Event-State Theory VIII: Mathematical Derivation of Pre-Causal Resonance

Event-State Theory (EST) makes a unique, falsifiable prediction: Pre-Causal Resonance (PCR), the existence of correlated vacuum fluctuations prior to a high-energy event. Previous work demonstrated this effect computationally. This paper provides the formal mathematical derivation. We show that the characteristic exponential ramp of the PCR signal is not a numerical artifact, but the necessary and unique solution to the discrete Euler-Lagrange equation for the EST cost function under a future boundary condition. By modeling the vacuum as a scalar field on a discrete causal timeline, we derive the exact shape of the precursor signal (φ ∝ eᵐᵗ). This establishes PCR as a core prediction of the Path of Causal Least Resistance and provides a concrete mathematical target for experimental verification in high-energy collider data. Significance: This derivation transforms the concept of 'retro-causal' influence from a metaphysical speculation into a rigorous, testable wave equation derived from the principle of least action. Status: Code for the numerical verification of this derivation is available in the linked GitHub repository.