{"id":"0e6ff280-ec2e-420e-8a69-2531e705215a","arxiv_id":"2607.07503","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Newton polygons and zig-zag path correspondences are extended from bipartite to non-bipartite dimer models on torus graphs, with real-rootedness proven for marginal polynomials of several graph families.","lead":"This paper proves that Newton polygons of dimer models are preserved when degree-3 vertices in bipartite torus graphs are replaced by triangles, and that marginal polynomials for these non-bipartite graphs are real-rooted. A generalist might read it because it extends a key tool from bipartite statistical mechanics to non-bipartite settings relevant to the Ising model.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Theorem 39's real-rootedness argument has a gap: the factorization into degree-1 factors is asserted but not proven for general isoradial graphs.","rationale":"The reader correctly identified that the isoradiality assumption is load-bearing for the Newton polygon equality (Theorem 22), and that Theorem 39's proof is not fully formalized. However, the reader's framing of the weakest assumption centers on isoradiality for the inclusion N(Γ_ΔS) ⊆ N(Γ), which is actually well-handled: Lemma 15 uses isoradiality cleanly, Remark 23 explicitly acknowledges the limitation, and the double inclusion proof (Lemmas 16 + 21) is the soundest part of the paper. The more pressing concern is that Theorem 39's real-rootedness proof contains a genuine gap: the claim that only two path configurations exist in the general isoradial setting is diagrammatically sketched but not proven. This is distinct from the isoradiality restriction itself — even under isoradiality, the argument is incomplete. The reader noted that 'Theorem 39 relies on a structural sketch about path choices that is not fully formalized,' which is the right instinct, but did not pinpoint that the specific unproven claim is the uniqueness of the two path configurations through the triangle-decorated zig-zag paths. The concrete examples (triangular lattice in Proposition 35, hexagonal Fisher graph in Theorem 38) are fully rigorous and provide real evidence for the phenomenon. The local moves (Section 6) are cleanly proven. The Newton polygon equality (Theorem 22) and corner graph result (Theorem 2) are solid. The verdict remains CONDITIONAL: the paper's core structural results hold, but Theorem 39 needs either a completed proof of the path-rigidity claim or a more honest scoping to the cases where it can be verified. Confidence in this assessment is moderate-to-high for the structural results and moderate for identifying the Theorem 39 gap, as it requires specialist knowledge of isoradial graph combinatorics to fully verify whether the path-rigidity argument extends.","tokens_in":27265,"tokens_out":1233,"duration_ms":674205,"concrete_test":"Construct a specific isoradial bipartite torus graph Γ with at least two zig-zag paths of homology (0,1), where the vertices in V(Z_i) ∩ S have varying local degrees and neighborhoods (not all degree 3, not all on the hexagonal lattice). Compute the modified Kasteleyn matrix K(z,w) for Γ_ΔS explicitly, extract the marginal polynomial P_ver(w) corresponding to the vertical side of N(Γ_ΔS), and verify: (1) that P_ver(w) factors into degree-1 polynomials in w, and (2) that for each i, the path polynomial P_{U_i⁻→U_i⁺}(w) has degree exactly 1. If any factor has degree > 1 or does not have a real root, the claim of Theorem 39 fails for this graph. A concrete starting point: take a 2×2 fundamental domain of the square lattice (isoradial, bipartite, with degree-4 vertices that are not in V₃) and modify it by inserting degree-3 vertices along one zig-zag path, then apply the triangle replacement","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 39 claims that for any isoradial bipartite torus graph Γ and any S ⊆ V₃, the marginal polynomials of N(Γ_ΔS) are real-rooted. The proof strategy is to factor P_ver(w) = c · P_{U₁⁻→U₁⁺}(w) · … · P_{U_r⁻→U_r⁺}(w), where each factor has degree at most 1. The key claim is: 'the only two ways in which these paths can be connected are u_{i,1}→u_{i,2},…,u_{i,k-1}→u_{i,k} or u_{i,1}→u_{i,k},…,u_{i,k-1}→u_{i,k-2}.' This is asserted by a diagrammatic argument (Figure 16, right) showing that a 'forbidden path' from u_{i,1} to u_{i,4} leaves no valid partner for u_{i,2}. However, this argument is only sketched for one specific forbidden configuration and is not proven in general. The paths from U_i⁻ to U_i⁺ live in the subgraph of Γ_ΔS induced by the triangle replacements of vertices in V(Z_i) ∩ S, and the claim that exactly two path configurations exist requires showing that no other combination of disjoint paths through these triangles is realizable (in the sense of Lemma 19). The author acknowledges this is 'the same argument we had in Lemma 37(c)' but that lemma is specific to the Fisher graph of the hexagonal lattice, where the local structure is rigid. For a general isoradial bipartite graph, the neighborhood of each vertex in Z_i can be arbitrary (subject to isoradiality), and the argument that the combinatorial rigidity persists is not formalized. If a third path configuration exists for some isoradial graph, the factor P_{U_i⁻→U_i⁺}(w) could have degree > 1, and real-rootedness would no longer follow from the factorization. This is the most load-bearing concern because Theorem 39 is one of the two main results cited in the abstract, and its proof is incomplete as presented.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper studies Newton polygons for the dimer model on two families of non-bipartite torus graphs: (1) graphs obtained by replacing degree-3 vertices of a bipartite isoradial graph with triangles (Γ_ΔS), and (2) corner graphs (C_Γ). The main results (Theorems 1/22 and 2) establish that N(Γ_ΔS) = N(Γ) for isoradial bipartite Γ, and N(C_Γ) = 2N(Γ), with primitive edge vectors given by zig-zag path homology classes. The proofs proceed by double inclusion: the forward inclusion (Lemma 16) uses a counting argument (Lemma 15) relying on isoradiality, while the reverse inclusion (Lemma 21) uses realizability of extremal homology classes (Lemmas 19, 20). Section 5 addresses real-rootedness of marginal polynomials for the triangular lattice (Proposition 35), the hexagonal Fisher graph (Theorem 38), and a general statement for isoradial bipartite graphs (Theorem 39). Section 6 introduces new local moves (λ-move, diagonal spider move, cross move) preserving dimer partition functions and characteristic polynomials.","tokens_in":28198,"tokens_out":1590,"duration_ms":348009,"significance":"The paper extends the Newton polygon / zig-zag path correspondence from the well-studied bipartite dimer model to two natural non-bipartite families. The triangle-replacement construction Γ_ΔS is directly motivated by the Fisher/Ising correspondence, making the results relevant to understanding spectral curves of non-bipartite dimer models. The explicit factorizations of marginal polynomial roots for the triangular and hexagonal Fisher lattices (Proposition 35, Theorem 38) are concrete and verifiable. The new local moves in Section 6, including the non-planar cross move and its connection to the cube recurrence, are a genuine addition to the combinatorial toolkit. The double-inclusion proof strategy is clean and the isoradiality assumption is properly identified as load-bearing (Remark 23).","major_comments":[{"comment":"Theorem 39 (Section 5): The proof claims that 'the only two ways in which these paths can be connected are u_{i,1}→u_{i,2},…,u_{i,k-1}→u_{i,k} or u_{i,1}→u_{i,k},…,u_{i,k-1}→u_{i,k-2}.' This is justified by a diagrammatic argument (Figure 16, right) showing one forbidden configuration, and the author notes this is 'the same argument we had in Lemma 37(c).' However, Lemma 37 is specific to the Fisher graph of the hexagonal lattice, where the local structure is rigid. For a general isoradial bipartite graph, the neighborhood of each vertex in Z_i can be arbitrary (subject to isoradiality), and the claim that exactly two path configurations exist is not formally proven. If a third configuration exists for some isoradial graph, the factor P_{U_i^-→U_i^+}(w) could have degree > 1, and real-rootedness would not follow. This is load-bearing for Theorem 39's central claim. The author should要么 (a","section":null},{"comment":"Lemma 15 (Section 3): The proof argues that if a zig-zag path Z has homology (0,d) with d≠0, it would intersect Z_1 more than once, contradicting isoradiality. However, the argument that 'such a path would necessarily intersect Z_1 more than once within the fundamental domain' needs more justification. A path with homology (0,d) is homologous to a vertical path, but it could potentially wind around the torus in a way that avoids Z_1 except at shared edges. The isoradiality condition forbids two distinct zig-zag paths from intersecting more than once in the universal cover, but the argument should explicitly connect the homology class (0,d) to the geometric necessity of multiple intersections with Z_1. This step is load-bearing for Lemma 15, which is in turn load-bearing for Lemma 16 and Theorem 22.","section":null}],"minor_comments":[{"comment":"Section 2.2, Definition 9: The definition of isoradiality for general (non-bipartite) graphs is given as 'no self-intersecting zig-zag paths and no two zig-zag paths intersect more than once.' This is sometimes called 'minimal' in the bipartite literature. The relationship between the non-bipartite isoradiality defined here and the standard rhombic embedding / isoradial embedding notion should be clarified.","section":null},{"comment":"Theorem 1 vs Theorem 22: Theorem 1 in the introduction states the result for 'isoradial bipartite torus graph,' while Theorem 22 in Section 3 uses 'isoradial bipartite graph.' These should be consistent. Also, Theorem 1 adds the claim about primitive edge vectors for 3-valent graphs, which appears as Corollary 24 in the body. The relationship should be stated explicitly.","section":null},{"comment":"Section 4.3 (Fisher graphs): The results N(Γ)⊆N(F_Γ)⊆2N(Γ) are stated without proof ('we omit the proofs of these results'). While the author notes reliance on previous techniques, for a journal publication these proofs should be included, or the results should be clearly labeled as conjectures/observations rather than propositions (Proposition 34).","section":null},{"comment":"Figure 16: The left and right panels use color (blue, red, green) to distinguish edge types and paths. For print accessibility, consider adding labels or patterns so the figure can be parsed without color.","section":null},{"comment":"Section 5, notation: The polynomials P_R, P_U are introduced in Section 5.1 for the triangular lattice and reused in Section 5.2 for the Fisher graph with potentially different meanings. Clarifying the scope of notation would help.","section":null},{"comment":"Proposition 42 (Section 6.1): The case analysis is thorough but lengthy. A summary table mapping each case to its contribution would improve readability.","section":null},{"comment":"Section 6.2: The connection to the cube recurrence and Kuo condensation is mentioned briefly. A reference to the specific form of Kuo condensation being invoked would strengthen this remark.","section":null},{"comment":"References: Reference [7] (de Tilière and Rey) is dated 2026 with arXiv number 2602.13526. This appears to be a forward-dated preprint. The author should verify this reference is accessible and correctly cited.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Theorem 39 is well-founded. The gap is not that the result is necessarily wrong, but that the proof for general isoradial graphs relies on an argument verified only for the hexagonal Fisher lattice. The author may be able to fix this by either restricting Theorem 39 to specific lattices (where it is proven) or by providing a general argument. The rest of the paper (Theorems 1/22, 2, and the local moves) appears sound. Proposition 34's unproven status is also worth flagging to the author—either prove it or reframe it. The paper is a strong contribution overall and should be publishable after revision."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and for identifying two points in the proofs that require additional justification. Both points concern gaps in the argumentation that we will address in revision: one in the proof of Theorem 39 (general isoradial case) and one in the proof of Lemma 15 (intersection count). We agree with the referee that both are load-bearing and will revise accordingly.","responses":[{"response":"The referee is correct that the argument as written does not adequately justify the claim for general isoradial bipartite graphs. The reference to Lemma 37(c) is insufficient because that lemma exploits the rigid local structure of the hexagonal Fisher graph, which does not transfer to the general isoradial setting. We have identified the key missing ingredient: the argument needs the fact that the curve γ_x intersects each zig-zag path Z_i exactly once, and that the triangles replacing vertices of S are sufficiently small so that γ_x crosses at most one edge in the corresponding portion of Γ_{ΔS}. This geometric constraint, combined with the requirement that the paths from U_i^- to U_i^+ must be disjoint and realizable within L(Γ_{ΔS}), is what forces the degree of P_{U_i^-→U_i^+}(w) to be at most one. The referee's concern about a potential third configuration is well-taken: in the current draft, we only exhibit one forbidden configuration (Figure 16, right) and do not systematically rule out all others. In the revision, we will provide a complete case analysis showing that any configuration other than the two stated would either (i) force a path to cross γ_x more than once, contradicting the degree-one bound, or (ii) create a vertex of degree 3 in the collection of loops, which is forbidden. We will also add a remark clarifying that the isoradiality condition enters through the edge-disjointness of zig-zag paths with the same homology class, which ensures the choices for different Z_i are independent. If, upon completing this analysis, we find that the general case cannot be fully justified, we will restrict Theorem 39 to the families where the proof is complete (triangular lattice, hexagonal Fisher graph, and graphs where all vertices of Z_i lie in S) and state the ","revision_made":"no","referee_comment":"Theorem 39 (Section 5): The proof claims that 'the only two ways in which these paths can be connected are u_{i,1}→u_{i,2},…,u_{i,k-1}→u_{i,k} or u_{i,1}→u_{i,k},…,u_{i,k-1}→u_{i,k-2}.' This is justified by a diagrammatic argument (Figure 16, right) showing one forbidden configuration, and the author notes this is 'the same argument we had in Lemma 37(c).' However, Lemma 37 is specific to the Fisher graph of the hexagonal lattice, where the local structure is rigid. For a general isoradial bipartite graph, the neighborhood of each vertex in Z_i can be arbitrary (subject to isoradiality), and the claim that exactly two path configurations exist is not formally proven. If a third configuration exists for some isoradial graph, the factor P_{U_i^-→U_i^+}(w) could have degree > 1, and real-rootedness would not follow. This is load-bearing for Theorem 39's central claim. The author should (a)"},{"response":"The referee identifies a genuine gap in the proof of Lemma 15. The claim that a zig-zag path with homology (0,d), d≠0, must intersect Z_1 more than once is stated without sufficient justification. The key observation is that after applying Lemma 14, the fundamental domain is chosen so that the vertical boundary crosses every edge of Z_1 and no other edges of Γ. A zig-zag path Z with homology (0,d) is homologous to d times the vertical generator, meaning it must cross the vertical boundary of the fundamental domain at least |d| times (with appropriate signs). Since the only edges crossing the vertical boundary are edges of Z_1, the path Z must share at least |d| edges with Z_1 in the fundamental domain. When we lift to the universal cover, each such shared edge gives rise to a distinct intersection between a lift of Z and a lift of Z_1. If |d| ≥ 2, this yields at least two intersections in the universal cover, contradicting isoradiality. If |d| = 1, then Z is homologous to Z_1, and one must argue that two distinct zig-zag paths with the same homology class (0,1) that share an edge must in fact coincide (since they both turn maximally left and right at each vertex), which would make them the same path, not two distinct paths. We agree that this chain of reasoning is not spelled out in the current draft and will add it explicitly in the revision.","revision_made":"no","referee_comment":"Lemma 15 (Section 3): The proof argues that if a zig-zag path Z has homology (0,d) with d≠0, it would intersect Z_1 more than once, contradicting isoradiality. However, the argument that 'such a path would necessarily intersect Z_1 more than once within the fundamental domain' needs more justification. A path with homology (0,d) is homologous to a vertical path, but it could potentially wind around the torus in a way that avoids Z_1 except at shared edges. The isoradiality condition forbids two distinct zig-zag paths from intersecting more than once in the universal cover, but the argument should explicitly connect the homology class (0,d) to the geometric necessity of multiple intersections with Z_1. This step is load-bearing for Lemma 15, which is in turn load-bearing for Lemma 16 and Theorem 22."}],"tokens_in":27271,"tokens_out":1272,"duration_ms":496639,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper extends the Goncharov–Kenyon zig-zag/Newton polygon correspondence to two families of non-bipartite graphs — triangle-replaced graphs and corner graphs — and proves real-rootedness of marginal polynomials for specific lattices. The structural results (Theorems 1/22 and 2/32) are clean and correct. The general real-rootedness claim (Theorem 39) has an incomplete proof at its load-bearing step. The local moves in Section 6 are correct and self-contained but are a secondary contribution. The reader's verdict of CONDITIONAL is right; the stress-test concern about Theorem 39 lands and is the central issue. The reader's MODERATE confidence is appropriate — the core arguments are checkable but Theorem 39 needs specialist verification that the paper does not currently provide. The reader correctly identifies the isoradiality restriction as necessary (Remark 23's counterexample confirms this). The reader's concern about omitted proofs in Section 4.3 is valid but minor — those results are presented as observations with explicit bounds, not as main theorems. The Proposition 34 genericity caveat is correctly flagged but is a minor issue since the main results do not depend on it. The stress-test concern about Theorem 39 is the real problem. The proof claims that for a general isoradial bipartite graph, the disjoint paths from U_i^- to U_i^+ admit exactly two configurations, each giving a degree-1 factor. The author argues this by a diagrammatic case analysis (Figure 16, right) showing that a 'forbidden path' from u_{i,1} to u_{i,4} leaves no valid partner for u_{i,2}. This argument is sketched for one forbidden configuration and then generalized by analogy to Lemma 37(c), but Lemma 37(c) is specific to the Fisher graph of the hexagonal lattice where the local structure is rigid. For a general isoradial bipartite graph, the neighborhood of each vertex in Z_i can be arbitrary (subject to isoradiality), and the claim that exactly two path configurations exist is not proven. If a third configuration exists for some isoradial graph, the factor P_{U_i^- -> U_i^+}(w) could have degree > 1, and real-rootedness would not follow. This is the most load-bearing concern because Theorem 39 is one of the two main results cited in the abstract, and its proof is incomplete as presented. The paper is for specialists in dimer models and integrable systems who work with Newton polygons and characteristic polynomials. Readers interested in the structural results get a complete, well-motivated treatment. Readers interested in the real-rootedness story get verified results for the triangular and Fisher lattices (Theorems 36, 38) but an incomplete general argument. The local moves (Lemmas 40, 41, 43) are cleanly proven and may be of independent interest to combinatorialists working on dimer equivalences. The paper deserves a serious referee. The structural results are correct and represent a genuine extension of known theory. The specific-lattice real-rootedness results are verified by direct computation. The general real-rootedness claim needs either a complete proof of the path-rigidity claim or a restriction to graphs where the local structure is sufficiently rigid. A referee with expertise in isoradial graphs and dimer characteristic polynomials should verify whether the path-configuration claim in Theorem 39 can be formalized or whether a counterexample exists.","headline":"Solid extension of Newton polygon / zig-zag correspondence to non-bipartite dimer models; Theorem 39's general real-rootedness proof has a genuine gap","tokens_in":28189,"tokens_out":827,"would_cite":false,"duration_ms":177128,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Non-bipartite dimer Newton polygons equal their bipartite origins","keywords":[],"falsifier":"A non-isoradial bipartite graph where replacing a degree-3 vertex with a triangle changes the Newton polygon (the author provides one in Remark 23), or a non-bipartite isoradial graph whose marginal polynomial has a non-real root.","tokens_in":27430,"feed_emoji":"🔺","tokens_out":1026,"duration_ms":132715,"temperature":0.7,"pith_summary":"The dimer model's algebraic geometry—Newton polygons, spectral curves, marginal polynomials—has been developed primarily for bipartite graphs, where a clean bipartite Kasteleyn matrix ensures all contributions to a given monomial share the same sign. This paper extends the theory to two families of non-bipartite graphs on the torus: graphs obtained by replacing degree-3 vertices of a bipartite graph with triangles (the blow-up operation, denoted Γ_ΔS), and corner graphs (C_Γ), where each vertex-face incidence becomes a vertex. The central discovery is that for isoradial bipartite graphs, the Newton polygon is invariant under the triangle blow-up: N(Γ_ΔS) = N(Γ) for any subset S of degree-3 vertices. For corner graphs, the polygon doubles: N(C_Γ) = 2N(Γ). In both cases, the primitive edge vectors of the Newton polygon are exactly the homology classes of the zig-zag paths. The paper then proves that the marginal polynomials (restrictions to boundary sides of the polygon) are real-rooted for these non-bipartite families, with explicit factorizations for the triangular lattice and the hexagonal Fisher graph. Finally, the paper introduces new local moves—the λ-move, diagonal spider move, and a non-planar cross move—that preserve the dimer partition function up to scale, extending the equivalence toolkit beyond the bipartite planar setting.","feed_headline":"Non-bipartite dimer Newton polygons equal their bipartite origins","feed_subtitle":"Triangle blow-ups and corner graphs on isoradial lattices preserve the polygon shape and yield real-rooted marginal polynomials.","key_machinery":"isoradial graph; zig-zag path; Newton polygon; characteristic polynomial; Kasteleyn matrix; triangle blow-up (Γ_Δ); corner graph (C_Γ); marginal polynomial; local moves (λ-move, diagonal spider move, cross move)","core_discovery":"The Newton polygon of a non-bipartite dimer model can be identical to that of its underlying bipartite graph, provided the graph is isoradial. The triangle blow-up operation Γ → Γ_ΔS preserves the polygon because zig-zag path homology classes are in bijection between the two graphs (Proposition 12), and the isoradiality condition ensures that boundary-crossing counts—which bound the polygon—are preserved (Lemma 15). For corner graphs, the polygon scales by a factor of 2 because each zig-zag path in Γ gives rise to two zig-zag paths in C_Γ with the same homology (Proposition 26). The marginal polynomials are real-rooted because the loop structure on boundary sides of the polygon forces a comb","pith_inferences":[],"forward_implications":["The equality N(Γ_ΔS) = N(Γ) means the phase diagram (smooth, frozen, rough regions) of the non-bipartite dimer model on Γ_ΔS is the same as that of the bipartite model on Γ, extending limit-shape analysis to non-bipartite settings.","Real-rootedness of marginal polynomials for non-bipartite graphs is a necessary condition for the spectral curves to be simple Harnack curves; this paper establishes that condition for these families, partially answering the open question of spectral curve classification in the non-bipartite case.","The new local moves (λ-move, diagonal spider move, cross move) enlarge the equivalence classes of graphs sharing a characteristic polynomial, meaning results about Newton polygons and partition functions transfer to a broader family of non-bipartite and even non-planar graphs.","The cross move's connection to the cube recurrence and Kuo condensation suggests a route to enumerating perfect matchings of non-planar analogs of the Aztec diamond.","The intermediate containment N(Γ) ⊆ N(F_Γ) ⊆ 2N(Γ) for Fisher graphs suggests a graded structure where the Newton polygon interpolates between the original and doubled polygon, depending on the decoration."],"fun_headline_variants":["Non-bipartite dimer Newton polygons match bipartite originals","Triangle blow-ups preserve Newton polygons via zig-zag homology","Corner graphs double Newton polygons under isoradiality","Marginal polynomials of non-bipartite dimers are real-rooted","Zig-zag path homology classes determine Newton polygon edges"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The isoradiality assumption on the underlying bipartite graph is load-bearing: it ensures zig-zag paths intersect at most once in the universal cover, which is used to bound boundary crossings and prove the polygon inclusion N(Γ_ΔS) ⊆ N(Γ). The author explicitly notes that this inclusion can fail for non-isoradial graphs, exhibiting a counterexample.","fun_headline_variants_meta":{"raw":{"variants":["Non-bipartite dimer Newton polygons match bipartite originals","Triangle blow-ups preserve Newton polygons via zig-zag homology","Corner graphs double Newton polygons under isoradiality","Marginal polynomials of non-bipartite dimers are real-rooted","Zig-zag path homology classes determine Newton polygon edges"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":643,"prompt_tokens":553,"completion_tokens":90,"prompt_tokens_details":null},"tokens_in":553,"tokens_out":90,"duration_ms":17771,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T08:49:28.463296+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A non-isoradial bipartite graph where replacing a degree-3 vertex with a triangle changes the Newton polygon (the author provides one in Remark 23), or a non-bipartite isoradial graph whose marginal polynomial has a non-real root.","supporting_citations":[],"review_version":1}