The Core Idea
One function. Three parameters. Every scale from Planck to cosmic.
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 between “fully quantum” and “fully classical.” The function must be bounded [0, 1], monotonic, and smooth. tanh arises naturally in mean-field models (e.g., the Ising model order parameter m = tanh(βJzm)). Other sigmoids (logistic, error function) share the same qualitative properties; tanh is a natural choice from the Landau theory family, not the only possible one.
Need an analogy first? · Full 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 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
γ « 1: Quantum Regime
Few correlated particles, strong coherence effects. Superposition, interference, entanglement. The domain of quantum mechanics.
γ ≈ 1: The Boundary
Phase transitions, chemistry, catalysis, biology. Where quantum meets classical. 1,703 phenomena cluster here at 89% validation rate.
γ » 1: Classical Regime
Many correlated particles, classical behavior. Galaxy dynamics, everyday physics. The domain of Newton and general relativity.
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