{"id":"76594ed1-46c1-4921-8e47-bb3be371ccc2","arxiv_id":"2608.05837","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A correspondence framework tracks Reeb space sheets across timesteps using range-space overlap and domain vertex support, and demonstrates persistent and changing structures on torus and molecular datasets.","lead":"Reeb space sheets are abstract pieces of a two-field dataset that can be followed across time by checking how much they overlap in value space. This paper develops that tracking idea into a visual analysis workflow and applies it to a synthetic torus and two molecular simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The range-space IoU match score is coordinate-dependent: monotone reparameterization of one scalar field changes sheet footprints, Jaccard values, and even top-N selection, so the claimed topological tracking may reflect arbitrary scaling choices.","rationale":"The reader's verdict (CONDITIONAL) identifies the heuristic, dense-sampling-dependent nature of q_R. My stress-test extends this: even under ideal dense sampling and clean data, q_R is not a topological invariant. The Reeb space's 'sheets' as quotients are invariant under monotone reparameterization of each scalar coordinate, because fibers and their connected components are unchanged; but every quantity used for matching and ranking — sheet footprint area, IoU, overlap — is defined in the fixed Euclidean range and changes under such reparameterization. The paper repeatedly grounds its claims in topological structures (sheets, Jacobi structure, persistence analog), yet the active score tracks geometry of the range image, not the quotient topology. This is not an internal inconsistency (the paper explicitly calls q_R a heuristic), but it makes the central claim 'sheets can serve as trackable topological structures' depend on an unstated choice of coordinates. The torus experiment would not reveal this because a symmetric deformation produces symmetric events under any reasonable parameterization; molecular NTO fields likewise give plausible chemistry, but similarity to known intervals is qualitative and could be robust to the defect. The check I propose is simple and decisive. If invariance fails, the framework is still a useful visual-analysis tool for fixed fields, but the paper should either restrict its claim to range-space footprint tracking or add an invariance/normalization step (e.g., use a coordinate-independent descriptor). The reader's conditional verdict remains the right call: accept only if this or an equivalent test is passed and the claim is scoped.","tokens_in":22838,"tokens_out":6968,"duration_ms":79912,"concrete_test":"On the torus dataset, replace the height field g by h_p(g)=sign(g)|g|^p for p ∈ {0.5, 2.0} (with p=1 as baseline) and rerun the full pipeline (Reeb space extraction, top-N=20 selection, q_R matching, event scores). Compare (i) the top event intervals for θ=0.5 and (ii) the temporal graph link structure around intervals 18–19, 50–51, and 81–82. If these intervals or the labeled merge/split events shift, appear, or disappear, the tracking is coordinate-dependent and the central claim 'trackable topological structures' is not supported; if they remain identical despite changed footprints, the concern is answered.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central inference is that q_R(S,S') = |S∩S'|/|S∪S'| (Sec. 4.3) is a meaningful correspondence score for Reeb-space sheets, making the sheets 'trackable topological structures' (Abstract). This is not invariant under monotone reparameterizations of the individual scalar coordinates. A strictly increasing map h applied to g leaves every fiber F^{-1}(a,b) and hence the Reeb space quotient unchanged as a topological object, but the sheet footprints in the bivariate range are transformed by (a,b) -> (a,h(b)). Areas, intersections, and unions change non-uniformly, so q_R between adjacent timesteps changes, as does the range-space area ranking used for top-N selection (Sec. 4.2). Consequently, the temporal graph, event score E_i(θ), and reported event intervals (e.g., 18–19, 50–51, 81–82 for the torus) are functions of the chosen coordinate system, not of the topological structure. The paper does not discuss this dependence. In molecular datasets the two NTO fields share units, but the range geometry is still not an intrinsic sheet invariant; nonlinear resealing of one field would alter results. Thus the evidence that 'Reeb space sheets can serve as trackable topological structures' is currently conditional on an unstated coordinate convention.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a framework for temporally tracking Reeb-space sheets in time-varying bivariate fields. At each timestep, Reeb spaces are computed and prominent sheets are selected by range-space area. Correspondences between consecutive timesteps are established with a range-space Jaccard (intersection-over-union) score, supplemented by raw domain-vertex overlap as supporting evidence. The resulting weighted temporal graph is visualized with a Sankey-style layout, and heuristic event scores are defined to highlight intervals of weak or ambiguous correspondence. The method is evaluated qualitatively on a synthetic bivariate torus sequence, on two MVK excited-state trajectories, and on a cis-stilbene trajectory, with an accompanying interactive prototype. The central claim is that Reeb-space sheets can serve as trackable topological structures for visual analysis.","tokens_in":23084,"tokens_out":3655,"duration_ms":39754,"significance":"If the claim holds, the paper fills a genuine gap: no existing method explicitly tracks Reeb-space sheets over time, and the proposed workflow is concrete, clearly described, and accompanied by a usable prototype and precomputed artifacts. The authors are also unusually explicit about the heuristic nature of their correspondence measure and about the method's limitations, which is commendable. The main significance, however, is conditioned on the range-space IoU score being a meaningful and sufficiently stable notion of sheet identity. Because that score is not invariant under monotone reparameterization of the individual scalar fields, the paper's stronger 'topological tracking' claim is currently overstated. The evaluation is entirely qualitative, which is acceptable for a visualization paper but does not by itself establish quantitative tracking reliability.","major_comments":[{"comment":"The range-space correspondence score q_R(S,S') = |S∩S'|/|S∪S'| is not invariant under monotone reparameterizations of the individual scalar coordinates. A strictly increasing map h applied to g leaves the Reeb space quotient R_F unchanged as a topological object, but transforms sheet footprints in the range by (a,b) -> (a,h(b)). Areas, intersections, and unions then change non-uniformly, so q_R, the top-N sheet selection by range-space area (Section 4.2), the event score E_i(θ) (Section 4.4), and the reported event intervals (e.g., 18–19, 50–51, 81–82 for the torus) all become functions of the chosen coordinate system. The paper does not discuss this dependence, even though the molecular datasets use NTO fields whose units and scaling are convention-dependent. This is load-bearing for the abstract claim that sheets are 'trackable topological structures'. Please either adopt coordinate-invariant descriptors, explicitly frame the results as conditional on a fixed range-space coordinate convention and demonstrate robustness under monotone rescaling, or soften the 'topological' language to 'range-space-geometric' tracking.","section":"Section 4.3, Eq. (1)"},{"comment":"The evaluation is entirely qualitative. The torus experiment demonstrates face validity by recovering expected symmetry and the merge/split at intervals 18–19 and 81–82, but no quantitative ground-truth comparison is provided: there are no tracking accuracy scores, no precision/recall for detected event intervals, no error bars, and no comparison against a baseline (e.g., domain-overlap-only tracking, centroid-based matching, or optimal-transport matching). The central claim that the method 'can track Reeb space sheets meaningfully over long sequences' (Section 7) rests on visual inspection of Figures 1, 3–9 and on agreement with previously reported chemical intervals. I ask for at least one quantitative sanity check on the torus dataset, where the expected event intervals are known, and ideally a small baseline comparison on the same dataset.","section":"Section 5"},{"comment":"The threshold sensitivity statement in the main text is stronger than the data support. The supplement's Table 1 shows that the top event interval for MVK state 2 shifts from 28.06–28.54 fs at θ=0.3 and 0.4 to 27.58–28.06 fs at θ=0.5–0.7, and for cis-stilbene the top interval changes from 243.82–244.31 fs at θ=0.3 to 288.21–288.33 fs at θ≥0.4. The text in Section 5.2 says the 27–30 fs interval is 'robustly highlighted,' and Section 5.3 calls the 288.21–288.33 fs interval 'stable under threshold variation.' These claims should be qualified with the observed shifts, and the main paper should summarize the sensitivity table rather than only placing it in the supplement.","section":"Section 4.4 and Supplement Table 1"}],"minor_comments":[{"comment":"The caption reads 'the100timestep sequence' with a missing space; should be 'the 100 timestep sequence'.","section":"Figure 1 caption"},{"comment":"The local graph in Figure 9 uses a link threshold of 0.41 for range-space similarity, but the choice of 0.41 is not explained. Please state how this value was selected and whether the conclusions are sensitive to it.","section":"Section 5.3 and Figure 9"},{"comment":"The definition of a generic bivariate field is given, but the paper never states whether all tested datasets satisfy genericity or how the Reeb-space computation handles violations (e.g., degenerate vertices). A brief remark would help reproducibility.","section":"Section 3"},{"comment":"Reference [9] is incomplete: it gives the title and authors but no publication venue, page numbers, or DOI. Reference [20] also lacks a venue; please complete the bibliographic entries.","section":"References"},{"comment":"The fourth limitation states that the continuing feature diagnostic is greedy and local, and the fifth notes the lack of evaluation on large-scale data. These are appropriate caveats, but consider moving the sensitivity caveat from the supplement into this section as well, since it directly affects the interpretation of the reported event intervals.","section":"Section 7, limitations"}],"recommendation":"major_revision","confidential_remarks":"The coordinate-dependence issue is the most serious concern. If the authors can add a robustness study under monotone reparameterization (even for the torus) or clearly reframe the contribution as range-space-geometric tracking rather than topological tracking, the paper would be publishable. The qualitative nature of the evaluation is acceptable for a vis venue, but the current abstract's 'trackable topological structures' phrasing is likely to draw criticism from topological-data-analysis reviewers. I also recommend the editors ask for the supplement's sensitivity table to be summarized in the main text, because the current text overstates threshold stability."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, the one thing to know: this is a useful first workflow for tracking Reeb space sheets across time, and it is more honest than most about its limitations. The core idea is simple—match sheets in adjacent timesteps by range-space Jaccard overlap, with domain vertex overlap as support—and as far as I can tell the novelty is real: no prior work defines temporal correspondences between Reeb space sheets. The torus test shows face validity, and the molecular case studies (MVK, cis-stilbene) pick out chemically plausible regions and intervals. I give them credit for the online prototype and for the explicit limitations section.\n\nThe soft spots are real but not fatal. The evaluation is entirely qualitative: no ground truth trajectory for the torus (even though one exists), no baseline comparison against, say, univariate tracking on each field or optical-flow matching, no error bars. The only numeric check is the sensitivity table for theta, and that actually shows the top interval changes for MVK state 2 and cis-stilbene. That is honest but weak.\n\nThe more serious concern, which the paper does not discuss, is that the primary similarity score q_R is coordinate-dependent. A monotone reparameterization of one scalar field leaves the Reeb space quotient unchanged as a topological object, but changes the sheet footprints in the range, the areas used for top-N selection, and all Jaccard values. So the \"topological\" tracking result can change under a monotone rescaling of one coordinate. The method is not tracking an intrinsic topological invariant; it is tracking range-space geometry under a particular coordinate convention. Since the data come with fixed units this is not fatal for the case studies, but the abstract's claim that \"Reeb space sheets can serve as trackable topological structures\" goes a bit beyond what the measure supports.\n\nThe citation pattern looks fine. They build on their own earlier analysis of the same molecular data, which is continuity rather than circularity. Computational complexity is reported, and the costs are reasonable.\n\nVerdict: the paper deserves a serious referee. I would ask the authors to (1) add a quantitative evaluation on the torus, e.g., compare extracted tracks against the known symmetry and merge/split intervals, (2) test coordinate-sensitivity explicitly by reparameterizing one field, and (3) add at least one baseline, even a simple one. With that, the contribution would be solid.","headline":"First Reeb-space-sheet tracking workflow, honestly presented but qualitatively evaluated and coordinate-dependent.","tokens_in":23591,"tokens_out":2527,"would_cite":true,"duration_ms":26235,"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":"The paper claims that Reeb-space sheets—the two-dimensional cells of a bivariate field's fiber-connectivity quotient—are trackable across time using range-footprint overlap plus domain support, demonstrated on a torus and…","keywords":["Reeb space","bivariate fields","feature tracking","time-varying data","topological data analysis","molecular visualization","intersection over union","temporal correspondence"],"falsifier":"Take a controlled sequence where a single known sheet moves in the bivariate range by more than its own width between consecutive frames while its spatial support stays intact: the range overlap score is zero, so the greedy track stops even though the feature persists. Running this with the torus pipeline by increasing the deformation between frames would show the exact sampling density at which sheet tracking fails, testing the paper's core assumption directly.","tokens_in":22637,"feed_emoji":"📊","tokens_out":11217,"duration_ms":94493,"temperature":0.7,"pith_summary":"The paper tries to establish that Reeb space sheets—the two-dimensional pieces of a topological quotient that records how connected components of fibers of a bivariate field change—can be treated as trackable features in time-varying data. This matters because bivariate fields arise in many scientific settings, and topological tracking tools for single scalar fields do not carry over to joint two-field structures. The method links sheets in consecutive timesteps using range-footprint intersection-over-union overlap as the primary similarity and raw domain vertex overlap as supporting evidence, then summarizes the correspondence graph with event scores and continuing-feature lifetimes. On a synthetic torus it recovers the expected symmetry and the merge/split events at intervals 18–19 and 81–82; on molecular dynamics data it produces persistent sheets tied to recognizable chemical regions and flags intervals consistent with earlier analyses. The paper presents the correspondence measure as a local, pairwise heuristic rather than a theoretical guarantee, and the temporal graph as a visualization aid rather than a conserved-flow model.","feed_headline":"Reeb-space sheets track features over time","feed_subtitle":"Range overlap plus domain support links sheets across time, exposing persistent features and change intervals.","key_machinery":"The central object is the Reeb space sheet: a two-dimensional cell of the quotient space $R_F = M/{\\sim}$ obtained by collapsing each connected component of every fiber $F^{-1}(a,b)$ to a point. Each sheet is stored as a polygonal footprint in the bivariate range, along with its range-space area and the set of regular domain vertices that support it. The argument is carried by a pairwise correspondence mechanism between consecutive timesteps: range-space intersection-over-union overlap serves as the active similarity score, raw domain vertex overlap serves as auxiliary support for filtering ambiguous matches, and an event score $E_i(\\theta)$ counts weak best continuations plus potential split and merge configurations to rank intervals of temporal change. A greedy lifetime diagnostic then follows the best above-threshold continuation from each sheet, producing the continuing-feature lists shown in the temporal graph.","core_discovery":"The paper's central claim is that Reeb space sheets can serve as trackable topological structures in time-varying bivariate fields. The authors compute the Reeb space of each bivariate field $F=(f,g)$ at every timestep, rank sheets by their range-space area, and retain the top $N$. Sheets in adjacent timesteps are then compared by the range-footprint overlap $q_R(S,S') = |S \\cap S'|/|S \\cup S'|$, supplemented by the raw overlap of their associated domain vertex sets $O(S,S')=|\\Omega_S \\cap \\Omega_{S'}|$. Because a sheet can split, merge, appear, disappear, and overlap many possible targets, the method keeps multiple candidate correspondences instead of forcing a one-to-one matching. The authors show that on the torus sequence the temporal graph is nearly symmetric and the event-score plot highlights the known central transition and the merge/split intervals, and that on MVK and cis-stilbene the flagged intervals coincide with chemically relevant periods reported in earlier work.","pith_inferences":["The same range-footprint overlap plus domain-support recipe could be applied to any sheet-like cell complex derived from a bivariate map—not only exact Reeb spaces but also coarser summaries—since only footprints and vertex supports are needed; the paper does not make this generalization.","Because event scores are cheap once correspondences are built, they could serve as a pre-filter that selects where to spend expensive Reeb-space computation or finer temporal sampling in a longer simulation; the paper uses them for visual ranking rather than for steering computation.","Replacing the greedy lifetime diagnostic with a jointly optimized assignment across all timesteps would turn persistent-sheet status from a local heuristic into a globally defined track; comparing the two on the torus ground truth would show how often greedy choices diverge.","The cis-stilbene domain-support drift suggests that range-space identity and spatial support can decouple; quantifying that decoupling per track could provide a new diagnostic for physical or chemical reorganization that the paper leaves for future work."],"forward_implications":["Reeb space sheets can be linked across long sequences: on MVK three sheets persist through all 83 timesteps and correspond to interpretable molecular regions near the oxygen atom, the C1–C2 bond, and C3.","Event-score peaks single out transition intervals such as torus intervals 18–19 and 81–82, and the cis-stilbene interval 288.21–288.33 fs that coincides with a pronounced rotation of the Reeb space in the range.","Because correspondences are pairwise and non-conserved, the temporal graph can represent splits, merges, appearances, disappearances, and geometric changes without imposing a one-to-one matching.","Domain overlap disambiguates range-space look-alikes: in the torus interval 18–19, sheets with nearly identical range footprints are resolved into the correct spatial continuations using shared domain vertices.","Event-score rankings are stable across tested thresholds for the main highlighted intervals, supporting their use as an entry point for exploring long sequences such as the 704-timestep cis-stilbene dataset."],"supporting_citations":[{"why":"Defines Reeb spaces of piecewise-linear mappings, the object whose two-dimensional sheets are tracked.","marker":"[15]"},{"why":"Supplies the Singular Arrange and Traverse algorithm used to compute the Reeb spaces and sheet geometries.","marker":"[19]"},{"why":"Supplies the arrange and traverse algorithm that produces the Reeb space cell complex and sheet footprints.","marker":"[20]"},{"why":"Defines natural transition orbitals, the two scalar fields forming the bivariate data in the molecular case studies.","marker":"[26]"},{"why":"Provides the time-varying molecular electronic structure datasets and the earlier continuous-scatterplot analysis whose highlighted intervals are compared against.","marker":"[40]"},{"why":"Reports intervals of importance in the MVK electron-density dynamics that the event-score peaks are checked against.","marker":"[52]"}],"fun_headline_variants":["Tracking Reeb-space sheets through time","Reeb-space sheets as trackable topology","Linking Reeb sheets across timesteps","Time-varying bivariate fields via Reeb sheets","Reeb sheets expose persistent structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that consecutive timesteps are sampled densely enough that a persistent sheet keeps roughly the same footprint in the bivariate range; if the field changes too quickly between frames, the overlap score falls below threshold and the track breaks, so the entire tracking result rests on temporal sampling density.","fun_headline_variants_meta":{"raw":{"variants":["Tracking Reeb-space sheets through time","Reeb-space sheets as trackable topology","Linking Reeb sheets across timesteps","Time-varying bivariate fields via Reeb sheets","Reeb sheets expose persistent structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000315,"raw_usage":{"total_tokens":1792,"prompt_tokens":958,"completion_tokens":834,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":779}},"tokens_in":574,"tokens_out":834,"duration_ms":6484,"temperature":1.0,"reasoning_tokens":779,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:33:54.690214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a controlled sequence where a single known sheet moves in the bivariate range by more than its own width between consecutive frames while its spatial support stays intact: the range overlap score is zero, so the greedy track stops even though the feature persists. Running this with the torus pipeline by increasing the deformation between frames would show the exact sampling density at which sheet tracking fails, testing the paper's core assumption directly.","supporting_citations":[{"cited_title":"Edelsbrunner, J","cited_arxiv_id":null,"evidence_quote":"Defines Reeb spaces of piecewise-linear mappings, the object whose two-dimensional sheets are tracked."},{"cited_title":"Hristov, I","cited_arxiv_id":null,"evidence_quote":"Supplies the Singular Arrange and Traverse algorithm used to compute the Reeb spaces and sheet geometries."},{"cited_title":"Hristov, D","cited_arxiv_id":null,"evidence_quote":"Supplies the arrange and traverse algorithm that produces the Reeb space cell complex and sheet footprints."},{"cited_title":"Sharma, T","cited_arxiv_id":null,"evidence_quote":"Provides the time-varying molecular electronic structure datasets and the earlier continuous-scatterplot analysis whose highlighted intervals are compared against."},{"cited_title":"Wetzels, T","cited_arxiv_id":null,"evidence_quote":"Reports intervals of importance in the MVK electron-density dynamics that the event-score peaks are checked against."}],"review_version":1}