Independent Composition and Quantum Limits in Transfinite Γ-Turing Machines

A Turing machine carries a computation forward through successive configurations, but continuation beyond infinitely many steps requires a rule assigning the configuration from which the next instruction can proceed. The Quantum Γ-Turing machine considered here retains a finite-dimensional register’s trace-norm limit when it exists, preserving the coherence needed by subsequent operations. Averaging a history that never settles permits another continuation, although registers independent at every finite stage can then acquire correlations through their shared timing. We characterise when an assigned state can replace its history without changing a later joint calculation. For assignments compatible with every fixed quantum operation, the largest composition error equals the mean trace-norm distance from the assigned state, and a partner carrying classical labels can approach that value. Independent composition with every partner holds precisely when temporal variance under the chosen mean vanishes. Requiring this under every shift-invariant mean gives uniform vanishing of average squared fluctuations, a condition that permits sparse excursions despite failure of ordinary convergence. For a programme repeatedly applying one fixed channel, independent composition is instead equivalent to ordinary convergence, with the exact error determined by its surviving nonstationary spectral component. Algebraically specified channels and initial states admit a finite decision procedure and certified bounds, while bounded repetitions of states preserve the stronger averaging condition and transfer its certificates to a simulation. These results connect transfinite continuation to finite verification for specified programmes and provide composition certificates that can be carried through a controlled simulation.