The Thales Variance Construction: A Geometric Diagnostic for Multi-Channel Variance Decomposition

Related work: De Jesus, Elias. (2026). The Thales Sphere Construction as a Cross-Domain Geometric Diagnostic for Quantum and Gravitational Variance Partitions. Zenodo. https://doi.org/10.5281/zenodo.18474674 This paper introduces the Thales variance construction, a purely geometric and model-agnostic diagnostic framework for analyzing how total variance is partitioned among two or more constrained channels. The method maps normalized channel amplitudes onto classical Thales geometries: a semicircle for two channels and the positive octant of a sphere for three channels. Within this construction, bounded geometric coordinates naturally emerge—arc-length measuring dominance balance and altitude measuring coupling efficiency—allowing regime boundaries, saturation limits, and identifiability constraints to be identified directly from data geometry rather than inferred from model fits. The framework is formulated as a coordinate system on the variance simplex and is explicitly diagnostic rather than dynamical. It neither introduces new physical parameters nor modifies existing theoretical models. Instead, it provides a unified geometric language that subsumes common scalar diagnostics such as ratios and principal-component eigenvalues, while retaining information about how variance partitions evolve as a function of a control parameter. The paper presents the two-channel and three-channel constructions, establishes their hierarchical nesting, and clarifies what information the method can and cannot extract. Applications are discussed across multiple domains—including cosmic microwave background power spectra, gravitational lensing configurations, and quantum-optical complementarity—demonstrating that the same geometric structure underlies diverse physical systems whenever observables share a constrained variance budget. The Thales variance construction thus serves as a general-purpose geometric ruler for multi-channel data analysis.