The non-reciprocal entropy–production bound: proven cases, the Q = I premise, an exact reduction, and an open residual

Abstract: Non-reciprocal couplings can make added dissipation raise the barrier that protects an attractor against rare escape — a protective reversal impossible at equilibrium. We ask, and bound, how strong this effect can be per unit of entropy production. For a linear stochastic system x=Lx+ε ξ, ⟨ξξ⊤⟩=Q, with Hurwitz L=−A+N (A=A⊤≻0, N=−N⊤), the barrier’s sensitivity to a dissipation increment M≽0 is the linear form ⟨M,K⟩ of a response operator K built from the stationary Gramian, and a reversal occurs iff λmaxK>0. Under isotropic noise Q=I we conjecture the universal bound supNλmaxK+/Si≤1/128 a13, a1=λminA, and prove it — sharp constant included — in the following cases: the value 1/128a13 and the saturating 7:1 geometry in the small-coupling limit in every dimension (an explicit copositivity certificate; unconditional for d≤4, otherwise conditional on a rank-two reduction we verify numerically), and the full two-mode case for all couplings. An exact Schur complement lowers the governing matrix inequality by one dimension, and we show the isotropy hypothesis is load-bearing — for general Q the ratio is unbounded, so Q=I is not a normalisation. A free-coordinate reduction recasts the general statement as a matrix inequality on a compact box and reduces d=3 to a matrix sum-of-squares certificate (whose existence the Hol–Scherer theorem guarantees where the matrix is strictly positive). The residual — d≥3 at all couplings — remains open; it is numerically unbroken over >107 configurations up to d=12. Ensemble simulations reproduce every prediction from trajectory data on circuit, optical, mechanical, and quantum realisations.