Threshold-Burst RF Harvesters: Stochastic First-Passage Closure, Leakage-Limited Optimization, and Sommerfeld Boundary Factorization

This manuscript derives a closed mathematical framework for threshold-burst radio frequency (RF) harvesters driven by boundary-conditioned available electromagnetic power. The system is modeled as a stochastic first-passage charging process coupled to a deterministic RLC burst release. The paper establishes several foundational operating laws, including: Inverse-Gaussian interval statistics in the constant-drift regime. Exact Ornstein-Uhlenbeck first-passage dynamics and leakage admissibility boundaries. Exact mean-power laws governing the maximal deterministic output. Rigorous spectral factorization via renewal theorems, proving that the total output spectrum separates cleanly into a single-burst envelope and a renewal timing spectrum. Multi-band aggregation, showing how upstream RF inputs accumulate mathematically. Finally, the input boundary layer is resolved analytically by deriving the available power from explicit contour deformations of Sommerfeld integral representations of Maxwell fields above finite-conductivity ground. The resulting architecture establishes conclusively that the optimal deterministic performance of the threshold-burst harvester is mathematically governed entirely by the upstream boundary conditions.