EP-sea Phase-Membrane Framework IV: Mass–Energy Equivalence, Quantum Probability, and Event-Dynamical Measurement Closure

We develop a quantum-interface closure for the EP-sea program by integrating out sub-observable degrees of freedom into an effective background term B that injects a finite set of response/noise parameters into the observable tier. Within this setting, the relativistic constant c is identified with the transverse Maxwell-sector characteristic speed, c 2 = (ε0µ0) −1 in homogeneous vacuum-like regimes, while longitudinal compression modes obey a distinct sound-law c 2 s = (∂p/∂ρ)s governed by stiffness-to-inertia, thereby avoiding any macroscopic preferred-frame reintroduction. Matter is modeled as a stable observable-tier phase-membrane defect whose rest energy is a free-energy gap E0 = ∆F, so that defining m ≡ ∆F/c 2 yields E0 = mc2 and the associated relativistic dispersion. On the quantum side, the complex amplitude ψ is not postulated as a prior wavefunction; it emerges as a phase-locked envelope variable of a tethered defect, where complex structure packages two orthogonal transverse/rotor degrees of freedom and an SU(2) double-cover constraint provides a geometric origin for spin-1/2. Probability and measurement are treated operationally: |ψ| 2 is an event-density (absorption/nucleation flux) whose continuity follows from stochastic defect dynamics with ℏeff ≡ 2mD(B); Born statistics arise as a frequency limit of Poisson/SDE detection events with rate Γ(x)|ψ(x, t)| 2 . The framework is falsifiable through (i) double-slit visibility scaling with environment noise Θ and detector coupling, (ii) Aharonov–Bohm phase consistency under minimal coupling, (iii) spin-1/2 (4π) behavior and the Pauli term with tree-level g = 2, and (iv) a mechanism-level interface to Lamb-shift/backreaction spectroscopy. Constancy of ℏeff in tested laboratory regimes imposes a leading-order scaling constraint on the background diffusion strength, e.g. D(B) ∝ 1/m for fixed species.