{"id":"9c581685-d040-4a84-b6bc-e5fb20aeebec","arxiv_id":"2511.21492","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The critical phase of the LYZ/dHYM equation on compact Kähler manifolds is solvable under the expected subsolution condition, resolving an open problem posed by Collins–Jacob–Yau and Li.","lead":"This paper proves that a fully nonlinear equation from mirror symmetry, the deformed Hermitian Yang-Mills equation, always has a smooth solution in the critical borderline case, which had been open. The result also yields new existence theorems for two classical Hessian-type equations in dimensions 3 and 4 under weaker conditions than before.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The scaling normalization in (52) of Theorem 4.1 is unjustified: setting sup−inf=1 forces the gradient bound to become C/(sup−inf)^{1/2}, which can exceed 1, and the subsequent comparison argument relies on the normalized gradient bound.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the scaling in (52) is not justified. I re-examined the scaling identity and confirmed it is not a consequence of the C^1 bound. This is not a disagreement with the field's consensus; it is an internal gap in the proof. The concern is genuinely load-bearing because Theorem 4.1 is the only device that rules out the mixed equation σ_{n−1}+σ_n=0 in the blow-up analysis; without it, the gradient estimate (and therefore Theorem 1.2 and Theorem 1.1) does not follow from the written argument. I do not see evidence of dishonesty or circularity, and the surrounding framework (reduction to supercritical case via Lin/Sun, Hessian estimate of Hou–Ma–Wu type) is plausible. The issue may well be fixable, e.g., by adjusting the comparison argument to work under a non-normalized gradient bound, but as written the paper contains a real gap. Hence the conditional acceptance remains appropriate; no verdict change is needed.","tokens_in":24036,"tokens_out":12404,"duration_ms":122590,"concrete_test":"Check whether the blow-up limit u_∞ in §4.4 satisfies sup u_∞−inf u_∞ ≥ 1. If not, explicitly compute |∇\\tilde v| for \\tilde v(z)=λ^{-2}v(λ z) with λ=(sup−inf)^{1/2} and verify it exceeds 1, contradicting (52). Then attempt to re-run the comparison argument in Case 2 of Theorem 4.1 using the un-normalized gradient bound |∇v|≤C; if the contradiction cannot be recovered without the simultaneous normalization sup v−inf v=1 and |∇v|≤1, the gap is real and the proof of the Liouville theorem needs repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1 (Liouville theorem for σ_{n−1}+σ_n=0) is the key tool used in Section 4.4 to exclude the blow-up limit u_∞ in the case 0<a_0<∞. In the proof, after assuming v is nonconstant, the authors set C_1=(sup v−inf v)^{1/2} and claim that \\tilde v(z)=C_1^{-2}(v(C_1 z)−inf v) satisfies |∇\\tilde v|≤1. But the equation is invariant only under scalings of the form v_λ(z)=λ^{-2}v(λ z). To normalize sup−inf=1 one must take λ=(sup v−inf v)^{1/2}; then |∇v_λ| = λ^{-1}|∇v| ≤ C λ^{-1}. The assertion |∇v_λ|≤1 therefore requires (sup v−inf v) ≥ C^2, which is not guaranteed by the hypotheses. In the application to u_∞, we know |∇u_∞|≤1, but sup u_∞−inf u_∞ may be arbitrarily small, so the claimed normalization can fail. Since the subsequent comparison/sub-solution construction in Case 2 (using 0≤v≤1 and |∇v|≤1 to control constants like 1/60, 9/10, 3/10) depends on these bounds, the proof of Theorem 4.1 is incomplete. Without Theorem 4.1, the gradient estimate and hence the uniform C^{2,α} estimate for u_t cannot be closed, leaving Theorem 1.1 unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to resolve the critical case of the LYZ/dHYM equation, i.e. the existence of smooth solutions to θ_ω(χ_u) = (n−2)π/2 on a compact Kähler manifold under the subsolution condition (5). The strategy is to perturb the critical phase by a small parameter t, obtain supercritical solutions u_t via Lin's theorem, and then prove uniform C^{2,α} estimates for u_t that are independent of t and of the phase gap. The main technical novelties are a Hou–Ma–Wu type Hessian estimate (Theorem 3.1) with uniform constants, and a new Liouville theorem for σ_{n−1}+σ_n=0 on C^n (Theorem 4.1), proved by lifting v to v+|z_{n+1}|^2. The paper also derives applications to the 3D Hessian equation σ_2=1 and the 4D Hessian quotient equation σ_3=σ_1 under weaker assumptions than in previous work.","tokens_in":24406,"tokens_out":21561,"duration_ms":219125,"significance":"If the proof is completed, this would settle an open critical case posed by Collins–Jacob–Yau and Li, and would strengthen the connection between the LYZ equation and Bridgeland stability. The uniform Hessian estimate with constants independent of (θ(t)−(n−2)π/2)^{-1} is an important technical step, and the reduction of the new Liouville theorem via the extra variable z_{n+1} is elegant and potentially useful beyond this paper. The applications improve earlier results by Sun and Székelyhidi. However, the proof of the Liouville theorem contains a load-bearing gap in its scaling normalization, and the most novel case is partly delegated to prior work; these issues need to be addressed before the main theorem is fully supported.","major_comments":[{"comment":"The scaling normalization in the proof of Theorem 4.1 is not justified as stated. From the hypothesis ∥v∥_{C^1(C^n)}≤C, the function \\tilde v(z)=C_1^{-2}(v(C_1 z)−inf v) with C_1=(sup v−inf v)^{1/2} has oscillation 1, but its gradient satisfies |∇\\tilde v|≤C C_1^{-1}. The displayed condition |∇\\tilde v|≤C_1 would require (sup v−inf v)≥C^2, which is not a consequence of the hypotheses. Since the subsequent argument in Case 2 uses the normalized bounds 0≤v≤1 and |∇v|≤1 (e.g. in the 'following [10,38]' step and in the Cartan-type lemma), this gap is load-bearing for the Liouville theorem, and Theorem 4.1 is used in §4.4 to rule out the nonconstant blow-up limit when 0<a_0<∞. A possible repair is to choose the scaling parameter λ=max(C,(sup v−inf v)^{1/2}), which gives |∇v_λ|≤1 and 0≤sup v_λ−inf v_λ≤1, and then rework the argument with those bounds; but as written the proof is incomplete.","section":"§4.3, Theorem 4.1, Case 1"},{"comment":"The proof of Case 1 is only sketched by 'Following the arguments in [38] and [10], we obtain...'. Since Theorem 4.1 is a new Liouville theorem and Case 1 is an essential part of its proof, this delegation is not adequate. The authors should either supply the full construction of v_∞, the verification that it is independent of z_n, and the contradiction with (53), or state and prove a precise lemma that covers this case with all hypotheses checked.","section":"§4.3, Case 1"},{"comment":"Theorem 1.1 asserts uniqueness of the smooth solution with sup_M u=0, but the proof in §4.5 only establishes existence via a subsequence limit. No uniqueness argument or reference is given. This can likely be fixed by a standard maximum principle: at a maximum of u_1−u_2 one has χ_{u_1}≤χ_{u_2}, whence θ(χ_{u_1})≤θ(χ_{u_2}); equality of the phases then forces equality of the Hermitian matrices and hence u_1−u_2 constant. The authors should include this argument or an explicit reference.","section":"Theorem 1.1 / §4.5"}],"minor_comments":[{"comment":"The statement 'χ^2∧ω>0 as a positive (2,2)-form' seems to be a typo: in dimension 3 χ^2∧ω is a (3,3)-form, while the correct subsolution condition appears in Corollary 5.1 as χ∧ω>0 as a (2,2)-form. Please harmonize.","section":"Corollary 1.1"},{"comment":"In the comparison principle, the displayed '≤−ϵn<0' after expansion is not literally correct: the expansion contains nonnegative lower-order terms whose coefficient is not simply ϵn. The contradiction still follows because w∈Γ_{n−1} makes the perturbed inequality strictly smaller than the subsolution inequality, but the formula should be corrected.","section":"Lemma 4.2"},{"comment":"The exponent in the Hölder estimate '|\\hat u_{t_i}|_{C^{1,2/3}}≤C' is unusual; please clarify whether this follows from Schauder estimates with the available L∞ bounds on Δ\\hat u_{t_i} and the C^0 bound, and state the relevant standard result.","section":"§4.4, Eq. (60)"},{"comment":"Typo: '4-dimesional' should be '4-dimensional'.","section":"§5.2"}],"recommendation":"major_revision","confidential_remarks":"The normalization gap in Theorem 4.1 is the main obstacle; I believe it is repairable along the lines of the suggested scaling λ=max(C,√S), but the current manuscript does not support the main theorem as written. Once this is fixed and the delegated Case 1 is supplied, the paper would be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper on an open problem that matters, and most of the machinery is sound, but the new Liouville theorem has a real gap. I wouldn't accept it yet; I would send it to a referee.\n\nWhat's genuinely new: the paper solves the critical LYZ phase θ=(n−2)π/2 under the subsolution condition, which Collins–Jacob–Yau and Li explicitly left open. The continuity method through supercritical equations is standard, but the paper gets the phase control and the uniform estimates right. The lifting trick v+|z_{n+1}|^2 turns σ_{n-1}+σ_n=0 into σ_n=0 on C^{n+1}; that's a nice idea and likely reusable. The 3D σ2=1 and 4D σ3=σ1 corollaries are credible, and the necessity arguments in Section 5 look correct. The complex Hessian estimate (Theorem 3.1) is long but coherent, and the isolation of the new Liouville theorem is honest.\n\nThe soft spot is Theorem 4.1. The proof's normalization (52) is not justified. For v with ||v||_{C^1}≤C, setting C1=(sup v−inf v)^{1/2} and \\tilde v(z)=C1^{-2}(v(C1z)−inf) gives |∇\\tilde v|≤C/C1, not ≤C1. The claimed bound needs (sup−inf)≥C^2, which is not in the assumptions. That matters because Case 2's comparison argument uses the normalized bounds 0≤v≤1, |∇v|≤1 to control everything from the constants 1/60, 9/10, 3/10 to the Dinew–Kołodziej/Székelyhidi subharmonicity step. Without a way to get those bounds, the Liouville theorem is unproved. And Theorem 4.1 is exactly what excludes the blow-up limit in Subcase 2.2 of the gradient estimate, so the chain from Theorem 4.1 to Theorem 1.1 is broken as written. There are also places where the proof says 'following [38] and [10]' without giving enough detail, which makes it hard to tell whether this is a small fix or a deeper issue.\n\nIs it fatal? Not obviously. The result may be true, and the rest of the architecture is plausible. But the central existence theorem currently rests on an unsupported lemma. A referee who knows [10,38] and the Liouville literature should be asked whether the normalization can be repaired—maybe by a different scaling, or by deriving sup−inf≥C^2 from the equation. If that can be fixed, this is a significant paper.\n\nAudience: people working on the deformed Hermitian–Yang–Mills equation, Hessian equations, and Bridgeland stability. It deserves peer review, not desk rejection, and the referee should be asked to focus on Theorem 4.1 first.","headline":"The paper attacks a real open problem and much of the proof is sound, but the new Liouville theorem has an unjustified scaling normalization that leaves the main theorem unsupported.","tokens_in":24927,"tokens_out":7350,"would_cite":false,"duration_ms":76910,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32W20","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Proven: smooth solution for critical LYZ equation under a subsolution","keywords":["LYZ equation","deformed Hermitian Yang-Mills","critical phase","subsolution","complex Hessian estimates","gradient estimates","Liouville theorem","Hessian quotient equations"],"falsifier":"Check the scaling normalization (52) directly: for any bounded C^1 function v with oscillation A and sup|∇v|=1, the rescaled function ṽ(z)=A^{-1}(v(√A z)−inf v) has oscillation 1 but gradient bound A^{-1/2}, which exceeds 1 when A<1. If such a v satisfies all other hypotheses of Theorem 4.1 (nonconstant, (n−1)-subharmonic, v+|z|² plurisubharmonic, weak solution and viscosity supersolution of σ_{n−1}+σ_n=0, with ∆v≤C weakly), then the Liouville theorem is false and the existence proof breaks at that point.","tokens_in":23921,"feed_emoji":"📐","tokens_out":11324,"duration_ms":102314,"temperature":0.7,"pith_summary":"The paper establishes existence and uniqueness of smooth solutions to the LYZ equation at its critical phase θ=(n−2)π/2 on any compact Kähler manifold, provided a subsolution satisfying a pointwise angular condition exists. This is the endpoint case of a family of equations previously solved only above the critical phase. The proof reaches the endpoint by solving nearby supercritical equations and showing their solutions stay uniformly bounded in C^{2,α} independently of how close the phase is to critical. A new Liouville theorem for the limiting equation σ_{n−1}+σ_n=0 on C^n supplies the missing gradient control. Success at the critical phase also yields the 3D Hessian equation σ_2=1 and the 4D Hessian quotient σ_3=σ_1 under weaker hypotheses than were previously available.","feed_headline":"Proven: smooth solution for critical LYZ equation under a subsolution","feed_subtitle":"The endpoint phase of the deformed Hermitian Yang-Mills equation is now solvable, unlocking 3D and 4D Hessian equations.","key_machinery":"Central object: the LYZ (deformed Hermitian Yang-Mills) equation θ_ω(χ_u)=θ, with θ_ω(χ_u)=Σ_{i=1}^n arctan λ_i and λ_i the eigenvalues of χ+√−1∂∂̄u with respect to ω. At the critical phase θ=(n−2)π/2 the target equation is approached by the supercritical family (6), and the proof's engine is a uniform complex Hessian estimate sup |√−1∂∂̄u_t|_ω ≤ C(1+sup|∇u_t|²), independent of t and of (θ(t)−(n−2)π/2)^{-1}. The gradient estimate then comes from rescaling a hypothetical blowing-up sequence; the limit v is a bounded nonconstant weak solution of σ_n=0, σ_{n−1}=0, or σ_{n−1}+σ_n=0. The new case is tamed by the lifting identity: v solves σ_{n−1}+σ_n=0 on C^n if and only if v+|z_{n+1}|² solves σ_","core_discovery":"On a compact Kähler manifold (M,ω), fix a closed real (1,1)-form χ whose total phase ∫_M (χ+√−1ω)^n has principal argument π. The paper's main theorem says that if some smooth u satisfies the critical subsolution condition min_j Σ_{i≠j} arctan λ_i(χ_u) > (n−3)π/2, then the critical LYZ equation θ_ω(χ_u)=(n−2)π/2 has a unique smooth solution normalized by sup u=0. The theorem reaches this endpoint by perturbing to supercritical phase θ(t)=(n−2)π/2+O(t), where solutions were already known to exist, and then proving C^{2,α} bounds on this family that stay uniform as t→0. The two new ingredients are a second-order bound sup |√−1∂∂̄u_t|_ω ≤ C(1+sup|∇u_t|²) that does not blow up at the critical ph","pith_inferences":["The normalization (52) in the Liouville proof is the load-bearing step; before relying on it, a reader should verify the rescaling claim, since the stated C^1 bound alone does not produce |∇v|≤(osc)^{1/2}.","The lifting identity suggests that other techniques from the homogeneous complex Monge-Ampère equation may transfer to the critical LYZ equation; for example, one might look for interior Hessian estimates for σ_{n−1}+σ_n=0 by working in one dimension higher.","The subcritical regime θ<(n−2)π/2 is likely to behave differently: known examples on R^n show C^{1,α} and Lipschitz solutions that are not smooth, so the compact-manifold subcritical case may genuinely admit singular solutions, making this endpoint result the sharp smooth boundary.","The applications reveal that the integral normalizations in dimensions 3 and 4 are necessary; this hints that the subsolution condition (5) may itself be necessary for existence in the critical case, though the paper does not prove the converse."],"forward_implications":["Under the stated subsolution condition, the critical LYZ equation has a unique smooth solution normalized by sup u=0.","The solutions of the approximating supercritical family converge in C^{2,α} to the critical solution, since the estimates do not depend on the distance to the critical phase.","In dimension 3, the critical equation is the Hessian equation σ_2(χ_u)=1, so the theorem solves it under the subsolution-type condition χ∧ω>0 plus integral normalizations, without requiring χ∈Γ_2(M).","In dimension 4, the critical equation is the Hessian quotient σ_3(χ_u)=σ_1(χ_u), solved under 3χ²∧ω−ω³>0 plus integral normalizations, weaker than χ∈Γ_3(M).","Solvability at the critical phase is a concrete step toward realizing the LYZ equation as a stability condition on the bounded derived category."],"fun_headline_variants":["Critical LYZ equation solved at endpoint phase","Smooth solution for critical LYZ equation found","Critical case of LYZ equation cracked","Endpoint phase of LYZ equation conquered","LYZ equation at critical phase now solvable"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing step is the normalization in (52): after rescaling, the paper assumes the gradient of v is no larger than the square root of v's total oscillation, and this inequality is not a consequence of the C^1 bound proved earlier; if it cannot be arranged, the Liouville theorem and with it the existence proof collapse.","fun_headline_variants_meta":{"raw":{"variants":["Critical LYZ equation solved at endpoint phase","Smooth solution for critical LYZ equation found","Critical case of LYZ equation cracked","Endpoint phase of LYZ equation conquered","LYZ equation at critical phase now solvable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1232,"prompt_tokens":697,"completion_tokens":535,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":479}},"tokens_in":441,"tokens_out":535,"duration_ms":5109,"temperature":1.0,"reasoning_tokens":479,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T20:00:59.688939+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the scaling normalization (52) directly: for any bounded C^1 function v with oscillation A and sup|∇v|=1, the rescaled function ṽ(z)=A^{-1}(v(√A z)−inf v) has oscillation 1 but gradient bound A^{-1/2}, which exceeds 1 when A<1. If such a v satisfies all other hypotheses of Theorem 4.1 (nonconstant, (n−1)-subharmonic, v+|z|² plurisubharmonic, weak solution and viscosity supersolution of σ_{n−1}+σ_n=0, with ∆v≤C weakly), then the Liouville theorem is false and the existence proof breaks at that point.","supporting_citations":[],"review_version":1}