{"id":"7a2f2145-22ae-4db0-ae3c-08591d5fd4f9","arxiv_id":"2607.23873","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For an unbounded open set of radii, the hard-sphere model on the hyperbolic plane has non-unique Gibbs measures at large activity, hence a phase transition.","lead":"The paper proves that the hard-sphere gas on the hyperbolic plane has a phase transition for an unbounded open set of sphere radii. This is the first rigorous continuum hard-sphere phase transition in any dimension, sidestepping the still-open Euclidean case by using hyperbolic packing geometry and annealed entropy.","discovery_kind":"new_application","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The entire separation argument (Theorem 6 → Theorem 2) reduces to uniqueness+periodicity of the optimal packing measures at the tight radii {r_n} and at r=∞; the uniqueness inputs are outsourced to prior work, including remarks in Kellerhals [46] rather than proved theorems.","rationale":"The reader's weakest_assumption named exactly the right load-bearing input, and my independent pass through the manuscript did not surface a stronger internal concern. The new content of this paper — the GNZ-defect/entropy-dissipation construction on BS-convergent quotients (§5), the near-optimal Gibbs measure construction with volume-independent activity threshold (§4), and the annealed-entropy separation of periodic measures from weak Poisson factors (§6) — is modular, explicit, and free of circularity or tunable parameters; the constants (D_opt(H²,r_n) = (3csc(π/n)−6)/(n−6), D_opt(H²,∞) = 3/π) are parameter-free and checkable. The residual risk is concentrated in cited packing theory that is 20–45 years old, peer-reviewed, and squarely in the first author's research area; the only wrinkle worth noting is that the uniqueness component is cited to remarks in [46] rather than to proved theorems, which is why the concrete test above is a genuine verification rather than a formality. This does not rise to the level of moving the verdict: the claim would only be damaged if the equality case of a classical, much-cited bound were mischaracterized, and the paper's architecture localizes that risk precisely (each radius independently; the (ρ,∞) strengthening specifically on the horoball case). I therefore recommend UNCHANGED (ACCEPT), with the equality-case re-derivation as the one check that would settle the residual risk.","tokens_in":42122,"tokens_out":11070,"duration_ms":295072,"concrete_test":"Independently re-derive the equality case of the simplex bound in dimension 2 at one tight radius, say r_7, and at r=∞: show that any isometry-invariant µ ∈ M_{r_7}(H²) with density_r(µ) = 3csc(π/7)−6 must be supported on the Isom(H²)-orbit of the {3,7}-tiling vertices. Route: equality in Böröczky's bound forces µ-a.e. packing's Delaunay cells to be regular triangles of side 2r_7, which rigidly forces the tiling; then argue as in Lemma 59 via ergodic decomposition. If a non-periodic optimizer exists at r_7, Theorem 6 fails there; if it fails at ∞, the \"(ρ,∞)\" strengthening of Theorem 2 is lost.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2 is a clean modular reduction: §4 builds a near-optimal Gibbs measure (Theorem 25), §5 builds a weak-Poisson-factor Gibbs measure (Theorem 32), and §6 shows lattice packings are not weak Poisson factors (Theorem 47). The reader correctly identified the soft spot: Lemma 59, the linchpin, requires that at the radii actually used — {r_n}_{n≥7} and ∞ — EVERY isometry-invariant optimizer of D_opt be periodic, so that Theorem 47 can contradict D_Pois = D_opt. This uniqueness is imported: Theorem 4 from [13] (self-citation), and the uniqueness assertions in §3.1 and §3.3 are supported in the text by \"the remark after Theorem 2.1 [46]\" and \"the remark after Proposition 2.2 in [46]\" — i.e., remarks in Kellerhals' paper on the simplicial density function, not full proved statements reproduced or re-checked here. If the equality case of Böröczky's simplex bound admitted a non-periodic invariant optimizer at some r_n, Lemma 59 would fail at that radius and the gap D_Pois < D_opt could vanish exactly where the argument needs it. Two mitigations: (i) measure-level uniqueness follows from packing-level uniqueness via ergodic decomposition (density is affine, so an optimal measure's ergodic components are all supported on optimal packings; packing uniqueness up to isometry then forces µ = µ_n), so the crux is genuinely the finite-dimensional rigidity of the equality case; (ii) the headline \"unbounded open set\" survives on the {r_n} alone (each contributes an open interval via USC of D_Pois + continuity of D_opt), while the stronger \"(ρ,∞) ⊂ R₂\" claim depends specifically on the horoball case (Theorem 22), making r=∞ the single most load-bearing radius for the full stated theorem. I found no internal gap in §4–§6: the entropy-dissipation chain (Lemmas 41–45), the BS-convergence lifting (Lemma 26), the volume-independent finite-volume estimate (Lemma 27), and the annealed-entropy separation (Propositions 56–58) all check out structurally, with cited machinery (H","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper proves the first phase transition for the continuum hard-sphere model in any geometry: on the hyperbolic plane there is an unbounded open set of radii R₂, containing an interval (ρ,∞), such that λ_u(H²,r) < ∞ for r ∈ R₂ (Theorem 2). The proof constructs two isometry-invariant (r,λ)-Gibbs measures of different densities at large λ. §4 (Theorem 25) builds a near-optimally-dense Gibbs measure by lifting high-activity finite-volume models on Benjamini–Schramm-convergent lattice quotients, with volume-independent activity thresholds (Lemma 27). §5 (Theorem 32) builds a Gibbs measure that is a weak Poisson factor by running empty-start spatial birth–death dynamics and driving the GNZ defect to zero via entropy/Fisher-information dissipation on quotients (Lemmas 42–45). §6 (Theorem 47) shows measures supported on lattice orbits are not weak Poisson factors: after reduction to free-group actions, lattice actions have annealed (sofic) entropy −∞ (Proposition 58) while weak Bernoulli factors have h_ann ≥ 0 (Proposition 57). Combined with uniqueness and periodicity of the optimal packing measures at the tight radii {r_n} and at ∞ (Theorems 4, 22), Lemma 59 yields D_Pois < D_opt on an open unbounded set (Theorem 6), hence non-uniqueness; Theorem 8 gives suboptimal completely saturated packings.","tokens_in":42488,"tokens_out":11525,"duration_ms":388087,"significance":"If correct, this is a landmark: the first proof of a phase transition in the hard-sphere model in any dimension or space, a problem open since Boltzmann and unresolved even in R²/R³. The argument is modular and largely self-contained, and it introduces two tools of independent interest: (i) entropy-dissipation control of the GNZ defect for continuum Glauber dynamics in a non-amenable setting, extending Holley–Stroock/Holley/Shriver ideas beyond amenability; (ii) the first use of annealed sofic entropy to separate structured from random-like Gibbs measures in statistical mechanics. The results are concrete and checkable: explicit tight radii with a closed-form optimal density, a corollary (Theorem 8) on completely saturated suboptimal packings that is false in R^d, and well-posed open problems (§8, e.g. Question 65 on random hyperbolic surfaces) that make the approach falsifiable in principle. The reliance on prior packing theory is clearly delineated. I verified the main reductions (§§4–7) in detail and found them sound; the soft spots are localized to imported uniqueness inputs and a definitional reconciliation, both addressable locally.","major_comments":[{"comment":"Lemma 59 — the linchpin of Theorems 6 and 2 — requires that at the radii used, EVERY invariant optimizer be the periodic measure. For finite tight radii this is Theorem 4, cited to [13] (adequate). But the supporting uniqueness statements in the text rest on weaker citations: §3.1 ('by uniqueness of the packing (see e.g., the remark after Theorem 2.1 [46])') and, critically, Theorem 22 ('This packing uniquely realizes the simplex bound (see, e.g., the remark after Proposition 2.2 in [46])'). The (ρ,∞) clause of Theorem 2 depends entirely on the horoball case, so its uniqueness input should not rest on remarks in [46]. Please state the equality-case rigidity of Böröczky's simplex bound (and its horoball analogue) as a lemma with a proof sketch or a theorem-level citation, and add the one-sentence ergodic-decomposition step from packing-level to measure-level uniqueness (density is affine,","section":"§3.1, §3.3 (Theorem 22); Lemma 59"},{"comment":"The manuscript uses two definitions of Poisson factor: Definition 5 (packings, via Isom-equivariant maps on Ω_T(X), i.e. marked Poisson processes on X×[0,T]) and Definition 16 (G-equivariant factors of a Haar–Poisson process on G). The separation argument silently identifies them: Lemma 59 bounds densities of Def-5 weak Poisson factors, while Theorem 47 (via Theorem 48 and Corollary 54) excludes Def-16 weak Poisson factors. Remark 17 only treats the unmarked case H² = G/K. What is needed is that every Def-5 weak Poisson factor is a Def-16 weak Poisson factor — i.e. that i.i.d. [0,T]-marks can be produced G-equivariantly from a Haar–Poisson process on G (e.g. using the Poisson configuration in the compact K-fibers). This is presumably routine, but as written the key lemma conflates two a priori different classes; please add an explicit reconciliation lemma.","section":"Definitions 5 vs 16; Lemma 59; Remark 17"}],"minor_comments":[{"comment":"Proposition 57 needs h_ann ≥ 0 for weak Bernoulli factors; the proof cites [48, Theorem 3.2] (completely positive sofic entropy of Bernoulli actions), and footnote 6 acknowledges that only non-negativity is needed but 'a short proof of that does not appear to be in the literature.' Since sofic entropy in [48] is a priori the quenched quantity, one sentence justifying that it yields the annealed statement for free groups (or the short direct argument alluded to) would close a small gap in the citation chain.","section":"§6.2, Proposition 57"},{"comment":"Definition 5 builds marks in [0,T], but the Glauber construction of §5.1 uses a Poisson process on X×R_+×R_+ (birth time and lifetime). A sentence noting that the time-t configuration is a factor of the process restricted to X×[0,t]×R_+ (and thinning the lifetime mark) would align Lemma 36 with Definition 5.","section":"§1.4 vs §5.1"},{"comment":"Definition 1: 'there are at least two distinct (r, λ)-Gibbs measure' → 'measures'. Similar number-agreement slips occur elsewhere (e.g. §1, 'there always exists at least one').","section":"Definition 1"},{"comment":"There are recurring typesetting artifacts: missing spaces ('inR 2', 'H d', 'onR d'), 'F act 12/38/39' running into the text, and the sentence break after (11) ('...1{s∈(t,t+ℓ]} andν s ∈M r(X) is the law of ηs'). Please proofread the source.","section":"Throughout"},{"comment":"Reference [43] (Jahnel–Köppl–Steenbeck–Zass) lacks a year and venue/arXiv identifier; please complete it.","section":"References"},{"comment":"The proof of Theorem 8 is a sketch relying on [9, Theorem 3.1]; it would help to state explicitly which lemmas of [9] (e.g. the Borel selection in [9, Lemma 4.1]) transfer verbatim to the weak-Poisson-factor setting and which need modification.","section":"§7, Theorem 8"},{"comment":"It may be worth stating in §7 that the argument gives λ_u(H², r) ≤ λ₀(r) with λ₀ from Theorem 25, and that no monotonicity of non-uniqueness in λ is claimed (cf. the discussion after Definition 1), to prevent misreading of the main theorem.","section":"§7, Theorem 2"}],"recommendation":"minor_revision","confidential_remarks":"The core structural inputs — periodic approximation of packing measures, the tight-radius optimizers, and the annealed-entropy theory — are drawn from the first author's prior body of work ([9]–[15]); this is appropriate given the tools needed, and the paper is unusually transparent about where each input comes from. My only residual concern is that the uniqueness of the horoball optimizer (Theorem 22), on which the headline '(ρ,∞)' clause rests, is supported by a remark citation in [46] rather than a theorem-level statement; the editor may wish to insist on the fix requested in Major Comment 1. Neither this nor the definitional reconciliation in Major Comment 2 appears to threaten the central claim. The result is a strong fit for a top journal in mathematical physics or probability, and I expect it to be influential."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first proof that the continuum hard-sphere model has a phase transition in any continuous space. They get λ_u(H²,r)<∞ on an unbounded open set of radii (including a ray (ρ,∞)), by building two isometry-invariant Gibbs measures of different densities for large λ.\n\nWhat is actually new is the combination: near-optimal Gibbs states lifted from high-activity models on BS-convergent lattice quotients (§4), a weak-Poisson Gibbs state from empty-start spatial birth-death dynamics with entropy-dissipation control of the GNZ defect (§5), and the annealed-entropy argument that the unique optimal lattice packings at the tight radii cannot be weak Poisson factors (§6). Continuity of D_opt then opens the countable tight set into an open set. The modular structure is clean; the finite-volume density estimate is volume-independent; the Glauber/GNZ estimates check out structurally.\n\nThe soft spot is real but proportionate and standard for this literature: the density gap D_Pois < D_opt (and thus non-uniqueness) rests on uniqueness-plus-periodicity of the optimizers at {r_n} and ∞, imported from Bowen–Radin and remarks in Kellerhals rather than re-proved. If equality cases of the simplex bound admitted non-periodic invariant optimizers, the separation would fail at those radii. Measure-level uniqueness follows from packing rigidity via ergodic decomposition, so the crux is the classical rigidity, not a hole inside §§4–6. The stronger claim that (ρ,∞) sits in the set leans hardest on the horoball case.\n\nNo circularity, no free parameters, citations are to the right packing and entropy sources. This is for people in rigorous stat mech, continuum Gibbs measures, and ergodic theory of packings. It deserves a serious referee. I would bring it to reading group and cite it if I touch hard spheres or hyperbolic continuum models.","headline":"First continuum hard-sphere phase transition, proved cleanly on H² by separating near-optimal lattice Gibbs states from weak-Poisson ones via annealed entropy.","tokens_in":42837,"tokens_out":501,"would_cite":true,"duration_ms":17351,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B26","82B21","52C17","37A35","60G55"],"pacs":[],"model":"grok-4.5","headline":"The hard-sphere model on the hyperbolic plane has a phase transition for an unbounded open set of radii.","keywords":["hard sphere model","phase transition","hyperbolic plane","Gibbs measures","sphere packing","Poisson factors","annealed entropy","Glauber dynamics"],"falsifier":"Exhibit, at one of the tight radii r_n, either a second isometry-invariant packing measure of the same optimal density or a weak Poisson factor that achieves that density; either would collapse the separation used to produce two Gibbs measures.","tokens_in":42485,"feed_emoji":"🔵","tokens_out":885,"duration_ms":16627,"temperature":0.7,"pith_summary":"For more than a century the hard-sphere model—equal non-overlapping balls whose only interaction is exclusion—has been expected to freeze or break symmetry in the Euclidean plane and in three-space, yet a mathematical proof of any phase transition there remains open. This paper proves that the same model on the hyperbolic plane does undergo a phase transition: for every radius belonging to an unbounded open set (including all sufficiently large radii), there is a finite activity threshold above which at least two distinct infinite-volume Gibbs measures exist. The argument constructs one Gibbs measure of near-optimal packing density from lattice packings and another of strictly lower density as a weak limit of Poisson factors obtained from Glauber dynamics; annealed entropy then shows that the unique densest lattice packing cannot itself be a weak Poisson factor, forcing the two measures apart. The result supplies the first rigorous confirmation of a hard-sphere phase transition in any constant-curvature geometry of dimension greater than one.","feed_headline":"Hard spheres freeze on the hyperbolic plane","feed_subtitle":"First proof of a phase transition for the classical hard-sphere gas in any dimension above one","key_machinery":"The density gap D_Pois(H²,r) < D_opt(H²,r) at the tight radii and nearby: lattice optimizers have annealed entropy −∞ while every weak Poisson factor has non-negative annealed entropy, so the high-density Gibbs measure built from lattices cannot coincide with the low-density Gibbs measure built from Glauber dynamics.","core_discovery":"There exists an unbounded open set R₂ of radii such that, for every r in R₂, the hard-sphere model on the hyperbolic plane admits at least two distinct isometry-invariant Gibbs measures once the activity is large enough; in particular R₂ contains an interval (ρ, ∞).","pith_inferences":["If the unique-optimizer property can be established for horoball packings in H³, the same argument would give a phase transition in three-dimensional hyperbolic space for large radii.","The gap between Poisson-factor density and optimal density quantifies how much “randomness” costs in non-amenable geometry and may bound the performance of local packing algorithms on random hyperbolic surfaces.","Failure of the Euclidean analogue is consistent with amenability: there the Poisson-factor density can reach the optimum, so the separation step is unavailable."],"forward_implications":["For every radius in the open set R₂ the activity threshold λ_u(H²,r) is finite, so uniqueness fails at large chemical potential.","There exist completely saturated packings of strictly sub-optimal density in the hyperbolic plane.","Density of invariant hard-sphere measures is not a function of activity alone once activity is large.","The same density-gap strategy yields a phase transition for all sufficiently large radii, including the horoball (infinite-radius) limit."],"fun_headline_variants":["Hard spheres show phase transition on hyperbolic plane","Phase transition proven for hard-sphere model in H²","Hard-sphere gas freezes on the hyperbolic plane","Multiple Gibbs measures for hard spheres in hyperbolic plane","Hard spheres undergo phase transition above one dimension"],"cache_read_input_tokens":128,"weakest_assumption_plain":"At a countable set of special radii the unique densest packing measure is a periodic lattice packing; if that uniqueness or periodicity failed, the density-gap argument would not start.","fun_headline_variants_meta":{"raw":{"variants":["Hard spheres show phase transition on hyperbolic plane","Phase transition proven for hard-sphere model in H²","Hard-sphere gas freezes on the hyperbolic plane","Multiple Gibbs measures for hard spheres in hyperbolic plane","Hard spheres undergo phase transition above one dimension"]},"model":"grok-4.5","effort":"low","cost_usd":0.002323,"raw_usage":{"total_tokens":847,"prompt_tokens":581,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":23228000,"prompt_tokens_details":{"text_tokens":581,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":192,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":581,"tokens_out":74,"duration_ms":4973,"temperature":1.0,"reasoning_tokens":192,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T13:43:22.584760+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit, at one of the tight radii r_n, either a second isometry-invariant packing measure of the same optimal density or a weak Poisson factor that achieves that density; either would collapse the separation used to produce two Gibbs measures.","supporting_citations":[],"review_version":2}