Finite-horizon propagation of a posteriori error certificates

Establish finite-horizon propagation of the one-step primaldual density-error estimate for the restarted Monge-growth neural H-JKO scheme by proving a suitable stability estimate for the exact H-JKO solution operator and rigorously controlling the per-step certificate gaps.

Background

Theorem D provides a one-step a posteriori bound for the density error of an admissible computed pair under positive-semidefinite interactions and globally feasible dual potentials. The bound is not automatically iterable because errors in successive approximate states may be amplified by the exact JKO solution operator, while numerical evaluation introduces per-step errors.

The paper explicitly identifies the missing ingredients for extending this one-step result to a finite time horizon: a stability estimate for the exact H-JKO map and rigorous control of the gaps at every step.

References

Finite-horizon propagation would require an additional stability estimate for $\mathcal S_\tau$ and rigorously controlled per-step gaps; no such propagation is proved here.

— A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs  (2610.07602 - Seo et al., 6 Oct 2026) in Remark 'Scope', Section 4.2 ('Theorem D: an a posteriori error estimate from a feasible dual pair')