Δ.72 Field Equation: A Unified Framework for Coherence Across Biological, Physical, and Cognitive Systems

Coherence emerges across biological, physical, and cognitive systems, yet no existing framework explains how coherence forms, collapses, and reorganizes across scales. This paper introduces the Δ.72 Field Equation, a unified three-layer model describing coherence through (1) empirical signal alignment, (2) a harmonic transition boundary at Δ.72, and (3) a deeper generative field architecture linking Reality Trajectory, Imagination Stream, and Arbitration Function.

The framework defines measurable coherence conditions, topological boundary behavior, resonance stabilization, and a field-level tensor mapping that predicts when systems cross from mechanical synchrony into harmonic coherence. All components are fully falsifiable, enabling empirical verification across biophysics, biosignals, AI decision arbitration, and cognitive stability.

Extended Summary

This work synthesizes three levels of coherence:

1. Layer 1 — Empirical Coherence Model
   Coherence is defined as a function of system–field alignment, phase coherence, and adaptive variability. This generates measurable predictions for biosignals (HRV, EEG, respiration patterns).

2. Layer 2 — Δ.72 Harmonic Threshold
   A mathematically stable transition boundary where coherence shifts from mechanical or chaotic modes into harmonic regimes. This threshold appears across models of adaptation, entrainment, and recovery.

3. Layer 3 — Generative Field Architecture
   A tensor-based formulation integrating:
   • Reality Trajectory (R)
   • Imagination Stream (I)
   • Arbitration Function (Ψ)
   These map to the full field Φ, which becomes harmonic when the Δ.72 boundary is reached.

Additional contributions include:
• κ-phase arbitration dynamics
• toroidal boundary geometry for coherence collapse and recovery
• a resonance engine predicting spontaneous alignment
• empirical falsifiability criteria
• fully reproducible figure set and replication instructions

This publication provides the first cross-domain, testable mathematical structure for coherence spanning biology, physics, cognition, and AI alignment.