What question did this study set out to answer?

The research aims to establish a robust framework for detecting rank-2 behavior in Laplace signals using spectral and structural analysis.

April 4, 2026Open Access

COS 45 A Detectability and Abstention Framework for Multi-Exponential Signals

Key Points

The research aims to establish a robust framework for detecting rank-2 behavior in Laplace signals using spectral and structural analysis.
Developed a falsifiable detection framework called COS45.
Defined three detection regimes: undetectable, borderline, and detectable.
Introduced a five-verdict system including a first-class outcome for intrinsic detectability limits.
Combined spectral deviation and Hankel structural metrics for signal assessment.
Identified a calibrated detection measure based on the signal-to-noise ratio.
Classified signals into three distinct categories based on their detectability characteristics.
Demonstrated that no global coercivity or equivalence is assumed within the framework.

Abstract

We present COS45, a falsifiable detection framework combining spectral dynamics and Hankel structure to assess rank-2 behavior in Laplace signals. Detection is governed by a single calibrated quantity: SNRₛpec = σ₃ (HW) / (σ√h) Three regimes are defined: <1: UNDETECTABLE (NON-CONCLUSIVE), 1–3: BORDERLINE, ≥3: DETECTABLE. The framework integrates two complementary channels: (i) spectral deviation Δ (t), (ii) structural deviation σ₃ (H). A five-verdict system is introduced, where NON-CONCLUSIVE is a first-class outcome reflecting intrinsic detectability limits. No global coercivity is claimed. No global equivalence between σ₃, Δ, and dist (F, V₂) is claimed. No pole reconstruction from (r, v) is claimed. COS45 does not perform inversion. It measures when inversion is justified. This framework does not impose a result. It measures when a result is justified.

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