{"id":"72b768eb-fbe6-4a99-806b-eb0705799b32","arxiv_id":"2411.17133","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Half-line singularities of viscosity solutions are removable for fully nonlinear elliptic PDEs with a Jacobi inequality, proven by a doubling argument; the single-side version is new.","lead":"A new proof shows that viscosity solutions to several geometric PDEs cannot be singular along a half-line: any singular set touching the domain boundary at a single point must be empty. The method is short, unifies three known examples, and removes a broader class of 'single side' singularities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Savin stability is asserted rather than verified: the uniform linearization and the closedness of the singular set are the load-bearing step, and the one-paragraph appendix does not establish them.","rationale":"The reader's weakest assumption correctly identifies the Savin stability lemma in the Appendix as the hinge of the proof. I agree that the paper does not write out uniform strict ellipticity, smallness thresholds, or a covering argument. My read adds a sharper point: the appendix linearizes around a smooth solution u, but at the point of use (Step 3) u is only known to be twice differentiable on sing(u)^c, not smooth. One must first use Savin on normalized functions to show that the twice differentiability set is open and u is smooth there; otherwise Step 1's assertion that sing(u) is closed and Step 3's local smooth convergence are circular. With this two-step argument filled in, the main doubling proof appears internally sound: the Jacobi inequality maximum principle in Step 2 has only a sign typo, and Step 4's use of (1.6) is legitimate. Since the gap is substantial but plausibly repairable, the conditional verdict is appropriate; I do not see a reason to reject. My concrete test targets exactly the missing verification.","tokens_in":7932,"tokens_out":30384,"duration_ms":300631,"concrete_test":"Fix x0∈sing(u)^c and its second-order Taylor polynomial q(y)=u(x0)+p·y+½y^T A y. For smooth approximants u_k, define w_k(y)=(u_k(x0+ry)−q(ry))/r^2 and the operator F_r(y,M,Q)=G(A+M, p+rQ)−G(A,p). Check explicitly for one of the claimed examples (e.g., the special Lagrangian or minimal-surface equation with its Jacobi function) that for small r and all large k, F_r is uniformly strictly elliptic, C^2 in (M,Q), with F_r(y,0,0)=0, so Savin's theorem gives uniform C^{2,α} bounds on w_k in B_{1/2}. Then verify that these bounds imply uk→u in C^{2,α} near x0 and that sing(u)^c is open. If the p-term rQ destroys uniformity or F_r(y,0,0)≠0, Step 3 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof's global control rests on Step 3, which claims uk→u in C^{2,α}_{loc}(sing(u)^c) by Savin's small perturbation theorem [S07, Thm 1.3]. To obtain this from the Appendix's linearized equation v_k=uk−u, one must check that F(x,M,p)=G(D^2u+M,Du+p)−G(D^2u,Du) satisfies Savin's hypotheses with constants uniform over the covering balls: uniform strict ellipticity near M=p=0, uniform C^2 bounds, and F(0,0,x)=0. This requires that u is already smooth (or at least C^2 with controlled derivatives) on the subdomain Ω, but that is exactly what is being proved at this stage. In particular, Step 1 asserts 'Since sing(u) is closed'; with sing(u) defined as the complement of twice differentiability, this is not automatic for a general viscosity solution and would follow from the same Savin argument. Thus the proof has a circular dependency: the uniform bound A in (2.5), and hence the entire doubling/Savin conclusion, rests on an unstated two-step argument (first apply Savin to u or to normalized u_k near a twice-differentiable point, then linearize). The paper's one-paragraph Appendix does not supply the needed verification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives a short proof of removability of singularities for viscosity solutions of fully nonlinear elliptic PDEs that satisfy two structural hypotheses: classical density (existence of uniformly convergent smooth approximating solutions) and a Jacobi inequality with the associated higher-order estimates. The main theorem, Theorem 1.4, states that if sing(u) is confined to one side of a box, then u is smooth in the interior of the box; Theorems 1.1–1.3 derive the corresponding segment and half-line cases for the Monge-Ampère equation and for the general equations, with the minimal surface and special Lagrangian equations as examples. The proof has four steps: (1) a geometric reduction of the singular set to a single side of a box; (2) a doubling inequality for b(D^2u,Du) obtained by comparing with a one-dimensional Green's function and using the Jacobi inequality; (3) an appeal to Savin's small perturbation theorem to pass from uniform convergence of smooth approximants to local C^{2,α} convergence on the regular set; and (4) higher-order estimates from the uniform bound on b. The paper is explicit that the statements for Monge-Ampère, minimal surface, and special Lagrangian equations are known results, and that its contribution is a new, unified proof.","tokens_in":8169,"tokens_out":62660,"duration_ms":572135,"significance":"If completed, the paper's method is of genuine interest: the doubling inequality in Step 2 is elegant and the contradiction at (2.3) is valid; the single-side formulation in Theorem 1.4 and the keyhole iteration idea in Example 1.1 are appealing; and the paper is candid that the target statements are known for its main examples, citing the classical sources (Caffarelli; Bombieri–De Giorgi–Miranda; Warren–Yuan). The general conditional theorem is a clean framework, and Remark 1.6 lists several recent equations with Jacobi-type inequalities to which the method might plausibly extend. The proof is short and readable. However, the load-bearing stability step (Step 3 and the Appendix) is not completed as written: the passage from uniform convergence to C^{2,α}_loc convergence on the regular set requires regularity of the limit solution u that is exactly what the theorem aims to establish. Because this step is essential for the uniform constant A in (2.5), the central proof is currently incomplete; the paper's value rests on closing this gap, either by a full stability argument or by a strengthened hypothesis.","major_comments":[{"comment":"The transition from uniform convergence u_k → u to local C^{2,α} convergence on the regular set is not justified. The stability proposition in the Appendix is stated under the hypothesis that u is smooth on the subdomain Ω, but Step 3 applies it with Ω a neighborhood of R inside sing(u)^c, where u is only known to be twice differentiable. The linearized operator F(M,p,x) = G(D^2u(x)+M, Du(x)+p) − G(D^2u(x),Du(x)) is not known to satisfy Savin's hypotheses — C^2 in (M,p) uniformly in x, uniform strict ellipticity near (M,p)=0, and F(0,0,x)=0 — on that set, because D^2u and Du are not known to be bounded or continuous there. Consequently the claim 'u_k → u in C^{2,α}_loc(sing(u)^c)' and the uniform bound sup_R b_k ≤ A in (2.5) are not established; the argument is circular in that it assumes the smoothness of u near R that Step 4 is supposed to produce. Even granting smoothness of u on Ω, the one-paragraph proof of Proposition 3.1 sketches rather than verifies the uniform-in-k rescaling, the flatness threshold of Savin's theorem, and the covering argument. This is the load-bearing step of the proof and needs either a full stability argument under the weaker hypothesis or an explicit strengthening of the hypotheses.","section":"Step 3 and §3 (Appendix)"},{"comment":"Step 1 uses 'Since sing(u) is closed' to conclude that the shrunken box B′ has R = ∂B′ \\ L0 compactly contained in sing(u)^c. For the generality claimed in Theorems 1.2 and 1.4 — any equation with classical density and a Jacobi inequality — closedness of the complement of the twice-differentiability set is not a consequence of Definition 1.1 and is not proved in the paper; the remark in §1.4 about Savin's theorem and Alexandrov-type results is an assertion rather than an argument. The property is known for the three model equations (Pogorelov for Monge-Ampère, and full interior regularity for minimal surface and special Lagrangian), but the conditional theorems as stated need closedness either proved from the hypotheses or added explicitly.","section":"Step 1 (proof of Theorem 1.4)"},{"comment":"The reduction 'We prove Theorem 1.4, since the others follow from this one' is not demonstrated for Theorems 1.1 and 1.2. When sing(u) is contained in a line segment with an endpoint in B1(0), the segment generically meets the boundary of any box B ⊂ B1 in two points on opposite sides, so the single-side hypothesis of Theorem 1.4 is not automatically satisfied; the half-line case advertised in the abstract requires a further argument (such as a last-singular-point or iteration argument along the segment, along the lines of Example 1.1), and none is supplied. The claimed reductions should be written out.","section":"§2 (opening) and §1.3 (Theorems 1.1–1.2)"}],"minor_comments":[{"comment":"The Jacobi-inequality computation in (2.2) is written for (1/η) after cancelling the positive factor W from the test function φ = W/η, but the cancellation is not displayed; writing the computation for φ explicitly would make the contradiction in (2.3) easier to verify.","section":"§2, Step 2 (Eqs. (2.2)–(2.3))"},{"comment":"The final sentence 'Varying x0, this shows sing(u) ∩ (B′)^0 = ∅, hence sing(u) ∩ B0 = ∅' is not justified, because B′ is a strictly smaller box than B (its right side is at b1 − r); the proof should run the argument for a family of boxes (for instance with r → 0) or otherwise cover B^0, since the conclusion for B^0 does not follow from the conclusion for a single (B′)^0.","section":"§2, Step 4"},{"comment":"The clause 'or ∂F(M,p)/∂M > 0 at all points (M,p) = (D^2u(x), Du(x))' restates the preceding strict ellipticity condition, and the 'or' is confusing; in addition, the inequality ∂F/∂M > 0 should be stated as positive definiteness of the matrix (F^ij).","section":"Definition 1.1"},{"comment":"The phrase 'the smooth, open set sing(u)^c' calls the set of twice-differentiability points the smooth set before smoothness is proved; using a neutral term such as 'regular set' (or proving the smoothness first) would avoid prejudging the conclusion of Step 4.","section":"Step 1"},{"comment":"There are typos, including 'viscos ity' in the Abstract and 'follow follow' in Remark 1.4; also, the reference [K87] appears in the bibliography but is not cited in the text.","section":"Abstract and Remark 1.4"},{"comment":"The proof would benefit from a precise definition of 'intersects only one side of B' as 'sing(u) ∩ ∂B is contained in the relative interior of a single side', since Step 1 uses exactly this stronger form.","section":"Theorem 1.4 / §1.3"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a short proof paper whose main theorems are known results for its three model equations, as the author explicitly acknowledges; the contribution is the doubling method. Given that the results are known, the strict correctness of the new proof is the decisive issue. The Step 3/Appendix stability gap is substantial and is the main obstacle; I believe it is repairable, either by supplying a genuine partial-regularity stability argument or by stating the needed regularity of u on sing(u)^c as an explicit hypothesis. For that reason I recommend major revision rather than rejection. I would also ask the author to spell out the segment-to-box reduction, since the abstract's headline claim of half-line removability depends on it, and to clarify the closedness of sing(u) in the general conditional theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nShort version: this is a promising proof announcement, not a finished paper. The core doubling argument is genuinely attractive, and the single-side condition results are new as far as I can tell. But the proof as written leans on two nontrivial facts that are asserted rather than established: that the singular set is closed (used in Step 1), and that Savin's small perturbation gives uniform smooth convergence of the approximating sequence (claimed in Step 3 with a one-paragraph appendix that doesn't do the work).\n\nFirst, what's good. The main observation—comparing with a 1D Green's function to get a doubling inequality that controls b(D^2u, Du) inside a box by its values on three sides—is clean and elegant. The contradiction at (2.3) is valid. The extension to singularities satisfying the single-side condition (Theorems 1.3, 1.4) is a real addition: I don't know of that statement in the literature for the general class. The paper is honest that the half-line removability for Monge-Ampère, minimal surface, and special Lagrangian is already known; the contribution is the unified proof and the new geometric condition.\n\nNow the soft spots. Step 1 says 'Since sing(u) is closed' without proof. For a general viscosity solution, the complement of twice-differentiability points need not be open; in fact, showing it is open is part of what Savin's theorem gives. For the specific equations like Monge-Ampère this is known, but for the abstract class in Theorem 1.2 it is not listed as an assumption. The paper mentions in Section 1.4 that Savin plus an Alexandrov theorem gives closedness, but that isn't tied into the proof.\n\nStep 3 is the bigger issue. The claim that uk→u in C^{2,α}_{loc}(sing(u)^c) requires applying Savin to the linearized equation for v_k = uk - u. That requires u to already be smooth (with bounded derivatives up to order two) on the subdomain—which is exactly what you're trying to prove. The right order is to first apply Savin to u itself to show the smooth set is open, then linearize. The paper's one-paragraph appendix assumes 'u is smooth on Ω' and then derives stability; it doesn't show that R sits in such a set. A referee should ask for a proper lemma with covering arguments and uniform ellipticity constants.\n\nIf those details are filled in, I'd believe the proof. As it stands, it's a plausible but unverified proof announcement. It deserves peer review—the technique is novel and the single-side theorem is worth a careful look—but I wouldn't cite it as settled until the gaps are addressed.","headline":"A genuinely new proof mechanism and a new single-side theorem, but the Savin stability step is asserted rather than verified; worth refereeing after the gaps are addressed.","tokens_in":8731,"tokens_out":6005,"would_cite":false,"duration_ms":60123,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35B65","35D40","35J96"],"pacs":[],"model":"deepseek-v4-flash","headline":"A viscosity solution whose singular set lies in a line segment with an endpoint inside the domain must be smooth everywhere, provided the equation has classical density and a Jacobi inequality.","keywords":["fully nonlinear elliptic equations","viscosity solutions","singular set removability","Jacobi inequality","classical density","doubling inequality","Monge-Ampère equation","special Lagrangian equation"],"falsifier":"Exhibit a viscosity solution of a fully nonlinear elliptic equation with classical density and a Jacobi inequality whose singular set is exactly a half-line segment with one endpoint inside the domain, for example $\\{x_1 \\ge 0, x_2=x_3=0\\}$ in dimension three; Theorem 1.2 predicts no such solution exists.","tokens_in":7699,"feed_emoji":"","tokens_out":11131,"duration_ms":89557,"temperature":0.7,"pith_summary":"This paper establishes a removability result for singularities of viscosity solutions to a broad class of fully nonlinear elliptic equations. It proves that if the set where a solution fails to be twice differentiable is contained in a line segment with one endpoint inside the domain, then that set is empty: the solution is smooth everywhere. The equations covered are those with two structural properties, classical density and a Jacobi inequality, and they include the Monge-Ampère, minimal surface, and special Lagrangian equations. The proof is short and unifies earlier results: a doubling inequality based on a one-dimensional Green's function forces a Jacobi quantity to be controlled by its boundary values, and a small perturbation stability theorem then upgrades uniform convergence to smooth convergence away from the singularity. The same mechanism removes singular sets satisfying a geometric single-side condition, such as a singularity touching the boundary at one point.","feed_headline":"Half-line singularities are removable for key nonlinear PDEs","feed_subtitle":"One doubling proof covers Monge-Ampère, minimal surface, and special Lagrangian equations.","key_machinery":"The key mechanism is a doubling inequality for the Jacobi quantity $b_k = b(D^2 u_k, Du_k)$ on a box $B' = \\{a_i \\le x_i \\le b_i\\}$. With $\\eta(x_1) = 1/(b_1-x_1) - 1/(b_1-a_1)$, the inequality says $\\sup_{B'} \\eta(x_1) b_k(x) \\le r^{-1} \\sup_R b_k(x)$, where $R$ is the boundary of $B'$ excluding the 'bad' side. It is proved by a maximum-principle contradiction: at an interior maximum, ellipticity and the Jacobi inequality force $0 > -\\eta''/\\eta^2 \\ge 0$. The single-side condition makes the exceptional side the only side that can touch the singular set, so $R$ lies in the smooth region; a small perturbation stability theorem then provides uniform bounds on $R$, and the higher bounds from the Jacobi inequality propagate smoothness inward.","core_discovery":"The central claim is Theorem 1.2: if $u \\in C(B_1(0))$ is a viscosity solution of $F(D^2u,Du)=0$ with classical density and a Jacobi inequality, and if $\\mathrm{sing}(u)$ is contained in a line segment with an endpoint in $B_1(0)$, then $\\mathrm{sing}(u)=\\emptyset$. Classical density means every viscosity solution is a uniform limit of smooth solutions; the Jacobi inequality means a positive function $b(D^2u,Du)$ satisfies $F^{ij}b_{ij} \\ge 2 F^{ij}b_i b_j/b$ along smooth solutions, and local bounds on $b$ imply all higher derivative bounds. Theorem 1.4 generalizes the conclusion: if $\\mathrm{sing}(u)$ intersects only one side of a closed box, then $\\mathrm{sing}(u)$ is empty inside that box. The Monge-Ampère theorem (Theorem 1.1) is a corollary, and the minimal surface and special Lagrangian equations are shown to satisfy the hypotheses.","pith_inferences":["The same doubling framework might extend to equations that only have an almost-Jacobi inequality or lack classical density, if the perturbation stability step can be replaced by a weaker compactness argument; this would widen the class beyond Monge-Ampère, minimal surface, and special Lagrangian.","The proof's reliance on a one-dimensional Green's function suggests a route to quantitative versions: the doubling inequality bounds the Jacobi quantity by boundary data, so one could hope for explicit modulus of continuity or derivative bounds near a removable segment.","A natural test case is whether half-plane singularities are removable in higher dimensions; the paper notes this is unclear, and the doubling method as written does not seem to handle it, which would be worth probing.","The single-side condition is essentially a transversality condition at the boundary; comparing with known full-line singular solutions suggests that some geometric restriction of this kind is necessary, and the method may help identify the optimal condition."],"forward_implications":["Monge-Ampère viscosity solutions cannot have a singular set contained in a line segment with an endpoint inside the domain; in particular half-line singularities are removable.","Minimal surface and special Lagrangian equations admit a unified proof of half-line singularity removal, based only on the Jacobi inequality and classical density rather than on equation-specific Hessian or gradient estimates.","For any equation satisfying the two structural hypotheses, any singularity that intersects only one side of a box is removable inside that box; iterating removes 'keyhole' singularities whose closure meets the boundary only once.","The proof yields interior smoothness of the limit solution from uniform bounds on the approximations, so the regularity conclusion is $C^{2,\\alpha}$ (and higher) at points previously allowed to be singular."],"supporting_citations":[{"why":"Supplies the small perturbation theorem used to pass from uniform convergence of smooth approximations to smooth convergence away from the singularity.","marker":"[S07]"},{"why":"Introduces the comparison with a one-dimensional Green's function that underlies the doubling inequality.","marker":"[SY23]"},{"why":"Provides the Jacobi inequality for the Monge-Ampère equation with $b = \\det(I+D^2u)^{1/(2n)}$.","marker":"[Y23]"},{"why":"Gives the Jacobi inequality for the minimal surface equation with $b=\\sqrt{1+|Du|^2}$ and the higher-bound consequences.","marker":"[GT77]"},{"why":"Provides the Jacobi inequality for the special Lagrangian equation with $b=(1+\\lambda_{\\max}(D^2u)^2)^{\\varepsilon(n)/2}$.","marker":"[WY14]"},{"why":"Together with the comparison principle, provides the classical solvability used to establish classical density for Monge-Ampère.","marker":"[CNS84]"},{"why":"Supplies the classical solvability for Monge-Ampère that, with comparison, yields classical density.","marker":"[K84]"},{"why":"Gives the classical solvability for the special Lagrangian equation used to obtain classical density.","marker":"[CNS85]"}],"fun_headline_variants":["Half-line singularities removed for several nonlinear PDEs","Doubling proof erases half-line singularities in PDEs","Monge-Ampère and minimal surface: singularities vanish on half-lines","New proof: half-line singularities removable in nonlinear PDEs","Removing half-line singularities in fully nonlinear equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the stability claim that near every point where the limit solution is smooth, the approximating smooth solutions have uniformly controlled derivatives, so estimates on the approximations transfer to the limit.","fun_headline_variants_meta":{"raw":{"variants":["Half-line singularities removed for several nonlinear PDEs","Doubling proof erases half-line singularities in PDEs","Monge-Ampère and minimal surface: singularities vanish on half-lines","New proof: half-line singularities removable in nonlinear PDEs","Removing half-line singularities in fully nonlinear equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000287,"raw_usage":{"total_tokens":1630,"prompt_tokens":833,"completion_tokens":797,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":711}},"tokens_in":449,"tokens_out":797,"duration_ms":7017,"temperature":1.0,"reasoning_tokens":711,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:35:32.716560+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a viscosity solution of a fully nonlinear elliptic equation with classical density and a Jacobi inequality whose singular set is exactly a half-line segment with one endpoint inside the domain, for example $\\{x_1 \\ge 0, x_2=x_3=0\\}$ in dimension three; Theorem 1.2 predicts no such solution exists.","supporting_citations":[],"review_version":1}