Joint-History Compression, Bilateral Retention, and the Consequence-Completeness Obstruction in Registration Theory

This technical research note investigates whether a retained daughter state in Registration Theory (RT) must preserve every consequential distinction in joint historical pairing that cannot be reconstructed from separately retained parental states. Using the previously published conservative recursive daughter-registration model, the investigation establishes an exact two-event exponential-retention counterexample. Two distinct joint histories possess identical parental weighted means and variances but opposite weighted cross-covariances, while the selected daughter rule produces identical three-component organizational states. The paper establishes the additive structure of the daughter's conserved-moment ledger and demonstrates how different merger histories can share the same active organizational representation. A second result constructs a symmetric, positive, cross-history-dependent extension of the daughter-allocation rule. This extension preserves the original three conservation laws while producing different downstream daughter capacities from otherwise identical organizational states. The investigation then establishes a precise consequence-completeness criterion: a retained-state representation is sufficient exactly when every admissible future host response factors through that representation. These results establish exact mathematical compression and conservation-level underdetermination. They do not demonstrate that physically consequential information is lost under canonical Trinity dynamics or establish a universal daughter-formation law. The remaining research objective is to determine whether ACRF-4 cross-retained states preserve the relevant joint-history distinctions and whether those distinctions remain observable through admissible recursive host interactions. Version 0.1 — October 24, 2026.