Stochastic Capacity Response in the SFIT Engine — Universal Scaling Hierarchy in Noise-Driven Dynamics

This paper analyzes the stochastic dynamics of the SFIT capacity engine under thermal driving, revealing a universal scaling hierarchy that emerges once the system crosses a sharp noise‑driven threshold. Contrary to the initial hypothesis, the engine does **not** exhibit Kramers‑like escape dynamics or Arrhenius activation. Instead, above a crossover temperature \(Tbath ≈ 0.0045\), the capacity‑closed pump is overwhelmed by thermal fluctuations, and the system enters a noise‑dominated regime characterized by robust, parameter‑independent scaling laws. In this regime, the mean local capacity scales linearly with bath temperature, \( Cloc ~ C_0(g) + k Tbath\), with a universal slope \(k = 55.6 ± 0.3\) independent of coupling \(g\) to within 0.6%. Field amplitudes scale as \(Tbath1/2\), and each quadratic energy component (kinetic, gradient, mass‑potential) independently satisfies generalized equipartition, scaling as \(Tbath^1\). The quartic term scales as \(Tbath^2\), consistent with Gaussian moment relations, demonstrating that the weak nonlinearity becomes dynamically irrelevant in the noise‑dominated regime. The paper provides a negative result (absence of metastable escape) and a positive result (a universal scaling hierarchy), supported by extensive diagnostics, convergence tests, and per‑component energy decomposition. These findings characterize the classical SFIT capacity engine’s stochastic behavior and clarify the conditions under which the engine produces regular pulsed dynamics versus noise‑dominated equipartition. Connections to the broader SFIT program and the Reconstruction Theorem are discussed but left for future work.