The Local Kernel and Global Bus: A Causal Ontology For Quantum Mechanics

THE LOCAL KERNEL AND GLOBAL BUS: A CAUSAL ONTOLOGY FOR QUANTUM MECHANICS

Resolving Non-locality Through Substrate Unity
John Gavel

1. The Local Kernel Equation (Strict Causality)

Every substrate node executes a purely local update rule:

\[ F_i(t + \tau_p) = \operatorname{sign} \left( \sum_{j \in \mathcal{N}(i)} w_{ij} F_j(t) - \theta_i \right) \tag{1} \]

where:

• \( F_i \in \{-1, +1\} \): Binary flow state at node \( i \)
• \( \mathcal{N}(i) = \{i-1, i+1\} \): 1D nearest neighbors (strict locality)
• \( w_{ij} \): Link weight (fixed by substrate geometry)
• \( \theta_i \): Local processing threshold ("busy-ness")

Key Principle: No node accesses information beyond its immediate neighbors. There is no "action at a distance" — only local tension minimization.

2. The Global Resource Constraint (Shared Bus Bandwidth)

While updates are local, the substrate operates under a global resource limit:

\[ \sum_{i \in \Omega} H_i(t) \leq \mathcal{C}_{\text{max}} \tag{2} \]

where:

• \( H_i(t) \): Handshake load of node \( i \) (proportional to \( |\theta_i| \))
• \( \Omega \): Coherence domain of size \( R_U \)
• \( \mathcal{C}_{\text{max}} = K^2 = 144 \): Total substrate capacity (from \( K=12 \) adjacency)

Emergent Non-locality: If node A increases its load \( H_A \) (e.g., by forming a massive motif), the available capacity for all other nodes in \( \Omega \) decreases instantly. This is not signaling — it’s resource contention in a shared hardware bus.

3. The Phase-Lock Equation (Emergent Synchronicity)

Adjacent nodes minimize tension through phase alignment:

\[ \Delta \phi_{ij} = \sin(\phi_i - \phi_j) \cdot \delta_{\text{eff}} \tag{3} \]

where \( \delta_{\text{eff}} \) is the effective coupling strength.

Over time, this drives \( \phi_i \to \phi_j \) for all \( i,j \in \Omega \), creating a phase-locked motif — a rigid structure where all nodes update in unison. This is the origin of particle stability and quantum coherence.

4. Apparent Velocity vs. Hardware Speed

Two distinct velocities emerge:

• Hardware speed (\( c \)): Propagation speed of a single bit-flip through the substrate
• Apparent velocity (\( v_{\text{app}} \)): Rate of correlated state change across a phase-locked motif

For a motif of size \( L \):

\[ v_{\text{app}} = \frac{L}{\tau_p} \gg c \quad \text{(since } L \gg a_p \text{)} \tag{4} \]

This explains "spooky" correlations: when Alice measures her particle, Bob's particle doesn't "receive a signal" — both are parts of a single phase-locked object that updates concurrently at the hardware clock rate \( 1/\tau_p \).

5. The TFP Wavefunction (Processing Density)

The quantum wavefunction is not a physical wave — it's a statistical description of substrate activity:

\[ \Psi(x,t) \equiv \sqrt{\rho_{\text{busy}}(x,t)} \cdot e^{i \phi(x,t)} \tag{5} \]

where:

• \( \rho_{\text{busy}}(x,t) \): Local handshake density (number of "busy" nodes per unit volume)
• \( \phi(x,t) \): Local phase offset relative to the global clock

Interpretation: \( |\Psi|^2 \) measures computational load; \( \arg(\Psi) \) measures clock skew. The "collapse" of \( \Psi \) is simply the transition from distributed processing to localized resolution.

6. Resolution of Quantum Paradoxes

Phenomenon	Standard QM Explanation	TFP Explanation
Entanglement	Non-local wavefunction collapse	Concurrent update of a single phase-locked motif
Bell Violations	Violation of local realism	Reveals pre-existing causal unity (no "realism" of separate particles)
Wave-Particle Duality	Complementarity principle	Distributed processing (wave) vs. localized resolution (particle)
Uncertainty Principle	Fundamental indeterminism	Trade-off between spatial resolution and temporal coherence in the substrate

7. Conclusion: One Table, Not Two Particles

Quantum non-locality is an illusion created by mistaking the appearance of separation for actual separation. In reality:

• The universe is a single, connected causal substrate
• "Particles" are stable patterns in this substrate
• "Measurements" are local interactions that reveal global structure

There is no "ghostly connection" — only one table. When you hit one end, the other end moves not by magic, but by physical continuity.