{"id":"bdf4c2ef-bc46-41d4-a0a2-29af2b51c500","arxiv_id":"2501.14666","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For odd dimensions with interaction powers (a,b)=(3,2-d) and even dimensions with (3,1-d), the unique energy minimizer is given explicitly, with d=1,2,3 in elementary functions.","lead":"This paper finds explicit formulas for the unique minimizers of a family of nonlocal interaction energies with power-law potentials. The key new idea is that, for special parameters, the nonlocal equilibrium condition becomes a local differential equation that can be solved exactly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof's terse induction step is valid, and the imported uniqueness theory is standard.","rationale":"I read the paper in good faith and checked the main proof steps. The explicit construction via power series is coherent; the recurrence (3.17) and matrix (3.23) are consistent. The terse induction in Theorem 3.1 is the only real shortcut, but it closes as described above. The dimension-reduction lemma 4.1 is correct, and Theorem 5.2's scaling argument verifies the EL condition. The cited qualitative theory (LIC, uniqueness) is standard and applicable. I therefore find no substantive objection that would change the ACCEPT verdict.","tokens_in":12936,"tokens_out":50095,"duration_ms":383614,"concrete_test":"Implement the power series (3.19)-(3.23) for d=5, solve det M(R)=0, pick the nullspace vector, and compute the residual max_{|x|<R} |W*ρ(x) - W*ρ(0)| on a fine grid; the induction step predicts residual ~ 1e-12. Also rerun the d=3 check with the closed-form (1.8). If residuals are not numerically zero, the construction fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central construction is sound. The only potentially under-justified step in Theorem 3.1 is the claim that the linear conditions M(R)c=0, which encode (3.10) at r=0, imply the full Euler-Lagrange equality (3.8) on the ball. However, this follows by a short induction: define H_k = A_{2k}|x|^{1-2k}*ρ - |S^{d-1}|Δ^kρ for k=0,...,(d-1)/2. Using Δ|x|^{1-2k} = (1-2k)(d-1-2k)|x|^{-1-2k} for k<m and the identity A_{2k}(1-2k)(d-1-2k)=A_{2k+2}, one obtains ΔH_k = H_{k+1} for k<m, and ΔH_m=0 from the ODE (3.12). Along with H_k(0)=0, radial harmonicity forces H_k≡0, giving (3.8). Thus the construction is valid. Proposition 2.1 is imported from [CS23, CDM16] and is applicable to (a,b)=(3,2-d) and (3,1-d); its hypotheses are met. No load-bearing concern remains.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies minimizers of the interaction energy E(ρ) = (1/2)∫∫ W(x−y) dρ(y) dρ(x) with W(x)=|x|^a/a − |x|^b/b on R^d. It obtains explicit global minimizers in two new parameter families: odd d with (a,b)=(3,2−d) and even d with (a,b)=(3,1−d). For odd d, the Euler−Lagrange identity W∗ρ=C0 on the support is differentiated (d+1)/2 times; the singular repulsive part is annihilated by successive Laplacians, yielding the local PDE Δ^{(d+1)/2}ρ=A∗ρ. A radial power-series ansatz then reduces the problem to a linear system M(R)c=0 for the power-series coefficients and a scalar equation det M(R)=0 for the support radius. Theorem 3.1 states that the unique positive root R whose nullspace produces a nonnegative density gives the unique minimizer up to translation. For even d, a new projection lemma (Lemma 4.1) shows that steady states in R^{d+1} project to steady states in R^d for a modified potential; after a rescaling this yields the minimizer in even dimensions (Theorem 5.2 and Corollary 5.3). Explicit elementary formulas are supplied for d=1,2,3.","tokens_in":13147,"tokens_out":23714,"duration_ms":209266,"significance":"If correct, these are new explicit global minimizers for power-law interaction energies, extending the previously known polynomial densities, spherical shells, and one-dimensional examples. The construction is genuinely explicit: all inputs are power-series coefficients and the solution of the scalar equation det M(R)=0, and no fitted quantity is fed back into the verification. The proof combines two solid ingredients: the imported LIC/existence/uniqueness theory in Proposition 2.1 and a new dimension-reduction lemma. The paper also honestly records its limitations, notably in Remark 3.2 and in the closing paragraph of Section 3. I checked the key distributional identities and the projection argument; the mathematics appears sound, and the paper is suitable for publication after minor clarifications.","major_comments":[],"minor_comments":[{"comment":"The induction step from M(R)c=0 to the full system (3.10) on B(0,R) is stated in one sentence: “one can use induction to show that (3.10) for these k are satisfied.” Since this is the central mechanism of the construction, please spell out the H_k argument: define H_k = A_{2k}|x|^{1−2k}∗ρ − |S^{d−1}|Δ^kρ, verify ΔH_k = H_{k+1} for k=0,…,m−1 and ΔH_m=0 from (3.12), and then use H_k(0)=0 together with radial harmonicity to conclude H_k≡0. Adding these three lines will make the proof self-contained and easier to verify.","section":"§3.2, proof of Theorem 3.1"},{"comment":"In the proof that Q[U]=0 implies U=0 on B^d(0,R), the sentence about preimages of zero-measure sets under dist(x∨v, x) should be justified more carefully: for fixed x with |x|>0 the map v↦dist(x∨v,x) is smooth with nonvanishing Jacobian away from the two critical points, so the coarea formula gives the required null-preimage property; the current phrasing is a little too casual for a measure-theoretic argument.","section":"§4, Lemma 4.1"},{"comment":"The entry [CCP15] contains a duplicated and garbled title (“Existence of compactly supported global minimisers for the interaction existence of compactly supported global minimisers for the interaction energy”); please restore the correct title.","section":"References"},{"comment":"The notation βR is used without a separator in equations such as (3.28) and (3.30); although the meaning is clear, typesetting it as βR with a thin space would improve readability.","section":"§3.3, equations (3.28)–(3.30)"}],"recommendation":"minor_revision","confidential_remarks":"This is a well-written and mathematically sound paper. The main theorem depends on imported LIC/uniqueness theory from [Lop19, CS23, CDM16], which is standard and correctly cited; the paper does not reprove that theory, and I do not see a circularity problem. The only request that goes beyond cosmetics is the expansion of the induction step in the proof of Theorem 3.1, which the authors can supply in a few lines. The paper is a good fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid paper, and it delivers more than the abstract promises. The explicit minimizers for (a,b)=(3,2-d) in odd dimensions and (3,1-d) in even dimensions are genuinely new — none of the previously known families listed in the introduction cover these exponents. The formulas for d=1,2,3 are explicit enough to plot, and the general construction is coherent.\n\nWhat I like: the Laplacian cascade is a nice idea. Taking successive Laplacians of the Euler-Lagrange condition turns the nonlocal equation into a local ODE, which can be solved by power series. The stress-test note checks out: the induction showing that (3.10) at r=0 implies the full Euler-Lagrange equality (3.8) on the ball is valid. The dimension-reduction lemma (Lemma 4.1) is a clean, reusable tool. The paper is also honest about what it imports — the uniqueness and EL-characterization theory from [CS23, CDM16] is standard and the hypotheses are met for these exponent pairs. The citation pattern is fine; the only self-citation, [CS23], is to the exact external result being used as a benchmark.\n\nSoft spots, in proportion: the main one is not a flaw. For general odd d, the \"explicit\" formula requires solving a transcendental equation and checking nonnegativity, and Remark 3.2 admits that uniqueness of the correct R in high dimensions is not proved. So in higher dimensions the result is semi-constructive rather than fully explicit. For d=1 and d=3 it is fully explicit, which is where the paper's examples land. The paper also leans on Proposition 2.1 without reproving it; if that theory had a gap, the constructed candidates would only be steady states rather than minimizers. But that theory is published and standard, so I do not count this against the paper.\n\nMinor caveat: the computations are long and not machine-checked, so there is room for algebra slips, but I traced the key induction and it holds.\n\nWho this is for: people working on aggregation equations, interaction energies, and explicit equilibria. It will be cited in that community. It deserves a serious referee; my recommendation is to send it out. I would accept it after minor revisions, mainly to sharpen Remark 3.2 and to add a sentence on how the transcendental equation is solved numerically in practice.","headline":"Solid explicit-minimizer result: the construction is sound, the new parameter families are genuinely new, and the reliance on cited uniqueness theory is appropriate.","tokens_in":13699,"tokens_out":1875,"would_cite":true,"duration_ms":18160,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","45E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for odd dimensions with potential exponents $(a,b)=(3,2-d)$ and even dimensions with $(a,b)=(3,1-d)$, the unique minimizer of the power-law interaction energy has an explicit formula, obtained by solving a local PDE…","keywords":["interaction energy","power-law potential","explicit minimizer","Euler-Lagrange condition","dimension reduction","radial symmetry","uniqueness","Laplacian"],"falsifier":"Compute for $d=5$ the entries of $M(R)$ from (3.23) using the recurrence (3.17) truncated at large order, find every positive root of $\\det M(R)=0$, and check whether the nullspace vector yields $\\rho_{R,c}\\ge 0$ on $B(0,R)$ and whether $W*\\rho_{R,c}$ is constant on the support. A high-resolution numerical minimization of the energy in $d=2$ compared with the explicit formula (1.10) would also settle the claim: any lower-energy competitor would contradict uniqueness.","tokens_in":12716,"feed_emoji":"📐","tokens_out":10393,"duration_ms":87244,"temperature":0.7,"pith_summary":"The paper derives explicit formulas for the unique minimizer (up to translation) of the interaction energy with power-law potential $W(x)=|x|^a/a - |x|^b/b$ in $d$ dimensions, for odd $d$ with $(a,b)=(3,2-d)$ and even $d$ with $(a,b)=(3,1-d)$. These are new families in the catalogue of explicit minimizers, where previously known examples included quadratic attractive cases, spherical shells, and one-dimensional power laws. In odd dimensions the key step is to apply the Laplacian repeatedly to the Euler-Lagrange condition $W*\\rho=C_0$ on the support, turning a nonlocal convolution condition into a local linear PDE whose radial power-series solutions are explicit; the support radius is then selected by a determinantal condition. In even dimensions the minimizer is the projection and rescaling of the minimizer in one higher dimension, which the paper proves by a new dimension-reduction lemma. In dimensions 1, 2, and 3 the formulas reduce to elementary functions, such as $\\rho_1(x)=C\\cosh(\\sqrt{2}\\,x)$ on its support.","feed_headline":"Power-law interaction minimizers now explicit in all dimensions","feed_subtitle":"Odd dimensions: repeated Laplacians turn the nonlocal condition into a solvable PDE; even dimensions follow by projection.","key_machinery":"The central object is the Laplacian ladder: for the chosen exponent pair, applying $\\Delta$ successively to the Euler-Lagrange equation $W*\\rho=C_0$ on the support produces the constant-coefficient linear PDE $\\Delta^{(d+1)/2}\\rho=A_*\\rho$, which is local and can be solved in radial coordinates by power series. Solving it yields the basis functions $u_{2k}$ and the coefficient matrix $M(R)$, whose determinant fixes the support radius; the same condition also picks the correct coefficients in the nullspace. For even dimensions, the projection operator $P$ (integration over the extra coordinate) and the averaging operator $Q$ are the central mechanisms: Lemma 4.1 shows that $W*F=C$ in dimension $d+1$ is equivalent to $\\tilde{W}*P[F]=C$ in dimension $d$, and Lemma 5.1 shows that power-law potentials are mapped by $Q$ to power-law potentials up to constants, up to the scaling factor $\\lambda$.","core_discovery":"The central claim is Theorem 3.1: for odd $d$, the unique minimizer of $E$ is the normalized radial function $\\rho_{R,c}$ supported on $B(0,R)$, built from the power series (3.19)-(3.20), where $R$ is the unique positive solution of $\\det M(R)=0$ whose associated coefficient vector lies in the one-dimensional nullspace and makes the density nonnegative. The matrix $M(R)$ encodes the repeated-Laplacian conditions evaluated at the origin, and its entries are explicit power series in $R$. For even $d$, Corollary 5.3 states that the minimizer is $\\tilde{\\rho}(x_1,\\ldots,x_d)=\\lambda^d \\int_{\\mathbb R} \\rho(\\lambda x_1,\\ldots,\\lambda x_d,x_{d+1})\\,dx_{d+1}$ with an explicit scaling $\\lambda$, so the even-dimensional minimizer is a projected and rescaled copy of the odd-dimensional one. The paper also writes the minimizers in dimensions 1, 2, and 3 in elementary closed form, with the support radii fixed by scalar equations such as $\\sqrt{2}R=\\coth(\\sqrt{2}R)$.","pith_inferences":["Editorial inference: the Laplacian-ladder mechanism should extend to powers $a>3$ whenever the successive derivatives of $|x|^a/a$ stay integrable and eventually become local; testing the same construction for $(a,b)=(5,2-d)$ against numerical minimizers would show whether the method is a general template.","Editorial inference: the dimension-reduction lemma is not tied to power-law potentials, so any new explicit minimizer in odd dimensions could immediately produce an even-dimensional minimizer through projection, provided the averaged potential retains the same structure.","Editorial inference: the explicit densities in dimensions 1, 2, and 3 are natural benchmark solutions for numerical solvers for aggregation equations, since they have known support radii, known energy values, and non-smooth behavior at the boundary that stress numerical methods."],"forward_implications":["In odd dimensions, the unique minimizer is real-analytic on its support, so its profile can be evaluated to arbitrary precision by truncating the defining power series.","In dimensions 1, 2, and 3, the elementary closed forms allow direct computation of the support radius and of the minimal interaction energy without numerical integration of the convolution.","In even dimensions, the minimizer is exactly a projection and rescaling of the minimizer in dimension $d+1$, so explicit minimizers propagate from one dimension to the next within this family.","Because these potentials satisfy linear interpolation convexity, verifying the Euler-Lagrange condition is sufficient, so the constructed densities are true global minimizers rather than merely steady states.","The same repeated-Laplacian procedure produces radial functions satisfying $W*\\rho=C$ on a ball for other integer-parity exponent pairs, although these are only steady states unless nonnegativity and convexity are independently verified."],"supporting_citations":[{"why":"Supplies the characterization that the minimizer is the only compactly supported probability measure satisfying the Euler-Lagrange condition, the key uniqueness input to the construction.","marker":"[CS23]"},{"why":"Supplies the regularity theory and the identity $\\Delta(|x|^a/a * \\rho)=|S^{d-1}|\\rho$ on the support that produces the local PDE after successive Laplacians.","marker":"[CDM16]"},{"why":"Establishes the linear interpolation convexity giving uniqueness and radial symmetry of minimizers for these power-law potentials.","marker":"[Lop19]"},{"why":"Proves existence of compactly supported global minimizers for the interaction energy, a foundational input to Proposition 2.1.","marker":"[CCP15]"},{"why":"Gives the Euler-Lagrange necessary condition (2.1) used to verify that the constructed radial functions are minimizers.","marker":"[BCLR13a]"}],"fun_headline_variants":["Power-law interaction minimizers: explicit and unique","Odd PDEs and even projections give explicit minimizers","Repeated Laplacians crack odd-dim minimizers","Even dims via projection of odd-dim solution","Explicit minimizers for power-law interaction energies"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the imported uniqueness theory applies to these exponent pairs, so any ball-supported radial density whose convolution with the potential is constant on its support must be the unique global minimizer; if that characterization failed, the constructed candidates would be only steady states.","fun_headline_variants_meta":{"raw":{"variants":["Power-law interaction minimizers: explicit and unique","Odd PDEs and even projections give explicit minimizers","Repeated Laplacians crack odd-dim minimizers","Even dims via projection of odd-dim solution","Explicit minimizers for power-law interaction energies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000934,"raw_usage":{"total_tokens":3994,"prompt_tokens":939,"completion_tokens":3055,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":2996}},"tokens_in":555,"tokens_out":3055,"duration_ms":19870,"temperature":1.0,"reasoning_tokens":2996,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:56:31.096979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute for $d=5$ the entries of $M(R)$ from (3.23) using the recurrence (3.17) truncated at large order, find every positive root of $\\det M(R)=0$, and check whether the nullspace vector yields $\\rho_{R,c}\\ge 0$ on $B(0,R)$ and whether $W*\\rho_{R,c}$ is constant on the support. A high-resolution numerical minimization of the energy in $d=2$ compared with the explicit formula (1.10) would also settle the claim: any lower-energy competitor would contradict uniqueness.","supporting_citations":[],"review_version":1}