Spectral-Shape Dependence of Recoverable-Null Geometry in Reduced Dynamical-Decoupling Spectroscopy: Horizon Maps and Design Rules for Colored Noise Floors

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 ωmin⁡ωmax⁡ 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.