{"id":"3876d51b-0c8f-491d-9491-f60c16cd277a","arxiv_id":"2608.06293","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the planar translating mean curvature equation with zero boundary data, convexity of the domain does not force convexity of sublevel sets, even in dimension two.","lead":"This paper constructs a smooth, bounded, uniformly convex disk in the plane whose solution to the translating mean curvature equation has a non-convex sublevel set. It settles a small open question about whether log(-u) must be concave for such solutions, in the negative.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing link is the expansion and remainder algebra in Theorem 4.2; a single wrong q-power or sign there would break the sandwich estimate, so those formulas should be independently checked.","rationale":"The reader's weakest assumption identifies exactly the same spot: the correctness of the residual and remainder bounds in Theorem 4.2, especially (4.6), (4.8), (4.9), and (4.39). My independent reading of the surrounding argument found no concrete algebraic error: the corrector w solves the linearized equation with the right sign, the cancellation of the τ-linear term is verified by the displayed formulas, the barrier coefficient estimates are internally consistent, and the final transfer of the O(q^{3/2}) midpoint defect through (4.42) is sound if Theorem 4.2 holds. The construction of the domain via two-moment smoothing is coherent, the comparison principle applies, and the root calculation in Theorem 5.1 correctly produces a positive midpoint gap of order q^{5/2} after scaling. Thus I do not see a demonstrated flaw that would change the verdict. The concern is a verification risk rather than a known error: the long algebra in Theorem 4.2 is load-bearing, and no machine or independent symbolic check is provided. A concrete symbolic verification would settle whether the concern lands. Until such a check is performed, the appropriate verdict remains the reader's ACCEPT, since the paper supplies exact identities and enough detail for an expert check, and the surrounding argument is strong.","tokens_in":21911,"tokens_out":27024,"duration_ms":272014,"concrete_test":"Use a computer algebra system to verify the algebra symbolically. (1) Expand F[V_q+sΨ_q] from (2.1), (2.7), (2.10), and (4.1), and confirm that the identity (4.6) holds exactly with E_s equal to the right-hand side of (4.35). (2) Substitute γ=π/2−qξ, y=π/2−qρ into the affine-width expansion and check that the τ-linear coefficient cancels identically and that τ^2R2+τ^3R3+τ^4R4 is bounded by C q^3 S using (4.16)–(4.17) and (4.24). (3) Verify each of the eight estimates in the table (4.39) by expanding the displayed groups in (4.35) and tracking powers of q symbolically. If all three checks reproduce (4.9), (4.8), and (4.39), the barrier inequalities (4.10) and (4.15) follow; any mismatch would pinpoint exactly where the sandwich argument fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 follows from the quantitative comparison (4.41)–(4.42), whose only nontrivial input is Theorem 4.2. In particular, the barrier inequalities (4.4) require three things: the residual bound |F[V_q]| ≤ C_R q^3 S in (4.9), the exact expansion (4.6) with the signed terms −sS−s^2X^2, and the remainder bound (4.8) with the table (4.39). The most delicate point is the cancellation of the τ-linear term in the affine-width expansion of F[V_q]: if the corrector w or the derivative formulas (4.19)–(4.23) contained a sign or coefficient error, F[V_q] would be O(q^{9/2}S) instead of O(q^3S), which would overwhelm the barrier term a_qS=O(q^3S). Likewise, the upper barrier requires the coefficient of B=s^2X^2 in (4.14) to be negative; this depends on the q-powers in (4.39) being exactly as claimed. The paper does give exact identities (4.26), (4.34), and (4.35), and a term-by-term table, which is strong evidence; but the estimates are long, combine many absorptions, and are not machine-checked or independently rederived. Since this algebra is the unique bridge from the model profile to the exact Dirichlet solution, it is the single most load-bearing assumption in the paper. I found no explicit error in reading through it, but the verification burden is concentrated there.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1: there exists a smooth bounded uniformly convex domain Ω ⊂ R² such that the unique zero-Dirichlet solution u of the planar translating mean curvature equation (1.1) has a nonconvex sublevel set, specifically {u < −log 2}. The construction starts from a long convex channel whose half-width is the near-critical grim-reaper profile γ_q(x) = arccos q + q^{9/4}x − q^7 x^2, and introduces an explicit corrector w so that the corrected profile V_q = g(y) − g(γ_q) + τ w(γ_q,y), τ = q^{9/2}, has residual O(q^3 sec^2 y). The channel is closed by circular caps with a two-moment smoothing, the Dirichlet solution u_q is obtained from Zhou's existence theorem, and explicit upper and lower barriers V_q ± (a_q + e_q)Ψ_q sandwich u_q with an error that is o(q^{3/2}) on the central channel. A quantitative midpoint defect of order q^{3/2} for V_q is then transferred to u_q, proving the nonconvex sublevel set.","tokens_in":22106,"tokens_out":36192,"duration_ms":422506,"significance":"The result is significant: if correct, it is the first counterexample to level-set convexity, equivalently to logarithmic concavity of log(−u), for the bounded zero-Dirichlet translating mean curvature equation in the plane, and it answers Wang's question in that bounded setting. The construction is genuinely explicit: the corrector w is given in closed form, the cancellation of the τ-linear term is exact, the residual estimates are organized as a finite table, and the final comparison error is quantitative rather than qualitative. The main residual risk is the long algebra in Theorem 4.2; I checked representative cancellations and q-powers and found them consistent, but the proof would be easier to certify if that computation were fully expanded or supplied in machine-checkable form.","major_comments":[],"minor_comments":[{"comment":"The definition of A(s) is internally inconsistent: the text states A(s) = tan s + log cos s, while immediately afterwards it asserts A'(s) = s sec^2 s and later uses the evenness of A. The correct definition used throughout the paper is A(s) = s tan s + log cos s; please fix the displayed formula.","section":"Section 2.3"},{"comment":"The phrase 'positive concave half-width f_q' should be qualified: f_q is positive in the interior of the interval and vanishes at the terminal points, where the boundary is smooth only as a parametrized circle, not as a graph y = f_q(x). The current wording could mislead a reader about endpoint regularity.","section":"Section 3.1"},{"comment":"The bounds for the remainder groups E3–E8 are stated with representative substitutions but not with the full term-by-term verification. I spot-checked the q-powers and the absorptions and found no error; for archival completeness, please include the complete derivation of the table, or provide a supplementary computation file.","section":"Section 4.2, Table (4.39)"},{"comment":"The notation O_{C^2(I)}(q^4) and o_{C^2(I)}(1) is introduced without definition; please define it explicitly in terms of the C^2(I) norm, since it is used in the crucial expansion (5.9)–(5.12).","section":"Section 5, Step 2"},{"comment":"There are minor formatting artifacts in the abstract and headings ('TRANSLA TING', 'CUR V A TURE') and a few other typographical slips; these should be cleaned up in the final version.","section":"General"}],"recommendation":"minor_revision","confidential_remarks":"The central proof is credible and the construction is explicit, but the paper concentrates a large amount of delicate algebra in Theorem 4.2. I found no error in spot checks, so I do not regard this as blocking acceptance; however, a supplementary machine-checked expansion would substantially reduce the verification burden for readers and referees. The paper fits the scope of math.AP and should be of interest to the PDE and mean curvature flow communities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper appears to deliver what it promises. It constructs a smooth, uniformly convex planar domain for which the zero-Dirichlet solution of the translating mean curvature equation has a nonconvex sublevel set, answering a question Wang left open. The structure is rigorous and the construction is explicit: a near-critical grim-reaper profile with a well-designed corrector, a two-moment smoothing lemma to close the channel with convex caps, and explicit barriers that transfer a quantitative midpoint defect to the exact solution. Proposition 2.1 is a nice, clean obstruction that explains why a naive slow-variable construction fails, and the corrected profile is genuinely new. The comparison argument is coherent, and the central estimate (4.42) gives the precise o(q^{3/2}) error needed to preserve the finite gap. The paper is honest about its scope: it concerns bounded Dirichlet solutions, not the entire-solution problem, and says so.\n\nThe soft spot is exactly where the stress-test note puts it: Theorem 4.2. The expansion (4.6), the residual bound (4.9), and the remainder table (4.39) are load-bearing. If a sign or a q-power is wrong in that algebra, the sandwich collapses. The paper does provide exact identities (4.26), (4.34), and (4.35), and a term-by-term table for the remainder groups, which is strong evidence. I read through the main steps and found no explicit error; the q-powers appear consistent. But this is not machine-checked, and the verification burden is real. A referee should re-derive at least (4.9) and the E8 estimates independently before trusting the result. That said, the density of the computation is not a flaw by itself. The paper gives enough detail for an expert check, which is the usual standard.\n\nThe citations look appropriate: external inputs are Zhou's existence theorem and Wang's question, both used as black boxes or motivation, and prior counterexamples by Hamel-Nadirashvili-Sire, Wang, and Zhang are clearly distinct. No circularity, no fitting to data.\n\nThis deserves peer review. It is a significant, concrete advance in a classical topic, and the central argument holds up on close reading. The main recommendation to the editor: send it to a PDE analyst willing to spend time on the algebra, and if possible ask for a supplementary note expanding the verification of Theorem 4.2. If that checks out, this is a clean counterexample that resolves an open question in the negative.","headline":"First counterexample to sublevel-set convexity for the planar translating mean curvature equation on a uniformly convex domain; the proof structure is sound, with the main verification burden concentrated in the heavy algebra of Theorem 4.2.","tokens_in":22749,"tokens_out":1583,"would_cite":true,"duration_ms":19112,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J93","35B06","53E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A convex domain can host a translator solution with a nonconvex sublevel set.","keywords":["translating mean curvature equation","nonconvex sublevel set","grim-reaper profile","logarithmic concavity","level-set convexity","Dirichlet problem","uniformly convex domain","barrier method"],"falsifier":"Recompute the exact identity (4.26) with the formulas (4.19)--(4.23) at a small concrete value such as $q=10^{-6}$, on the midline $y=0$ and at $y=\\gamma_q(0)/2$, and compare $F[V_q]$ with the claimed $Cq^3\\sec^2 y$ bound; also evaluate the remainder estimate (4.8) for the barrier perturbation. If the leading $q^3$ term fails its stated size or sign, or the remainder grows faster than $C_M q^{1/4}(|s|S+s^2X^2)$, the comparison gap of order $q^{3/2}$ used in Theorem 4.3 and Corollary 4.4 would collapse.","tokens_in":21612,"feed_emoji":"📐","tokens_out":14793,"duration_ms":151140,"temperature":0.7,"pith_summary":"This paper constructs a smooth bounded uniformly convex domain in the plane for which the unique zero-Dirichlet solution of the translating mean curvature equation has a nonconvex sublevel set, namely $\\{x\\in\\Omega:u(x)<-\\log 2\\}$. If the construction is correct, it answers in the negative the logarithmic-concavity question for $\\log(-u)$ in the bounded zero-Dirichlet setting, a question left open in reference [12]. The proof builds a corrected grim-reaper profile in a long convex channel, closes the channel with strictly convex caps, and uses explicit upper and lower barriers plus the comparison principle to show that the true solution inherits a finite midpoint defect of order $q^{3/2}$ from the model profile. The upshot is that convexity of the domain alone does not force convexity of the sublevel sets for this equation, already in two dimensions on a smooth uniformly convex domain.","feed_headline":"Convexity of a domain does not force convex level sets for translators","feed_subtitle":"The counterexample answers an open logarithmic-concavity question for zero-Dirichlet translator solutions.","key_machinery":"The engine of the proof is the corrected near-critical grim-reaper profile $V_q(x,y)=g(y)-g(\\gamma_q(x))+\\tau w(\\gamma_q(x),y)$, with $g(s)=-\\log\\cos s$, $\\gamma_q(x)=\\arccos q+q^{9/4}x-q^7x^2$, $\\tau=q^{9/2}$, and the explicit corrector $w(\\gamma,y)=A(y)-A(\\gamma)+\\frac{\\tan^2\\gamma}{2}(y\\tan y-\\gamma\\tan\\gamma)$, $A(s)=s\\tan s+\\log\\cos s$. The corrector cancels the complete term linear in the squared modulation speed, so that the operator $F[u]=(1+u_y^2)u_{xx}-2u_xu_yu_{xy}+(1+u_x^2)u_{yy}-(1+u_x^2+u_y^2)$, which has the same sign as the equation's left-hand side minus one, leaves residual $O(q^3\\sec^2 y)$ on the corrected profile. The transverse barrier $\\Psi_q(x,y)=A(\\gamma_q(x))-A(y)$ is positive inside the channel and vanishes on its lateral boundary; adding or subtracting multiples of $\\Psi_q$ with coefficients $a_q=Kq^3$ and $e_q=C_E|\\log q|\\cosh(\\varepsilon_0qx)/\\cosh(\\varepsilon_0q\\Lambda_q)$ produces super- and subsolutions whose signs are controlled by the exact expansion (4.6)--(4.9). Finally, the level-root curve $y=r_q(x)$, defined by $V_q(x,r_q(x))=-\\log 2$, is shown to have $r_q''(x)\\asymp q^5>0$, which is the local geometric source of the nonconvexity.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.1: for the equation $\\operatorname{div}(Du/\\sqrt{1+|Du|^2})=1/\\sqrt{1+|Du|^2}$ on a bounded smooth uniformly convex domain $\\Omega$, with $u=0$ on $\\partial\\Omega$, the sublevel set $\\{x\\in\\Omega:u(x)<-\\log 2\\}$ need not be convex. The proof constructs a one-parameter family of uniformly convex domains $\\Omega_q$ whose central part is a long channel of grim-reaper half-width $\\gamma_q(x)=\\arccos q+q^{9/4}x-q^7x^2$, closed by circular caps joined by a two-moment smoothing that preserves strict concavity. On this domain the authors define the corrected profile $V_q(x,y)=-\\log(\\cos y/\\cos\\gamma_q(x))+q^{9/2}w(\\gamma_q(x),y)$, with $w$ chosen so that the leading error of the slowly modulated grim reaper cancels exactly, and they prove that $V_q$ satisfies the equation to residual $O(q^3\\sec^2 y)$. Explicit barriers of the form $V_q-a_q\\Psi_q$ and $V_q+(a_q+e_q)\\Psi_q$ lie on opposite sides of the equation, so the comparison principle gives $|u_q-V_q|\\le o(q^{3/2})$ near the center. The level curve of $V_q$ at height $-\\log 2$ has positive second derivative of order $q^5$, producing a midpoint gap of order $q^{3/2}$; because the comparison error is much smaller than the gap, the real solution has the same defect and its sublevel set is nonconvex.","pith_inferences":["Because the comparison only needs the values of $u_q-V_q$ at three points, the same corrected-profile and barrier scheme should work for any smooth strictly convex cap completion whose junction tangency prevents boundary interference; this geometric flexibility may simplify adaptations to related quasilinear equations.","The corrector cancels only the leading linear term in the squared modulation speed; pushing the modulation scale further or adding a second corrector might produce larger level-set defects or reveal the next obstruction, a check that could be done numerically from the explicit formulas.","A higher-dimensional analogue would require a multidimensional version of the grim-reaper reduction and of the convexity obstruction in Proposition 2.1; the present method suggests where a corrector would need to be inserted, though that extension is not attempted here."],"forward_implications":["If $\\log(-u)$ were concave, every sublevel set $\\{u