Inner-Shell Raman X-Gate Tradeoffs for a Neutral ^171Yb Nuclear Qubit at Fixed Optical Power

We study single-qubit X-gate feasibility for a ^171Yb nuclear-spin qubit encoded in the ^3P_0(F=1/2) manifold and driven by Raman coupling through an inner-shell-excited J=2 intermediate. The design objective is explicitly constrained: optical intensity is fixed at 1\,\mathrmW/cm^2, and the output must report gate-time versus target-infidelity operating points at 10^-2, 10^-3, 10^-4, and 10^-5. We develop a hybrid formal-empirical methodology that combines (i) a convex surrogate optimality certificate for detuning selection, (ii) a robust feasibility-floor certificate for impossibility regions, and (iii) a posterior chance-constrained table for uncertainty-aware decision support. The resulting evidence indicates a stable, non-provisional operating regime at 10^-2 with a best deterministic gate time of 0.1661\,\mu\mathrms and dominant leakage contribution. After targeted posterior uncertainty ablations (model-discrepancy and noise-floor scaling), posterior confidence intervals widen materially and strict-threshold recommendations at 10^-3 and below become infeasible in both deterministic and posterior modules. Those strict rows remain provisional because robust-floor diagnostics still classify 10^-3 as potentially feasible under conservative assumptions. Beyond this specific gate, the study connects atomic-structure uncertainty management to broader neutral-atom processor planning and high-accuracy metrology workflows.