Electrons as Waves of the Elastic Lattice: An RTE-Based Route to the Klein-Gordon and Schrödinger Equations

The Elastic Three-Dimensional Lattice (RTE) model — proposed in Colombo (2026a) and formalized in Papers 1–4 — has until now been applied to gravitational and kinematic phenomena. This paper proposes an extension to quantum mechanics. We show that, when the RTE wave equation is supplemented by a massive-mode rest-frequency postulate (ω₀ = mc²/ℏ), it leads to a Klein-Gordon-type equation; taking the non-relativistic limit then yields the Schrödinger equation. A field redefinition (χ = √K̃ × u) is demonstrated to eliminate the first-derivative coupling term ∇(ln K̃)·∇u, producing an equation with a multiplicative effective potential V_eff(r). The standard hydrogen atom energy levels are recovered as a consistency check once the Coulomb potential is supplied externally. Physical interpretations are proposed for wave-particle duality, wave function collapse (elastic auto-focus), and the complex structure of ψ as the two phases of the bifasic cycle. The paper distinguishes systematically between what is mathematically derived, what is physically proposed, and what remains open.