A complete conditional recovery of gravity

9. What the complete construction establishes

This native source sector realizes the entire massless spin-two bootstrap and recovers nonlinear Einstein–matter gravity. Support, literal preparation, raw histories, path operations, projection, inverse and admissible variations each have their own definition. The off-shell stress and residual identities retain the derivative self-current, and the cubic coupling reaches the full nonlinear endpoint. This preserves Deser's achievement and supplies the extra work required to lift its operations rather than only its final equations.

Two improvements reduce the separate kinematic assumptions. Multiplicative transport that is smoothly differentiable at constant paths has a velocity-linear generator, with controlled finite-resolution and noise bounds. The nonadditive immersed-path example shows why the hypothesis at constants matters. The unrestricted affine parent dispenses with imposed torsion-free kinematics. Its equations eliminate all nonprojective distortion, and the density-normalized readout removes the remaining projective ambiguity. The projective Palatini kernel is established mathematics; its integration with the source observable and quotient stress extends the off-shell parent while preserving the full metric endpoint.

The construction also gives a restricted scalar selection theorem, derives gauge identities from the chosen laws and fixes the cosmological term within the flat-vacuum Einstein–Hilbert family. These deductions have distinct sources. Admission gives the support and preparation compatibility. Regularity, naturality, the quadratic spin-two law, total-current coupling and the observable contract supply further premises. The combination and exact recovery do not show that the target-neutral core chooses four dimensions, Lorentzian signature, the coupling or a unique matter law. The explicit competing realizations show precisely what selection still has to accomplish.

The next problem is to find a target-neutral principle that constrains the raw action and observable class strongly enough to exclude those competitors. It needs content before the Einstein endpoint is specified. The exact return maps and discriminating alternatives in this construction provide a standard against which to test it.