Inverse Geometry and Identifiability of Protocol Ratios in Dynamical-Decoupling Spectroscopy

Inverse Geometry and Identifiability of Protocol Ratios in Dynamical-Decoupling Spectroscopy This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Building on reduced observation geometry, it asks the inverse question: given an observed protocol ratio, what spectral information is uniquely reconstructible, what is only weakly constrained, and what is fundamentally unrecoverable? The analysis is formulated in terms of the Jacobian of the reduced observation map and the associated Fisher information matrix. The paper shows that the inverse problem separates into three qualitatively distinct sectors. In the regular sector, the ratio map is locally invertible and reconstruction accuracy is controlled by a Cramér–Rao bound. For the pure power-law family S (ω) =Aω^α, the ratio R (α) =2/ (1+2^-α) is strictly monotone, admits a closed-form inverse, and yields an explicit conditioning profile through the slope ∂_αR. In the degenerate sector, exemplified by the white-noise family S (ω) =A, the protocol ratio is exactly pinned to R=1 for all amplitudes. The Fisher information for the amplitude direction vanishes identically, so amplitude reconstruction from ratio data alone is impossible. This gives an exact no-go theorem for identifiability from protocol ratios in the white-noise case. In the branched sector, non-injectivity of the inverse map produces intrinsic ambiguity. Using the mixed family S (ω; α1, α2, w) =wAω^α1+ (1−w) Aω^α2, the paper shows both global non-injectivity and local fold branching. Along an explicit symmetric slice, the ratio develops a fold at Rc=4/3, so nearby observed values admit two distinct inverse branches. These results establish protocol ratios as inverse-geometric probes with sharply defined domains of identifiability and intrinsic limits of spectral reconstruction. The framework clarifies the difference between unique inference, exact information collapse, and multi-valued inversion, and provides a basis for future extensions to multi-ratio, mixed-spectrum, and frequency-resolved inverse design. V2: This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Interpreting the reduced observation map through Fisher information, we show that the inverse problem separates into three sectors: regular (locally invertible with Cramér–Rao bounds), degenerate (exact collapse with vanishing Fisher information), and branched (non-injective with multiple solutions). Using power-law and mixed spectral families, we derive explicit inversion formulas, a white-noise no-go theorem, and fold-induced branching. The framework establishes protocol ratios as inverse-geometric probes with sharply defined limits of spectral reconstruction.