Text Has Curvature

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 Schrödinger 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.

When we read, the words before and after a gap both shape what we expect to fill it. Sometimes the two sides agree and settle on a single clear meaning; other times they pull apart, leaving several plausible readings. This paper asks whether that interplay gives language a kind of hidden "shape" — much as a landscape can be flat or curved — and whether we can measure it directly from text itself instead of assuming it in advance. We introduce Texture, a tool that reads this shape at every word using an ordinary, already-trained language model, with no retraining required. At each position it outputs a single signed number: positive where the surrounding context converges on one meaning (we call this "focus"), and negative where it branches into competing alternatives ("fan-out"). Using careful statistical tests, we first show that everyday text really is "curved" in this sense — and that scrambling the words makes the effect disappear, confirming it reflects genuine meaning rather than a quirk of our method. Because the signal is cheap to compute, it is also practical. We use it to help large language models handle long documents: it picks which parts of a long input are most worth keeping when space is limited, and flags when the model should pause to look up extra information. On standard benchmarks this improves answer quality while using far less context — pointing toward more efficient and focused AI systems.