Dimensional Stratification of Galaxy Mass Discrepancy

This revised manuscript presents a conservative probe-coordinate diagnostic for galaxy mass-discrepancy measurements, replacing the earlier claim of empirically supported (e^{1/n}) dimensional anchors with a stricter distinction between algebraic markers, aperture geometry, and mass-ratio measurements. The central thesis is that rotation curves and strong lensing should not automatically be collapsed into a single scalar discrepancy ratio, because they probe different observational coordinates of the gravitational field. Rotation curves measure volumetric dynamical response through velocity, radius, and enclosed mass, while strong lensing measures projected aperture and surface-density geometry. The framework therefore treats (e^{1/n}) values as candidate diagnostic coordinates, not as derived force laws or physical constants. A revised audit of the SLACS strong-lens sample shows that (\sqrt e=e^{1/2}) is not supported as a projected mass-discrepancy anchor. Aperture-matched projected mass ratios in the available posterior-predictive model lie near (2.4)–(2.7), far from (\sqrt e). The directly observable aperture ratio (R_{\rm eff}/R_{\rm Ein}) has a median near (1.7), but this is a broad, sample-conditioned aperture-geometry feature rather than a data-selected (\sqrt e) constant. Nearby values such as (1.70), (7/4), and (e/\phi) describe the central aperture ratio at least as well as (\sqrt e). The SPARC rotation-curve result is also reframed conservatively. The morphology gradient remains the stronger empirical feature: later-type and irregular galaxies show systematically larger velocity-level discrepancy ratios across stellar mass-to-light assumptions. However, the relation to (e^{1/3}) is treated as exploratory rather than as evidence for a physically selected volumetric threshold. The revised manuscript therefore retains the useful diagnostic principle—mass-discrepancy ratios are probe-dependent and coordinate-sensitive—while retiring the stronger claim that (\sqrt e) or (e^{1/3}) are established empirical anchors. The Thales construction and (e^{1/n}) hierarchy are kept as algebraic reference markers only. No modification of general relativity is proposed, and no claim is made that dark matter is eliminated. The paper’s strongest falsifiable prediction is that future matched samples containing both strong lensing and resolved kinematics should continue to show probe-dependent ratio structure, but not necessarily at the specific (e^{1/n}) markers.