Measurable Social Geometry

Methodological preprint. Self-contained: takes no position on the substantive social theses that have motivated geometric language, and depends on no prior work. Includes exact computation with published code. Claims that social or cognitive space is "curved" — that some transitions are downhill and others uphill, that atomisation is a geometric property — are common, intuitive, and almost always unfalsifiable as stated. This paper does not argue for such claims; it specifies what would make them measurable, in four steps. (1) A constraint most such models violate. A Riemannian metric induces a symmetric distance by an elementary argument (reversing the path parameter leaves the arc-length integrand invariant, since the metric is a quadratic form). Therefore no Riemannian metric can represent an asymmetric transition cost — cheap descent, expensive ascent — which is precisely the phenomenon these models exist to express. Three well-posed alternatives are given (Finsler metric; metric plus drift field; gradient flow), with the metric-plus-drift form recommended because it preserves the standard Riemannian toolkit for the symmetric part, separates two claims that informal models conflate, and yields a drift field estimable from panel data. (2) A derived rather than posited metric. If population states are probability distributions over measured behaviours, the Fisher information matrix supplies a metric that is not chosen by the modeller, is unique up to scale among metrics invariant under sufficient statistics, and is estimable from data. A worked closed-form case shows the substantive consequence: an equal eight-percentage-point change is 2.24 times larger a move in the space of distributions near a floor (0.10 → 0.02) than mid-range (0.50 → 0.42), so percentage-point arithmetic systematically understates change in populations whose rates approach a boundary. (3) A computable rather than gestural curvature. Ollivier–Ricci curvature on empirical graphs is already validated as a fragility indicator on weighted financial networks, where curvature declines precede crashes. (4) A worked counterexample that refutes the naive hypothesis. Two graphs are constructed with identical node count, identical edge count, and hence identical density, differing only in that one contains a cut edge. Exact Ollivier–Ricci curvature is computed with the 1-Wasserstein distance solved as a linear program. Result: mean curvature does not distinguish the fragile graph and points weakly in the wrong direction (+0.1970 robust versus +0.2121 fragile); average clustering also points the wrong way (0.562 versus 0.750); only the lower tail identifies the difference, with the bridge attaining exactly the theoretical extremum κ = −1.000 against −0.500 for its counterpart. Fragility is a local property of the weakest connection, and averaging destroys it. Consequently any hypothesis of the form "curvature declines as social structure degrades" must pre-specify the functional — minimum, low quantile, or mass below zero — and never the mean; hypotheses that do not are not yet testable, and a study reporting mean curvature can return a null while the structure it sought is plainly present. Three limits of ensemble inference follow, each with its constructive fix: strong dependence, where non-convergence near criticality becomes a second measurement via early-warning signals; the ecological fallacy, whose fix is multilevel modelling rather than a caveat; and reflexivity, which makes every prediction time-indexed and regime-conditional. Five pre-registrable predictions are stated with named falsifiers, including a machine-side test runnable without any demographic panel. The paper closes with a governance proposal — indicators of this family must be published as diagnostics, aggregate-only, with open code, and must never be used as optimisation objectives, since the cheapest way to raise a curvature tail statistic is to engineer the ties it counts rather than the resilience it was built to detect — and with four openings for attack that the author does not answer.