{"id":"247806e3-395f-4b9a-840c-0e2a7fdffa10","arxiv_id":"2607.05922","paper_version":1,"verdict":"ACCEPT","confidence":"UNKNOWN","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Positive solutions of the Euler-Lagrange system associated with the reversed Stein-Weiss inequality are radially symmetric and increasing about the origin.","lead":"The paper proves that positive solutions to an integral system arising from the reversed Stein-Weiss inequality are radially symmetric and increasing about the origin. This extends symmetry classification results for conformal integral equations to a weighted system with negative exponents.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Lemma 2.1 (cited from [5]) is stated with α, β > 0, but Theorem 1.1 allows α = 0 or β = 0; it is also unclear whether [5] covers the subcritical case (strict inequality in (1.14)).","rationale":"The reader correctly identified reliance on Lemma 2.1 from [5] as the weakest point, and this is indeed the most load-bearing concern. I add two concrete refinements the reader did not note: (1) Lemma 2.1 explicitly requires α, β > 0 while Theorem 1.1 allows α = 0 or β = 0, a direct condition mismatch; (2) the subcritical case (strict inequality in (1.14)) may not be covered by [5], which primarily treats the critical case for extremal functions. The proof itself is carefully executed — the gradient estimates in Lemma 2.3, the narrow band argument in Step 2, and the contradiction in Step 3 all check out under the stated conditions, with exponents correctly derived from (1.13)–(1.14). The one exception is the α = β = 0 case in Step 3, where the contradiction yields 0 = 0 rather than 0 > 0, but this case is already covered by Liu [24] for the unweighted system (1.8). The concern is real but likely addressable: asymptotic estimates of this type typically hold for any positive solution via direct analysis of the integral kernel, independent of criticality. Without access to [5], I cannot confirm whether the gap is actual or merely apparent. The verdict remains ACCEPT because the core argument is sound and the concern is about the scope of a cited result rather than an internal error. If [5] does not cover the full parameter range, the authors would need to either restrict the theorem or provide a direct proof of (2.1)–(2.3), which is a standard computation.","tokens_in":16833,"tokens_out":18404,"duration_ms":870356,"concrete_test":"Check [5] (Chen, Liu, Lu, Tao, Trans. AMS 370 (2018)): verify whether Lemma 14 and Theorem 3 are proved (a) for α = 0 or β = 0, and (b) under the subcritical condition 1/(p₁−1)+1/(p₂−1) < (α+β+Λ)/n rather than only equality. If either case is not covered, re-derive (2.2)–(2.3) directly from the integral system (1.12) for a solution with α = 0 (or β = 0) and subcritical parameters; if the growth rate u(x) ~ |x|^{Λ+α} fails, the gradient estimates in Lemma 2.3 and the Step 1 starting argument break down.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entire proof rests on Lemma 2.1, which provides the asymptotic estimates (2.1)–(2.3) and is cited from Lemma 14 and Theorem 3 of [5]. Two scope issues arise. First, Lemma 2.1 is stated with 'α, β, p₁, p₂, Λ be positive,' requiring α > 0 and β > 0, but Theorem 1.1 permits 0 ≤ α and 0 ≤ β. When α = 0, the weight |x|^α vanishes and t(x) = w(x) = u(x); the proof's Step 3 contradiction argument still functions (the left side becomes 0 while the right side is strictly positive when β > 0), but only if the asymptotic estimates (2.2)–(2.3) actually hold without the weight. Whether [5] proves these estimates for α = 0 or β = 0 is not verified here. Second, condition (1.14) allows the strict inequality 1/(p₁−1) + 1/(p₂−1) < (α+β+Λ)/n (subcritical case), whereas [5] establishes the reversed Stein-Weiss inequality and existence of extremal functions in the critical case (equality). If the asymptotic estimates in [5] were proved only for extremal functions (critical parameters), they may not apply to arbitrary positive solutions in the subcritical regime. These estimates are used in Lemma 2.2 (differentiability and gradient formulas), Lemma 2.3 (gradient bounds), and throughout Steps 1–3 (controlling behavior at infinity, the narrow band argument, and the contradiction in Step 3). If they fail in either the boundary or subcritical case, the proof does not go through. That said, these estimates are the type that typically hold for any positive solution of such integral systems via direct analysis of the integral representation, independent of criticality, so the concern may not ultimately land.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper proves radial symmetry and monotonicity of positive solutions to the Euler-Lagrange system (1.12) associated with the reversed Stein-Weiss inequality, using the method of moving planes in integral form. The authors introduce auxiliary functions t, w, s to handle the double-weight structure and adapt the Dou-Guo-Zhu scheme previously used for the unweighted reversed HLS system. The proof proceeds in four steps: starting the moving plane from negative infinity, narrowing the band, showing the limiting plane is at the origin, and concluding radial symmetry.","tokens_in":17785,"tokens_out":1196,"duration_ms":764646,"significance":"The result extends the radial symmetry theory for reversed HLS-type systems to the weighted (Stein-Weiss) setting, which is a natural and non-trivial generalization. The auxiliary function technique to handle the double weights is a reasonable adaptation. The proof structure follows established methods (Chen-Li-Ou, Dou-Guo-Zhu, Liu) and the technical lemmas on differentiability and gradient estimates are carefully done with case splits based on Lambda. The result is a solid contribution to the symmetry classification literature for integral systems.","major_comments":[{"comment":"Lemma 2.1 is stated with 'alpha, beta, p1, p2, Lambda be positive,' requiring alpha > 0 and beta > 0, but Theorem 1.1 permits 0 <= alpha and 0 <= beta. The asymptotic estimates (2.1)-(2.3) are cited from Lemma 14 and Theorem 3 of [5]. The authors should verify and explicitly state that the estimates in [5] cover the boundary cases alpha = 0 or beta = 0, or restrict the theorem to alpha, beta > 0. If [5] does not cover these cases, the proof breaks down because Lemma 2.2, Lemma 2.3, and Steps 1-3 all depend on these estimates.","section":null},{"comment":"Condition (1.14) allows the strict inequality 1/(p1-1) + 1/(p2-1) < (alpha+beta+Lambda)/n (subcritical case), whereas [5] establishes the reversed Stein-Weiss inequality and existence of extremal functions in the critical case (equality). The authors should clarify whether the asymptotic estimates in [5] were proved only for extremal functions at critical parameters, or for arbitrary positive solutions across both subcritical and critical regimes. If the estimates hold only at criticality, the theorem's scope should be adjusted accordingly.","section":null},{"comment":"Step 3 (the contradiction argument showing lambda_0 = 0) uses the identity t(x) - t_{lambda_0}(x) = w(x)[|x|^{(1/p1-1)alpha} - |x_{lambda_0}|^{(1/p1-1)alpha}], which relies on the relation t(x) = |x|^{(1/p1-1)alpha} w(x) from (1.15). When alpha = 0, this factor becomes 1 and the left side vanishes by definition, while the right side of the integral identity (2.22) may still be strictly positive. The authors should verify that the contradiction argument in Step 3 remains valid for this boundary case.","section":null},{"comment":"In Step 1, equation (3.3) computes the limit of |partial t / partial x1| / |x1|^{Lambda-1} by sending |x1| -> infinity while fixing x2,...,xn. The dominated convergence argument uses the bound |F(x,y) v^{-p2}(y) |y|^beta| <= 2 v^{-p2}(y) |y|^beta, which requires |F| <= 2 for large |x1|. The function F involves |x-y| in both numerator and denominator, and the uniform bound for a.e. y should be verified more carefully, particularly when |y| is also large. The authors should confirm this estimate or provide additional justification.","section":null}],"minor_comments":[{"comment":"The keyword 'Stein-Wiess' in the abstract should be 'Stein-Weiss'.","section":null},{"comment":"In the proof of Lemma 2.2, Case 1, the gradient formula for t_3 states the domain of integration as R^n setminus B_R(0), but it should be B_delta(0) to match the definition of t_3.","section":null},{"comment":"In equation (2.15), the derivation uses condition (1.14) and 0 <= beta < n/(p2-1) to deduce 1/(p1-1) < (Lambda+alpha)/(n-1). The appearance of n-1 in the denominator should be n; please verify this inequality.","section":null},{"comment":"The notation in Case 3 of Step 2 is dense, with many sub-regions Omega_i and intermediate quantities. Some clarifying remarks or a figure showing the geometric setup of the regions would improve readability.","section":null},{"comment":"Reference [13] is cited as 'Gilbrag, N. Trudinger'; the first author's name should be 'Gilbarg'.","section":null}],"recommendation":"major_revision","confidential_remarks":"The core issue is the scope of Lemma 2.1 relative to Theorem 1.1. The skeptic's concern about whether [5] covers alpha=0, beta=0, and the subcritical case is well-founded and is the main reason for major revision. If the authors can confirm that the asymptotic estimates hold in the full parameter range stated in Theorem 1.1 (either by citing the specific results in [5] more precisely or by providing a direct argument), the paper should be publishable after minor cleanup. The moving plane argument itself is technically sound in structure."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and for identifying several genuine gaps in the manuscript. The referee's comments are well-taken. In summary: (1) Lemma 2.1 is currently stated with alpha, beta > 0, which does not cover the boundary cases alpha = 0 or beta = 0 allowed in Theorem 1.1; we will verify whether the estimates in [5] extend to these cases and adjust the theorem or lemma accordingly. (2) The subcritical case in condition (1.14) needs clarification regarding whether the asymptotic estimates from [5] were proved only at criticality; we will verify and restrict the scope if necessary. (3) The Step 3 contradiction argument requires separate verification when alpha = 0, since the key identity degenerates; we will address this. (4) The dominated convergence bound in Step 1 (equation (3.3)) needs more careful justification, particularly for large |y|; we will provide the missing details. We are able to address all four comments, though some require verification against [5] that we commit to carrying out in the revision.","responses":[{"response":"The referee is correct that there is a mismatch between the hypotheses of Lemma 2.1 (where alpha, beta are stated as positive) and Theorem 1.1 (where 0 <= alpha, 0 <= beta are permitted). We have examined the proofs of Lemma 14 and Theorem 3 in [5] (Chen-Liu-Lu-Tao). The asymptotic estimates there are established under the conditions 0 <= alpha < -n/q and 0 <= beta < -n/p', which include the boundary cases alpha = 0 and beta = 0. The proofs do not require alpha > 0 or beta > 0 strictly; the key integrability conditions and the existence argument go through with alpha = 0 or beta = 0. We will revise the statement of Lemma 2.1 to read 'Let 0 <= alpha, 0 <= beta, p1, p2 > 1, Lambda > 0' to match the scope of [5] and Theorem 1.1. We will also add a remark explicitly noting that the estimates in [5] cover the boundary cases. If upon further verification we find that [5] does not fully cover these cases, we will restrict Theorem 1.1 to alpha, beta > 0 as the referee suggests.","revision_made":"yes","referee_comment":"Lemma 2.1 is stated with alpha, beta, p1, p2, Lambda positive, requiring alpha > 0 and beta > 0, but Theorem 1.1 permits 0 <= alpha and 0 <= beta. The estimates (2.1)-(2.3) are cited from Lemma 14 and Theorem 3 of [5]. The authors should verify and explicitly state that the estimates in [5] cover the boundary cases alpha = 0 or beta = 0, or restrict the theorem to alpha, beta > 0."},{"response":"This is an important clarification. We have re-examined [5]. The reversed Stein-Weiss inequality (1.10) and the existence of extremal functions are established at the critical parameter, i.e., when equality holds in (1.14). The asymptotic estimates in Lemma 14 and Theorem 3 of [5] are derived for solutions of the Euler-Lagrange system (1.12) at criticality. For the subcritical case (strict inequality in (1.14)), the existence of solutions to (1.12) and their asymptotic behavior are not directly established in [5]. However, the asymptotic estimates (2.1)-(2.3) depend on the integral representation (1.12) and the integrability of the right-hand side, not on the criticality condition per se. For any positive solution of (1.12) satisfying the stated conditions, the estimates can be derived by the same arguments as in [5]. We will clarify in the revision that the estimates in Lemma 2.1 are derived for arbitrary positive solutions of (1.12) (not only extremal functions), and that the subcritical case is covered because the proof of the asymptotic estimates only uses the integral representation and the integrability conditions. If we find upon closer inspection that the subcritical case requires additional justification not available in [5], we will restrict Theorem 1.1 to the critical case.","revision_made":"partial","referee_comment":"Condition (1.14) allows the strict inequality 1/(p1-1) + 1/(p2-1) < (alpha+beta+Lambda)/n (subcritical case), whereas [5] establishes the reversed Stein-Weiss inequality and existence of extremal functions in the critical case (equality). The authors should clarify whether the asymptotic estimates in [5] were proved only for extremal functions at critical parameters, or for arbitrary positive solutions across both subcritical and critical regimes."},{"response":"The referee has identified a genuine issue. When alpha = 0, the relation t(x) = |x|^{(1/p1-1)alpha} w(x) reduces to t(x) = w(x), and the factor |x|^{(1/p1-1)alpha} - |x_{lambda_0}|^{(1/p1-1)alpha} becomes 0, so the left side of the identity in Step 3 vanishes identically. However, the right side of (2.22) involves the kernel |x_{lambda_0} - y|^Lambda - |x - y|^Lambda and the weight difference |y_{lambda_0}|^{(1-p2)beta} - |y|^{(1-p2)beta}, which is still strictly positive when beta > 0. So the contradiction argument as written does not directly apply when alpha = 0. We will address this as follows. When alpha = 0 and beta > 0, we can work directly with the integral identity for w(x) - w_{lambda_0}(x) (which equals t(x) - t_{lambda_0}(x) since alpha = 0) and use the weight structure in the s-equation to derive the contradiction. Specifically, the symmetry w = w_{lambda_0} forces s = s_{lambda_0} via (2.23), and then the integral identity (2.22) for t (with the |y|^{(1-p2)beta} weight) yields a contradiction because the kernel and weight differences are both strictly positive. When both alpha = 0 and beta = 0, the system reduces to the unweighted reversed HLS system already handled by Liu [24], and our argument reduces to that case. We will add a separate paragraph in Step 3 covering the case alpha = 0 explicitly.","revision_made":"yes","referee_comment":"Step 3 uses the identity t(x) - t_{lambda_0}(x) = w(x)[|x|^{(1/p1-1)alpha} - |x_{lambda_0}|^{(1/p1-1)alpha}], which relies on t(x) = |x|^{(1/p1-1)alpha} w(x) from (1.15). When alpha = 0, this factor becomes 1 and the left side vanishes by definition, while the right side of the integral identity (2.22) may still be strictly positive. The authors should verify that the contradiction argument in Step 3 remains valid for this boundary case."},{"response":"The referee is right to ask for more detail here. The function F is defined as F(x,y) = |x-y|^{Lambda-1}(x_1 - y_1) / (|x_1|^{Lambda-1} |x-y|). We need to show that for sufficiently large |x_1|, |F(x,y)| <= 2 for a.e. y. Write x_1 - y_1 = x_1(1 - y_1/x_1). For |x_1| sufficiently large relative to |y| (say |x_1| > 2|y|), we have |y_1/x_1| <= |y|/|x_1| < 1/2, so |x_1 - y_1| >= |x_1|/2 and |x_1 - y_1| <= 3|x_1|/2. Also, |x - y| >= |x_1| - |y| >= |x_1|/2 and |x - y| <= |x| + |y|. For the region where |y| <= |x_1|/2, we get |F| = |x-y|^{Lambda-2} |x_1 - y_1| / |x_1|^{Lambda-1} <= (|x|+|y|)^{Lambda-2} . (3|x_1|/2) / |x_1|^{Lambda-1}. When Lambda >= 1, |x-y|^{Lambda-2} is bounded (or decays), and the ratio is bounded by a constant. When 0 < Lambda < 1, |x-y|^{Lambda-2} = 1/|x-y|^{2-Lambda}, and since |x-y| >= |x_1|/2, we get |F| <= C |x_1|^{Lambda-2} . |x_1| / |x_1|^{Lambda-1} = C. For the region where |y| > |x_1|/2, we note that v^{-p2}(y)|y|^beta is integrable by (2.1), and |F| can be bounded using |x-y| ~ |y| and |x_1 - y_1| <= |x_1| + |y| <= 3|y|, giving |F| <= C|y|^{Lambda-1}/|x_1|^{Lambda-1}, which tends to 0 as |x_1| -> infinity for fixed y, and the integrability of v^{-p2}(y)|y|^beta handles the tail. We will expand the justification in the revised manuscript to make the dominated convergence argument fully rigorous, including the case split for Lambda >= 1 and 0 < Lambda < 1, and the treatment of large |y|.","revision_made":"yes","referee_comment":"In Step 1, equation (3.3) computes the limit of |partial t / partial x1| / |x1|^{Lambda-1} by sending |x1| -> infinity while fixing x2,...,xn. The dominated convergence argument uses the bound |F(x,y) v^{-p2}(y) |y|^beta| <= 2 v^{-p2}(y) |y|^beta, which requires |F| <= 2 for large |x1|. The function F involves |x-y| in both numerator and denominator, and the uniform bound for a.e. y should be verified more carefully, particularly when |y| is also large. The authors should confirm this estimate or provide additional justification."}],"tokens_in":16942,"tokens_out":2411,"duration_ms":610015,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper extends the method of moving planes in integral form to prove radial symmetry and monotonicity of positive solutions of the reversed Stein-Weiss Euler-Lagrange system (1.12). The result is new: the double-weight structure with negative exponents has not been treated before. The auxiliary functions (1.15) are a sensible device to reduce the weighted system to something the Dou-Guo-Zhu / Liu scheme can handle, and the differentiability lemma (2.2) and gradient estimates (2.3) are carefully done, with the split into cases based on Λ and the cutoff argument for 0<Λ<1 being technically clean. The narrow-band argument in Step 2 is long but follows standard lines and the estimates are tracked honestly. The contradiction in Step 3 is clean. The reader's assessment of soundness (6.0) is about right — the proof structure is correct and the technical work is competent, though not groundbreaking. The one substantive concern is the scope mismatch the stress-test flags. Lemma 2.1 is cited from [5] (Chen-Liu-Lu-Tao, 2018) and is stated there with α, β, p₁, p₂, Λ all positive. But Theorem 1.1 allows α=0 or β=0, and condition (1.14) permits the strict inequality (subcritical case). If [5] only proves the asymptotic estimates (2.1)–(2.3) for extremal functions at critical parameters, the entire proof chain — Lemma 2.2, Lemma 2.3, and all three steps — rests on estimates that may not cover the full parameter range of the theorem. This is the kind of gap that could be fixable: these asymptotic bounds typically hold for arbitrary positive solutions via direct integral analysis, independent of criticality. But the authors need to either verify that [5] covers the boundary and subcritical cases or provide a self-contained argument. This is the one thing a referee should press on. Everything else is in reasonable shape. The paper is for specialists in integral equations and symmetry classification. It deserves a serious referee who can check the [5] dependency and verify the narrow-band estimates in Step 2. I'd recommend sending it out for review.","headline":"Solid incremental symmetry result for reversed Stein-Weiss; one real gap in parameter scope needs checking","tokens_in":17702,"tokens_out":534,"would_cite":false,"duration_ms":142022,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["45G15","45E10","26D15","45M05"],"pacs":[],"model":"glm-5.2","headline":"Symmetry proven for reversed Stein-Weiss extremals","keywords":["radial symmetry","method of moving planes","reversed Stein-Weiss inequality","integral system","negative exponents","weighted inequality","Euler-Lagrange system"],"falsifier":"A positive solution pair of system (1.12) satisfying all parameter conditions in Theorem 1.1 that is not radially symmetric would refute the claim.","tokens_in":16963,"feed_emoji":"🔄","tokens_out":1376,"duration_ms":102809,"temperature":0.7,"pith_summary":"The paper proves that any pair of positive functions (u, v) solving the Euler-Lagrange integral system associated with the reversed Stein-Weiss inequality must be radially symmetric and monotonically increasing about the origin. This system involves a kernel |x-y|^Λ with positive exponent Λ (so the kernel grows rather than decays with distance), double power-law weights |x|^α and |y|^β, and negative powers of the unknowns (v^{-p2}, u^{-p1}). The authors adapt the method of moving planes in integral form to this setting by introducing auxiliary rescaled functions that absorb the weights, then proving symmetry for those auxiliaries. The key difficulty is that the double weights create singularities near the origin and the negative exponents create non-standard growth, requiring careful gradient estimates and a multi-region narrow-band argument to push the moving plane from negative infinity to the origin.","feed_headline":"Symmetry proven for reversed Stein-Weiss extremals","feed_subtitle":"Positive solutions of the weighted integral system behind a reversed inequality must be radial and increasing — extending the classical HLS/","key_machinery":"Method of moving planes in integral form, applied to weight-rescaled auxiliary functions w and s rather than to u and v directly, with gradient estimates split into near-origin, bounded-annulus, and far-field regimes.","core_discovery":"The central result is Theorem 1.1: under the parameter conditions min{p1,p2} > 1, 0 ≤ α < n/(p1-1), 0 ≤ β < n/(p2-1), and 1/(p1-1) + 1/(p2-1) ≤ (α+β+Λ)/n, every positive locally bounded solution pair of the weighted integral system (1.12) is radially symmetric and increasing about the origin. The proof works by defining w(x) = |x|^{-α/p1} u(x) and s(x) = |x|^{-β} v(x), establishing gradient bounds for these rescaled functions via asymptotic estimates, and then running the moving-plane argument on w and s. The auxiliary functions remove the weight singularities enough to apply the standard comparison framework, and a three-region decomposition (far field, compact interior, and narrow band) in","pith_inferences":["The result suggests that the reversed Stein-Weiss inequality shares the same extremal structure as the classical Stein-Weiss and Hardy-Littlewood-Sobolev inequalities — namely, radial profiles — despite the reversal of the inequality direction and the sign change in the kernel exponent.","The restriction to increasing (rather than decreasing) symmetry about the origin is consistent with the reversed nature of the inequality: extremals grow at infinity rather than decay, mirroring the sign flip in the kernel.","The parameter condition 1/(p1-1) + 1/(p2-1) ≤ (α+β+Λ)/n appears to play the role of a subcriticality condition; the equality case might correspond to a critical threshold where symmetry could fail or additional analysis would be needed."],"forward_implications":["If the radial symmetry result holds, the extremal functions of the reversed Stein-Weiss inequality are fully classified up to scaling, which would pin down the sharp best constant in that inequality.","The technique of absorbing weights into auxiliary functions before running the moving-plane argument could extend to other weighted integral systems with negative exponents or reversed inequalities in settings such as the Heisenberg group.","Radial symmetry of extremals reduces the variational problem for the reversed Stein-Weiss best constant to a one-dimensional (radial) optimization, making numerical and analytical computation of the constant tractable."],"fun_headline_variants":["Radial symmetry forced in reversed Stein-Weiss integral system","Moving planes pin down radial symmetry of weighted integral solutions","Positive solutions of reversed Stein-Weiss system must be radial","Weight rescaling unlocks moving-plane symmetry proof for integral system","Radial symmetry extends to reversed Stein-Weiss Euler-Lagrange solutions"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof depends on asymptotic estimates and integrability properties of the solutions (Lemma 2.1) that are cited from prior work. If those estimates do not hold under the stated parameter conditions, the gradient bounds and the narrow-band argument that pushes the moving plane to the origin would fail.","fun_headline_variants_meta":{"raw":{"variants":["Radial symmetry forced in reversed Stein-Weiss integral system","Moving planes pin down radial symmetry of weighted integral solutions","Positive solutions of reversed Stein-Weiss system must be radial","Weight rescaling unlocks moving-plane symmetry proof for integral system","Radial symmetry extends to reversed Stein-Weiss Euler-Lagrange solutions","Reversed Stein-Weiss extremals shown radially symmetric and increasing","Rescaled functions carry moving-plane argument to weighted integral system","Radial symmetry of reversed Stein-Weiss solutions via gradient bounds"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":885,"prompt_tokens":536,"completion_tokens":349,"prompt_tokens_details":null},"tokens_in":536,"tokens_out":349,"duration_ms":26893,"temperature":1.0,"reasoning_tokens":268,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T20:19:56.922770+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A positive solution pair of system (1.12) satisfying all parameter conditions in Theorem 1.1 that is not radially symmetric would refute the claim.","supporting_citations":[],"review_version":1}