The Physical Metric as Normalized Registration Geometry

This paper is Paper 4 of the five-paper completion spine for the Registration Tensor program. Paper 1 defined the registration tensor object, Paper 2 moved it into coordinate-free tensor geometry, and Paper 3 established a Lorentzian registration-signature form L_reg = -alpha tau otimes tau + h_reg. The present paper addresses the next structural step: how a Lorentzian registration-signature form becomes a physical metric candidate through normalization, units, and calibration.

The central object is the physical normalization map N_phys, which sends the Lorentzian registration-signature form L_reg to a normalized physical metric candidate g_reg. In local form, this is written as g_reg_mu_nu = Omega^2(x) Lhat_reg_mu_nu, where Lhat_reg_mu_nu is the convention-normalized registration-signature form and Omega^2(x) is a positive calibration factor. The normalization fixes scalar-clock scale, transverse record scale, relative clock/ruler normalization, and remaining conformal freedom.

The strongest claim is that, given a smooth nondegenerate Lorentzian registration-signature form L_reg and an admissible physical normalization map N_phys, one obtains a physical metric candidate g_reg = N_phys[L_reg]. This completes the physical metric stage of the five-paper spine.

The paper does not claim that the map from A_mu_nu to g_mu_nu is finished. It does not derive general relativity, gravity, mass, photon physics, or observational confirmation. It defines a normalized registration-geometric metric candidate and leaves the final field-to-metric map for Paper 5.