NSk--Dirac Local First-Order Spinorial Geometry, the Lorentzian Dirac Operator, and the Einstein--Dirac Stress--Enery Tensor in the NSk/ψ Program

NSk--Dirac establishes the local first-order spinorial apparatus of the NSk/ψ programme. The module builds the local Clifford algebra, spinor field, tetrad, spin connection, spinorial covariant derivative, and the Lorentzian Dirac operator on the exact local tetrad background exported by NSk--Einstein. The core of the module separates the covariant part from the realization-level Hamiltonian form. The exact Dirac operator does not require a separate evolution parameter; the EinsteinTimeContract is consumed only in the local weak-field Hamiltonian branch. The module also contains a Lorentzian Dirac–Lichnerowicz-type identity, understood as a local algebraic-differential identity, without exporting elliptic or spectral consequences specific to the Riemannian case. In the variational part, the module defines the spinorial stress–energy tensor as a variational/tetrad tensor, type-compatible with the Einstein equations, rather than as a raw canonical Noether tensor. For variable effective mass, the module makes explicit the need for a Bianchi gate: the spinorial tensor alone is then not a legal standalone source for the Einstein equation without an additional sector carrying the missing divergence. The module also contains a flat compatibility branch: on Minkowski data and in the constant-mass subbranch, the local Dirac operator takes the standard flat form. This is a reference variant, not a mandatory reduction requirement for all branches of the programme. Global spinorial legalization belongs to NSk--Spin, while further use of the Dirac operator belongs to downstream modules, in particular Einstein, Minkowski, Wightman, Yang–Mills, and future quantum field theory modules.