Assemble the recovery theorem

The local sections, comparison identities, matter maps and observational reductions now form one compatible reconstruction. The cumulative theorem states exactly which operations and target classes are covered.

18. The cumulative reconstruction theorem

The preceding sections have used one admitted support and literal preparation, one smooth geometric calculus, and explicit primitive matter extensions. The physical inputs and operation domains have not been moved into the admission predicate. This permits a precise cumulative statement.

Theorem 18.1 (Conditional source–readout recovery of the 1916 chain)

Fix a smooth four-dimensional event manifold with local coordinate patches, the Lorentz interval convention (2.5), the compatible torsion-free comparison sector, and the physical premises identified in the relevant sections: non-null interval stationarity and test-particle interpretation; Einstein’s vacuum and total-energy coupling laws; the specified symmetric matter stress or fluid and electromagnetic constitutive variables; and the stated boundary and approximation protocols. Then the following hold.

1. Every local target metric and primitive matter tuple in the stated domains has a source realization over $Y_\star$, with explicit reverse maps. The metric fibre is exactly local Lorentz-frame freedom; matter transforms simultaneously, and potential gauge freedom has the separately stated local and boundary scope.

2. Tensor products, contractions, index changes, compatible differentiation, curvature and its contractions are recovered by commuting operation identities. The raw nonmetricity, torsion, contact and Maxwell defects remain explicitly available before selecting their zero sectors.

3. Fixed-endpoint non-null curve variation and the chosen fixed-chart field variation are recovered in both directions, including their allowed test directions and boundary terms. In a unit-volume chart the field variation and source Ricci contraction recover the selected vacuum equations. Early trace restriction gives a different and strictly weaker variational problem.

4. The source canonical current recovers Einstein’s gravitational energy array with its coordinate dependence. The same-realization total-energy addition recovers the full coupled field equation and the original double-divergence, exchange and matter-conservation operations. General-chart volume defects and finite-region fluxes are retained.

5. Fluid constitutive readouts recover the original density convention and an invertible energy/Euler split. Electromagnetic readouts recover both Maxwell systems, stress and the complete transfer identity, without claiming an unsupplied charge-motion law or an inverse from stress alone.

6. Finite-jet recovery yields the stated Newtonian reduction under its weak, slow, quasi-static and material assumptions. The original static rod and clock coefficients and asymptotic bending are recovered with the documented printed-source qualifications. The explicitly labelled spherical completion and cited orbit dependency yield the leading perihelion expression and the proved same-family remainder bounds.

7. Source solutions and target solutions of each selected local system correspond. This does not assert that arbitrary constitutive choices admit solutions, that an initial-value solution is unique, or that a finite set of observations determines every native or physical source.

Proof.

Theorem 3.1 supplies local metric sections and classifies their fibres. Equations (4.1), (9.1) and (12.1) supply the primitive inverse maps. Uniqueness of the comparison connection makes these maps compatible with frame changes and overlaps. Proposition 4.1 and Theorems 4.3 and 6.1 establish the algebra and differential operations, including curvature before any field equation is imposed.

Equations (5.2)–(5.4) recover the curve operation. Theorem 7.1, the surjective variation lift and (7.10) recover field stationarity, with the chart order and boundary conditions kept fixed. Invertibility of $E,q$ and the four-dimensional trace-reversal involution prove the two-way vacuum and coupled field implications. The original current and double-divergence chain is given by (8.2)–(8.5) and Theorem 10.1; (10.12) and (10.15) recover the matter statements. The independent curvature contraction proves their generally covariant residual form.

Theorems 11.1, 12.1 and 12.2, together with (12.8), recover the primitive material operations. Theorem 13.1 transports finite-order calculations under the stated norms; Sections 13–17 perform the specific reductions, optical variation and regularized integrals. Theorem 16.1 retains the endpoint term and Theorem 17.1 proves the orbital remainder.

Apply Lemma 2.1 along these finite dependency chains. Starting with a target solution, choose the local calibration section, construct its comparison, and invert the primitive matter maps; the recovered identities make the corresponding source residuals vanish. Conversely, a zero source residual projects to the target one by the invertible slot maps. Compatible overlap records extend local metric and tensor statements to their atlas descriptions. Chart-tagged energy arrays are not improperly glued as tensors, and no global gauge or potential conclusion is added. The source discrepancies in Appendix A are explicitly excluded from literal simultaneous equivalence and replaced only by the proved, separately labelled equation-consistent statements.

$\square$