A Geometric Coincidence: How a Deterministic Global Phase Model Reproduces the CHSH Value Through Trigonometric Averaging

Important: This repo does not simulate ±1 Bell outcomes; it computes a bounded field correlation $E_{\text{raw}}=\langle A\,B\rangle$ with $A\cdot B\in[-1,1]$ and then reports the rescaled value $E=-2\,E_{\text{raw}}$ (so the printed CHSH $S$ applies to that constructed $E$, not directly to $\langle A B\rangle$).

Note: The model reproduces the quantum correlation $−\cos(2θ)$ not by magic, nor by nonlocality but by the continuous trigonometric structure of rotational symmetry. It does not use discrete outcomes $(±1)$ but continuous values.

Model Definition

The model uses two observables defined as:

• $A(a, λ) = \cos(2(a − λ))$
• $B(b, λ) = −\cos(2(b − λ))$

The correlation function is then:

$E(a,b) = ⟨A(a,λ) \cdot B(b,λ)⟩ = −\frac{1}{2} \cos(2(a−b))$

The factor of two required to match quantum mechanics $E(a,b) = −\cos(2(a−b))$ arises from the inherent averaging of a product of cosine waves over a uniform hidden phase:

$⟨\cos(x)\cos(y)⟩ = \frac{1}{2} \cos(x−y)$

This is not a trick — it is Fourier arithmetic.

Key Points

1. No violation of Bell's theorem: The model does not simulate discrete measurement outcomes, only the underlying continuous correlation function.
2. Geometric interpretation: The $−\cos(2θ)$ dependence is not arbitrary, it's the direct consequence of SU(2) rotational symmetry.
3. Local structure: The math is local in structure; one wave, two measurements, no signaling.

Pedagogical Implications

While quantum mechanics correctly predicts this correlation and Bell's theorem is valid, the pattern is often presented as evidence of "spooky action at a distance." However, this model shows that the correlation might instead be evidence of symmetry.

Interpretation

Perhaps the deeper structure isn't disconnected particles, but a single resonant field whose correlations are written in the language of geometry. We don't need to reject QM to ask: What else might the math be trying to tell us?

Footnote: While the mathematical formalism appears local, the model fundamentally relies on a globally shared hidden variable λ making it nonlocal without signaling. The uniform averaging of λ is a key assumption not derived from first principles and its role in reproducing quantum correlations deserves scrutiny. The pedagogical implication that "spooky action" might stem more from correlation patterns than fundamental nonlocality does not address quantum mechanics discrete measurement outcomes or probabilistic framework.