The Core Idea
One function. Three parameters. Every scale from Planck to cosmic.
Think of it as a dimmer switch from sparse/independent to dense/collective. Feed it the local density of a system, and it outputs a number between 0 (sparse/independent) and 1 (dense/collective). The same switch, the same scale, works for a single atom and a galaxy cluster.
tanh, in one picture. tanh is just an S-shaped curve — flat, then steep, then flat again. Nothing happens at low density, a small change flips the behavior in the middle, and past the knee more density changes almost nothing. That's the whole “dimmer switch.”
Drawn with γ = 2 (the galaxy-scale pin — refuted on SPARC data, see the caveat below) on a logarithmic density axis. Note the two honest markers: the curve's true midpoint (C = 0.5) sits near 0.32·ρcrit, and at ρcrit itself C ≈ 0.88 — ρcrit is a saturation knee, not a critical point.
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 from sparse/independent to dense/collective. The function must be bounded [0, 1], monotonic, and smooth. tanh's form recalls the Ising model self-consistency equation m = tanh(βJzm), but the resemblance is only visual: C(ρ) has no feedback loop (ρ goes in, C comes out), so the Ising analogy carries no motivating force. Other sigmoids (logistic, error function, Hill) satisfy the same four constraints. tanh is a phenomenological choice, not a derived result. Note: C(ρ) is real-analytic everywhere on its domain — no singularity, no free-energy functional, and no length scale anywhere in the equation — so it has no critical point and therefore no critical exponents (β, ν) to compare. Asking for them is a category error, not a near-miss with Landau theory; the analogy is motivational only.
Need an analogy first? · Why tanh? (motivation, not 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 2/√Ncorr? (motivation, not derivation) →
Why log? (Honest answer: it's cosmetic)
The traditional motivation was compression: density spans 80+ orders of magnitude (from interstellar gas at 10−24 g/cm³ to neutron stars at 1014 g/cm³), and the logarithm squeezes that range into something the tanh can work with. But an exact identity retires that story. With x = ρ/ρcrit:
tanh(γ·ln(1+x)) ≡ [(1+x)2γ − 1] / [(1+x)2γ + 1] — exactly.
The equation is an ordinary rational (Hill-type) saturation function of (1+x)2γ; the tanh∘ln decomposition is notation, not mechanism. Any saturating rational function “spans 80 orders of magnitude” for free — the ln does no work that the exponent 2γ doesn't undo. So the Hill functions and Naka–Rushton curves named below are not merely similar alternatives; the flagship equation is one of them, exactly. (Identity stated 2026-07-10. Provenance: derived independently by two successive internal review passes, 2026-07-09 and 2026-07-10 — the audit machinery caught it; the identity then took a day to reach this page. No external derivation is on record.)
What It Predicts
γ « 1: Collective Regime
Many correlated particles (large Ncorr → small γ = 2/√Ncorr). BECs, superconductors. Despite being quantum systems in the real-world sense, these score C ≈ 0 here — the S-curve is so flat that typical densities leave C near zero. This is why “coherence” in Synchronism is not quantum coherence.
γ ≈ 1: The Boundary
Phase transitions, chemistry, catalysis, biology. Where the S-curve has intermediate steepness. 1,703 phenomena cluster here at 89% boundary-consistency rate.
Pending null model: density-monotonic targets produce r > 0.95 from trivial polynomial fits. Chemistry correlations may reflect monotonic data structure, not framework specificity.
γ » 1: Independent Regime
Few correlated particles (Ncorr ≈ 1 → large γ). Ideal gas, independent stars in a galaxy. Steepest S-curve — C rises quickly with density. Galaxy dynamics is here (stars treated as independent, Ncorr=1, γ=2).
The data say the same thing, independently (added 2026-08-08). Written as a Hill function the exponent is the Hill index n = 2γ. Fit freely to SPARC, the galaxy sector prefers γ ≈ 0.489, i.e. n ≈ 0.98 ≈ 1 — and n = 1 is the non-cooperative limit of a Hill function, the value that means no collective effect. The framework's whole distinctive claim is that density drives a collective transition. Its own best fit sets cooperativity to zero. That is the criticality retraction above arriving a second time, through the data rather than through the algebra. At exactly γ = 1/2 the identity closes: C(x) = x/(x+2) = μsimple(x/2), MOND's simple interpolating function — so “the fit converges to MOND” is not an empirical coincidence but algebra (see the RAR entry and For Researchers).
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