Scope and status of the convex-normal, volume-time and temporal-splitting material

This note fixes the scope of the page's convex-normal, volume-time and splitting content and the status of its claims; it proves nothing by itself.

• Model conditions versus physical readings. Strictly convex normal neighbourhoods (Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points), the local causality inside them (Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic), the auxiliary metric, the volume time function and its Cauchy levels (The Geroch volume function: a continuous strictly increasing time function whose levels are Cauchy sets), the equivalence of the Cauchy-set property with global hyperbolicity (The existence of a Cauchy set implies strong causality and compact causal diamonds) and the smooth temporal function with its complete orthogonal flow (Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow) are mathematical statements about a time-oriented smooth Lorentzian four-manifold. One may optionally interpret the Cauchy levels as predictability slices, the temporal function as a global simultaneity coordinate, or the flow lines as a preferred observer family, but each interpretation requires separate physical adoption. Adopted spacetime model and classical domain of the gravitational framework adopts the spacetime model and classical domain only; it does not adopt these readings. They are not proved here, and nothing on this page is compared with any observation.

• What is proved here. Conditional deductions inside the stated classes of models and curves: the local causal cone and displacement-length calculus of Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic, the locally finite cone-bump and weak-step constructions (Cone bumps: for every point of a Cauchy level there is a smooth bump with past timelike gradient on its positive region below that level, Strip functions: every inner strip carries a smooth function with past timelike gradient, constant below and above the outer levels), the continuous volume time (The Geroch volume function: a continuous strictly increasing time function whose levels are Cauchy sets), the local Cauchy neighbourhoods (Local Cauchy neighborhoods in a conically bounded product chart) and the smooth Cauchy time function with Cauchy levels, complete flow of the normalized field $X=\mathrm{\nabla }T\mathrm{/}g(\mathrm{\nabla }T,\mathrm{\nabla }T)$ and orthogonal product (Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow). Each theorem states its hypotheses, including the choice principles it uses, and each is a statement about the explicitly displayed model.

• Deferred claims. No statement of this page asserts existence, uniqueness, stability or maximality of solutions of the Einstein field law, nor the existence of a maximal globally hyperbolic development of given initial data; those are the objects of the later GR14 and GR15 pages of the track. The splitting proved here is conditional on global hyperbolicity and supplies the geometric setting those pages consume.

• Empirical status. No experiment, measurement or observational uncertainty occurs anywhere in the page's deductions. The empirical qualifications of the framework are carried by the postulate items and by the report pages of the track, not by these mathematical results.