{"id":"6b7e638e-8f31-4751-ada0-7fae8056584d","arxiv_id":"2608.01065","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For convex solutions of the special Lagrangian curvature potential equation, boundary curvature blows up if and only if the boundary limiting-phase gap collapses, with optimal rate δ^{-1}.","lead":"This paper proves sharp boundary estimates for the special Lagrangian curvature potential equation, showing that convex solutions can blow up at the boundary exactly when a limiting phase gap collapses, at the optimal rate. It introduces a new compatibility condition for mixed second derivatives and gives an exact formula for the normal second derivative in terms of the boundary phase gap.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Theorem 1.7's conditional structure is stated precisely, and the proofs of Theorems 1.4, 1.5, and 1.8 are internally consistent.","rationale":"The reader's ACCEPT verdict is justified. I focused on Theorem 1.7 as the strongest claim. Its proof is conditional in a clearly stated way: the constant depends on Z, and the equivalence is asserted only for families satisfying (1.12) uniformly. This is not circular, because Theorem 1.4 independently establishes Z under MRC, and Theorem 1.8 constructs a family with Z uniform while delta* tends to zero. I checked the recession barrier argument in Lemma 3.2, the Killing-field comparison in Theorem 1.4, and the sign of the mean-curvature maximum principle in Proposition 5.1; each step is coherent. The only genuine limitation is that MRC is a sufficient condition whose converse is proved only at the recession level for a restricted barrier class, and the paper explicitly flags this in Remarks 3.4 and 5.4. This limits the scope of applications but does not contradict the main theorems. No load-bearing concern lands, so an honest non-finding is appropriate.","tokens_in":23475,"tokens_out":25966,"duration_ms":247629,"concrete_test":"As a verification step independent of the paper's derivations, recompute the complex Schur-complement identity (4.2)-(4.8) on random positive semidefinite block data: choose n=3, P=1, arbitrary q and s with |p| <= 1, random M >= 0, z, and r_nu >= 0; form C_t and D^2u, compute F(A_t[u]), L_t, alpha_t, beta_t, and delta_t by direct numerical eigenvalue evaluation, and check that r_nu - beta_t - alpha_t cot(delta_t) is machine-zero and that 1 <= alpha_t <= C(P,Z), |beta_t| <= Xi(P,Z). This directly tests the identity on which both (1.13) and (1.14) rest.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the paper as proving the mixed derivative estimate under MRC (Definition 1.2), an exact Schur-complement identity (Theorem 1.5), and a global equivalence (Theorem 1.7) for families satisfying the uniform control (1.12), which explicitly includes the mixed boundary derivative bound Z. The weakest point is indeed MRC: Remark 2.7 and Proposition 2.6 show the full linearized separation J can have the wrong sign, and Theorem 3.3 proves necessity of MRC only within the fixed zero-order barrier class (3.12); Remark 3.4 concedes that failure of MRC need not produce actual solutions with unbounded mixed derivatives. However, this is a stated limitation, not a hidden flaw. Theorem 1.7 does not assume MRC; it assumes Z, and Theorem 1.8 verifies Z directly for the sharpness family. I found no algebraic error in the Schur-complement identity, the alpha/beta bounds of Lemma 4.2, or the mean-curvature maximum principle. The lower bound (1.14) correctly gives cot(delta*) minus a constant because alpha >= 1, and the upper bound (1.13) follows from the boundary Hessian estimate and Proposition 5.1. Thus the central claim holds under its stated hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation F(A[u])=θ, where A[u] depends on both Du and D^2u. The central new object is the mixed recession compatibility condition (MRC), imposed only on doubly degenerate recession jets. Under (MRC), Theorem 1.4 establishes a uniform mixed tangential-normal boundary Hessian bound; Theorem 1.5 gives an exact complex Schur-complement identity u_{\\nu\\nu}=\\beta+\\alpha\\cot\\delta with explicit coefficients and 1\\le\\alpha\\le C, |\\beta|\\le C; Theorem 1.7 converts this into a global curvature estimate and the equivalence sup|D^2u_j|\\to\\infty \\iff \\delta^*(u_j)\\to 0 for uniformly controlled families; Theorem 1.8 constructs smooth radial solutions attaining the optimal \\delta^{-1} rate with uniformly controlled lower-order boundary data but divergent C^4 boundary traces; Proposition 6.4 shows a strict lower subsolution need not force strict convexity.","tokens_in":23720,"tokens_out":18876,"duration_ms":187449,"significance":"If the results are correct, they provide sharp boundary recession criteria for a gradient-dependent curvature equation, going beyond the Hessian-phase theory where subsolution separation alone suffices. The paper's strengths are its explicitness: no fitted parameters appear, constants are displayed with their dependencies, the Schur-complement identity is exact rather than asymptotic, and the conditional role of (MRC) is stated openly. The MRC condition is genuinely load-bearing for Theorem 1.4, and the paper demonstrates in Remark 2.7 and Proposition 2.6 that it is not automatic from strict phase subsolution; Theorem 3.3 and Remark 3.4 honestly limit its sharpness to the fixed zero-order barrier class. Theorem 1.7, by contrast, assumes the mixed derivative bound Z as part of its controlled-data hypothesis, so the global equivalence is correctly framed as conditional. The sharpness family of Theorem 1.8 and the rank-loss counterexample of Proposition 6.4 strengthen the paper's claim that the identified mechanisms are both necessary and optimal within the stated scope.","major_comments":[],"minor_comments":[{"comment":"The abstract and page 1 contain the typo 'bothDu' where 'both Du' is intended; this should be corrected in the final version.","section":"Abstract and page 1"},{"comment":"The statement of Theorem 1.5 uses \\alpha_t and \\beta_t without restating their definitions; since they are defined only in the preceding paragraph, I recommend adding a pointer or restating the definitions in the theorem statement for readability.","section":"Section 4, before Theorem 1.5"},{"comment":"Condition (5.9) writes M(x,s)\\ge 0 under the infimum, which is easy to misread as a property of the gap; I suggest defining the set explicitly, for example {x\\in\\partial\\Omega, s\\in J_x, M(x,s)\\ge 0}.","section":"Section 5, Corollary 5.3"},{"comment":"Remark 3.4 is the key interpretive caveat for the mixed estimate, but it appears only after Theorem 3.3; consider moving or at least cross-referencing it from the discussion of Theorem 1.4 so that readers understand the scope of (MRC) before the main estimate is used.","section":"Section 3, Remark 3.4"},{"comment":"Figure 2.1 is referenced only as a support-function illustration and has no descriptive caption text; either add a short caption and explanation or remove the figure, since the argument is already fully written out.","section":"Figures"}],"recommendation":"minor_revision","confidential_remarks":"I have no concerns about novelty disclosure, citation patterns, or fit with the journal's scope. The MRC condition is a new, somewhat abstract hypothesis, but the authors are unusually explicit about what it does and does not imply; the stress-test concern about MRC does not, on my reading, amount to an internal inconsistency or a hidden assumption. The paper can be accepted after the minor presentation revisions listed above."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead Huang-Jiang arXiv:2608.01065. Verdict: referee it. The new material is real: the mixed recession compatibility condition (Definition 1.2), the exact complex Schur-complement identity (Theorem 1.5) giving u_nu = beta + alpha cot delta with 1 <= alpha <= C, and the sharpness family (Theorem 1.8) attaining the delta^{-1} rate while keeping C^1, boundary C^3, and mixed derivatives bounded. The paper also proves a clean global equivalence (Theorem 1.7): under uniform (P,K_phi,Z) control, sup |D^2u| blows up iff the realized boundary gap delta*(u) collapses. The proofs are laid out in detail: the barrier construction in Section 3, the phase-lift determinant argument in Section 4, and the mean-curvature maximum principle in Section 5 are all sound as far as I can tell. I checked the Schur-complement algebra and the alpha/beta bounds; no error.\n\nThe soft spot is exactly where the authors put it. The mixed derivative estimate (Theorem 1.4) rests on (MRC), and (MRC) is not automatic: Remark 2.7 shows a strict phase subsolution can have wrong-sign linearized separation, and Proposition 2.6 shows the joint map is not concave. The necessity theorem (Theorem 3.3) only holds within the fixed zero-order barrier class (3.12); Remark 3.4 concedes that failure of (MRC) need not produce actual solutions with unbounded mixed derivatives. That is a stated limitation, not a hidden flaw, but it means the paper does not prove that (MRC) is necessary in the class of solutions. If I wanted to use Theorem 1.7 for a specific family, I would need to verify the mixed bound Z directly. The sharpness family does verify it, but the verification is a bit compressed: the derivative bounds in Step 4 (|f'''|, |f^(4)|) are sketched rather than fully expanded. I would ask the authors to expand that step or provide a table of derivative bounds.\n\nAlso worth noting: Example 4.5 correctly shows the normal gap alone cannot control double-normal when the mixed block grows, so (MRC) is not redundant. The radial example and the rank-loss model (Proposition 6.4) are useful. The paper is honest about what it does not do.\n\nTake it seriously. It deserves a careful referee, not a desk reject. If the referee asks for one thing, make it the full expansion of the sharpness-family derivative bounds.","headline":"Solid boundary regularity paper: exact phase-gap identity and sharp blow-up criterion, with the main technical condition (MRC) honestly scoped.","tokens_in":24247,"tokens_out":1818,"would_cite":true,"duration_ms":15138,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35B45","53A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Boundary curvature blow-up for convex solutions of the special Lagrangian curvature potential equation is governed exactly by the collapse of a boundary limiting-phase gap, with optimal rate $\\delta^{-1}$.","keywords":["special Lagrangian curvature potential equation","mixed recession compatibility","boundary Hessian estimates","tangent cone at infinity","curvature blow-up","convex graphs","Dirichlet problem","limiting-phase gap"],"falsifier":"Compute the boundary mixed derivative $D^2u_\\delta(\\xi,\\nu)$ for the radial sharpness family of Theorem 1.8 on the fixed ball as $\\delta\\to 0$; the proof claims it stays uniformly bounded while $u_{\\nu\\nu}$ grows like $\\delta^{-1}$. If any evaluation, numerical or analytic, shows the mixed derivative also diverging under (MRC), the sharp split between the two recession mechanisms in Theorem 1.4 and Remark 5.4 fails. Conversely, the decisive counterexample to Theorem 1.7 would be a controlled convex family with $\\delta^*(u_j)$ bounded below by a positive constant but $\\sup_\\Omega|D^2u_j|\\to\\infty$; such a family would have to realize the unbounded mixed jets of Example 4.5 as actual boundary Hessians.","tokens_in":23244,"feed_emoji":"📐","tokens_out":9472,"duration_ms":75388,"temperature":0.7,"pith_summary":"The paper aims to establish boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation $F(A[u])=\\theta$, the graph-curvature analogue of the classical special Lagrangian equation. Because the curvature matrix $A[u]$ depends on both $Du$ and $D^2u$, a phase subsolution does not supply the full linearized separation needed for mixed derivative control; the paper introduces a new condition, mixed recession compatibility (MRC), imposed only on doubly degenerate level jets, and proves that it yields a uniform mixed derivative bound that is sharp for the fixed zero-order barrier class. For the double-normal direction the paper proves an exact complex Schur-complement identity $u_{\\nu\\nu}=\\beta+\\alpha\\cot\\delta$ with $1\\le\\alpha\\le C$ and $|\\beta|\\le C$, where $\\delta$ is the actual boundary limiting-phase gap. The global consequence is that, within uniformly controlled families, curvature blows up if and only if that gap collapses, at the optimal rate $\\delta^{-1}$. A sympathetic reader would care because this converts a boundary blow-up question into a single geometric quantity that can be read off from the boundary phase.","feed_headline":"Curvature blows up exactly when a boundary phase gap collapses","feed_subtitle":"The Hessian grows at the optimal rate 1/δ as the realized limiting-phase gap δ shrinks to zero.","key_machinery":"The machinery has two independent parts. The mixed recession compatibility condition (MRC) is imposed on doubly degenerate recession jets, boundary jets where the Hessian tends to infinity while both quadratic coercive terms $a(Dd,Dd)$ and $Q=(p-Du)^\\top a(p-Du)$ tend to zero, and it requires a uniform positive margin for the full linearized separation $J-\\tau\\ell$ on all such jets; it is what makes the concavified barrier $v=\\Phi_\\mu(u-\\underline u)+\\tau d-Nd^2$ strictly negative under the linearized operator, producing the uniform boundary bound on $D^2u(\\xi,\\nu)$ in Theorem 1.4. The double-normal part is an exact complex Schur complement: writing $C_t+\\sqrt{-1}D^2u$ in tangential-normal blocks and eliminating the normal block gives $\\delta_t=\\arctan(\\alpha_t/(r_\\nu-\\beta_t))$ with explicit $\\alpha_t,\\beta_t$, which yields the pointwise identity $r_\\nu=\\beta_t+\\alpha_t\\cot\\delta_t$ without any asymptotic expansion. A maximum principle for the mean curvature of convex constant-phase graphs then carries boundary Hessian control to the whole graph, and a rank-one limiting formula computes the boundary gap in the sharpness examples.","core_discovery":"The central discovery is an exact quantitative control of boundary Hessian blow-up on the convex branch $D^2u\\ge 0$ of the equation $F(A[u])=\\theta$. At a boundary point the paper computes the phase gap $\\delta_t=L_t-F(A_t[u])$ in closed form as $\\arctan(\\alpha_t/(r_\\nu-\\beta_t))$, equivalently $r_\\nu=\\beta_t+\\alpha_t\\cot\\delta_t$, where $r_\\nu=u_{\\nu\\nu}$ is the double-normal Hessian entry, $\\alpha_t\\ge 1$ and $|\\beta_t|$ are bounded in terms of the gradient and mixed derivative bounds, and $L_t$ is the limiting phase obtained by sending the double-normal entry to $+\\infty$ while holding tangential and mixed blocks fixed. At $t=1$, setting $\\delta^*(u)=\\min_{\\partial\\Omega}(B(x,u_\\nu(x))-\\theta)$, Theorem 1.7 concludes that every $(P,K_\\varphi,Z)$-controlled convex solution satisfies $\\sup_\\Omega|D^2u|+\\sup_\\Omega|A[u]|\\le C(1+\\cot\\delta^*(u))$ and $\\sup_\\Omega|D^2u|\\ge\\cot\\delta^*(u)-C$, so for a uniformly controlled family, $\\sup_\\Omega|D^2u_j|\\to\\infty$ if and only if $\\delta^*(u_j)\\to 0$. The paper also shows that the mixed derivative estimate requires a genuinely new hypothesis rather than a phase subsolution, and that the rate $\\delta^{-1}$ is attained by smooth radial solutions.","pith_inferences":["Editorial extension: the exact Schur-complement identity is pointwise in $t$, so the same block calculation should apply to other gradient-dependent curvature equations whose composite curvature map is not jointly concave; the natural test is whether the resulting identity still yields a two-sided bound with an explicit $\\alpha\\ge 1$.","Editorial extension: because (MRC) is stated at the level of level jets, the sharpness of the barrier class leaves open whether the doubly degenerate jets are dynamically reachable by actual solutions; a construction of a genuine solution family realizing the bad jets would clarify whether the condition is necessary as well as sufficient.","Editorial extension: the equivalence (1.15) suggests a computable boundary diagnostic for numerical solvers: monitor the realized gap $\\delta^*(u)$ on the boundary; if it approaches zero while the interior Hessian stays bounded, that would conflict with the lower bound in (1.14) and indicate a mixed-block effect left out of the controlled-data class.","Editorial extension: the rank-loss model in Proposition 6.4 indicates that any strict-convexity existence theory for this boundary problem needs a quantitative lower curvature estimate, since a strict lower subsolution alone cannot preserve full rank of $D^2u$."],"forward_implications":["For any family of convex solutions satisfying uniform $C^1$ and boundary mixed derivative bounds, $\\sup_\\Omega|D^2u_j|\\to\\infty$ if and only if the realized boundary limiting-phase gap $\\delta^*(u_j)\\to 0$, with the same equivalence for the curvature norm $|A[u_j]|$.","The double-normal derivative obeys explicit two-sided bounds $\\cot\\delta - C \\le u_{\\nu\\nu} \\le C(1+\\cot\\delta)$, so the blow-up rate $\\delta^{-1}$ is optimal and is attained by smooth radial solutions.","The mixed derivative estimate holds uniformly along the whole vertical graph homotopy $G_t=\\theta$ whenever (MRC) holds with a uniform margin, giving a priori boundary Hessian bounds of the form $C_0+C_1\\cot\\sigma$ under a normal-window phase gap.","On the convex branch with $0<\\theta<\\pi/2$, the boundary gap satisfies $\\delta^*\\ge\\pi/2-\\theta>0$, so boundary curvature blow-up on the convex branch is a genuinely high-phase phenomenon.","A strict lower subsolution does not force strict convexity: there are smooth data with a strict convex subsolution whose unique convex solution is only rank-one."],"supporting_citations":[{"why":"Establishes the calibrated-geometry origin of the special Lagrangian phase, the context in which the curvature potential equation sits.","marker":"[8]"},{"why":"Introduces special Lagrangian curvature, the geometric object whose principal curvatures are the eigenvalues of $A[u]$.","marker":"[19]"},{"why":"Provides the foundational Dirichlet theory for graph curvature equations that the boundary second derivative estimates build on.","marker":"[2]"},{"why":"Introduces the finite asymptotic limit obtained by sending the normal Hessian eigenvalue to infinity, the precursor of the limiting-phase gap used here.","marker":"[22]"},{"why":"Proves the interior Hessian estimate for convex solutions of the classical special Lagrangian equation, the interior counterpart whose boundary analogue the paper establishes.","marker":"[3]"},{"why":"Supplies interior Hessian and gradient estimates for special Lagrangian curvature equations, identifying phase and convexity as the regularity sources carried to the boundary here.","marker":"[17]"},{"why":"Treats graph-curvature boundary estimates with first-order drift terms, the technical context for the drift-corrected boundary curvature in (MRC).","marker":"[13]"},{"why":"Computes the asymptotic interior and optimal viscosity pseudoconvexity for the special Lagrangian theory including the graph-curvature operator, background for the limiting phase.","marker":"[9]"}],"fun_headline_variants":["Hessian blow-up exactly when boundary phase gap collapses","Boundary phase gap controls curvature blow-up at optimal rate","Sharp recession criteria: blow-up rate 1/δ as gap vanishes","No subsolution needed for mixed derivative bound in SLCPE"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that on every boundary point where the two standard quadratic control terms both vanish, a certain linearized-separation quantity must stay positive by a fixed margin; the paper shows this does not follow automatically from the equation or from a strict phase subsolution.","fun_headline_variants_meta":{"raw":{"variants":["Hessian blow-up exactly when boundary phase gap collapses","Boundary phase gap controls curvature blow-up at optimal rate","Sharp recession criteria: blow-up rate 1/δ as gap vanishes","No subsolution needed for mixed derivative bound in SLCPE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3425,"prompt_tokens":1093,"completion_tokens":2332,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":2261}},"tokens_in":709,"tokens_out":2332,"duration_ms":16372,"temperature":1.0,"reasoning_tokens":2261,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:16:59.837214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the boundary mixed derivative $D^2u_\\delta(\\xi,\\nu)$ for the radial sharpness family of Theorem 1.8 on the fixed ball as $\\delta\\to 0$; the proof claims it stays uniformly bounded while $u_{\\nu\\nu}$ grows like $\\delta^{-1}$. If any evaluation, numerical or analytic, shows the mixed derivative also diverging under (MRC), the sharp split between the two recession mechanisms in Theorem 1.4 and Remark 5.4 fails. Conversely, the decisive counterexample to Theorem 1.7 would be a controlled convex family with $\\delta^*(u_j)$ bounded below by a positive constant but $\\sup_\\Omega|D^2u_j|\\to\\infty$; such a family would have to realize the unbounded mixed jets of Example 4.5 as actual boundary Hessians.","supporting_citations":[{"cited_title":"Harvey and H","cited_arxiv_id":null,"evidence_quote":"Establishes the calibrated-geometry origin of the special Lagrangian phase, the context in which the curvature potential equation sits."},{"cited_title":"Smith,Special Lagrangian curvature, Math","cited_arxiv_id":null,"evidence_quote":"Introduces special Lagrangian curvature, the geometric object whose principal curvatures are the eigenvalues of $A[u]$."},{"cited_title":"Caffarelli, L","cited_arxiv_id":null,"evidence_quote":"Provides the foundational Dirichlet theory for graph curvature equations that the boundary second derivative estimates build on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the finite asymptotic limit obtained by sending the normal Hessian eigenvalue to infinity, the precursor of the limiting-phase gap used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the interior Hessian estimate for convex solutions of the classical special Lagrangian equation, the interior counterpart whose boundary analogue the paper establishes."},{"cited_title":"Jiao and Z","cited_arxiv_id":null,"evidence_quote":"Treats graph-curvature boundary estimates with first-order drift terms, the technical context for the drift-corrected boundary curvature in (MRC)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the asymptotic interior and optimal viscosity pseudoconvexity for the special Lagrangian theory including the graph-curvature operator, background for the limiting phase."}],"review_version":1}