Companion Technical Note: Part III's Mathematical Apparatus and the Operational Architecture of Register: A Formal-Compatibility Study

This companion technical note evaluates whether the mathematical apparatus of Mathematical Framework for Multi-Scale Recursive Dynamics, Part III of the prior series, can coherently support the register architecture developed in Universal Recursive Dynamics Papers A and B. It is a formal-compatibility review, not a derivation. The note evaluates two candidate directions: tensor or product structure over Part III's parameter space, for register axes, intersectionality, and cross-system mutual conditioning; and saturation-derivative structure, for register expansion as a derived quantity. It concludes that neither direction is sufficient alone and proposes a three-apparatus synthesis: cross-scale coupled dynamics for the cascade structure, per-scale saturation indices for threshold events, and parameter-space trajectory for post-threshold register expansion. The note is careful about its limits. It does not claim to derive the framework mathematically. It names a central information discrepancy: Part III's information parameter operates in a Shannon-style sense, while the framework's broader use of information remains not yet formally specified. This discrepancy is identified as a frontier any future formal work must address. Pressure-test points and interpretive moves are flagged explicitly throughout.