Deterministic Quantum Gravity from Entanglement Compression: An Einstein-Extension of ECT

Deterministic Quantum Gravity from Entanglement Compression: An Einstein-Extension of ECT establishes the formal structure linking Entanglement Compression Theory (ECT) to the Einstein Field Equations. ECT begins from the causal axiom existence = motion = oscillation, unifying quantum mechanics and general relativity through deterministic compression dynamics. This version supersedes the preliminary outline released as Version 1 and presents the full mathematical formulation of the Einstein-extension of Entanglement Compression Theory (ECT). This work is part of a four-paper theoretical set that defines the core structure of ECT: Theory of Derived Probability and Entanglement Compression (Lawrence, 2025, https://doi.org/10.5281/zenodo.15786696) establishes the foundational framework of ECT, deriving probability weights, stability, and measurement structure from deterministic wave dynamics under compression. Unified Derivation of Probability, Curvature, and Compression Geometry in Entangled Systems: A Tensor-Formalism Extension of ECT (Lawrence, 2025, https://doi.org/10.5281/zenodo.17349900) develops the covariant tensor formalism of compression geometry, introducing the symmetric compression tensor and controlled metric deformation used to couple compression dynamics to spacetime structure. The Oscillation Principle (Lawrence, 2025, https://doi.org/10.5281/zenodo.17058692) formalizes the causal axiom existence = motion = oscillation, providing the ontological basis for the wave dynamics underlying all subsequent ECT constructions. The present manuscript extends this framework by coupling the compression geometry developed in the tensor-formalism paper directly to the Einstein field equations. Curvature emerges as a geometric response to sustained compression gradients of the universal wave field, yielding a closed, covariant formulation without introducing stochastic postulates or quantized geometry. Compression is treated as a derived physical response rather than an independent variational degree of freedom. Its effects enter through a real, multiplicative functional of the wave amplitude, producing controlled contributions to the stress–energy tensor and a leading-order effective metric deformation. Within this formulation, general relativity and nonrelativistic quantum mechanics are recovered as explicit limiting regimes rather than postulated structures. Key results include: a covariant compression-modified wave equation with conserved probability current, deterministic derivation of Born weights as energy partition across resolvable channels, a geometric response law linking compression gradients to spacetime curvature, controlled recovery of Einstein gravity and Schrödinger dynamics as limits, existence, regularity, and spectral discreteness of stationary compression–curvature modes, and leading-order, falsifiable predictions for gravitational lensing residuals, Shapiro delay drift, and compression-modified cosmological dynamics. This manuscript serves as the canonical gravitational extension of Entanglement Compression Theory, completing the mathematical linkage between oscillation, compression, probability, and spacetime geometry. All claims are explicitly scoped, derivations are stated under clear hypotheses, and no probabilistic or quantization postulates are assumed beyond the underlying wave dynamics. Supplementary numerical work and simulations demonstrating compression-driven structure formation under the Primordial Wave Equation are available through linked datasets and the Compression Theory Institute website (https://compressiontheoryinstitute.org).