{"id":"66ff5d2e-ef74-4912-ad84-afd482363331","arxiv_id":"2608.08728","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A normalized entropy, computed from sheaf residuals that compare neighboring vector states in edge-aligned frames, is proposed as a descriptor of how vector fields are organized on geometric graphs and is demonstrated on proteins, brains, traffic, and power grids.","lead":"This paper introduces SheafIQ, a score for how coordinated the arrows on a network are: it compares each pair of neighboring vectors in the local geometry of their connection and measures how spread out the mismatches are. The score is demonstrated on proteins, brain scans, traffic, and power grids, where it is said to reveal organization that ordinary network statistics miss.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Even when q=d, the sheaf residual t_ij in Eq. (3) reduces to the ordinary edge difference s_j - s_i, and the r_ij actually used in applications is not a sheaf residual; the sheaf-theoretic content of SheafIQ is therefore unsupported.","rationale":"The reader's weakest assumption was the q=d requirement and the undefined node vectors in the brain application. That is a data-availability and specification concern. The concern raised here is more fundamental and internal: even in regimes where q=d holds, the sheaf residual defined by Eq. (1) and Eq. (3) collapses algebraically to the ordinary Euclidean difference s_j - s_i, so the geometric edge frame contributes nothing to the residual norm. The paper's applications then switch to r_ij in Eq. (5), which is not a sheaf residual and is introduced with an invariance statement that is at best under-specified. This does not necessarily invalidate every empirical result, because r_ij is a concrete, computable statistic, but it invalidates the claim that SheafIQ is quantifying sheaf-theoretic incompatibility and that the edge-associated coordinate system is doing the work. The issue is testable by a short symbolic expansion, and if it lands, the theoretical framing and the interpretation of all application results need substantial revision. Since the reader already returned a conditional verdict, the final recommendation is unchanged: the authors should either use the actual sheaf residual with a nontrivial restriction map or present r_ij as an explicitly heuristic edge statistic and re-derive the information-theoretic claims for it.","tokens_in":22225,"tokens_out":11846,"duration_ms":143093,"concrete_test":"Expand Eq. (3) symbolically using Eq. (1) and verify whether t_ij = s_j - s_i. If confirmed, recompute SheafIQ on the METR-LA dataset or on a synthetic vector field using the full sheaf residual ||t_ij|| in place of r_ij; if the temporal patterns or correlations change materially, the published H_SheafIQ depends on the non-sheaf r_ij. Also evaluate the two-node case u=(1,0), s_i=(0,1), s_j=(0,-1), where r_ij=0 but ||t_ij||=2, to confirm that the adopted residual fails to measure a vector incompatibility that the sheaf residual detects.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Even when q=d, the central residual loses its advertised geometric meaning. Substituting Eq. (1) into Eq. (3) gives t_ij = [(s_j^T u - s_i^T u)u; P_perp(s_j) - P_perp(s_i)] = s_j - s_i. Hence ||t_ij|| = ||s_j - s_i|| and the edge direction u cancels; 'local compatibility' is exactly equality of the original vectors. The claimed edge-induced coordinate system is only a block decomposition of the same vector, not a transport of information into a common frame. The applications do not use this residual: Section 2.3.3 adopts r_ij in Eq. (5), which replaces the perpendicular vector difference by a difference of perpendicular magnitudes. That statistic is not a sheaf coboundary and is not derived from the sheaf axioms; e.g., on an edge with u=(1,0), s_i=(0,1), s_j=(0,-1), r_ij=0 even though t_ij=(0,-2). Thus the residual energy field, O(G,S), and H_SheafIQ measure an ad hoc partial statistic rather than sheaf incompatibility. The central claim of a unified sheaf-theoretic information measure is therefore unsupported, and the cross-domain conclusions must be re-derived for r_ij as an independent edge statistic.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces SheafIQ, a descriptor intended to quantify the global organization of vector-valued states on geometrically embedded graphs. The construction proceeds by attaching a cellular sheaf to the graph, using edge-induced restriction maps to represent neighboring node vectors in a common edge stalk, defining a sheaf residual per edge, converting residual norms into edge energies, and finally taking the Shannon entropy of the normalized edge-energy distribution as the SheafIQ value. Propositions 1 and 2 establish standard entropy bounds and scale invariance for this distribution. The paper reports applications to protein dynamics, Alzheimer's disease functional networks, urban traffic data, and power grids, with claims that SheafIQ captures organizational information beyond topology-only and magnitude-only descriptors, supported by negative controls, baseline regressions, and multi-dataset replication.","tokens_in":22556,"tokens_out":6446,"duration_ms":77609,"significance":"If the sheaf-theoretic interpretation were valid, the paper would offer a general, parameter-free descriptor applicable to many scientific domains; the extensive empirical sections, including randomization controls in proteins, traffic regression residuals, and power-grid fault experiments, are a genuine strength and demonstrate care in validation. However, the significance as stated is not supported: the central sheaf residual reduces to the ordinary edge difference, the descriptor actually used in all applications is an ad hoc nonlinear statistic that is not a sheaf coboundary, and the brain application does not specify the vector field at all. Proposition 1 and Proposition 2 are correct, but they hold for any probability distribution over edges and therefore do not depend on the sheaf construction. As a result, the claimed novelty of a unified sheaf-theoretic information measure collapses, and the empirical findings must be reinterpreted as properties of a heuristic edge statistic, not of sheaf-theoretic incompatibility.","major_comments":[{"comment":"The sheaf residual t_ij is mathematically identical to the ordinary vector difference s_j - s_i. Substituting Eq. (1) into Eq. (3) gives the first block ((s_j - s_i)^T u_ij)u_ij and the second block (s_j - s_i) - ((s_j - s_i)^T u_ij)u_ij, so t_ij = s_j - s_i and ||t_ij|| = ||s_j - s_i||. The edge direction u_ij cancels completely. Consequently, the claimed 'edge-induced coordinate system' does not provide any geometric comparison beyond the Euclidean difference already used in graph signal processing, and the statement in Section 2.3.1 that the residual 'quantifies the local incompatibility ... relative to the underlying graph geometry' is unsupported. The sheaf construction is therefore equivalent to a trivial constant sheaf with identity-style restrictions, and the central theoretical novelty is not present.","section":"Section 2.2.2, Eq. (1); Section 2.3.1, Eq. (3); Section 2.3.2"},{"comment":"The edge measure actually used in all downstream quantities — residual energy, the measure in Eq. (8), the probability in Eq. (9), the entropy in Eq. (10), and Definition 2 — is r_ij from Eq. (5), not the sheaf residual norm ||t_ij|| from Eq. (4). The statistic r_ij is not a sheaf coboundary and is not obtained from any cellular sheaf: it replaces the perpendicular vector difference by a difference of perpendicular magnitudes, which is a nonlinear operation. As a concrete counterexample, for u=(1,0), s_i=(0,1), and s_j=(0,-1), one has r_ij=0 while t_ij=(0,-2). Thus an edge can have zero residual energy under the application's statistic while being maximally incompatible under the stated sheaf definition. All application conclusions in Sections 3.1–3.5 therefore concern an ad hoc partial statistic, and the claims that SheafIQ quantifies 'sheaf incompatibility' or 'sheaf residual organization' are not justified. The authors must either re-derive every result for r_ij as an independent descriptor, with no appeal to sheaf theory, or replace r_ij by a genuinely sheaf-derived residual and rerun the applications.","section":"Section 2.3.3, Eq. (5); Section 2.4; Section 2.5.2"},{"comment":"The brain-network application never specifies the node vectors or their dimension q, and no node coordinates or edge directions are defined. The SheafIQ framework requires q=d and an explicit geometric embedding with coordinates x_i to construct u_ij and the restriction maps of Eq. (1). If the node objects are regional time series, connectivity profiles, or any objects other than vectors in the same Euclidean space as the node coordinates, then the residual, energy, and entropy are not defined. The reader cannot verify the validity of any SheafIQ value reported in Section 3.3, and the central assumption of the method is unstated. This is a load-bearing omission because all subsequent hotspot and enrichment analyses inherit it.","section":"Section 3.3"},{"comment":"The power-grid application states that 'bus voltage magnitudes and phase angles obtained from AC power-flow analysis were treated as node signals (or node vectors)', but this does not specify a concrete vector construction. A phase angle is a scalar, and a magnitude-angle pair is not automatically a vector in R^d for the graph embedding; no node coordinates are defined for the IEEE 118-bus system. Without a precise specification of s_i, the embedding dimension d, and the edge directions, Eq. (1) and all derived quantities are undefined. The same ambiguity applies to the IEEE 300-bus generalization in Supplementary Section 6.3.","section":"Section 3.5"}],"minor_comments":[{"comment":"In the proof, the symbol n is used for |E| without being defined; the statement should explicitly set n=|E| at the start.","section":"Section 2.5.2, proof of Proposition 1"},{"comment":"The sentence attributing the rotational invariance of r_ij to Eq. (2) is imprecise: Eq. (2) states equality of restriction maps under sign reversal of u_ij, while the claimed rigid-motion invariance of r_ij requires a separate argument and is not generally true if the parallel component changes under rotation or translation.","section":"Section 2.3.3, Eq. (5)"},{"comment":"The caption for Figure 2(c) uses the phrase 'Normalized Information Quantity', which is not the term defined in Definition 2; the notation should be consistent with H_SheafIQ throughout.","section":"Section 3.1, Figure 2 caption"}],"recommendation":"reject","confidential_remarks":"The theoretical core of the paper does not support its central claim: the sheaf residual is trivial, and the applied measure is not a sheaf residual. The empirical sections are extensive and include useful controls, but they validate a different, unnamed heuristic descriptor. A resubmission that honestly frames the edge statistic without sheaf-theoretic claims, clearly specifies the vector constructions in every application, and re-derives the conclusions for that statistic could be of interest, but such a revision would change the identity of the paper and would still need to demonstrate that the entropy of this particular edge weighting is more than a re-expression of existing variation measures."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: the stress-test note holds up. Substituting (1) into (3) gives t_ij = [(parallel diff); (perp diff)] = s_j - s_i in the sense that the two blocks sum to that vector, and the norm is ||s_j - s_i||. The edge direction cancels. So the sheaf residual carries no geometric information beyond ordinary Euclidean distance. Worse, the paper doesn't even use t_ij in the applications: Section 2.3.3 replaces the perpendicular vector difference with a magnitude difference and calls it r_ij (Eq. 5). That statistic is not a sheaf coboundary and is not justified by sheaf theory. The claim that SheafIQ is a unified sheaf-theoretic information measure therefore lacks support.\n\nWhat the paper does well: the entropy of the normalized residual energy field is cleanly defined, scale-invariant, and the two propositions are standard but correct. The empirical sections are extensive, with multiple datasets and some genuinely useful negative controls — DHFR randomization, power-grid global scaling, traffic regression residuals. The limitations in Section 4 are honest. If you strip away the sheaf vocabulary, r_ij-based entropy is a perfectly usable edge-weight statistic for measuring how concentrated vector disagreements are across a network.\n\nSoft spots, in order: (1) the theoretical framing — this is the load-bearing one; (2) the brain application never defines the node vectors or states q, so the q=d premise is unverifiable there; (3) no code or data shipped, so the results are hard to reproduce; (4) many analysis thresholds are hand-chosen (top-5, 10 Å, etc.), with robustness checks covering only some of them. The rigid-motion invariance justification for r_ij also looks wrong unless you mean something very specific by 'rigid motion of local configurations'.\n\nWho this is for: a reader looking for a new empirical descriptor of spatial concentration in vector-valued network data might find the applications informative. But as a sheaf-theoretic framework, the paper overstates its case. I wouldn't cite it in its current form, and the sheaf terminology should be dropped or the residual redefined to something that genuinely depends on edge orientation.\n\nRecommendation: it deserves a serious referee, but with clear instruction to check the central derivation. The errors are fixable in a major revision that reframes the contribution; if the authors are unwilling to do that, I'd reject.","headline":"The sheaf machinery is decorative: the statistic actually used in applications is not a sheaf residual, and the sheaf residual itself reduces to Euclidean distance.","tokens_in":23034,"tokens_out":3912,"would_cite":false,"duration_ms":42170,"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":"This paper claims that the organization of vector fields on geometric graphs reduces to the normalized Shannon entropy of a sheaf residual energy field, and that this single scale-invariant descriptor exposes structure in proteins, brain…","keywords":["cellular sheaf","vector field on graph","graph entropy","residual energy","geometric graph","network organization","Shannon entropy","information quantification"],"falsifier":"On a fixed geometric graph with $|E|>1$, engineer two vector fields with identical total residual energy but with one field's residual energy placed on a single edge and the other spread uniformly; the framework predicts SheafIQ values of 0 and 1 respectively, so a failure to approach these bounds would falsify the entropy computation. For the complementary-information claim, regress $\\hat{H}$ on total residual energy, mean signal magnitude, and a spectral graph entropy across many random vector fields on one graph: residuals near zero would show SheafIQ carries no information beyond the descriptors it claims to complement.","tokens_in":22020,"feed_emoji":"🕸️","tokens_out":9282,"duration_ms":94868,"temperature":0.7,"pith_summary":"This paper proposes SheafIQ, a method that turns the pattern of local mismatches between neighboring vectors on a graph into a single normalized entropy between 0 and 1. The idea is to express each node's vector in the coordinate frame of each incident edge, splitting it into parts parallel and perpendicular to the edge, so that disagreements between neighbors are measured relative to the geometry of the connection rather than in a global coordinate system. The size of each disagreement becomes an edge energy; after normalizing those energies to a probability distribution, ordinary Shannon entropy divided by the log of the number of edges summarizes how concentrated or spread out the local incompatibilities are. The paper argues that this one number captures organizational information about the vector field that graph topology alone and signal magnitude alone do not, and it demonstrates the claim on protein motions, Alzheimer's brain networks, urban traffic, and power grids.","feed_headline":"One entropy score orders vector fields on geometric graphs","feed_subtitle":"A normalized Shannon entropy of sheaf residuals tracks protein mutations, brain disease, traffic, and grid faults.","key_machinery":"The load-bearing object is a cellular sheaf on the geometric graph whose edge stalks are $\\mathbb{R}^{2d}$ and whose restriction maps are the edge-induced coordinate transformations of Eq. (1): a node vector is sent to its signed projection onto the edge direction together with its orthogonal complement, where $u_{ij}$ is the unit direction of the edge. This map puts both endpoint vectors into a common frame, so the sheaf residual measures genuine local incompatibility rather than an ordinary coordinate difference. From the residual norms an edge energy field $\\varepsilon_{ij}=r_{ij}^2$ is built, normalized to a probability measure over edges, and the final descriptor is the Shannon entropy of that measure divided by $\\log|E|$. The machinery's work is to separate magnitude, captured by total residual energy, from organization, captured by the relative distribution, and to make the comparison scale-invariant and graph-size-invariant.","core_discovery":"The central claim is that a vector field on a geometrically embedded graph has a well-defined organizational signature: the normalized entropy of its sheaf residual energy distribution. For each edge, the two endpoint vectors are projected onto the edge direction and its orthogonal complement inside the edge stalk $\\mathbb{R}^{2d}$; the difference of those projections is the sheaf residual. Squaring a stable scalar version of that residual gives an edge energy, the energies are normalized to a probability measure, and $\\hat{H}=H/\\log|E|$ is the SheafIQ score. The paper proves scale invariance under uniform rescaling of all energies and establishes the bounds $0\\le \\hat{H}\\le 1$, with 0 meaning all incompatibility is concentrated on one edge and 1 meaning it is spread uniformly. Across the four application domains, the paper reports that this score and its associated residual hotspots reveal structure that conventional graph and signal descriptors miss.","pith_inferences":["Shannon entropy is permutation-invariant over edges, so SheafIQ cannot distinguish a clustered arrangement of high-residual edges from a scattered arrangement with the same histogram; a spatial variant weighted by graph distance would be a direct testable extension of the paper's own 'organization' language.","In the brain application (Section 3.3), the node vectors are never defined and q is never stated; if the node objects are time series or connectivity profiles rather than vectors in the same $\\mathbb{R}^d$ as the AAL coordinates, the residual, energy, and entropy in that section are undefined, and the claim there should be read as conditional on the $q=d$ premise.","The paper uses $r_{ij}$, which suppresses directional variation of the perpendicular components, rather than the full residual norm $\\|t_{ij}\\|$; comparing both versions on the same datasets would reveal how much rotational information the current score discards.","The parallel/orthogonal split suggests a natural decomposition into gradient-like and curl-like residual energies; one could define separate 'parallel entropy' and 'orthogonal entropy' to characterize flow versus rotational organization on the graph."],"forward_implications":["Vector fields on identical graphs can now be distinguished: where conventional graph entropy returns the same value for the same topology, SheafIQ changes with the organization of the node vectors.","Because the score is scale-invariant and normalized by $\\log|E|$, organizations can be compared across graphs and datasets of different sizes on a common [0,1] scale.","Low SheafIQ localizes incompatibility: in the applications it flags mutation hotspots, AD-associated brain regions, congested traffic states, and critical power-grid buses and lines, while high SheafIQ indicates broadly distributed residual energy.","The response to localized electrical faults is monotone in severity and nearly absent for coherent global changes, suggesting the score isolates truly local organizational disruptions.","Edge-wise residual energies remain available, so SheafIQ is not just a scalar summary: it also produces interpretable hotspot maps and communication backbone structures."],"supporting_citations":[{"why":"Supplies the spectral theory of cellular sheaves and the notion of global-section compatibility that SheafIQ's residual is built upon.","marker":"(27)"},{"why":"Supplies the sheaf Dirichlet energy formulation and the motivation for edge-stalk coordinate systems in graph neural settings.","marker":"(29)"},{"why":"Supplies the sheaf coboundary-operator view that identifies the residuals as components of the sheaf coboundary acting on the concatenated node vectors.","marker":"(35)"},{"why":"Supplies the quadratic potential function whose norm aggregates local sheaf inconsistencies into a global energy.","marker":"(36)"},{"why":"Provides the conventional graph-entropy baseline that SheafIQ explicitly contrasts with, since topology-only entropies ignore vector-field organization.","marker":"(12)"},{"why":"Provides the anisotropic network model used to turn protein residue fluctuations into displacement vector fields.","marker":"(48)"},{"why":"Supplies the METR-LA and PEMS-BAY traffic datasets and their sensor-network construction.","marker":"(72)"},{"why":"Supplies the IEEE 118- and 300-bus power system models and AC power-flow analysis used for the grid experiments.","marker":"(74)"},{"why":"Supplies the ADNI resting-state fMRI data for the Alzheimer's disease brain-network analysis.","marker":"(64)"}],"fun_headline_variants":["One entropy score quantifies vector fields on graphs","SheafIQ: entropy of sheaf residuals orders vector fields","A single number ranks vector-field organization on geometric graphs","Entropy of edge residuals orders vector fields on geometric graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every node vector lives in the same Euclidean space as the geometric embedding ($q=d$), so the edge direction $u_{ij}$ defines a valid frame for the parallel/orthogonal decomposition; in the brain application this premise is not demonstrated, and if it fails the entire residual-energy construction is undefined.","fun_headline_variants_meta":{"raw":{"variants":["One entropy score quantifies vector fields on graphs","SheafIQ: entropy of sheaf residuals orders vector fields","A single number ranks vector-field organization on geometric graphs","Entropy of edge residuals orders vector fields on geometric graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000834,"raw_usage":{"total_tokens":3604,"prompt_tokens":875,"completion_tokens":2729,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":2664}},"tokens_in":491,"tokens_out":2729,"duration_ms":23732,"temperature":1.0,"reasoning_tokens":2664,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:26:11.492886+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a fixed geometric graph with $|E|>1$, engineer two vector fields with identical total residual energy but with one field's residual energy placed on a single edge and the other spread uniformly; the framework predicts SheafIQ values of 0 and 1 respectively, so a failure to approach these bounds would falsify the entropy computation. For the complementary-information claim, regress $\\hat{H}$ on total residual energy, mean signal magnitude, and a spectral graph entropy across many random vector fields on one graph: residuals near zero would show SheafIQ carries no information beyond the descriptors it claims to complement.","supporting_citations":[],"review_version":1}