Advancing the unification of probability, curvature, and quantum emergence through Entanglement Compression Theory (ECT).
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Deterministic Quantum Gravity from Entanglement Compression
An Einstein-Extension of Entanglement Compression Theory (ECT)
1. Purpose and Scope
This paper develops a deterministic extension of Entanglement Compression Theory that incorporates spacetime curvature as a derived, compression-driven geometric response. Its aim is not to quantize gravity in the conventional sense, nor to introduce stochastic collapse, hidden variables, or semiclassical coupling to expectation values. Instead, it establishes a single dynamical framework in which oscillation, compression, probability, and curvature arise from the same underlying wave dynamics.
The work closes a structural gap left open by earlier ECT results. Prior papers derived probability weights, continuity, and stability from compression-regulated oscillatory motion on fixed geometry. The present extension promotes compression gradients to geometric significance, showing how curvature emerges locally from the same amplitude structure that governs quantum behavior. General relativity and quantum mechanics appear as controlled limiting regimes of a unified deterministic system rather than as fundamentally incompatible theories.
2. Foundational Commitments
The framework is built on the Oscillation Principle: physical degrees of freedom persist as sustained oscillatory motion constrained by stabilizing compression. Motion is not assumed to occur within spacetime as a passive background. Instead, spacetime structure responds dynamically to persistent compression gradients in the universal wave field.
The theory is formulated on a globally hyperbolic Lorentzian spacetime equipped with a preferred foliation, allowing first-order-in-time evolution while retaining covariant geometric structure in the metric sector. Probability is not postulated as fundamental. It arises deterministically through energy partition across resolvable modes, as established in earlier ECT work. Compression is not an independent field but a derived functional of the wave amplitude, defined through regularized logarithmic density and its covariant derivatives.
3. Compression-Driven Geometric Response
Curvature enters the theory through a covariant compression tensor constructed from the LUWF density. The symmetric compression Hessian
Cμν(sym) = ∇μ∇ν(−lnερ₁)
encodes how sustained amplitude gradients generate local geometric response. No exchange particles or independent gravitational degrees of freedom are introduced.
Spacetime responds through a leading-order effective metric deformation gμνeff = gμν + κ̃L*²Cμν(sym), which acts as a diagnostic geometry governing propagation and curvature effects. In regions of uniform density, compression gradients vanish and the effective metric reduces exactly to the background metric. Classical Einstein gravity is recovered smoothly in this weak-compression limit.
The metric deformation is not introduced as a new dynamical field equation but as a controlled geometric response derived from the same compression structure that regulates wave evolution. This effective-response stance ensures locality, second-order derivative control, and covariant conservation without introducing higher-derivative pathologies.
4. Field Equations and Conservation
The gravitational sector is organized through a diffeomorphism-invariant action used as a consistency and bookkeeping device rather than as a generator of the full coupled dynamics. Metric variation yields generalized Einstein equations of the form
Gμν + Λgμν = κTμν[Ψ],
where all non-gravitational contributions arise from the universal wave field and its compression-derived response.
Compression does not carry an independent stress-energy tensor. Apparent compression-induced energy density or pressure reflects geometric response encoded in the effective metric and its curvature. Covariant conservation follows from the Bianchi identity together with the imposed wave evolution, not from additional postulates. Energy and probability remain locally conserved under the same continuity structure established in the quantum sector.
5. Stationary Structure and Quantization
The coupled compression-curvature system admits stationary solutions in which oscillation, compression, and geometry enter equilibrium. These configurations correspond to discrete spacetime resonances determined by the spectral properties of a stationary compression-curvature operator. Quantization emerges from elliptic spectral structure and stability criteria rather than from measurement axioms or intrinsic randomness.
Particle-like behavior is interpreted as persistent compression-supported modes of the universal wave field. The framework does not attempt to enumerate the particle content of quantum field theory but demonstrates how discreteness, stability, and probability weights arise from deterministic geometry-coupled dynamics.
6. Limiting Regimes and Recovery of Known Physics
When compression gradients vanish, all compression-derivative contributions disappear and the field equations reduce exactly to classical general relativity coupled to a free wave sector. In weak-curvature regimes where back-reaction is negligible, the wave dynamics reduce to the Schrödinger equation on an approximately fixed background. These limits arise directly from the structure of the theory and do not require auxiliary assumptions.
The framework therefore does not replace general relativity or quantum mechanics. It embeds both as asymptotic descriptions within a single deterministic system whose intermediate regime is governed by compression-regulated oscillation.
7. Significance and Continuity
This work completes the first closed loop of Entanglement Compression Theory. Probability, continuity, curvature, and spacetime response are shown to originate from one oscillatory-compression mechanism rather than from parallel postulates. The Einstein extension establishes gravity as a geometric expression of compression rather than as an independent interaction layered atop quantum structure.
Together with the prior ECT series, this paper defines a coherent deterministic alternative to probabilistic quantum gravity programs. Its validation path emphasizes internal consistency, controlled limits, reproducible numerics, and falsifiable geometric effects rather than institutional endorsement.
Source: Lawrence, W.A. (2025). Deterministic Quantum Gravity from Entanglement Compression: An Einstein-Extension of ECT. Zenodo. https://doi.org/10.5281/zenodo.17538477
ECT Canonical Series:
Theory of Derived Probability and Entanglement Compression |
The Oscillation Principle |
Unified Derivation of Probability, Curvature, and Compression Geometry in Entangled Systems |
Mathematical Foundations of Derived Probability and ECT |
Deterministic Quantum Gravity from Entanglement Compression: An Einstein-Extension of ECT