Conditional Conjugate-Ledger Reconstruction in Registration Theory: Host-Relative Quadratic Selection, Rank-One Pure Registrations, and Lorentzian Moment Geometry

This paper develops a conditional mathematical reconstruction connecting conservative bilateral daughter registration, host-relative invariance, and Lorentzian moment geometry within Registration Theory (RT).The investigation establishes three principal results.First, a generalized family of conservative daughter-registration laws demonstrates that bilateral exchange, additive conservation, and consequential retention do not independently require quadratic variance. An explicit countermodel establishes this limitation.Second, the Host-Translation Selection Theorem proves that requiring retained historical consequences to remain invariant under arbitrary translations of a host's organizational origin uniquely selects a quadratic retention law within the specified class of twice-differentiable convex generators.Third, the Conjugate-Ledger Reconstruction Theorem establishes that a symmetric two-channel registration ledger, subject to determinant-one covariance, possesses a unique rank-one pure-registration representation up to normalization. Bilateral contact between linearly independent pure registrations then produces a strictly positive determinant, yielding a nondegenerate Lorentzian quadratic form on the three-dimensional moment state space.These findings provide an exact mathematical bridge between conjugate registration, bilateral contact, and consequentially retained organization.The paper explicitly distinguishes established model-level theorems from conditional representation assumptions. Deriving the symmetric-square ledger directly from primitive Trinity axioms, identifying physically admissible host transformations, and connecting the resulting geometry to operational spacetime remain foundational research objectives.