Distinguishability of Q5 Deviations via Phase-Cycling Invariants

T73 develops a phase-cycling discriminator for Q5-consistent deviations that cannot be identified from a single interference scan. Building on T72, which shows that any first-harmonic residual can be absorbed by refitting the ordinary fringe parameters, T73 demonstrates that the lost information can survive in the constrained transformation law across multiple controlled phase settings. The theorem considers four scans separated by externally controlled \(\pi/2\) phase offsets and replaces independent per-scan refitting with a single jointly constrained fit across all four settings. For these offsets, the ordinary fringe model occupies one two-dimensional phase-cycling chirality subspace, while the opposite chirality subspace is exactly orthogonal to it. Consequently, the joint fit removes the ordinary \(J^{-1}\)-chirality component while preserving the orthogonal \(J\)-chirality component unattenuated. This yields the phase-cycling invariant \[ \mathcal I_{Z_4} = \sum_{k=0}^{3} \left|v_{k+1}-Jv_k\right|^2, \] where \(v_k\) is the residual quadrature vector extracted after the joint fit. A nontrivial signal detectable by this construction must exhibit the surviving quarter-turn transformation law \(v_{k+1}=Jv_k\), giving \(\mathcal I_{Z_4}\approx0\) with nonzero residual amplitude. The result is a single-chirality test rather than a choice between two equivalent rotation signs, because the opposite chirality is structurally absorbed by the joint null model. T73 does not establish that Q5's physical residual necessarily carries the surviving chirality. That identification requires a separate residual-to-observable bridge connecting the later Q5 residual structure to the experimental phase-cycling coordinates. Nor does the theorem establish immunity to correlated experimental systematics; those require separate noise and control analysis. T73 establishes the mathematical distinguishability mechanism and operational phase-cycling protocol under the stated common-reference and joint-fit assumptions. Status: solid for the exact chirality separation, joint-fit discriminator, and \(\mathbb Z_4\) phase-cycling invariant under the stated four-offset protocol; conditional on a common external phase reference and jointly constrained null model; unresolved as to whether the physical Q5 residual occupies the surviving chirality; no claim of experimental detection or uniqueness against all correlated systematics.