Additive Moment Characters and the Phase-Lift Obstruction in Registration Theory

This research paper advances the mathematical foundations of Registration Theory (RT) by investigating whether conservative bilateral daughter registration can support an additive conjugate phase structure relevant to the emergence of quantum action. Using the existing recursive daughter-registration model, the paper establishes that its nonlinear daughter-construction rules become exactly additive when expressed through a three-component moment ledger consisting of transferable capacity, the first organizational moment, and the combined second moment. It then proves a conditional classification theorem: continuous scalar phase generators that compose additively under admissible daughter mergers are linear combinations of the three conserved moment coordinates. Imposing invariance under shifts of the organizational-coordinate origin selects a unique generator proportional to transferable capacity, up to an undetermined constant. A second result identifies a phase-lift obstruction: arbitrary fractional budget transfers cannot consistently divide a bare circle-valued phase without additional information. An unwrapped additive phase or equivalent retained structure resolves this mathematical ambiguity. Together, these results establish a conditional mathematical bridge between bilateral registration, conservation, conjugate phase composition, and a possible universal energy–frequency relationship. The paper does not claim to derive Planck's constant, physical quantum phase dynamics, or a completed quantum theory. Instead, it identifies precise mathematical conditions under which Registration Theory could support a universal action scale and distinguishes established model-level theorems from unresolved physical reconstruction problems. The accompanying research archive provides editable LaTeX source, symbolic verification, computational audits, and supporting documentation. Version 0.1 — October 20, 2026.