The Thales Lagrangian as a Regime-Aware Predictive Instrument: Incircle Hit Rate, Feuerbach Gap Sensitivity, and Mass-Stratified Performance Across 107 Disk Galaxies

We report the first dynamical extension of the Thales partition framework from static geometric diagnostics to predictive integration. For any conserved two-channel system (a + b = 1), the framework assigns four dimensionless diagnostics: resistance, stability bandwidth (incircle radius), coupling, and active coupling. A Thales Lagrangian — whose kinetic term is the Poincaré metric induced by the semicircle embedding and whose effective potential is calibrated from 27 SPARC galaxies — is integrated outward from the central baryon fraction to predict rotational velocities at four disk scale lengths for 107 held-out galaxies. Three results emerge. First, 85.0% of predictions fall within the incircle of the observed partition, matching the 83.7% natural incircle occupancy of the SPARC sample. The predictive success rate equals the geometric stability rate. Second, the Feuerbach gap (distance from incircle to nine-point circle) correlates with absolute prediction error at rs = +0.415 (p = 8.7 × 10⁻⁶), establishing it as a forecastability parameter: wide-gap systems are harder to predict, narrow-gap systems are easier. Third, prediction accuracy is strongly mass-dependent: 0.6% RMS for log Mb > 10.5, 6.4% for 9.5–10.5, and 12% for 8–9.5, spanning two orders of magnitude as predicted by the diminishing-returns property of the incircle–mean identity r_in = h/(1 + √(1+2h)). Low-mass failures are interpreted as steep-regime amplification of calibration errors, not hidden. The framework is presented as a bounded, scale-invariant diagnostic overlay whose error structure is controlled by its own stability architecture. No modification of established dynamics is proposed.