Temporal Matrix Scale Invariance and the Classification of Tipping Points

Abstract

We introduce temporal matrix scale invariance (tMSI), a mathematical structure for the two-time correlation kernel of a multivariate observable. A kernel $C(t,t')$ satisfies tMSI of order $\alpha$ if $C(kt, kt') = k^{-\alpha}C(t,t')$ for all $k>0$; this condition holds near a tipping point, where the divergence of the coherence time produces temporal scale freedom. By a kernel factorization theorem, every tMSI kernel separates into a power-law envelope $(tt')^{-\alpha/2}$ and a shape function $F(t/t')$ diagonalized by the Mellin transform. This reveals a decoupling of two independent exponents: the dynamical exponent $\alpha$, carried by the envelope, and the spectral relaxation exponent $\beta$, determined by the eigenvalue decay of the finite-dimensional truncation. Their equality $\alpha = \beta$ characterizes a simple critical point; their inequality $\alpha \neq \beta$ is the signature of temporal multicriticality. We provide a classification of tipping points. The Landau quartic coefficient $a_4$ is given exactly by $a_4 = p^2 + q^2 - 2\lambda pq - g^2_{\alpha\alpha\beta}\Gamma(\sigma_\alpha, \sigma_\beta)$, where $\lambda = 2\sqrt{\sigma_\alpha\sigma_\beta}/(\sigma_\alpha+\sigma_\beta) \in (0, 1]$, $g_{\alpha\alpha\beta}$ is the three-point structure constant, and $\Gamma > 0$ is in explicit closed form. The transition is continuous for $a_4 > 0$, tricritical for $a_4 = 0$, and discontinuous for $a_4 < 0$. The simple critical point $\alpha = \beta$ is maximally fragile: any nonzero operator mixing drives $a_4 < 0$, placing the synchronized state generically at the edge of catastrophe. The framework yields a matrix-valued early warning diagnostic, computable from a multivariate time series without knowledge of the underlying equations, that classifies an approaching tipping point as recoverable or catastrophic. Applications to epilepsy and acute myocardial infarction are discussed.