Source-free smooth maximal globally hyperbolic Einstein–Maxwell development

Given: Countable Choice and Dependent Choice, connected smooth constrained source-free data, the fixed covariant model and its complete two-form data.

Draft model Electrovacuum tensor Cauchy system fixes the equations, stress, constants and pullback conventions. The initial $F$ data in the Statement specify all six independent components on $\mathbb{R}N\oplus T\mathrm{\Sigma }$, with no global potential requirement. Its item file is listed in batch 6; its status is draft and its precheck is pending; this input remains conditional.

Draft constraint result Gauss-Codazzi and Maxwell initial constraints assumes an existing smooth source-free solution near a smooth spacelike embedded hypersurface and proves that its induced data satisfy the four Einstein and Maxwell constraints in the selected $K,e,b$ convention. It proves necessity, not existence from constrained data. Its item file is listed in batch 6; its status is draft and its precheck is pending; this input remains conditional.

Draft identity Maxwell stress algebra and covariant conservation gives ${\mathrm{\nabla }}^a{T}_{ab}=-F_{bc}{j}^c$ for smooth fields satisfying $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$. The exact Hodge conversion is ${\mathrm{\nabla }}_a{F}^{ab}=\frac{1}{3!}{\epsilon }^{bacd}(d{\star }_{g}F{)}_{acd}=\frac{1}{\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}}{\mathrm{\partial }}_a(\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}{F}^{ab}),{\epsilon }_{abcd}:=({\mathrm{v}\mathrm{o}\mathrm{l}}_g{)}_{abcd},$ with indices raised by $g^{-1}$. It follows from metric-compatibility ${\mathrm{\nabla }}_a{\epsilon }_{bcde}=0$, $({\star }_{g}F{)}_{cd}=\frac{1}{2}{\epsilon }_{cdef}{F}^{ef}$, the antisymmetrized covariant derivative, and ${\epsilon }^{bacd}{\epsilon }_{cdef}=-2({\delta }_e^b{\delta }_f^a-{\delta }_f^b{\delta }_e^a)$; the coordinate divergence follows from the trace Christoffel identity and antisymmetry of $F$. Thus $d{\star }_{g}F=0$ supplies ${\mathrm{\nabla }}_a{F}^{ab}=0$, and the model's $dF=0$ supplies the other hypothesis, so the stress identity applies with $j=0$ before Einstein recovery. Its item file is listed in batch 6; its status is draft and its precheck is pending; this input remains conditional.

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data proves smooth local existence and geometric uniqueness for this exact covariant model and complete data on arbitrary smooth initial slices. Its proof verifies the local symmetric-hyperbolic reduction, full model recovery, stress conservation and constraints, conditional on the actual CP01 draft statements [L1]–[L3], whose files are present but whose prechecks are pending although their IDs are listed in the batch-6 manifest.

Causal and Cauchy-surface inputs for smooth electrovacuum developments records the causal, Cauchy-surface and choice interfaces used in the globally hyperbolic development construction, with their exact hypotheses.

Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality defines a development, a data-fixing development embedding, and maximality by universal embedding into a smooth globally hyperbolic development. It makes no completeness requirement.

Existence and uniqueness of the smooth maximal globally hyperbolic development of constrained initial data in the selected matter sectors proves the set-sized gluing, quotient, universality and uniqueness result for any fixed admissible covariant model once constrained local existence and geometric uniqueness hold at every spacelike restart. The actual inspected carrier is published with verification precheck pending. Its proof is sourced to Sbierski, whose original theorem is vacuum-only; therefore the current GR theorem's additional-model clause applies here only after [F1] discharges the EMG local obligations.