{"id":"410c0aa9-5c07-4061-8801-6a970f4ea61f","arxiv_id":"2607.23592","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Convexity and critical phase are necessary for SLCE interior estimates: dim-2 Lipschitz non-C¹ and C^{1,1/3}-sharp curvature blow-up, plus a dim-3 subcritical gradient-jump viscosity solution.","lead":"Three explicit counterexamples show that interior curvature and C¹ estimates for the special Lagrangian curvature equation fail without convexity or the critical phase. The constructions pin down where recent a priori theory is sharp and cannot be relaxed.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The 3D collar step (Lemma 5.2) is the right place to probe, but the x-dependent perturbation cancels exactly on ∂K; the residual risk is one compressed second-normal-derivative identity, not the perturbation. Verdict unchanged.","rationale":"I agree with the reader that the only place a hidden error could live is the 3D section, specifically Lemma 5.2–5.3; the 2D constructions are elementary, fully explicit, and constant-tracked (Remarks 3.1 and 4.1 give simultaneous admissible choices), and the viscosity vacuity arguments in Theorem 1.1 (impossibility of touching from above at the |z|-corner; k₁ ≤ −τ forcing F_Θ < 0 for lower test functions, using arctan(−cot Θ) + π/2 = Θ) check out line by line, as does the C^{1,1/3} scaling in Theorem 1.2 (z ≍ τεs + s³, R′ ≍ ε^{1/2} at z ≍ ε^{3/2}, c_Θ = (6/(τK₀))^{1/3}). Where I diverge from the reader is the mechanism of the risk: the weakest_assumption frames it as the x-dependent congruence P(x)^{1/2} perturbing the Mooney–Savin operator, but that dependence cancels exactly on ∂K in the normal-derivative comparison, because v and Φ_λ agree to full second order there (strict ellipticity in νν pins the one undetermined entry). The true residual soft spot is narrower: the one-line identity ∂_{νν}(det D²v) = γκ_ξω in the tangent-kernel case and the unstated compactness interpolation between the two cases, both of which I traced to be correct in structure but which are compressed enough that a sign error would be invisible without redoing the computation. This is a correctness-risk refinement, not a new objection, and it does not rise to the level of changing the verdict: every orientation, jump sign, and viscosity inequality I checked is consistent, and the construction has the self-correcting redundancy of multiple independent sign mechanisms (ω < 0 from the strict phase minimum, γ > 0 from rank two, κ_ξ > 0 from strict convexity of the sublevel set). Hence agreement \"partial\" (same location, different precise sub-step) and verdict UNCHANGED — ACCEPT with low correctness risk stands, with the Lemma 5.2 sign identities as the single worthwhile verification target.","tokens_in":20578,"tokens_out":12729,"duration_ms":190873,"concrete_test":"Verify the two sign identities of Lemma 5.2 for the explicit Φ_λ, retaining all suppressed terms. Concretely: for λ near λ₀ = ½tan(Θ*/2), compute ∂K = {ϑ_λ = Θ*}, the kernel field ξ(x) of D²Φ_λ, and ω = −∂_νϑ_λ/F_{νν}; check ∂_ν det = γ(ξ·ν)²ω ≤ 0 on ∂K. At any boundary point where ξ·ν = 0, recompute ∂_{νν} det = γκ_ξω keeping the ∂_ξν and second-fundamental-form cross terms (noting that at the poles (0,0,±x₃), D²Φ_λ = diag(2λ/(1+x₃), 2λ/(1−x₃), 0), so ξ = e₃ = ν there — the easy case; tangency points, if they exist, are the real test). Then evaluate det D²v on a thin exterior collar via the second-order Taylor expansion on a fine grid of ∂K. If det goes positive anywhere, the inertia (+,+,−) fails and Theorem 1.3 collapses; if it stays uniformly negative, the concern is fully settled. As a cheap warm-up, a computer-algebra re-evaluation of D²ϑ_λ(0) in (5.2) confirms the nondegenerate mi","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for Theorem 1.3 is Lemma 5.2's claim that det D²v < 0 on a uniform exterior collar of the analytic core K. Tracing the chain: (i) full Hessians of v and Φ_λ agree on ∂K — correct, because all second derivatives except the pure normal one are fixed by the Cauchy data, and F is strictly elliptic in the νν entry (F_{νν} > 0 by (5.1)), so the equation F = Θ* pins v_{νν} = (Φ_λ)_{νν}; (ii) ω := v_{ννν} − (Φ_λ)_{ννν} = −∂_νϑ_λ/F_{νν} < 0 — here the explicit x-dependence of F cancels exactly on ∂K, because the x-slot and the full Hessian slot both agree between v and Φ_λ there. So the reader's flagged mechanism (spatial dependence destabilizing the Mooney–Savin structure) is neutralized by an exact cancellation, not merely a smallness argument. The genuinely compressed step is (iii): at boundary points where the kernel ξ of D²Φ_λ is tangent to ∂K, the proof asserts ∂_{νν}(det D²v) = γκ_ξω < 0 in one line, suppressing terms involving ∂_ξν and tangential third derivatives. Those terms do cancel between v and Φ_λ (full Hessians and all tangentially-differentiated third derivatives agree on ∂K, and det D²Φ_λ ≡ 0 identically kills Φ-side normal derivatives), so the identity is correct in substance — but it is the single place where a sign or second-fundamental-form error could hide, and a sign flip there propagates: det D²v > 0 on part of the collar would destroy the inertia (+,+,−) needed for Lemma 2.4 and the strict monotonicity of T₃ behind the fibre collapse. Finally, the \"by compactness\" uniform-collar claim silently interpolates between the two cases: it works only because a(x) := ∂_ν det = γ(ξ·ν)²ω ≤ 0 on all of ∂K and b(x) := ∂_{νν}det is negative on the zero set of a. That is true here, but it is not written out and depends on the same identity (iii). Nothing I traced contradicts the claims: the jump sign (5.3) is consistent with T₃ decreasing and yields the concave corner making the supersolution condition vacuous; the w_k = w − x₃²/k subsolution","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper constructs three explicit counterexamples for the special Lagrangian curvature equation (SLCE) ∑ arctan κ_i = Θ, aimed at showing that the hypotheses of the interior a priori estimates of Qiu–Zhou [20] — convexity and the critical phase Θ = (n−2)π/2 — are necessary. Theorem 1.1: in 2D, for every 0 < Θ < π/2 and every corner slope α > 0, a rotational constant-Gauss-curvature surface is offset to signed distance t = −τ (τ = cot Θ); the admissible post-focal branch is written as a one-sided graph and reflected across the focal line, yielding a Lipschitz viscosity solution with an exact corner, classical off the line, with the viscosity inequalities verified directly by test functions (subsolution vacuous, supersolution via the normal-curvature bound k_y ≤ −τ). Theorem 1.2: a family of smooth admissible 2D solutions with Jacobi factor J_{s,ε}(0) = −τε converging to a quadratic zero produces uniformly C¹-bounded graphs with |D²u_ε(0)| ≍ ε⁻¹, a uniform C^{1,1/3} bound, and blow-up of every C^{1,β} seminorm for β > 1/3; the limit has the exact focal profile (3c_Θ/4)|z|^{4/3}. Theorem 1.3: in 3D subcritical phase, a Mooney–Savin-type construction for the Legendre-dual operator F(x, D²w) = tr arctan(P(x)^{1/2}D²wP(x)^{1/2}) produces a Lipschitz viscosity solution whose gradient jumps across a compact analytic surface; the new difficulty relative to [15] is the explicit x-dependence via P(x)^{1/2}, handled through a strictly convex analytic core K = {ϑ_λ ≤ Θ*}, a Cauchy–Koval","tokens_in":20921,"tokens_out":6860,"duration_ms":121159,"significance":"If the constructions hold up — and after detailed checking I believe they do — this paper settles the sharpness of the Qiu–Zhou interior theory for the SLCE and identifies three genuinely distinct obstruction mechanisms (reflected first-order focal singularity, smooth higher-order focal concentration, and Bellman–Legendre collapse). Particular strengths worth naming: the examples are fully explicit and constructive, with all constants tracked in closed form (Remarks 3.1 and 4.1 give non-optimized but completely explicit admissible radii); Theorem 1.2 is not merely a failure of estimates but pins down the exact critical exponent 1/3 with an explicit limiting profile and constant c_Θ = (6/τK_0)^{1/3}, a falsifiable, parameter-free prediction; the viscosity conditions on the singular sets are verified by direct test-function arguments rather than appeal to stability alone (Theorem 1.1), and the 3D construction adapts Mooney–Savin to an operator with genuine x-dependence, with the potentially dangerous perturbation shown to cancel exactly on the core boundary. The paper also complements the Nadirashvili–Vlăduţ, Wang–Yuan, and Mooney–Savin singular-solution literature by carrying it ove","major_comments":[{"comment":"The sign det D²v < 0 on the exterior collar is the load-bearing input for the inertia (+,+,−), Lemma 2.4, and the fibre collapse in Lemma 5.3; a sign error here would invalidate Theorem 1.3. In the case where the kernel direction ξ of D²Φ_λ is tangent to ∂K, the identity ∂_{νν}(det D²v) = γκ_ξω < 0 is derived in a single sentence. I checked the identity and believe it is correct in substance — tangentially differentiated third derivatives of v and Φ_λ agree on ∂K because the full Hessians and the Cauchy data agree, the Φ-side of term II vanishes since det D²Φ_λ ≡ 0 identically, and the x-dependence of F cancels exactly on ∂K since the x-slot and Hessian slot coincide there. But the manuscript suppresses precisely these cancellation arguments (why term II vanishes, i.e. why G_{νν,kl} = 0 unless (k,l) = (ξ,ξ), and why differentiating v_{νν} − Φ^λ_{νν} = 0 twice along the boundary geodesic","section":"§5, Lemma 5.2 (proof, tangent-kernel case)"}],"minor_comments":[{"comment":"The displayed formula for the normal curvature in the ∂_y direction is typographically garbled (missing fraction bar): it should read k_y = u_{yy}/(√(1+|Du|²)(1+u_y²)). Similar formatting artifacts occur elsewhere (e.g., 'notC 1' in the abstract, the double period at the end of the statement of Lemma 5.2).","section":"§2.3, Eq. (2.5)"},{"comment":"The claim that H is a 'distance-expanding global diffeomorphism onto its image' is justified by det DH = cof(D²w)_{33} > 0 together with uniform closeness of DH to diag(2λ, 2λ, 1). Strictly, positive Jacobian plus closeness gives a local diffeomorphism; the global injectivity on U_0 uses the quantitative uniform closeness (e.g., a lower bound on the minimal singular value of DH). One sentence making this explicit would close a small gap.","section":"§5, Lemma 5.3 (proof)"},{"comment":"In the subsolution approximation w_k = w − x_3²/k, the phrase 'by the strict ellipticity of F_{ij} evaluated on the rank-one segment' is opaque. Presumably one integrates d/dt F(x, D²w − t x_3²·(·)) along the segment joining D²w and D²w_k using F_{33} > 0; please spell this out.","section":"§5, proof of Theorem 1.3"},{"comment":"Reference [20] (Qiu–Zhou) is cited only as '2024' with no journal or arXiv identifier; since the paper's stated purpose is to demonstrate sharpness of that work's hypotheses, full bibliographic data should be supplied at revision.","section":"References"},{"comment":"I verified the computation (5.2) independently via the spectral formula (including the divided-difference term, which contributes −16λ³/(1+4λ²) to the 11-entry); it is correct. Since this positivity is what makes the core K a compact strictly convex analytic body, a line indicating that the mixed/pair terms from ∂_kS(0) vanish or vanish in the (1,1)-entry would help the reader.","section":"§5, Lemma 5.1, Eq. (5.2)"},{"comment":"In the blow-up estimate for [Du_ε]_{C^{0,β}}, the displayed lower bound on |R'_ε(z♯_ε)| is typeset ambiguously ('≥ 1/2τε1/2'); it should read ≥ ε^{1/2}/(2τ). Also, it is worth stating explicitly that the quotient is evaluated between z = 0 and z♯_ε using (u_ε)_z(0,0) = 0.","section":"§4, proof of Theorem 1.2"},{"comment":"Figures 1–5 are genuinely helpful and the 'not to scale' disclaimers are appropriate. Consider adding one numerically computed meridian profile (the ODE r'' + K_0 r = 0 is explicit) to replace the 'local asymptotic model' curves in Figure 1.","section":"Figures 1–5"}],"recommendation":"minor_revision","confidential_remarks":"The first author is a co-author of the Qiu–Zhou estimates [20] whose sharpness this paper establishes; the self-citation pattern is therefore intrinsic to the result rather than promotional, and I see no novelty-disclosure issue. The manuscript is a good fit for a journal in fully nonlinear PDE / geometric analysis. The single point I would want the editor to insist on is an expanded proof of the determinant sign in Lemma 5.2 (major comment 1): I verified the identity independently and believe it is correct, but the published argument should not require the reader to redo it."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: Qiu–Zhou’s interior theory for the special Lagrangian curvature equation really does need convexity and the critical phase. The paper supplies three constructive counterexamples that close that question.\n\nWhat is new is the geometry. In 2D they take a rotational surface of constant positive Gauss curvature, push it by the parallel distance −τ, and keep the post-focal branch. Reflection across the focal line gives an explicit Lipschitz viscosity solution that is not C¹ (Thm 1.1), with a clean expansion and a short test-function argument on the singular line. The same family, smoothed so the Jacobi factor vanishes quadratically, produces a sequence of smooth admissible graphs with uniform C¹ bound, curvature blowing up at one point, and the sharp threshold C^{1,1/3} (Thm 1.2). The limiting meridian profile is exactly |z|^{4/3}. Constants are tracked in the remarks; the branch condition is verified by the elementary Lemma 2.5. That part is elementary, self-contained, and hard to argue with.\n\nThe 3D subcritical example adapts Mooney–Savin’s Bellman–Legendre collapse. The dual operator carries explicit x-dependence through P(x)^{1/2}, but on the analytic core boundary the Hessians and the x-slot agree, so the dependence cancels exactly when comparing normal derivatives. The residual risk is one compressed second-normal-derivative identity for det when the kernel is tangent to ∂K; the cancellations are real (full Hessians and tangential third derivatives match, det Φ ≡ 0), but they are written in one line. If that sign flipped, the inertia and fibre collapse would fail. Nothing I checked contradicts the claim, and the subsequent Legendre jump and viscosity perturbation are standard.\n\nCitations are appropriate: they cite their own estimates as the target to be shown sharp, and the classical parallel-surface / front literature and Mooney–Savin correctly. No circularity.\n\nThis is for people who work on fully nonlinear geometric PDE or calibrated geometry and care about the precise range of a priori estimates. It deserves a serious referee. I would accept it for peer review and would cite the 2D examples myself.","headline":"Three explicit SLCE counterexamples that make Qiu–Zhou’s convexity and critical-phase assumptions necessary; 2D is clean and sharp, 3D is a controlled Mooney–Savin adaptation with one compressed collar identity.","tokens_in":21948,"tokens_out":560,"would_cite":true,"duration_ms":10357,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","53C42","35B45","49L25","53A10"],"pacs":[],"model":"grok-4.5","headline":"Three counterexamples show that convexity and the critical phase are strictly necessary for interior regularity of the special Lagrangian curvature equation.","keywords":["special Lagrangian curvature equation","viscosity solutions","a priori estimates","focal sets","Lipschitz singularities","parallel surfaces","interior regularity","critical phase"],"falsifier":"Verify by direct computation that the reflected post-focal parallel-surface graph satisfies the viscosity inequalities for the constant-phase equation along the singular line, and that retaining the full P(x)^{1/2} congruence in the three-dimensional dual still yields a strict negative vertical derivative outside the core and a well-defined gradient jump after Legendre transform.","tokens_in":21325,"feed_emoji":"📐","tokens_out":917,"duration_ms":40398,"temperature":0.7,"pith_summary":"The special Lagrangian curvature equation requires the principal curvatures of a hypersurface to have arctangents summing to a fixed phase. Recent interior theory gives curvature control for smooth graphical solutions when the graph is convex or the phase is critical, plus gradient estimates for every constant phase. This paper constructs three counterexamples proving those structural hypotheses cannot be removed. In two dimensions it builds an explicit Lipschitz viscosity solution with a genuine corner, and a family of smooth admissible solutions with uniformly bounded C¹ norm whose curvature blows up, with every uniform C^{1,β} bound failing for β > 1/3. In three dimensions and subcritical phase it builds a Lipschitz viscosity solution whose gradient jumps across an analytic surface. The examples isolate focal degeneration of parallel surfaces and dual-map fibre collapse as real geometric obstructions, not artefacts of proof technique.","feed_headline":"SLCE regularity fails without convexity or critical phase","feed_subtitle":"Three constructions give Lipschitz corners, curvature blow-up at Hölder 1/3, and gradient jumps.","key_machinery":"Parallel surfaces of constant positive Gauss curvature and their focal sets in two dimensions, and a dual Bellman–Legendre collapse of a rank-two phase core in three dimensions. The same Jacobi factor that makes the parallel map lose rank forces a principal curvature of the SLCE surface to blow up; after a coordinate change the dual gradient map collapses vertical fibres onto an analytic surface, which a local Legendre transform turns into a gradient jump.","core_discovery":"The convexity assumption and the critical-phase restriction used for interior a priori estimates of the special Lagrangian curvature equation are strictly necessary. Without them there exist Lipschitz viscosity solutions that fail to be C¹, and sequences of smooth admissible solutions with uniform C¹ control whose second derivatives become unbounded, with sharp two-dimensional Hölder threshold exactly 1/3 in the focal direction.","pith_inferences":["The same focal Jacobi-factor mechanism is likely to obstruct curvature estimates for other fully nonlinear hypersurface equations whose linearization sees the normal map.","Because the constant-phase equation already produces singularities, the related optimal-transport problem with relativistic cost can lose regularity without any freedom in the densities.","A direct next test is whether critical-phase nonconvex solutions in dimension three still enjoy C¹ or curvature bounds, or whether a different obstruction appears."],"forward_implications":["Interior curvature estimates for the SLCE cannot follow from positive phase and uniform C¹ control alone on the nonconvex branch.","The exponent 1/3 is the exact borderline Hölder regularity for the two-dimensional higher-order focal degeneration.","Subcritical-phase Lipschitz viscosity solutions of the three-dimensional SLCE need not become C¹.","Adding flat variables extends the smooth two-dimensional curvature-blow-up examples to every dimension n ≥ 2 while keeping the numerical phase fixed."],"fun_headline_variants":["SLCE needs convexity and critical phase: three sharp counterexamples","Lipschitz non-C¹ SLCE solution from post-focal constant-curvature branch","Smooth SLCE solutions: C¹ bounded yet curvature blows up, Hölder 1/3 sharp","Subcritical 3D SLCE: Mooney-Savin Lipschitz solution with gradient jump","Convexity and critical phase are necessary for SLCE a priori estimates"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"In the three-dimensional dual construction, the explicit spatial dependence of the curvature operator is only a small perturbation near the origin and does not destroy the determinant sign change, the inertia, or the vertical-fibre collapse needed for the gradient jump.","fun_headline_variants_meta":{"raw":{"variants":["SLCE needs convexity and critical phase: three sharp counterexamples","Lipschitz non-C¹ SLCE solution from post-focal constant-curvature branch","Smooth SLCE solutions: C¹ bounded yet curvature blows up, Hölder 1/3 sharp","Subcritical 3D SLCE: Mooney-Savin Lipschitz solution with gradient jump","Convexity and critical phase are necessary for SLCE a priori estimates"]},"model":"grok-4.5","effort":"low","cost_usd":0.003359,"raw_usage":{"total_tokens":1099,"prompt_tokens":700,"num_sources_used":0,"completion_tokens":92,"cost_in_usd_ticks":33588000,"prompt_tokens_details":{"text_tokens":700,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":307,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":700,"tokens_out":92,"duration_ms":6253,"temperature":1.0,"reasoning_tokens":307,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T18:03:02.869249+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Verify by direct computation that the reflected post-focal parallel-surface graph satisfies the viscosity inequalities for the constant-phase equation along the singular line, and that retaining the full P(x)^{1/2} congruence in the three-dimensional dual still yields a strict negative vertical derivative outside the core and a well-defined gradient jump after Legendre transform.","supporting_citations":[],"review_version":1}