The Partition Rectangle Diagnostic for Gravitational Coupling Regimes

This technical note introduces the Partition Rectangle Diagnostic, a geometric method for classifying gravitational coupling regimes using only the baryon fraction of a system. Starting from the Thales partition a=m_b, b=1-m_b, the construction generates a right triangle inscribed in a semicircle and an associated rectangle whose aspect ratio \mathcal{R}=\sqrt{(1-m_b)/m_b}. The key identity \mathcal{R}^2 = M_X/M_b shows that the squared aspect ratio is exactly the missing-to-observed baryon ratio, providing a geometric reinterpretation of the baryonic mass–halo mass relation discussed by McGaugh et al. (2026). The aspect ratio maps gravitational systems onto a continuous geometric trajectory ranging from baryon-deficient dwarf galaxies (\mathcal{R}\gg1) to equipartition at the horizon (\mathcal{R}=1). Along this trajectory the partition rectangle passes through several geometrically distinguished thresholds corresponding to specific right-triangle templates, including the photon sphere (\mathcal{R}=\sqrt{3}), the torus partition (\mathcal{R}=e/\varphi), and the EFE self-coupling fixed point (\mathcal{R}\approx1.466). These thresholds define a geometric classification of gravitational coupling depth with no free parameters. The framework is connected to the Contextual Diagonal Extraction (CDE) method, which predicts cross-channel correlation transitions at these geometric thresholds. Using SPARC galaxy data, the diagnostic shows that the rectangle aspect ratio organizes systems across the baryonic mass–halo mass ladder and provides a simple estimator for dark matter mass based on baryonic observables alone. Overall, the partition rectangle reframes the missing-baryon problem as a geometric shape problem: the dark-to-visible mass ratio corresponds directly to the squared aspect ratio of a partition rectangle, linking empirical scaling relations to a simple Thales construction.