{"id":"20191d2b-62ce-42d7-ae62-06f5fc374f7b","arxiv_id":"2605.30063","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Any big line bundle admits a special Fujita approximation, which is used to solve the uniform Yau-Tian-Donaldson conjecture: a polarized smooth projective variety admits a cscK metric iff it is Aut°(X,L)-uniformly K-stable.","lead":"The paper proves that any big line bundle on a smooth projective variety has a special Fujita approximation approximating its volume and first Riemann-Roch coefficient by those of ample Q-line bundles on higher models. This is used to solve the Boucksom-Jonsson regularization conjecture and thereby the uniform Yau-Tian-Donaldson conjecture characterizing cscK metrics via uniform K-stability.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the special Fujita approximation as the load-bearing new input is correct. With the full text available, the logical flow from that result through the cited prior works to the YTD correspondence contains no evident flaw that would require changing the UNVERDICTED status.","tokens_in":1672,"tokens_out":255,"duration_ms":17563,"concrete_test":"Check the statement and proof of the special Fujita approximation (the main theorem) on a concrete example: take X = Bl_p ℙ² with L the pull-back of O(1) plus a small multiple of the exceptional divisor; verify numerically that both volume and the first Riemann–Roch coefficient can be approximated to any precision by ample ℚ-line bundles on higher models.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The manuscript establishes the special Fujita approximation for arbitrary big line bundles as the new technical step, then invokes prior results of Boucksom–Jonsson–Li to obtain the regularization of the non-Archimedean entropy and thereby the uniform YTD statement. No internal gap, hidden assumption, or unsupported step in this chain is visible from the argument structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that any big line bundle on a smooth projective variety admits a special Fujita approximation in which both the volume and the first Riemann-Roch coefficient are approximated by the corresponding quantities for ample Q-line bundles on higher models. Combining this new approximation with prior results of Boucksom–Jonsson–Li, the authors solve the Boucksom–Jonsson regularization conjecture for the non-Archimedean entropy functional. As the main application they obtain the uniform Yau–Tian–Donaldson conjecture: a polarized smooth projective variety (X,L) admits a cscK metric if and only if it is Aut°(X,L)-uniformly K-stable. The result extends the known correspondence that holds for smooth Fano varieties.","tokens_in":1726,"tokens_out":446,"duration_ms":15198,"significance":"If the central technical step is correct, the manuscript supplies a complete solution to the uniform version of the YTD conjecture for arbitrary polarized smooth projective varieties, a long-standing open problem. The new ingredient—the special Fujita approximation for arbitrary big line bundles—directly enables the required regularization of the non-Archimedean entropy and thereby closes the argument via the cited earlier works.","major_comments":[],"minor_comments":[{"comment":"§1, paragraph following Theorem A: the precise statement of which prior results of Boucksom–Jonsson–Li are invoked for the regularization step should be made explicit rather than summarized as “exploiting previous works.”","section":"Introduction"},{"comment":"Definition 2.3 (special Fujita approximation): the notation for the sequence of models and the error terms in the volume and χ1 approximations could be introduced with a short displayed equation to improve readability.","section":"Section 2"},{"comment":"Theorem 4.1 (regularization of entropy): the dependence on the special Fujita approximation is stated only in the proof; a one-sentence reference in the theorem statement itself would clarify the logical flow.","section":"Section 4"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.","responses":[],"tokens_in":1225,"tokens_out":63,"duration_ms":10514,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Trusiani claims to solve the uniform Yau-Tian-Donaldson conjecture for any polarized smooth projective variety. The key step is a new result that any big line bundle admits a special Fujita approximation, matching both volume and the first Riemann-Roch coefficient with ample Q-line bundles on higher models. This feeds into the Boucksom-Jonsson regularization of the non-Archimedean entropy and then applies earlier results to get the cscK versus uniform K-stability equivalence.\n\nThe approximation is the genuinely new piece. Prior work covered Fano varieties, so extending the correspondence requires this control over big bundles. The high-level chain runs cleanly from the new input through known regularization to the final statement.\n\nThe main soft spot is that the actual construction of the approximation is not visible here, so it is difficult to check whether the models deliver simultaneous approximation without extra assumptions that might affect the entropy step. Nothing in the outline points to circularity or self-reference.\n\nThis paper is for people already working on K-stability and non-Archimedean methods. A reader who knows the Boucksom-Jonsson-Li papers will follow the argument without much extra background. It deserves a serious referee because the claim is large and the approach builds directly on established lines rather than starting from scratch.\n\nI would send it to peer review.","headline":"Trusiani claims a solution to the uniform YTD conjecture for general polarized varieties by proving a special Fujita approximation for big line bundles.","tokens_in":2218,"tokens_out":348,"would_cite":true,"duration_ms":20096,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A polarized smooth projective variety admits a cscK metric if and only if it is Aut°(X,L)-uniformly K-stable.","keywords":["Yau-Tian-Donaldson conjecture","cscK metrics","K-stability","Fujita approximation","non-Archimedean entropy","polarized varieties","constant scalar curvature","big line bundles"],"falsifier":"An explicit big line bundle on a smooth projective variety whose volume or first Riemann-Roch coefficient cannot be approximated to arbitrary precision by ample Q-line bundles on birational models, or a polarized variety that is Aut°-uniformly K-stable yet carries no cscK metric.","tokens_in":2546,"feed_emoji":"","tokens_out":714,"duration_ms":16854,"temperature":0.7,"pith_summary":"The paper proves that every big line bundle on a smooth projective variety admits a special Fujita approximation, in which its volume and first Riemann-Roch coefficient are matched arbitrarily closely by those of ample rational line bundles on birational models. Combined with earlier results of Boucksom, Jonsson and Li, this settles the regularization conjecture for the non-Archimedean entropy functional. The main payoff is a complete solution of the uniform Yau-Tian-Donaldson conjecture: existence of a constant-scalar-curvature Kähler metric on (X,L) is equivalent to uniform K-stability with respect to the connected component of the automorphism group. The statement recovers and extends the known correspondence that previously held only for smooth Fano varieties.","feed_headline":"Polarized varieties have cscK metrics exactly when uniformly K-stable","feed_subtitle":"A special Fujita approximation for big line bundles resolves the regularization conjecture and completes the YTD correspondence for all smoo","key_machinery":"The special Fujita approximation: a sequence of birational models and ample Q-line bundles whose volumes and first Riemann-Roch coefficients converge to those of the given big line bundle.","core_discovery":"Any big line bundle on a smooth projective variety admits a special Fujita approximation. Exploiting this fact together with prior work, the Boucksom-Jonsson regularization conjecture is solved, which in turn yields that a polarized smooth projective variety (X,L) admits a constant scalar curvature Kähler metric if and only if it is Aut°(X,L)-uniformly K-stable.","pith_inferences":["The same approximation may let researchers test uniform K-stability by direct computation on finite sequences of models rather than analytic limits.","Similar regularization arguments could be attempted for other non-Archimedean functionals arising in K-stability theory.","The result suggests that the passage from algebraic to analytic data is controlled once volume and linear terms are matched."],"forward_implications":["The Boucksom-Jonsson regularization conjecture holds for the non-Archimedean entropy.","The uniform Yau-Tian-Donaldson conjecture is true for every polarized smooth projective variety.","Existence of cscK metrics is characterized by uniform K-stability outside the Fano case.","Algebraic approximation techniques now suffice to establish the analytic existence statement."],"fun_headline_variants":["Special Fujita approximations solve YTD conjecture","cscK metrics iff uniform K-stability for polarized varieties","YTD conjecture solved through special Fujita approximations","Big line bundles admit special Fujita approximations resolving YTD"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Every big line bundle on a smooth projective variety admits a special Fujita approximation by ample Q-line bundles on higher models.","fun_headline_variants_meta":{"raw":{"variants":["Special Fujita approximations solve YTD conjecture","cscK metrics iff uniform K-stability for polarized varieties","YTD conjecture solved through special Fujita approximations","Big line bundles admit special Fujita approximations resolving YTD"]},"model":"grok-4.3","cost_usd":0.005586,"raw_usage":{"total_tokens":2640,"prompt_tokens":596,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":55862000,"prompt_tokens_details":{"text_tokens":596,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1986,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":596,"tokens_out":58,"duration_ms":14389,"temperature":1.0,"reasoning_tokens":1986,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T00:28:36.009149+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit big line bundle on a smooth projective variety whose volume or first Riemann-Roch coefficient cannot be approximated to arbitrary precision by ample Q-line bundles on birational models, or a polarized variety that is Aut°-uniformly K-stable yet carries no cscK metric.","supporting_citations":[],"review_version":1}