Four-Point Thales Regression and Structural Arc Length: A Geometric Diagnostic Method with Application to CMB Data

This paper introduces a purely geometric diagnostic framework for analyzing trajectories in normalized binary partition systems. The framework consists of two complementary tools: Four-Point Thales Regression, which samples the mediator variable h=\sqrt{ab} at four structurally selected points and fits a cubic polynomial to characterize trajectory shape; and Structural Arc Length Analysis, which defines a dimensionless coordinate measuring accumulated geometric change along the constraint manifold.

The methods are diagnostic rather than dynamical: they describe trajectory geometry without invoking causal mechanisms or replacing domain-specific physical models. Structural markers—including apex points, efficiency crossings at \chi = 1/e, and response-ratio thresholds R = (e+1)/(e-1)—are identified directly from geometry and used to define intrinsic structural periods.

Applied as a proof of concept to the Planck 2018 CMB TT power spectrum, the trajectory spans nearly the full mediation capacity of the constraint manifold. Distinguished geometric points are found to coincide within sub-multipole precision on both rising and falling phases, and normalized structural arc length exhibits a strong correlation with multipole progression (r = 0.995). Four-point cubic regression robustly yields negative quadratic curvature across multiple sampling strategies, indicating a concave elevation–valley trajectory, with interpolation errors at the \sim 2\% level.

The framework provides a model-independent geometric scaffold applicable to any system governed by conserved binary partitions, offering a compact way to characterize structural organization, symmetry, and progression without asserting physical causation.