Inflection Geometry of Partition Dynamics: A Coordinate-Invariant Diagnostic Applied to the Radial Acceleration Relation

This work develops a coordinate-invariant diagnostic for two-component partition dynamics and applies it to the radial acceleration relation (RAR) in disk galaxies. Starting from general dynamics of the form da/du = f(a) under the observational mapping u = \sqrt{\bar{g}/g_\dagger}, \lambda = \log_{10}\bar{g}, we derive an exact factored expression for the second derivative d^2a/d\lambda^2. We show that the curvature inflection condition reduces to a fully coordinate-independent equation: f'(a^*) \int_0^{a^*} \frac{ds}{f(s)} = -1, in which all observational parameters cancel identically. The inflection location is therefore determined solely by the intrinsic dynamical structure of f(a), not by coordinate choice. For the power-law family f(a) = (1-a)^n, the inflection admits the closed-form solution: a^*(n) = 1 - n^{-1/(n-1)}, which is continuous, strictly monotonically decreasing, and surjective from (0,\infty) onto (0,1). This establishes that no logarithmic coordinate mapping of the form du/d\lambda \propto u imposes a geometric ceiling on the inflection location. The exponent–inflection mapping provides a structural classification of partition dynamics, distinguishing logistic, MOND-like, sub-MOND, and super-MOND classes without reference to specific gravitational field theories. The framework is presented as a model-independent structural diagnostic rather than a proposal of new fundamental physics.