Input-to-state stability of the forced Homeostatic Background Processor

Prove that, under bounded forcing, the velocity-equivalent internal state of the Homeostatic Background Processor is input-to-state stable to a ball around its learned rest state.

Background

The paper establishes contraction for the implicit diffusive kernel and stability conditions for individual branches, but the forcing includes learned, state-dependent terms and the analysis does not prove the resulting input-to-state stability property. The authors state only that ISS is expected under bounded forcing.

A proof would clarify whether bounded interoceptive and external drives keep the Homeostatic Background Processor within a controlled neighborhood of its equilibrium, particularly when the learned forcing depends on the field state.

References

Under bounded forcing, the VEI is expected to be ISS to a ball around $h\ast$ by a standard argument \citep{sontag1989smooth}, which we do not prove here and on which nothing below depends.

— Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Control  (2608.24319 - Arrabal-Campos et al., 25 Aug 2026) in Section 4, Remark on the convex mixture and runtime certification (Stability analysis)