{"id":"87fb5dea-96f3-4dd0-8d52-f3a7c7612172","arxiv_id":"2412.03997","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Centered spheres minimize the weighted free energy for all volumes only under extra monotonicity of both weights; otherwise the second-variation condition psi''+g' >= 0 can fail to select global minimizers.","lead":"The paper studies when centered balls minimize a weighted perimeter-plus-potential energy with radial densities, and shows that the natural local stability condition is not sufficient in higher dimensions. It proves classification and quantitative stability when both the density and the potential are monotone increasing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.2's displayed hypothesis (4.3) is not implied by the Theorem 1.8 hypotheses, but the proof only needs the weaker condition in Remark 4.3, so the gap is repairable.","rationale":"The reader correctly identified that Proposition 4.2's displayed condition (4.3) is not a consequence of monotone increasing strict admissibility, and the concrete arctan example is valid. This is a genuine flaw in the written proof of Theorem 1.8, because the cited proposition is invoked at the radii r_k and its stated hypotheses are not met. However, the paper itself contains the needed repair: Remark 4.3 states a weaker sufficient condition, and the subsequent proof uses (4.3) only to assert the positivity of Lambda. Under the theorem's monotonicity hypotheses, that weaker condition holds automatically, so the contradiction argument in Theorem 1.8 can be made rigorous with a modest amendment. I therefore do not regard this as a counterexample to the main theorems, but the false statement in Proposition 4.2 and Remark 4.3 must be corrected before acceptance. The classification result Theorem 1.5 is not directly affected by this particular gap, though its proof is long and still depends on technical lemmas imported from [11] and [15]. The reader's conditional verdict is the appropriate level of confidence: the paper is promising and likely correct, but the stability proof needs a written repair and a fuller verification of the imported machinery.","tokens_in":36930,"tokens_out":16219,"duration_ms":143390,"concrete_test":"Re-derive the branch of Proposition 4.2 in which the spherical mean of u is nonnegative, replacing (4.3) by the weaker condition (R^2 psi' + R(n-1))(psi'+g) + R(n-1)psi' + R^2(psi''+g') + (n-1)(n-2) > 0. Compute this expression for n=2, psi = arctan(r), g(r) = r + C and verify it is positive for all R > 0; then trace the proof of Theorem 1.8 to confirm that no other use of (4.3) occurs. If the expression is positive and no other use occurs, the concern is a repairable statement-level error rather than a counterexample to the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing soft spot is in the quantitative stability half. Theorem 1.8 is proved by applying Proposition 4.2 at the radii r_k produced in the contradiction argument of Section 4.2. As stated, Proposition 4.2 requires condition (4.3): psi'(R) > -[(n-2)(n-1)]/[R^2(psi'(R)+g(R))] + (n-1)R. For n=2 this reduces to psi'(R) > R. The monotone increasing strictly admissible hypotheses do not imply this: take psi(r) = arctan(r) and g(r) = r + C with C > 0; then psi'(R) = 1/(1+R^2) < R for all sufficiently large R, while psi''(R)+g'(R) = 1 - 2R/(1+R^2)^2 > 0, so the pair satisfies the Theorem 1.8 hypotheses and (4.3) fails. Hence the application of Proposition 4.2 to r_k is not justified exactly as written, and Remark 4.3's assertion that (4.3) is automatic is false. The gap is not fatal, because the proof uses (4.3) only to assert positivity of the displayed constant Lambda, and the weaker condition stated later in Remark 4.3 is automatic for monotone increasing weights and gives the same positivity. Thus Theorem 1.8 appears recoverable, but the written statement and proof need amendment.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the free-energy isoperimetric problem in R^n with perimeter density f=e^ψ and external potential g, both radial, under a volume constraint with respect to f. After establishing existence, boundedness, and regularity of minimizers, it computes the first and second variations and shows that centered spheres are stationary and stable exactly when ψ''+g'≥0. The main results are: (i) a complete one-dimensional classification; (ii) two counterexamples in n≥2 showing that ψ''+g'>0 plus a minimum of ψ at the origin does not suffice for global optimality of centered spheres; (iii) a global uniqueness theorem for centered spheres when ψ and g are both monotone increasing and strictly admissible (Theorem 1.5); and (iv) a sharp quadratic stability inequality in that monotone class (Theorem 1.8). The proof of (iii) is an adaptation of Chambers' proof of the log-convex density conjecture, and the proof of (iv) follows the Fuglede-type scheme of Fusco–La Manna.","tokens_in":37260,"tokens_out":19338,"duration_ms":151092,"significance":"If the proofs are completed, the paper gives a fairly complete answer to the question of when the stability condition ψ''+g'≥0 is also sufficient for global optimality of centered spheres, and it provides a quantitative rigidity statement in the monotone regime. The two counterexamples are instructive and show that the one-dimensional behavior does not persist in higher dimensions. The main classification is a substantial adaptation of Chambers' argument to the case with a potential, and the quantitative stability result is a natural and useful strengthening. The paper also contributes elementary existence and regularity results for this free-energy functional. The proofs are mostly self-contained and the statements are precise, with the qualification that the quantitative stability half currently contains two repairable gaps, detailed below. The counterexamples and the classification argument are, in my reading, sound.","major_comments":[{"comment":"Condition (4.3) in Proposition 4.2 is not implied by the hypotheses of Theorem 1.8, and the assertion in Remark 4.3 that (4.3) is automatically satisfied for monotone increasing strictly admissible weights is false. For example, take n=2, ψ(r)=arctan(r), and g(r)=r+C with C>0. Then ψ is increasing, g is increasing, and ψ''(r)+g'(r)=1-2r/(1+r^2)^2>0 for all r>0, so the weights satisfy the assumptions of Theorem 1.8. However, for n=2 the right-hand side of (4.3) reduces to R, while ψ'(R)=1/(1+R^2)<R for all sufficiently large R, so (4.3) fails. Since the proof of Theorem 1.8 in Section 4.2 applies Proposition 4.2 at the radii r_k produced by Lemma 4.14, with r_k converging to the arbitrary radius R>0, the proof of Theorem 1.8 is not justified as written. This is repairable: the proof of Proposition 4.2 only requires the weaker condition stated in Remark 4.3, namely (R^2ψ'(R)+R(n-1))(ψ'(R)+g(R))+R(n-1)ψ'(R)+R^2(ψ''(R)+g'(R))+(n-1)(n-2)>0, which is automatically satisfied when ψ'≥0, g≥0 and ψ''+g'>0. The proposition should be restated with that weaker hypothesis and the remark amended.","section":"Section 4.1 (Proposition 4.2 and Remark 4.3)"},{"comment":"The proof of Theorem 1.8 uses Lemma 4.7 in the case of large asymmetric difference, but Lemma 4.7 requires the hypothesis ψ(r)>ψ(0). This is not a consequence of the assumptions 'ψ,g monotone increasing strictly admissible' if 'monotone increasing' is understood in the standard non-decreasing sense, which the paper's own usage in Proposition 1.4 and Proposition 2.17 suggests. For instance, ψ≡0 and g(r)=r are monotone increasing and strictly admissible, yet ψ(r)=ψ(0) for every r. The contradiction argument of Lemma 4.7 would still need a separate justification in this degenerate case; as written, the application of Lemma 4.7 in the proof of Theorem 1.8 is not covered by its hypotheses. This is a further gap in the stability theorem, though it is also local and repairable, either by treating the constant-ψ case separately (e.g., via [43] when ψ≡0) or by a limiting argument from strictly increasing perturbations.","section":"Section 4.2 (Theorem 1.8 proof, Lemma 4.7)"}],"minor_comments":[{"comment":"The statement of Proposition 2.2 uses condition (2.2), but Remark 2.3 and the proof refer to 'condition (2.3)' and 'thanks to condition (2.3)'; these references should be to (2.2).","section":"Section 2.1 (Proposition 2.2 and Remark 2.3)"},{"comment":"The estimate e^{M/n}-e^{M/(2n)} ≥ M/n used near the end of the proof is not correct as written; the correct lower bound is e^{M/n}-e^{M/(2n)} ≥ M/(2n). The conclusion remains valid because the second term on the right-hand side of (2.25) still has a factor ε^{n-1} and can be made small, but the displayed constant in the perimeter estimate should be adjusted.","section":"Section 2.4 (Proposition 2.16)"},{"comment":"The last display of Proposition 3.2 reads Pf(F⋆ξ) ≤ Pf(Fξ); the right-hand side should be Pf(F), not Pf(Fξ).","section":"Section 3.1 (Proposition 3.2)"},{"comment":"In the statement of Lemma 4.14, the phrase 'for each i ∈ N' is unused and appears to be a leftover; the graph u_k should be indexed by k, and the regularity statement 'C^{1,α} for all α<1/2' should be stated consistently.","section":"Section 4.1 (Lemma 4.14)"},{"comment":"In the lower bound for h, the expression ℓ′ + ℓ r + (n-1)/r should read ℓ′ + ℓ(r + (n-1)/r); the factor ℓ is missing in front of (n-1)/r.","section":"Section 2.3 (Theorem 1.3 proof)"},{"comment":"The displayed computation of ˜H''_1(0) contains unreadable OCR artifacts (the strings '/bracehtipupleft/bracehtipdownright/...'); these should be cleaned up in the LaTeX source.","section":"Section 3.2 (Lemma 3.9(2))"}],"recommendation":"major_revision","confidential_remarks":"The reader's report correctly identifies the condition (4.3) problem; I agree with the skeptic that the gap is repairable by using the weaker condition stated in Remark 4.3. The classification part (Theorem 1.5 and the counterexamples) appears sound in my reading. The quantitative stability theorem is the weakest part of the paper as written, and in addition to the (4.3) issue, there is a second gap concerning the use of Lemma 4.7 when ψ(r)=ψ(0). Both are local and fixable, so I recommend major revision rather than rejection. The paper is otherwise well-written and contains several interesting results that merit publication after the stability section is repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper answers a natural question — whether ψ′′+g′ ≥ 0 suffices for centered spheres to minimize the free energy P_f + G_f — and the answer is no. The two counterexamples (Propositions 2.16 and 2.17) are genuinely new and reasonably clean: the stability condition holds, ψ has its minimum at the origin, and a displaced ball still beats the centered one at intermediate volumes. That alone makes the paper worth reading for anyone in weighted isoperimetric problems.\n\nThe positive results are also real. Theorem 1.5, the classification under monotone increasing ψ and g, is a careful adaptation of Chambers' log-convex density proof to a functional with a potential. Much of the machinery is imported, from Chambers and from Boyer–Brown–Chambers–Loving–Tammem, but the adaptation is the contribution, and it appears to hold together. The one-dimensional characterization (Theorem 1.2) is new and sharp, and the large-volume result (Theorem 1.3) is a solid minor extension of the Kolesnikov–Zhdanov calibration.\n\nThe soft spot is Theorem 1.8, the quantitative stability estimate. Proposition 4.2 states condition (4.3), and Remark 4.3 claims it is automatic under the stated hypotheses. That claim is false: with n = 2, ψ(r) = arctan(r), g(r) = r + C, the hypotheses of Theorem 1.5 hold but (4.3) fails for large R. So the application of Proposition 4.2 at the radii r_k in the contradiction argument of Section 4.2 is not justified as written. The stress-test note is right that this is repairable: the proof only needs the weaker condition stated inside Remark 4.3, which does follow from monotone increasing weights. So Theorem 1.8 likely survives with a corrected statement and proof. There are also some inconsistent displayed formulas in Proposition 2.16 and Lemma 3.9 — minor, but they should be fixed.\n\nThe citation pattern is fine. Heavy reliance on [11] and [15] is legitimate because those are published proofs; the counterexamples and the classification are not circular.\n\nWho this is for: people working on weighted isoperimetric problems, free boundary problems with external potentials, and quantitative stability. It deserves a serious referee. I would send it out, asking the authors to fix the (4.3)/Remark 4.3 issue and clean up the displayed formulas. The main classification and the counterexamples look solid.\n\nRecommendation: engage with it — referee it if you get the chance, and read the counterexamples either way.","headline":"Counterexamples genuinely answer Q1 in the negative; the monotone classification is a serious Chambers adaptation; the stability theorem has a repairable but real gap in Proposition 4.2/Remark 4.3.","tokens_in":37793,"tokens_out":4339,"would_cite":true,"duration_ms":35340,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q20","49Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that, for monotone increasing strictly admissible radial weights, centered spheres uniquely minimize the free energy at every volume, and the energy gap controls the squared weighted distance to the ball.","keywords":["weighted isoperimetric problem","free energy minimizers","radial density","spherical symmetrization","quantitative stability","isoperimetric profile","monotone weights","nearly spherical sets"],"falsifier":"Evaluate condition (4.3) along the radii $r_k$ produced by Lemma 4.14 for a monotone strictly admissible pair; in $n=2$ with $\\psi(r)=\\arctan r$ the condition reduces to $\\psi'(r_k)>r_k$, which fails for all $r_k>1$, so if the contradiction argument reaches such radii the proof of Theorem 1.8 as written cannot invoke Proposition 4.2.","tokens_in":36674,"feed_emoji":"🔵","tokens_out":12221,"duration_ms":101223,"temperature":0.7,"pith_summary":"The paper studies minimizers of a free-energy functional $E(F)=P_f(F)+G_f(F)$ on $\\mathbb{R}^n$, where $f=e^\\psi$ weights the perimeter and $g$ weights the bulk potential, with both $\\psi$ and $g$ radially symmetric. The central claim is a complete answer to the question of when centered spheres are the unique global minimizers under a fixed weighted volume. Local stability of the spheres is equivalent to $\\psi''+g'\\ge 0$, but the paper shows this condition alone is not sufficient in dimension $n\\ge2$, constructing two counterexamples. Adding the assumption that both $\\psi$ and $g$ are monotone increasing (with strict admissibility, meaning $\\psi''+g'>0$ for $r>0$, together with the standing growth and regularity hypotheses) forces the centered spheres to be the unique minimizers for every volume. From this classification the paper derives an explicit isoperimetric profile and a sharp quadratic stability inequality, in which the energy excess of a same-volume competitor controls the squared $f$-weighted symmetric difference to the ball.","feed_headline":"Monotone weights force centered spheres to be unique minimizers","feed_subtitle":"A single local curvature condition is not enough; adding monotonicity gives the full classification and a sharp stability bound.","key_machinery":"The classification argument runs through spherical symmetrization, which replaces any competitor by a spherically symmetric set with no larger energy and the same weighted volume. The boundary of the symmetrized minimizer is described by a plane curve $\\gamma$ solving the constant weighted mean curvature equation (3.3), and the proof decomposes $\\gamma$ into an upper curve, a lower curve, and a curl curve; a comparison argument shows the lower curve bends faster than the upper one, so it cannot cross the symmetry axis, and any curling contradicts the tangent restriction, forcing $\\gamma$ to be a centered circle. A separate calibration argument, using a radial vector field with an explicit multiplier, gives large-volume uniqueness under $\\kappa$-uniform admissibility. For stability, the key machinery is a second-order expansion on nearly spherical sets, writing $\\partial E=\\{Rx(1+u(x))\\}$, expanding the energy gap to second order, and controlling it from below by $\\|u\\|^2_{L^2}$; the local estimate is then connected to global sets through almost-minimizers, uniform boundedness, and a compactness argument.","core_discovery":"The central discovery is that the stability condition $\\psi''+g'\\ge 0$, which characterizes local minimality of centered spheres, is not enough to guarantee global optimality once a potential term $g$ is present. In dimension $n\\ge2$, the paper constructs $\\kappa$-uniformly admissible weights with $\\psi$ minimized at the origin for which non-centered balls beat the centered ball at intermediate volumes, so monotonicity of the two weights is needed. Under monotone increasing strictly admissible hypotheses, Theorem 1.5 establishes that centered spheres uniquely solve the volume-constrained problem for every volume; Corollary 1.6 records the resulting isoperimetric profile identity, and Theorem 1.8 states a sharp quadratic stability bound $E(E)-E(B_R)\\ge c|E\\triangle B_R|_f^2$ for all sets of the same weighted volume.","pith_inferences":["A natural extension not pursued in the paper is to lower the regularity requirement on the monotone weights; the comparison structure of the curve argument suggests that a density/approximation argument could preserve the classification for merely $C^2$ monotone strictly admissible weights.","The counterexamples occupy the intermediate-volume regime, which is the natural frontier: small volumes concentrate near minima of $e^\\psi$, large volumes are forced to centered balls by calibration, and the failure of global optimality is a genuinely intermediate-volume phenomenon; a one-parameter family of weights could be tested numerically across this transition.","For the stability theorem, the proof would become unconditional if the local estimate of Proposition 4.2 were stated directly under the weaker condition displayed in Remark 4.3 and that condition were shown to hold at the radii produced by the compactness argument; making this substitution is a concrete open step.","The sharpness of the quadratic exponent via ellipsoids suggests that the optimal constant in (1.4) is governed by the lowest eigenvalue of the second variation; computing it for a model potential such as a linear field would give an explicit quantitative bound for droplet shapes in a gravitational field."],"forward_implications":["For any monotone increasing strictly admissible pair $(\\psi,g)$, the volume-constrained variational problem is fully solved: every minimizer is a centered ball, so further analysis of the free energy can be restricted to balls.","The explicit isoperimetric profile formula $E'(v)=g(r)+\\psi'(r)+(n-1)/r$ with $r=\\Phi^{-1}(v)$ makes it possible to detect where the profile is concave or convex and to locate volumes at which centered uniqueness may fail.","The two counterexamples show that numerical or variational methods that rely only on $\\psi''+g'\\ge0$ to certify global optimality will fail in dimension $n\\ge2$; monotonicity checks must be added.","The sharp quadratic stability inequality implies quantitative convergence of any minimizing sequence to the centered ball in the $f$-weighted $L^1$ distance, with the exponent 2 optimal since ellipsoidal deformations saturate the bound.","In the large-volume regime, $\\kappa$-uniform admissibility alone guarantees that centered spheres of radius larger than $\\sqrt{n+2}/\\kappa$ are the unique minimizers."],"supporting_citations":[{"why":"Supplies the spherical-symmetrization and upper/lower/curl curve comparison method that the classification proof adapts to the weighted free energy.","marker":"[15]"},{"why":"Provides the calibration lemma and explicit multiplier used for large-volume uniqueness of centered spheres under uniform admissibility.","marker":"[45]"},{"why":"Establishes existence of isoperimetric regions in $\\mathbb{R}^n$ with density plus the monotonicity and mean-convexity tools used in the curve analysis.","marker":"[57]"},{"why":"Provides the nearly-spherical second-order stability template, the almost-minimizer framework, and the compactness lemmas on which the proof of Theorem 1.8 is built.","marker":"[33]"},{"why":"Gives the detailed lemmas for the generating curve of isoperimetric regions in density $r^p$, which are adapted for general monotone weights.","marker":"[11]"},{"why":"Proves the analogous large-volume and free-energy profile results in the case $\\psi\\equiv0$, used as the benchmark for Theorems 1.3 and 1.8.","marker":"[43]"},{"why":"Supplies the quantitative isoperimetric theory and the ellipsoid computation showing that the quadratic exponent in the stability estimate is sharp.","marker":"[31]"}],"fun_headline_variants":["Local sphere test not enough; monotone weights fix it","Monotone radial weights guarantee global sphere minimizers","Sharp stability bound for monotone radial energy","Counterexample: local sphere optimality fails without monotonicity","Monotone weights: the missing key to global sphere minima"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stability proof requires that at every radius $r_k$ selected by the compactness argument the local estimate's condition (4.3) holds, meaning $\\psi'(r_k)>-\\frac{(n-2)(n-1)}{r_k^2(\\psi'(r_k)+g(r_k))}+(n-1)r_k$; this domination is not implied by the monotone strictly admissible hypotheses, and for $n=2$, $\\psi(r)=\\arctan r$ it fails for every $r>1$.","fun_headline_variants_meta":{"raw":{"variants":["Local sphere test not enough; monotone weights fix it","Monotone radial weights guarantee global sphere minimizers","Sharp stability bound for monotone radial energy","Counterexample: local sphere optimality fails without monotonicity","Monotone weights: the missing key to global sphere minima"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000984,"raw_usage":{"total_tokens":4113,"prompt_tokens":819,"completion_tokens":3294,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":3215}},"tokens_in":435,"tokens_out":3294,"duration_ms":53164,"temperature":1.0,"reasoning_tokens":3215,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:53:43.775368+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate condition (4.3) along the radii $r_k$ produced by Lemma 4.14 for a monotone strictly admissible pair; in $n=2$ with $\\psi(r)=\\arctan r$ the condition reduces to $\\psi'(r_k)>r_k$, which fails for all $r_k>1$, so if the contradiction argument reaches such radii the proof of Theorem 1.8 as written cannot invoke Proposition 4.2.","supporting_citations":[{"cited_title":"Proof of the log-convex density conjec ture","cited_arxiv_id":null,"evidence_quote":"Supplies the spherical-symmetrization and upper/lower/curl curve comparison method that the classification proof adapts to the weighted free energy."},{"cited_title":"On isoperimetric se ts of radially symmetric measures","cited_arxiv_id":null,"evidence_quote":"Provides the calibration lemma and explicit multiplier used for large-volume uniqueness of centered spheres under uniform admissibility."},{"cited_title":"Existence of isoperimetric regions in Rn with density","cited_arxiv_id":null,"evidence_quote":"Establishes existence of isoperimetric regions in $\\mathbb{R}^n$ with density plus the monotonicity and mean-convexity tools used in the curve analysis."},{"cited_title":"Some weighted isoperimetri c inequalities in quan- titative form","cited_arxiv_id":null,"evidence_quote":"Provides the nearly-spherical second-order stability template, the almost-minimizer framework, and the compactness lemmas on which the proof of Theorem 1.8 is built."},{"cited_title":"Isoperimetric Regions in Rn with Density rp","cited_arxiv_id":null,"evidence_quote":"Gives the detailed lemmas for the generating curve of isoperimetric regions in density $r^p$, which are adapted for general monotone weights."},{"cited_title":"The quantitative isoperimetric inequality and related topics","cited_arxiv_id":null,"evidence_quote":"Supplies the quantitative isoperimetric theory and the ellipsoid computation showing that the quadratic exponent in the stability estimate is sharp."}],"review_version":1}