A Native Entangling Gate Inside a Protected Code, and Its Price

A register of n channels, each a complex function on the eight ticks of the ledger’s cycle, is updated once per tick: shift by one tick, then keep the modes the environment records. On the subspace where every channel is antisymmetric under a half cycle (the odd-mode code, of dimension 4^n) the update is a pure shift for every environment: nothing is recorded. Inside that code there is a native entangling gate, the quarter turn of a two-channel interaction obtained by conjugating one channel rotation with the ledger’s entangler; on the code’s free bits it is a controlled flip. It is a word of fourteen flat tokens: three entanglers, ten braid rotations through three fixed angles, one entangler. The price is exact. A tick removes exactly the weight a state carries outside the code and posts exactly when that weight is positive. Along the word the exposed weight is ½, 1, ½ after the entanglers, ½ after each rotation block, 0 at the end. A tick at any of those four moments, and one at the close, keep a quarter of the weight and leave the exact gate on the survivors: a heralded gate with success probability one quarter.