{"id":"c24f3c2c-44ba-407b-a291-4a038f94ae2d","arxiv_id":"2505.08384","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new mixed convex integration technique gives a fresh proof that scalar curvature can be almost prescribed by a C0-close metric, recovering Lohkamp's theorem.","lead":"This paper introduces a mixed convex integration method that combines first- and second-order corrugation loops to solve a class of semilinear second-order partial differential relations. It uses the method to give a new proof of Lohkamp's theorem on almost prescribing scalar curvature on any Riemannian manifold of dimension at least three.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing k=1 base case in Lemma 4.1 leaves the P0-to-P1 induction step unproved.","rationale":"The reader's weakest assumption points to Lemma 4.1 and the global gluing argument, which is the right neighborhood. I partially agree, but I do not think the delicate radial-approximation step in Appendix C is the real problem: after replacing tau_+ by a smooth tau with tau_R > tau, the radial map has Jacobian positive because lambda + r lambda_r = (1 - mu) + mu tau_R/tau + r mu'(tau_R/tau - 1) > 0 and lambda > 0, so it is a diffeomorphism. The displayed sign inconsistency in the flat-torus scalar formula for the i >= 4 directions is also harmless in this construction: those derivatives are O(1/N) by Proposition 2.2(ii), so the erroneous cross terms contribute O(1/N^2) to the scalar curvature and are absorbed by the O(1/N) error. The genuine hole is the transition from P0 to P1: it requires Lemma 4.1 for k = 1 and P_{-1}, neither of which exists in the paper. Because this is a localized, easily patched gap rather than a flaw in the curvature estimates or the loop constructions, the appropriate verdict remains CONDITIONAL, and my read does not change the reader's verdict.","tokens_in":29114,"tokens_out":52546,"duration_ms":497318,"concrete_test":"Write out a k = 1 version of Lemma 4.1 (Sigma_1 = empty, Phi: {0} x [0,1] -> Int(D^1) with Phi({0} x {0,1}) in any prescribed neighborhood of the two boundary points) and then re-derive the P0-to-P1 induction step in Section 4, replacing the invalid invocation of Lemma 4.1 with k = 1 and of P_{-1}. If this 1D base cannot be supplied, the induction is genuinely incomplete; if it can, the gap is purely expository and the central claim survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is not the smooth approximation in Appendix C (that step is defensible: with tau_R > tau and mu nondecreasing, the radial map has Jacobian (lambda + r lambda_r) lambda^{k-1} > 0), but the induction architecture in Section 4. Theorem 1.1 is obtained from property (P_d) with d = n, and (P_d) is proved by induction on d. The step from d = 0 to d = 1 handles 1-simplices, yet it invokes Lemma 4.1 with k = d + 1 = 1 and then property P_{d-1} = P_{-1}. Lemma 4.1 is stated and proved only for k >= 2 (Appendix C starts with k = 2), and P_{-1} is undefined. Thus, as written, the proof does not cover 1-dimensional faces of the triangulation. This is not a failure of the geometric idea: a k = 1 version of Lemma 4.1 is trivial (Sigma_1 = empty, T^0 x [0,1] embeds in (-1,1) with endpoints in any prescribed neighborhoods), and a separate P1 base case can be supplied. But the manuscript does not provide it, so the central claim is not fully established as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a mixed convex integration method for a class of semilinear second-order partial differential relations and uses it to give a new proof of Lohkamp's theorem: on any smooth Riemannian manifold of dimension n >= 3, any smooth metric can be C^0-approximated by a smooth metric whose scalar curvature lies strictly between Scal_{g0} - k - epsilon and Scal_{g0} - k for any prescribed positive function k. The core tool is Proposition 2.4, a corrugation criterion built from the integral-loop operator, applied to scalar curvature through explicitly constructed loops. The torus cases are proved in Section 3, and Section 4 assembles the general manifold case by induction over a triangulation, using a thick-torus embedding lemma in the simplex-by-simplex construction.","tokens_in":29307,"tokens_out":11092,"duration_ms":108817,"significance":"If the proof is completed, the paper gives a genuinely new proof of an important theorem of Lohkamp and introduces a reusable technique for semilinear second-order differential relations that are not amenable to classical convex integration. The loop constructions are explicit and parameter-free rather than fitted, the local asymptotic expansions are clean, and the overall geometric strategy is well organized. The central claim is not a new theorem but a new proof method, and that method is the paper's main contribution.","major_comments":[{"comment":"The induction proving property (P_d) does not cover the step from d = 0 to d = 1 as written. The proof assumes (P_{d'}) for all d' in {0,...,d} and then, for a (d+1)-simplex, invokes Lemma 4.1 with k = d+1 and property (P_{d-1}). For d = 0 this requires Lemma 4.1 with k = 1, but Lemma 4.1 is stated only for k in [2,n], and property P_{-1} is undefined. Since the proof of Theorem 1.1 requires P_n and the induction starts at P_0, the chain is broken unless a k = 1 version of Lemma 4.1 is supplied and P_{-1} is either defined or the first step is handled separately. This is patchable, but it is a genuine gap in the manuscript as it stands.","section":"Section 4, induction step"},{"comment":"The displayed scalar curvature formula for the flat torus contains a sign error in the cross terms for i >= 4: the final sum reads (partial_i h_2)^2 + (partial_i h_3)^2 - partial_i h_2 partial_i h_3, but the general diagonal-metric formula in Proposition A.4 gives + partial_i h_2 partial_i h_3 inside the bracket. Thus the displayed identity is false as an exact formula. Under the corrugation construction the error is O(1/N) and may be absorbable in the epsilon thickening, but the proof of Proposition 3.1 as written relies on this exact identity to define the relation R, so the identity must be corrected or the argument must explicitly carry the error term.","section":"Section 3.1, displayed scalar curvature formula"}],"minor_comments":[{"comment":"The word 'taylored' should be 'tailored'.","section":"Section 2, first paragraph"},{"comment":"The statement of Proposition A.5 says 'Assume that g is diagonal of the form (A.7)', but equation (A.7) defines \\bar g = \\bar A^{tr} \\bar A, which is not diagonal; the intended hypothesis is that \\bar g has the form (A.7) and the perturbation \\hat g has the diagonal form (A.8).","section":"Appendix A.4, Proposition A.5"},{"comment":"In Proposition 3.5, the compact set C is introduced in the statement and then used both as the compact neighborhood C in the proof; this is not an error but the reuse of the symbol C for two different sets can confuse the reader.","section":"Section 3.3, notation"},{"comment":"The notation R_{sigma, partial_1, partial^2_1} is introduced without a precise definition of the subscripts; please clarify that these denote the dependence on the 2-jet components that are fixed in the slice.","section":"Section 2.4, geometric interpretation"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized and fixable: add a k=1 base case for Lemma 4.1 or prove P_1 separately, and correct the sign in the flat-torus scalar curvature formula. I do not see circularity or fitted parameters in the arguments; the issues are internal completeness issues rather than a flaw in the overall method."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper does something genuinely new—a mixed first/second-order corrugation scheme that solves a non-ample semilinear second-order relation—and uses it to give a fresh proof of Lohkamp's scalar curvature theorem. The asymptotic expansion in Proposition 2.2 is clean, the loops are explicit and parameter-free, and the thick-torus perturbation is a nice piece of machinery. The authors are upfront that the theorem itself is Lohkamp's; their contribution is the method. I think the method is worth taking seriously.\n\nThat said, the proof has a structural gap. The induction for property (P_d) does not cover the step from d=0 to d=1. Lemma 4.1 is stated and proved only for k>=2, but the step to 1-simplices needs the k=1 case. The stress-test note is right: a trivial version exists (Sigma_1 empty, an embedded interval in (-1,1)), but it is not in the manuscript. As written, Theorem 1.1 is not fully proved. This is a small fix, not a deep one, but it is a real omission.\n\nThere is also a sign error in the displayed flat-torus scalar curvature formula (and in the corresponding R term) for the i>=4 cross term: the coefficient of partial_i h2 partial_i h3 should be -2, not +2. In the actual construction the term vanishes because the base function f is zero, so the loop identities are unaffected. But the formula is wrong and should be corrected. A referee will catch it.\n\nThe smooth approximation of Psi_R in Appendix C is sketched rather than proved. It looks defensible—the Jacobian argument works—but the manuscript would be stronger with a few lines of detail.\n\nThe citation pattern is honest; [12] is independently published, and the loops are not fitted to the conclusion. No circularity.\n\nNet: this is a serious paper that deserves a serious referee. The core asymptotic analysis is sound, the method is new, and the gaps are mechanical. I would send it to peer review, with the referee asked to verify the k=1 base case and the sign. After those patches, it will be a solid contribution to the h-principle literature.","headline":"New mixed convex integration proof of Lohkamp's theorem; the method is real, but the induction has a missing k=1 base case that needs to be patched.","tokens_in":29874,"tokens_out":7343,"would_cite":true,"duration_ms":61363,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that on any smooth manifold of dimension at least three, any smooth metric can be perturbed slightly so its scalar curvature lies below and arbitrarily close to a prescribed negative target, and it derives this…","keywords":["scalar curvature","convex integration","mixed corrugation","semilinear second-order PDE relations","Riemannian metrics","h-principle","integral-loop operator","metric gluing"],"falsifier":"Run the construction on the two-dimensional torus, where the conclusion is known to fail: the loop equation of Proposition 2.4 should then have no zero-mean periodic solution with the chosen diagonal ansatz, and exhibiting that failure would confirm the mechanism rather than a technical gap. Separately, examine the least verified premise by taking the proposed codimension-two set $\\Sigma_3$ in the unit disc and attempting to construct the embedded torus cylinder it is supposed to allow; an explicit neighborhood of $\\Sigma_3\\cup\\partial D^3$ that blocks such an embedding would falsify the gluing step.","tokens_in":28854,"feed_emoji":"📐","tokens_out":9138,"duration_ms":87969,"temperature":0.7,"pith_summary":"The paper establishes a broad flexibility principle for scalar curvature: on any smooth Riemannian manifold of dimension at least three, any smooth metric can be perturbed by an arbitrarily small $C^0$ change so that the new scalar curvature is strictly lower than the original scalar curvature minus a prescribed positive function, while staying within an arbitrary tolerance of that lowered value. This recovers, by a different route, a theorem previously established by other means. The interest is the method: a mixed convex integration scheme that handles semilinear second-order differential relations in which the highest-order term appears linearly, a setting not covered by classical convex integration. If the method is sound, it provides a general template for solving such relations whenever the relevant loop equations admit zero-mean periodic solutions.","feed_headline":"New proof: scalar curvature is almost prescribable on any manifold","feed_subtitle":"The new metric stays arbitrarily close to the old one while its scalar curvature lands in the prescribed window.","key_machinery":"The central mechanism is the mixed corrugation process: starting from a function $f$, one defines $F_N(x)=f(x)+N^{-1}\\mathrm{Int}(\\gamma_x)(N x_i)+N^{-2}\\mathrm{Int}^2(\\delta_x)(N x_i)$, where $\\mathrm{Int}$ is the integral-loop operator that returns the mean-zero primitive of a periodic loop and $\\gamma_x,\\delta_x$ are smooth families of loops. Proposition 2.2 gives asymptotic formulas showing that the first derivative gains $\\gamma_x(Nx_i)$, the pure second derivative gains $N\\dot\\gamma_x(Nx_i)+\\delta_x(Nx_i)$, and mixed second derivatives gain transverse derivatives of $\\gamma_x$. Proposition 2.4 converts these formulas into an $\\varepsilon$-approximate solution of the semilinear relation, provided the loops satisfy three conditions: zero mean, membership in the kernel of the linear coefficient $L$, and a loop equation $L(x)\\cdot(\\sigma^2_{11}+\\delta_x(t))+R(x,\\dots)=0$ for every $t$. The scalar curvature application chooses the loops explicitly, for example $\\gamma_x(t)=\\sqrt{(\\tilde k(x)+\\bar\\epsilon)/\\alpha_x}\\cos(2\\pi t)$, with $\\delta_x$ adjusted to have zero mean.","core_discovery":"The central claim is Theorem 1.1: for every smooth Riemannian manifold of dimension $n\\ge 3$, every smooth metric $g_0$, every smooth positive function $k$, and every $\\varepsilon>0$, there exists a smooth metric $g_\\varepsilon$ with $\\mathrm{Scal}_{g_0}-k-\\varepsilon<\\mathrm{Scal}_{g_\\varepsilon}<\\mathrm{Scal}_{g_0}-k$ and $\\|g_\\varepsilon-g_0\\|_{C^0_g}<\\varepsilon$. The paper proves this by treating the scalar curvature equation as a semilinear second-order relation of the form $L(x)\\cdot \\partial_1^2 F + R(x,\\text{lower-order jet terms})=0$, after a diagonal deformation of the metric in a frame adapted to $g_0$. A general criterion, Proposition 2.4, solves such relations by inserting two periodic loop families into a corrugation process; the torus case is then globalized simplex by simplex. The proof thereby avoids the covering-and-superposition technique of the earlier argument and replaces it with a local curvature calculus plus a geometric gluing lemma.","pith_inferences":["The positivity of $k$ is not an accident: the zero-mean condition on the loop $\\delta$ forces $\\tilde k(x)+\\epsilon$ to be nonnegative, as the paper notes in Remark 3.2. A natural extension, not pursued by the authors, is to test whether allowing the metric perturbation to use more than two diagonal directions removes that restriction for sign-changing $k$.","The same two-loop splitting should apply to other geometric relations whose highest-order term is linear, such as prescribing a component of the Ricci tensor or prescribing scalar curvature within a fixed conformal class; the paper does not make this claim.","Lemma 4.1, the embedding lemma used to globalize the construction, is independent of the curvature estimates. The proof would be easier to check if that lemma were replaced by an explicit construction; isolating its first nontrivial case, $k=3$, would separate the geometric gluing content from the PDE mechanism."],"forward_implications":["Every smooth metric on an $n$-dimensional manifold with $n\\ge 3$ can be approximated arbitrarily well in the $C^0$ norm by a metric whose scalar curvature lies strictly between $\\mathrm{Scal}_{g_0}-k-\\varepsilon$ and $\\mathrm{Scal}_{g_0}-k$.","The semilinear criterion of Proposition 2.4 becomes a reusable tool: any second-order relation that is affine in the top derivative and admits zero-mean loop pairs satisfying the three conditions can be solved in the same $\\varepsilon$-thickened sense.","The proof recovers the known scalar-curvature flexibility result without the measure-covering superposition argument used previously, replacing it with a local curvature calculus and a simplex-by-simplex gluing scheme.","Because the metric perturbations are controlled in the $C^0$ norm at every step, the theorem tolerates arbitrary initial metrics and arbitrary local variations of the prescribed positive function $k$.","The strict inequalities and arbitrary $C^0$ closeness give the result an h-principle-like character: local geometry imposes no obstruction to lowering scalar curvature in dimension at least three, in contrast to the two-dimensional obstruction."],"supporting_citations":[{"why":"supplies the theorem being reproved and the stronger form in which the metric is unchanged on a closed subset.","marker":"[7]"},{"why":"introduces the corrugation process without integration that the mixed two-loop formula adapts.","marker":"[12]"},{"why":"provides the convex integration framework and explains why the scalar curvature relation is not amenable to classical convex integration.","marker":"[4]"},{"why":"establishes the negative-Ricci-curvature flexibility that the scalar curvature result extends to the almost-prescription window.","marker":"[6]"}],"fun_headline_variants":["New proof: almost prescribe scalar curvature on any manifold","Mixed convex integration: new proof of scalar curvature result","Almost prescribing scalar curvature via mixed convex integration","New method: almost prescribe scalar curvature on manifolds","Almost prescribable scalar curvature: new proof"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The global gluing step requires Lemma 4.1, which asserts that after removing a codimension-two set from a disc, a thickened torus times an interval can still be embedded with its two ends in any prescribed neighborhood of the disc boundary; if that geometric embedding cannot be arranged, the local metric perturbations cannot be assembled into a metric on the whole manifold.","fun_headline_variants_meta":{"raw":{"variants":["New proof: almost prescribe scalar curvature on any manifold","Mixed convex integration: new proof of scalar curvature result","Almost prescribing scalar curvature via mixed convex integration","New method: almost prescribe scalar curvature on manifolds","Almost prescribable scalar curvature: new proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00059,"raw_usage":{"total_tokens":2691,"prompt_tokens":791,"completion_tokens":1900,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":407,"completion_tokens_details":{"reasoning_tokens":1837}},"tokens_in":407,"tokens_out":1900,"duration_ms":15500,"temperature":1.0,"reasoning_tokens":1837,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:01:36.011324+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the construction on the two-dimensional torus, where the conclusion is known to fail: the loop equation of Proposition 2.4 should then have no zero-mean periodic solution with the chosen diagonal ansatz, and exhibiting that failure would confirm the mechanism rather than a technical gap. Separately, examine the least verified premise by taking the proposed codimension-two set $\\Sigma_3$ in the unit disc and attempting to construct the embedded torus cylinder it is supposed to allow; an explicit neighborhood of $\\Sigma_3\\cup\\partial D^3$ that blocks such an embedding would falsify the gluing step.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the theorem being reproved and the stronger form in which the metric is unchanged on a closed subset."},{"cited_title":"Theilli` ere","cited_arxiv_id":null,"evidence_quote":"introduces the corrugation process without integration that the mixed two-loop formula adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the convex integration framework and explains why the scalar curvature relation is not amenable to classical convex integration."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the negative-Ricci-curvature flexibility that the scalar curvature result extends to the almost-prescription window."}],"review_version":1}