The work output of a thermodynamic cycle is the area enclosed by the curve in the pressure-volume diagram. This is a global property — you need the whole loop to compute it. A Carnot engine running between two temperatures performs work proportional to the area of its rectangle in PV space. You cannot ask how much work is being done “at” a particular point on the cycle without reference to the rest of the path.
Bittner (arXiv:2603.22559) shows that you can. The area in the PV diagram and the area in the TS (temperature-entropy) diagram are both projections of a single canonical two-form on the thermodynamic state manifold. This two-form is a local geometric object — it has a value at every point. The work done by an infinitesimal cycle is proportional to the mixed curvature of the equilibrium energy surface, U_SV, which can be expressed entirely in terms of measurable susceptibilities: heat capacity, compressibility, thermal expansion.
The reframing converts a global integral (area enclosed by a finite cycle) into a local field (curvature at a point). Any sufficiently small cycle encloses an amount of work proportional to the local curvature. Large cycles are built up from these local contributions, but the contributions are not equal — the curvature varies across the state space, concentrating work generation in regions where the energy surface bends most sharply.
What this reveals is that the PV diagram was always a projection that obscured locality. The work wasn't distributed uniformly around the cycle. Some segments contributed more curvature than others. The integral hid this, reporting only the total. The geometry recovers what the diagram collapsed.
The connection extends to nonequilibrium paths. Stochastic trajectories on the thermodynamic manifold generate work along each step, and the Jarzynski equality — which relates free energy differences to the exponential average of work along irreversible paths — acquires a geometric interpretation as the average of these local curvature contributions across an ensemble of fluctuating trajectories.
The area was never just arithmetic. It was geometry all along.