Linearized Regimes of Hull Mechanics. Wave, Klein–Gordon, Schrödinger, and Dirac Structures as Projective Mode Equations

This article investigates the linearized regimes of hull mechanics and reconstructs familiar physical mode equations as projective regime-forms of stabilized hull dynamics. Building on the structure formula of hull mechanics, it considers small deviations Φ=Φ_0+εψ around stationary hull regimes and derives the general linearized hull-mode equation ∂_t^2ψ−v_max^⁡2Δψ+U′′(Φ_0)ψ=0. The paper interprets the wave equation as free difference continuation, the Klein–Gordon structure as mode dynamics in a stability-curved hull landscape, the Schrödinger structure as a nonrelativistic slow-envelope approximation, and the Dirac-type structure as the projective regime of internally oriented hull modes. These equations are not treated as independent ontological foundations, but as linearized projective regimes of a deeper operator-based architecture of stabilized difference.