The Coherence Function

Phenomenological Ansatz — tanh motivated, not derived

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.

C(ρ) = tanh(γ · ln(ρ/ρ_crit + 1))

The Synchronism coherence function

New to the notation? Walk through it step by step → — a 6-step interactive that defines every symbol (C, ρ, ρ_crit, γ, ln, tanh) in plain language.

This maps presence to coherence(a dimensionless number between 0 and 1 that measures how collectively elements behave — from sparse/independent (C ≈ 0) to dense/collective (C ≈ 1)).

Expert note (physicist terminology)

“Coherence” here means density-driven classicality/collectivity, not quantum phase coherence. BCS superconductors and BECs land at C ≈ 0 at all physical densities because their large N_corr → tiny γ → flat tanh curve — not because they lack macroscopic phase coherence. See the glossary warning.

Inputs and Outputs

Input: presence — compatible structural elements within the system's MRH. Physical density is one form of presence.

Output: coherence (0 = sparse/independent, 1 = dense/collective)

γ, ρ_crit

Parameters: coupling strength, saturation knee (reference density — not a critical point; C(ρ_crit,γ=2)≈0.88)

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

Several compander functions satisfy all four constraints: logistic, erf, arctan, Hill, and tanh. tanh is the chosen form, motivated historically by its appearance in the Ising mean-field self-consistency equation m = tanh(βJzm) — the same shape. 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. The tanh shape is motivated by the Ising analogy, not derived from it; it is a phenomenological choice from the compander family (μ-law, Hill, logistic, tanh), not the uniquely forced form.Empirical caveat: Landau-universality critical exponents (β, ν, α, δ, η) are off by ~2× in practice — the Landau analogy is motivational, not an accurate prediction of universality class. See Honest Assessment. See Parameter Derivations for the complete derivation vs. motivation distinction.

Saturation note: At γ = 2, C(ρ) saturates within ~1 decade of ρ_crit(C(10·ρ_crit) ≈ 0.9999). The coherence transition is sharp — more like a phase transition than a smooth interpolation. Each system has its own ρ_crit, 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: Sparse/Independent

Low density or flat tanh (large N_corr). Elements act independently — sparse gases, dilute quantum systems, quantum computing platforms. Note: BCS superconductors also land here due to tiny γ (≈6×10⁻⁴), not because they are incoherent — they are maximally quantum-phase-coherent, but C measures density-collectivity, not quantum phase coherence. Warning: do not read C≈0 as "quantum" or "wave-like" — C is classicality in the density/collective sense, the opposite of the quantum-coherence axis.

C → 1: Dense/Collective

High density relative to ρ_crit, moderate N_corr. Elements act together as a collective — dense matter, crystal lattices, everyday macroscopic physics. Newton's laws work reliably here. Galaxy dynamics (N_corr≈1, γ=2) lives in this regime. Note: high C does not mean "classical particle" in the quantum-decoherence sense — it means dense/collective in the density axis this equation measures.

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.

What Class of Function Is C(ρ)?

C(ρ) is a compander — a logarithmic companding function, not an order parameter. This is the settled self-identification used on the Parameter Derivations page and in the Honest Assessment.

The compander class includes: μ-law audio compression (telephone networks), Hill function (oxygen binding to hemoglobin), Naka-Rushton equation (retinal response), Kubo susceptibility (stat-mech near criticality). All map a wide dynamic range onto a bounded output using a sigmoidal curve. C(ρ) does the same: density spanning 80+ orders of magnitude → coherence in [0, 1].

Key implication: Companders are purely evaluative (input → output). They cannot encode universality classes, critical exponents, or spontaneous symmetry breaking — those require a self-consistency loop. The Ising mean-field equation m = tanh(βJzm) is in a different structural class: it is implicit (m appears on both sides) and supports a phase transition. C(ρ) evaluates C from ρ directly with no feedback. These are different objects. The Landau analogy is motivational, not a claim of class membership.

Scope Limit: Universality Classes

C(ρ) cannot encode universality-class structure. Real phase transitions are classified by (spatial dimension d, order-parameter dimension n, symmetry group) — these determine whether a system belongs to the 3D-Ising universality class (α = 0.110), 3D-XY (α = −0.015), or mean-field (α = 0). C(ρ) has no inputs encoding d, n, or symmetry. It therefore cannot reproduce critical exponents or distinguish universality classes — a one-parameter scalar coherence function is structurally blind to this information.

This is a scope limit, not a parameter-tuning failure. C(ρ) describes sigmoidal interpolation between coherent and decoherent regimes. It does not claim to predict critical exponents, universality-class membership, or quantitative crossover structure. The failures documented in Honest Assessment (critical exponents 2× off, T_c 6.5× wrong) are consequences of applying the function outside this scope, not evidence that the function itself is wrong within scope.

Derivation History

Session #64-65: γ = 2.0 motivated by 6D phase space degrees of freedom — see Parameter Derivations for why “motivated, not derived” is the correct framing
Session #66: tanh form motivated by the Ising self-consistency equation (tanh appears in the fixed-point, not as a derivation from it)
Session #67: Information-theoretic compression argument
Session #87-91: Connected to cosmological parameters (a₀, Σ₀)

Next: The γ Parameter →Try It: Coherence Explorer

Related Concepts

The γ Parameterγ = 2/√N_corr: why 2, why √N Critical Densityρ_crit = A V_flat²: the transition point Phase Transitionsγ < 1, γ ≈ 1, γ > 1 regimes