Population Capacity and the Geometry of Thermomajorization

We characterize quasiclassical state conversion under thermal operations through population capacities: the maximum population that thermal operations can concentrate on each set of energy levels. These capacities form a normalized, nondecreasing submodular rank function whose base polytope is exactly the thermally reachable population region. Their bounds therefore provide a complete system of accessibility constraints. Every nested chain of capacities can be simultaneously saturated by a reachable state, yielding the maximum expectation value of any prescribed observable, including the system Hamiltonian. For a prescribed target, its thermomajorization prefixes suffice to certify exact conversion through at most $n-1$ binary work-extraction probes. For approximate conversion, we construct a distinguished reachable state from active capacity constraints using a weighted principal sequence in $O(n^2 \log n)$ operations. At fixed comparison weights, this state minimizes every weighted convex separable residual cost and realizes a common first contact with the corresponding target-centered smoothing regions. Finally, the optimal trace-distance error equals the largest capacity advantage of the target over the initial state. This gap is attained at a target prefix and equals the largest positive optimized energy-storage deficit among the same binary probes, divided by their energy gap. Thus a common population capacity geometry determines accessibility, observable optimization, conversion certificates, and universal approximation.