Multi-Ratio Identifiability and Local Completeness in Dynamical-Decoupling Spectroscopy This work studies how inverse identifiability changes when several independent protocol ratios are measured simultaneously in dynamical-decoupling spectroscopy. Building on earlier reduced-observation and single-ratio inverse frameworks, the paper formulates a multi-ratio map R (θ) ∈ Rᵐ with Jacobian J ∈ R^m×n and Fisher matrix F = JT Σ^-1 J on an n-dimensional spectral family. The central question is when observable enrichment restores local identifiability, and what obstructions remain even after Fisher-rank recovery. The paper proves a generic local-completeness theorem: for an n-dimensional spectral family, fewer than n independent observables are generically insufficient for local identifiability, whereas n observables are generically sufficient whenever the Jacobian has full rank. This establishes mₘin = n as the sharp threshold for generic local completeness and separates rank recovery from stronger notions such as global injectivity or practical conditioning. These general results are then demonstrated in an explicit three-parameter mixed spectral family S (ω; α1, α2, w) = wAω^α1 + (1 − w) Aω^α2, using an octave-separated sequence of protocol ratios. In this family, one ratio generically gives rank one, two ratios generically give rank two, and three ratios generically give rank three, confirming the sharpness of the completeness bound in a concrete model. The paper also identifies a symmetry-protected obstruction. Along a distinguished symmetric locus, all exchange-symmetric ratios satisfy ∂w R = 0 and ∂α1R = ∂⏐₂R, forcing the Jacobian to remain rank one no matter how many symmetric ratios are added. Thus multi-ratio enrichment does not automatically remove local branching: some fold obstructions survive as a consequence of symmetry rather than insufficient observable number. To resolve this obstruction, the paper introduces asymmetric ratios that probe the two spectral components at different effective sampling scales. Replacing at least one symmetric ratio by an asymmetric one lifts the symmetry protection and restores full rank on the symmetric locus. The resulting picture yields a three-layer inverse structure: generic local completeness through rank recovery, symmetry-protected local branching on special loci, and residual global non-injectivity beyond local analysis. Overall, the work extends the inverse geometry of protocol ratios from the scalar setting to vector-valued observations and provides a precise observable hierarchy for spectral reconstruction in dynamical-decoupling spectroscopy. It clarifies that adding more observables can generically restore local identifiability, but that symmetry-protected obstructions require symmetry-breaking observables rather than mere multiplicity. Description (v2) We extend the inverse geometry of dynamical-decoupling protocol ratios from the scalar to a multi-ratio setting. For an nnn-parameter spectral family, we prove that fewer than nnn observables are generically insufficient for local identifiability, while nnn independent ratios achieve generic local completeness when the Jacobian has full rank. In a three-parameter mixed spectral model, we explicitly demonstrate rank recovery (1→2→3) with increasing ratios. We further identify a symmetry-protected obstruction: on a Z2Z₂Z2-invariant locus, exchange-symmetric ratios enforce rank collapse and preserve fold branching. This obstruction is lifted only by asymmetric ratios that break the symmetry. The results establish a three-layer inverse structure: generic local completeness, symmetry-protected differential obstruction, and residual global non-injectivity.
Hiroyuki Shioiri (Tue,) studied this question.