{"id":"3cce26e9-9607-4e2a-bb21-ebf49e68be76","arxiv_id":"2606.19200","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Holomorphic tensors on products of algebraic cones are invariant under contractions, implying invariance under Reeb flows on products of Sasaki manifolds via an explicit cone embedding.","lead":"The paper proves invariance of holomorphic tensors on quotients of products of algebraic cones under Zariski closures of contractions (for dim >=2), and extends this via an embedding to show that holomorphic tensors on products of Sasaki manifolds are invariant under Reeb field flows. A smart generalist might read it for new symmetry results linking algebraic geometry techniques to special metrics in complex and contact geometry.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Embedding of Sasaki cones into normal varieties must preserve tensor structures and Reeb-to-contraction correspondence for claim to follow","rationale":"The reader's weakest assumption identifies precisely the embedding step as the transfer point whose correctness is required for the central Sasaki claim. The algebraic cone result itself appears self-contained for its stated setting. Because the embedding details and the required equivariance/tensor correspondence are not visible in the provided abstract, the load-bearing risk remains at that interface; this does not alter the provisional UNVERDICTED status.","tokens_in":1554,"tokens_out":372,"duration_ms":33528,"concrete_test":"In the Sasaki manifolds section, extract the explicit embedding map; verify whether it is shown to be holomorphic, whether each Reeb field maps to a contraction whose Zariski closure is contained in the acting group, and whether a holomorphic tensor on the Sasaki product pulls back to an algebraic holomorphic tensor whose invariance under the algebraic action restricts to Reeb invariance on the product of links.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The algebraic result shows invariance of holomorphic tensors on quotients of the product cone C under the Zariski closure of contractions (when factors have dim ≥2). The Sasaki claim requires an explicit embedding of each Sasaki cone into a normal variety such that: (i) the product of embedded cones carries an algebraic group action whose contractions correspond to the two Reeb fields, (ii) holomorphic tensors on the product of Sasaki manifolds correspond to those on the algebraic quotient, and (iii) invariance under the algebraic Zariski closure implies invariance under the real Reeb flows on the links. The abstract states that the embedding is given and the application is direct, but provides no verification that the embedding is equivariant or that the tensor spaces match under restriction to the links.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the product C of two algebraic cones equipped with algebraic contractions. It proves that any holomorphic tensor on a quotient of C by a group containing a contraction on both factors is invariant under the Zariski closure of the contraction, provided the factors have dimension at least 2. It then constructs an explicit embedding of the cone of a Sasaki manifold into a normal variety and applies the algebraic result to conclude that any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields.","tokens_in":1703,"tokens_out":391,"duration_ms":24852,"significance":"If the embedding construction is verified to be equivariant and to preserve the relevant tensor structures and Reeb-contraction correspondence, the result would provide a concrete bridge between algebraic geometry results on cone quotients and Sasakian geometry, yielding new invariance statements for holomorphic tensors on products of Sasaki manifolds. The algebraic invariance theorem itself appears to rest on standard tools of the field and could be of independent interest for studying holomorphic sections on quotients of cone products.","major_comments":[{"comment":"The section on Sasaki manifolds: the central application rests on an explicit embedding of each Sasaki cone into a normal variety such that (i) the product of embedded cones carries an algebraic group action whose contractions correspond to the two Reeb fields, (ii) holomorphic tensors on the product of Sasaki manifolds correspond to those on the algebraic quotient, and (iii) invariance under the algebraic Zariski closure implies invariance under the real Reeb flows on the links. The manuscript states that the embedding is given and the application is direct, but provides no explicit verification that the embedding is equivariant or that the tensor spaces match under restriction to the links; this correspondence is load-bearing for the Sasaki claim.","section":"Sasaki manifolds section"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying the need for explicit verification in the Sasaki manifolds application. We agree that the correspondence between the algebraic embedding and the Sasakian structures requires detailed checks to be fully rigorous, and we will revise the manuscript to supply them.","responses":[{"response":"We acknowledge that while the manuscript constructs an explicit embedding of each Sasaki cone into a normal variety, the verification steps for equivariance of the embedding with respect to the algebraic group action, the precise identification of holomorphic tensor spaces under restriction to the links, and the translation of Zariski-closure invariance to real Reeb-flow invariance were stated as direct consequences without expanded checks. In the revised version we will insert a new subsection that carries out these verifications explicitly: (i) confirming that the embedding intertwines the algebraic contractions with the Reeb vector fields, (ii) showing that the restriction map induces an isomorphism between the relevant spaces of holomorphic tensors on the product of links and on the algebraic quotient, and (iii) explaining why invariance under the Zariski closure of the contractions implies invariance under the real one-parameter Reeb flows. This addition will make the load-bearing correspondence fully transparent.","revision_made":"yes","referee_comment":"The section on Sasaki manifolds: the central application rests on an explicit embedding of each Sasaki cone into a normal variety such that (i) the product of embedded cones carries an algebraic group action whose contractions correspond to the two Reeb fields, (ii) holomorphic tensors on the product of Sasaki manifolds correspond to those on the algebraic quotient, and (iii) invariance under the algebraic Zariski closure implies invariance under the real Reeb flows on the links. The manuscript states that the embedding is given and the application is direct, but provides no explicit verification that the embedding is equivariant or that the tensor spaces match under restriction to the links; this correspondence is load-bearing for the Sasaki claim."}],"tokens_in":1254,"tokens_out":413,"duration_ms":14495,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is that any holomorphic tensor on the product of two Sasaki manifolds is invariant under the Reeb flows. It gets there by first proving an algebraic statement: on the quotient of the product of two cones (each of dimension at least 2) by a group that includes contractions on both factors, holomorphic tensors are invariant under the Zariski closure of those contractions. Then it supplies an explicit embedding of a Sasaki cone into a normal variety and applies the algebraic fact.\n\nThe algebraic invariance step looks like standard material once the setup is fixed. The new piece is the embedding and the claim that it lets the Reeb flows correspond directly to the algebraic contractions so that the invariance transfers to the links.\n\nThe soft spot is exactly that transfer. The embedding must be equivariant enough that holomorphic tensors on the Sasaki product restrict to the algebraic quotient in a way that preserves the invariance, and the real Reeb vector fields must match the contractions on the quotient. The abstract presents this as direct, but the stress-test note correctly flags that without checking equivariance and the restriction to the links the implication does not automatically follow. If the full text supplies a clean verification of those points the argument holds; if it only sketches the embedding the claim rests on an unexamined step.\n\nNo obvious circularity or invented objects appear. The paper is aimed at people who work with Sasaki manifolds or with holomorphic tensors on cones and want an algebraic route to invariance statements. A reader already familiar with the algebraic cone literature will see the reduction quickly.\n\nIt is worth sending to a referee who can check the embedding construction and the precise correspondence between the geometric and algebraic sides. The result is narrow but cleanly stated, so a serious review makes sense.","headline":"The paper reduces Reeb-flow invariance for holomorphic tensors on Sasaki products to an algebraic cone result via an explicit embedding, but the correspondence between the algebraic and geometric structures is the part that needs verification.","tokens_in":2175,"tokens_out":433,"would_cite":false,"duration_ms":14793,"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":"Any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields.","keywords":["holomorphic tensors","algebraic cones","Sasaki manifolds","Reeb fields","invariance under contractions","Zariski closure","normal varieties","cone embeddings"],"falsifier":"A holomorphic tensor on the product of two Sasaki manifolds (for example the product of two three-spheres) that varies nontrivially along a Reeb orbit would falsify the claim.","tokens_in":2455,"feed_emoji":"","tokens_out":642,"duration_ms":23004,"temperature":0.7,"pith_summary":"The paper proves an invariance result for holomorphic tensors on quotients of products of algebraic cones under group actions that include contractions on both factors, provided the factors have dimension at least two; the tensors must be fixed by the Zariski closure of the contraction. It then constructs an explicit embedding of the cone of a Sasaki manifold into a normal variety. This embedding transfers the algebraic invariance directly, yielding the conclusion that holomorphic tensors on products of Sasaki manifolds remain fixed by Reeb field flows. A sympathetic reader would care because the result imposes a strong rigidity condition on the holomorphic geometry of these products and reduces questions about their tensors to orbitwise constancy.","feed_headline":"Tensors on Sasaki products fixed by Reeb flows","feed_subtitle":"Algebraic cone quotients yield invariance under contractions; an explicit embedding transfers it to show Reeb preservation for dimensions at","key_machinery":"The explicit embedding of the cone of a Sasaki manifold into a normal variety, which carries holomorphic tensors and allows the algebraic cone invariance result to apply directly to the Sasaki setting.","core_discovery":"Any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields. The proof proceeds by first establishing the corresponding invariance for quotients of products of algebraic cones under contractions, then using an explicit embedding of each Sasaki cone into a normal variety to reduce the Sasaki case to the algebraic one.","pith_inferences":["The same embedding technique might transfer other algebraic invariance statements to contact or Sasakian settings.","Products of more than two Sasaki manifolds could inherit similar invariance under simultaneous Reeb flows.","The result constrains the possible holomorphic sections of tensor bundles on these products, potentially simplifying computations of their cohomology."],"forward_implications":["Holomorphic tensors on such products must be constant along Reeb orbits when each factor has dimension at least two.","The invariance extends to the Zariski closure of the contraction actions on the algebraic cones.","The result applies to quotients by any group containing contractions on both cone factors.","Tensors on the product are determined by their values on a transverse slice to the Reeb flows."],"fun_headline_variants":["Reeb flows fix tensors on Sasaki products","Algebraic cones imply Reeb invariance for Sasaki tensors","Tensors invariant under Reeb flows on manifold products","Contractions preserve tensors on algebraic cone products"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The explicit embedding of the Sasaki cone into a normal variety preserves the holomorphic tensor structures so that the algebraic invariance result applies without additional adjustment.","fun_headline_variants_meta":{"raw":{"variants":["Reeb flows fix tensors on Sasaki products","Algebraic cones imply Reeb invariance for Sasaki tensors","Tensors invariant under Reeb flows on manifold products","Contractions preserve tensors on algebraic cone products"]},"model":"grok-4.3","cost_usd":0.003697,"raw_usage":{"total_tokens":1847,"prompt_tokens":521,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":36974500,"prompt_tokens_details":{"text_tokens":521,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1269,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":521,"tokens_out":57,"duration_ms":9889,"temperature":1.0,"reasoning_tokens":1269,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T19:05:45.295776+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A holomorphic tensor on the product of two Sasaki manifolds (for example the product of two three-spheres) that varies nontrivially along a Reeb orbit would falsify the claim.","supporting_citations":[],"review_version":1}