering combinatorial, topological, geometric, and algebraic perspectives on data over scales. Comparative analysis reveals that while PH offers computational efficiency and intuitive visualization, PL provides enhanced geometric sensitivity, and PCA delivers rich algebraic interpretability. Together, these methods form a hierarchy of mathematical representations, enabling explainable and generalizable AI for real-world data.