The thermodynamic geometry of a statistical model treats the space of equilibrium states as a Riemannian manifold. The metric is the Fisher information matrix — the second derivative of the entropy with respect to the thermodynamic parameters. Curvature of this manifold encodes the strength and structure of correlations: flat regions correspond to non-interacting systems, and curvature divergences signal phase transitions where correlations become infinite.

Tsallis nonextensive statistics generalizes the Boltzmann-Gibbs framework by replacing the standard logarithmic entropy with a power-law generalization controlled by a parameter q. At q = 1, the standard framework is recovered. Away from q = 1, the statistics assigns different weights to rare events — more weight for q > 1, less for q < 1 — creating a deformed probability landscape.

In the one-dimensional Blume-Capel model — a spin-1 system with single-ion anisotropy — the Tsallis deformation reshapes the curvature profile of the thermodynamic manifold. The curvature singularities at phase transitions shift position as q changes: the critical temperature moves, the curvature exponents change, and the geometric signatures of the different phases (ordered, disordered, quadrupolar) deform continuously.

The deformation is systematic: small changes in q produce small changes in the curvature profile, allowing the Boltzmann-Gibbs limit to be approached continuously. But the approach is not uniform — different features of the curvature profile converge at different rates, revealing which aspects of the correlation structure are most sensitive to the choice of statistical framework.

The information geometry makes the deformation visible. In the space of thermodynamic parameters, the Tsallis generalization literally changes the shape of the manifold — stretching some directions, compressing others, moving the singular points where correlations diverge.

The statistics you choose changes the geometry you see. The geometry is the statistics, made visible.