<b>Retained History and Operational Frequency Transfer in Registration Theory: A Conditional Clock-Transport Identity and Calibration-Invariant Test</b>

This technical research note extends the investigation of joint-history compression and consequence completeness in Registration Theory (RT) toward operational clock measurements, signal propagation, and frequency transfer. Building on the conservative recursive daughter-registration model, the paper examines whether differences in retained historical pairing can influence the relationship between emission intervals, reception intervals, and locally registered frequencies. A conditional clock-transport identity is derived using emitter and receiver reference-cycle counts, a differentiable signal-arrival map, and accumulated signal phase. The analysis distinguishes changes in arrival spacing from phase evolution, constant transport delays, and amplitude effects. The paper establishes a calibration-invariant comparison between two signal-transfer channels sharing matched endpoint reference states. It also connects this operational framework to RT's consequence-completeness criterion, identifying the conditions under which historically distinct registrations could produce distinguishable future measurements. These results provide a mathematical framework for testing whether retained registration history has observable consequences for frequency transfer and potentially cosmological redshift. The paper does not claim to derive physical clock dynamics, cosmological expansion, or the Hubble tension from Trinity axioms. The outstanding objective is to derive the required signal-transport and reference-clock dynamics from canonical RT registration rules.