NSk--Einstein Local Lorentzian Spacetime, the Einstein Field Equation, and the Einstein--Dirac Package in the the NSk/ψ Program

NSk--Einstein establishes the local Lorentzian spacetime apparatus of the NSk/ψ program. The module develops the exact local geometric track: Lorentzian metric, interval, proper time, causal structure, Levi-Civita connection, Einstein tensor, local field equation, and the local Einstein-Hilbert and Einstein-Hilbert-Dirac variational principles. A central feature of the module is the separation of two logically distinct inputs. The first is the exact geometric input consumed from NSk--Geo and realized as Einstein.ExactLorentzianInput. The second is the weak-field realization readout consumed from NSk--Gravity as Einstein.WeakFieldReadoutInput, used to construct the local time readout, the g00 component, redshift, and the NewtonMatch condition. The weak-field branch is not the foundation of the entire module, but a specific readout, calibration, and compatibility-testing regime. For coupling to matter, the module defines the local Einstein tensor, the Bianchi identity, the local source equation for admissible stress-energy tensors, and the legal consumption of the spinorial stress-energy tensor exported by NSk--Dirac in the conservative case. After closing the Einstein-Hilbert variation and the tetrad variation of the Dirac action, the module contains the local Einstein-Hilbert-Dirac variational principle for constant effective mass. Its critical points are locally equivalent to the coupled Einstein-Dirac system: the Einstein field equation with spinorial source, the Dirac equation, and the adjoint Dirac equation. The export to NSk--Dirac is carried by the exact local tetrad record Einstein.ExactDiracTetradInput and the package Core.Einstein.DiracMetricInput. Global spinorial legalization of this local package belongs to NSk--Spin and is controlled by the vanishing of the second Stiefel-Whitney class. The module prepares the formal background for downstream modules of the programme: NSk--Spin, NSk--Minkowski, matter sectors, gauge/Yang-Mills fields, black holes, cosmology, and future quantum field theory modules.