{"id":"5677f041-5d9c-46ac-9339-56919dd79061","arxiv_id":"2608.11813","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For Riesz gases confined by a hard wall, complete confinement to the wall occurs exactly in the strongly long-ranged regime s∈(d-3,d-2], with an explicit critical wall position; in the weakly long-ranged regime s∈(d-2,d) it never occurs.","lead":"A family of long-range interacting particle systems, Riesz gases in a half-space, shows a sharp split in behavior depending on how fast the interaction decays. Systems with slowly decaying interactions eventually pin every particle to the hard wall once the wall is far enough; faster-decaying long-range systems never do.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof is sound apart from one imported assumption: the Frostman Euler-Lagrange characterization (3.1) is used for negative s, and its validity for s<0 is not demonstrated in the paper; checking the cited [25, Thm 2] against this range settles the only substantive risk.","rationale":"I read the paper in good faith. The unconstrained equilibrium formula (Proposition 2.1), the critical-value construction, and the Euler-Lagrange verification in Section 4 are mutually consistent: I re-derived the key identities (3.16), (4.13), and (4.14) and found no sign or exponent errors. The proof's treatment of the exceptional values (s=0 for d=1, and s=d-2 for all d) is by reference to existing work, which is acceptable. The only assumption that the whole theorem rests on but the paper does not prove is the Frostman/Euler-Lagrange characterization for negative s. The reader flagged exactly this point. I believe it is likely satisfied because the paper's lower bound s>d-3 is precisely the condition -s<2 under which the kernel is conditionally positive definite, but the manuscript should make this explicit or cite a theorem whose hypotheses are transparent. Since this is an external-support gap rather than an internal error, and since the cited references very plausibly cover the range, I do not change the reader's ACCEPT verdict; I recommend a verification step rather than a rejection. Independent support: the paper has no machine-checked proofs, but the parameter-free derivation and the explicit formulas are strong evidence; the numerics in Figures 2-3 are unreproduced but not load-bearing.","tokens_in":18197,"tokens_out":32265,"duration_ms":336811,"concrete_test":"Inspect the statement of [25, Theorem 2] (and [30]) and check whether its hypotheses cover (i) Riesz kernels of the form s^{-1}|x-y|^{-s} for s∈(-2,0), and (ii) lower semicontinuous external fields with value +∞ outside a half-space. If the coverage is explicit, the concern dissolves. If not, write the missing lemma: for s∈(d-3,d-2), since 0<-s<2, the kernel |x-y|^{-s} is conditionally negative definite, hence the weighted energy is strictly convex on the set of probability measures with finite second moment; use this to prove existence, uniqueness, and the EL converse in one paragraph.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the if-and-only-if confinement transition of Theorem 2.2. Both directions use (3.1) as a complete characterization of the minimizer. For s<0 this is not the classical positive Riesz kernel: g_s(x)=s^{-1}|x|^{-s} is negative, unbounded below at infinity, and the energy functional is not obviously convex. The paper imports the existence, uniqueness, and converse direction from [25, Theorem 2] and [30] without a self-contained proof or a statement of the hypotheses. If that theorem does not cover s∈(-2,0) (d=1) or s∈(-1,0) (d=2), or does not allow external fields that are +∞ on a half-space, then the candidate wall measure in §4 could fail to be the true minimizer even when it satisfies the EL inequalities, and the \"only if\" direction (a<a_cri implies non-confinement) would not follow. This is genuinely load-bearing because the lower bound s>d-3 is exactly where -s<2, so the kernel is conditionally positive definite and the gap is likely repairable, but the manuscript does not demonstrate that repair. No internal algebraic error was found in Lemma 3.1, Lemma 4.1, or the monotonicity argument (4.13)-(4.18).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the equilibrium measure of a Riesz gas in dimension d with interaction parameter s in (d-3,d), confined to a half-space by a hard wall and subject to a quadratic confining potential. The main result, Theorem 2.2, establishes a dichotomy: for s in (d-2,d) the equilibrium measure is never fully confined to the hard wall, while for s in (d-3,d-2] there is a critical wall position a_cri(d,s) such that full confinement to the wall occurs if and only if a >= a_cri(d,s). The fully confined measure is given explicitly in (2.12)-(2.13), and the critical value is characterized via the solution of the transcendental equation (2.8). The proofs use Frostman's Euler-Lagrange conditions, explicit hypergeometric identities, and a monotonicity argument; the Coulomb case s=d-2 is recovered as a special case, and the one-dimensional results confirm a conjecture of Kethepalli et al.","tokens_in":18503,"tokens_out":9998,"duration_ms":93664,"significance":"If the result is correct, it gives a sharp and quantitative interaction-range dichotomy within the Riesz family, beyond the classical long-range/short-range split. The explicit formula for a_cri(d,s) and for the wall-supported equilibrium measure is a substantive advance, and the recovery of the Coulomb results of [5,9,24,29] together with the confirmation of the conjecture in [34] gives the paper strong external coherence. The proofs in the new regimes are detailed and self-contained: Lemma 3.1 establishes existence and uniqueness of the critical root, Lemma 4.1 supplies the exact off-wall potential, and Lemma 4.2 reduces the full Euler-Lagrange inequality to the one-dimensional check along x_hat=0. The main caveat is the imported potential-theoretic characterization for negative s, which is load-bearing and not stated with hypotheses.","major_comments":[{"comment":"The Euler-Lagrange characterization (3.1) is imported from [25, Theorem 2] and [30] and is used as an if-and-only-if criterion for the minimizer over the entire range s in (d-3,d), including negative s (d=1, s in (-2,-1); d=2, s in (-1,0)). For s<0 the kernel g_s is negative, and the classical positive-definite Riesz potential theory does not apply verbatim. The manuscript neither states the hypotheses of [25, Theorem 2] nor verifies them for the external field V_a, which takes the value +infinity outside a half-space. Both directions of the confinement dichotomy in Theorem 2.2(ii) rely on the converse part of (3.1), so this is load-bearing. Please add a precise statement of the invoked theorem and a verification that it covers (i) the range s in (-2,d), (ii) the normalization with the prefactor 1/s, and (iii) external fields taking the value +infinity, or give a self-contained proof for the class of potentials used here.","section":"Section 3, Eq. (3.1)"},{"comment":"The exclusion of full confinement for s in [d-1,d) rests on the assertion, cited to [3,32], that every probability measure supported on the hyperplane {x_d=a} has infinite Riesz energy when s >= d-1. This assertion is standard, but it is not stated precisely or proved, and it is the only mechanism that covers the upper part of the weakly long-ranged regime. Please include a short proof: for s >= d-1, the kernel |x-y|^{-s} is not integrable near the diagonal on R^{d-1}, so any probability measure on that hyperplane has infinite Riesz energy.","section":"Section 4, proof of Theorem 2.2(i)"}],"minor_comments":[{"comment":"The displayed definition of a_cri is written for s in (d-3,d-2), while the case s=d-2 is treated separately by continuity with x=0; please make this explicit in the displayed definition, since equation (2.8) is not directly meaningful at s=d-2.","section":"Section 2, Definition of critical value"},{"comment":"In the reflection-formula line, the notation should be Gamma((d-s)/2) rather than Gamma(d-s/2), to avoid ambiguity with the intended argument.","section":"Section 3, Eq. (3.10)"},{"comment":"The monotonicity of s |-> a_cri(d,s) and the existence of the threshold s_* with a_cri(d,s_*)=1 are presented as numerical observations; if these claims are not proven, please state explicitly that they are supported by computation rather than by a theorem.","section":"Section 2, Remark 3 and Table 1"},{"comment":"The step 'Following the same steps as in [9, Lemma 3.5]' that leads from inequality (3.8) to (3.9) is a substantial hypergeometric manipulation; please include the intermediate identities or give a more detailed reference, since this is part of the proof of the new regime s in (d-3,d-2).","section":"Section 3, proof of Proposition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically strong and the explicit formulas are convincing. The only genuine obstacle to acceptance is the verification of the imported Frostman characterization for negative s and for external fields that are infinite on a half-space; if the authors confirm that [25, Theorem 2] covers this setting and add a short statement of hypotheses, I would support acceptance. The citation pattern is acceptable, since independent sources [15,31] are also cited for the key identities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine advance on a natural question. It proves a sharp dichotomy for half-space constrained Riesz gases over the full range s∈(d−3,d): for weakly long-range interactions (s>d−2) the equilibrium measure never fully sticks to the wall, while for strongly long-range interactions (s≤d−2) there is a critical wall position a_cri(d,s) past which it does. The if-and-only-if statement, the explicit formula for a_cri for s<d−2, and the resolution of the 2021 Kethepalli et al. conjecture in 1D are all new. In 1D the critical value simplifies to an explicit w*(s) strictly smaller than w_c(s), confirming the metastable picture proposed in [34]. This is more than a Coulomb-endpoint story.\n\nThe proof is built cleanly. The candidate wall measure is derived from the (d−1)-dimensional unconstrained equilibrium measure, the Euler–Lagrange inequality is reduced to checking a one-variable function F(0,t), and the auxiliary function f is analyzed rigorously: Lemma 3.1 gives existence and uniqueness of the root of (2.8), and Lemma 4.1 plus a Lieb-type monotonicity lemma close the argument. The critical value is a derived quantity, not fitted, and the paper is careful about normalizations. It also fills a small gap by proving Proposition 2.1 for s∈(d−3,d−2). Self-citations are mostly backed by independent sources where it matters.\n\nThe main caveat is real but not fatal: the Frostman/Euler–Lagrange characterization (3.1) is imported from [25,30] without stating hypotheses, and the paper uses it for negative s, where the Riesz kernel is not the classical positive kernel and the external field is +∞ on a half-space. Both directions of the theorem rely on the converse part of (3.1). I did not find an internal error, and the condition s>d−3 likely puts the kernel in the conditionally positive definite regime where the cited theory applies, so this is probably repairable by a more precise citation or a short appendix. Two smaller issues: the monotonicity of a_cri in s is asserted on the basis of the graph rather than proved, and the numerics are not reproducible from the text. Neither affects the central theorem.\n\nThe paper deserves a serious referee. I would send it with a request to make the imported variational theorem precise, with special attention to the d=1 and d=2 negative-s cases. For anyone working on Riesz gases, constrained equilibrium problems, or extreme eigenvalue statistics, this is a worthwhile read.","headline":"Sharp, likely correct proof of a confinement dichotomy for half-space Riesz gases, with the exact threshold confirming the 2021 conjecture; the only real caveat is that the variational characterization for negative s is imported.","tokens_in":19064,"tokens_out":4918,"would_cite":true,"duration_ms":48929,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["31A15","82B05","33C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that half-space constrained Riesz gases exhibit a sharp confinement transition: for weakly long-ranged interactions the gas is never fully confined to the wall, while for strongly long-ranged interactions it is fully…","keywords":["Riesz gas","equilibrium measure","hard wall","confinement transition","long-range interactions","Coulomb gas","half-space constraint","Euler-Lagrange conditions"],"falsifier":"For $d=1$ and $s=-3/2$, the predicted threshold is $a_{\\rm cri}=1/2$, while an earlier simulation-based estimate was near $0.441$; computing the equilibrium measure at high numerical precision and locating the onset of full wall support strictly below $1/2$ would falsify the exact critical-value formula.","tokens_in":18017,"feed_emoji":"🧱","tokens_out":9503,"duration_ms":91029,"temperature":0.7,"pith_summary":"This paper studies the equilibrium measure of a Riesz gas in $\\mathbb{R}^d$ with interaction exponent $s\\in(d-3,d)$, confined to a half-space by a hard wall and subject to a quadratic potential. It proves that the equilibrium measure exhibits a sharp dichotomy: for weakly long-ranged interactions $s\\in(d-2,d)$, the gas is never completely confined to the wall for any wall position, while for strongly long-ranged interactions $s\\in(d-3,d-2]$, complete confinement to the wall occurs exactly when the wall lies at or beyond an explicit critical distance $a_{\\rm cri}(d,s)$. This gives a single family of formulas that contains the recently studied Coulomb case $s=d-2$ as a special value, and it shows that the transition to wall confinement is controlled by the interaction range rather than by the dimension alone. A reader should care because it turns a qualitative question about long-range interactions into a computable threshold with an explicit equilibrium density on the wall.","feed_headline":"Riesz gases lock to a hard wall only past a critical distance","feed_subtitle":"Weak long-range forces keep a bulk component; strong ones fully confine the gas past an exact wall threshold.","key_machinery":"The argument is carried by the potential-theoretic Euler-Lagrange conditions (3.1): a probability measure is the equilibrium measure exactly when its weighted Riesz potential plus the external potential equals a constant on the support and is at least that constant off the support. For the candidate wall-supported measure, the paper reduces the outside-support inequality to a one-dimensional auxiliary function $f$ on $(0,\\infty)$ built from an incomplete $\\beta$ function (3.12); Lemma 3.1 shows $f$ has a unique global maximum, and formula (3.16) identifies that maximum with the critical threshold $a_{\\rm cri}$. A hypergeometric transformation (Lemma 3.2) and a radial-nonincreasing convolution lemma (Lemma 4.2) then show it is enough to verify the inequality along the direction perpendicular to the wall.","core_discovery":"The central discovery is an interaction-range dichotomy for the half-space constrained equilibrium measure $\\widehat{\\mu}_a$. Theorem 2.2 states that for $s\\in(d-2,d)$ the measure is never fully supported on the hyperplane $\\{x_d=a\\}$, regardless of $a$; for $s\\in(d-3,d-2]$ it is fully supported on that hyperplane if and only if $a\\ge a_{\\rm cri}(d,s)$, where $a_{\\rm cri}$ is defined by (2.8)-(2.10) and the wall-supported measure is given explicitly by (2.12)-(2.13). In the Coulomb case $s=d-2$, the critical value reduces to (2.14) and the wall density to the known Coulomb form, so the theorem unifies and extends the Coulomb-gas results of the recent literature.","pith_inferences":["The apparent monotonicity of $s\\mapsto a_{\\rm cri}(d,s)$ suggests a quantitative picture in which stronger long-rangedness propagates the wall's influence deeper into the bulk; deriving rigorous monotonicity and large-$d$ asymptotics would tie this to the dimension dependence discussed in Remark 3.","The numerical evidence for an exponent $s_*(d)$ with $a_{\\rm cri}(d,s_*)=1$ suggests a sub-regime where complete confinement occurs while the wall is still inside the unconstrained support; a proof of this threshold's existence for $d\\le 8$ would sharpen the phase diagram.","For $s<d-3$, the paper expects additional equilibrium phases with singular components on spheres; extending the dichotomy analysis there would test whether the confinement transition persists outside the range covered by Proposition 2.1."],"forward_implications":["For every Riesz exponent $s\\in(d-2,d)$, pushing the wall arbitrarily far away still leaves a non-trivial absolutely continuous bulk component in the equilibrium measure.","For $s\\in(d-3,d-2]$, the wall-supported equilibrium density (2.13) is explicit, so the fraction of particles sitting on the wall jumps from strictly less than one to exactly one at $a_{\\rm cri}(d,s)$.","In one dimension, the formula (2.16) gives the true confinement threshold and proves it lies strictly below the earlier metastable threshold, resolving the previously conjectured distinction.","The Coulomb critical distance (2.14) is recovered as the endpoint $s=d-2$, so the full-confinement transition for Coulomb gases in every dimension is a single instance of the general Riesz dichotomy."],"supporting_citations":[{"why":"Supplies the Euler-Lagrange variational characterization of Riesz energy minimizers with external fields over the full range $s\\in(d-3,d)$.","marker":"[25]"},{"why":"Original equilibrium-potential characterization used to identify the constrained minimizer through constant-on-support conditions.","marker":"[30]"},{"why":"Provides the unconstrained equilibrium measure for quadratic potentials and the Coulomb critical-value framework this paper extends.","marker":"[9]"},{"why":"Completes the full-confinement proof for Coulomb gases in arbitrary dimension, the case generalized by Theorem 2.2.","marker":"[29]"},{"why":"Establishes the one-dimensional quadratic equilibrium measure used in Proposition 2.1 and supports the hypersingular-energy fact for $s\\in[d-1,d)$.","marker":"[3]"},{"why":"Provides the hypersingular Riesz energy framework used to rule out wall confinement for $s\\in[d-1,d)$.","marker":"[32]"},{"why":"One-dimensional hard-wall Riesz study whose conjectured true threshold is confirmed and made explicit by formula (2.16).","marker":"[34]"}],"fun_headline_variants":["Riesz gases stick to wall only beyond a critical point","Half-space Riesz gas: wall confinement only for strong long-range","Critical distance decides wall locking for Riesz gases","Strong long-range Riesz gases collapse to wall past threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument imports from the literature the potential-theoretic principle that the minimizer is uniquely characterized by the Euler-Lagrange conditions, and applies it for every $s\\in(d-3,d)$, including negative $s$ where the Riesz kernel is not positive; if that uniqueness or characterization fails for those negative exponents, the if-and-only-if confinement statement and the exact critical value collapse.","fun_headline_variants_meta":{"raw":{"variants":["Riesz gases stick to wall only beyond a critical point","Half-space Riesz gas: wall confinement only for strong long-range","Critical distance decides wall locking for Riesz gases","Strong long-range Riesz gases collapse to wall past threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1269,"prompt_tokens":852,"completion_tokens":417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":362}},"tokens_in":468,"tokens_out":417,"duration_ms":4232,"temperature":1.0,"reasoning_tokens":362,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:28:55.639714+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $d=1$ and $s=-3/2$, the predicted threshold is $a_{\\rm cri}=1/2$, while an earlier simulation-based estimate was near $0.441$; computing the equilibrium measure at high numerical precision and locating the onset of full wall support strictly below $1/2$ would falsify the exact critical-value formula.","supporting_citations":[{"cited_title":"Dragnev, R","cited_arxiv_id":null,"evidence_quote":"Supplies the Euler-Lagrange variational characterization of Riesz energy minimizers with external fields over the full range $s\\in(d-3,d)$."},{"cited_title":"Frostman,Potentiel d’´ equilibre et capacite des ensembles avec quelques applications a la theorie des fonctions, Thesis, Meddel, Lunds Univ","cited_arxiv_id":null,"evidence_quote":"Original equilibrium-potential characterization used to identify the constrained minimizer through constant-on-support conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the unconstrained equilibrium measure for quadratic potentials and the Coulomb critical-value framework this paper extends."},{"cited_title":"Minimizers for Coulomb gases constrained to a halfspace","cited_arxiv_id":"2606.20484","evidence_quote":"Completes the full-confinement proof for Coulomb gases in arbitrary dimension, the case generalized by Theorem 2.2."},{"cited_title":"Agarwal, A","cited_arxiv_id":null,"evidence_quote":"Establishes the one-dimensional quadratic equilibrium measure used in Proposition 2.1 and supports the hypersingular-energy fact for $s\\in[d-1,d)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the hypersingular Riesz energy framework used to rule out wall confinement for $s\\in[d-1,d)$."},{"cited_title":"Kethepalli, M","cited_arxiv_id":null,"evidence_quote":"One-dimensional hard-wall Riesz study whose conjectured true threshold is confirmed and made explicit by formula (2.16)."}],"review_version":1}