Phase Coherence Condition

The Phase Coherence Condition sets the reference coherence C⨶, about 0.8, at which the coherence factor (C/C⨶)^β of the Combined Threshold Function reaches 1; the synchronization a stable form needs is set by Θ ≥ 1 as a whole, not by C⨶ alone. It is the second of the three reference scales that normalize Θ, between the density scale and the information scale. Substrate elements each carry a phase; when those phases line up closely enough across the ensemble, the collective behavior becomes coherent enough for a stable structure to hold. The three criteria are combined in the Combined Threshold Function, where the coherence ratio C/C⨶ enters with exponent β = 1, and as coupling rises past its critical value the coherence grows toward 1. The formal statement is |⟨e^i(φ_j − φ_k)⟩| ≥ C⨶, where | | is the magnitude (the Phase-Locking Strength R⊠), ⟨ ⟩ is the ensemble average, e^i(φ_j − φ_k) the complex phase factor between the jth and kth substrate elements, φ_j and φ_k their phases and C⨶ the critical coherence parameter, approximately 0.8. All quantities are dimensionless [∅]; the book's check is consistent. It connects to Phase Coherence, the Critical Density Threshold, Information Density Criterion, Combined Threshold Function, Form Emergence Condition and Critical Coupling.