Geometric Threshold Structure in Gravitational Systems: A Relational Framework for Coupling Cost

We present a unified information-geometric analysis of galactic dynamics, gravitational lensing, Type Ia supernova light curves, and cosmological expansion within the Relational Thermodynamic Layered Index (RTLI) framework, supported by a methodological approach referred to as Heuristic Calculus (HC). RTLI characterizes physical systems through layered coherence regimes governed by dimensionless thresholds, rather than through model-specific dynamical assumptions. Using dimensionless observables, we identify a sequence of thermodynamic-coherence thresholds associated with structural emergence across astrophysical systems. Analysis of galaxy rotation curves from the SPARC maximum-disk dataset reveals a statistically significant, morphology-ordered gradient in the velocity ratio V_{\mathrm{obs}}/V_{\mathrm{bar}}, transitioning from unity in early-type galaxies to a characteristic scale e^{1/3} \approx 1.396 in irregular systems. This scale is shown to be reciprocal to the e^{-1/3} decay factor independently observed in Type Ia supernova energy release, indicating a shared three-dimensional volumetric coupling geometry rather than a direct dynamical interaction. We further demonstrate that gravitational lensing efficiency and cosmological expansion can be described using a single quadratic geometric cost function centered at the fixed point \sqrt{e}, allowing partial reconciliation of local and global measurements of the Hubble constant without modifying General Relativity or introducing new degrees of freedom. Within a variance-geometric (Fisher-information) formulation, electromagnetic coupling and cosmological expansion occupy orthogonal parameter directions in the broken-symmetry regime, with their shared appearance of a threshold near \eta \approx 1.7 arising from projection onto a common organizational attractor rather than direct physical coupling. A key result is the derivation of a coupling cost \delta = (\sqrt{e} - \varphi)\, e \approx 0.083, which quantitatively accounts for the offset between symmetric (\eta = 1) and matter-dominated (\eta \approx 1.7) coherence regimes. This cost follows from the near-coincidence of fundamental mathematical constants and is not fitted to observational data. The RTLI framework yields multiple falsifiable predictions across galaxy morphology, lensing efficiency, supernova energetics, and null correlations between electromagnetic and cosmological parameters. These results suggest that several long-standing astrophysical tensions may reflect underlying thermodynamic-coherence organization rather than missing components or modified gravitational dynamics.