Heat a magnet toward its Curie temperature and something strange happens right at the edge: patches of alignment appear at EVERY size at once, from tiny to system-spanning, and the correlation length — how far order reaches — diverges to infinity. At the critical point a system is scale-free, and wildly different materials share the exact same critical exponents. This is the phase transition ScaleSpaceSynth lets you feel. Slide the temperature toward Tc.
At a continuous (second-order) phase transition the correlation length ξ — the distance over which fluctuations are coupled — DIVERGES as ξ ∝ |T−Tᶜ|⁻ᶠ, so fluctuations appear at all scales and the system becomes scale-invariant. Thermodynamic quantities follow power laws with critical exponents (ν, β, γ…) that are UNIVERSAL: they depend only on symmetry and dimension, not microscopic details, so a magnet and a fluid at their critical points behave identically. This universality is explained by [[the-renormalization|renormalization]] fixed points. A fail-loud self-check throws unless ξ grows without bound as T→Tᶜ. ◆ real critical phenomena, node-verified.
A mean-field power-law ξ∝|T−Tᶜ|⁻ᶠ is shown; exact exponents differ by dimension/universality class. The divergence of the correlation length and universality are the exact content — the reason the whole scale ladder rhymes.