Gravitation as an Emergent Covariant Relaxation and Phase-Dynamics Framework: Induced Einstein-Hilbert Action, Lorentz Duality, and Cosmological Constraints

We present a rigorous foundational framework for emergent gravitation driven by a universal equilibrium-seeking principle (the Law of Striving for Equilibrium). We demonstrate that the Einstein-Hilbert action naturally arises as an induced low-energy effective action via heat-kernel expansion of a structural relaxation operator acting on a localized spatial imbalance scalar $\Delta$. The scalar $\Delta$ represents an invariant measure of localized energy-density deviations from a maximally symmetric background equilibrium state. Utilizing the Palatini variational framework, we establish exact connection and metric emergence. We prove three central theorems: (i) the Emergence of Geodesic Motion as a spatial relaxation restoring force, (ii) the Carrier-Envelope Covariance Theorem enforcing four-wavevector orthogonality under Lorentz transformations, and (iii) the Relaxation Stability Theorem proving that under specific spectral dampening parameters, the system mathematically evolves toward non-oscillatory minimization of the imbalance gradient ($\dot{\Delta} \le 0$). Finally, using current observational bounds from Planck 2018, DESI 2024, and Pantheon+ data, we map the Chevallier-Polarski-Linder dark energy parameters directly onto the cosmic relaxation parameter $\xi$, establishing a strict upper limit of $\vert{}\xi\vert{} \le 1.4 \times 10^{-3}$ at the 95% confidence level.

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