Lorentzian Signature from Registration Order and Transverse Record Geometry

This paper is Paper 3 of the five-paper completion spine for the Registration Tensor program. Paper 1 defined the registration tensor as a coordinate-covariant contravariant tensor object, and Paper 2 moved it into coordinate-free form as T_reg in Gamma(Sym^2(TM)). The present paper addresses the next structural step: Lorentzian signature.

The paper emphasizes that the covariance-derived registration tensor T_reg does not by itself automatically imply Lorentzian signature. Instead, Lorentzian structure is obtained from a signed registration decomposition involving a scalar-clock or registration-order one-form tau, a positive transverse record geometry h_reg on ker(tau), and a positive scale factor alpha. The central signed form is L_reg = -alpha tau otimes tau + h_reg.

In four dimensions, when tau is nonzero, ker(tau) is three-dimensional, h_reg is positive definite on the transverse record subspace, and alpha > 0, the resulting registration-signature form has Lorentzian signature (-,+,+,+). The paper gives the corresponding linear-algebra criterion and proof in an adapted basis.

The strongest claim is that a Lorentzian registration-signature structure follows from one scalar-clock/order direction together with positive transverse record geometry. The paper does not claim that L_reg is already the physical metric, does not derive gravity, mass, photon physics, or g_mu_nu, and does not complete the map from A_mu_nu to g_mu_nu. Those remain the subjects of the final two papers in the five-paper completion spine.