Recovering WKB and EBK Quantization from an Action-Phase Reinterpretation of the Semiclassical Standing-Wave Detection Law in Extended Phase Space

This preprint introduces the phase-space action-phase (PSAP) framework as a conservative semiclassical reformulation of one-dimensional bound-state quantum mechanics. Its starting point is the standing-wave optical detection law, whose familiar sinusoidal phase dependence can be rewritten as dependence on a dimensionless action phase S/ℏ. Motivated by this observation, the manuscript treats the action phase as the natural orbit coordinate on extended phase space and models the observable probability density of interaction as a classical probability density of presence envelope multiplied by a bounded kernel of the action phase. The same action-phase structure is then generalized from the optical standing-wave setting to material particles. Within classically allowed regions, this construction recovers the standard WKB spatial-density structure, while phase closure around a closed orbit yields the usual EBK quantization condition. Soft turning points and forbidden regions are treated using the standard Airy/WKB uniform approximation together with analytic continuation to complexified extended phase space. In its present scope, the framework does not claim new semiclassical numerical predictions; its contribution is organizational and interpretive, recasting familiar WKB/EBK structure in terms of action-phase modulation. The paper also includes an appendix discussing a wrapped cumulative distribution function (CDF/CCDF) heuristic which models the cyclic interaction propensity arising from monotonically accumulated action in classically allowed regions.