The Theory of Derived Probability and Entanglement Compression

Entanglement Compression Theory (ECT) is a deterministic framework that unifies quantum mechanics, general relativity, and cosmology. It holds that probability and curvature, the two organizing principles of modern physics, are complementary outcomes of a single law: the compression of an entangled oscillatory field that sustains all physical structure. From this process arise both the geometry of spacetime and the statistical behavior of quantum systems.

This page outlines the paper The Theory of Derived Probability and Entanglement Compression, which establishes the mathematical foundation of that claim. It derives Born weights, continuity, and curvature from one deterministic equation—the Primordial Wave Equation (PWE)—and shows that “chance” is not fundamental but a record of how energy divides under compression.

0. Background and Purpose

Every physical theory defines what it treats as fundamental. Quantum mechanics assumes irreducible randomness, and general relativity assumes intrinsic curvature. ECT rejects both assumptions. It treats them as emergent consequences of a deterministic entangled field whose compression and relaxation generate the patterns observed as particles, forces, and spacetime geometry.

The aim of this first paper is to demonstrate that the probabilistic rule of outcomes (the Born rule) and the geometric rule of motion (curvature) can both be derived from one continuous law, without stochastic collapse or hidden variables. When an entangled field compresses, its energy divides deterministically among available modes according to local geometric tension. What appears as “probability” is the statistical trace of that division.

The governing field is the Lawrence Universal Wave Function (Ψ_L), which evolves under the Primordial Wave Equation:

iħ ∂tΨ = (−αd Δ + V(x,t) + β ℂ[Ψ]) Ψ .

The PWE joins the familiar Schrödinger dispersion term with a real, multiplicative compression operator ℂ[Ψ], which describes how amplitude density shapes local curvature and time asymmetry. Under the usual measure assumptions—normalization, σ-additivity, and noncontextuality—the equation reproduces Born probabilities as a deterministic result of energy conservation. Detailed proofs of continuity and global well-posedness appear in Mathematical Foundations of Derived Probability and ECT (DOI 10.5281/zenodo.17071135).

Where conventional quantum theory inserts randomness, ECT introduces compression. From this single substitution , causality, curvature, and probability follow as deterministic outcomes. This first paper establishes the mechanism; later works extend it to tensor formalism, geometric coupling, and gravitational tests.

1. Measurement and Derived Probability

In conventional quantum mechanics, measurement stands apart from the rest of physics. It is treated as a discontinuity that converts a continuous wave function into a discrete outcome through probabilistic collapse. Entanglement Compression Theory removes that exception. It treats measurement as a standard episode of field evolution: when an apparatus interacts with a system, the two form a single entangled field whose amplitude distribution compresses and redistributes energy among all resolvable outcome channels. The fractions of total energy that emerge in those channels appear, from an observer’s limited view, as probabilities.

This physical process of compression replaces the abstract postulate of collapse. As the combined system undergoes local contraction in configuration space, its normalization and total energy remain conserved while amplitude density reorganizes according to geometric tension. Each resolvable channel of compression corresponds to a projection operator P_i acting on the total field Ψ. Because compression is conservative, the share of energy reaching each channel is proportional to its squared amplitude in the L² sense:

pi = ⟨Ψ, Pi Ψ⟩ .

This expression, familiar as the Born rule, is not assumed but derived. Appendix A.6 of the paper shows that when compression acts locally, the measure axioms of normalization, σ-additivity, and noncontextuality are sufficient to enforce equality between energy share and amplitude weight. Under these conditions, probability serves as a bookkeeping record of deterministic energy division within the compressing field. There is no hidden variable, no stochastic trigger, and no observer-induced collapse. The appearance of chance reflects the coarse-grained limit of continuous energy redistribution.

The result establishes that probability is not fundamental. It is an emergent statistic of deterministic compression—a macroscopic shadow cast by the microscopic conservation of amplitude and curvature. Quantum randomness is neither mysterious nor acausal; it is the statistical residue observed when the continuous geometry of energy flow is resolved into discrete measurement outcomes.

2. The Primordial Wave Equation and the Compression Operator

The dynamics of Entanglement Compression Theory are governed by the Primordial Wave Equation (PWE). It generalizes the linear Schrödinger framework by introducing a deterministic, real compression term that couples amplitude to curvature. The equation is written as

iħ ∂tΨ = (−αd Δ + V(x,t) + β ℂ[Ψ]) Ψ .

Here α_d = ħ²/(2m_eff) controls dispersion, V(x,t) represents the external potential, and βℂ[Ψ] introduces self-organizing compression. The operator ℂ[Ψ] is real and multiplicative, making the equation nonlinear yet deterministic and ensuring that the energy operator remains Hermitian under regularity conditions. Conventional quantum mechanics adds a stochastic postulate to explain outcome statistics; the PWE embeds that structure directly in the field’s dynamics. Compression modifies the effective geometry in which Ψ evolves.

To maintain smoothness wherever ρ₁ = |Ψ|² ≠ 0 and to avoid singularities at low densities, ℂ[Ψ] takes a regularized logarithmic form:

ℂ[Ψ](x) = −c₀ lnε(ρ₁(x)/ρ₀ + δ) .

Small parameters ε and δ regularize the operator at vanishing amplitude, c₀ defines the compression scale, and ρ₀ sets a reference density. This form guarantees differentiability and bounded curvature response across all regions of the domain. In dense regimes, compression gradients generate measurable curvature and phase asymmetry. In the weak-compression limit (β → 0), the nonlinear term vanishes and the equation reduces continuously to the standard Schrödinger equation. ECT therefore recovers verified physics where compression effects are negligible and extends it into regimes where curvature and probability share a single origin.

2a. Mathematical Summary (Continuity, Normalization, and Well-Posedness)

The Primordial Wave Equation preserves global normalization, enforcing

∫ |Ψ(x,t)|² dx = 1 .

Differentiating the PWE and using the Hermitian structure yields the continuity law

∂tρ₁ + ∇·j = 0, where j = (ħ/meff) Im(Ψ*∇Ψ) .

This condition ensures conservation of total energy flux through any closed surface. Under bounded time derivative of the information functional ∂_tℐ and Lipschitz continuity of ℂ[Ψ], the initial-value problem is globally well-posed in H¹ for spatial dimension d ≤ 3. These conditions, proved in Mathematical Foundations of Derived Probability and ECT (DOI 10.5281/zenodo.17071135, §4–5), guarantee that Ψ evolves deterministically and smoothly without collapse or singular divergence. The resulting conservation laws—energy, normalization, and curvature flux—form the mathematical backbone supporting later derivations of Born weights, curvature coupling, and transport.

ECT preserves the mathematical discipline of quantum mechanics while reinterpreting its probabilistic postulates as consequences of a continuous, deterministic compression process. The PWE defines the complete dynamical core from which all subsequent results—derived probability, geometric deformation, and gravitational correspondence—follow.

3. Curvature from Compression

In general relativity, curvature is sourced by stress-energy: matter tells spacetime how to curve, and curvature tells matter how to move. Entanglement Compression Theory lowers that principle to a deeper level. It holds that what general relativity describes as mass-energy is an emergent signature of informational curvature—gradients in amplitude density within the universal wave function. Compression generates these gradients naturally: as an oscillatory field contracts, its amplitude becomes unevenly distributed, producing measurable geometric tension.

This local geometric response is represented by the compression tensor, defined in symmetric form as

C(sym)μν = ∇μ∇ν(−lnερ₁) ,

where ρ₁ = |Ψ|² is the local amplitude density and ln_ε is the regularized logarithm that maintains smoothness across low-density regions. This tensor functions as an informational analogue of the stress-energy tensor in relativity: it encodes how spatial and temporal curvature arise from internal gradients of compression rather than from external mass-energy sources.

The effective spacetime metric that governs transport and optical propagation follows directly:

geffμν = gμν + κ̃ L*² C(sym)μν .

Here g_μν is the background metric, κ̃ sets the coupling between compression geometry and spacetime curvature, and L_* defines the characteristic compression scale. When gradients are small, this deformation is negligible and the standard geodesic equations of general relativity are recovered exactly. Where compression varies strongly—near energetic boundaries, phase interfaces, or quantum-optical nodes—the corrections become finite and measurable.

The tensor C^(sym)_μν plays two roles: it supplies curvature to the effective metric and it determines how amplitude information propagates through space and time. Light rays and probability flux both follow the deformed null structure of g^eff_μν. As shown in Mathematical Foundations of Derived Probability and ECT (§6.2) and extended in Unified Derivation of Probability, Curvature, and Compression Geometry (§3–4), contraction of this tensor with ray vectors reproduces the Eikonal and WKB transport equations in the small-deformation limit. Standard optical paths therefore appear as first-order approximations to deterministic compression geometry.

At higher gradients, ECT predicts small but measurable deviations: phase-shift asymmetries, residual lensing, interferometric drift floors, and fractional timing offsets that scale with κ̃ L_*²∇²lnρ₁. These signatures provide falsifiable tests of the theory’s geometric predictions. A null result below the predicted thresholds constrains the compression coupling directly, while a confirmed residual would establish curvature as an emergent property of informational flow rather than an independent attribute of spacetime.

Curvature is not imposed by matter; it arises from it’s own organization. Matter, geometry, and measurement trace back to the same origin—the compression of an oscillatory continuum whose gradients sculpt the structure of reality.

4. The Compression–Balance Invariant and Effective c(x)

Every oscillatory system balances two opposing drives: dispersion, which spreads energy outward, and compression, which concentrates it. Entanglement Compression Theory formalizes this balance through the ratio of kinetic to compressive energy densities, expressed as the compression–balance invariant:

U(x) := T(x) / C(x), with c(x)² = U(x) .

Here T(x) represents the local kinetic or dispersive tension, proportional to |∇Ψ|², and C(x) denotes the curvature or compression potential derived from the operator ℂ[Ψ]. Their ratio defines a dimensionless invariant that measures the dynamic equilibrium of the field at each point in space. When U(x) = c², the field is in harmonic balance: dispersion exactly offsets compression, and propagation proceeds at the standard light speed. Where U(x) differs from c², the local geometry exhibits small effective shifts in propagation rate, refractive index, or phase velocity.

These departures do not imply that nature’s constants vary freely; they describe context-effective changes in the local metric generated by compression gradients. Because the effective metric is

geffμν = gμν + κ̃ L*² C(sym)μν,

the invariant U(x) measures how much the local compression tensor contributes to that deformation. In slowly varying regions, ∇U ≈ 0 and the effective speed c(x) equals the universal constant c. Where amplitude changes rapidly—near optical boundaries, gravitational wells, or strong field interactions—U(x) departs slightly from c², producing measurable phase lags or timing offsets. These deviations remain bounded and reversible: when compression relaxes, U → c² again.

ECT identifies four parameters that set the scale and strength of these effects: κ̃ (compression–geometry coupling), L_* (characteristic compression length), m_C (effective compression mass), and λ_C (characteristic wavelength). Together they define the transition region where quantum-optical and gravitational effects overlap. Because U(x) connects the kinetic and curvature sectors directly, it provides a quantitative link between microscopic dispersion and macroscopic spacetime structure.

In the linear limit—small κ̃L_*² and weak gradients—U(x) stabilizes to c², recovering classical relativistic transport. At higher compression gradients, small corrections appear, measurable as fractional variations in interferometric phase, pulsar timing, or high-energy spectral drift. The invariant functions as both a local diagnostic of compression geometry and a direct experimental handle on ECT’s coupling constants.

Conceptually, U(x) is the field’s self-regulating equilibrium constant. It maintains coherence by balancing outward kinetic dispersion against inward geometric pressure. Where the balance holds, spacetime is smooth and uniform; where it fluctuates, curvature and information flow intertwine, revealing the dynamics behind the apparent constancy of the speed of light.

5. Emergence and Structural Hierarchy

Entanglement Compression Theory defines emergence as a physical cascade, not a metaphor. From the simplest oscillatory motion arise successive layers of structure that compose the observable universe. Each stage preserves the rules of the one below it while adding a new form of stability. The progression is deterministic and recursive: oscillation creates interference, interference generates curvature, and curvature forms geometry. The laws of physics are the long-lived equilibria of that compression flow.

The hierarchy proceeds through the following stages:

• Oscillation: establishes continuity and flux conservation. The universal field Ψ_L oscillates in complex phase, ensuring smooth propagation of local interactions.
• Interference: superposition of oscillations forms interference patterns—localized amplitude modulations that encode energy density variations while conserving total normalization.
• Compression: interference generates gradients in amplitude density ρ₁ = |Ψ|². These gradients drive real compression terms ℂ[Ψ], introducing tension and directionality into the field and initiating deterministic energy partition.
• Curvature: double gradients of −lnρ₁ form the symmetric compression tensor C^(sym)_μν, the seed of geometric curvature. Spacetime is the smoothed geometry of these compression gradients.
• Metric: the effective metric g^eff_μν = g_μν + κ̃L_*²C^(sym)_μν governs transport, redshift, and lensing. Classical laws—relativity, optics, and field dynamics—emerge as mean-field limits of this geometry.

Because every layer inherits its structure from the same governing equation—the Primordial Wave Equation—quantum and geometric behavior remain coherent. Energy division, probability weights, and curvature share a common origin in compression dynamics. In regions of equilibrium, the system reproduces conventional physics; under strong gradients, new behaviors appear predictably and without randomness.

This cascade forms a hierarchy of descriptive scales rather than distinct theories. At one end is pure oscillation, the continuous substrate of existence; at the other is geometry, the stabilized form of compression flow Between them, information organizes itself into particles, fields, and forces. ECT makes this structure explicit, showing that emergence is the natural mode of the universe.

6. Relation to the Tensor Formalism

This paper introduces the compression tensor as the geometric imprint of amplitude gradients, linking local information density to curvature. Its purpose is to show that curvature originates from measurable compression geometry: the second derivatives of −lnρ₁ define the shape of space itself. Geometric deformation follows directly from amplitude structure rather than from external stress–energy sources.

Formally, curvature is represented by the symmetric second-derivative form

C(sym)μν = ∇μ∇ν(−lnρ₁) ,

which encodes how compression modifies local geometry. The corresponding effective metric is

geffμν = gμν + κ̃L*²C(sym)μν .

In weak-compression domains this form reproduces general relativity exactly. Under strong gradients it predicts small, testable departures. These curvature responses arise intrinsically from compression dynamics; no auxiliary matter field or scalar potential is required. The result is a geometrically closed model in which space, curvature, and energy division emerge from the same deterministic substrate.

The full tensor construction, including coupling coefficients, small-deformation expansions, and transport corrections, is developed in two companion works: Mathematical Foundations of Derived Probability and ECT: Well-Posedness and Gravitational Tests (§6.2), and the tensor-formalism extension Unified Derivation of Probability, Curvature, and Compression Geometry in Entangled Systems (§§3–4). Those analyses show that contraction of C^(sym)_μν with ray vectors reproduces the Eikonal and WKB transport equations in the small-deformation limit and quantify the finite corrections that distinguish ECT from general relativity at high compression gradients.

7. Outlook Toward the Einstein Field Equations

The Theory of Derived Probability and Entanglement Compression establishes how curvature arises from compression but does not yet present the full field-theoretic framework linking the Primordial Wave Equation to Einstein’s equations. That step—treating compression as an independent field C(x) with derivatives C_μν = ∇_μ∇_νC and embedding it within a unified action functional—is developed in the companion work Tensor-Formalism and Curvature Geometry (DOI 10.5281/zenodo.17349900). There, variation of the total action

S = ∫ d⁴x √|g| [ Lgrav + LΨ + LC + Lint ]

produces coupled equations for Ψ and C and recovers the Einstein Field Equations in the weak-field limit. This formulation completes the conceptual bridge outlined here: the deterministic mechanism that yields Born weights also generates curvature in a covariant field description.

8. Falsifiability and Current Experimental Targets

Entanglement Compression Theory is constructed for empirical verification. Its parameters—κ̃, L_*, m_C, and λ_C—map directly onto measurable quantities that govern curvature coupling, compression length, and characteristic oscillation rates. The framework defines explicit numerical domains where deviations from general relativity or standard quantum optics should appear. A verified null result below these thresholds constrains the parameters immediately, narrowing or excluding portions of the model’s range. ECT is therefore a falsifiable physical theory, not a conceptual reformulation.

Because the compression tensor modifies transport, refraction, and coherence at the geometric level, the theory’s predictions extend from laboratory interferometers to astrophysical observations, spanning more than twelve orders of magnitude. Representative experimental targets include:

• Astrophysical tests: microlensing residuals (δθ ≈ 1–3 μas) across large density contrasts, and timing asymmetries in pulsar or quasar delays (Δt/t ≈ 10⁻⁸–10⁻⁷). Both probe cumulative curvature deviations over astronomical distances.
• Solar-limb propagation: context-effective light-speed variations, Δc/c ≈ (1–3)×10⁻⁶, along steep compression gradients near the solar limb. These geometry-linked shifts distinguish ECT curvature from plasma-dispersion effects.
• Interferometry: baseline-independent phase drift floors Δφ ≈ 10⁻¹⁸ rad·m⁻¹ in cryogenic Michelson or LIGO-class instruments, produced by residual compression curvature along vacuum paths.
• Cavity QED systems: coherence-threshold offsets γ_c ≈ (2–5)×10⁻⁵ at critical photon densities, expressing the compression–dispersion balance of confined oscillatory fields.
• Cold-atom and BEC experiments: sub-ppm dephasing-time shifts under controlled gradient traps, directly probing the U(x) = T/C invariant and its influence on phase transport.

Each regime translates compression parameters into directly observable quantities. Together these tests define a falsifiability ladder that links quantum interference to cosmological curvature, ensuring empirical accountability at every scale.

8a. Quantitative Summary

The magnitudes above follow directly from the compression–geometry coupling term κ̃L_*² and the curvature scale embedded in the Primordial Wave Equation. Each predicted deviation, whether angular deflection, phase drift, or timing offset, scales monotonically with local compression curvature. Any experiment probing compression gradients therefore defines a numerical bound on the same parameter set.

A null result below the expected signal tightens the upper limit of κ̃L_*², constraining the admissible compression strength. A consistent deviation observed across independent domains—such as matching phase shifts in interferometry and gravitational lensing—would empirically fix this parameter window and confirm the deterministic curvature response predicted by ECT. Cross-domain correlation is the decisive test: the same coupling must reproduce identical ratios from laboratory to astrophysical scales.

Because the compression operator ℂ[Ψ] is dimensionless in the α–β formalism, all derived observables retain consistent energy units and dimensional closure. This guarantees that thresholds remain physically comparable across optical, atomic, and gravitational regimes, allowing coordinated experimental validation through a unified parameter set rather than arbitrary fitting constants.

9. Where ECT Fits

Entanglement Compression Theory does not replace existing physics; it defines the context in which it operates. When compression is negligible or smoothly varying, the Primordial Wave Equation reduces to the linear Schrödinger form, and the effective metric g^eff_μν reduces to g_μν. Quantum mechanics and general relativity emerge as large-scale equilibria of the same deterministic substrate.

ECT explains why these equations succeed and where their limits appear. Where information-density gradients steepen—inside interferometers, near gravitational lenses, within coherent cavities, or across cosmological contrasts—the compression tensor introduces small but cumulative corrections. These corrections are intrinsic, not ad hoc: they follow directly from the constants that govern amplitude partitioning and energy conservation. Probability weights and spacetime curvature therefore represent two aspects of a single compression process.

In practice, ECT unifies quantum mechanics and relativity without altering their mathematical form. It provides the deterministic field dynamics that make both possible. At smooth scales it is indistinguishable from conventional theory; at sharp gradients it delivers quantitative, falsifiable predictions. Its role is integrative: to show that coherence, curvature, and probability are expressions of the same compression law governing the physical world.

10. Addressing Common Objections

“Fringe or speculative” label

This characterization reflects citation status, not scientific merit. The ECT papers define closed-form operators, explicit unit systems, and falsifiable quantitative predictions. All parameters are constrained within narrow, testable ranges, distinct from the unconstrained assumptions of nonphysical models.

“Unclear compression operator”

The operator ℂ[Ψ] is explicitly defined in regularized logarithmic form and analyzed for differentiability, continuity, and boundedness. It conserves normalization and reduces exactly to linear quantum mechanics in the limit β → 0, ensuring both mathematical closure and physical recoverability.

“Variable constants conflict with data”

ECT alters only effective constants through the local invariant U(x) = T/C. Fundamental constants remain fixed. Contextual variations arise solely from geometric compression, producing bounded and reversible deviations. In validated domains U → c² and α remains constant, fully consistent with existing precision limits.

“No peer-reviewed verification”

The theory is directly testable across multiple experimental classes—lensing catalogs, pulsar timing arrays, precision interferometry, and cavity QED systems. Because predictions are parameterized, every null result tightens the model’s bounds, and every concordant observation cross-validates it. ECT’s falsifiability is intrinsic to its design.

11. Summary of Proofs and Appendices

The appendices of The Theory of Derived Probability and Entanglement Compression present the formal mathematics supporting each result summarized above. Appendix A.6 derives Born weights from deterministic energy partition, proving that statistical probabilities arise from conserved flux division. Appendices A.19–A.22 establish global well-posedness of the Primordial Wave Equation under bounded time flux and Lipschitz-continuous compression, confirming continuity and finite energy flow. Appendix A.21 links the compression tensor to optical geometry through explicit Eikonal and WKB analyses, while Sections G and S tabulate falsifiability templates, dimensional parameters, and experimental bounds.

Together these results demonstrate that probability and curvature are not separate postulates but complementary outcomes of a single deterministic wave law that unites information, geometry, and energy through compression.

Source: Lawrence, W.A. (2025). The Theory of Derived Probability and Entanglement Compression. Zenodo. https://doi.org/10.5281/zenodo.15786696