{"id":"877b8d85-97b7-450e-9507-b8e64c2c01d2","arxiv_id":"2607.07594","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Under Hankel-DMDc, lithium-ion SOC emerges as the marginally stable Koopman mode nearest the unit circle and yields a usable SOC-sensitive observable after min-max scaling.","lead":"A data-driven Koopman analysis of battery voltage and current shows that state of charge appears as the slowest, nearly unit-circle dynamical mode. This offers battery management systems a parameter-light, interpretable way to track remaining charge from ordinary measurements.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.5","headline":"SOC mode identification and quantitative accuracy rest on an unvalidated near-unit eigenvalue and min-max scaling of one HPPC trajectory.","rationale":"The Reader correctly isolates the weakest link: reliance on a single globally linear operator in a high-dimensional Hankel space whose near-unit eigenvalue is labeled the SOC mode and then affinely scaled. That assumption is load-bearing for both the physical interpretation and the reported accuracy numbers. My concern sharpens the same point with the concrete numerical details already present in the paper (λ = 1.000001, d = 2000, min-max step, unrealistically small RMSE) and proposes a falsifying experiment that would settle whether the spectral identification is robust. No stronger internal inconsistency appears; the voltage-reconstruction results and the theoretical integrator argument remain intact. Consequently the verdict stays CONDITIONAL and confidence remains moderate pending independent verification.","tokens_in":10743,"tokens_out":617,"duration_ms":6633,"concrete_test":"Re-identify the DMDc operator on a second, independent HPPC (or drive-cycle) trajectory recorded at a different temperature or after mild aging; extract the eigenvalue nearest unity and its modal coordinate, apply the original min-max map (or re-fit only the affine map), and recompute Table 2. If the nearest eigenvalue moves by more than ~10^{-4} from 1, or if SOC RMSE rises above ~0.5 %, the spectral labeling and quantitative claim do not generalize.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that SOC 'naturally emerges' as the slowest marginally stable Koopman mode (eigenvalue nearest the unit circle) of a single globally linear DMDc operator learned in a 2000-dimensional Hankel space, and that its modal coordinate, after min-max normalization over the operating range, yields quantitative SOC with RMSE 0.0043 % (Section 2.5, Eq. 9; Section 3.4, λ_SOC = 1.000001; Table 2). The theoretical link to the continuous-time integrator pole (Eqs. 6–8) is sound, but the numerical identification is not: the reported eigenvalue is slightly outside the unit circle (unstable), no uniqueness or isolation argument is given among the 2000 modes, and quantitative SOC still requires a data-dependent affine map rather than a parameter-free observable. Moreover the SOC RMSE figures are two orders of magnitude smaller than typical laboratory voltage/current noise and Coulomb-counting drift, raising the possibility that the modal coordinate is essentially a rescaled version of the same charge integral used as the reference. Without an independent hold-out trajectory, aging/temperature variation, or an ablation that freezes the operator and tests only the mode-selection step, the claim that the spectrum itself supplies a robust SOC observable remains under-supported.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proposes a data-driven SOC estimation framework based on Dynamic Mode Decomposition with control (DMDc) applied to Hankel time-delay embeddings of terminal voltage, with applied current as the control input, using HPPC measurements. After identifying a linear lifted operator, eigen-decomposition is used to argue that SOC appears as the slowest, marginally stable Koopman mode (eigenvalue nearest the unit circle), consistent with the continuous-time integrator pole of charge conservation. The corresponding modal coordinate, after min–max normalization over the operating SOC range, is reported as an SOC estimate (RMSE 0.0043%), while the same model reconstructs terminal voltage with RMSE 0.0131 V, outperforming a 2RC-ECM EKF and Coulomb counting on the same HPPC cycle.","tokens_in":11110,"tokens_out":1386,"duration_ms":28665,"significance":"If the spectral identification of SOC as a robust, physically grounded Koopman mode holds under broader conditions, the work would offer a useful middle path between ECM/Kalman estimators (which require repeated parameter identification) and black-box learning methods (which often lack interpretability). The theoretical link from charge conservation (pole at s=0) to a discrete eigenvalue at unity is standard and correctly stated, and framing SOC as a marginally stable observable is a clear, falsifiable spectral claim. Voltage reconstruction performance is credible and supports the utility of Hankel–DMDc for input–output battery modeling. The contribution is primarily methodological and interpretive rather than a fully validated BMS-ready estimator; its lasting value depends on whether mode selection and quantitative SOC mapping remain reliable beyond a single calibrated HPPC trajectory.","major_comments":[{"comment":"Quantitative SOC is obtained only after min–max normalization of the modal coordinate over the experimental operating SOC range (Section 2.5, Eq. 9; Section 3.5). This is an affine calibration that requires knowledge of the SOC bounds (or equivalent end-point labels). The abstract and Table 2 present RMSE 0.0043% as if the spectrum alone supplies a quantitative SOC observable; the manuscript should clearly separate (i) spectral mode identification from (ii) the subsequent data-dependent scaling, and report accuracy of the raw modal coordinate (or a fixed calibration) rather than only the post-normalized signal. The conclusion already flags this limitation; it should be reflected in the claims and tables.","section":null},{"comment":"Table 2 reports SOC RMSE of 0.0043% (DMDc), 0.0047% (Coulomb counting), and 0.0078% (EKF). These values are orders of magnitude below typical laboratory SOC estimation errors and below what sensor noise and Coulombic-efficiency uncertainty usually allow on multi-hour HPPC profiles. The reference is described as “derived from experimental discharge capacity data,” which is often itself an ampere-hour integral of the same current. The paper must specify exactly how the reference SOC is constructed, whether it is independent of the Coulomb-counting baseline, and why all three methods achieve sub-0.01% RMSE. Without that, the performance comparison does not support the claim of a superior SOC-sensitive observable.","section":null},{"comment":"Section 3.4 identifies mode 457 with λ_SOC = 1.000001 among 2000 modes as the unique SOC mode because it is “closest to the unit circle.” The reported eigenvalue lies slightly outside the unit circle (mildly unstable), with no discussion of numerical tolerance, isolation gap to the next-nearest eigenvalues, or sensitivity to Hankel dimension, SVD truncation rank, or train/test split. A load-bearing claim that SOC “naturally emerges” requires evidence that this selection is unique and stable (e.g., eigenvalue gap plot, ablation over embedding dimension/rank, and at least one independent trajectory or operating condition).","section":null},{"comment":"All identification and evaluation use a single HPPC trajectory (≈75% train / held-out remainder of the same test; Sections 3.1–3.3). There is no second cell, temperature, aging state, or drive-cycle hold-out that freezes the learned operator and tests only mode selection and the SOC coordinate. Given the free parameters (embedding dimension d=2000, SVD truncation, min–max bounds), the central claim that the Koopman spectrum itself supplies a robust SOC observable remains under-supported for generalization beyond this dataset.","section":null}],"minor_comments":[{"comment":"Table 1: EKF voltage MAE is listed as 0.0007 V while RMSE is 0.0439 V. Such a large RMSE–MAE gap is possible only with rare extreme outliers; please verify the numbers and, if correct, briefly explain the residual structure (e.g., pulse spikes).","section":null},{"comment":"Figure 3A caption refers to “long-timescale capacity fade behavior” for the SOC mode; the experiment is a single HPPC cycle and does not measure capacity fade. Align the caption with the charge-conservation / OCV interpretation used in the text.","section":null},{"comment":"Notation: continuous-time eigenvalue is written λ_c = 0 and discrete λ_d = 1 (Eqs. 7–8), while the identified value is 1.000001; state consistently whether “closest to unity” or “on the unit circle” is the selection criterion.","section":null},{"comment":"Section 2.6 and Eq. (14): EKF noise covariances are given without tuning rationale or sensitivity; a short note would help readers reproduce the baseline.","section":null},{"comment":"References [2] and [3] share nearly identical generic titles (“Review of battery management systems”); expand bibliographic detail for traceability.","section":null},{"comment":"Abstract and keywords: “0.0043%” should be reconciled with whatever unit convention is adopted after addressing the reference-SOC definition (percentage points vs. absolute SOC fraction).","section":null}],"recommendation":"major_revision","confidential_remarks":"The spectral framing is interesting and publishable in principle, but the reported SOC errors look like a red flag for either a near-tautological reference or post-hoc scaling that should not be advertised as estimation accuracy. I would not accept without a clear, independent reference definition and at least one robustness check on mode selection. Scope is appropriate for eess.SY / battery control venues if those issues are fixed."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news here is that a standard Hankel-DMDc pipeline on HPPC voltage/current cleanly surfaces the SOC integrator as the eigenvalue nearest the unit circle (they report 1.000001), and the associated modal coordinate tracks reference SOC after a simple min-max map. Voltage reconstruction is also better than their 2RC-EKF (0.0131 V vs 0.0439 V RMSE). That spectral identification is the new piece; the tools themselves (DMDc, delay embedding, Koopman) are textbook.\n\nThey do the theory cleanly. The continuous-time pole at s=0 maps to discrete eigenvalue 1 via the sampling relation, so mode selection by proximity to the unit circle has external grounding rather than pure data fitting. The workflow figure and the eigenvalue plot make the argument easy to follow. Citations to Schmid, Proctor, Brunton, Mezic, Takens, and the battery literature look appropriate; no obvious gaps or padding.\n\nSoft spots are real but not fatal. Quantitative SOC still needs a data-dependent affine map over the operating range, so it is not a parameter-free observable. The reported SOC RMSE of 0.0043% (and Coulomb 0.0047%) is two orders of magnitude tighter than typical lab noise and drift; that almost certainly reflects the min-max scaling against the same charge integral used as reference rather than true predictive power. Only one HPPC trajectory is shown, the operator is globally linear in a 2000-dimensional lift, and the eigenvalue sits slightly outside the unit circle. No aging, temperature, or multi-cell hold-out. Code and data are not released. These are the usual early-stage limitations, not internal contradictions.\n\nThis is for people who already work on data-driven BMS or Koopman methods for electrochemical systems. It will not reorganize the field, but it is a concrete, interpretable demonstration that deserves a serious referee rather than a desk reject. I would bring it to reading group if we are talking about spectral methods for batteries this month, and I would cite the spectral identification claim once the numbers are stress-tested. Send it out.","headline":"Solid application of Hankel-DMDc that recovers the charge-conservation integrator as a near-unit eigenvalue; useful for BMS people, but the 0.0043% SOC RMSE and single-trajectory min-max scaling keep it provisional.","tokens_in":11711,"tokens_out":539,"would_cite":true,"duration_ms":5965,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"State of charge appears as the battery's slowest, marginally stable Koopman mode, recovered from voltage and current alone without circuit parameters.","keywords":["state of charge estimation","Dynamic Mode Decomposition with control","Koopman operator","Hankel embedding","lithium-ion batteries","battery management system","marginally stable mode"],"falsifier":"Re-run the identical Hankel-DMDc pipeline on a second cell of the same chemistry under a different temperature or after measurable aging; if the eigenvalue nearest unity no longer tracks reference SOC (or another mode becomes closer to the unit circle), the spectral identification claim fails.","tokens_in":11611,"feed_emoji":"🔋","tokens_out":813,"duration_ms":9196,"temperature":0.7,"pith_summary":"Lithium-ion batteries are nonlinear and change with temperature, load, and age, so conventional SOC estimators either keep re-fitting circuit parameters or abandon physical meaning. This paper shows that lifting terminal voltage into a high-dimensional Hankel space and applying Dynamic Mode Decomposition with control recovers a linear operator whose spectrum already contains the battery's intrinsic timescales. The mode whose discrete eigenvalue sits nearest the unit circle is precisely the integrator that charge conservation demands; its modal coordinate, after a simple min-max map, is an SOC estimate. On HPPC data the method reconstructs voltage to 0.0131 V RMSE and SOC to 0.0043 % RMSE, beating both Coulomb counting and a 2RC extended Kalman filter, all without ever writing down resistances or capacitances. A sympathetic reader cares because the same spectral picture that yields the estimate also explains why SOC is the slowest, non-decaying mode.","feed_headline":"Battery SOC is the slowest mode on the unit circle","feed_subtitle":"Hankel-DMDc recovers it from voltage and current alone, no circuit parameters needed","key_machinery":"Hankel-DMDc operator: voltage is stacked into a delay-embedded Hankel matrix so that the nonlinear battery map becomes approximately linear; DMDc then yields the pair (A,B) whose eigen-decomposition of A produces the Koopman eigenvalues and modes that separate SOC from faster polarization dynamics.","core_discovery":"When battery voltage and current from HPPC testing are lifted by Hankel delay embedding and the resulting input-output map is identified by DMDc, eigen-decomposition of the learned state-transition matrix isolates SOC dynamics as the unique slowest, marginally stable mode (eigenvalue closest to 1). The associated modal coordinate, after min-max normalization over the operating range, supplies a quantitatively usable SOC estimate without any explicit equivalent-circuit parameter identification.","pith_inferences":["If mode mixing is the main obstacle, sparse or orthogonalized Koopman dictionaries could further separate SOC from hysteresis and diffusion without enlarging the Hankel dimension.","Adaptive re-identification of the DMDc operator on a sliding window would turn the present offline spectral snapshot into an online BMS estimator that tracks aging.","The same marginally stable mode idea may transfer to other electrochemical storage devices (fuel cells, supercapacitors) whose charge conservation likewise produces an integrator pole."],"forward_implications":["SOC can be read directly from the spectrum of a data-driven operator rather than from an equivalent-circuit model that must be re-identified whenever temperature or age changes.","Voltage reconstruction and SOC estimation become a single eigen-decomposition step once the Hankel-DMDc operator is known.","Long-term drift of the same eigenvalue or its associated mode may later serve as a spectral signature of capacity fade.","The same workflow can be re-applied to other chemistries or packs without rewriting physics-based state equations."],"fun_headline_variants":["SOC is the slowest unit-circle mode from Hankel-DMDc","Battery SOC emerges as the near-unit Koopman eigenvalue","Hankel-DMDc isolates SOC as marginally stable mode","Voltage and current alone yield SOC as slowest DMDc mode","No ECM needed: SOC is the integrator mode on the unit circle"],"cache_read_input_tokens":128,"weakest_assumption_plain":"A single linear operator learned from one HPPC trajectory in a 2000-dimensional delay space remains faithful enough that the eigenvalue nearest unity can be unambiguously labeled the SOC mode and simply rescaled by min-max to give quantitative state of charge.","fun_headline_variants_meta":{"raw":{"variants":["SOC is the slowest unit-circle mode from Hankel-DMDc","Battery SOC emerges as the near-unit Koopman eigenvalue","Hankel-DMDc isolates SOC as marginally stable mode","Voltage and current alone yield SOC as slowest DMDc mode","No ECM needed: SOC is the integrator mode on the unit circle"]},"model":"grok-4.5","effort":"low","cost_usd":0.005754,"raw_usage":{"total_tokens":1537,"prompt_tokens":771,"num_sources_used":0,"completion_tokens":91,"cost_in_usd_ticks":57540000,"prompt_tokens_details":{"text_tokens":771,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":675,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":771,"tokens_out":91,"duration_ms":7487,"temperature":1.0,"reasoning_tokens":675,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T18:33:40.585245+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Re-run the identical Hankel-DMDc pipeline on a second cell of the same chemistry under a different temperature or after measurable aging; if the eigenvalue nearest unity no longer tracks reference SOC (or another mode becomes closer to the unit circle), the spectral identification claim fails.","supporting_citations":[],"review_version":2}