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.
Read this before Step 1: one of the terms you are about to tour is known-broken. The γ = 2/√Ncorr rule is sign-inverted against real finite-size scaling — more correlated systems should sharpen transitions, but this formula flattens them (it files BCS superconductors, among nature's sharpest transitions, at the flattest end). The step for γ repeats this in its honesty note; it is stated here first so you don't absorb the formula as-is and get the correction only at the end.
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 | The local mass/energy density, the same definition Step 1 and every executed test use. The universal input. (The ontology pages read ρ more broadly, as “presence” within a relevancy boundary; that broader reading has never been operationalized.) |
| ρcrit | Reference density — a saturation knee, not a critical point (the “crit” in the name is historical). Important: not the midpoint of C, and where it sits is set by γ — C(ρcrit) = tanh(γ·ln 2). At γ = 2 that is 0.88 (near saturation); at the SPARC-fitted γ ≈ 0.49 it is 0.33 (below the half-way point). Always quote the γ: a bare “0.88” silently assumes the pin the RAR refuted at ΔBIC = +184. |
| γ | Controls sigmoid sharpness. The original proposal was γ = 2/√Ncorr, with Ncorr = number of particles moving together, so one particle would give γ=2 (sharp) and a million γ=2×10⁻³ (flat). That proposal failed: its sign is inverted against real collective physics (audited-negative; see the red box below). Where γ is used today it is fitted, not computed from Ncorr. |
| 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). Note: the ln is cosmetic here — tanh(γ·ln(1+x)) is exactly [(1+x)2γ−1]/[(1+x)2γ+1], a Hill-type rational function (see Step 3). |
| tanh | Hyperbolic tangent — an S-shaped saturation function mapping any real number to (−1, +1); with this equation's argument (always ≥ 0) the output C stays in [0, 1). Like a dimmer switch: input near 0 → output near 0; input very large → output near 1. |
Revision note
This legend used to define ρ as “Presence — the density of compatible elements within the system's relevancy boundary”, which disagreed with Step 1 directly below it. It also called ρcrit “a characteristic scale parameter”. Both rows now use the definitions from Step 1 and the Glossary.Step 1 of 6 (this tool's own tour — not a learning-path step)
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.
Two independent failures in the formula's motivation: (1) Sign inverted. 1/√N is a fluctuation width that shrinks as N grows (sharper collective behavior) — but sitting in the rate slot of tanh, larger Ncorr gives aflatter curve. A BCS superconductor (Ncorr ≈ 107, among the sharpest real transitions) gets γ ≈ 6×10−4, the flattest possible curve. An ideal gas (Ncorr = 1, no real transition) gets γ = 2, the sharpest. Backwards at the level of the physics, not just the prefactor. (2) CLT self-contradiction. The 2/√N formula invokes the Central Limit Theorem, which requires independent variables — but Ncorr is defined as the count of correlated ones. The statistical motivation undercuts itself before you reach the sign. See the γ Calculator for the full reductio with interactive BCS/ideal-gas examples →