{"id":"d977df75-c513-438b-8a0e-e8c8c4f0b097","arxiv_id":"2412.12586","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For a fractional Keller-Segel model at the mass-critical diffusion exponent, the authors claim a sharp critical mass defined by a Hardy-Littlewood-Sobolev type constant.","lead":"This mathematics paper analyzes a model of cell motion where diffusion and long-range attraction compete. It claims to find the exact mass threshold separating infinite-time survival from finite-time blow-up for a family of fractional interaction potentials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharp critical mass is not established: Proposition 4's asserted C_* equals the HLS constant is incompatible with its own maximizer existence proof, so M_* shifts.","rationale":"The reader's weakest assumption is correctly identified: the equality of C_* with the HLS constant in (39) is the pivot of the sharp threshold. My reading of Proposition 4 confirms this concern and strengthens it. The proof of Proposition 4 does not establish the value of the optimal constant; Step 1 only supplies an upper bound. Step 2 establishes (if correct) that a maximizer exists. These two facts together are inconsistent with (39), because attaining the value C_HLS would require equality in both the HLS inequality and the L1-Lm interpolation inequality, whose equality sets are disjoint. Moreover, the proposition explicitly asserts that the maximizer is compactly supported, and a compactly supported function cannot be an HLS maximizer, so its J-value is strictly below C_HLS. Therefore the sharp constant - and hence M_* via (12) - is not the explicitly stated value. Theorems 8 and 9 may be salvageable with the true VHLS constant, but the paper's quantitative classification at the stated threshold is not. I therefore keep the reader's REJECT verdict.","tokens_in":11791,"tokens_out":17001,"duration_ms":145687,"concrete_test":"Verify the equality-case incompatibility: show that equality in HLS (18) forces u to be the Lorentzian profile (20), while equality in the L^1-L^m interpolation used in (40) forces u to be constant a.e. on its support. Since no nonzero admissible u satisfies both, the composite bound (40) is strict for every admissible u with positive HLS term. Combined with Proposition 4's asserted existence of a maximizer, this proves C_* < C_HLS. For a concrete numerical illustration, evaluate R = J(U_HLS)/C_HLS in d=4, s=1; the ratio is strictly below 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's sharp threshold M_* is defined through the constant C_* in (39), and Theorems 8-9 assert blow-up for M > M_* and global existence for M < M_*. Proposition 4, however, does not prove that the optimal constant in the VHLS inequality (11) equals the standard HLS constant. Step 1 of the proof only gives the upper bound C_* <= C(d,s) by composing HLS (18) with the L^1-L^m interpolation. Equality in that composite bound would require a single function to attain equality in both inequalities: HLS equality forces the non-compactly supported Lorentzian profile (20), while equality in the interpolation step forces the function to be constant a.e. on its support. No nonzero admissible function satisfies both conditions. Moreover, Proposition 4 itself asserts the existence of a maximizer U and states that U is compactly supported; a compactly supported function cannot attain HLS equality, so its J-value is strictly below C(d,s). Hence the sharp constant C_* is strictly smaller than the value in (39), and the critical mass in (12) is correspondingly too small. The proof's compactness argument does not repair the value: at best it establishes that a maximizer exists, which makes the claimed equality with the HLS constant internally inconsistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the aggregation-diffusion equation (1) with porous-medium diffusion exponent m = 2 - 2s/d, fractional interaction (−∆)^s φ = u, and 2 < 2s < d. This is the mass-critical regime in which the scaling preserves the L1 norm. The authors propose a sharp critical mass M* defined through the optimal constant C* of a variant of the Hardy-Littlewood-Sobolev inequality (the VHLS inequality, Eq. (11) and Proposition 4). They claim finite-time blow-up for initial mass M > M* and global existence for M < M* (Theorems 8 and 9), with local existence and blow-up criteria in Theorem 3. The proofs combine HLS inequalities, L1-Lm interpolation, compactness and rearrangement arguments for maximizers of the VHLS functional, and energy/second-moment arguments for the evolution.","tokens_in":12018,"tokens_out":12422,"duration_ms":109635,"significance":"If the sharp constant and the threshold M* were correctly identified, the paper would provide a clean mass dichotomy for a fractional Keller-Segel system in the mass-critical exponent, extending known results for the classical case and for other ranges of m. The overall strategy is attractive: free-energy dissipation, second-moment evolution, and a variational characterization of the threshold are natural tools, and the paper makes explicit use of the correct-looking scaling and interpolation structure. However, the central load-bearing step — the identification of the optimal constant C* in Proposition 4 — is not proved and is in fact incompatible with the paper's own maximizer existence claim. Since the value of M* in Eq. (12) is directly defined through C*, the main dichotomy as stated is not established. The paper also contains a concrete algebraic error in the computation of the free energy in Theorem 8. These are not cosmetic issues: they concern the exact threshold that constitutes the paper's main contribution.","major_comments":[{"comment":"The equality C* = C(d,s) is not established, and the proof given in Step 1 only yields the upper bound C* ≤ C(d,s). Equality in the composed inequality (40) would require one function to attain equality simultaneously in the HLS inequality and in the L1-Lm interpolation. HLS equality forces the function to be of the Lorentzian form (20), which is not in L1(R^d) when d > 2s and is not compactly supported; equality in the interpolation step forces a function that is essentially two-valued on its support. No admissible function satisfies both conditions. Moreover, Proposition 4 itself asserts that the maximizer U is compactly supported, and such a function cannot attain HLS equality. Therefore the sharp constant C* is strictly smaller than the HLS constant (or, at minimum, the asserted value is unsupported). Since M* in Eq. (12)/(52) is defined through the value in Eq. (39), the claimed critical mass and the statements of Theorems 8 and 9 do not follow.","section":"Section 3, Proposition 4, Eqs. (38)–(41)"},{"comment":"The compactness argument for the existence of a maximizer is not valid as written. The bound (47) gives u_j(R) ≤ G(R) with G(R) ≈ R^{-d/m} for large R. For p = 2d/(d+2s), a direct computation shows that G is not in L^p(R^d) when d > 2s, because the decay exponent satisfies (d/m) · p < d. Consequently, the claimed dominated-convergence step for ω(u_j) → ω(U) lacks a valid dominating function. The argument also does not prove that the limit U is compactly supported, although Proposition 4 asserts this and Proposition 5 uses supp(U) in an essential way. Thus the proof of Proposition 4 is incomplete independently of the constant-value issue.","section":"Section 3, Step 2 of Proposition 4, Eqs. (45)–(50)"},{"comment":"The computation of the free energy of u0 = (M/M*)U is algebraically incorrect. The interaction term scales quadratically in the amplitude: ∫ φ_{u0} u0 dx = (M/M*)^2 ∫ φ_U U dx, not (M/M*) ∫ φ_U U dx. The correct expression is F(u0) = (||U||_m^m/(m-1))[(M/M*)^m − (M/M*)^2], up to the standard constant conventions. The displayed factorization in Eq. (65) is therefore false. Since m = 2 − 2s/d ∈ (1,2), the corrected expression is still negative for M > M*, so Theorem 8 may be repairable, but the proof as written contains a genuine mathematical error in a main theorem.","section":"Section 4.1, Eq. (65)"}],"minor_comments":[{"comment":"The sentence 'global solutions exist for M > M*' contradicts Theorem 9, which proves global existence for M < M*, and Theorem 8, which proves blow-up for M > M*; the words 'greater than' and 'less than' are interchanged.","section":"Introduction, after Eq. (12)"},{"comment":"The term involving ∫ φ dx should be handled after using the definition of φ with zero integral; as written, the expression is hard to follow and should be made explicit that the mass-derivative term vanishes.","section":"Proposition 5, Eq. (57)"},{"comment":"The proof of Theorem 3 establishes L^r bounds for finite r but Definition 1 requires u ∈ L∞(0,T; L1 ∩ L∞(R^d)); the gap between the stated regularity and the obtained estimates should be clarified or the definition adjusted.","section":"Section 2, Theorem 3 and Definition 1"},{"comment":"The notation for the exponent of the L^r norm in the equality at the end of (23) is ambiguous; please write the exponent as (r+1)/(1+(2s−2)/d) or an equivalent unambiguous form.","section":"Eq. (23) and surrounding text"},{"comment":"The reference list contains many entries that are not cited in the text (for example items [28]–[41]), which should be either cited or removed.","section":"References"}],"recommendation":"reject","confidential_remarks":"The central claim of the paper is the explicit formula for the sharp critical mass M*, and that formula depends entirely on the asserted optimal constant C* in Proposition 4. Since the proof of that proposition is internally inconsistent — the constant cannot equal the HLS constant while a compactly supported maximizer exists — the main result is not merely missing a technical justification; the stated threshold is not supported by the manuscript. Correcting this would require a new variational analysis of the sharp constant, which is outside the scope of a revision. For this reason I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Couple of things you should know. The paper targets a real gap—the mass-critical case m = 2 − 2s/d with s > 1 for the fractional Keller-Segel equation—and the proof strategy is the standard one: energy estimates, second moment, and a variational constant. If the constant were right, Theorems 8 and 9 would give a clean dichotomy. But the central claim is not established. Proposition 4 asserts the optimal constant C_* in the VHLS inequality equals the standard HLS constant (39). The proof only gives the upper bound by composing HLS with L^1–L^m interpolation. Equality in that composite bound would need a single function to be simultaneously a Lorentzian (for HLS equality) and constant on its support (for the interpolation equality). No such function exists. Worse, the paper itself claims the maximizer U is compactly supported, and a compactly supported function cannot attain HLS equality. So C_* is strictly smaller than the value in (39), and the critical mass M_* defined through it is too small. The maximizer existence proof also has a gap: pointwise convergence of the radially symmetric normalized sequence doesn't preserve the L^1 and L^m norms; the argument implicitly requires norm preservation to conclude J(U) = C_*, but that is not shown. These are not cosmetic issues. The threshold value is the main quantitative output of the paper, and it is wrong as stated.\n\nThere is also a minor internal inconsistency: the introduction says global existence holds for M > M* and blow-up for M < M*, which is the reverse of Theorems 8–9 and the abstract. The abstract itself is fine.\n\nWhat the paper does well: the local existence and blow-up criteria follow known regularization arguments, the second-moment computation is clean, and the structure of the proof is sensible. The flaw is localized to Proposition 4, but since that proposition carries the sharp constant, the main theorems do not stand.\n\nWho is this for? People working on aggregation-diffusion equations might find the approach worth knowing, but they should not cite the threshold. I would send it to a referee because the constant issue is subtle enough to require expert scrutiny, but my own verdict is reject unless the authors can replace (39) with a correct value or drop the claim of sharpness.","headline":"The claimed sharp critical mass rests on an unproven and likely false equality of constants in Proposition 4; the threshold value is not established.","tokens_in":12538,"tokens_out":6159,"would_cite":false,"duration_ms":50726,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","35B44","35R11","35A23","35Q92"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sharp critical mass separates spreading from blow-up in a fractional Keller-Segel model.","keywords":["Keller-Segel","critical mass","fractional Laplacian","porous medium diffusion","Hardy-Littlewood-Sobolev inequality","finite time blow-up","global existence","free energy"],"falsifier":"Evaluate the quotient $J(U)=\\omega(U)/(\\|U\\|_{L^1}^{2s/d}\\|U\\|_{L^m}^m)$ for the explicit HLS maximizer $U(x)=(1+|x|^2)^{-(d+2s)/2}$ with $2<2sM_*$ has $F<0$, and the identity $dm_2/dt=2(d-2s)F(u)$ turns that negative energy into finite-time blow-up of the second moment and then of $L^r$ norms.","core_discovery":"The paper's central claim is that for $m=2-2s/d$ with $2<2sM_*$, there exists an initial datum—a rescaling of the maximizer of the modified Hardy-Littlewood-Sobolev inequality—whose free energy is negative, so the second moment decreases to zero in finite time and the $L^r$ norm blows up. The constant $C_*$ is the optimal constant of the modified Hardy-Littlewood-Sobolev inequality (11), and the extremals are radially symmetric, non-increasing, compactly supported functions; at exactly $M=M_*$ these extremals are stationary solutions of the equation.","pith_inferences":["Editorial: the numerical value of $M_*$ rests on the unproved equality of the VHLS and HLS optimal constants; if the true VHLS constant is strictly smaller, the threshold moves but the global/blow-up dichotomy itself survives.","Editorial: the paper leaves the critical case $M=M_*$ open; by analogy with the classical Keller-Segel critical-mass problem, one expects infinite-time aggregation or stationary behavior at exactly the threshold, but this is not established here.","Editorial: a numerical maximization of the functional $J$ over compactly supported radial functions could test the sharp constant directly and would give a practical check of the predicted threshold.","Editorial: the same free-energy plus second-moment structure could be used to locate critical masses for other fractional interaction kernels, since the only input needed is a sharp interpolation inequality of the same form."],"forward_implications":["For every initial mass below $M_*$, the $L^m$ norm of the solution remains bounded for all time, so the regularized approximations converge to a global weak solution.","For every mass above $M_*$, there exist smooth initial data with finite second moment whose second moment hits zero in finite time; consequently the $L^r$ norm blows up before collapse.","The threshold is explicit and computable from $d$, $s$, and the optimal VHLS constant: $M_*=[2/((m-1)C_*c_{d,s})]^{d/(2s)}$.","At exactly $M=M_*$, the maximizer of the VHLS inequality is a stationary solution of the equation, linking the dynamical threshold to compactly supported steady states.","The paper also reports that in the range $01$."],"supporting_citations":[{"why":"Supplies the HLS inequality, its maximizers, the rearrangement and compactness tools, and the GNS inequality used in Sections 2 and 3.","marker":"[21]"},{"why":"Provides the neighboring-regime threshold result and the stationary-solution characterization used to identify VHLS maximizers with steady states.","marker":"[2]"},{"why":"Gives the local-existence framework and the energy/second-moment strategy adapted in Theorems 3 and 8.","marker":"[4]"},{"why":"Supplies the critical-mass analysis and the maximal-existence-time theorem used to convert $L^m$ bounds into global existence.","marker":"[5]"},{"why":"Gives the diffusion-dominated global-existence result and the approximation lemma used in the regularization argument.","marker":"[11]"},{"why":"Supplies the regularization convergence arguments used to pass from the epsilon-approximation to weak solutions.","marker":"[26]"}],"fun_headline_variants":["Critical mass threshold flips Keller-Segel to blow-up","Subcritical mass lives, supercritical dies in Keller-Segel","Sharp mass bound decides global existence vs collapse","Keller-Segel: mass cutoff separates spread from blow-up","Mass critical exponent dictates finite-time blow-up"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical value of the threshold assumes, without proof, that the best constant in the modified Hardy-Littlewood-Sobolev inequality is exactly the best constant in the classical one; if that equality is false, the claimed threshold shifts.","fun_headline_variants_meta":{"raw":{"variants":["Critical mass threshold flips Keller-Segel to blow-up","Subcritical mass lives, supercritical dies in Keller-Segel","Sharp mass bound decides global existence vs collapse","Keller-Segel: mass cutoff separates spread from blow-up","Mass critical exponent dictates finite-time blow-up"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1613,"prompt_tokens":899,"completion_tokens":714,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":634}},"tokens_in":515,"tokens_out":714,"duration_ms":6699,"temperature":1.0,"reasoning_tokens":634,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:57:56.650509+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the quotient $J(U)=\\omega(U)/(\\|U\\|_{L^1}^{2s/d}\\|U\\|_{L^m}^m)$ for the explicit HLS maximizer $U(x)=(1+|x|^2)^{-(d+2s)/2}$ with $2<2s