Advancing the unification of probability, curvature, and quantum emergence through Entanglement Compression Theory (ECT).
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Unified Derivation of Probability, Curvature, and Compression Geometry in Entangled Systems
A Tensor-Formalism Extension of Entanglement Compression Theory (ECT)
1. Introduction and Purpose
This paper extends Entanglement Compression Theory into tensor form, uniting its statistical, geometric, and physical foundations.
It establishes a single covariant structure in which probability, curvature, and compression share the same mathematical origin.
The formulation begins from the Primordial Wave Equation and demonstrates that the Born-weights theorem, the continuity relations, and the deformation of the effective metric are not independent assumptions but linked results of the same compression dynamics.
Its objective is to place these relations on firm analytic ground and to show that a deterministic wave equation can generate both quantum probability and geometric curvature within a unified tensor framework.
2. Framework and Key Definitions
The tensor formalism begins from the Primordial Wave Equation,
iħ ∂tΨ = −αd∇²Ψ + β ℂ[Ψ] Ψ,
where αd = ħ² / (2 meff).
The compression operator ℂ[Ψ] is real and multiplicative, defined on the one-body density and regularized by a small parameter ε to preserve local Lipschitz continuity.
From this operator, the symmetric compression tensor is constructed as
Cμν(sym) = ∇μ∇ν(−lnερ₁),
which captures curvature and anisotropy within the evolving field.
The effective metric is written as gμνeff = gμν + κ̃ L*² Cμν(sym), and its linearized inverse as gμνeff = gμν − κ̃ L*² C(sym)μν. The background metric remains fixed and non-dynamical, while gμνeff serves as a diagnostic field that translates local compression into measurable geometric deformation.
These constructions extend the analytic foundation developed in Theory of Derived Probability and Entanglement Compression and apply the continuity and oscillation behavior formalized in The Oscillation Principle. Together they define the formal bridge between probability density, compression strength, and curvature geometry within the unified structure of Entanglement Compression Theory.
3. Born-Rule Integration and Continuity
The Born-weights theorem is included within the main text under explicit axioms A0-A5,
which define normalization, noncontextuality, regularity, σ-additivity, and factorization.
Under these assumptions, the probability weights
pi = ⟨Ψ, Pi Ψ⟩
arise directly from the Primordial Wave Equation without external statistical postulates.
The continuity lemma is proven for real, multiplicative compression, giving
∂t|Ψ|² + ∇·J = 0 with J = (ħ / meff) Im(Ψ* ∇Ψ).
Conservation of norm and local no-signaling follow as corollaries.
These results link probability conservation to the same geometric consistency that governs curvature through the symmetric compression tensor. The treatment refines the probability derivation presented in Theory of Derived Probability and Entanglement Compression and carries forward the continuity and propagation structure developed in The Oscillation Principle. Together they demonstrate that the quantum measure, the continuity law, and curvature diagnostics emerge from one deterministic system rather than parallel assumptions.
4. Tensor Formalism and Metric Deformation
The tensor extension of Entanglement Compression Theory introduces a geometric layer that links probability flow to curvature through measurable deformation. The starting point is the symmetric compression tensor,
Cμν(sym) = ∇μ∇ν(−lnερ₁),
which describes how spatial variations in density produce second-order curvature within the Lawrence Universal Wave Function. Each component of this tensor represents a local measure of compression curvature, formed directly from derivatives of the normalized one-body density ρ₁.
To connect this curvature to geometry, the effective metric is defined by a first-order perturbation of the background metric, gμνeff = gμν + κ̃L*²Cμν(sym). Here, κ̃ is a dimensionless coupling and L* sets the characteristic compression scale. This relation interprets the tensor field as a source of geometric response: compression produces curvature, and curvature feeds back into the propagation of the wave function through gμνeff.
The corresponding inverse metric follows from the first term of the Neumann expansion, gμνeff = gμν − κ̃L*²C(sym)μν. This approximation holds while the deformation amplitude obeys the small-deformation bound ∥κ̃L*²C(sym)∥op < 1. The bound maintains the Lorentzian signature of spacetime and guarantees that higher-order corrections remain convergent. Physically, it limits the compression energy density so that curvature remains a measurable but non-singular deviation.
Under this regime, the tensor formalism ensures that probability density, continuity, and curvature share a single covariant representation. The metric deformation does not introduce a new field; it is a diagnostic reflection of compression geometry within the same deterministic system. This approach extends the analytic base developed in Theory of Derived Probability and Entanglement Compression and connects it to the propagation structure established in The Oscillation Principle. Together these results provide a consistent geometric interpretation of the Primordial Wave Equation and establish the groundwork for coupling compression geometry to the Einstein field framework.
The resulting picture is compact: the compression tensor defines curvature, the effective metric records it, and the Lorentzian bound secures stability. Each step remains derivable from the same deterministic law, confirming that geometry in ECT is not an imposed structure but an emergent property of compression itself.
5. Eikonal and Transport Structure
The geometric-optics limit of the Primordial Wave Equation is obtained by introducing the WKB ansatz
Ψ = A eiS / ħ,
where A(x, t) is a slowly varying amplitude and S(x, t) is a rapidly varying phase.
Substituting this form into the equation
iħ ∂tΨ = −αd∇²Ψ + β ℂ[Ψ] Ψ
and collecting powers of ħ gives a systematic expansion that separates classical motion from quantum transport.
At leading order, the ħ0 terms yield a Hamilton-Jacobi relation on the effective geometry, geffμν ∂μS ∂νS = 0. This equation defines the null surfaces of the effective metric and therefore the causal structure of propagation. The compression tensor enters implicitly through geffμν, producing measurable shifts in the null cone when local curvature changes. In geometric-optics diagnostics, these shifts correspond to minute variations in propagation delay and interference phase, providing a direct test of curvature induced by compression.
The next order, proportional to ħ1, produces a transport equation for the amplitude, ∇μ(A² ∂μS) = 0. This represents conservation of probability flux along the trajectories defined by S(x, t). Amplitude modulation therefore encodes local changes in compression intensity and curvature. Because the compression operator ℂ[Ψ] is real and multiplicative, energy flow and probability current remain locally conserved under the same continuity condition established in earlier sections.
Together these relations show that the eikonal and transport structure of the Primordial Wave Equation reproduces a deterministic, covariant form of geometric optics. The null-cone geometry of geffμν defines the causal domain, while the amplitude equation governs energy and probability transport within that domain. Observable cone shifts and transport deviations become the measurable signatures of compression-induced curvature. This analysis extends the wave-propagation results of The Oscillation Principle and links them to the geometric corrections defined here, providing a coherent experimental path for testing Entanglement Compression Theory at the level of field propagation and signal delay.
6. Falsifiability and Experimental Parameters
The tensor formalism of Entanglement Compression Theory defines a finite set of measurable and tunable parameters that determine its experimental reach.
The primary quantities are the coupling constant κ̃, the compression length scale L*, the phase-frequency ratio θf, the spatial regularization scale εx, the effective mass meff, the nonlinear coefficient β, and the calibration windows ηi used for device normalization.
Each parameter controls a specific physical effect, making the framework directly testable.
The product κ̃L*² sets the magnitude of geometric deformation through the symmetric compression tensor. Variation of this product within the bound ∥κ̃L*²C(sym)∥op < 1 determines the measurable cone shift in geometric-optics experiments. The parameter θf governs the phase compression rate and can be constrained by high-precision interferometry. The small-scale regulator εx fixes the spatial cutoff for compression gradients and can be explored through diffraction-limited optical or matter-wave setups.
The effective mass meff links the curvature response to inertial scaling, providing a measurable bridge between material composition and geometric deformation. The coefficient β defines the nonlinear response strength of the compression term ℂ[Ψ] and is recoverable from transport-law deviations measured in controlled amplitude-phase propagation. The calibration parameters ηi identify reference states for reproducibility across instruments and density profiles.
Predicted experimental signatures include: (1) null-cone shifts proportional to κ̃L*², (2) transport-law corrections observable through phase-velocity dispersion, and (3) probability-distribution bounds where |pi − si| < 4ε / (κ̃L*²). These observables provide falsifiable limits that distinguish deterministic compression dynamics from conventional quantum indeterminacy.
Together the parameter set defines a clear experimental program: control L* and κ̃ to tune curvature strength, measure phase delay through θf, verify probability stability through β and ηi, and confirm geometric consistency by monitoring cone deformation. This falsifiability structure continues the predictive discipline established in Theory of Derived Probability and Entanglement Compression and extends it into the geometric domain developed here, closing the loop between mathematical formulation and measurable effect.
7. Significance and Continuity
The tensor formalism establishes a covariant and falsifiable connection between compression dynamics and curvature.
It completes the mathematical chain initiated in
Theory of Derived Probability and Entanglement Compression,
extended through the analytic structure formalized in
Mathematical Foundations of Derived Probability and ECT,
and consolidated by the propagation behavior characterized in
The Oscillation Principle.
Together these works define a coherent framework in which probability, continuity, and curvature emerge from the same deterministic compression law.
By extending the Primordial Wave Equation into tensor geometry, the present formulation provides the missing link between quantum probability and spacetime curvature. Compression, previously expressed as a scalar interaction, now carries geometric meaning through the symmetric tensor Cμν(sym) and its metric response gμνeff. This establishes a clear route from local density gradients to measurable curvature effects, closing the conceptual gap between microscopic probability structure and macroscopic geometry.
The resulting theory is deterministic, covariant, and testable. It unifies the statistical and geometric sectors under a single wave-dynamic law and provides a pathway toward the Einstein-extension framework that will couple compression geometry to gravitational curvature. The tensor formalism therefore represents not only the completion of the first ECT series but also the foundation for its continuation into relativistic and cosmological regimes.
Source: Lawrence, W.A. (2025). Unified Derivation of Probability, Curvature, and Compression Geometry in Entangled Systems: A Tensor-Formalism Extension of Entanglement Compression Theory (ECT). Zenodo. https://doi.org/10.5281/zenodo.17349900
ECT Canonical Series:
Theory of Derived Probability and Entanglement Compression |
The Oscillation Principle |
Mathematical Foundations of Derived Probability and ECT