Spectral Analysis and Low-Rank Structural Transitions Emergent from Dimensionless Causal Metric Networks under SRE Dynamics

This study investigates a coordinates-free relational matrix framework governed by Status-Relational Entropy Dynamics. Operating under the protocol of the SRE User Guide, we simulate a zero-dark-energy cosmological boundary coupled with an evolving variable speed of light parameterized by the global baud rate ceiling defined by Planck metrics. We demonstrate that the discrete progression of a one-dimensional sequence breaks the low-rank degeneracy of the inner-product feature space. By implementing a mathematically rigorous double centering operator, we isolate authentic structural dimensions from macroscopic metric expansion artifacts. The computational model is evaluated via an empirical pipeline driven by un-scalized spectroscopic dataset slices of three hundred galaxies from the Sloan Digital Sky Survey. The empirical results reveal a structural dichotomy: while the intrinsic algebraic network rank exhibits an abrupt discrete leap from two to four near the early epoch and subsequently bifurcates into a five-dimensional subspace at late times due to dense large-scale galactic anchors, the macroscopic emergent dimension remains perfectly locked at a four-dimensional dead-lock via a strict holographic observational projection matrix. Concurrently, tracking the first-order discrete difference of the maximum eigenvalue maps a discrete multi-shock spectrum, revealing isolated twin phase mutation peaks matching the geometric decoupling thresholds.