Multimode Affinity Geometry for Finite Categorical Systems

Paper 6- Multimode Affinity Geometry for Finite Categorical Systems presents a mathematical framework for studying relationships between systems that have a finite number of possible outcomes. In many datasets, a relationship is summarized by only one number. However, a single number can hide important structure. This work represents the relationship as a small matrix of interaction modes, so different parts of the relationship can be studied separately. What this paper does The framework can: measure the overall strength and shape of an interaction; separate a complex interaction into different mathematical modes; show when a single-number summary is sufficient and when important information is lost; study how hidden or unobserved factors can change the observed interaction; identify which parts of an interaction remain measurable when some hidden effects are allowed; compare interaction patterns across different datasets using matrix and singular-value based quantities. The paper also gives exact mathematical results for combining interactions in parallel and in series, and provides a detailed treatment of the three-outcome (qutrit) case. Where can it be useful? This framework may be useful in problems involving: categorical and contingency-table data; multi-level statistical systems; qutrit and other finite-dimensional quantum data; hidden-variable or latent-factor analysis; comparison of interaction patterns across different experiments or platforms; deciding whether a complex relationship can safely be reduced to one scalar value. The methods are demonstrated using cross-platform qutrit-style datasets, including photonic, superconducting/transmon, and optical-clock examples. Important interpretation This is primarily a mathematical and statistical framework. An observed affinity matrix does not, by itself, identify the physical cause of an interaction or prove a particular hidden mechanism. The paper therefore separates what can be identified directly from the data from what requires additional physical assumptions. Reproducibility The calculations and reproducibility tools for this work are included in QDL Research Suite v1.6.0: Software DOI: 10.5281/zenodo.23032918 Paper DOI: 10.5281/zenodo.23034646