Elastic Spacetime with Scale-Dependent Coupling (ESSC) v19: Low-Dimensional Conditional Structure of Closure in Galactic Systems

This work presents ESSC v19 (Elastic Spacetime with Scale-Dependent Coupling), a structural framework for understanding how closure-related observables are realized in galactic systems. We demonstrate that closure is not represented by a single deterministic value, but instead forms a statistical distribution across systems. Standard deterministic models—including linear proportionality, power-law relations, and logit-linear transformations—fail to reproduce the observed distribution, both at the level of point predictions and distributional properties such as quantiles and tails. To resolve this, we introduce a geometric baseline quantity, the closure fraction: f2π ≈ 2π H r / V which represents a minimal condition for maintaining closed trajectories under cosmological expansion. We further define a structural factor: S = f_true / f2π and show that closure can be decomposed into a geometric baseline and a multiplicative structural component. The distribution of S is well described by a lognormal form, indicating multiplicative structure. A central result of this work is that the residual is not independent noise, but condition-dependent: ε ~ P(ε | f2π, R_norm) This leads to a generative model: f_true = f2π · exp( μ(f2π, R_norm) + ε ) which reproduces the full observed distribution, including central values, quantiles, and tail behavior. Importantly, we find that the residual structure is effectively low-dimensional. Conditioning on two variables (f2π, R_norm) captures the structure, while additional variables provide only marginal improvement. This indicates that closure is governed by a low-dimensional conditional structural field. The key conceptual result is: Closure is not deterministic, but a conditionally structured realization around a geometric baseline. Within ESSC, π encodes the ideal geometric condition for closure, while observable structures arise from its distributed realization in a low-dimensional structural space. This work does not introduce new forces or modify existing dynamical theories. Instead, it provides a structural interpretation of observed variability as a condition-dependent generative process.