{"id":"87537c04-2804-45b4-9d79-e9c4b8565246","arxiv_id":"2608.08707","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every Riemannian metric with scalar curvature at least κ on a closed manifold, or on an open manifold when κ≤0, is a local Burnett-class limit of metrics with scalar curvature exactly κ, with sharp Hölder closures.","lead":"This paper proves that on closed manifolds, and on open manifolds when the target curvature is nonpositive, every smooth metric with scalar curvature at least κ can be approximated locally uniformly, with first derivatives uniformly bounded, by smooth metrics whose scalar curvature is exactly κ. The result confirms and strengthens a conjecture about the Riemannian side of Burnett's compactness program in general relativity.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The almost-prescription engine (Proposition 3.3) inherits the two-scale cancellation from unpublished [1, Prop 2.2]; a gap there would break the proof of Theorem 1.2, though the paper's own Lemma 3.2 appears to supply the computation.","rationale":"I read the proof in good faith and followed the main chain of reductions: almost-prescription (Proposition 3.3), conformal correction (Lemmas 2.2, 4.1, 4.4, and 6.1), and diagonal choices. I checked the local two-scale algebra, the parabolicization argument, the sub- and supersolution iteration, and the spectral contraction in Section 6. I found no clear internal inconsistency. The weakest point, as the reader notes, is the provenance of the two-scale corrugation. I partially agree with the reader: the paper gives a full proof of Lemma 3.2, so the dependence on [1] is less severe than a black-box citation, but because the paper explicitly attributes the construction to [1, Prop 2.2] and does not isolate the imported algebra, the central claim remains conditional on the correctness of [1]. I do not see a stronger internal concern; the compressed compactness steps in Section 4 are standard and the cited estimates are appropriate. I would keep the CONDITIONAL verdict: accept pending independent verification of the two-scale cancellation or a direct comparison of Lemma 3.2 with [1].","tokens_in":20361,"tokens_out":39704,"duration_ms":377930,"concrete_test":"Re-derive the cancellation identity in Lemma 3.2 without consulting [1]: substitute the loops from (13)-(14) into identity (6), collect all terms proportional to N and to the 2*partial_n(gamma) terms, and verify they cancel identically, leaving an O(N^{-1}) error whose constant is uniform in s. This recomputation should then be compared against [1, Proposition 2.2]; if the identities match and no hidden assumption is used, the external-dependence concern is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 is the load-bearing core: Proposition 3.3 (the parameterized almost-prescription) is used in every case of Theorem 1.2, and it is proved by iterating Lemma 3.2. Lemma 3.2 is introduced as 'the local counterpart of [1, Proposition 3.5] and is based on the two-scale corrugation in [1, Proposition 2.2].' The proof defines the loops gamma and sigma explicitly and derives the O(N^{-1}) cancellation, so this is not an unexamined black box. However, the cancellation identity, i.e. the final displayed line of the proof of Lemma 3.2, is exactly the content of [1, Prop 2.2], and the manuscript does not state which identities are being imported from [1] nor prove them independently. If [1] contained a sign error or a missed mean-zero condition in the two-scale loop pair, the claimed approximation Scal_{h_F} = Scal_h - a^2 + O(N^{-1}) would fail. In that case the inductive error estimates (19)-(20) in Proposition 3.3 break, and each proof in Sections 4, 5, and 6, all of which start from the endpoint of Proposition 3.3, would no longer be supported. The paper's own computation appears to be self-contained, but the explicit attribution to an unreviewed preprint leaves a genuine, localized correctness risk that no other part of the paper addresses.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"Let M be a connected smooth n-manifold, n≥3. The paper proves (Theorem 1.2) that, if M is closed or κ≤0, every smooth metric g0 with Scal_{g0}≥κ is a limit, in the local Riemannian Burnett compactness class (C^0_loc convergence plus locally uniform W^{1,∞} bounds), of smooth metrics with constant scalar curvature κ. It then derives the C^{0,α}_loc closure identity {Scal_g=κ}^{C^{0,α}_loc}={Scal_g≥κ} for α<1, notes that α=1 is sharp, and at κ=0 states that this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk. The proof is organized as a parameterized almost-prescription theorem (Proposition 3.3), built on the two-scale convex-integration construction of [1], followed by global conformal corrections: eigenvalue and parabolicity arguments for κ=0, constant barriers for κ<0, and a nonresonant rescaling plus contraction argument for closed manifolds with κ>0.","tokens_in":20654,"tokens_out":13954,"duration_ms":147749,"significance":"If correct, this is a substantial and surprising flexibility result: it identifies the full scalar-curvature relaxation allowed by Gromov's C^0-stability theorem with the closure in the first-order Burnett compactness class, and it settles the Riemannian reverse-Burnett conjecture at κ=0 in a strengthened form. The paper is honest and careful in several places: the local scalar-curvature identity is computed explicitly, the uniform W^{1,∞} bounds are tracked, the open case κ>0 is openly left unresolved, and the sharpness at α=1 is proved. The main risk is structural rather than internal: the parameterized almost-prescription in Section 3 is explicitly built on the two-scale corrugation of the unpublished preprint [1], and the final cancellation identity in Lemma 3.2 is exactly the content of [1, Proposition 2.2]. If that external result contained a gap, Proposition 3.3 and all later sections would be unsupported.","major_comments":[{"comment":"The construction is introduced as 'based on the two-scale corrugation in [1, Proposition 2.2]', and the cancellation identity at the end of the proof of Lemma 3.2 is exactly the content of that proposition. Since [1] is an unpublished preprint, the correctness of Theorem 1.2 is structurally dependent on an external result that is neither stated as a separate lemma nor proved independently. The manuscript's own computation with (13) appears self-consistent, but it does not say which identities are imported from [1, Prop. 2.2] or verify them here. A sign error or a missed mean-zero condition in the imported two-scale cancellation would destroy the O(N^{-1}) estimate, and then the inductive error estimates (19)-(20) in Proposition 3.3, and hence every case of Theorem 1.2, would fail. I recommend proving the two-scale cancellation independently as a lemma, or at minimum stating the imported identities precisely and verifying them in this paper.","section":"§3, Lemma 3.2 and the preceding paragraph"},{"comment":"The construction of the limit u_i for fixed i is compressed: the diagonal subsequence is asserted, and the uniformity of the Harnack constants as the exhaustion parameter j varies is not explicitly justified. The later claim that every subsequential limit is 1 also relies on positivity and local H^1 boundedness of u_i, which are established only after the Harnack propagation step. These steps are standard, but they are load-bearing for the open case κ=0, so the argument should be expanded into a fully explicit proof.","section":"§4.2, Lemma 4.4"}],"minor_comments":[{"comment":"The proof asserts that α_s has a uniform positive lower bound on the spatial support of a; this should be justified explicitly from compactness of supp_x a and uniform equivalence of the family (h_s).","section":"§3, Lemma 3.2"},{"comment":"The sentence that the decrements Scal_{q_i}+m^{-1}β_i are bounded in C^0_loc is literally true only on K_{i+1}; the finitely many small i for which a given compact set is not contained in K_{i+1} are harmless, but the text should say this so that the uniform assertion in Proposition 3.3 is applied correctly.","section":"§4.2, proof of Theorem 1.2 for open M, κ=0"},{"comment":"The proof uses continuity of the first eigenvalue λ_1(g_{i,s}) in s to locate the root s_i; this is standard, but a one-line justification via min-max and the C^0 closeness (24) would make the argument self-contained.","section":"§4.1, closed case κ=0"},{"comment":"The convergence λ_ℓ(-Δ_{h_i})→λ_ℓ(-Δ_g) and the uniform lower bound λ_ℓ(-Δ_{h_i})≥c λ_ℓ(-Δ_g) are used to obtain the spectral gap; they follow from min-max and uniform equivalence of metrics, but stating the comparison lemma explicitly would improve readability.","section":"§6, Lemma 6.1"}],"recommendation":"major_revision","confidential_remarks":"The only substantive risk is the unpublished dependence on [1]. If the journal prefers to avoid reliance on non-peer-reviewed preprints, the author should be asked to make the two-scale cancellation self-contained in an appendix. The rest of the proof appears coherent, and the remaining issues are local expansions rather than fundamental gaps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Wan's paper. The headline result is real: for closed manifolds, or any manifold with κ≤0, every smooth metric with Scal≥κ is a limit in the local Riemannian Burnett class of metrics with Scal=κ, with uniform W^{1,∞} bounds. The corollary—the C^{0,α} closure of M_κ equals M_{≥κ} for every α<1—is sharp and genuinely surprising. At κ=0 this settles the Riemannian reverse-Burnett conjecture of Huneau and Luk. The proof splits into a local almost-prescription step (Section 3) and a global conformal correction; both are executed carefully. I checked the conformal corrections for κ<0 and the spectral argument for κ>0, and they look sound. The local step is load-bearing. Lemma 3.2 computes the two-scale cancellation explicitly, and the O(N^{-1}) bound follows from the definitions. But the final cancellation identity is exactly [1, Prop 2.2], an unpublished arXiv preprint. The paper states what it imports, which is honest, but a gap there would break every case of Theorem 1.2. My own reading of the computation suggests it is self-contained enough that the risk is low, yet I would want [1] checked or the key identity proved in an appendix before calling this definitive. The soft spots elsewhere are minor: Lemma 4.4's diagonal subsequence and Harnack propagation are sketched, and Lemma 4.1's uniform normalization is terse. These are standard and I did not find a real gap. The κ>0 open case is clearly flagged. Who should read this: anyone working on scalar curvature stability, Gromov's C^0 program, or high-frequency limits in GR. It deserves a serious referee: the main theorem is important, the proof is detailed, and the external dependence, while real, is disclosed and localized. I would accept it for review and, conditional on verifying [1], likely endorse publication.","headline":"A strong, likely correct proof of the Riemannian reverse-Burnett conjecture that deserves refereeing, with the main caveat being a disclosed but load-bearing dependence on an unpublished preprint.","tokens_in":21176,"tokens_out":2092,"would_cite":true,"duration_ms":22620,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","35J60","58J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every smooth metric with scalar curvature at least $\\kappa$ is a local Burnett-class limit of smooth metrics with scalar curvature exactly $\\kappa$; on closed manifolds this holds for every real $\\kappa$, and on…","keywords":["scalar curvature","Burnett compactness","conformal Laplacian","reverse Burnett conjecture","prescribed scalar curvature","convex integration","W^{1,∞} convergence","C^0 stability"],"falsifier":"Check Lemma 3.2 on a flat three-torus with a constant positive function $a$: compute the scalar curvature of the two-scale perturbation $h_F$ for increasing frequency $N$ and verify that $\\|\\mathrm{Scal}_{h_F}-\\mathrm{Scal}_h+a^2\\|_{C^0}$ tends to zero while $\\|h_F\\|_{W^{1,\\infty}}$ stays bounded independently of $N$; an uncancelled $O(1)$ term or a derivative blow-up would falsify the local almost-prescription and with it Theorem 1.2.","tokens_in":20145,"feed_emoji":"📐","tokens_out":14438,"duration_ms":131016,"temperature":0.7,"pith_summary":"This paper proves a flexibility theorem for scalar curvature in a natural compactness regime for Riemannian metrics: sequences converge locally uniformly in $C^0$ while their first derivatives stay locally uniformly bounded, so the derivatives only converge in a weak sense and quadratic expressions can relax. In that class, every smooth metric with scalar curvature at least $\\kappa$ is the limit of smooth metrics with scalar curvature exactly $\\kappa$, whenever the manifold is closed or $\\kappa\\le 0$; for $\\kappa<0$ and a complete target metric, the approximating metrics can be chosen complete. Combined with the $C^0$-stability theorem for scalar-curvature lower bounds, this gives the equality of the $C^{0,\\alpha}_{\\mathrm{loc}}$-closure of the exactly-$\\kappa$ set with the at-least-$\\kappa$ set for every $\\alpha\\in(0,1)$, and the paper proves that $\\alpha<1$ is sharp. At $\\kappa=0$, this is the Riemannian reverse-Burnett conjecture, strengthened by the uniform local $W^{1,\\infty}$ bound and by the sharp Hölder exponent.","feed_headline":"Scalar-curvature bounds become exact equalities under Burnett limits","feed_subtitle":"It shows how much scalar-curvature information survives when only C^0 limits and bounded first derivatives are visible.","key_machinery":"Two mechanisms carry the argument. The first is a parameterized almost-prescription of scalar curvature: Proposition 3.3 produces a smooth one-parameter family $g_s$ starting at a given metric with $\\mathrm{Scal}_{g_s}\\approx \\mathrm{Scal}_g-s^2k$, while staying arbitrarily close in $C^0$ and locally bounded in $W^{1,\\infty}$; it is built from a two-scale corrugation in which one oscillation scale creates a signed quadratic term and a second scale cancels the unwanted affine term. The second is a global conformal correction via the conformal Laplacian $L_h=-c_n\\Delta_h+\\mathrm{Scal}_h$ with $c_n=4(n-1)/(n-2)$, using the conformal formula $\\mathrm{Scal}_{u^{4/(n-2)}h}=u^{-(n+2)/(n-2)}L_h u$. Existence of the positive factor $u$ is obtained case-by-case: by continuity of the first eigenvalue of the conformal Laplacian, by parabolicity with a weighted spectral gap, by constant sub- and supersolutions, or by rescaling away a resonance and running a contraction argument.","core_discovery":"The central claim is that the scalar-curvature inequality $\\mathrm{Scal}_g\\ge\\kappa$ is exactly the relaxation of the equation $\\mathrm{Scal}_g=\\kappa$ under local Riemannian Burnett compactness. For a connected $n$-manifold with $n\\ge 3$, if $M$ is closed or $\\kappa\\le 0$, every $g_0$ with $\\mathrm{Scal}_{g_0}\\ge\\kappa$ is the local uniform limit of smooth metrics $\\hat g_i$ satisfying $\\mathrm{Scal}_{\\hat g_i}=\\kappa$ and locally uniformly bounded in $W^{1,\\infty}$. The paper shows that the failure of the first derivatives to converge strongly carries the entire scalar-curvature defect: the divergence part of the curvature-density formula passes to the limit, while the quadratic part is what produces the inequality. It also proves sharpness: if a sequence of exactly-$\\kappa$ metrics converges in $C^0$ and its derivatives converge strongly in $L^2_{\\mathrm{loc}}$, the limit again has scalar curvature exactly $\\kappa$, so the Hölder exponent $\\alpha<1$ in the closure identity cannot be improved to Lipschitz.","pith_inferences":["Because the equality is sharp at the Hölder scale, intermediate closures should be visible if the compactness assumption is interpolated between $W^{1,\\infty}$ and strong $W^{1,2}$; one could test whether $W^{1,p}$-bounded sequences with large $p$ produce a family of strict intermediate sets between $\\{\\mathrm{Scal}=\\kappa\\}$ and $\\{\\mathrm{Scal}\\ge\\kappa\\}$.","The construction for $\\kappa=0$ on open manifolds parabolicizes the metric and does not preserve completeness; a natural extension is to ask whether complete scalar-flat approximants exist for every complete metric of nonnegative scalar curvature.","The same two-scale cancellation is likely to apply to other equations whose curvature density splits into a divergence term plus a quadratic term in first derivatives, such as the time-symmetric constraints, which would connect this Riemannian flexibility statement to the initial-data side of the reverse Burnett problem.","For $\\kappa>0$, the nonresonant rescaling shows that the obstruction on open manifolds is spectral rather than local; an extension would be to seek positive-$\\kappa$ examples on open manifolds where the essential spectrum of the conformal Laplacian is pushed away from $\\kappa/(n-1)$."],"forward_implications":["The closure identity $\\overline{\\{g:\\mathrm{Scal}_g=\\kappa\\}}^{C^{0,\\alpha}_{\\mathrm{loc}}}=\\{g:\\mathrm{Scal}_g\\ge\\kappa\\}$ holds for every $\\alpha\\in(0,1)$, and the same equality holds in the $C^0_{\\mathrm{loc}}$-topology; on closed manifolds the topologies may be taken globally.","At $\\kappa=0$, every smooth metric of nonnegative scalar curvature on a connected manifold of dimension $n\\ge 3$ is a local Riemannian Burnett limit of scalar-flat metrics, settling the reverse-Burnett conjecture in strengthened form.","For $\\kappa<0$, the approximating metrics can be chosen complete whenever the original metric is complete.","The sharpness statement gives $\\overline{\\{g:\\mathrm{Scal}_g=\\kappa\\}}^{C^{0,1}_{\\mathrm{loc}}}=\\{g:\\mathrm{Scal}_g=\\kappa\\}$, so strong $C^{0,1}$ limits do not relax the equation.","For $\\kappa>0$, the theorem covers closed manifolds, while the case of open manifolds is explicitly left open because the global conformal correction is not available."],"supporting_citations":[{"why":"Supplies the two-scale corrugation and scalar-curvature cancellation, the external proposition on which Lemma 3.2 and every subsequent deformation path depend.","marker":"[1]"},{"why":"Provides the interior elliptic estimates, maximum principles, and Schauder theory used to upgrade convergence of conformal factors and to run eigenvalue and contraction arguments.","marker":"[6]"},{"why":"Establishes the C^0-stability inclusion that, paired with Theorem 1.2, yields the closure identity.","marker":"[9]"},{"why":"Formulates the Riemannian reverse-Burnett conjecture whose κ=0 case is proved and strengthened here.","marker":"[11]"},{"why":"Gives the invariant divergence-form decomposition of scalar curvature used to show weak convergence of derivatives and the sharpness of the C^{0,1} closure.","marker":"[17]"},{"why":"First proved the unparameterized almost-prescription theorem, the endpoint of Proposition 3.3 underlying the flexibility construction.","marker":"[18]"},{"why":"Supplies the ground-state alternative for nonnegative Schrödinger operators used to obtain the weighted spectral gap in the open κ=0 case.","marker":"[20]"}],"fun_headline_variants":["Scalar curvature inequalities are Burnett limits of equalities","Equality metrics are dense in scalar curvature bounds","Sharp scalar curvature flexibility in Burnett class","Scalar curvature equalities approximate all lower bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the local two-scale corrugation construction taken from reference [1] and used without proof is correct; if that construction cannot actually lower scalar curvature by approximately $s^2k$ while keeping the metrics $C^0$-close and first-derivative-bounded, then Proposition 3.3, and with it the whole theorem, would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Scalar curvature inequalities are Burnett limits of equalities","Equality metrics are dense in scalar curvature bounds","Sharp scalar curvature flexibility in Burnett class","Scalar curvature equalities approximate all lower bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001539,"raw_usage":{"total_tokens":6182,"prompt_tokens":994,"completion_tokens":5188,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":5128}},"tokens_in":610,"tokens_out":5188,"duration_ms":40259,"temperature":1.0,"reasoning_tokens":5128,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:26:44.208108+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check Lemma 3.2 on a flat three-torus with a constant positive function $a$: compute the scalar curvature of the two-scale perturbation $h_F$ for increasing frequency $N$ and verify that $\\|\\mathrm{Scal}_{h_F}-\\mathrm{Scal}_h+a^2\\|_{C^0}$ tends to zero while $\\|h_F\\|_{W^{1,\\infty}}$ stays bounded independently of $N$; an uncancelled $O(1)$ term or a derivative blow-up would falsify the local almost-prescription and with it Theorem 1.2.","supporting_citations":[{"cited_title":"Almost prescribing scalar curvature by mixed convex integration","cited_arxiv_id":"2505.08384","evidence_quote":"Supplies the two-scale corrugation and scalar-curvature cancellation, the external proposition on which Lemma 3.2 and every subsequent deformation path depend."},{"cited_title":"Gilbarg and N","cited_arxiv_id":null,"evidence_quote":"Provides the interior elliptic estimates, maximum principles, and Schauder theory used to upgrade convergence of conformal factors and to run eigenvalue and contraction arguments."},{"cited_title":"Gromov,Dirac and Plateau billiards in domains with corners, Cent","cited_arxiv_id":null,"evidence_quote":"Establishes the C^0-stability inclusion that, paired with Theorem 1.2, yields the closure identity."},{"cited_title":"Huneau and J","cited_arxiv_id":null,"evidence_quote":"Formulates the Riemannian reverse-Burnett conjecture whose κ=0 case is proved and strengthened here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the invariant divergence-form decomposition of scalar curvature used to show weak convergence of derivatives and the sharpness of the C^{0,1} closure."},{"cited_title":"Lohkamp,Scalar curvature and hammocks, Math","cited_arxiv_id":null,"evidence_quote":"First proved the unparameterized almost-prescription theorem, the endpoint of Proposition 3.3 underlying the flexibility construction."},{"cited_title":"Pinchover and K","cited_arxiv_id":null,"evidence_quote":"Supplies the ground-state alternative for nonnegative Schrödinger operators used to obtain the weighted spectral gap in the open κ=0 case."}],"review_version":1}