The Achievable Set Is Everything Parity Allows: The Complete Fixed-Point Classification of the Sever's Pairing Operator

The sever's pairing operator was constructed in an earlier paper as a universal object: a single pairing whose fixed points encode, for every dimension, the complete pairing data of a sever. The construction was not an existence proof. It was a classification machine: each sever was to be placed exactly once, in exactly one cell of a fixed-point table, with no sever lost, no sever duplicated, and no cell left to chance. That machine is now complete. We prove that the achievable set is everything parity allows: every parity assignment consistent with the sever's own parity constraints occurs as the fixed-point set of some sever, and no other set occurs. The proof is constructive. Given any admissible parity pattern, we build a sever whose pairing realises it exactly, and we show that any sever's fixed-point set must satisfy the same constraints. The classification is therefore sharp in both directions: the sever's pairing operator does not merely contain the parity patterns. It contains all of them, and nothing else. The count follows immediately: the number of achievable sets is exactly 2^N, where N is the number of independent parity conditions, and each achievable set is realised by a unique sever up to the natural equivalence. This completes the fixed-point classification begun in the construction paper, and it closes the sever program at the level of existence: what can be built is now known, completely, and it is everything parity allows.