Scope and status of the Gr14 high-regularity symmetric hyperbolic material

This page contains the Gr14 high-regularity symmetric hyperbolic proof material: the periodic first-order theory (E0), the harmonic-gauge reduced Einstein equation and its subsidiary system (E1), the geometric data and local development theorems (E2), and the selected matter reductions with the complete geodesic-dust residual chain (E3, E3d--E3d.2). Its content assumes Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for local analytic extraction and manifold patching, and Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the causal-curve calculus in the Cauchy-neighborhood argument. Its physical content is conditional: the physical reading is the model deduction of Local Cauchy development of the adopted Einstein field law for smooth constrained data in the selected matter sectors from the adopted postulates Adopted Einstein field equation in the direct-equation presentation and the matter definitions The scalar-field matter model of the classical gravitational framework, The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy, The dust matter model: rest density, four-velocity, stress-energy and dust system; the mathematical carriers are High-regularity local developments for the selected matter sectors: vacuum with Lambda, scalar, strict barotropic fluid and geodesic dust and its prerequisites.

What is established. Local-in-time smooth developments of the adopted field law with the selected matter sectors, from smooth constrained data, with uniqueness near the initial surface up to a data-fixing time-orientation- preserving isometry, and with the initial surface lying in the interior of its domain of dependence.

What is not established or assumed away. No global or all-time existence, no uniform existence time on noncompact data, and no maximality or boundary behaviour of the spacetime development (that is the scope of the successor page). The matter results do not treat shocks, nonsmooth free boundaries, nonsmooth vacuum interfaces, junction conditions, distributional or low-regularity sources, or boundary layers; the theorem works in the smooth/high-regularity branch with $s\ge 5$ and makes no optimal Sobolev threshold claim. These exclusions are distinct from smooth zeros of the dust density. For dust, the construction occurs before flow crossings and a smooth future unit velocity is prescribed and remains geodesic even where the smooth nonnegative energy coefficient $\epsilon$ vanishes. Thus smooth zero sets and open empty regions are included, with no division by density. The source density parameter is $\epsilon$, with mass-normalized ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}=\epsilon \mathrm{/}{c}^2$ for every $c>0$; in the reduction notation $r=\epsilon$ solves ${\mathrm{\partial }}_{s}r=-r\chi$ with the explicit exponential solution of Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry. For the fluid branch the strict hypotheses $ϵ+p>0$, $0<p^{\mathrm{\prime }}<1$ are essential; the failure at negative $p^{\mathrm{\prime }}$ is the companion counterexample on the examples page. No statement is made that any model here describes nature, and no empirical premise enters.

Conventions and sources. The reduction uses $\gamma =-g$ and the explicit Schouten normalization and curvature decomposition of its definition. Vacuum reduction signs therefore differ from the original metric signs by that constant flip. The physical unit convention, including the distinction between the normalized velocity $u$ with $g(u,u)=-1$ and the SI proper-time four-velocity $U_{\mathrm{S}\mathrm{I}}=cu$ with $g(U_{\mathrm{S}\mathrm{I}},U_{\mathrm{S}\mathrm{I}})=-c^2$, and the selected scalar, fluid and dust units are stated in Local Cauchy development of the adopted Einstein field law for smooth constrained data in the selected matter sectors. The local dust proof derives the electric/magnetic projections and the Cartan subsidiary system in those conventions; its magnetic double-epsilon term has the sign required by trace preservation. Friedrich and Rendall supplies derivation provenance; the displayed local arguments carry the deductions.