Contextual Scalar Partitioning and Coherence in Thales Geometry

This paper presents a proof-of-concept framework for contextual scalar partitioning in two-dimensional Thales geometry, with explicit emphasis on negative results and observability limits. Four scalar partitions are defined—vertex-local (central angle), edge-local (leg ratio), dual (diagonal duality invariant), and coherence (constraint deviation)—and their informational roles are analyzed both analytically and computationally.

For exact Thales configurations with isotropic measurement noise, we prove and verify that the three shape-encoding partitions are perfectly redundant, encoding the same one-dimensional shape degree of freedom. Aggregation of these scalars yields no improvement in estimation accuracy; optimal combination weights are equal, and RMSE gains are effectively zero. This establishes a principled limit: scalar aggregation cannot exceed the information content of a single injective observable in a one-dimensional shape space.

The framework’s primary contribution lies elsewhere. The coherence scalar provides genuinely new information, measuring displacement normal to the constraint manifold rather than position along it. This enables reliable detection and signed diagnosis of constraint violations. When extended to time-dependent trajectories, the framework reveals temporal instantiation: geometric structure concentrates at specific constraint-crossing moments, while time-averaging degrades observability. Phase alignment sharpens inference; averaging obscures it.

We emphasize that contextual scalar partitioning is an observability and diagnostic framework, not an optimization method. Its value lies in clarifying dimensional limits, diagnosing noise anisotropy, revealing ensemble structure, and distinguishing shape information from constraint fidelity. The Thales setting serves as a controlled testbed, establishing baseline expectations for applying similar ideas to higher-dimensional geometric or physical systems.