Outcome-Resolved Mid-Circuit Measurements as Exact Gauge Constraints

Self-consistent quantum tomography is invariant under similarity transformations that can leave all observed circuit probabilities unchanged. Mid-circuit measurements (MCMs) add a qualitatively different object to this problem: an outcome-resolved quantum instrument has both a classical label and a post-measurement quantum state, so each outcome branch must itself remain completely positive under any admissible representation change. We study an exactly solvable residual-gauge model in even dimension d = 2r ≥ 4. After a fixed two-qubit Clifford control set reduces the trace-preserving similarity freedom to the scalar family B_c = D + c(id − D), we consider a symmetric two-sector Lüders instrument with label-confusion probability p and isotropic state-and-label replacement probability η. For the transformed branches J_{s,c} = B_c J_s^{η,p} B_c^{-1}, we prove exact necessary-and-sufficient complete-positivity criteria for both positive and negative scalar gauges. For c > 0 the condition is (1 − η)|1 − 2p||c − c^{-1}| ≤ η, while for c < 0 it is (1 − η)|1 − 2p|(|c| + 2 + |c|^{-1}) ≤ η. The result follows from an explicit Choi-spectrum calculation. It yields a closed positive gauge interval, an exact threshold for the appearance of negative gauges, and a logarithmic gauge radius |log c| ≤ arsinh(K/2), with K = η/[(1 − η)|1 − 2p|]. If the outcome label is discarded, the averaged channel is invariant under every B_c, so the additional gauge information disappears exactly. Two lag correlations determine K within the assumed model and therefore give an observable conditional gauge interval; a Hoeffding bound gives a finite-sample outer interval. We also derive an explicit outer interval when each instrument branch is known only within a Choi-Frobenius error budget. Finally, a Lüders-versus-measure-and-prepare countermodel shows that repeated outcome strings alone do not validate the quantum state continuation. A targeted literature audit finds substantial prior art on gate-set-tomography gauge freedom, instrument tomography, MCM learnability, and gauge-aware physicality intervals, including a closely related 2026 three-measurement protocol. The contribution is therefore deliberately narrow: the exact higher-dimensional full-complete-positivity boundary for the stated instrument family and its derived consequences, not the general idea that physicality can restrict tomography gauge.