{"id":"55c1ec01-30bb-4c68-b59a-5f4e52d8adc1","arxiv_id":"2606.30792","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Local polynomial convexity at an isolated singularity suffices for Carleman approximation on unions of transverse totally real subspaces, with additional conditions for three planes in C^2 and Lipschitz graphs.","lead":"The paper proves that local polynomial convexity at the origin for unions of finitely many transverse totally real subspaces suffices for Carleman approximation. A smart generalist might read it to see how isolated singularities affect holomorphic approximation in several complex variables.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption directly matches the hypotheses required by the strongest_claim in the abstract. The paper's additional results on three planes and Lipschitz graphs are presented as separate contributions and do not appear to prop up the main sufficiency statement.","tokens_in":1578,"tokens_out":215,"duration_ms":33248,"concrete_test":"Confirm that the main sufficiency theorem (as stated in the introduction or §1) invokes only local polynomial convexity plus the three listed conditions, with no additional global polynomial convexity or non-transversality hypotheses appearing in the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim states that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension suffices for Carleman approximation. The listed conditions (transversality, maximal dimension, isolated singularity) are explicitly required by the claim, and no internal gap, unstated hypothesis, or failure of a supporting lemma is apparent from the argument structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to prove that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is sufficient for Carleman approximation. It provides new conditions for the polynomial convexity of the union of three transverse totally real planes in C^2, sufficient conditions for the union of two Lipschitz graphs to be polynomially convex and to allow Carleman approximation, and a family of surfaces in C^2 with a hyperbolic complex point that permits Carleman approximation.","tokens_in":1609,"tokens_out":279,"duration_ms":27347,"significance":"If the central sufficiency result holds, it would extend Carleman approximation theory to certain isolated singularities in several complex variables by linking it to local polynomial convexity under transversality and maximality conditions. The additional results on three planes, Lipschitz graphs, and hyperbolic points would supply concrete criteria in C^2, potentially useful for further work on approximation on singular sets.","major_comments":[],"minor_comments":[{"comment":"The abstract states the main sufficiency result but does not indicate whether the proofs rely on any previously unpublished lemmas or reductions that would require separate verification.","section":null}],"recommendation":"uncertain","confidential_remarks":"Full manuscript text was referenced as available but not supplied in the review package; only the abstract could be examined, preventing any technical verification of the derivations or examples."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their review and summary of the manuscript. The recommendation is listed as uncertain, but the report contains no specific major comments or questions to address point by point. We appreciate the referee's recognition of the potential significance of linking local polynomial convexity to Carleman approximation under the stated transversality and maximality conditions. If any concrete concerns arise, we are happy to provide further clarification or revisions.","responses":[],"tokens_in":1108,"tokens_out":100,"duration_ms":14581,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result states that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is enough for Carleman approximation. They add explicit conditions for three such planes in C^2, a sufficient condition for two Lipschitz graphs, and a family of surfaces with hyperbolic complex points that also work.\n\nThe paper does a reasonable job of spelling out these conditions and tying them to the isolated singularity setup. The Lipschitz graph part and the hyperbolic examples give concrete cases that could be checked against known examples in the literature.\n\nThe soft spots are the narrow scope and the modest step forward. Everything requires transversality, maximal dimension, and an isolated singularity, which keeps the results inside a small corner of several complex variables. How much of the three-plane and graph conditions is genuinely new versus a direct extension of earlier polynomial convexity work is not obvious from the abstract alone, though the claims are presented as new.\n\nThe logic in the stated results lines up without internal contradictions or unstated gaps that jump out. This is aimed at specialists already working on Carleman approximation and polynomial convexity for singular sets. A reader in that niche might pick up a usable condition or two.\n\nI would send it for peer review. The claims are precise enough for referees to evaluate against the existing literature.","headline":"This paper gives a few narrow sufficiency conditions for Carleman approximation on unions of transverse totally real subspaces and Lipschitz graphs in C^2, but the advance stays incremental and specialized.","tokens_in":2065,"tokens_out":345,"would_cite":false,"duration_ms":35203,"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":"Local polynomial convexity at the origin suffices for Carleman approximation on unions of finitely many transverse totally real subspaces of maximal dimension.","keywords":["Carleman approximation","polynomial convexity","totally real subspaces","isolated singularity","Lipschitz graphs","hyperbolic complex points","several complex variables"],"falsifier":"An explicit union of transverse totally real subspaces of maximal dimension that is locally polynomially convex at the origin yet fails to admit Carleman approximation.","tokens_in":2479,"feed_emoji":"📐","tokens_out":637,"duration_ms":31722,"temperature":0.7,"pith_summary":"The paper establishes sufficiency of local polynomial convexity at an isolated singularity for Carleman approximation on unions of transverse totally real subspaces. It supplies additional conditions that guarantee polynomial convexity for the union of three such planes in C squared and for unions of two Lipschitz graphs. A concrete family of surfaces in C squared that contain a hyperbolic complex point is shown to satisfy the approximation property. A sympathetic reader would care because the result enlarges the class of sets on which continuous functions can be approximated by holomorphic functions even when smoothness fails at one point.","feed_headline":"Local polynomial convexity suffices for Carleman approximation","feed_subtitle":"Unions of transverse totally real subspaces with an isolated singularity at the origin admit approximation by holomorphic functions under th","key_machinery":"The local polynomial convexity condition at the isolated singularity, applied to transverse totally real subspaces of maximal dimension.","core_discovery":"The authors prove that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is sufficient for Carleman approximation. They give new conditions for the polynomial convexity of the union of three transverse totally real planes in C squared. They also provide a sufficient condition on the union of two Lipschitz graphs for Carleman approximation together with sufficient conditions for such unions to be polynomially convex, and they exhibit a family of surfaces in C squared with a hyperbolic complex point that allows Carleman approximation.","pith_inferences":["The sufficiency result may extend without change to unions involving more than three subspaces in dimensions higher than two.","Similar local-convexity criteria could apply to approximation questions on other isolated singularities that are not unions of linear subspaces.","Explicit parametrizations of the surfaces with hyperbolic points could be used to test whether the approximation rate can be made quantitative."],"forward_implications":["Carleman approximation holds for any such union once local polynomial convexity at the origin is verified.","The union of three transverse totally real planes in C squared is polynomially convex under the new conditions supplied.","Unions of two Lipschitz graphs admit Carleman approximation under the stated sufficient condition.","The exhibited family of surfaces in C squared with a hyperbolic complex point permits Carleman approximation."],"fun_headline_variants":["Polynomial convexity allows Carleman approximation","Transverse subspaces admit Carleman approximation","New conditions for convexity of three planes","Lipschitz graphs allow Carleman approximation","Hyperbolic points allow Carleman approximation"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The subspaces must be transverse, of maximal dimension, and the singularity at the origin must be isolated.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial convexity allows Carleman approximation","Transverse subspaces admit Carleman approximation","New conditions for convexity of three planes","Lipschitz graphs allow Carleman approximation","Hyperbolic points allow Carleman approximation"]},"model":"grok-4.3","cost_usd":0.00701,"raw_usage":{"total_tokens":3199,"prompt_tokens":575,"num_sources_used":0,"completion_tokens":51,"cost_in_usd_ticks":70099500,"prompt_tokens_details":{"text_tokens":575,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2573,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":575,"tokens_out":51,"duration_ms":23220,"temperature":1.0,"reasoning_tokens":2573,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T01:38:38.363338+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit union of transverse totally real subspaces of maximal dimension that is locally polynomially convex at the origin yet fails to admit Carleman approximation.","supporting_citations":[],"review_version":1}