Protected-Code Recoverability as a Diagnostic for Spin-Coupled Gravitational Channels

Abstract

Protected subspaces are usually introduced to suppress noise. Here they are used as diagnostic filters for gravitational channel structure. The question is not whether a gravitationally motivated environment causes decoherence, but what error algebra it induces on a protected code and its nearby error shell. We first formulate a recoverability diagnostic for collective versus constituent-resolving channels. For the code $\C=\operatorname{span}\{\ket{01},\ket{10}\}$, collective dephasing obeys $P_\C(Z_1+Z_2)P_\C=0$, while an unmatched spin-coupled response obeys $P_\C(q_1Z_1+q_2Z_2)P_\C=(q_1-q_2)Z_L$. Thus matched response is protected and unmatched response becomes logical dephasing. We then work this out in a weak-field axial-torsion model drawn from the propagating-torsion sector of quadratic Poincare gauge gravity. Under an even-$N$ CPMG sequence, the unmatched recoverability contrast samples a torsion spectral density at a tunable control frequency. A thermal or quasi-static axial background consistent with present spin-gravity and torsion bounds appears far too small for discovery, while a narrowband nonequilibrium bath remains only a conditional enhancement window. Finally, we describe three upgrade paths: $n>3$ hybrid-code design, a model-derived two-block axial exchange term, and a bath-commutator protocol. These are routes toward stronger nonclassicality tests, not results already achieved. The uploaded PDF contains the fully typeset abstract and mathematical notation.