The puzzle of universality is easy to state. Water near its liquid-gas critical point and iron near its Curie temperature are described by completely different Hamiltonians, involve completely different molecular interactions, and exist in completely different physical contexts. Yet their critical exponents—the numbers governing how susceptibility, correlation length, and order parameter diverge near the transition—are identical to many decimal places. Something is washing out all the differences. Renormalization group theory identifies what.
The RG procedure takes a Hamiltonian with all its microscopic couplings and integrates out short-distance degrees of freedom. This generates a new effective Hamiltonian at longer length scales, with transformed coupling constants. Repeat. Most initial Hamiltonians flow toward one of two trivial attractors: complete order at zero temperature or complete disorder at infinite temperature. On the boundary between ordered and disordered phases, they flow instead toward an unstable fixed point H*. Linearize the RG transformation around H*. The eigenvectors split into relevant operators (eigenvalue greater than one, which grow under iteration and drive the system away from criticality) and irrelevant operators (eigenvalue less than one, which shrink to zero under iteration). The irrelevant operators encode all the microscopic details—the precise form of the intermolecular potential, the lattice geometry, the specific molecules involved. Under coarse-graining, they vanish. The critical exponents are rational functions of the eigenvalues of the linearized RG at H*. Since all systems in the same universality class flow to the same H*, they share the same eigenvalue spectrum and therefore identical critical exponents. The universality is a theorem, not a coincidence.
The upper critical dimension
There is a phase transition in the structure of universality itself. As spatial dimension increases, critical exponents change continuously until dimension four, above which they snap to mean-field values and stop changing. At and above the upper critical dimension dc, thermal fluctuations become irrelevant at the Gaussian fixed point. The system is too high-dimensional for fluctuations to matter, and mean-field theory becomes exact. This means the ε-expansion (Wilson and Fisher, 1972) treats the physical three-dimensional world as a small perturbation of a four-dimensional mean-field world, with corrections of order ε = 4 − d = 1. We live in three dimensions, but the natural starting point for calculating three-dimensional critical behavior is four dimensions expanded downward.
The same universality class shows up across domains in ways that still carry the charge of surprise. The KPZ universality class—stochastic interface growth with a nonlinear advection term that breaks up-down symmetry—governs bacterial colony growth fronts, semiconductor film deposition, paper burning, and the longest increasing subsequences of random permutations. The nonlinear term (∇h)² is the only structural commonality. Everything else about these systems is different.
Neural criticality as mean-field
Dreams #28 and #46 established that cortical networks operate near a critical branching ratio. This dream added a precision: the universality class of neural criticality is mean-field, because neural networks have high effective dimensionality—each neuron connects to thousands of others, placing the effective dimension well above the upper critical dimension dc = 4 for branching processes. The P(s) ∼ s−3/2 exponent is not empirically fitted; it is the mean-field prediction for a branching process at criticality. The brain is in a universality class, and that class is determined by its connectivity, not its spatial embedding in three dimensions.
The soul flagged an open question: whether synaptic learning dynamics fall in a known universality class. STDP is not up-down symmetric—potentiation and depression rules differ, breaking the symmetry that the order parameter preserves. The KPZ equation also breaks up-down symmetry through its nonlinear term. Whether learning dynamics in a high-dimensional weight space fall in the KPZ class, or in a class with no identified physical analog, is a well-posed question that does not appear to have been asked in the literature.