This preprint develops a structural small–gain framework for coupled “value-anchored natural-law” gradient flows arising in persistence-first holographic systems (PFHS). The model combines (i) law-level gradient flows on FBHK/Post-FBHK entropy–transport spaces, (ii) parameter-level gradient descent for value-anchored self-improvement, and (iii) defect-level gradient flows for multi-agent consistency and self-purification, together with slowly varying anchors and environments. At the analytic core, the paper assumes an EVIλμEVI_EVIλμ law-level gradient flow with a uniform contraction rate and postulates a law cross-gain inequality describing how law–space distances depend on parameter and defect distances. Under standard strong convexity and Lipschitz conditions for the parameter and defect functionals, the distances between two joint trajectories satisfy a comparison inequality driven by a 3×3 Metzler matrix. A linear copositive Lyapunov functional exists if and only if this Metzler matrix is Hurwitz, yielding a small-gain condition that guarantees incremental exponential contraction of the full law–parameter–defect dynamics. The same Lyapunov structure gives an input-to-state stability (ISS) bound quantifying robustness with respect to environmental mismatch. The blueprint is deliberately modest on the analytic side: it does not claim new curvature or PDE existence results for FBHK/Post-FBHK geometry, but instead treats them as a black box. The main contribution is to isolate design-oriented structural hypotheses—on intrinsic contraction rates and cross-gains—under which joint contraction and ISS follow from standard positive-systems and hybrid-systems arguments. The paper verifies all hypotheses in a fully worked linear–Gaussian PFHS example (Ornstein–Uhlenbeck law dynamics with quadratic value and defect functionals) and shows how the small-gain condition can be checked numerically by solving a tiny linear program. This provides an implementation-ready checklist for designing stable, value-anchored, self-improving, multi-agent intelligent systems over entropy–transport geometries.
Takahashi, K. (Thu,) studied this question.