{"id":"5502adbd-e82e-4300-aa73-9cdfe0e3aa23","arxiv_id":"2607.27441","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"On any compact toric contact manifold of Reeb type, T-invariant CR structures realize both positive and negative CR Yamabe invariants, via a polytope PDE reduction.","lead":"Toric contact manifolds can carry CR structures whose Yamabe energy is positive, zero, or negative. The paper reduces the torus-equivariant CR Yamabe problem to a PDE on a polytope and proves sign-changing examples exist in every dimension.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the order condition in Case 2 as the sole technical caveat and notes that it is automatic in dimension 3. Because that condition can always be arranged by a compactly-supported high-order perturbation (preserving the Guillemin boundary asymptotics), the existence statements of Cors 1.5–1.6 go through in all dimensions. The remainder of the argument—dictionary to the polytope PDE, integration-by-parts formula for EH, test-function limits giving limsup Y≤0 or −∞, and the appeal to SE+Obata for the positive sign—is standard toric calculus and contains no further load-bearing soft spot. The long symbolic examples of §6 are illustrative rather than essential. Consequently the reader’s ACCEPT verdict (moderate confidence) stands without adjustment.","tokens_in":38163,"tokens_out":572,"duration_ms":209379,"concrete_test":"Explicitly construct the path of Example type above on the simplest higher-dimensional Delzant polytope (e.g., the standard simplex for S^5 or CP^2-bundle) with m=n: verify that φ_t remains a normalised symplectic potential for all t<T, compute the crease integral of H_t^{11} against a fixed concave PL test function, and confirm numerical divergence to −∞ as t→T−, matching Cor 4.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Thm 1.4 and Cors 1.5–1.6) hold. The only explicitly flagged hypothesis—the vanishing-order condition α>n−1 on ⟨v_T,(Hess φ_T)v_T⟩ in Case 2—is realizable in every dimension: start from any φ_0∈S_o, subtract a compactly supported non-negative bump ρ whose Hessian is a high even power |x_1|^{2m} (m>(n−1)/2) in one direction inside a ball in P°, and scale until the first degeneracy time T; the resulting path stays in S for t<T, meets the boundary conditions of Prop 2.4, and produces exactly the required order. Thus the energy can be driven to −∞, yielding Y^T_CR<0. Positive sign follows from the existence of a Sasaki–Einstein structure (FOW09) together with the known CR-Obata uniqueness for that structure; the zero value is then the remaining intermediate sign on the connected space of toric CR structures. No hidden analytic gap undermines the polytope reduction or the test-function arguments.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the T-equivariant CR Yamabe problem on compact co-oriented contact manifolds of Reeb type carrying a fixed-point-free torus action of maximal dimension. Using the Guillemin–Abreu–Martelli–Sparks–Yau dictionary, the authors reduce T-invariant constant Tanaka–Webster scalar curvature to a nonlinear elliptic boundary-value problem (Prop. 1.2 / Eq. (2)) for a pair (symplectic potential φ, conformal factor f) on the labelled moment polytope P. They derive an explicit formula for the reduced Einstein–Hilbert functional EH(φ,f) and prove (Thm. 1.4) that along paths of normalised symplectic potentials that either escape to infinity with strictly convex leading term or hit a controlled degeneracy of the Hessian, the equivariant CR Yamabe energy Y^T_CR tends to a non-positive value or to −∞ respectively. Corollaries assert that every such contact toric manifold admits a T-invariant CR structure of negative Yamabe invariant, and that transversally Fano ones realise all three signs. Further results include a convex-hull obstruction for cscS structures (Prop. 1.8), an extension of EH to piecewise-linear geodesic ribbons via an action functional (Prop. 1.7 / §5), and a collection of explicit cscTW and sign-changing examples (§6).","tokens_in":38449,"tokens_out":1372,"duration_ms":47499,"significance":"The work supplies the first systematic dimensional reduction of the equivariant CR Yamabe problem in the toric setting and gives concrete, computable control on the sign of Y^T_CR. The answer to the question of Sung–Takeuchi on the existence of compatible CR structures of both positive and negative Yamabe invariant on a fixed contact structure is of clear interest. The reduced functional I(f) and its link to toric K-stability (Question 5.3) open a natural bridge between CR Yamabe theory and the existing Donaldson–Futaki / weighted cscK literature. The explicit non-Sasaki cscTW examples on the trivial Hirzebruch surface and the sign-changing deformations in §§6.1–6.3 are useful concrete data. The core analytic arguments (test-function estimates with piecewise-linear creases and radial powers) are standard and transparent once the toric dictionary is in place.","major_comments":[{"comment":"Corollary 1.5 asserts existence of a T-invariant CR structure with Y^T_CR < 0 in every dimension, yet Case 2 of Theorem 1.4 requires the quantitative vanishing-order hypothesis α > n−1 on ⟨v_T, (Hess φ_T) v_T⟩ along the crease (Lemma 4.1 / Cor. 4.2). The paper only remarks that the condition is automatic when dim N = 3. An explicit construction (or a short existence argument) of paths in S_o(P) that realise this order in all dimensions is needed for the corollary to hold as stated; without it the higher-dimensional claim rests on an unverified hypothesis.","section":"Theorem 1.4 (Case 2), Corollary 1.5, Lemma 4.1 / Corollary 4.2"},{"comment":"Corollary 1.6 claims the existence of a T-invariant CR structure with Y^T_CR = 0 on every transversally Fano toric contact manifold. Positive sign follows from the Sasaki–Einstein metric (FOW09) together with CR-Obata uniqueness; negative sign follows from Cor. 1.5. The zero value is not constructed and is not justified by an intermediate-value argument: the paper does not establish continuity of the map φ ↦ Y^T_CR(φ) on S_o(P) (or on any path connecting a positive structure to a negative one). Either a continuity statement or an explicit zero-energy example should be supplied.","section":"Corollary 1.6, §1.1"}],"minor_comments":[{"comment":"Proposition 1.2: the phrase “the value of the of Tanaka-Webster scalar curvature” contains a duplicated article; also the constant C should be identified with the value of EH up to the usual positive normalisation factor already used later in the text.","section":"Proposition 1.2"},{"comment":"Figure 1 is a helpful schematic but the regions labelled Y_CR ≤ 0 / ≥ 0 / < 0 are not rigorously delimited by the theorems; a caption clarifying that the figure is only heuristic would avoid over-interpretation.","section":"Figure 1 (§4)"},{"comment":"Several typographical artefacts appear in the extracted text (“T oric”, “Y amabe”, “W e will use”, stray spaces in section headings). A careful proof-reading pass is recommended before final submission.","section":"Throughout"},{"comment":"In §5 the passage from the smooth action functional A^χ_s(t) to the singular Monge–Ampère expression (17) invokes Rudin’s theorems on Lebesgue decomposition; a one-sentence reminder that the singular part of MA(φ_∞) is supported on the crease set (hence has vanishing Radon–Nikodym derivative) would make the justification self-contained.","section":"§5, equations (16)–(17)"},{"comment":"Question 1.9 is well-motivated by the examples of §6.2 and §6.4; it would help the reader if the authors briefly indicated whether any of the explicit non-Sasaki cscTW solutions constructed in §6.2.3 could serve as counter-examples once the EH = EH_min hypothesis is dropped (they already show they are not absolute minimisers).","section":"Question 1.9, §6.2.3"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a natural and solid continuation of the authors’ earlier work (LLS23/25, BHLTF). The two gaps flagged as major comments are local and appear readily fixable; once addressed the paper meets the standard for acceptance. No concerns about novelty disclosure or scope."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The headline result is clean: on any compact toric contact manifold of Reeb type you can find T-invariant CR structures with negative equivariant Yamabe energy, and on transversally Fano ones you get all three signs. That settles the question from Sung–Takeuchi inside the toric setting, which is what most people in this corner actually work with.\n\nWhat is new is the dictionary. They reduce the T-equivariant cscTW equation to a boundary-value problem on the moment polytope (Prop. 1.2), write the Einstein–Hilbert energy as an explicit integral involving the inverse Hessian and a boundary measure, then drive that energy to ≤0 or −∞ along two families of paths of symplectic potentials (Thm 1.4). The test functions are elementary—piecewise-linear creases and radial powers—and the estimates are standard toric calculus (Guillemin–Abreu plus the Don02/AM19 integration-by-parts). Section 5 extends EH to piecewise-linear ribbons via an action functional and ties it to the algebraic picture in LLS; that is useful even if the K-stability conjectures stay open. The explicit non-Sasaki cscTW families on CP¹×CP¹ and the Hirzebruch surfaces in §6 are concrete and show that cscS alone does not force a Yamabe minimizer.\n\nThe only real soft spot is Case 2 of Thm 1.4: in dimension >3 you need the Hessian degeneracy to vanish faster than order n−1 along a hyperplane so the crease integral diverges. The authors flag it. It is realizable by a compactly supported bump, so the negative-sign corollary still holds, but a referee will want that written down. Question 1.9 (whether an EH-minimizing cscS is the unique T-Yamabe minimizer) is left open; the examples only show the answer is no without the EH=EHmin assumption. The long symbolic calculations in §6 were not independently re-run here, but they are of the usual admissible/Calabi type and look consistent.\n\nMath and citations are in order—no circularity, self-cites are background. This is for people already in toric Sasaki/CR Yamabe; it will not travel far outside that circle. Still, it is a genuine advance with usable tools. I would send it to referees without hesitation.","headline":"Solid toric reduction that answers the sign question for CR Yamabe invariants; main caveat is a mild higher-dim hypothesis that is fixable.","tokens_in":39145,"tokens_out":593,"would_cite":true,"duration_ms":15347,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","58E11","53C18","32Q15","53D10","32V20"],"pacs":[],"model":"grok-4.5","headline":"On any compact toric contact manifold of Reeb type, there are T-invariant CR structures whose CR Yamabe invariants take negative values, and when the manifold is transversally Fano they take all three signs.","keywords":["CR Yamabe problem","toric Sasaki manifolds","Tanaka-Webster scalar curvature","Einstein-Hilbert functional","symplectic potentials","moment polytope","equivariant Yamabe invariant"],"falsifier":"Construct an explicit path of symplectic potentials on a polytope of dimension greater than one whose Hessian vanishes only to order n−1 or lower at an interior degeneracy, and check whether the reduced energy still tends to minus infinity; if it remains bounded the case-2 claim fails.","tokens_in":38997,"feed_emoji":"📐","tokens_out":1126,"duration_ms":26108,"temperature":0.7,"pith_summary":"The paper studies the torus-equivariant CR Yamabe problem on compact co-oriented contact manifolds that carry a fixed-point-free torus action of Reeb type. In the toric case the problem reduces to a boundary-value elliptic PDE for a pair of functions on the moment polytope: a normalised symplectic potential that encodes the CR structure and a positive conformal factor. By evaluating a reduced Einstein–Hilbert energy along explicit paths of potentials, the authors prove that the equivariant CR Yamabe energy can be driven to minus infinity or to non-positive values. Consequently every such manifold admits a T-invariant CR structure with negative Yamabe invariant, and every transversally Fano example admits structures with negative, zero and positive invariants. The same reduction supplies many explicit constant Tanaka–Webster curvature examples and shows that constant-scalar-curvature Sasaki structures sit at the boundary of the convex hull of constant-curvature conformal structures.","feed_headline":"Toric contact manifolds admit CR structures of every Yamabe sign","feed_subtitle":"A polytope PDE shows the equivariant CR Yamabe energy can be driven negative, zero or positive.","key_machinery":"The reduced CR Einstein–Hilbert functional EH(φ,f) = (2∫_∂P f^{-n} dσ + n∫_P f^{-n-1} ⟨H_φ, Hess f⟩ dx) / (∫_P f^{-n-1} dx)^{n/(n+1)}, defined on pairs of a normalised symplectic potential φ and a positive function f on the moment polytope P; its critical points are precisely the T-invariant constant Tanaka–Webster structures, and its limiting behaviour along paths of potentials controls the sign of the equivariant Yamabe energy.","core_discovery":"For any compact toric contact manifold of Reeb type the T-equivariant CR Yamabe problem is equivalent to an elliptic boundary-value problem on the moment polytope. Along paths of symplectic potentials that become singular or go to infinity, the reduced Einstein–Hilbert energy tends to non-positive values or to minus infinity, so the manifold always carries T-invariant CR structures of negative CR Yamabe invariant; when it is transversally Fano it also carries structures of zero and positive invariant.","pith_inferences":["The same polytope reduction should let one decide whether a constant-scalar-curvature Sasaki metric that realises the global minimum of the Einstein–Hilbert energy on the Reeb cone is automatically a T-invariant CR Yamabe minimiser in its conformal class.","Extending the energy continuously to weakly convex or piecewise-linear potentials would turn the question of the sign of the toric CR Yamabe invariant into a purely convex-geometric problem on the moment polytope.","The appearance of both signs on a fixed contact structure suggests that the ordinary (non-equivariant) CR Yamabe invariant may also change sign under deformation of the CR structure, even without torus symmetry."],"forward_implications":["Every compact toric contact manifold of Reeb type admits a T-invariant CR structure with negative CR Yamabe invariant.","Every transversally Fano toric contact manifold admits T-invariant CR structures realising all three signs of the CR Yamabe invariant.","Constant-scalar-curvature Sasaki structures cannot lie in the interior of the convex hull of constant Tanaka–Webster structures in the same conformal class.","The equivariant Yamabe energy is controlled by a simpler functional I(f) on concave positive functions on the polytope, linking it to K-stability of the toric Sasaki manifold.","Explicit non-Sasaki constant Tanaka–Webster examples exist that are not absolute minimisers of the Einstein–Hilbert energy in their T-invariant conformal class."],"fun_headline_variants":["Toric contacts admit T-invariant CR structures of every Yamabe sign","Equivariant CR Yamabe reduces to elliptic PDE on the moment polytope","Paths of potentials drive CR Yamabe energy to any sign on toric contacts","All CR Yamabe signs achieved via toric CR structures on contact manifolds","Moment polytope PDE proves every Yamabe sign for toric CR structures"],"cache_read_input_tokens":128,"weakest_assumption_plain":"In higher dimensions the energy is proved to go to minus infinity only when the Hessian of the limiting potential vanishes to order strictly higher than the dimension of the polytope along a hyperplane through an interior point; without that quantitative vanishing the integral against the test function need not diverge.","fun_headline_variants_meta":{"raw":{"variants":["Toric contacts admit T-invariant CR structures of every Yamabe sign","Equivariant CR Yamabe reduces to elliptic PDE on the moment polytope","Paths of potentials drive CR Yamabe energy to any sign on toric contacts","All CR Yamabe signs achieved via toric CR structures on contact manifolds","Moment polytope PDE proves every Yamabe sign for toric CR structures"]},"model":"grok-4.5","effort":"low","cost_usd":0.0064,"raw_usage":{"total_tokens":1537,"prompt_tokens":651,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":64004000,"prompt_tokens_details":{"text_tokens":651,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":805,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":651,"tokens_out":81,"duration_ms":11420,"temperature":1.0,"reasoning_tokens":805,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T00:47:35.106253+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct an explicit path of symplectic potentials on a polytope of dimension greater than one whose Hessian vanishes only to order n−1 or lower at an interior degeneracy, and check whether the reduced energy still tends to minus infinity; if it remains bounded the case-2 claim fails.","supporting_citations":[],"review_version":1}