Classical and relativistic physics treat gravitational interaction either as action at a distance between static masses, or as a geometric property of curved spacetime. However, the nature of inertia, the equality of inertial and gravitational mass, the mechanisms of momentum transfer in vacuum, and the origin of quantum and gauge structures remain dualistic postulates.The present work proposes the Local Energy Deficit Model (LEDM) — a deductive mechanical approach based on the concept of a dynamic elastic energy medium (vacuum substrate) in a state of elastic thermodynamic equilibrium. Developing the ontological principle of the objective reality of space and time, the author transitions from a purely geometric description of trajectories to the physics of spatial pressure gradients in a real medium. This allows for a consistent mechanical interpretation of action at a distance, unification of macroscopic gravity with optical phenomena, and extension of the model to quantum and gauge dynamics.Part I. Classical Mechanics of the Elastic Substrate1.Derivation of Newton's law. The gravitational force is derived from the pressure gradient of the elastic medium, created by the superposition of energy deficit fields of individual bodies. It is shown that the background energy density of vacuum ρu2080 completely cancels out in the final expression for weak fields. The absorption coefficient is fixed as σ = 16πGρu2080, ensuring dimensional and quantitative consistency.2.Orbital precession. In the nonlinear synergistic regime of the medium, an additional cubic force (1/r³) arises, leading to secular pericenter shift. The precession formula coincides with empirical data for Mercury (43u2033 per century). The normalization of the nonlinear modulus γ/ρu2080 = 1 receives microscopic justification.3.Optical and cosmological consequences. Gravitational lensing, the Shapiro effect, and cosmological redshift are interpreted as consequences of changes in optical density and interaction of photon wave packets with the elastic medium. The broadening (stretching) of Type Ia supernova light curves is explained without invoking geometric expansion of empty space.4.Casimir effect. The mechanism of the Casimir force is interpreted as a result of pressure difference between the external medium and the shielded gap; an explicit regularization transition is provided.Part II. Quantum and Gauge Extension5.Derivation of the Schrödinger equation. It is shown that the Schrödinger equation can be obtained from the stochastic wave equation of MLDE as its non-relativistic limit via the envelope approximation procedure. Energy quantization in bounded regions arises as a consequence of boundary conditions for waves in the elastic medium.6.Hierarchy of gauge structures. A three-level scheme for introducing charge has been developed:Level 1 — minimal gauge coupling (charge as an external sink parameter);Level 2 — complex field δu with U(1) symmetry; charge as a Noether current; relation of |ψ|² to the density of conserved charge in the envelope approximation;Level 3 — SU(3) color generalization; structural analogies with confinement and limitations of the envelope approximation applicability.7.Fundamental limitations of the model. It is emphasized that MLDE remains a local theory and, according to Bell's theorem, does not explain the violation of Bell inequalities. This fundamental limitation of the model clearly delineates its applicability boundaries.Keywords: Model of Local Energy Deficit (MLDE); vacuum elasticity; strain tensor; orbital precession; Shapiro effect; Casimir effect; gravitational lensing; cosmological redshift; stationary Universe; Schrödinger equation; envelope approximation; U(1); SU(3); Bell's theorem. IntroductionIn the absence of material objects, physical space represents an elastic energy medium in elastic thermodynamic equilibrium: points of the continuum exchange equal momenta, and the average vacuum pressure remains constant. This equilibrium is maintained through background isotropic elastic oscillations of the medium — an analogue of quantum zero-point oscillations.The model proposed in this work differs fundamentally from historical predecessors (Descartes, Euler, Le Sage). In it, the energy medium is considered not as viscous, but as ideally elastic, in dynamic thermodynamic equilibrium. It is the elasticity of the medium, not its viscosity, that is the source of the pressure gradient, and therefore of the attractive force. This allows not only to strictly derive Newton's law, but also to explain the secular precession of orbits as a nonlinear effect in the elastic substrate — without violating the stability of trajectories.The methodological foundation of MLDE rests on four principles:1.Logical deduction of dynamic forces from kinematics.2.Objective reality of the medium and dynamic nature of mass.3.Formation of radial energy deficit (gravitational wells).4.Sub-nuclear additivity of quantized forces and the equivalence principle.Structure of the work. This article consists of two parts. Part I develops the classical mechanics of the elastic vacuum substrate: derivation of Newton's law, orbital precession, optical and cosmological effects, Casimir effect. Part II extends the model to quantum and gauge dynamics: derivation of the Schrödinger equation, hierarchy of U(1) and SU(3) gauge structures, discussion of fundamental limitations (Bell's theorem).