Volume 12, Issue 1, pages 497-589
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 $T(x^\mu)$ whose gradient selects preferred directions and whose Hessian governs admissible microscopic support [7–10]. The central mechanism—the \emph{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 $v_2(p_T)$ and ALICE $R_{\mathrm{FB}}(p_T)$) [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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