TFP Causal CHSH Theorem: A Complete Proof

By John Gavel

This theorem proposes that violations of the Bell inequality (the CHSH value \(S\)) are not due to non-locality (“spooky action”), but are a direct, geometric consequence of the causal overlap between the experiment’s components, as defined by a specific effective causal speed \(c_\text{eff}\).

Key Definitions and Setup

Let the experiment be set up relative to the Source O event.

• Source (O): Spacetime point (0,0). The origin where a coherent parent cluster splits into two anti-correlated clusters.
• Alice (E_A): Measurement event (L, T_A). Alice’s detector is physically rotated to angle \(\theta_a\) at position L and time T_A.
• Bob (E_B): Measurement event (-L, T_B). Bob’s detector is physically rotated to angle \(\theta_b\) at position -L and time T_B.
• Hidden Variable (\(\lambda\)): The initial flow state of the parent cluster. In TFP, this is operationally defined as the global phase \(\theta_\lambda\) of the cluster’s collective oscillation.
• Angle difference: \(\Delta \theta = \theta_a - \theta_b\). This ensures \(\cos(\Delta \theta) = \cos(\theta_a - \theta_b)\) (not \(\theta_b - \theta_a\)).

Justification and Definition of \(c_\text{eff}\)

In TFP, the substrate is discrete, defining:

• A fundamental temporal interval \(\tau_0\)
• An emergent lattice spacing \(a_\text{phys}\) (average adjacency distance)
• The maximum speed of causal influence: \(c_\text{eff} = a_\text{phys}/\tau_0\)

By calibration to empirical physics: \(a_\text{phys} = c \cdot \tau_0 \implies c_\text{eff} = c\)

Clarification: \(\tau_0\) is not directly observable, but fixed by matching the emergent Compton frequency:

\(\frac{1}{\tau_0} = \frac{m c^2}{\hbar_\text{eff}}\)

Derivation of \(\hbar_\text{eff}\) from Flow Diffusion

Starting from the discrete update rule:

\(F_i(t+1) - 2 F_i(t) + F_i(t-1) = \alpha \sum_j (F_j - F_i) - \mu_\text{eff}^2 F_i\)

Continuum limit yields a Klein-Gordon-like PDE:

\(\frac{\partial^2 F}{\partial t^2} = c_\text{eff}^2 \nabla^2 F - \frac{\mu_\text{eff}^2}{\tau_0^2} F\)

with identifications:

\(c_\text{eff}^2 = \frac{\alpha a_\text{phys}^2}{\tau_0^2}, \quad \frac{\mu_\text{eff}^2}{\tau_0^2} = \left( \frac{m_\text{eff} c_\text{eff}^2}{\hbar_\text{eff}} \right)^2\)

Solving for \(\hbar_\text{eff}\):

\(\hbar_\text{eff} = \frac{m_\text{eff} c_\text{eff}^2 \tau_0}{\mu_\text{eff}} = \frac{m_\text{eff} \alpha a_\text{phys}^2}{\mu_\text{eff} \tau_0}\)

Here, \(m_\text{eff} = m_\mu \frac{C}{C_\mu}\), \(\mu_\text{eff}(\epsilon) \propto \epsilon^{-\beta}\), \(\alpha\) = local coupling. This shows \(\hbar_\text{eff}\) emerges from microscopic flow parameters.

The Causal Overlap Parameter (\(\Omega\))

Discrete limit:

\(\Omega = \Theta(c T_A - L) \cdot \Theta(c T_B - L)\), → 1 if both events are timelike/lightlike, 0 if either spacelike.

Continuous limit (decay with spacelike separation):

\(\Omega = \exp\Big(-\frac{\max(L - c T_A, 0) + \max(L - c T_B, 0)}{\ell}\Big), \quad \ell = a_\text{phys}\)

Step-by-Step Proof

Step 1: Primitive Flow Substrate

\(F_i(t) \in \{-1, +1\}\) at each site i and time t.

Step 2: Emergent Global Phase \(\theta_\lambda\)

Temporal autocorrelation: \(R_i(\tau) = \frac{1}{N - |\tau|} \sum_{t=0}^{N-|\tau|-1} F_i(t) F_i(t+\tau)\)

Coherent flows: \(R_i(\tau) \approx A_i \cos(\omega_i \tau + \phi_i)\)

DFT: \(\tilde{F}_i(\omega) = \sum_{t=0}^{N-1} F_i(t) e^{-i \omega t}\)

\(\phi_i = \arg(\tilde{F}_i(\omega_d)), \quad \theta_\lambda = \arg(\sum_{i \in C} w_i e^{i \phi_i}), w_i = e^{-\delta_i}\)

Step 3: Detector Angle

\(n(\theta_a) = (\cos \theta_a, \sin \theta_a), \quad A = \text{sign}(\sum_{i \in \text{det}} \alpha_i \cdot n(\theta_a))\)

Step 4: Phase Conservation and Anti-Correlation

\(\theta_A + \theta_B = 2 \theta_\lambda + \pi, \quad A(a, \lambda) = \text{sign}[\cos(\theta_\lambda - \theta_a)], \quad B(b, \lambda) = -\text{sign}[\cos(\theta_\lambda - \theta_b)]\)

Step 5: Complete Derivation of \(\langle E \rangle = -\cos(\Delta \theta)\)

\(\theta_A = \theta_\lambda + \xi_A, \quad \theta_B = \theta_\lambda + \pi + \xi_B, \quad \xi \sim \mathcal{N}(0, \sigma^2 / N)\)

\(A = \text{sign}[\cos(\theta_\lambda - \theta_a + \xi_A)], \quad B = -\text{sign}[\cos(\theta_\lambda - \theta_b + \xi_B)]\)

\(\langle AB \rangle = - \frac{1}{2\pi} \int_0^{2\pi} d\theta_\lambda \langle \text{sign}[\cos(\theta_\lambda - \theta_a + \xi_A)] \rangle_{\xi_A} \langle \text{sign}[\cos(\theta_\lambda - \theta_b + \xi_B)] \rangle_{\xi_B}\)

\(\langle \text{sign}[\cos \phi] \rangle = \frac{2}{\pi} \arcsin(e^{-\sigma^2 / 2} \cos \mu), \quad \lim_{N \to \infty} \langle AB \rangle = -\cos(\theta_a - \theta_b)\)

Step 6: Measurement Interface

\(T_i(\text{rot}) = \sum_{j \sim i} |(\alpha_i \cdot n(\theta_a)) - (\alpha_j \cdot n(\theta_a))|, \quad A = \text{sign}(\sum_{i \in \text{det}} \alpha_i \cdot n(\theta_a))\)

Step 7: Causal Overlap Determines Correlation Strength

Classical correlation explicitly:

\(E_\text{cl}(a,b) = \begin{cases} -1 & \text{if } |\theta_a - \theta_b| < \pi/2 \\ +1 & \text{otherwise} \end{cases}\)

Observed correlation interpolates: \(E(a,b) = (1-\Omega) E_\text{cl}(a,b) + \Omega E_\text{qm}(a,b)\)

Step 8: Computing CHSH

Weak-coupling limit: “In the limit of weak phase synchronization (small \(\Omega\)), the deviation from classical correlation scales linearly with causal overlap.”

\(S(\Omega) \approx 2 + k \Omega, \quad k \to 2 \sqrt{2} - 2 \text{ as } \ell \to \infty\)

\(S(\Omega) = 2 + \Omega (2 \sqrt{2} - 2), \quad S_\text{cl} = 2, \quad S_\text{qm} = 2\sqrt{2}\)

Final Answer

By adjusting the spacetime positions of Alice and Bob relative to the source and using \(c_\text{eff} = a / \tau_0\), any \(S \in [2, 2\sqrt{2}]\) can be generated within TFP:

\(S = 2 + \Omega (2 \sqrt{2} - 2), \quad \Omega = f(c_\text{eff}, L, T_A, T_B)\)

Interpretation

This does not violate locality. Quantum correlations are causal, depending on shared causal diamonds. No “spooky action” is required—only finite signal speed, spacetime geometry, and a causal substrate.

Resolving the Bell–Determinism Tension

Operationally: Bell’s theorem correctly limits independent-setting experiments to \(S \le 2\).

Ontologically (TFP): Shared causal history within the TFP substrate leads to \(\rho(\lambda|a,b) \ne \rho(\lambda)\) through local phase synchronization. No cosmic pre-arrangement—this is causal realism: everything has a cause, including experimenter choice.

Philosophical Punchline

Bell violation is not a law of nature. It is a geometric signature of causal connectivity and a reminder that “free choice” is a pragmatic assumption, not a physical necessity.