The Registration Tensor: From Measure-Valued Records to a Coordinate-Covariant Field

This paper is Paper 1 of the five-paper completion spine for the Registration Tensor program. It consolidates the measure-valued registration operator and same-packet covariance tests into a proposed final registration tensor object. Starting from the registration measure nu_reg = pi_#(W_reg dmu), the paper defines a finite-scale family of local covariance objects computed on same-packet neighborhoods of the registration measure.

The directly tested object is contravariant, written T_reg^{ab}. It is proposed as the atlas-compatible refinement limit, when that limit exists, of same-packet local covariance tensors derived from the measure-valued registration operator. The paper distinguishes this object from a covariant dual geometry, which may be constructed only when the tensor is nondegenerate and appropriately normalized.

The evidence summarized includes the synthetic same-packet test, the actual reconstructed Julia registration-measure test, the robust Julia packet test across 24 center/transform cases, and the finite-scale tensor-candidate criterion. The paper completes the first stage of the five-paper completion spine by defining the registration tensor object and its finite-scale-to-limit criterion.

The paper does not yet claim coordinate-free tensor geometry in full generality, Lorentzian signature, a physical metric, gravity, mass, photon physics, or the map from A_mu_nu to g_mu_nu. Those remain the subjects of the remaining four papers in the completion spine.