Ordering and Quantifying Textual Cohesion via Semantic, Geometric and Statistical Structure

We propose a semantic, geometric, and statistical framework for quantifying and ordering textual cohesion in long-form discourse. Sentences are embedded into a semantic similarity graph and Ollivier–Ricci curvature is used to extract sentence- and document-level structural profiles, represented as step functions on a normalized rhetorical-time axis. On this functional space we define the Weighted Utopia Index (wUI), a corpus-relative measure of weighted shortfall from an upper-envelope profile under a dominance-type ordering. The rhetorical-time weighting function is learned self-supervised: we generate controlled sentence-order perturbations with known ordinal coherence degradation and estimate the weight parameters via an ordered probit model on a training split. We evaluate ordering recovery on held-out State of the Union speeches using rank correlations, pairwise and adjacent ordering accuracy, and violation-localization diagnostics with bootstrap uncertainty. Across these criteria, wUI systematically outperforms embedding-only adjacent-similarity baselines, while a Nash-type aggregation provides an interpretable semantic–structural trade-off score. An application to later-period speeches illustrates how the method yields interpretable cohesion rankings and curvature-profile diagnostics without requiring external annotations.