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 ₌₎₍₎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∗_*α∗ and the O (ρ−1) O (^-1) O (ρ−1) vs. O (ρ−2) O (^-2) O (ρ−2) classification, remain unchanged.
Hiroyuki Shioiri (Tue,) studied this question.