The Coherence Function

Before the math: coherence measures how collectively particles behave. Ice cubes have high coherence (rigid crystal lattice). Steam has low coherence (random motion). This function quantifies that spectrum for any system, at any scale.

This maps presence to coherence(a dimensionless number between 0 and 1 that describes how quantum or classical a system is).

Inputs and Outputs

Why This Specific Function?

1. The Compression Requirement

The physical state of any system lives in a high-dimensional space: magnitude, direction, temporal structure, spatial correlations, interference patterns. But the quantum/classical distinction is binary. You need a function that compresses high-dimensional information into a bounded scalar. This is an information-theoretically necessary compression (Session #67).

2. Why tanh?

The compression function must satisfy four properties:

• Bounded [0, 1]: Coherence can't exceed unity or go negative
• Monotonic: Higher presence → higher coherence (no paradoxical inversions)
• Smooth saturation: Gradual approach to limits (enables field equations)
• Handles extremes: ρ → 0 gives C → 0, ρ → ∞ gives C → 1

From mean-field theory, tanh is the unique sigmoid that arises naturally from these constraints. It's not a choice; it's a consequence of the physics.

3. Why log?

Presence spans enormous ranges — in the astrophysical case, density alone covers 80+ orders of magnitude (interstellar gas at ~10⁻²⁴ g/cm³ to neutron stars at ~10¹⁴ g/cm³). The logarithm maps this range into something tanh can differentiate between. The “+1” inside the log prevents divergence at ρ = 0.

What C = 0 and C = 1 Mean

Derivation History

• Session #64-65: γ = 2.0 derived from 6D phase space degrees of freedom
• Session #66: tanh form derived from mean-field theory
• Session #67: Information-theoretic compression argument
• Session #87-91: Connected to cosmological parameters (a₀, Σ₀)