## Abstract This work extends an effective phase–based framework in which relativistic dynamics, mass, spin, charge, measurement, and correlations are described through phase propagation and constitutive response in an effective medium. The framework preserves the macroscopic Hamiltonian structure established in the companion works while clarifying its microphysical implications and limitations. The effective constitutive Hamiltonian is written as (x, p) =K (x) \, F\! (C^ij (x) pᵢ pⱼK² (x) ), \ where \ (K (x) \) is the local rigidity scale and \ (C^ij (x) \) is the anisotropic propagation tensor. In spherical symmetry, ^ij (x) =cᵣ² (x) rⁱ rʲ+c_² (x) (^ij- rⁱ rʲ). \ To interpret particle families as admissible localized excitations of the medium, the paper introduces a radial eigenvalue problem based on an effective wave action, [= dt\, d³x (r) c²|ₜ|²-Bᵣ (r) |ᵣ|²-B_ (r) |_ |²r²-U (r) ||². \] For stationary configurations, \ (t, r, ) =e^-i tR_ (r) Y ₌ (), \ the radial equation becomes \1r²ddr (r² Bᵣ (r) dR₃ₑ) -B_ (r) (+1) r²R_ (r) +[A (r) c²²-U (r) _ (r) =0. \] The associated rest energies and masses are ₙ = ₙ, ₙ = Eₙc². \ The framework therefore interprets particle species as admissible localized modes whose spectra depend on constitutive rigidity profiles, confinement structure, and boundary conditions. Electromagnetic coupling follows from local phase-convention invariance. Under \ (x) (x) + (x), \ the covariant phase gradient is _ =_ -qA_, \ with the gauge transformation __+q_. \ This yields the minimal substitution __-qA_. \ Combined with Hamiltonian dynamics, this reproduces the Lorentz-force structure, dt=-ₓ H (x, ) +q (E+ ẋ B). \ Measurement is interpreted as a local constraint on admissible phase configurations rather than as collapse or branching. Randomness arises from the underdetermination of local anchoring conditions prior to measurement, while probabilities emerge as statistical weights over admissible configurations, () of admissible configurations compatible with outcome. \ Entanglement is described through a single global phase configuration subject to multiple local constraints. Bell inequalities are violated because the framework rejects factorized hidden variables and pre-existing local properties, while preserving strict no-signaling. The theory remains explicitly effective and incomplete. It constrains Hamiltonian dispersion, gauge coupling, inertial response, and locality properties, but leaves open the microscopic origin of rigidity, confinement, particle masses, and coupling constants. Possible falsifiable directions include high-frequency dispersion, cutoff effects, strong-field deviations, and spectral constraints on admissible localized modes.
Thomas Jublot (Wed,) studied this question.