Parity Observation and Flow-Relative Fixed-Point Capacities: An Observation-Flow Formulation of Loopback Observation Theory

This paper develops an observation-flow formulation of Loopback Observation Theory (LOT), focusing on parity observation and flow-relative fixed-point capacities in finite observation models. The central claim is that a loopback invariant should not be treated as an absolute scalar attached to an observation instrument alone. Rather, it appears only after specifying an observation instrument, a record partition, a preserved relation, an observation flow, and, where relevant, a decoder or retrieval rule. The main example is a three-body odd-parity observation model. In this model, record-only parity characters are degenerate, while a logical-inclusive parity relation can survive independent bit-flip noise. The surviving relation does not by itself determine a capacity: under a synchronized-flip mixing flow the fixed-point capacity vanishes, whereas under an even-flip preserving flow a nonzero binary-channel capacity remains. The paper then generalizes this split to an n-body parity family, where the parity of the record-flip subset classifies whether the relevant relation is preserved or destroyed within the specified flow family. The paper also provides auxiliary classical shadows, numerical no-go observations for simple scalar candidates, and comparisons with nearby work on decohered quantum error-correction dualities, quantum partial information, and reference/pointer distinctions in measurement theory. The contribution is intentionally limited. It is not a new theory of quantum error-correction phase transitions, not a replacement for existing duality frameworks, and not a new quantum PID. Its aim is to formulate, within finite observation models, how fixed-point capacity depends on the chosen observation flow and the relation tested under that flow.