Existence and uniqueness of the smooth maximal globally hyperbolic development of constrained initial data in the selected matter sectors

Given: Countable Choice, Dependent Choice, connected complete constrained initial data of the specified fixed admissible model, and at least one development, with the actual branch local existence and geometric uniqueness/restart obligations supplied.

The set-sized family and the quotient: the family $\mathcal{X}$ of canonical developments over $\mathrm{\Sigma }\times \mathbb{R}$ is a set, the generated identification relation on the disjoint union is given by the pairwise maximal common developments, and the quotient $Q$ is Hausdorff, locally Euclidean and smooth with a Lorentzian metric, time orientation and matter fields, all member maps being diffeomorphisms onto open sets preserving the structure; these maps are data-fixing isometries onto their images, and every causal curve in $Q$ inextendible in $Q$ meets the initial surface exactly once (Existence of a universal development: the quotient of the set-sized family of canonical developments along pairwise maximal common developments clauses (a)-(d)).

Second countability of $Q$: there is a countable dense subset of $Q$, and from it and the convex normal exponential charts a countable basis of $Q$ is produced; hence $Q$ is second countable (A countable dense subset of the universal development by rational timelike exponential steps from a dense subset of the initial surface, An explicit rational-exponential basis for the universal development quotient).

Every development embeds into $Q$: for every development $(M,g,N,j,\psi )$ of the data there is a data-fixing development embedding into $Q$ (Existence of a universal development: the quotient of the set-sized family of canonical developments along pairwise maximal common developments clause (e)).

Rigidity and uniqueness of data-fixing maps: a data-fixing isometric immersion between developments is injective and a diffeomorphism onto its open image; on a connected development two data-fixing isometric immersions coincide; and every data-fixing self-isometry of a connected development is the identity (A data-fixing isometric immersion of developments is injective, hence a development embedding; uniqueness of data-fixing embeddings clauses (b)-(c), Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps clause (a)).

Connectedness of members and of $Q$: if $\mathrm{\Sigma }$ is connected, every member $U_{\alpha }$ is connected, and $Q$ is connected. Indeed, each member contains $j_0(\mathrm{\Sigma })$ and is a development, so a maximal timelike geodesic through any point is inextendible in that member and meets its Cauchy surface. Its connected image lies in the component of that point, proving every component meets $j_0(\mathrm{\Sigma })$ (Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal clause (b), Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a)5); the member maps identify these initial surfaces, so the union $Q$ of the images is connected.

Development axioms of $Q$: with [L2] the space $Q$ satisfies every clause of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a), i.e. it is a time-oriented second countable smooth Lorentzian four-manifold with a Cauchy embedding of $\mathrm{\Sigma }$, the field equations, and the induced data; hence $(Q,\bar{g},\bar{N},\bar{j},\bar{\psi })$ is a development of the data.