From Brownian Motion to Black Holes: Chronoscalar Transport via the Hessian Flip

Abstract

We develop a single covariant transport principle that spans nonrelativistic Brownian motion, relativistic dissipative hydrodynamics, heavy–ion anisotropic flow, and black–hole horizon dynamics [1–6]. The framework is built on a physical scalar ordering field T(xμ) whose gradient selects preferred directions and whose Hessian governs admissible microscopic support [7–10]. The central mechanism—the Hessian flip—maps negative–curvature directions of T (microscopic stabilization corridors) into enhanced macroscopic dissipation channels [11–14]. This yields a covariant anisotropic constitutive law that is hyperbolic, entropy producing, and regularizing, providing a geometric closure for Navier–Stokes–type evolution and its membrane–paradigm dual [15–19]. The same ordering geometry deforms the path–integral measure, producing a “yaw” of propagator weights and an explicit derivation of Tsallis non–extensive statistics (q > 1) from diffusion on a curved T background [20–24]. We connect these structures to quantitative heavy–ion observables (CMS v2(pT ) and ALICE RFB(pT )) [25–29] and formulate theorem–level horizon results, including a Regular Core Theorem and a Sub–Extremal Spin Theorem [30–34].

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