{"id":"d7456add-6bbf-4bb9-abdf-4b955b36a765","arxiv_id":"2509.08761","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For essentially convex interaction kernels, a threshold condition on an auxiliary density guarantees existence, uniqueness, and compact support of minimizers in R^d and curved half-spaces; a counterexample shows the threshold fails for superharmonic kernels.","lead":"This paper proves a sufficient and almost necessary condition for the existence of unique minimizers of interaction energies with external potentials, for a class of strongly repulsive kernels including Riesz potentials. A new 'height functional' viewpoint connects energy minimizers to maximizers of the essential infimum of the generated potential.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof is internally coherent; the load-bearing risk is that Lemma 1.14 (SPD, hence uniqueness and Euler-Lagrange sufficiency) depends on two results from the author's unpublished preprint [Shua], which are neither stated nor proved here.","rationale":"I read the paper as a serious attempt to give a sharp existence criterion. The height-functional method is novel and the microscopic-diffusion estimate (Lemma 3.5) is carefully quantified. I checked the main proof chain: Lemma 2.1's representation formula; Theorem 3.2's compactness via mass redistribution; Theorem 3.3's Euler-Lagrange condition for the height functional; Lemma 4.1's comparison argument; and the curved half-space handling in Lemma 4.3. I found no internal error. The equal-sign caveat in Proposition 1.17 and the superharmonic counterexample are explicit and honest. The only place where the proof outsources a critical fact is Lemma 1.14, where the Fourier-representability and cross-energy finiteness come from [Shua]. If those results are correct, the paper's central claim is solid; if not, the uniqueness and sufficiency parts fail. This is why I flag it as the single load-bearing concern rather than Delta W > 0, which the paper itself shows is necessary via Theorem 1.20. Since the reader already noted the reliance on unpublished preprints and the paper's claims are otherwise well supported, I do not change the verdict: the concern is a verification risk, not a demonstrated flaw.","tokens_in":27237,"tokens_out":40619,"duration_ms":260025,"concrete_test":"Independently verify [Shua, Theorem 3.4]: for a W satisfying Definition 1.12, take a compactly supported signed measure mu with E_W[|mu|] < infinity and prove (2.36) directly from Lemma 2.1 by convolving mu with a mollifier and passing to the limit; if the double integral in (2.36) diverges or differs from E_W[mu], Lemma 1.14 collapses. A lighter check: read arXiv:2503.09948 and confirm Theorem 3.4's assumptions cover the growth bounds (1.23)-(1.25) and the local integrability of W-hat, and that Theorem 3.12's argument works for signed measures of nonzero total mass. If both pass, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.15's central mechanism is: Lemma 1.14 makes W CSPD (via positive Fourier transform), and Lemma 1.10 then turns the height-functional maximizer's Euler-Lagrange condition into a unique energy minimizer. Lemma 1.14's proof is thin exactly where it matters: it asserts that because W-hat is locally integrable (Lemma 2.1), [Shua, Theorem 3.4] yields the Fourier representation (2.36) for compactly supported signed measures with finite energy, and then that 'proceeding similarly to [Shua, Theorem 3.12]' gives E_W[mu+ + mu-] < infinity for a general finite-energy signed measure. Both steps are external and unpublished. If [Shua, Thm 3.4] has hidden hypotheses not met by the potentials in Definition 1.12 (e.g., W-hat integrability at infinity, or a stronger decay condition), the identity (2.36) could fail exactly for the measures used in the microscopic-diffusion argument. Lemma 4.4 similarly leans on [Shua] for cross-energy finiteness. This is not an observed contradiction, but it is the least-secure load-bearing dependency in the chain; a failure there would break uniqueness and the minimizer identification, whereas the Delta W > 0 assumption is explicitly tested by Theorem 1.20 and is not in question.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the existence and uniqueness of minimizers of the interaction energy E_{W,U} with an external potential on the space of probability measures. The main result, Theorem 1.15, gives a sufficient condition for existence and uniqueness when W is 'essentially convex' (ΔW>0 on R^d\\{0} with appropriate growth estimates): if there is a compactly supported probability density ρ7 whose generated potential W*ρ7+U has supremum on its support strictly below U∞, then E_{W,U} has a unique compactly supported minimizer on R^d or on certain half-space-type domains. The proof introduces a height functional, proves the existence of its maximizer under a subcritical condition, derives an Euler-Lagrange equation for maximizers via a 'microscopic diffusion' operation that uses ΔW>0, and then shows that the maximizer satisfies the sufficient Euler-Lagrange condition for the energy minimizer. The paper also proves an 'almost necessary' converse (Proposition 1.17), a counterexample for superharmonic potentials (Theorem 1.20), a simple sufficient condition for general potentials (Theorem 1.4), and an improvement of the known existence result without external potentials (Theorem 1.1).","tokens_in":27502,"tokens_out":30530,"duration_ms":242265,"significance":"If the main result is correct, it provides a clean and essentially sharp condition for existence of minimizers for a natural class of singular repulsive interaction potentials, including Riesz potentials and anisotropic analogues, and it is a notable advance over previous sufficient conditions. The height-functional method and the microscopic-diffusion comparison are interesting technical contributions. The paper is largely self-contained for its main analytic estimates: Lemma 2.1 gives an explicit representation of essentially convex potentials via their Laplacian, and the proofs of Lemmas 3.4 and 3.5 are complete. A significant caveat is that key Fourier-representation results used in Lemma 1.14 and Lemma 4.4 are cited from the author's unpublished preprint [Shua]; those results are load-bearing for the uniqueness and Euler-Lagrange sufficiency, so the reader cannot fully verify the proof without access to that preprint.","major_comments":[{"comment":"The conclusion that essentially convex potentials are SPD depends on [Shua, Theorem 3.4] for the Fourier representation (2.36) and on [Shua, Theorem 3.12] for the reduction from general finite-energy signed measures to compactly supported ones. Neither theorem is stated in the present paper, so the hypotheses cannot be checked; in particular, Ŵ is only locally integrable and not integrable at infinity, and it is not clear from the text that the definition of 'Fourier representable at level 0' in [Shua] applies to this class. Since Lemma 1.14 is used to obtain uniqueness and to justify Lemma 1.10, this is a load-bearing dependency. Please either state these theorems with full hypotheses or include self-contained proofs in an appendix.","section":"Section 2, proof of Lemma 1.14"},{"comment":"The proof of the non-existence criterion invokes [Shua, Theorems 3.4 and 3.10] to justify the Fourier representation of the energy along linear interpolations and the finiteness of the cross-energy. As in Lemma 1.14, this makes the counterexample of Theorem 1.20 depend on an unpublished preprint. For the particular Riesz-type potential in Theorem 1.20 the Fourier representation is classical and can be proved directly; the manuscript should either provide such a proof or quote the precise statement used.","section":"Section 4, Lemma 4.4"}],"minor_comments":[{"comment":"In the proof, 'Since U is continuous in a neighborhood of S' uses the letter S without defining it in this lemma; it should be 'a neighborhood of supp ρ1' (or of the domain D).","section":"Lemma 4.3"},{"comment":"The word 'complimentary' in the abstract should be 'complementary'.","section":"Abstract"},{"comment":"The reference [CCP15] contains a duplicated title fragment: 'Existence of compactly supported global minimisers for the interaction existence of compactly supported global minimisers for the interaction energy.' This should be corrected.","section":"References"},{"comment":"The sentence 'This idea was first introduced in the author's work with Wang [SW, Lemma 2.4] in a one-dimensional setting' is clear, but the phrase 'in proof of Theorem 3.3' earlier in the paragraph is grammatically awkward; consider rewording.","section":"Section 1.4"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the dependence on the author's own preprint [Shua] for load-bearing Fourier-representation results. If the author can state or prove the needed theorems, the paper would be acceptable. The reader's report recommends acceptance, but I believe the citation dependency should be addressed before the paper is published."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper gives a sufficient and almost necessary condition for existence and uniqueness of minimizers of interaction energies with external potential, for essentially convex (subharmonic) potentials. The genuinely new idea is the height functional: instead of minimizing energy directly, maximize the essential infimum of the generated potential, then use the Euler-Lagrange equation for the maximizer to identify the energy minimizer via strict convexity. That is a real conceptual step, and it is executed carefully. The main theorem covers R^d and reasonable half-space domains, and Proposition 1.17 is an honest converse up to mollification and equality at infinity. The counterexample for superharmonic Riesz potentials (Theorem 1.20) shows the Delta W > 0 assumption is not idle, and the construction is explicit and convincing. The paper also removes a monotonicity assumption from the CCP15 existence proof, a smaller but useful improvement.\n\nThe main caveat is exactly the one the stress test flags. Lemma 1.14 (essentially convex implies SPD) leans on two results from the author's unpublished preprint [Shua] to get from the explicit Fourier representation, which is proven here, to the energy identity for finite-energy signed measures. The present paper computes W-hat > 0 and gives locally integrable bounds; what is missing is the bridge from that to Fourier representability at level 0 for signed measures. If [Shua] has hidden hypotheses, that step would break, and with it uniqueness and the Euler-Lagrange sufficiency. I have not found an actual contradiction, and the claim is plausible, but it is a real external dependency. The same reliance appears in Lemma 4.4. The microscopic diffusion idea from [SW] is much less concerning because Lemma 3.5 is fully proved here. Minor regularity caveats in Proposition 1.17 are acknowledged and seem benign.\n\nThe paper is for people working on nonlocal interaction energies, potential theory, or calculus of variations. It does not reorganize the field, but it fills a real gap: no general necessary-and-sufficient existence criterion existed for this class of kernels with arbitrary external fields. I would send it to peer review. The referee should be asked to check whether the borrowed results from [Shua] are actually available and whether the present paper's statements are sufficient; ideally the author should state them in an appendix.","headline":"Solid, genuinely new existence theory for interaction energies with external potentials; the main theorem is well-supported, but one load-bearing step depends on an unpublished preprint of the author.","tokens_in":28034,"tokens_out":2253,"would_cite":true,"duration_ms":161754,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J45","31B15","35A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For essentially convex interaction potentials, one test-density inequality guarantees a unique, compactly supported minimizer of the interaction-plus-external energy.","keywords":["interaction energy","external potential","Riesz potentials","essentially convex","height functional","Euler-Lagrange condition","existence of minimizers","strictly positive definite"],"falsifier":"Compute the derivative $d(W\\ast\\phi_t)/dt$ in Corollary 4.5 for a superharmonic power-law $W(x)=-|x|^b/b$ ($d\\ge3$, $2-d<b<0$): the proof needs this derivative to be $\\le -c<0$ on $1\\le t\\le2$, $|x|\\le2$, because that is what makes $\\sup_{B(0;2)}(W\\ast\\rho_7+\\alpha U)<0$ while $E_{W,\\alpha U}$ has no minimizer. If the derivative can vanish for some such $b$, the sharpness claim about $\\Delta W>0$ is unsupported; and a single essentially convex example satisfying (1.29) whose energy infimum is not attained would refute Theorem 1.15 outright.","tokens_in":27004,"feed_emoji":"⚖️","tokens_out":22545,"duration_ms":186260,"temperature":0.7,"pith_summary":"This paper gives a sufficient and almost necessary condition for the existence of a minimizer of the interaction energy $E_{W,U}[\\rho]=\\tfrac{1}{2}\\int\\int W(x-y)\\,d\\rho(y)d\\rho(x)+\\int U(x)\\,d\\rho(x)$ over probability measures on $\\mathbb{R}^d$ or on half-space-type domains. For a class of subharmonic potentials called essentially convex — $\\Delta W>0$ away from zero, which includes the repulsive Riesz potentials $|x|^{-s}$ with $\\max\\{0,d-2\\}<s<d$ — the condition is: some compactly supported test density $\\rho_7$ has $\\sup_S(W\\ast\\rho_7+U)<U_\\infty$, where $U_\\infty$ is the limiting value of the external potential. If that holds, the energy has a unique minimizer and it is compactly supported; conversely, any minimizer with equilibrium constant below $U_\\infty$ can be recovered as such a test density, so the condition is almost necessary. The paper also shows the sign condition on $\\Delta W$ is essential: for superharmonic power-law potentials with $2-d<b<0$ in $d\\ge3$, the same threshold inequality can hold while no minimizer exists. A reader should care because it reduces a difficult variational existence question to one explicit inequality on a single test density.","feed_headline":"One inequality guarantees unique minimizers of interaction energy","feed_subtitle":"For Riesz-type repulsions, a compact ground state appears as soon as one test density beats the far-field potential.","key_machinery":"The load-bearing object is the height functional $H_S[\\rho]=\\operatorname{ess\\,inf}_{x\\in S}V[\\rho](x)$, with generated potential $V[\\rho]=W\\ast\\rho+U$. The Euler-Lagrange inequalities of Lemma 1.5 — $V[\\rho]\\le C_0$ on $\\operatorname{supp}\\rho$ and $V[\\rho]\\ge C_0$ almost everywhere — are precisely the optimality conditions for maximizing $H_S$, so once $\\sup_{\\rho}H_S[\\rho]<U_\\infty$ guarantees a maximizer, Lemma 1.10 turns that maximizer into the unique energy minimizer. Lemma 4.1 converts the test-density condition $\\sup_S(W\\ast\\rho_7+U)<U_\\infty$ into this subcritical-height condition by building an auxiliary potential $U_7=-W\\ast\\rho_7$ on $\\overline S$ and invoking strict positive definiteness. The decisive internal mechanism is Lemma 3.5, 'microscopic diffusion': replacing a small mass $\\sigma$ by its uniform average over a ball of radius $\\delta$ produces a mean-zero perturbation whose generated potential at distance $\\ge3\\delta$ increases by at least $\\delta^2\\inf\\Delta W$; this is strictly positive exactly because $\\Delta W>0$, and it forces the height maximizer's support into $\\overline S$ and yields the almost-everywhere lower bound. Around this, Lemma 2.1 represents $W$ via the Newtonian kernel and $\\Delta W$, yielding $\\widehat W>0$ and hence strict positive definiteness (Lemma 1.14), which is the source of uniqueness; Lemma 4.3 handles curved half-spaces by mollifying and translating along the boundary using the modulus of continuity of $U$ and the boundary function $\\Phi$.","core_discovery":"The central result, Theorem 1.15, asserts: let $W$ satisfy (W0)(W1) and be essentially convex, i.e. $W\\in C^2(\\mathbb{R}^d\\setminus\\{0\\})$, $\\Delta W>0$ on $\\mathbb{R}^d\\setminus\\{0\\}$, with growth estimates (1.23)-(1.25); let $U$ be continuous with $\\lim_{|x|\\to\\infty}U(x)=U_\\infty\\in\\mathbb{R}$; and let $D$ be $\\mathbb{R}^d$, a halfline in $d=1$, or a curved half-space $\\{x_d\\ge\\Phi(\\hat x)\\}$ in $d=2$. If there exists a compactly supported locally integrable probability density $\\rho_7\\in M(D)$ with $\\operatorname{supp}\\rho_7=\\overline{S}$ for some open set $S$ whose boundary has Lebesgue measure zero, with $W\\ast\\rho_7$ continuous and $\\sup_S(W\\ast\\rho_7+U)<U_\\infty$, then $E_{W,U}$ has a unique minimizer in $M(D)$, and this minimizer is compactly supported. Proposition 1.17 shows the hypothesis is almost necessary: any minimizer whose equilibrium constant $C_0$ satisfies $C_0<U_\\infty$ and whose generated potential $W\\ast\\rho_8$ is continuous can be translated and mollified to produce a $\\rho_7$ with the stated properties. Theorem 1.20 demonstrates that the hypothesis $\\Delta W>0$ cannot simply be dropped: for $d\\ge3$ and $W(x)=-|x|^b/b$ with $2-d<b<0$, there is a smooth radial $U$ with limit $0$ and a smooth radial $\\rho_7$ supported in a ball such that $\\sup_{B(0;R)}(W\\ast\\rho_7+U)<0$, yet $E_{W,U}$ has no minimizer.","pith_inferences":["The height-functional equivalence suggests a general variational recipe: whenever the Euler-Lagrange inequality characterizes $E_{W,U}$ minimizers, one can try to maximize $\\operatorname{ess\\,inf}V[\\rho]$ instead; the comparison argument of Lemma 4.1 should carry over to kernels that are only conditionally strictly positive definite, not necessarily SPD, so long as a locally integrable test densit","The microscopic-diffusion estimate is quantitative: the gain in the generated potential away from a perturbed mass is at least $\\delta^2\\inf\\Delta W$ over a relevant annulus. That explicit lower bound could be used to prove stability estimates for minimizers under perturbations of $W$ or $U$, or to design numerical schemes that exploit such local rearrangements.","The paper leaves the equal-sign case $C_0=U_\\infty$ open. A natural next step is to decide whether mass can escape along level sets of $V$ in that critical case; if not, the 'almost necessary' statement in Proposition 1.17 could be upgraded to a full necessary-and-sufficient characterization.","The superharmonic counterexample in Theorem 1.20 uses pure power-laws with $2-d<b<0$; a testable extension is to add a small essentially convex component to such a $W$ and ask how large the convex component must be before the threshold condition again guarantees existence, which would quantify how much positivity of $\\Delta W$ is needed."],"forward_implications":["For Riesz-type repulsive potentials $W(x)=|x|^{-s}$ with $0<s<d$, existence and uniqueness of a compactly supported minimizer in the presence of any continuous external potential with a finite limit are reduced to checking one explicit inequality on one test density.","Uniqueness is automatic whenever the minimizer exists, because essentially convex potentials are strictly positive definite; the external potential breaks translation invariance, so no additional convexity or symmetry assumptions are needed for uniqueness.","The sufficient condition is stable under small perturbations of $U$: since the inequality $\\sup_S(W\\ast\\rho_7+U)<U_\\infty$ is strict, it persists for any continuous perturbation of $U$ that is uniformly small, so the existence conclusion holds on an open set of external potentials.","For curved half-spaces in dimension two, the theorem covers boundaries given by a continuous profile $\\Phi$, so mass concentration on the boundary is compatible with existence; the proof supplies the needed mollification-and-translation argument for such domains.","The complementary results broaden the toolbox: Theorem 1.4 gives a simple sufficient condition for general potentials satisfying only (W0) and (U0), and Theorem 1.1 improves the known no-external-potential existence theorem by removing the extra monotonicity assumption and keeping a uniform bound on the support diameter."],"supporting_citations":[{"why":"Introduced the microscopic-diffusion operation in a one-dimensional setting (Lemma 2.4 there); Lemma 3.5 here generalizes it to R^d and relies on it to increase the generated potential away from the perturbed mass.","marker":"[SW]"},{"why":"Supplies the Fourier-representability theorem (Theorem 3.4 there) used to convert the representation formula (2.3) into strict positive definiteness of essentially convex W, which yields uniqueness.","marker":"[Shua]"},{"why":"Provides the Euler-Lagrange sufficiency criterion for E_W without external potentials (Theorem 2.4) and the convolution-approximation lemma (Lemma 2.5) that lets a.e. inequalities be upgraded to a true minimizer; also supplies the (W1) condition.","marker":"[CS23]"},{"why":"Defines strictly and conditionally strictly positive definite kernels (Definition 4.2.5) with the companion lemmas needed for the linear-interpolation convexity argument.","marker":"[BHS19]"},{"why":"Gives the previous sufficient condition for Riesz potentials with external fields (Theorem 2.4(i)) and the refined Euler-Lagrange characterization (Theorem 2.1) that the main theorem generalizes.","marker":"[DOSW23]"},{"why":"Establishes the necessary-and-sufficient threshold condition for existence of minimizers of E_W without external potential, the baseline that the external-potential criterion extends.","marker":"[SST15]"},{"why":"Proved existence and size bounds for compactly supported global minimizers of E_W under an extra monotonicity assumption; Theorem 1.1 removes that assumption via a new no-large-gap estimate.","marker":"[CCP15]"},{"why":"Supplies the classical Frostman/Euler-Lagrange inequality for minimizers, cited as the template for Lemma 1.5.","marker":"[ST13]"}],"fun_headline_variants":["One density check guarantees a unique minimizer for Riesz interactions","Test density dipping below far-field potential forces unique ground state","Sufficient and almost necessary condition yields existence and uniqueness","Existence of interaction minimizers decided by a single inequality","Unique energy minimizer appears when a test density beats the far field"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the interaction potential $W$ has a strictly positive Laplacian away from the origin, which the paper calls essential convexity; the proof's microscopic-diffusion step raises the generated potential only under that sign condition, and without it the same threshold condition can fail to produce a minimizer.","fun_headline_variants_meta":{"raw":{"variants":["One density check guarantees a unique minimizer for Riesz interactions","Test density dipping below far-field potential forces unique ground state","Sufficient and almost necessary condition yields existence and uniqueness","Existence of interaction minimizers decided by a single inequality","Unique energy minimizer appears when a test density beats the far field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00061,"raw_usage":{"total_tokens":2931,"prompt_tokens":1131,"completion_tokens":1800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":747,"completion_tokens_details":{"reasoning_tokens":1717}},"tokens_in":747,"tokens_out":1800,"duration_ms":13150,"temperature":1.0,"reasoning_tokens":1717,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:00:00.718513+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the derivative $d(W\\ast\\phi_t)/dt$ in Corollary 4.5 for a superharmonic power-law $W(x)=-|x|^b/b$ ($d\\ge3$, $2-d<b<0$): the proof needs this derivative to be $\\le -c<0$ on $1\\le t\\le2$, $|x|\\le2$, because that is what makes $\\sup_{B(0;2)}(W\\ast\\rho_7+\\alpha U)<0$ while $E_{W,\\alpha U}$ has no minimizer. If the derivative can vanish for some such $b$, the sharpness claim about $\\Delta W>0$ is unsupported; and a single essentially convex example satisfying (1.29) whose energy infimum is not attained would refute Theorem 1.15 outright.","supporting_citations":[],"review_version":2}