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. From information theory (Session #67): compressing a high-dimensional state to a binary quantum/classical decision requires exactly these properties. tanh is the unique sigmoid function that arises from mean-field theory.
Need an analogy first? · Full derivation →
Why γ = 2/√Ncorr?
The factor of 2 comes from phase space dimensionality (6D: 3 position + 3 momentum, contracted to 3 effective). Ncorr is the number of particles moving as a correlated unit — the only physically measurable quantity needed.
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.