RTLI Variance-Geometry Diagnostics Across the Hubble Sequence: Analysis of Early-Type and Late-Type Galaxy Structure

We apply the Relational Thermodynamic Layered Index (RTLI) variance–geometry framework to galaxy structure across the full Hubble sequence, using two complementary datasets: resolved surface brightness profiles for 12 early-type NGC galaxies and global structural parameters for 175 disk galaxies from the SPARC catalog. Structural organization is quantified via the eigenvalue ratio λₑ = σ₁²/σ₂², which measures the degree of variance concentration into a dominant mode. An empirically identified transition corridor, λₑ ∈ [√e, 7/4] ≈ [1.649, 1.750], marks the boundary between distributed and integrated variance regimes. Early-type galaxies exhibit extreme super-critical behavior (λₑ ≈ 17–113), reflecting single-mode dominance characteristic of dynamically relaxed, bulge-dominated systems. Late-type galaxies are overwhelmingly sub-critical (94.5% with λₑ < 1.5), indicating variance distributed across multiple structural components such as disks, spiral arms, gas, and star-forming regions. Only three galaxies in the entire SPARC sample occupy the corridor, all intermediate spirals (Sb–Sbc), suggesting that the corridor captures morphological transition states rather than stable endpoints. Gas fraction anti-correlates with λₑ (r = −0.17, p ≈ 0.055), linking variance integration to evolutionary state: gas-rich systems remain structurally distributed, while gas-poor systems exhibit integrated variance. Residual analysis after Sérsic profile subtraction reveals coherent substructure in early-type galaxies (bars, rings, halos), indicating that corridor transitions operate at sub-galactic scales rather than at the level of the global profile. These results demonstrate that RTLI variance–geometry diagnostics recover the classical Hubble sequence through purely statistical measures of structural organization, independent of visual morphology or parametric model assumptions. The framework provides a scale-aware, morphology-independent tool for diagnosing dynamical state, structural complexity, and evolutionary progression in galaxies.