Text Has Curvature
TL;DR
This work argues that text does indeed have curvature, and shows how to detect it, define it, and use it, and establishes a text-native curvature paradigm, making Texture practically useful.
Abstract
Does natural language text have an intrinsic curvature? Language is increasingly modeled in curved geometries - hyperbolic spaces for hierarchy, mixed-curvature manifolds for compositional structure - yet a basic scientific question remains unresolved: what does curvature mean for text itself, in a way that is native to language rather than an artifact of the embedding space we choose? We argue that text does indeed have curvature, and show how to detect it, define it, and use it. To this end, we propose Texture, a text-native, word-level discrete curvature signal, and make three contributions. (a) Existence: We provide empirical and theoretical certificates that semantic inference in natural corpora is non-flat. (b) Definition: We define Texture as a signed two-axis curvature of the word-in-context belief field - the differential of reconciliation between prefix and suffix - measuring, via a debiased Schr\"odinger transport divergence, whether adding context from one side contracts the semantic effect of context from the other side (focus, positive) or expands it into competing continuations (fan-out, negative). (c) Utility: Texture is actionable: it serves as a general-purpose measurement and control primitive enabling geometry without geometric training; we instantiate it on two representative tasks, improving long-context inference through curvature-guided compression and retrieval-augmented generation through curvature-guided routing. Together, our results establish a text-native curvature paradigm, making Texture practically useful.