Action-Quantized Scar Multiplets and Dissipative Cat States from Fractal Period-Locking in the Open Quantum Baker Map

We open the quantum baker map with a position window of width $w$ and center $q_c$. Classical pruning tiles the $(q_c,w)$ plane into diamonds where the $T$ leading resonances form root-of-unity multiplets. At $w=1\mathrm{/}2$ one periodic orbit survives for almost every $q_c$, and the multiplets are quantized without free parameters: their modulus $2^{-1\mathrm{/}2}$ saturates the pressure bound and their phase is the orbit action. They form cycling multicomponent cat states, and the opening becomes one mid-circuit measurement, enabling spectroscopy with $\sim {10}^2$ shots on quantum processors.