CESD-X IV: The Directional Framework \u2014 Galaxy Clusters as Thermodynamic Distributions, and a Geometry-plus-Severity Classifier

This release contains the manuscript, figure set, directional-analysis data products, and supporting materials for CESD-X IV: The Directional Framework. Building on the CESD-X Foundation Paper and Papers II-III, this work extends the (TSI, ACI) state space of Paper I from a single-point representation to a directional framework in which a galaxy cluster is represented as a thermodynamic distribution organized by three complementary, separately measurable observables: Geometry (a blind image-space axis finder and its radial coherence discriminant), Severity (a feature-radius-free, geometry-free density-profile slope), and Architecture (the distribution of per-sector structural complexity). Applied to four clusters spanning the dynamical regimes (A2256, A2142, A1795, Perseus) from archival Chandra event data, the architecture moments are found not to separate the regimes once a centering systematic intrinsic to multi-component mergers is included (0 of 6 pairs), while geometry plus severity separate all six demonstrated regime pairs and survive the same systematic. A geometry-free severity scalar reproduces the per-sector ordering (Spearman rank correlation of unity) and, under bootstrap and centering perturbation, preserves that ordering in 100 percent of statistical draws and 90 percent of combined draws. The directional severity and per-sector architecture quantities are explicit proxies for the Paper I four-term and state-sequence indices; the four-term severity index could not be computed per sector under spectral projection, and this is recorded. The results are a demonstration on four clusters (one per regime): the claim is that the three legs can be separated and severity can be made geometry-free, not that they always separate across the population, which is reserved for subsequent work. This work extends, and does not revise, Papers I-III. The companion data package includes the axis-finder and coherence outputs, per-sector index tables, moment-convergence and centering-systematic results, geometry-free severity values, and the ordering-robustness draws underlying Section 7.3.