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.
Hiroyuki Shioiri (Tue,) studied this question.