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 density-keyed test use (the galaxy RAR fits and the Cassini squeeze key C on acceleration instead). 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 | Sets the low-density slope and how fast C approaches 1, not a sharpness: no γ narrows the 10–90% width below 1.17 dex. 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 not an independent feature — tanh(γ·ln(1+x)) is exactly [(1+x)2γ−1]/[(1+x)2γ+1], a Hill-type rational function (see Step 3). But it is not cosmetic: it turns the tail into a power law, 1−C ≈ 2x−2γ, and that is what makes γ = ½ equal MOND's simple μ. Without it, tanh(γx) has an exponential tail and leaves the simple-μ family, with no power-law return. It would still have the right limits for a MOND μ. (Until 2026-11-07 this note said “the ln is cosmetic”; a graduate simulated-visitor persona corrected it.) |
| 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 8 (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 density-keyed test uses exactly this physical density, and no broader form has ever been operationalized. Not every executed test is density-keyed: the galaxy RAR fits, the γ = 2 pin, TEST-09/10 and the Cassini squeeze key C on acceleration instead (see Coherence Function, "which C"). This walkthrough uses the density definition. This sentence said "every executed test" until 2026-10-04.)
Density is the input the framework states. In galaxies that is the refuted version: fed density, the equation loses to the same equation fed acceleration by ΔBIC +2843 and is excluded again by lunar laser ranging; the form that fits rotation curves takes acceleration, MOND's own variable (Glossary: Which C? Which γ?). This tour follows the equation as stated.
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 →