From Robotics to Chip Design: The Universal Equilibrium Between Recursive Functional Amplification and Curvature-Induced Constraint — The Route to Optimization

This paper presents a unifying optimization principle spanning robotics and semiconductor chip design. Rather than treating optimization as a domain-specific problem, it shows that stable high-performance systems consistently emerge at an equilibrium between recursive functional amplification (e.g. gain, feedback, reuse, density) and curvature-induced constraint (e.g. geometry, confinement, surface area, thermal and material limits). The framework is formalized using a simple scaling functional and applied to two major engineering challenges: the semiconductor interconnect and thermal scaling wall, and the stability–efficiency tradeoff in robotic feedback control. Minimal quantitative models and a reproducible Python implementation demonstrate how the equilibrium condition predicts observed design “knees” and optimal operating regions. The result is a domain-agnostic design framework that explains why certain architectures recur across technologies and provides a practical, pre-simulation guide for co-designing function and geometry. No new device physics or control laws are assumed; instead, the paper exposes a shared structural principle underlying optimization across scales.