Observable-Horizon Optimization in Reduced DD Spectroscopy: Floor-Resilient Frequency Geometries and Robust Protocol Design

This work solves the inverse design problem for multi‑ratio dynamical‑decoupling (DD) spectroscopy in the presence of a flat noise floor. For spectra of the form S (ω) =Aω−α+B, p=B/A, the asymptotic robustness of reduced inverse reconstruction is governed by the prefactor C (α, ωk) =σmin⁡ (J (1) ), the smallest singular value of the leading‑order Jacobian coefficient matrix. We derive an exact analytic expression for the column structure of J (1) and show that its row geometry is controlled by the sensitivity levers λk (α) =log⁡ωk+δ (α), which depend logarithmically on frequency. The angular separation of these levers determines the asymptotic robustness, yielding a transparent design principle: maximize lever spread over the accessible frequency range. We map the full design space of three‑frequency protocols within a fixed range ωmin⁡, ωmax⁡ and identify the horizon‑optimal protocol family for each spectral slope α. The optimal geometry concentrates sampling toward the extremes of the accessible range, leaving the center undersampled; the gain over log‑uniform placement exceeds a factor of two for α≥1. We further construct a robust protocol for cases where the target slope is unknown, showing that the lever‑spread principle extends cleanly to broadband optimization. Together with the collapse taxonomy of earlier work, these results close the design side of reduced DD spectroscopy: they identify when floor‑induced information compression is inevitable, and how it can be minimized through protocol geometry. V2: Refined the geometric interpretation of the leading-order Jacobian structure by replacing the previous “lever-spread” formulation with a weighted row-geometry criterion. Corrected the asymptotic lever expressions and row-vector notation, clarified the α-dependent crossover mechanism, and improved the treatment of the α→0 limit. Added extensive numerical verification of the asymptotic scaling law, Jacobian convergence, finite-ρ optimality consistency, and robust broadband protocol optimization.