The resolution gap in many degrees of freedom

Gaussian coarse-graining at resolution σ turns the conserved Gibbs entropy of a Hamiltonian ensemble into a coarse-grained entropy that grows if and only if the resolution coarsens faster than a critical rate R. For one degree of freedom, the best Gaussian state stays below the largest strain rate κ by a resolution gap proportional to σ. This article asks how the gap depends on the number of degrees of freedom, for mechanical systems H = |p|²/2 + V(q). We prove that the averaged strain felt by a Gaussian state depends only on its spread in configuration space, so that spreading along momentum directions is free, and that the best Gaussian state has a gap of at most Γσ + O(σ²) with the matrix formula Γ = 2 tr[(B^½ C B^½)^½]. The gap does not grow with the number of degrees of freedom as such: uncoupled degrees of freedom cost nothing, and a coupled one costs in proportion to how strongly the maximal strain depends on it. For two coupled pendulums Γ is explicit. Numerically, the formula is the optimum over all Gaussian states in thirteen systems, to a relative 4 × 10⁻⁸; six of them were predicted before they were optimized, with the predictions recorded and hashed in advance. Mixtures of Gaussians gain only 40–55% as much in two degrees of freedom as in one. Code and verification (479 + 64 checks): https://github.com/neginyan/resolution-gap-many-dof (doi:10.5281/zenodo.23370396).