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Reproducibility gap: key dynamical model and simulation details are missing or ambiguous (Sec. 2.1), making headline quantitative claims (e.g., mean $T_H\approx 0.76\,\text{s}$, range $[0.60,0.92]\,\text{s}$, regression/correlation in Sec. 3.3) non-auditable. The manuscript does not clearly specify the governing equation (e.g., $m\ddot x+b\dot x+kx=0$ vs forcing), the exact parameterization used to generate the 20 oscillators (ranges/distributions for $m,b,k$, any constraints such as fixed $\omega_n$), initial conditions $x(0),v(0)$, solver/integration method, time step/sampling rate, and number of discrete time points used in sensitivity calculations.
Recommendation: In Sec. 2.1, explicitly state: (a) the ODE (and whether any forcing/noise is present); (b) the initial conditions and how they are chosen; (c) whether solutions are analytic or numerical and, if numerical, the solver, $\Delta t$, and simulation duration; (d) the discrete sampling grid for $E(t)$ and sensitivities; and (e) how the 20 oscillator parameter sets are generated (ranges/distributions, any coupling/constraints, and resulting ranges of $\omega_n$ and $\zeta$). Provide a parameter table (main text or supplement) so Sec. 3.1–3.3 statistics can be independently reproduced.
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Central methodological ambiguity: $\partial E/\partial b$ (and in general “Jacobian of $E$”) is not defined/computed in a mathematically checkable way (Sec. 2.2–2.3). As written, Eq. (1) has no explicit dependence on $b$, so $\partial E/\partial b\neq 0$ only through the implicit dependence $x(t;b),v(t;b)$. The paper does not state whether derivatives are computed via sensitivity ODEs, closed-form differentiation, automatic differentiation, or finite differences; nor does it discuss numerical error, step-size selection, smoothing, or stability. This affects $S_b(t)$, $R(t)$, and therefore $T_H$ (Sec. 3.1–3.2).
Recommendation: In Sec. 2.2, define $E(t;m,b,k)=\tfrac12 m v(t;m,b,k)^2+\tfrac12 k x(t;m,b,k)^2$ and explicitly state that derivatives are total derivatives through the state: e.g., $\tfrac{\partial E}{\partial b}=m v\,\tfrac{\partial v}{\partial b}+k x\,\tfrac{\partial x}{\partial b}$. Then specify how $\tfrac{\partial x}{\partial b},\tfrac{\partial v}{\partial b}$ (and similarly for $m$) are obtained: (i) forward sensitivity equations (include the augmented ODEs in-text or appendix), or (ii) finite differences (scheme, perturbation size, convergence checks), or (iii) AD (tooling + validation). Also clarify whether $\partial E/\partial m$ includes both the explicit term $\tfrac12 v^2$ and the implicit dependence of $x,v$ on $m$.
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Dimension/unit inconsistency: the sensitivity norm $S(t)=\|J(t)\|_2$ and the ratio $R(t)=S_b(t)/S_m(t)$ compare/aggregate derivatives with different physical units (Sec. 2.2–2.3; Eq. (3)/(6), Eq. (4)/(5); Figures 1–2). As a result, both the magnitude of $S(t)$ and the threshold $R(t)=1$ are parameterization- and unit-dependent, undermining the quantitative meaning of “dominance” and comparability across oscillators.
Recommendation: Replace raw derivatives with dimensionless or uncertainty-weighted quantities. Options include: (a) relative sensitivities $\tilde S_m=\left|\tfrac{m}{E}\tfrac{\partial E}{\partial m}\right|$, $\tilde S_b=\left|\tfrac{b}{E}\tfrac{\partial E}{\partial b}\right|$; (b) log-sensitivities $\left|\partial \ln E/\partial \ln \theta\right|$; or (c) error-propagation/FIM-style weighting with parameter prior uncertainties $\sigma_m,\sigma_b$: $\mathrm{Var}[E(t)]\approx J(t)\Sigma_\theta J(t)^\top$. Redefine $T_H$ using equality of *comparable* (dimensionless/weighted) contributions and update Figures 1–2 axes/units accordingly.
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“Identifiability/information” claims are not yet supported by an explicit measurement model, noise model, or estimation procedure (Sec. 1, Sec. 2.2–2.4, Sec. 3.1, Sec. 4). Local sensitivity magnitude is related to parameter influence but is not equivalent to structural/practical identifiability, nor to “information” in the Fisher sense. Moreover, the discussion does not address that later times may have higher *relative* damping sensitivity but lower SNR due to decay, which can reduce practical information under noise (big-picture concern raised by the current framing in Sec. 4).
Recommendation: Either (A) reframe the paper consistently as *time-varying local sensitivity of the energy output* and soften language in Abstract/Sec. 1/Sec. 4 (avoid “fundamental limit,” “operational limit,” etc.), or (B) add a minimal inference layer: assume a noise model for the observable ($E$ or $x,v$), compute a time-window Fisher information (or cumulative information) for $m$ and $b$, and/or run a parameter-estimation experiment (fit $m,b$ on early vs late windows) showing estimation variance/bias changes around $T_H$. If retaining the term “Information Horizon,” justify it via an information measure (even in a simplified setting).
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Observability/measurement realism: it is unclear what is assumed measured and how $E(t)$ is formed when $m$ is unknown (Sec. 1–2). In many setups, $E(t)$ is not directly measured; it is derived from $x(t),v(t)$ and parameters (including $m$ and $k$). If $m$ is the unknown, constructing $\tfrac12 m v^2$ already requires $m$, creating a circularity unless $E$ is treated as a model-predicted quantity rather than a direct measurement.
Recommendation: Clarify the observation model in Sec. 2.1–2.2: is the ‘output’ assumed to be (i) directly measured energy (with what sensor/proxy), (ii) energy computed from measured $x,v$ using nominal parameters, or (iii) purely model-based analysis of $E$ as a derived signal? If the goal is system identification, consider also presenting results for more standard observables ($x(t)$, $v(t)$, or envelope/decay rate) or discuss explicitly why energy is the chosen output and what is gained/lost compared to displacement-based identification.
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The operational definition of $T_H$ and the definitions of $S_m, S_b$ are inconsistent/underspecified (Sec. 2.2 vs Sec. 3.1), affecting the existence and timing of the horizon. The manuscript alternates between absolute-value and signed derivatives for $S_m, S_b$; does not specify how $T_H$ is extracted on a discrete time grid; and does not describe handling for multiple crossings, non-crossing cases, or numerical oscillations in $R(t)$.
Recommendation: Unify the definition of $S_m,S_b$ across Sec. 2.2–2.3 and Sec. 3.1 (preferably magnitudes if the goal is “dominance”). In Sec. 2.3, provide an explicit algorithm for $T_H$: e.g., “the smallest sampled time where $R(t)\ge 1$, with linear interpolation between neighboring samples,” plus rules for multi-crossings (first sustained crossing; smoothing/hysteresis), and for non-crossing trajectories (report as missing and discuss). Report variability for $T_H$ (SD/CI) in Sec. 3.1 and mention any atypical cases.
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Scope limitations are not sufficiently reflected in conclusions: the study uses 20 noise-free simulations and focuses on only $m$ and $b$ while stiffness $k$ is present in both dynamics and energy (Sec. 2.1–2.2, Sec. 3, Sec. 4). The exclusion of $k$ is not justified, and robustness to parameter ranges, initial conditions, or measurement noise is not examined, yet conclusions are phrased broadly.
Recommendation: In Sec. 2.2 and Sec. 4, explicitly justify excluding $k$ (e.g., assumed known/calibrated; focus on inertial vs dissipative). Ideally add $\partial E/\partial k$ and discuss whether multiple crossovers/horizons appear when $k$ is included. Add at least one robustness check: more oscillators ($n\gg 20$), alternative parameter distributions (including very low and near-critical damping), alternative initial conditions/amplitudes, and a simple measurement-noise experiment to test whether $T_H$ remains meaningful when amplitude decays.
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Positioning within existing literature is thin, making novelty and interpretation of “Information Horizon” unclear (Sec. 1–4). There is no dedicated Related Work section tying the approach to established sensitivity analysis for ODEs, structural/practical identifiability, observability windows, or Fisher-information-based time-window design.
Recommendation: Add a short Related Work subsection (end of Sec. 1 or as Sec. 2.5) covering: (i) local/global sensitivity analysis in dynamical systems, (ii) identifiability/observability for oscillatory second-order systems, and (iii) time-window selection/optimal experimental design via Fisher information. In Sec. 1 and Sec. 4, position $T_H$ explicitly as a heuristic crossover metric (unless upgraded to an information-based measure per above).