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.
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.
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) ∈ R^m with Jacobian J ∈ R^{m×n} and Fisher matrix F = J^T 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 m_min = 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 ∂{α1}R = ∂_{α2}R, 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 Z2\mathbb{Z}_2Z2-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.
Genuine Collapse, Apparent Flattening, and Observable Horizons in Inverse Spectral Reconstruction This paper studies two distinct mechanisms by which inverse spectral reconstruction can fail. In one case, the forward observation map loses rank and information is destroyed at the geometric level; this is termed genuine collapse. In the other, the map remains full-rank but its weakest reconstruction direction is compressed below practical observational reach; this is termed apparent flattening. The paper introduces the corresponding notion of an observable horizon, defined by the competition between geometric compression and finite measurement noise. The analysis is carried out for the spectral family S(ω) = Aω^α + B, using scale-independent ratio observables motivated by dynamical-decoupling spectroscopy. Writing ρ = B/A, the paper derives the Jacobian of the ratio map with respect to (α, ρ), relates its weakest singular direction to the Fisher information matrix, and shows that practical identifiability is governed by the smallest singular value of the reduced observation map. For the minimal two-frequency ratio model, the paper obtains an exact closed-form formula for the singular value s(J) and proves a classification theorem for its large-ρ decay. Away from a distinguished fragility locus α = α*, the singular value decays generically as O(ρ^−1). On the fragility locus, the leading response cancels and the decay accelerates to O(ρ^−2), causing observability to deteriorate more rapidly. The paper also proves that, in this minimal model, no genuine-collapse locus exists at any finite parameter value: all finite-parameter information loss is therefore of the apparent-flattening type. This leads to an observable-horizon map in the (α, ρ) plane. Lowering observational noise expands the visible region and pushes the horizon outward, but it does not move the fragility locus or the asymptotic geometric boundary. The resulting picture distinguishes a movable observational wall from an immovable geometric wall and provides a concrete criterion for probe-frequency selection: choose frequencies so that the fragility value α* lies outside the target spectral-exponent range. Overall, the paper gives a geometric and statistical characterization of information loss in floor-dominated inverse spectral reconstruction. It clarifies the difference between exact geometric impossibility and noise-limited practical impossibility, and formulates observable horizons as the natural interface between them. V2:This version refines the interpretation and presentation of the single-ratio model without altering the main analytical results. Clarified the Fisher structure: the single-ratio model is rank-1, and observability is formulated via the scalar response norm s(J)s(J)s(J) rather than a two-parameter Fisher minimum eigenvalue. Introduced the scalar observable horizon explicitly as a local detectability (sensitivity) threshold, removing ambiguity with estimation-based notions. Strengthened the framing as a scalar sensitivity prototype for multi-ratio Fisher geometry (treated in the companion paper). Corrected minor issues, including numerical consistency (αmono\alpha_{\rm mono}αmono) and full alignment between figures, code, and text. Improved wording to avoid overstatements and clarify scope (e.g., scalar-response interpretation). The analytical results, including the fragility locus α∗\alpha_*α∗ and the O(ρ−1)O(\rho^{-1})O(ρ−1) vs. O(ρ−2)O(\rho^{-2})O(ρ−2) classification, remain unchanged.
This work formulates dynamical-decoupling (DD) spectroscopy as a reduced observation problem. Rather than treating measured coherence responses as direct probes of the full environmental noise spectrum, the paper introduces a protocol-dependent map from parametrized spectral families to a small set of observables and analyzes its local Jacobian structure. The main result is a geometric framework in which spectral directions are classified into observable, weakly observable, and compressed sectors according to the singular-value structure and effective rank of the reduced map. Within this framework, plateau behavior and fixed protocol ratios arise when the reduced image becomes effectively low-dimensional, while abrupt transitions are interpreted as switching between response sectors with different local observability structure. A running power-law example is used to illustrate how amplitude and slope deformations are transmitted differently into protocol observables, and how protocol ratios can become locally insensitive without implying full spectral universality. The analysis emphasizes that DD protocols access environmental structure only through compressed, protocol-shaped images of the underlying spectrum. This provides a unifying interpretation of several familiar DD phenomena, including plateau regions, fixed-ratio structures, and sharp protocol-level transitions, in terms of reduced observation geometry rather than direct spectral reconstruction. v2 :This work formulates dynamical-decoupling spectroscopy as a reduced observation map from parameterized noise spectra to protocol-level observables. Using the Jacobian of this map, we classify spectral directions into observable, weakly observable, and compressed sectors via an effective-rank criterion. Plateau behavior and fixed protocol ratios are shown to arise from geometric compression of the reduced image, combined with a proportional-locking condition, rather than from full spectral universality. Abrupt transitions are interpreted as switching between response sectors with different local observability structures. The framework provides a unified geometric interpretation of sensitivity, compression, and invariance in DD-based noise spectroscopy.
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)=logωk+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.
This paper extends the recoverable‑null and observable‑horizon framework of reduced dynamical‑decoupling (DD) spectroscopy to colored noise floors. Earlier work (Papers Q6–Q7) established that a flat floor B produces a recoverable near‑null: the reduced Jacobian remains full‑rank at every finite floor ratio ρ=B/A, and the weakest reconstruction direction becomes unobservable only when its singular value falls below the measurement noise, defining an observable horizon. Here we replace the flat floor by a shape‑dependent floor ρ Sfloor(ω;ν) and compute the horizon map ρ∗(α;ν) for three canonical spectral families: a flat floor, a 1/f-like floor, and a Lorentzian floor parameterised by a knee frequency ωc. The main findings are: Shape‑dependent horizon deformation: each floor family produces a qualitatively distinct deformation of the horizon map. A flat floor yields a smooth baseline; a 1/f-like floor introduces a sharp but finite minimum at α=1; a Lorentzian floor shifts the horizon‑gain region across the α-axis as ωc varies. 1/f resonance collapse: when the floor exponent matches the signal exponent (α=1), the ln(ωk)-amplified component of the α-column of the leading Jacobian J(1) is suppressed by the prefactor 1−21−α, which vanishes at α=1. This produces a strong recoverable‑null degeneracy and a sharp horizon minimum. Lorentzian knee as a design parameter: the knee frequency ωc determines which α sector benefits from the floor. An intermediate knee near the geometric mean ωminωmax achieves the broadest horizon gain (up to ∼3.5× near α≈0.9). Together with Papers Q6 and Q7, these results complete the three‑part analysis of recoverable‑null geometry under coarse‑grained floors in the DD series. Floor spectral shape emerges as a geometrically active design variable, providing a full toolkit for horizon optimisation in reduced DD spectroscopy under realistic colored noise conditions. v2: Clarified the distinction between recoverable near-nulls and exact structural nulls in reduced DD spectroscopy under colored noise floors. Added the exact 1/f resonance result at α=1, where the observable map loses the logρ direction identically, together with revised horizon geometry, Lorentzian knee-frequency analysis, normalization conventions, convergence checks, and improved figure annotations for structural rank loss.