Curvature Deficit as Energy on a Discrete Closed Surface

I describe a discrete relational model in which space is a finite, closed, triangulated 2-manifold and in which no quantity is placed on the vertices by hand. The only local quantity is the combinatorial curvature deficit κ(v) = 6 − deg(v). By the discrete Gauss–Bonnet theorem its sum over the manifold is fixed by topology, Σκ = 6χ = 12 − 12g, so the total is a conserved charge rather than a tunable parameter. I argue that the quantity which actually drives the dynamics is not Σκ but Σκ², which is not conserved and decreases under the smoothing move. Two operations are considered: a vertex split (mitosis), which preserves both χ and Σκ exactly and locally, and a handle-cutting surgery, which changes Σκ by exactly +12 for a cross-section of any length. Stating the splitting rule as descent on Σκ² makes an unexpected feature explicit: the dynamics has frozen states at arbitrarily high genus — a uniformly hyperbolic triangulation of degree 7 to 11 admits no admissible move at all, the smallest example being the Klein quartic. Only surgery can leave such a state. The paper also reports a finite combinatorial result on orientation patterns of 4-cycles, in which exactly one of the four possible types admits a rolling move whose square reverses all orientations. This is a report on work in progress, not a completed theory. The framework does not have Lorentz invariance, a metric, or quantum amplitudes, and no engine implementing the formulation presented here has been built; a section is devoted to stating these gaps explicitly. The deposit contains the paper and nine short Python scripts reproducing every combinatorial check reported in it. Each runs in a few seconds. Keywords: discrete spacetime, combinatorial curvature, Regge calculus, discrete Gauss–Bonnet, triangulated manifold, emergent geometry, causal sets, quantum gravity, spectral dimension, topological surgery