Equation Anatomy
A term-by-term tour of C(ρ) — what each piece means and why it was chosen. Note: tanh and γ = 2/√Ncorr are motivated choices, not derived results (there is no derivation to walk through — only choices to examine). See the caveat blocks in each step.
Variables in this equation — defined before we start:
| C(ρ) | Coherence — a number from 0 (sparse/independent) to 1 (dense/collective). The output we're computing. ⚠ Physicist note: C here measures collective ordering, not quantum phase coherence — BEC/BCS condensates have low C by this measure. |
| ρ | Presence — the density of compatible elements within the system's relevancy boundary. The universal input. |
| ρcrit | A characteristic scale parameter for the system. Important: not the midpoint of C. At γ=2, C(ρcrit) ≈ 0.88 — ρcrit is near saturation, not the half-way point. |
| γ | 2/√Ncorr — controls sigmoid sharpness. Ncorr = number of particles moving together. One particle: γ=2 (sharp). A million: γ=2×10⁻³ (flat). |
| ln | Natural logarithm — grows slowly. Doubling x doesn't double ln(x). A number 1,000× bigger comes out only ~7 units bigger (ln(1000) ≈ 6.9). This compression lets one equation span 80 orders of magnitude of physical density. |
| tanh | Hyperbolic tangent — an S-shaped saturation function that maps any real number to (0, 1). Like a dimmer switch: input very negative → output near 0; input very large → output near 1. |
Step 1 of 6
Step 1: Start with density
Everything begins with density. In a galaxy, ρ is the baryon density profile. In chemistry, it's the number density of particles. The same starting point everywhere. (Definition note: the ontology pages define ρ more broadly as "presence within a relevancy boundary" — but every executed test uses exactly this physical density, and no broader form has ever been operationalized. This walkthrough uses the definition the tests use.)
Density is the universal input to the coherence function.