The Coherence Function
Phenomenological Ansatz — tanh motivated, not derivedBefore 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 arises naturally from these constraints — the Ising-model self-consistency equation m = tanh(βJzm) has the same form. However, Synchronism's C(ρ) is not the Ising equation. The Ising result is a self-consistency loop where m appears on both sides. C(ρ) evaluates directly with no feedback loop — ρ goes in, C comes out. This distinction matters: the tanh shape is motivated by mean-field theory, not derived from it. Any sigmoid satisfying the four constraints above (logistic, erf, arctan, Hill) would have been an equally valid choice. tanh is the most natural choice given the Landau-theory connection, but not the uniquely forced one. See Parameter Derivations for the complete derivation vs. motivation distinction.
Saturation note: At γ = 2, C(ρ) saturates within ~1 decade of ρcrit(C(10·ρcrit) ≈ 0.9999). The coherence transition is sharp — more like a phase transition than a smooth interpolation. Each system has its own ρcrit, so what is universal is the form of the crossover, not its location.
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
C → 0: Quantum
Superposition maintained. Wave-like behavior. Interference possible. Systems in this regime show non-classical correlations. This is where quantum computing operates.
C → 1: Classical
Definite positions. Particle-like behavior. No interference. Everyday physics. Newton's laws work here. Galaxy dynamics lives in this regime.
Relationship to C = f(γ, D, S)
The consciousness and measurement pages of this site use a second form of coherence: C = f(γ, D, S) where D is decoherence and S is self-modeling. Both are called “coherence” (C) but they are not obviously the same observable. The relationship — whether there is a function ρ = g(γ, D, S) that reduces the parametric form to the density-based form — has not been derived. If the reduction exists, the “one equation” framing is vindicated. If it does not, C(ρ) and C = f(γ, D, S) are two different observables sharing a symbol. See research proposal dual_C_symbol_ambiguity_and_bridge_derivation.md.
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)