Tunneling and Bound States from Persistence-Weighted Calculus

This paper analyzes quantum tunneling and bound states as empirical stress tests for a persistence-based comparison framework whose structure is fixed independently. Rather than postulating wave mechanics or operator dynamics, the analysis takes as given a small set of structural ingredients: admissible variation, accumulated loss under recomposed change, and a persistence requirement selecting stable continuations. Within this framework, persistence-weighted path amplitudes and an associated quadratic stability operator arise as derived objects. The paper examines what is forced when these objects are applied to barrier penetration and confinement, and which standard quantum-mechanical results are recovered only as correspondence limits under additional conditions. Tunneling is treated as a local suppression problem, while bound states are analyzed in terms of closure and stability rather than operator postulates. Explicit worked examples demonstrate how WKB exponents and discrete spectra emerge in entropy-flat limits, and how controlled deviations arise when admissibility or curvature conditions are relaxed. No new principles are introduced. The results isolate the structural footing of familiar semiclassical constructions and delineate a clear window of falsifiability for persistence-based comparison beyond standard wave-mechanical representations.