{"id":"e70dc071-a44a-45f7-956b-5b818584edc5","arxiv_id":"2607.11296","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In complex dimension three, a Nakai–Moishezon-type numerical criterion for the complex 2-Hessian equation is equivalent to existence of a smooth 2-admissible representative and thus to solvability.","lead":"A numerical Nakai–Moishezon-type test for the complex 2-Hessian equation is shown to yield a Gauduchon class, and in complex dimension three it is equivalent to solvability of that equation. The result settles related conjectures of Murakami and Székelyhidi in dimension three and extends to a boundary setting.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Abstract-only review: the load-bearing step converting the Nakai–Moishezon numerical criterion into a Gauduchon class (and thence into 2-admissibility in dim 3) cannot be audited, so the equivalence and settled conjectures remain unverified.","rationale":"The Reader correctly flags that the entire chain of consequences rests on the uninspectable conversion of a numerical criterion into a Gauduchon class. Because only the abstract is available, no deeper technical flaw can be isolated and no stronger objection can be raised; the honest posture is therefore to leave the verdict UNVERDICTED and to restate the same load-bearing concern. The concrete test is simply to examine that conversion step once the full text appears. No ad-hominem or theatrical language is warranted; the limitation is purely epistemic.","tokens_in":1966,"tokens_out":491,"duration_ms":3822,"concrete_test":"Obtain the full preprint and verify the proof that the Nakai–Moishezon numerical condition implies the existence of a Gauduchon metric in the relevant class (the abstract’s first stated result). Check whether the argument uses only the numerical hypothesis or tacitly assumes extra positivity/closedness that is not guaranteed by the numerical data; if the latter, re-examine whether the dim-3 equivalence still holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper’s central claim is an equivalence, in complex dimension three, between a Nakai–Moishezon-type numerical criterion for the complex 2-Hessian equation and the existence of a smooth 2-admissible representative (hence solvability). The abstract states that the numerical criterion “produces a Gauduchon class,” which is then used as the bridge to 2-admissibility. Without the full text one cannot inspect the analytic estimates, positivity arguments, or closedness/positivity properties that convert the numerical condition into a Gauduchon class on a compact Kähler manifold. If that conversion fails (or holds only under extra unstated hypotheses), the dimension-three equivalence collapses and the claimed resolutions of Murakami’s and Székelyhidi’s conjectures do not follow. This is precisely the weakest link identified by the Reader; no independent verification is possible from the abstract alone.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims that a Nakai–Moishezon-type numerical criterion associated with the complex 2-Hessian equation produces a Gauduchon class on a compact Kähler manifold. In complex dimension three this numerical condition is asserted to be equivalent to the existence of a smooth 2-admissible representative and therefore to solvability of the 2-Hessian equation. As consequences the authors claim to settle the corresponding conjectures of Murakami (complex Hessian equation) and of Székelyhidi (Hessian quotient equation) in dimension three, and they also announce a boundary version of the same results.","tokens_in":2168,"tokens_out":630,"duration_ms":11423,"significance":"If the arguments hold, the work would supply a genuine numerical criterion for solvability of the complex 2-Hessian equation in dimension three and would resolve two well-known conjectures in that dimension. Such a criterion would be a substantial contribution to the analytic theory of fully nonlinear equations on compact Kähler manifolds and would parallel the classical Nakai–Moishezon theorem in a non-linear setting. The boundary extension would further enlarge the geometric scope of the results.","major_comments":[{"comment":"Only the abstract is available for review. The central load-bearing step—that the stated Nakai–Moishezon-type numerical criterion produces a Gauduchon class—cannot be inspected. All subsequent claims (the dimension-three equivalence with 2-admissibility and solvability, the resolution of Murakami’s and Székelyhidi’s conjectures, and the boundary version) rest on this conversion. Without the analytic estimates, positivity arguments, and closedness properties that effect the conversion, soundness cannot be verified.","section":"Abstract (full manuscript unavailable)"},{"comment":"The abstract asserts an equivalence in complex dimension three between the numerical criterion and the existence of a smooth 2-admissible representative. The precise hypotheses under which this equivalence holds, the role of the ambient Kähler class, and the a-priori estimates that close the argument are invisible from the abstract alone; any hidden restriction would immediately limit the claimed resolution of the two conjectures.","section":"Abstract (full manuscript unavailable)"}],"minor_comments":[{"comment":"The abstract is clear and well-written, but a complete referee report requires the full text, including statements of the precise numerical criterion, the definition of the associated Gauduchon class, and the statements of the conjectures being settled.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review. A definitive recommendation (accept / revise / reject) is impossible until the complete manuscript is supplied. The load-bearing conversion from numerical criterion to Gauduchon class is precisely the step that must be audited; if it is sound the paper is likely of high interest, but that cannot be confirmed from the abstract."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that the paper asserts an equivalence, in complex dimension three, between a Nakai–Moishezon-type numerical criterion for the complex 2-Hessian equation and the existence of a smooth 2-admissible representative (hence solvability). As direct consequences it claims to prove Murakami’s conjecture for the complex Hessian and Székelyhidi’s for the Hessian quotient, both restricted to dim 3, plus a boundary version. It also claims the numerical condition produces a Gauduchon class on a compact Kähler manifold in general.\n\nWhat is actually new is that dim-3 equivalence and the resulting resolution of the two named conjectures. The abstract presents a clean existence/equivalence theorem that sits squarely inside the existing program of numerical criteria for Hessian-type equations. If the proofs check out, this supplies a usable concrete tool rather than a rehash.\n\nThe soft spot is exactly the one the stress-test flags: we have only the abstract. The load-bearing step is the conversion of the numerical positivity condition into a Gauduchon class, which then feeds the 2-admissibility argument in dimension three. Without the a-priori estimates, positivity arguments, or closedness properties, that bridge cannot be audited. If it fails or needs unstated extra hypotheses, the equivalence and the settled conjectures collapse. That is a real uncertainty, not a manufactured one; everything else hangs on it. Circularity burden looks low for a pure existence result with no free parameters.\n\nThis is for specialists in fully nonlinear geometric PDE and complex Hessian equations on Kähler manifolds. They will get value from the statements and will want to see the proofs. It is important enough inside the subfield, and the conjectures are known enough, that it deserves a serious referee rather than a desk reject. Send it out.","headline":"Claims to settle Murakami and Székelyhidi conjectures for the 2-Hessian in complex dim 3 via a Nakai–Moishezon criterion, but we only have the abstract so the key analytic bridge is unchecked.","tokens_in":2807,"tokens_out":518,"would_cite":false,"duration_ms":15469,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","32W20","32Q15"],"pacs":[],"model":"grok-4.5","headline":"In complex dimension three, a Nakai–Moishezon-type numerical criterion is equivalent to solvability of the complex 2-Hessian equation.","keywords":["complex 2-Hessian equation","Nakai–Moishezon criterion","Gauduchon class","2-admissible representative","compact Kähler manifolds","Murakami conjecture","Székelyhidi conjecture","boundary version"],"falsifier":"An explicit compact Kähler threefold on which the stated Nakai–Moishezon-type numerical criterion holds but no smooth 2-admissible form exists (or the 2-Hessian equation fails to admit a smooth solution).","tokens_in":2810,"feed_emoji":"📐","tokens_out":564,"duration_ms":3784,"temperature":0.7,"pith_summary":"The paper shows that a Nakai–Moishezon-type numerical criterion associated with the complex 2-Hessian equation produces a Gauduchon class on a compact Kähler manifold. In complex dimension three this numerical condition becomes equivalent to the existence of a smooth 2-admissible representative and therefore to the solvability of the 2-Hessian equation. As direct consequences the corresponding conjectures of Murakami (for the complex Hessian equation) and of Székelyhidi (for the Hessian quotient equation) hold in dimension three. A parallel set of statements is established in the boundary setting. The result therefore converts a purely numerical positivity condition into an analytic existence theorem in low dimension, giving a concrete test for when the 2-Hessian equation can be solved.","feed_headline":"Numerical test settles 2-Hessian solvability in dimension three","feed_subtitle":"A Nakai–Moishezon-type criterion becomes equivalent to existence of a smooth 2-admissible form.","key_machinery":"The Nakai–Moishezon-type numerical criterion associated with the complex 2-Hessian equation, which is shown to produce a Gauduchon class that, in dimension three, is equivalent to 2-admissibility and solvability.","core_discovery":"A Nakai–Moishezon-type numerical criterion for the complex 2-Hessian equation produces a Gauduchon class; in complex dimension three this criterion is equivalent to the existence of a smooth 2-admissible representative and hence to solvability of the 2-Hessian equation, settling the corresponding conjectures of Murakami and of Székelyhidi in that dimension.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Nakai-Moishezon test yields Gauduchon class for 2-Hessian","In complex dim three, numerical criterion solves 2-Hessian","Dim-three 2-Hessian solvability via Nakai-Moishezon criterion","Murakami and Székelyhidi conjectures settled in dim three","Smooth 2-admissible form iff numerical criterion holds in dim 3"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that the numerical criterion produces a Gauduchon class, which is the analytic bridge used to reach 2-admissibility and solvability in dimension three.","fun_headline_variants_meta":{"raw":{"variants":["Nakai-Moishezon test yields Gauduchon class for 2-Hessian","In complex dim three, numerical criterion solves 2-Hessian","Dim-three 2-Hessian solvability via Nakai-Moishezon criterion","Murakami and Székelyhidi conjectures settled in dim three","Smooth 2-admissible form iff numerical criterion holds in dim 3"]},"model":"grok-4.5","effort":"low","cost_usd":0.004902,"raw_usage":{"total_tokens":1347,"prompt_tokens":651,"num_sources_used":0,"completion_tokens":105,"cost_in_usd_ticks":49020000,"prompt_tokens_details":{"text_tokens":651,"audio_tokens":0,"image_tokens":0,"cached_tokens":384},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":591,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":651,"tokens_out":105,"duration_ms":4404,"temperature":1.0,"reasoning_tokens":591,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T02:08:43.856086+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit compact Kähler threefold on which the stated Nakai–Moishezon-type numerical criterion holds but no smooth 2-admissible form exists (or the 2-Hessian equation fails to admit a smooth solution).","supporting_citations":[],"review_version":1}