{"id":"1ed4249d-ce48-4d04-af82-6472054a8575","arxiv_id":"2608.11262","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Tracking how spurious pole-zero pairs move across Padé approximation orders lets an iterative algorithm flag inconsistent data points and reconstruct the underlying function while preserving genuine analytic structures.","lead":"Scientists often fit data with rational functions, and the method in this paper turns the small glitches that appear in those fits into signals that reveal which data points are wrong. The result is an automated way to repair noisy datasets while keeping genuine physical features, which is useful for particle physics and other fields that rely on measured or simulated data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Recurrent-doublet classification is the load-bearing premise, but the paper neither proves it nor isolates it from the generic Stieltjes-enforcing correction; without a no-doublet baseline the central diagnostic claim is unsubstantiated.","rationale":"The reader identified the unproven recurrence heuristic as the weakest assumption; my read agrees and sharpens it. The paper's central claim is that Froissart-doublet dynamics provide a general diagnostic for localized data inconsistencies. What must be true is that recurrent doublets specifically mark inconsistent points, not merely that some poles appear. The paper never proves this, and its own text acknowledges the classification is dynamical, not causal. More importantly, the validation protocol cannot distinguish the doublet signal from the effect of replacing flagged points with a smooth Stieltjes fit. For the tested Stieltjes function, such replacement is likely to improve MAE regardless of the selection rule, so the reported 56–99.9% improvements do not establish the mechanism. I do not see circularity: the controlled examples use external ground truth, and the code is promised as supplementary material, which is genuine support. The issue is evidentiary isolation. The concrete ablation test would settle whether the doublet statistic carries the claimed information. If it does, the paper is conditional but promising; if it does not, the central claim is not established. I therefore leave the reader's CONDITIONAL verdict unchanged, with the added condition that the no-doublet baseline be reported.","tokens_in":17773,"tokens_out":8001,"duration_ms":89707,"concrete_test":"Run a three-way ablation on the Sec. 3.1 Stieltjes samples (ρ = 2.5, n = 5, 15, 25; precision 1e-4) over at least 200 random noise-node realizations: (A) the published pole-vote algorithm; (B) the same iterative Stieltjes correction but with the same number of nodes selected uniformly at random; (C) the same correction with nodes selected by largest absolute residual from a low-order Stieltjes fit. Report the mean and spread of relative MAE improvement. If (A) does not clearly beat (B) and (C), or if for a single corrupted point a pole is not reproducibly localized at that node across successive Padé orders, the doublet-recurrence premise is not demonstrated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"All correction decisions rest on the Sec. 2 classification: doublets that recur across Padé orders are 'associated with inconsistencies in the input data,' while unstable doublets are finite-information effects. The paper itself calls this distinction 'not causal but dynamical' and provides no theorem, no quantitative definition of recurrence, and only adjustable thresholds (currentTolerance 0.45 Δx, vote threshold >1, diagOrders 5–12 in Appendix A). The deeper problem is that the validations do not isolate the diagnostic value of this split. The correction step flags pole-vote nodes and replaces their values with the selected Stieltjes component under positivity/convexity constraints. For a positive, decreasing, convex Stieltjes function such as log(1+x)/x, this replacement can reduce the MAE even if nodes are chosen by an unrelated rule. Table 2 and Sec. 4 include no ablation with randomly selected nodes or nodes chosen by ordinary residual size, so the reported improvements may come from the Stieltjes refit rather than from Froissart-doublet dynamics. Likewise, the Breit-Wigner preservation is attributed to stability of complex poles across orders, but no statistic is given that separates that stability from spurious pole stability caused by a localized systematic bump. Consequently, the central claim that doublet dynamics identify localized inconsistencies is not yet supported by the evidence presented.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Padé-based iterative algorithm for reconstructing the analytic structure of a function from finite, noisy datasets. The central idea is that Froissart doublets appearing in diagonal Padé sequences should be treated as diagnostic objects: unstable doublets are associated with finite-information effects, while recurrent doublets indicate localized inconsistencies in the input data. The algorithm votes on pole recurrences to identify anomalous nodes, corrects those nodes using a selected Stieltjes component, and iterates until the pole structure stabilizes. The method is validated on Stieltjes functions, on pseudo-experimental data with Gaussian and Breit-Wigner distortions, and on a few holomorphic functions, with the full implementation released as a Mathematica notebook in a GitLab repository.","tokens_in":18032,"tokens_out":5204,"duration_ms":57218,"significance":"If the central premise is correct, the paper offers a practical, largely model-independent tool for separating genuine analytic features from finite-information and systematic effects, which would be useful across experimental and lattice-data analyses. The strengths of the manuscript are its concrete algorithmic implementation, open code, and a broad set of test cases including realistic pseudo-experimental ensembles. The reported improvements are large (MAE reductions up to about 99.9% in controlled examples), and the Breit-Wigner/Gaussian contrast is a sensible test of the method's diagnostic power. However, the load-bearing classification of Froissart doublets is heuristic, and the validation does not currently isolate the contribution of that classification from the effect of the Stieltjes refit.","major_comments":[{"comment":"The central premise of the paper is that recurrent Froissart doublets across Padé orders correspond to localized data inconsistencies, whereas unstable doublets are finite-information effects. This premise is asserted rather than demonstrated: the manuscript states that the distinction is 'not causal but dynamical,' and its operational implementation relies on adjustable thresholds (currentTolerance 0.45 Δx, vote threshold >1, diagOrders 5–12 in Table 6). Since every correction decision in the algorithm is conditional on this split, the paper needs either a proof, a rigorously quantified definition of recurrence, or a direct validation showing that the recurrence criterion correlates with injected inconsistencies rather than with other features of the data.","section":"Sec. 2 and App. A.2"},{"comment":"The controlled validation consists of single 'representative' realizations. No ensemble averages, standard deviations, or seed dependence are reported, and the text itself concedes that other configurations with the same density and noise count 'may yield even better improvement.' Consequently the reported improvements, ranging from 62.31% to 99.90%, cannot be distinguished from favorable sampling. The authors should repeat the noise-generation procedure many times and report the distribution of MAE improvements, not just one example per configuration.","section":"Sec. 3.1, Table 2"},{"comment":"The reported improvements do not isolate the diagnostic role of Froissart-doublet dynamics. The correction step replaces flagged nodes with the selected Stieltjes component under positivity and convexity constraints. For a positive, decreasing, convex Stieltjes function such as log(1+x)/x, such a replacement can reduce the MAE even if the flagged nodes are chosen by an unrelated rule. A necessary control is a baseline that corrects the same number of randomly selected nodes, or nodes chosen by ordinary residual magnitude, using the same Stieltjes refit. Without such an ablation, the improvements in Table 2 could be attributed to the Stieltjes refit rather than to the recurrence classification.","section":"Sec. 3, Eq. (2)"},{"comment":"The preservation of the Breit-Wigner resonance is attributed to the stability of complex poles across Padé orders, but no statistic is provided that quantifies this stability or separates it from the stability that a localized systematic bump could also produce in Padé approximants. The reported RMSE and pull values show that the resonance is not removed, but they do not demonstrate that the algorithm's pole-stability criterion was the operative cause. A null test with a non-resonant localized feature of comparable width and height, analyzed with the same pole-stability metric, would strengthen the claim.","section":"Sec. 4.2"},{"comment":"The reported post-reconstruction MAE for the holomorphic example g(z)=(z+1)^{3/2} with n=5 noisy points is 8.0×10^-6, which is identical to the value reported in Table 2 for the Stieltjes example log(1+z)/z with the same density and noise count, despite the two functions having very different scales over the interval [0,10]. Please verify this number and report the correct value; as printed, it suggests a copy error in a validation table that is central to the holomorphic-extension claim.","section":"Table 3, first row"}],"minor_comments":[{"comment":"There are typographical errors in the text, including 'doubltes' in the abstract and 'without lose of generality' in Section 2.","section":"Abstract and Sec. 2"},{"comment":"The figure captions describe the perturbation as 'additive noise,' while Eq. (4) defines multiplicative noise f(z_i)(1+ε_i). Please make the terminology consistent.","section":"Fig. 1 captions and Eq. (4)"},{"comment":"The function tan^2(x) is listed as a holomorphic test function, but it has real poles at π/2 + kπ inside the interval [0,10]. Either choose a genuinely holomorphic example or state explicitly that the sampling domain excludes those poles.","section":"Sec. 5.2"},{"comment":"The MAE is defined with respect to the exact analytic function, which is unknown in realistic applications. Please state explicitly that this metric is used only for validation in controlled settings.","section":"Sec. 3, MAE definition"},{"comment":"The algorithm is described as requiring no model for the origin of the inconsistencies, yet many parameters, including sigma, prec, currentTolerance, and vote threshold, are dataset-dependent. A brief discussion of sensitivity to these parameters would help users apply the method to new datasets.","section":"App. A.2, Tables 5-6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the open-code practice is a plus. The main risk is that the headline claim -- that Froissart-doublet dynamics identify localized inconsistencies -- is under-supported by the current validation. An ablation with random-node or residual-based baselines, together with ensemble statistics for the controlled tests, would address the core concern. If those are added, I would be willing to consider a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the iterative use of Froissart-doublet recurrence to locate bad data points is a genuinely new idea, and the paper is worth engaging seriously. But the validation as written does not yet show that the doublet dynamics, rather than the generic Stieltjes refit, is what does the work.\n\nWhat is new: prior work treated Froissart doublets as artifacts to filter or as statistical indicators; this paper instead tracks their positions along Padé sequences, uses recurrence across orders to flag inconsistent nodes, and iteratively corrects those nodes while preserving Stieltjes (or holomorphic) structure. That is a real departure, and the algorithm is concrete enough to implement. The tests on log(1+x)/x with multiplicative noise show large MAE reductions (62–99.9%), and the contrast between suppressing a Gaussian bump and preserving a Breit-Wigner resonance is a sensible sanity check. The pseudo-experimental setup with Poisson statistics is a good effort to mimic realistic data.\n\nWhere the soft spots are. The central premise—recurrent doublets signal localized inconsistencies, unstable ones are finite-information effects—is asserted in Sec. 2 as a dynamical classification ('not causal but dynamical') and is never proven or tied to a theorem. The quantitative definition of 'recurrent' is just voting with adjustable thresholds (currentTolerance 0.45 Δx, vote threshold >1, diagOrders 5–12), so the algorithm has several free knobs. That alone would be okay if the validation isolated the mechanism; it does not. Table 2 and Fig. 1 are single 'representative' realizations with no error bars. There is no comparison against a baseline where nodes are chosen randomly or by plain residual size. For a smooth, convex Stieltjes function, replacing selected data points with the best Stieltjes fit will reduce MAE regardless of how those points are selected, so the reported improvements do not establish that doublet dynamics is the causal ingredient. The Breit-Wigner preservation is also anecdotal: no statistic separates stable complex poles of a resonance from pole stability that a localized systematic bump might produce. Finally, the abstract promises an open GitLab repository, but the text never gives the link or repository identifier; the appendix references 'the associated GitLab repository' without a URL. For a methods paper, that is a serious reproducibility gap.\n\nWho benefits: people in hadron spectroscopy, lattice QCD, and scattering analyses who already use Padé approximants for analytic continuation and want a practical tool to separate systematic distortions from physics. The idea has enough potential that I would send it to a good referee with the expectation of major revision: add ensemble statistics, baselines (random-node and residual-based node selection), a sensitivity scan over thresholds, and the actual repository link. The central heuristic also needs a crisper statement of what recurrence means quantitatively, or at least empirical support from a designed experiment that varies the node-selection rule. Not a desk reject; not yet a definitive method.","headline":"Genuinely new iterative doublet-dynamics algorithm, but the validation does not yet isolate the doublet mechanism from the Stieltjes refit—send to peer review with major revision required.","tokens_in":18579,"tokens_out":3224,"would_cite":false,"duration_ms":33000,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Recurring spurious poles flag bad data points","keywords":["Analyticity","reconstruction","experimental data","lattice data","Padé sequences","Froissart doublets","Stieltjes functions","holomorphic functions"],"falsifier":"Take a known Stieltjes function and add a single controlled systematic bump with amplitude below the statistical noise floor. If the resulting Froissart doublet wanders across orders and the algorithm leaves the bump in place, the recurrence criterion fails in the weak-signal regime. Conversely, place a genuine complex-conjugate pole pair with very small residue so its doublet appears nearly stationary; if the algorithm corrects it away, the criterion mistakes a true singularity for an inconsistency.","tokens_in":17500,"feed_emoji":"🧮","tokens_out":8328,"duration_ms":81439,"temperature":0.7,"pith_summary":"This paper claims that the spurious pole–zero pairs known as Froissart doublets, usually discarded as artifacts of Padé approximation, can instead be read as diagnostics of where a finite dataset is analytically inconsistent. The authors build an iterative algorithm that tracks these doublets along a sequence of Padé approximants: doublets that reappear at nearly the same location mark data points to correct, while doublets that wander are treated as harmless finite-information noise. In controlled tests on Stieltjes functions, the correction reduces mean absolute error by 56% to 99.9%, depending on noise level and sampling density. In pseudo-experimental examples, a smooth Gaussian distortion is suppressed while a Breit-Wigner resonance is preserved, and the method extends to general holomorphic functions. If the recurrence heuristic holds, this gives a model-independent way to clean experimental or lattice data before extracting its analytic structure.","feed_headline":"Recurring spurious poles flag bad data points","feed_subtitle":"An iterative Padé-sequence algorithm fixes noisy measurements while preserving genuine resonances.","key_machinery":"The central object is the three-family classification of poles and zeros observed across successive diagonal Padé approximants $P_N^N$: unstable Froissart doublets (spurious pole–zero pairs) as finite-information effects, recurrent Froissart doublets as localized data inconsistencies, and stable poles compatible with the underlying analytic function. The algorithm carries the argument by splitting each approximant as $P_N^N(x)=P_M^M(x)+P_{N-M-1}^{N-M}(x)$, assigning recurring poles to the nearest sampling node, voting on nodes across the sequence, and iteratively correcting the highest-vote nodes toward the selected Stieltjes component subject to positivity and convexity constraints.","core_discovery":"The central claim is that the dynamics of Froissart doublets along a Padé sequence separate three families: unstable doublets caused by finite information, recurrent doublets caused by localized inconsistencies in the input data, and stable poles compatible with the underlying analytic structure. From this classification, the paper derives a reconstruction algorithm that splits each Padé approximant into a Stieltjes (or holomorphic) component and a noise component, uses a voting system to rank grid nodes by how often they host a recurring pole, and iteratively updates the top-ranked nodes toward the analytic component while preserving Stieltjes positivity and convexity. The paper reports that this procedure removes localized systematic distortions such as a Gaussian bump, leaves genuine complex-conjugate poles such as a Breit-Wigner resonance in place, and keeps statistical fluctuations consistent with the underlying analytic behavior.","pith_inferences":["Inference: the recurrence criterion could be used before any correction as a data-quality audit, flagging grid nodes that deserve re-measurement rather than numerical adjustment.","Inference: a natural stress test is correlated or clustered noise, since the synthetic tests use isolated random nodes; the voting tolerance would need to distinguish a genuine localized systematic from a patch of correlated fluctuations.","Inference: the stable Stieltjes order reached after convergence is an empirical estimate of how much analytic information the dataset really supports, which could inform where to stop a Padé sequence in future fits.","Inference: the same dynamics could check whether two datasets of the same observable are mutually compatible, flagging regions where they cannot share a common analytic sequence."],"forward_implications":["A dataset with localized systematic distortions can be corrected without modeling the source of the distortion, because the criterion is analytic rather than statistical.","Genuine physical structures with complex-conjugate poles, such as a Breit-Wigner resonance, survive the procedure, while smooth non-analytic deformations such as Gaussian bumps are suppressed.","The method applies beyond Stieltjes functions: for holomorphic functions, real poles inside the sampling region are treated as noise, while complex-conjugate or Szegő-curve structures are assigned to the analytic component.","The reconstructed dataset remains on the original sampling grid and keeps its statistical uncertainties, so it can be used as input to later fits or analytic continuations.","Finite sampling density and numerical precision shift the critical Padé order at which Froissart doublets appear, so the algorithm's behavior depends on the data's precision and density."],"supporting_citations":[{"why":"Supplies the Padé-approximant formalism and the Stieltjes-function convergence, positivity, and convexity properties the correction step enforces.","marker":"[1]"},{"why":"Identifies the spurious pole–zero pairs, the Froissart doublets, that the method reinterprets as diagnostic objects.","marker":"[2]"},{"why":"Shows that noise in input data organizes into localized pole–zero structures, the empirical basis for the three-family classification.","marker":"[3]"},{"why":"Defines the critical Padé order and connects finite precision and sampling density to the emergence of Froissart doublets, grounding the finite-information family.","marker":"[33]"}],"fun_headline_variants":["Froissart doublets turn artifacts into data-quality probes","Iterative Padé method removes noise, preserves resonances","Padé sequence dynamics highlight bad input points","Spurious pole patterns unmask inconsistent data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a spurious pole–zero pair (a Froissart doublet) which reappears at nearly the same spot across successive Padé orders signals a genuine localized inconsistency in the data, while a doublet that moves is only finite-information noise; this distinction is asserted rather than proved and is validated on a handful of constructed examples.","fun_headline_variants_meta":{"raw":{"variants":["Froissart doublets turn artifacts into data-quality probes","Iterative Padé method removes noise, preserves resonances","Padé sequence dynamics highlight bad input points","Spurious pole patterns unmask inconsistent data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000739,"raw_usage":{"total_tokens":3268,"prompt_tokens":883,"completion_tokens":2385,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":2323}},"tokens_in":499,"tokens_out":2385,"duration_ms":21829,"temperature":1.0,"reasoning_tokens":2323,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:21:59.805265+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a known Stieltjes function and add a single controlled systematic bump with amplitude below the statistical noise floor. If the resulting Froissart doublet wanders across orders and the algorithm leaves the bump in place, the recurrence criterion fails in the weak-signal regime. Conversely, place a genuine complex-conjugate pole pair with very small residue so its doublet appears nearly stationary; if the algorithm corrects it away, the criterion mistakes a true singularity for an inconsistency.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Padé-approximant formalism and the Stieltjes-function convergence, positivity, and convexity properties the correction step enforces."},{"cited_title":"Froissart,Approximation de Pad´ e","cited_arxiv_id":null,"evidence_quote":"Identifies the spurious pole–zero pairs, the Froissart doublets, that the method reinterprets as diagnostic objects."},{"cited_title":"Bessis, J","cited_arxiv_id":null,"evidence_quote":"Shows that noise in input data organizes into localized pole–zero structures, the empirical basis for the three-family classification."},{"cited_title":"Noise Effects on Pade Approximants and Conformal Maps","cited_arxiv_id":"2208.02410","evidence_quote":"Defines the critical Padé order and connects finite precision and sampling density to the emergence of Froissart doublets, grounding the finite-information family."}],"review_version":1}