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 $C(t,t^{\prime })$ satisfies tMSI of order $\hat{I}\pm$ if $C(kt,k{t}^{\prime })=k^{-\hat{I}\pm }C(t,t^{\prime })$ 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 $(t{t}^{\prime }{)}^{-\hat{I}\pm /2}$ and a shape function $F(t/t^{\prime })$ diagonalized by the Mellin transform. This reveals a decoupling of two independent exponents: the dynamical exponent $\hat{I}\pm$, carried by the envelope, and the spectral relaxation exponent ${\hat{I}}^2$, determined by the eigenvalue decay of the finite-dimensional truncation. Their equality $\hat{I}\pm ={\hat{I}}^2$ characterizes a simple critical point; their inequality $\hat{I}\pm \ne {\hat{I}}^2$ is the signature of temporal multicriticality. We provide a classification of tipping points. The Landau quartic coefficient $a_4$ is given exactly by $a_4=p^2+q^2-2\hat{I}»pq-g_{\hat{I}\pm \hat{I}\pm {\hat{I}}^2}^2\hat{I}“(\ddot{I}{ƒ}_{\hat{I}}\pm ,\ddot{I}{ƒ}_{\hat{I}}^2)$, where $\hat{I}»=2\sqrt{\ddot{I}{ƒ}_{\hat{I}}\pm \ddot{I}{ƒ}_{\hat{I}}^2}/(\ddot{I}{ƒ}_{\hat{I}}\pm +\ddot{I}{ƒ}_{\hat{I}}^2)\in (0,1]$, $g_{\hat{I}\pm \hat{I}\pm {\hat{I}}^2}$ is the three-point structure constant, and $\hat{I}“>0$ 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 $\hat{I}\pm ={\hat{I}}^2$ 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.