Operational Registers and Reciprocity-Preserving Morphisms v0.1

FRC uses several legitimate coherence routes, including phase order, the von Mises mean resultant, quantum purity, basis-dependent off-diagonal interference, and platform-specific composites. This paper formalizes how those routes may share a framework without being declared the same observable. A weak reciprocity map preserves the pulled-back ledger one-form up to a nonzero multiplier. A strong affine register morphism additionally supplies a state or model map and a commuting ledger square. Both classes compose; the strong class forms a category with explicit identities, associativity, exact-preserving and signed subcategories, and an invertible groupoid. One-form preservation alone is shown insufficient by an exact counterexample. The von Mises family supplies a nontrivial strong morphism into the phase-distribution register: the state-space inclusion commutes exactly with the entropy and mean-resultant ledger, but is not an isomorphism. A qubit no-go theorem proves that purity and fixed-basis off-diagonal coherence admit no single-valued ledger adapter under the identity state map, even though both are valid operational routes. Present FRC registers therefore form a typed formal category populated by a sparse graph of verified arrows, not one proven physical equivalence class.