Thales Arc-Length as a Geometric Coordinate for Quantum Complementarity: Experimental Evidence from Photon Interferometry

We present a geometric reparameterization of quantum complementarity data using the arc-length of a Thales semicircle as a natural coordinate on the constraint manifold defined by the Englert relation V^2 + D^2 \leq 1. Applying this transformation to atom-interferometry measurements from Schwindt et al. (1999), which span the full range of wave–particle complementarity, reveals a previously hidden geometric structure: the corridor boundaries at amplitude ratios \lambda = 1/e and \lambda = e are conjugate, occurring at identical visibility, distinguishability, and equal arc-length displacement from the symmetric apex. While the standard V(D) representation saturates the Englert bound and shows no distinguished interior structure, the arc-length representation V(\Delta s) exposes a symmetric transition region corresponding to an intermediate compatibility class previously identified through principal component analysis (PCA) diagnostics in two-channel systems. The observed correspondence does not imply deviations from quantum mechanics or the introduction of new constants; rather, it provides a geometric organization of existing experimental results and clarifies why specific partition ratios require additional physical constraints beyond generic complementarity. This work positions Thales arc-length as a diagnostic coordinate that unifies experimental complementarity data with constraint-geometry analyses, suggesting a systematic framework for comparing diverse interferometric systems without modifying established theory.