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Inconsistent exponential decay model and damping-rate definition across the manuscript (factor-of-two mismatch). Sec. 1 and Sec. 2.3 describe energy decay as $E(t) = E_0 \exp(-2 \gamma t)$ (and Eq. (3) appears to use $\exp(-2 \gamma_{\text{obs}} t)$), while Sec. 3.2 states fitting $E_{\text{corrected}}(t) = E(0) \exp(-\gamma_{\text{obs}} t)$. This ambiguity obscures whether $\gamma$ refers to amplitude decay or energy decay, and whether $\gamma_{\text{obs}}$ is directly comparable to $\gamma_{\text{theory}} = b/(2m)$ (Sec. 1; Sec. 2.3; Sec. 3.2).
Recommendation: Adopt one consistent convention and apply it uniformly in Sec. 1, Sec. 2.3, Eq. (3), and Sec. 3.2. For example: start from the equation of motion $m\, \ddot{x} + b\, \dot{x} + k x = 0$ (underdamped case), note amplitude $\propto \exp(-\gamma t)$ with $\gamma = b/(2m)$, and therefore energy $\propto \exp(-2 \gamma t)$. If instead you fit $E(t) = E_0 \exp(-\lambda t)$, explicitly define $\lambda = 2\gamma$ and compare $\lambda_{\text{obs}}$ to $\lambda_{\text{theory}} = b/m$. Update all reported values/units accordingly so $\gamma_{\text{obs}}$ and $\gamma_{\text{theory}}$ are unambiguously comparable.
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Noise/measurement model is under-specified, and the derivation/validity conditions for $\Delta E_{\text{noise}}$ are not stated explicitly. The correction implicitly assumes additive, zero-mean, stationary noise and that cross-terms vanish in expectation (e.g., $\mathbb{E}[x \eta_x]=0$), but these assumptions are not formalized; correlations (including between $x$ and $v$ channels) can change the bias structure (Sec. 2.2; Sec. 3.1).
Recommendation: In Sec. 2.2, explicitly write the measurement model (e.g., $x_m = x + \eta_x$, $v_m = v + \eta_v$) and state assumptions needed for the bias term: $\mathbb{E}[\eta] = 0$, $\operatorname{Var}[\eta]$ constant in time (stationarity over the relevant interval), and zero correlation with the signal (or at least $\mathbb{E}[x \eta_x]=0$, $\mathbb{E}[v \eta_v]=0$). Show the short expectation calculation leading to $\mathbb{E}[E_m]=\mathbb{E}[E_{\text{true}}]+\Delta E_{\text{noise}}$ in the late-time regime. State whether you assume $\operatorname{Cov}(\eta_x, \eta_v) = 0$ and comment on what changes if it is nonzero.
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Velocity treatment is potentially unrealistic for experiments and may invalidate the assumed constant-bias form if $v$ is obtained by differentiating noisy displacement. The manuscript appears to assume $v(t)$ is directly available and independently noisy, but in many settings $v$ is computed numerically from $x$, which amplifies high-frequency noise and introduces temporal correlation and $x$–$v$ noise coupling (Sec. 2.1–2.2).
Recommendation: Clarify in Sec. 2.1–2.2 how $v$ is generated/measured in simulation: (i) independently simulated noisy velocity channel, or (ii) numerical differentiation of noisy $x$. If (i), add a limitation statement about applicability to experiments where $v$ is not directly measured. If (ii) (or to broaden applicability), include an additional validation case where $v$ is estimated from noisy $x$ (finite differences / Savitzky–Golay / filtering), report how $\sigma_v^2$ is estimated consistently, and discuss how the bias formula and detrending should be adapted under correlated/colored differentiation noise.
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Key assumptions behind selecting the late-time window ($t > 15$ s) and using local linear detrending are not justified or stress-tested. The method assumes negligible physical motion and stationary noise in that window, but there is no sensitivity analysis to cutoff choice, residual motion, drift, or colored noise; this limits confidence for real data (Sec. 2.2; Sec. 3.1; Sec. 4).
Recommendation: In Sec. 2.2 and Sec. 3.1, justify $t > 15$ s relative to the slowest decay in the simulated set (e.g., report max time constant / remaining theoretical energy fraction at 15 s for all oscillators). Add a sensitivity study varying the late-time start and/or window length (e.g., $t > 10, 12, 15, 18$ s) and report resulting variation in $\Delta E_{\text{noise}}$ and $\gamma_{\text{obs}}$ residual statistics (mean/SD, MAE). Add at least one stress-test where the late-time segment contains (a) low-amplitude residual oscillation and/or (b) slowly drifting variance or colored noise, and summarize failure modes and practical checks (e.g., stationarity diagnostics) in Sec. 4.
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Clipping negative corrected energies to zero introduces censoring/nonlinearity that can bias exponential fits, especially when many late-time points are clipped (the manuscript also hints this may explain the residual mean). The fit protocol (time range, weighting, inclusion/exclusion of clipped points) is not described in enough detail to assess bias (Sec. 2.2; Sec. 2.3; Sec. 3.2).
Recommendation: Make the fitting protocol explicit in Sec. 2.3: fitting interval, objective (linear-in-log vs nonlinear least squares), weights, bounds/initialization, and how points with $E_{\text{corrected}} \leq 0$ (or clipped zeros) are handled. Quantify clipping frequency across oscillators/SNR (Sec. 3.2). Add a small ablation: compare $\gamma_{\text{obs}}$ when (i) clipping+fit all points, (ii) no clipping but fit only points with $E_{\text{corrected}}>0$, and/or (iii) fit only up to a cutoff where SNR remains above a threshold. Use this to give actionable guidance for practitioners on avoiding fit bias at low SNR.
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Validation scope is narrow and does not include natural baselines/competitors; this weakens the bigger-picture claim of broad experimental usefulness. The current evaluation uses only 20 idealized simulated linear oscillators, and does not compare to simple alternatives such as subtracting the late-time mean energy floor, fitting only early-time points, or fitting an exponential-with-offset model $E(t) = E_0 \exp(-2\gamma t)+C$ (Sec. 3.1–3.2; Sec. 4).
Recommendation: In Sec. 3.2, add baseline comparisons alongside the proposed method: (a) subtract $\operatorname{mean}(E_{\text{total}})$ over the same late-time window; (b) fit only early-time data above an SNR threshold; (c) directly fit $E_{\text{total}}(t)$ to $E_0 \exp(-2\gamma t)+C$ (or $\exp(-\lambda t)+C$) and compare bias/variance and parameter identifiability. Report the same residual metrics as Table 1 for each baseline. If feasible, broaden validation with (i) more diverse parameter ranges and SNRs, and (ii) at least one realism stress-test (e.g., mild nonlinearity, two-mode response, time-varying damping, colored/nonstationary noise). If expansion is not feasible, temper claims in Sec. 4 and clearly delimit applicability to the tested conditions.
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Reproducibility is limited by missing implementation details: simulator setup (integrator, time step/sampling rate, initial conditions), parameter distributions/ranges ($m,k,b$), exact noise statistics (distribution, $\sigma_x$, $\sigma_v$, whiteness/coloredness, whether shared across oscillators), detrending algorithm details, and optimizer settings for the fit (Sec. 2.1–2.3).
Recommendation: Augment Sec. 2.1–2.3 (or add an appendix) with a concise but complete specification: sampling rate/time step; numerical solver; ranges/distributions for $m,k,b$ and resulting $\gamma_{\text{theory}}$; initial conditions; whether noise is added to $x/v$ before computing energy; noise distribution and correlation properties; detrending procedure (global vs sliding window, window length/overlap, edge handling); and fitting algorithm (model form, optimizer, initialization, bounds, convergence criteria). Consider adding a parameter table and stating whether code/synthetic data will be released.