Exact versus Recoverable Null in Reduced DD Spectroscopy under Coarse-Grained Floorsa

This work analyzes how a flat noise floor reshapes the reduced geometry of multi‑ratio dynamical‑decoupling (DD) spectroscopy. For spectra of the form S (ω) =Aω−α+B, p=B/A, we study the conditioning of the reduced Jacobian with respect to the natural reduced coordinates θ= (α, log⁡p). In the floor‑dominated regime, the smallest singular value obeys a universal asymptotic law σmin⁡ (J) ∼C (α, ωk) p−1, where the prefactor C depends strongly on spectral slope and protocol geometry. This produces a protocol‑dependent observable horizon and identifies frequency configurations that remain resilient deepest into the floor‑dominated regime. The main conceptual result is a geometric distinction between genuine white‑noise collapse and coarse‑grained flattening. Both exhibit the same p−1 decay, but they differ in rank structure: exact white noise (A=0) produces an exact null through true rank loss, coarse‑grained floors (A>0) retain full rank for all finite p, producing only a recoverable near‑null. Thus genuine white noise destroys reduced spectral information, while coarse‑grained floors merely compress it beyond an observable horizon. This establishes an exact‑versus‑recoverable‑null dichotomy in reduced DD spectroscopy and clarifies how apparent flattening can be distinguished from genuine collapse at the level of reduced geometry. V2: This version clarifies the distinction between recoverable near-null and exact null regimes by explicitly separating interior coarse-grained behavior (A>0A>0A>0) from the boundary model (A=0A=0A=0). The analysis is reformulated in natural coordinates (α, log⁡ρ) (, ) (α, logρ), ensuring balanced Jacobian scaling and a consistent interpretation of rank structure. Figure definitions and captions have been corrected for full consistency with the theoretical framework. New numerical validations confirm: (i) the universal scaling σmin⁡ (J) ∼C ρ−1_ (J) C\, ^-1σmin (J) ∼Cρ−1, (ii) collapse of the prefactor near frequency degeneracies, and (iii) observable-horizon scaling ρhor∼C/σobs₇₎ₑ C/₎₁ₒρhor∼C/σobs. These updates strengthen the interpretation of identifiability limits as a rank-structure phenomenon rather than an asymptotic scaling effect.