A resolution gap for the coarse-grained arrow of time in nonlinear Hamiltonian flows

Gaussian coarse-graining at resolution σ turns the conserved Gibbs entropy of a Hamiltonian ensemble into a coarse-grained entropy S_C. For an isotropic resolution that coarsens at rate r = σ̇/σ, every state has a critical rate R such that dS_C/dt ≥ 0 if and only if r ≥ R. For linear flows the supremum of R over all states equals κ, the largest symmetric strain rate of the flow, and it has been conjectured that R ≤ κ for every state of a nonlinear flow. Here we derive an exact decomposition of R into a law term, a covariance between the local Fisher matrix of the coarse-grained density and the local strain, and a posterior Stein defect. It shows that a violation of the conjecture requires either regions in which the local Fisher matrix has a negative eigenvalue, or a positive Stein defect. We prove the conjecture for all Gaussian states, in any dimension. For flows with one degree of freedom the best Gaussian state is a zero-width segment through the point of maximal strain, and the resolution gap of Gaussian states is κ − R = 2√(2βκ) σ + O(σ²), where β measures how fast the strain decreases away from its maximum; we prove this exactly for the pendulum and for quadratic strain, and as an upper bound on the gap in general. For the pendulum the gap is σ − (7/8)σ² + (65/96)σ³ − …, with exact rational coefficients. Gaussian states are not extremal: mixtures arranged as a fan of segments that cross at the point of maximal strain exceed the best Gaussian state by about 5% of its gap, and in a flow whose stretching direction rotates, bent needles that follow the rotation reduce the gap by about 40%. In every case examined the gap remains proportional to σ. For the pendulum we prove, from a weighted Cramér–Rao inequality, a state-wise lower bound κ − R ≥ Θ′σ/(1 + Θ′σ/2) − ε̃ − D. It gives a uniform gap under explicit conditions, and on all 268 stored pendulum states with σ ≤ 10⁻² it is at least 0.898σ, against a smallest observed gap of 0.946σ. A uniform gap would follow from a bound on how fast the coarse-grained Fisher matrix can vary along the contracting direction, together with control of the valleys and of the Stein defect. All statements and numbers are reproduced by openly available code and are cross-checked by independent numerical methods.