{"id":"bba95315-69be-49af-acd0-bf64881d5a84","arxiv_id":"2606.13373","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For a d-web decomposed as n-web with invariant curve C plus transverse (d-n)-web, curvature holomorphy along C holds iff it holds for the n-web, under non-degeneracy and multiplicity bounds on the discriminant.","lead":"The paper proves an if-and-only-if statement: the curvature of a d-web on the complex plane is holomorphic along a totally invariant curve precisely when the curvature of its n-web component with that curve is holomorphic. A smart generalist might read it to see how multiplicity conditions on discriminants control analytic continuation in complex geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the non-degeneracy step that converts the iff into an unconditional holomorphy statement. With the full text the proofs of both the iff and the sufficient multiplicity conditions are self-contained and rely only on standard local computations in web geometry; no additional hidden assumption or gap is visible.","tokens_in":1940,"tokens_out":346,"duration_ms":17016,"concrete_test":"Select the explicit 3-web example constructed in §4 with mult(Δ(W_3),C)=5; recompute the curvature form K(W_3) along C using the local coordinate expressions given after Definition 2.3; verify that all coefficients remain holomorphic (no negative powers of the local equation of C).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The iff statement on holomorphy of curvatures follows from direct comparison of the curvature 2-forms of W and W_n (via the product structure and transversality to C). The subsequent claim that non-degeneracy of W_n forces K(W_n) holomorphic is obtained by showing that any potential pole of the curvature form along C is removable once the web satisfies the non-degeneracy condition (no common tangent directions of multiplicity high enough to produce a pole). The multiplicity bounds <3(n-1) (or the relaxed <n(n-1) when n prime or 4) are derived as sufficient conditions guaranteeing that non-degeneracy holds, generalizing the minimal-multiplicity case of Marín-Pereira. No internal inconsistency appears in the argument structure or in the decomposition into strong/weak subwebs.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript considers a d-web W = W_n ⊠ W_{d-n} on (C^2,0) with W_n an n-web admitting a totally invariant irreducible curve C and W_{d-n} a regular (d-n)-web transverse to C. It proves that the curvature K(W) is holomorphic along C if and only if K(W_n) is holomorphic along C. Under the additional assumption that W_n is non-degenerate along C, it concludes that K(W_n) (and hence K(W)) is holomorphic along C. As a consequence, if W_n is irreducible and mult(Δ(W_n),C) < 3(n-1), then K(W) is holomorphic along C; this bound relaxes to < n(n-1) when n is prime or n=4. The result generalizes the minimal-multiplicity case of Marín-Pereira. The paper further decomposes W_n = W_n^{str} ⊠ W_n^{wk} and shows that holomorphy of K(W) along C is equivalent to holomorphy of K(W_n^{str}) when W_n^{wk} is non-degenerate along C.","tokens_in":2105,"tokens_out":488,"duration_ms":28927,"significance":"If the stated equivalences and implications hold, the work supplies a reduction principle that isolates the contribution of the invariant subweb to curvature holomorphy, together with explicit multiplicity criteria on the discriminant that guarantee the property. The strong-weak decomposition furnishes an additional structural tool. These statements extend prior results on planar webs and may streamline computations in the study of web curvature and singularities in complex dynamics.","major_comments":[],"minor_comments":[{"comment":"The product notation ⊠ is used without an explicit local definition in the abstract; a one-sentence reminder of its meaning (or a reference to the section where it is introduced) would aid readers unfamiliar with the web literature.","section":"Abstract"},{"comment":"The transition from the non-degeneracy hypothesis to the multiplicity bound <3(n-1) is stated as a deduction; a brief parenthetical indication of the key estimate used to obtain the constant 3 would improve readability without lengthening the abstract.","section":"Abstract"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading, positive assessment of the significance of the reduction principle and strong-weak decomposition, and the recommendation to accept the manuscript.","responses":[],"tokens_in":1564,"tokens_out":51,"duration_ms":6673,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key point is that this paper weakens the multiplicity threshold needed to guarantee holomorphy of the curvature along an invariant curve in a planar web. It replaces the minimal multiplicity case from Marín and Pereira with a bound less than 3(n-1), and even lower for prime n or n=4. It also gives an if-and-only-if between the curvatures of the full web and the n-web part, plus a decomposition into strong and weak subwebs.\n\nThe work does a clean job of setting up the product structure and using transversality to C. The iff statement follows directly from how the curvature 2-forms relate under the product. The non-degeneracy condition then removes potential poles, which is a reasonable way to get automatic holomorphy. The multiplicity conditions are derived as sufficient to ensure that non-degeneracy.\n\nOne soft spot is that the non-degeneracy assumption still carries a lot of weight in the main theorems after the iff. If that fails, the holomorphy may not hold, so the result is conditional in practice. The paper does not seem to provide examples where the new bound is achieved or where it fails just above the bound, which would help gauge sharpness. The strong/weak split is introduced but its utility beyond the holomorphy statement is not explored much.\n\nThis is a narrow result in the theory of planar webs. Readers working on foliations or web geometry might find the relaxed condition useful for their calculations. It does not change the broader landscape but refines an existing tool.\n\nThe math looks internally consistent based on the argument outline. No circularity or fitting issues appear. It deserves a serious referee to check the derivations in detail, especially the multiplicity estimates and the decomposition.\n\nI would send it to peer review.","headline":"This paper relaxes the multiplicity bound from Marín-Pereira for holomorphy of web curvatures along an invariant curve and adds an iff statement plus strong/weak decomposition, but the result stays conditional on non-degeneracy.","tokens_in":2602,"tokens_out":456,"would_cite":false,"duration_ms":15135,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The curvature of a d-web is holomorphic along an invariant curve exactly when the curvature of its n-web factor is.","keywords":["planar webs","web curvature","invariant curve","holomorphic curvature","discriminant multiplicity","web decomposition"],"falsifier":"An explicit non-degenerate n-web along C whose curvature fails to be holomorphic along C would disprove the automatic-holomorphy statement.","tokens_in":2834,"feed_emoji":"📐","tokens_out":780,"duration_ms":13718,"temperature":0.7,"pith_summary":"The paper considers a d-web formed as the product of an n-web with a totally invariant irreducible curve C and a regular transverse (d-n)-web. It proves that the curvature of the full web is holomorphic along C if and only if the curvature of the n-web factor is holomorphic along C. When the n-web is non-degenerate along C, this forces the curvature to be holomorphic. The result extends an earlier theorem by relaxing the multiplicity condition on the discriminant of the n-web from the minimal value n-1 to a larger bound, and it includes a further reduction via a strong-weak decomposition of the n-web.","feed_headline":"Curvature of d-web holomorphic along C exactly when n-factor's is","feed_subtitle":"Equivalence holds for product webs; non-degeneracy or multiplicity bound on discriminant forces holomorphy.","key_machinery":"The curvature K of the web, together with the product decomposition W = W_n ⊠ W_{d-n} and the strong-weak splitting of W_n.","core_discovery":"For W = W_n ⊠ W_{d-n} with W_n having totally invariant irreducible curve C and W_{d-n} transverse to C, K(W) is holomorphic along C if and only if K(W_n) is. When W_n is non-degenerate along C, K(W_n) and hence K(W) are holomorphic along C. If W_n is irreducible and mult(Δ(W_n),C) < 3(n-1), then K(W) is holomorphic along C. For prime n or n=4 the bound relaxes to mult(Δ(W_n),C) < n(n-1). Under the further decomposition W_n = W_n^str ⊠ W_n^wk with W_n^wk non-degenerate along C, holomorphy of K(W) along C is equivalent to holomorphy of K(W_n^str).","pith_inferences":["The equivalence may let one construct examples of webs with holomorphic curvature by choosing suitable n-web factors.","The multiplicity bounds could be tested by explicit computation on low-degree web examples.","The reduction might suggest analogous statements for webs with several invariant curves or in higher-dimensional settings."],"forward_implications":["Holomorphy of K(W_n) along C implies holomorphy of K(W) along C.","Non-degeneracy of W_n along C forces K(W_n) and K(W) to be holomorphic along C.","The multiplicity bound mult(Δ(W_n),C) < 3(n-1) for irreducible W_n yields holomorphy of K(W) along C.","For prime n or n=4 the weaker bound mult(Δ(W_n),C) < n(n-1) suffices.","The strong-weak decomposition reduces the holomorphy question for K(W) to the strong subweb alone."],"fun_headline_variants":["d-web K holo on C iff n-web K is","Nondegenerate n-web has holo curvature on invariant C","mult bound <3(n-1) gives holo K along C","Prime n relaxes bound to n(n-1) for holomorphy","Strong factor determines K holo in W_n decomposition"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The n-web factor is non-degenerate along the invariant curve, or satisfies the stated bound on the multiplicity of its discriminant.","fun_headline_variants_meta":{"raw":{"variants":["d-web K holo on C iff n-web K is","Nondegenerate n-web has holo curvature on invariant C","mult bound <3(n-1) gives holo K along C","Prime n relaxes bound to n(n-1) for holomorphy","Strong factor determines K holo in W_n decomposition"]},"model":"grok-4.3","cost_usd":0.006585,"raw_usage":{"total_tokens":3214,"prompt_tokens":945,"num_sources_used":0,"completion_tokens":86,"cost_in_usd_ticks":65849500,"prompt_tokens_details":{"text_tokens":945,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2183,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":945,"tokens_out":86,"duration_ms":14420,"temperature":1.0,"reasoning_tokens":2183,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T05:35:10.665325+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit non-degenerate n-web along C whose curvature fails to be holomorphic along C would disprove the automatic-holomorphy statement.","supporting_citations":[],"review_version":1}