Dimensional Stratification of Galaxy Mass Discrepancy: A Relational Thermodynamic Analysis

Latest diagnostic tool: De Jesus, Elias. (2026). Complementary Methods for Equilibrium Diagnostics: Geometric Ratio Analysis and Variance-Based Deviation Measures. Zenodo. https://doi.org/10.5281/zenodo.18406494 Researchers interested in applying or extending diagnostic methods of this type are welcome to reach out. Correspondence regarding potential diagnostic tools may be directed to dejesuselias10@gmail.com. We investigate the longstanding galaxy mass discrepancy using a geometric diagnostic framework within the Relational Thermodynamic Layered Index (RTLI) formalism. Rather than treating the discrepancy as a single scalar quantity, we test whether it exhibits dimensional stratification—systematically different characteristic ratios depending on the effective dimensionality of the observational probe. We introduce an $n$-dimensional geometric cost function, V_n(R) = A_n (R - e^{1/n})^2, where $R$ is the ratio of observed to baryonic dynamical tracers and $n$ encodes the probe dimensionality. Analysis of 175 SPARC rotation curves (3D volumetric probes) reveals a robust morphology-ordered threshold at $R \approx \sqrt[3]{e} \simeq 1.40$, with late-type and irregular galaxies preferentially approaching or exceeding this scale. This gradient persists across all tested stellar mass-to-light ratios and is statistically significant ($p < 10^{-11}$). Independently, SLACS strong gravitational lensing data (2D projected probes) show mass ratios clustering near $R \approx \sqrt{e} \simeq 1.65$, with a mean $M_{\rm total}/M_* = 1.45$. A cross-domain comparison rejects the hypothesis of a single universal discrepancy scale at overwhelming significance ($p < 10^{-21}$), demonstrating that lensing and kinematic probes yield systematically different characteristic ratios. We further identify a normalization-invariant barrier structure: the transition from $R = 1$ to $R = \sqrt[3]{e}$ contains approximately 94% of the integrated geometric cost, explaining why most galaxies occupy the high-cost regime below the volumetric threshold. The framework does not modify General Relativity or eliminate dark matter, but provides a falsifiable diagnostic predicting that identical galaxies probed by lensing and rotation will yield distinct mass ratios corresponding to their respective dimensional anchors.