The Core Idea

One function. Three parameters. Every scale from Planck to cosmic.

Think of it as a dimmer switch from sparse/independent to dense/collective. Feed it the local density of a system, and it outputs a number between 0 (sparse/independent) and 1 (dense/collective). The same switch, the same scale, works for a single atom and a galaxy cluster.

C(ρ) = tanh(γ · ln(ρ/ρcrit + 1))
ρ
Presence: local density — how many relevant elements are packed in a given region (e.g. stars per cubic light-year, atoms per unit volume)
C
Coherence: 0 = sparse/independent, 1 = dense/collective. ⚠ Not quantum coherence — superconductors score low here.
γ
2/√Ncorr: coupling strength
ρcrit
A × Vflat²: saturation knee (not a critical point)

Why These Specific Choices?

Why tanh?

tanh is an S-shaped curve that smoothly transitions from 0 to 1 — think of it as a dimmer switch from sparse/independent to dense/collective. The function must be bounded [0, 1], monotonic, and smooth. tanh is motivated by the Ising model self-consistency form m = tanh(βJzm), but C(ρ) is not a self-consistency equation — ρ goes in, C comes out, with no feedback loop. Other sigmoids (logistic, error function, Hill) satisfy the same four constraints. tanh is a phenomenological choice, not a derived result. Note: Landau-universality critical exponents (β, ν, etc.) are off by ~2× in practice — the analogy is motivational, not literal.

Need an analogy first? · Why tanh? (motivation, not derivation) →

Why γ = 2/√Ncorr?

The 1/√Ncorr dependence resembles central-limit-theorem scaling (fluctuations ~ 1/√N), which is generic statistics for correlated ensembles. Ncorr (number of correlated particle units) is the physically measurable quantity. The factor of 2 is motivated by phase-space arguments (6D contracted to 3 effective) but should be understood as a motivated ansatz rather than a rigorous derivation.

Why 2/√Ncorr? (motivation, not derivation) →

Why log?

Density spans 80+ orders of magnitude (from interstellar gas at 10−24 g/cm³ to neutron stars at 1014 g/cm³). The logarithm compresses this range into something the tanh can work with.

What It Predicts

Audit note: 6 of 6 badges previously labeled “Validated” on this site have been demoted to Reparametrization on closer review. These regime cards have not yet been independently audited — treat all current badges as Reparametrization-pending-audit until shown otherwise. See honest assessment.

γ « 1: Collective Regime

Many correlated particles (large Ncorr → small γ = 2/√Ncorr). BECs, superconductors. Despite being quantum systems in the real-world sense, these score C ≈ 0 here — the S-curve is so flat that typical densities leave C near zero. This is why “coherence” in Synchronism is not quantum coherence.

Reparametrization — regime definition

γ ≈ 1: The Boundary

Phase transitions, chemistry, catalysis, biology. Where the S-curve has intermediate steepness. 1,703 phenomena cluster here at 89% boundary-consistency rate.

Pending null model: density-monotonic targets produce r > 0.95 from trivial polynomial fits. Chemistry correlations may reflect monotonic data structure, not framework specificity.

Reparametrization — null model pending

γ » 1: Independent Regime

Few correlated particles (Ncorr ≈ 1 → large γ). Ideal gas, independent stars in a galaxy. Steepest S-curve — C rises quickly with density. Galaxy dynamics is here (stars treated as independent, Ncorr=1, γ=2).

Reparametrization — regime definition
What C(ρ) actually is: A logarithmic compander — in the family of μ-law encoders, Hill functions, and Naka–Rushton curves — that maps log-density onto a [0,1] saturation curve. The “Ising motivation” is conceptual: C(ρ) is not a self-consistency equation (ρ goes in, C comes out, no feedback). Critical exponents (β, ν) are off by 2× — the diagnostic result that rules out C(ρ) as a Landau-theory continuum order parameter. See honest assessment.

Choose Your Path

The Math

Dive into the equation: derivations, proofs, parameter origins

The Evidence

See it tested against 14,760 galaxies

The Chemistry

Explore 1,703 phenomena at the γ ≈ 1 boundary

The Failures

Where the theory falls short and what that teaches us

The Foundations

Four axioms everything else flows from — including what Intent actually is

Related Concepts

FundamentalsThe four foundations — everything else flows from theseThe Coherence FunctionC(ρ) = tanh(γ ln(ρ/ρ_crit + 1))The γ Parameterγ = 2/√N_corr: why 2, why √NScale InvarianceFrom Planck to cosmic: 80 orders of magnitude