Brain Criticality — Phase Transition at the Edge of Chaos via the Latent Framework

Many neural systems are hypothesized to operate near a critical point between ordered and disordered dynamics, balancing sensitivity and stability. Testing this hypothesis on finite data requires metrics that survive estimation noise and connect to established statistical physics models.

The Latent framework contributes scalar diagnostics that remain comparable across experiments once measurement embeddings are fixed, complementing scaling-collapse analyses that demand large dynamic ranges.

This paper pairs a mean-field Curie–Weiss scaffold (§2) with a Lean 4 lemma bundle (§3) and Monte Carlo diagnostics on finite spin systems (§4). The Latent Number \(\rho\) used in §2.4–§4 is a spectral compressibility ratio built from sampled covariance eigenvalues (see numerical_validation.py); it is not the abstract real parameter \(\rho\) that appears in the inequality lemma critical_balance_rho_one inside the formal file. The effective dimension \(N^\ast\) counts directions needed for a fixed cumulative variance threshold in that same embedding.

Thirty-six kernel-verified theorems in elysium/fields/bio_brain_criticality/platonic.py (zero user axioms in the domain state) organize into six narrative groups: order-parameter constraints, subcritical-style bounds, supercritical-style bounds, critical-window inequalities, information-processing lemmas, and cross-domain bridges to Ising and epidemic imagery. §3 states how those lemmas sit in the story; it is not a full mean-field derivation on the page.

Numerical regression uses \(N=50\) sparse random symmetric couplings (not the all-to-all Hamiltonian of §2). Fourteen scripted checks pass; \(\rho>1\) holds on the three reference temperatures, while susceptibility need not peak at the nominal \(T_c\) scale when coupling is heterogeneous—the scan exhibits a \(\chi\) maximum on the sampled grid (see §4).