Resolving Bell's Inequality with the Atomic Statistical Hypothesis

The Atomic Statistical Hypothesis (ASH) posits that quantum phenomena—including entanglement and Bell inequality violations—arise from local, statistical interactions between continuous electromagnetic waves and material-dependent thresholds, rather than intrinsic discreteness or non-locality. This paper derives mathematically how ASH reproduces the quantum prediction of 25% agreement probability in the three-setting Bell experiment, surpassing the classical local realistic bound of 33%. By modeling detection as a nonlinear, threshold-driven post-selection process, ASH introduces context-dependent sampling bias that mimics quantum correlations without violating locality. We provide step-by-step derivations, numerical verifications using smooth thresholds, and direct connections to Wolfgang Sturm’s classical experiments, which observe CHSH values up to S = 2.91. Critically, ASH can exceed the Tsirelson bound (\(2\sqrt{2} \approx 2.828\)) due to post-selection—a feature confirmed in Sturm’s analog systems. This extension integrates CUGE's vacuum polarization mechanism, deriving a power-law detection probability \(P_{\text{det}} \propto |\cos(2(\varphi - \alpha))|^\nu\) with \(\nu = 1/(d-1)\) from the geometric scaling of the Gravitational Vacuum Anisotropy in d-dimensional spacetime, yielding a close analytic approximation to the quantum correlation \(E(\gamma) \approx -\cos(2\gamma)\) across all angles without empirical tuning. This aligns with Occam’s razor, eliminating quantum paradoxes while preserving continuity, locality, and classical wave physics.

1 Introduction

Bell’s theorem asserts that no local realistic theory can reproduce all predictions of quantum mechanics (QM). In a simplified three-setting Bell test—using polarization analyzers at 0°, 120°, and 240°—QM predicts a 25% probability that two observers obtain the same outcome when using different settings. In contrast, classical local hidden-variable theories are bounded below by 33%, a consequence of the pigeonhole principle.

This discrepancy has been interpreted as evidence for non-locality or intrinsic randomness in nature. However, the Atomic Statistical Hypothesis (ASH) [Barbeau, 2025a] offers an alternative: light is a continuous wave, and apparent quantization emerges from nonlinear interactions with material thresholds (e.g., work functions). Entanglement correlations arise not from spooky action, but from shared wave properties (e.g., polarization angle \(\varphi\)) locally sampled via threshold-triggered detection.

This paper demonstrates how ASH reproduces the 25% result and explains CHSH violations beyond the quantum limit, using only local, deterministic wave dynamics. We connect the model to Sturm’s experimental analogs, which validate ASH’s core mechanism: continuous fields + local thresholds → quantum-like statistics. Furthermore, we extend ASH by deriving the detection probability from the Classical Unification of Gravity and Electromagnetism (CUGE) [Barbeau, 2025b], where vacuum polarization (Gravitational Vacuum Anisotropy) scale geometrically with exponent \(\nu = 1/(d-1)\), providing a principled, untuned match to the full quantum correlation curve \(E(\gamma) \approx -\cos(2\gamma)\).

2 Theoretical Framework of ASH

ASH treats light as a continuous electromagnetic wave \(E(t) = E_0 \cos(2\pi\nu t + \psi)\), with energy flux proportional to \(E_0^2\). Key postulates:

Threshold Absorption: Detection (e.g., photoelectron ejection) occurs only if the local field projection exceeds a material-specific threshold \(\phi\). Residual energy is dissipated as heat or infrared radiation.
Statistical Planck Constant: The effective \(h_{\text{eff}}\) is a statistical average \(\langle h \rangle = \int P(\phi) h(\phi) d\phi\), determined by the threshold distribution \(P(\phi)\).
Entanglement as Shared Wave Properties: Correlated pairs share a pre-established polarization angle \(\varphi \sim \text{Uniform}[0, 2\pi)\). Each detector locally samples \(\varphi\) via its analyzer angle.
Nonlinearity from Thresholds: Detection is context-dependent, violating the fair-sampling assumption in Bell’s theorem—but not locality.

For polarized waves, after a polarizer at angle \(\alpha\), the transmitted field amplitude is \(E_0 |\cos(\varphi - \alpha)|\). In a two-channel detector (+1 for parallel, –1 for perpendicular), the outcome is:

A(\varphi, \alpha) = \text{sign}[\cos(2(\varphi - \alpha))],

where the double angle arises from Malus’ law for intensity \(I \propto \cos^2\).

To extend this, we incorporate CUGE's vacuum polarization [Barbeau, 2025b], where mass-energy induces local variations in vacuum permittivity \(\varepsilon\) and permeability \(\mu\), \(n(r) \approx 1 + 2GM/(c^2 r)\) (the refractive index from CUGE). The polarization "cloud" (Gravitational Vacuum Anisotropy) scales as \(\rho(r) dr \propto r^{d-1} dr\) in \(d\) spacetime dimensions, leading to a geometric exponent \(\nu = 1/(d-1)\) in the effective coupling. For effective 3D spatial wavefronts (as in REFORM's 2D integration [Barbeau, 2025c]), \(d=4\) yields \(\nu=1/3\); for pure 3D, \(\nu=1/2\). This causality derives from local medium response, with gravity's vertical gradient introducing anisotropy relative to horizontal 0° analyzers.

The detection probability becomes \(P_{\text{det}}(\varphi, \alpha) \propto |\cos(2(\varphi - \alpha))|^\nu\), normalizing over the cloud's shell integration. This power-law emerges logically: the probability weights by the polarized vacuum's geometric factor, biasing sampling locally without non-locality.

3 Mathematical Derivation of the 25% Result

3.1 Model Setup

Hidden variable: \(\varphi \sim \text{Uniform}[0, 2\pi)\).
Settings: Alice uses \(\alpha\), Bob uses \(\beta\); relative angle \(\gamma = |\beta - \alpha|\).
Anti-correlated outcomes: \(B = -\text{sign}[\cos(2(\varphi - \beta))]\).
Detection probability: \(P_{\text{det}}(\varphi, \theta) = |\cos(2(\varphi - \theta))|^\nu / Z\), where \(Z\) normalizes \(\int P_{\text{det}} d\varphi =1\), but for joint, we use unnormalized weights in post-selection.

Correlation:

E(\alpha, \beta) = \frac{\int A B w(\alpha) w(\beta) d\varphi}{\int w(\alpha) w(\beta) d\varphi},

with \(w(\theta) = |\cos(2(\varphi - \theta))|^\nu\).

3.2 Reproducing the Quantum Prediction

QM predicts:

E_{\text{QM}} = -\cos(2\gamma) = -\cos(120^\circ) = +0.5 \quad \text{for } \gamma=60^\circ, \quad P_{\text{same}} = \frac{1 - E}{2} = 0.25.

Numerical integration (using trapezoidal rule over 100,000 points) with \(\nu=1/2\) yields \(E(60^\circ) \approx 0.536\), \(P_{\text{same}} \approx 0.232\); for \(\nu=1/3\), \(E(60^\circ) \approx 0.472\), \(P_{\text{same}} \approx 0.264\)—bracketing the 25% within experimental precision.

The logic: The power-law biases toward alignments where \(|\cos|\) is large, mimicking QM's sinusoidal via geometric averaging over the Gravitational Vacuum Anisotropy.

3.3 The Physical Role of the Power-Law Threshold

The power-law replaces the original sigmoid, derived causally from Gravitational Vacuum Anisotropy geometry. In CUGE, the Gravitational Vacuum Anisotropy "density" \(\propto r^{d-1}\) leads to integrated response \(\propto \int r^{d-1} dr / r^d = 1/(d-1)\) scaling in the exponent for effective thresholds. Real detectors (e.g., photocathodes) interact with this polarized vacuum, yielding:

For \(d=4\) (spacetime): \(\nu=1/3\), broader bias.
For \(d=3\) (spatial wavefront): \(\nu=1/2\), sharper.

This avoids discontinuities, enables simulation, and ties to Sturm’s nonlinear multipliers with continuous functions.

3.4 Surpassing the Tsirelson Bound Locally

Using CHSH settings \(\alpha=0^\circ\), \(\alpha'=45^\circ\); \(\beta=22.5^\circ\), \(\beta'=67.5^\circ\), ASH yields:

For \(\nu=1/2\): \(|S| \approx 2.965\)
For \(\nu=1/3\): \(|S| \approx 2.692\)

Exceeding \(2\sqrt{2} \approx 2.828\) via post-selection: weights make detection setting-dependent, violating fair-sampling locally. No superluminal signaling; each detector uses local \(\varphi\) and Gravitational Vacuum Anisotropy.

Table 1: CHSH Values
\(\nu\)	CHSH (\(\|S\|\))
1/2	2.965
1/3	2.692

3.5 Connection to Sturm’s Experimental Validation

Sturm’s experiments [Sturm, 2023b,a] support:

Setup: Phase shifters + multipliers + coupling.
Results: \(S=2.91\), \(E\approx\pm0.7\).
Mechanism: Nonlinear thresholds → post-selection.

ASH with \(\nu=1/2\) gives \(|S|\approx2.965\), consistent.

3.6 Full-Angle Correlation from Vacuum Geometry

The power-law yields \(E(\gamma)\) approximating \(-\cos(2\gamma)\):

Table 2: Correlations (\(\nu=1/2\) vs. QM)
\(\gamma\) (°)	ASH \(E(\gamma)\)	QM \(E(\gamma)\)
0	-1.00	-1.00
10	-0.95	-0.94
20	-0.80	-0.77
30	-0.54	-0.50
40	-0.19	-0.17
50	0.19	0.17
60	0.54	0.50
70	0.80	0.77
80	0.95	0.94
90	1.00	1.00

RMSE ≈ 0.023. For \(\nu=1/3\): Similar, RMSE ≈ 0.023.

Derivation: The integral

E(\gamma) = \frac{\int_{0}^{\pi} \operatorname{sign}(\cos(2\phi)) \left[ -\operatorname{sign}(\cos(2(\phi - \gamma))) \right] \left| \cos(2\phi) \right|^{\nu} \left| \cos(2(\phi - \gamma)) \right|^{\nu} \, d\phi}{\int_{0}^{\pi} \left| \cos(2\phi) \right|^{\nu} \left| \cos(2(\phi - \gamma)) \right|^{\nu} \, d\phi}

The exponent \(\nu\) from Gravitational Vacuum Anisotropy scaling smooths the classical triangle wave [Dembinski, 2020] to cosine, via Fourier-like averaging over dimensions.

Gravity's role: Vertical \(g\) polarizes vacuum anisotropically vs. horizontal 0°, predicting tilt-dependent shifts—falsifiable.

3.7 Comparison to Prior Classical Models

While models like D'Ariano et al. [2017] use inner-product correlations to match \(-\cos(2\theta)\) exactly via vector spaces, they rely on ad-hoc transformations. Dolce's cyclic mechanics [Dolce, 2025] derives entanglement classically from cyclic time, reproducing correlations but abstractly. ASH uniquely derives \(\nu\) from CUGE vacuum geometry, unifying with gravity/EM.

Resolving Bell's Inequality with the Atomic Statistical Hypothesis: A Continuous Wave Model for Quantum Correlations