{"id":"1fe524b5-dfd2-4874-8ab2-c735a8d4b9a1","arxiv_id":"2411.15793","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp two-sided estimates are stated for even and odd Jacobi heat kernels on double cones and for even kernels on hyperboloids; the final comparison step in the proof is not justified.","lead":"This paper derives explicit sharp upper and lower bounds for Jacobi heat kernels on double cones and hyperboloids, building on closed formulas by Xu and sharp estimates by Nowak, Sjogren, and Szarek. If correct, the bounds give precise control of diffusion in weighted orthogonal polynomial spaces, but the proof contains an unproved comparison step at its core.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'After noticing' reduction in Section 3 is not uniform: near the outer boundary with antipodal points, I2/(π−arccos(I1+I2)) ≫ max{I2,τ} for intermediate τ, so the claimed (I2∨τ)^−γ factor is not comparable.","rationale":"The reader's weakest assumption is exactly the load-bearing defect: the proof reduces the heat kernel to a factor (I2/(π−arccos(I1+I2))+τ)^−γ and then replaces it by (I2∨τ)^−γ. The explicit antipodal-boundary example shows this replacement fails by an unbounded factor for τ between ε and √ε, which is an admissible range of heat times. Because the final theorems state the simplified factor uniformly in all points and τ∈(0,1], the central claim is not merely missing a routine estimate; it is false as stated for every γ>0. The method is credible and the reduction via Lemmas 2.1–2.5 is otherwise sound, so a revised version could replace the boundary factor by I2/(π−arccos(I1+I2))+τ and possibly recover sharp estimates; but the current paper's headline results cannot be accepted. All corollaries inherit the gap, and the hyperbolic corollaries inherit it through the change of variables, so the issue is global rather than localized to one technical lemma.","tokens_in":14597,"tokens_out":12317,"duration_ms":112468,"concrete_test":"Compute the disputed ratio for the one-parameter family ε=2^{−n}: set t=s=1−ε, x=t e1, y=−s e1, and τ=ε^{3/4}. Then verify numerically that R_ε := (I2/(π−arccos(I1+I2))+τ)^−γ / (I2∨τ)^−γ behaves like C ε^{γ/4}, tending to 0 for γ>0. If the 'After noticing' step were valid, R_ε would remain bounded away from 0 and ∞ uniformly; the numerical check directly disproves the asserted comparability and confirms the gap in both Theorem 1.6 and the corresponding step in Theorem 1.12.","verdict_should_be":"REJECT","load_bearing_attack":"Both principal proofs rest on the step in Section 3, after Lemma 2.5, where the factor (I2/(π−arccos(I1+I2))+τ)^−γ is replaced by (I2∨τ)^−γ with the phrase 'After noticing that'. This comparability is not uniform on V0^(d+1). Take t=s=1−ε, x=t e1, y=−s e1. Then I1=−(1−ε)^2, I2=2ε−ε^2, so I1+I2=−1+4ε−2ε^2 and ψ=arccos(I1+I2)=π−2√(2ε)+O(ε^{3/2}). Hence I2/(π−ψ)=√(ε/2)+O(ε^{3/2}), whereas I2=2ε+O(ε^2). Choosing τ=ε^{3/4}, one has ε<τ≪√(ε/2), so (I2/(π−ψ)+τ)^−γ≈ε^{−γ/2}, while (I2∨τ)^−γ=ε^{−3γ/4}. Their ratio is ≈ε^{γ/4}→0 for every γ>0. Thus the claimed uniform comparability is false, and the stated formulas in Theorems 1.6 and 1.12 are not established; the analogous assertion in the proof of Theorem 1.12 that π−arccos ξ(1,v) is comparable to a constant fails in the same regime. Since all corollaries inherit this boundary factor, the central sharp-estimate claims are not supported as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies even and odd Jacobi heat kernels associated with Xu's orthogonal polynomial frameworks on the surface and solid double cone, the hyperboloid, and the paraboloid. The main results (Theorems 1.6 and 1.12, and Corollaries 1.7, 1.13, 1.17, 1.21) assert explicit two-sided bounds of the form τ^{-d/2}(π−arccos(I1+I2)+τ)^{-γ-d/2+1/2}(I2∨τ)^{-γ} exp(−arccos²(I1+I2)/(4τ)) uniformly for τ∈(0,1]. The proofs express the conic kernels through the Jacobi interval heat kernel via Lemmas 2.1 and 2.2 and then apply estimates due to Nowak, Sjögren, and Szarek. The paper also honestly records the obstructions for odd hyperboloid kernels and for the paraboloid settings. However, the decisive final comparison in Section 3 is asserted without proof and is not uniform, so the main claims are not established.","tokens_in":14951,"tokens_out":12521,"duration_ms":102266,"significance":"If the estimates were correct, they would be a valuable extension of the sharp Jacobi and spherical heat kernel bounds of NSS21 to double cones and hyperboloids, with explicit dependence on the boundary parameters. The initial reduction to the interval kernel is transparent, the parameter bookkeeping is mostly careful, and the paper correctly identifies the known obstructions for the paraboloid and the odd hyperboloid. The self-cited work [HK23] is not used as a load-bearing input. However, the central uniform comparison fails, and the stated theorems do not follow from the proof.","major_comments":[{"comment":"The displayed comparison after Lemma 2.5, namely (I2/(π−arccos(I1+I2))+τ)^{−γ} ≃ (I2+τ)^{−γ} ≃ (I2∨τ)^{−γ}, is not uniform on V0^{d+1}. Fix a unit vector e1 and take t=s=1−ε, x=t e1, y=−s e1 with 0<ε≪1. Then I1=−(1−ε)^2, I2=2ε−ε^2, and I1+I2=−1+4ε−2ε^2, so π−arccos(I1+I2)=2√(2ε)+O(ε^{3/2}). For τ=ε^{3/4}, we have I2/(π−arccos(I1+I2)) ≈ √(ε/2) and I2∨τ = τ, so the ratio of (I2/(π−arccos(I1+I2))+τ)^{−γ} to (I2∨τ)^{−γ} is approximately ε^{γ/4}, which tends to 0 for every γ>0. Thus the two sides are not comparable, and the proof of Theorem 1.6 fails at this step. Corollaries 1.7, 1.17 and 1.21 inherit the gap.","section":"Section 3, proof of Theorem 1.6"},{"comment":"The statements that 'π−arccos ξ(1,v) is comparable to a constant' and that 'π−arccos ξ(1,1) is comparable to a constant' are false. On V^{d+1}, take t=s=1, x=e1, y=−e1; then I1=−1 and I2=I3=0, so ξ(1,1)=−1 and π−arccos ξ(1,1)=0. In the near-boundary regime t=s=1−ε, x=(1−ε−δ)e1, y=−(1−ε−δ)e1 with δ=ε^{3/4}, one has I3≈2δ and π−arccos ξ(1,1)≈2√(ε+δ), so for τ=ε^{3/4} the factor (I3/(π−arccos ξ(1,1))+τ)^{−µ} is not comparable to (I3∨τ)^{−µ}. The same nonuniformity affects the replacement involving I2. Therefore the proof of Theorem 1.12 fails, and Corollary 1.13 is not established.","section":"Section 3, proof of Theorem 1.12"}],"minor_comments":[{"comment":"The weight notation in formula (1.10) omits the parameter μ: it should read P^O_n(w_{β,γ,μ}; ...) = ... P^E_n(w_{β+1,γ,μ}; ...).","section":"Equation (1.10)"},{"comment":"The last two rows of Table 1 both label the domain ~V^{d+1}_0; one of them should presumably be the solid paraboloid ~V^{d+1}.","section":"Table 1"},{"comment":"The text refers to 'the Jacobi heat kernel on B^d', but Lemma 2.3 concerns the Jacobi kernel on the interval [−1,1]; the wording should be corrected.","section":"Paragraph before Lemma 2.3"},{"comment":"The opening line fixes '(x,y),(y,s)∈V^{d+1}' instead of '(x,t),(y,s)∈V^{d+1}'.","section":"Proof of Theorem 1.12"}],"recommendation":"reject","confidential_remarks":"The paper is a competent assembly of known tools, and the gap is localized to one comparison, but that comparison is essential and the advertised theorems are false as stated. A revised version might replace the factors (I2∨τ)^{−γ} and (I3∨τ)^{−µ} by the quantities that actually emerge from Lemma 2.5, namely (I2/(π−arccos(I1+I2))∨τ)^{−γ} and the analogous expression for I3, but that would change the statements of Theorem 1.6, Theorem 1.12, and all corollaries, and would require a fresh uniformity analysis. Given that the central claim is not supported, rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me be direct: the central theorems are not proven as stated. The derivation is mostly clean, and the author honestly flags the paraboloid limitations, but the 'After noticing' comparability in Section 3 is asserted without proof and, as written, false uniformly. Near the outer boundary, take t=s=1−ε, x=t e1, y=−s e1, τ=ε^{3/4}. Then I2≈2ε, π−arccos(I1+I2)≈2√(2ε), so I2/(π−ψ)≈√(ε/2). For γ>0, (I2/(π−ψ)+τ)^−γ≈ε^{−γ/2} while (I2∨τ)^−γ=ε^{−3γ/4}; the ratio tends to 0. So the claimed equivalence in Theorem 1.6 fails. The same false step appears in the proof of Theorem 1.12 when asserting π−arccosξ(1,v) is comparable to a constant; that fails in the same regime. Every corollary inherits the gap.\n\nWhat is good: the paper is clearly written, the reduction to the Jacobi interval kernel via Lemmas 2.1 and 2.2 is correct, the parameter bookkeeping is careful, and the negative observations about the paraboloid are honest and useful. The architecture is exactly what NSS21 does, applied to Xu's domains, and the author credits that properly.\n\nThe flaw is load-bearing, not a minor typo. But it may be repairable: perhaps the boundary factor should keep the I2/(π−arccos) term, or the stated comparability is only valid in a different range. A referee should ask for a corrected statement or a rigorous proof of that step. As it stands, the sharp estimates for the double cone and hyperboloid are not established.\n\nI would recommend sending this to peer review rather than desk-rejecting: the error is subtle, not sloppiness, and a motivated referee or author could fix it. But I would not cite the main theorems until that step is repaired.","headline":"The central 'After noticing' comparability in Section 3 is non-uniform, so the main sharp estimates for the double cone and hyperboloid are not established, though the paper is otherwise well structured and likely fixable.","tokens_in":15426,"tokens_out":3052,"would_cite":false,"duration_ms":24337,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K08","33C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves genuinely sharp two-sided estimates for even and odd Jacobi heat kernels on double cones, their surfaces, and hyperbolic counterparts, with explicit formulas.","keywords":["Jacobi heat kernel","double cone","hyperboloid","paraboloid","sharp estimates","orthogonal polynomials","heat kernel bounds"],"falsifier":"Set $d=2$, $\\gamma=1$, take antipodal unit vectors $x=-y$ on $V_0^3$ with $t=s=\\sqrt{1-\\varepsilon}$ and $\\tau=\\varepsilon^{3/4}$; then $I_1=-1$, $I_2=\\varepsilon$, and $\\pi-\\arccos(I_1+I_2)\\asymp\\sqrt{\\varepsilon}$, so the asserted comparability would claim $(\\sqrt{\\varepsilon})^{-1}\\asymp(\\varepsilon^{3/4})^{-1}$, which fails as $\\varepsilon\\to0$. Evaluating the two sides numerically at these parameters settles the uniformity question.","tokens_in":14373,"feed_emoji":"","tokens_out":8529,"duration_ms":70669,"temperature":0.7,"pith_summary":"This paper seeks genuinely sharp two-sided estimates for the even and odd Jacobi heat kernels on the surface and interior of the multidimensional double cone, and for the even kernel on the hyperboloid and its surface. Genuinely sharp means the kernel is trapped between constant multiples of one explicit expression, uniformly in the two points and in the heat time $\\tau$. The main results, Theorems 1.6 and 1.12, give such expressions: a Gaussian in $\\arccos(I_1+I_2)$ (with an extra variable $I_3$ in the solid case), multiplied by a power of $\\tau$ and by factors $(I_2\\vee\\tau)^{-\\gamma}$ and $(I_3\\vee\\tau)^{-\\mu}$. The odd kernels are obtained by multiplying by $st$, and the hyperbolic results follow by a change of variables. The paraboloid cases are left open, because their orthogonal polynomials are not eigenspaces of a single diffusion operator with degree-only eigenvalues.","feed_headline":"Sharp two-sided heat kernel bounds found for double cones","feed_subtitle":"Explicit formulas now pin down the even and odd Jacobi heat kernels on cones, hyperboloids, and their surfaces.","key_machinery":"The carrying device is an integral representation: the double-cone heat kernel is written as an integral, against a $\\beta$-type measure $d\\Pi_{\\gamma-\\frac12}(v)$ (and a second such integral in the solid case), of the classical one-dimensional Jacobi heat kernel evaluated at a linear argument $\\xi(v)=I_1+vI_2$ or $\\xi(u,v)=I_1+vI_2+uI_3$. Sharp interval estimates for that kernel, a monotonicity reduction of the integration range, and a boundary evaluation lemma for integrals of $\\exp(-\\arccos^2(A+Bw)/(4\\tau))$ convert the integral into the closed-form boundary expression stated in the theorems.","core_discovery":"On the surface of the double cone $V_0^{d+1}$, the even Jacobi heat kernel $h^\\mathrm{E}_\\tau$ is comparable, uniformly for $\\tau\\in(0,1]$, to $\\tau^{-d/2}(\\pi-\\arccos(I_1+I_2)+\\tau)^{-\\gamma-d/2+1/2}(I_2\\vee\\tau)^{-\\gamma}\\exp(-\\arccos^2(I_1+I_2)/(4\\tau))$, and for $\\tau>1$ it is comparable to $1$. On the solid double cone, the even kernel is comparable to $\\tau^{-d/2-1/2}(\\tau+\\pi-\\arccos(I_1+I_2+I_3))^{-\\gamma-\\mu-d/2}(I_2\\vee\\tau)^{-\\gamma}(I_3\\vee\\tau)^{-\\mu}\\exp(-\\arccos^2(I_1+I_2+I_3)/(4\\tau))$. The odd kernels obey the same bounds with an additional factor $st$, and the hyperbolic analogues follow by replacing $t,s$ with $\\sqrt{t^2-\\rho^2},\\sqrt{s^2-\\rho^2}$.","pith_inferences":["Beyond the paper: the closed forms suggest a geometric reading of $\\arccos(I_1+I_2)$ (and $\\arccos(I_1+I_2+I_3)$) as a distance-like angle between the two points, so on these domains the short-time heat kernel has the familiar Gaussian shape with power-law corrections whose exponents are the weight parameters $\\gamma$ and $\\mu$.","Beyond the paper: the same integral-reduction scheme should apply to any domain whose reproducing kernel is a one- or two-fold integral of a univariate special function with a linear argument and whose eigenvalues depend only on the total degree; the paraboloid fails the second condition.","Beyond the paper: one testable extension is to check numerically whether the stated bound remains uniform when one point approaches the cone boundary while the other is held fixed and $\\tau$ scales with the boundary distance; the current proof leaves that regime open."],"forward_implications":["For $\\tau\\in(0,1]$, the even kernel on $V_0^{d+1}$ decays like $\\tau^{-d/2}(I_2\\vee\\tau)^{-\\gamma}$ times the Gaussian $\\exp(-\\arccos^2(I_1+I_2)/(4\\tau))$, uniformly in both points.","The odd kernels on both double cones are bounded by the same expressions with an extra factor $|st|$, so they vanish near the cone tips.","The hyperboloid estimates are the double-cone estimates after the change $t\\mapsto\\sqrt{t^2-\\rho^2}$, $s\\mapsto\\sqrt{s^2-\\rho^2}$.","For $\\tau>1$, every treated kernel is uniformly comparable to $1$.","The paraboloid settings are not covered: the diffusion operator there has eigenvalues depending on both the total degree $n$ and the internal degree $m$."],"supporting_citations":[{"why":"Supplies the double-cone and hyperboloid domains, weights, reproducing kernels, and the closed-form expressions that the estimates start from.","marker":"[Xu21]"},{"why":"Provides the genuinely sharp estimates for the classical Jacobi heat kernel on $[-1,1]$ and the general reduction strategy used here.","marker":"[NSS21]"},{"why":"Establishes the sharp spherical heat kernel estimates that the interval arguments extend.","marker":"[NSS18]"},{"why":"Supplies earlier Jacobi heat kernel techniques that feed into the sharp interval estimates.","marker":"[NS13]"},{"why":"Defines the paraboloid settings whose eigenvalue structure blocks the method.","marker":"[Xu23]"},{"why":"Provides the general orthogonal-polynomial and heat-kernel framework used to define the kernels.","marker":"[DX14]"}],"fun_headline_variants":["Sharp Jacobi heat kernel bounds on double cones and hyperboloids","Two-sided Jacobi heat kernel estimates on double cones","Sharp decay for Jacobi heat kernels on double cones","Jacobi heat kernels: sharp bounds on double cones","Double cone Jacobi kernels: sharp two-sided estimates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the assertion, made without proof in Section 3, that $\\big(I_2/(\\pi-\\arccos(I_1+I_2))+\\tau\\big)^{-\\gamma}$ is uniformly comparable to $(I_2\\vee\\tau)^{-\\gamma}$; near the boundary $t,s\\to1$ with $I_1+I_2\\to-1$ the two sides differ by an unbounded factor for intermediate $\\tau$.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Jacobi heat kernel bounds on double cones and hyperboloids","Two-sided Jacobi heat kernel estimates on double cones","Sharp decay for Jacobi heat kernels on double cones","Jacobi heat kernels: sharp bounds on double cones","Double cone Jacobi kernels: sharp two-sided estimates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00132,"raw_usage":{"total_tokens":5351,"prompt_tokens":898,"completion_tokens":4453,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":4374}},"tokens_in":514,"tokens_out":4453,"duration_ms":30758,"temperature":1.0,"reasoning_tokens":4374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:56:48.277947+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $d=2$, $\\gamma=1$, take antipodal unit vectors $x=-y$ on $V_0^3$ with $t=s=\\sqrt{1-\\varepsilon}$ and $\\tau=\\varepsilon^{3/4}$; then $I_1=-1$, $I_2=\\varepsilon$, and $\\pi-\\arccos(I_1+I_2)\\asymp\\sqrt{\\varepsilon}$, so the asserted comparability would claim $(\\sqrt{\\varepsilon})^{-1}\\asymp(\\varepsilon^{3/4})^{-1}$, which fails as $\\varepsilon\\to0$. Evaluating the two sides numerically at these parameters settles the uniformity question.","supporting_citations":[],"review_version":1}