Theory of Expansion-Induced Metric Tension (TEIM)

TEIM has stopped being a phenomenological model and emerged as a mathematically closed mechanical theory of the universe. From a covariant Lagrangian fixed by four fundamental constants—and with no free parameters left to tune—the same machinery that reproduces the ΛCDM distance-redshift relation now analytically deduces the universal galactic acceleration scale (a₀ = 1. 19 × 10⁻¹⁰ m/s²) from first principles. This exact theoretical prediction fits the rotation curves of 153 SPARC galaxies without assuming any dark matter halo profile and without per-galaxy tuning. Furthermore, this identical continuum extends flawlessly into the strong-field interior of black holes, predicting a finite Yield Horizon rY = √ (Rₛ L₀) that safely resolves the singularity. One framework. Four constants. From the cosmological horizon, down to the galactic scale, straight into the core of a black hole. No dark matter. No phenomenological fitting. No singularities swept under the rug. Abstract We formulate the Theory of Expansion-Induced Metric Tension (TEIM), a covariant variational framework that models the universe not as an empty geometric stage, but as a relativistic continuous medium capable of storing and relaxing elastic memory. In this framework, we mathematically demonstrate that the phenomenon phenomenologically labeled as dark energy emerges exactly as the macroscopic mechanical relaxation of the space-time fabric itself: a strained continuum discharging stored potential energy. Simultaneously, what is usually interpreted as dark matter arises as an unavoidable structural phase transition—an organized relational tension response. To formulate this covariantly, we combine a material coordinate description of the medium with the Schutz-Brown formalism for relativistic fluids. The theory partitions into two sectors: a global volumetric strain field s, encoding stored elastic memory via a dynamically relaxable internal module ϖ, and a local relational mode ψ, which acts as the structural susceptibility sector. We derive the exact equations of motion, the exact stress-energy tensor, and the fundamental Noether identity linking the macroscopic description to the underlying material pullback. We prove that the divergence of the fundamental tensor rigorously ensures temporal energy conservation and completely cancels non-linear gradient sources. In exact FLRW, the relational sector remains on its symmetric branch. We establish the profound mechanical thermodynamics of the continuum, proving via a Lyapunov function (dρₛ/dt + 3H (ρₛ + pₛ) = 0) that the exact continuity equation dictates a conservative relaxation rather than irreversible bulk viscosity. For a logarithmic-hyperbolic elastic potential, U (s) = U_⋆ ln (cosh (s / s_⋆) ), the background possesses no exact de Sitter fixed point but admits a slow-roll quasi-attractor. We analytically demonstrate that the elastic energy reservoir decays monotonically in an expanding universe, precluding eternal homogeneous cycles, and we derive the deterministic sequence of early Hubble freezing, late-time thawing, transient acceleration, and damped overshoot. Exact analytical marginalization against DESI DR1 BAO and Pantheon+SH0ES, combined with direct numerical optimization over the background parameters, confirms that TEIM accurately mimics the flat ΛCDM distance-redshift relation. Furthermore, we establish the Primordial Freeze-Out Theorem, proving that the strain velocity is exponentially suppressed by radiation (O (e⁴ᴺ) ), which guarantees exact preservation of the acoustic scale at background level and, under the potential-freeze criterion derived later, perturbative preservation of the primordial transfer sector. We then develop the structural regime of ψ. At linear order on exact FLRW, δψ is strictly auxiliary, while δs is the unique propagating mode, yielding an exact subhorizon scale-dependent coupling Gₑff (k, a). In the nonlinear structural regime, we formulate the birth of active halos as a local supercritical Lyapunov–Schmidt bifurcation problem, identifying the trivial symmetric state as a tachyonic false vacuum whenever the principal susceptibility eigenvalue crosses zero. By invoking the canonical Maxwell-Eshelby identity and the universal exponential-disk projection, we analytically derive the global acceleration scale a₀TEIM and the projection coefficient CTEIM entirely from the four fundamental elastic constants, yielding an exact a priori match to the SPARC empirical scan. We analytically solve the active structural family ρ_α, proving through exact Hypergeometric and Gamma functions that α = 1 is the unique, structurally privileged topological attractor yielding flat rotation curves and isothermal-like lensing deflections. We further prove a coarse-graining theorem demonstrating that shell-like elementary activations yield centrally supported macroscopic halos after isotropic convolution. Finally, we establish that the framework natively satisfies stringent relativistic constraints: the SVT decoupling theorem guarantees exact light-speed propagation of gravitational waves (cGW = c), and an explicit 1PN weak-field exact radial solution proves that the scalar gradients decouple exactly from baryonic matter, securing the Parameterized Post-Newtonian limit (γ = 1, β = 1). TEIM thus stands as a fully closed mechanical framework of a universe that remembers, relaxes, and structurally condenses.