{"id":"e7ffd8ab-4a94-4d36-8d6c-be73ec962ea4","arxiv_id":"2506.08115","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The heat kernel of the fractional Laplacian with Hardy potential in each angular momentum channel is comparable to the kernel without the potential, multiplied by two known weight factors.","lead":"This mathematics paper finds precise formulas describing how the fractional Laplacian with a Hardy (inverse-distance) potential spreads heat over time in each angular momentum channel. The result gives two-sided estimates that match, so it provides exact rates, which will help analyze relativistic atoms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Form equality for η<0 rests on the cited 3G estimate (4.71); it appears internally consistent, so this is a verification burden rather than a detected error.","rationale":"The reader's weakest assumption correctly identifies Theorem 4.2(2) as the load-bearing bridge and the η<0 case, especially the dependence on Lemmas 4.15-4.17 and the 3G inequalities, as the most delicate part. I agree with that assessment. My reading of the proof, however, suggests that the argument is internally consistent: Lemma 4.16's bound, combined with Lemma 4.15 and the scaling (4.68), gives the required o(t) behavior for (4.73), and the domain equality D(E_{ζ,η})=D(I_{ζ,η}) then follows by Fatou and dominated convergence. No concrete error emerged from re-checking the estimates. I flag two secondary issues that the reader did not mention: Lemma 4.16's statement omits the factor Ψ_ζ(η) (harmless since constants absorb it), and Theorem 4.2(2) states ζ∈(-1/2,∞) even though the proof of the η<0 case, Theorem 4.18, covers only ζ∈[0,∞). The latter is a genuine overclaim, but it does not affect Theorem 1.1 because ζ=(d_ℓ-1)/2≥0 for all d∈N and ℓ∈L_d. The proposed numerical check of (4.71) in a minimal covered case would settle whether the cited 3G inequality actually delivers the required bound in the regime used; until then, the residual risk is exactly as the reader stated. The verdict ACCEPT remains appropriate, with a recommendation to correct the ζ-range overclaim in a revision.","tokens_in":33854,"tokens_out":18976,"duration_ms":199998,"concrete_test":"Set ζ=1/2, α=1, η=-1/2 (so d=2, ℓ=0, a channel covered by Theorem 1.1). Compute the first Duhamel term p^{(1,D)}_t(r,s)=∫_0^t dτ∫_0^∞ dz z^{2ζ} p^{(1)}_ζ(τ,r,z) z^{-α} p^{(1)}_ζ(t-τ,z,s) by numerical quadrature, using the explicit kernel (3.10) for p^{(1)}_{1/2}. Compare against the right-hand side of (4.71) for r=1, s=2, t=10^{-3},10^{-4},10^{-5}, with G_0 and \\tilde G computed from (4.66)-(4.67). If the inequality fails at any point, the estimate behind Lemma 4.17 is invalid and the η<0 form equality would not transfer; if it passes, the cited 3G inequality is corroborated in the regime used for the main theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bridge is Theorem 4.2(2): for α<2 and η<0, the quadratic form I_{ζ,η} generated by the ground-state representation equals the Dirichlet form E_{ζ,η} of the Hardy-perturbed Bessel kernel p^{(α)}_{ζ,η}. This identification transfers the known bounds (3.18) to Theorem 1.1. For the channels used in the main theorem, ζ=(d_ℓ-1)/2≥0, the critical step is Lemma 4.17, whose limit (4.72) is obtained by showing the first Duhamel term (4.73) is o(t). That reduction uses Lemma 4.16, whose estimate (4.71) is proved in Appendix B.2 via the 3G inequality [BM25, Theorem 3.1] for ζ≥0 and [BJM24, Lemma 2.5] for ζ<0, together with Lemma 4.15. I did not find a concrete error in the derivation: after scaling, G_0(t,r,s)=O(t), so (4.71) yields p^{(1,D)}_t=O(t^2) for fixed r≠s, and (4.73) indeed vanishes. The concern is therefore one of verification burden: the decisive estimate is imported from the authors' prior work rather than proved here, and the statement of Lemma 4.16 omits the coupling factor Ψ_ζ(η). In addition, Theorem 4.2(2) claims ζ∈(-1/2,∞), but the cited proof for η<0, Theorem 4.18, is stated only for ζ∈[0,∞); since d_ℓ≥1 this does not affect Theorem 1.1, but it is an unsupported overclaim in the paper as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves sharp, two-sided heat kernel bounds for the Hardy operator (−Δ)^{α/2} − κ|x|^{−α} in L²(R^d) restricted to a fixed angular momentum channel ℓ. Theorem 1.1 asserts that for α∈(0,2]∩(0,d+2ℓ) and η∈(−S,(d_ℓ−α)/2] with κ=Φ^{(α)}_{d_ℓ}(η), the heat kernel e^{−tL_{κ,ℓ}}(r,s) is comparable to (1∧r/t^{1/α})^{−η}(1∧s/t^{1/α})^{−η} e^{−tL_{0,ℓ}}(r,s), with explicit formulas for the unperturbed kernel in both the cases α<2 and α=2. The proof reduces Theorem 1.1 to an identification (Theorem 4.2(2)) between the ground-state-representation form I_{ζ,η} and the form E_{ζ,η} generated by the Schrödinger-perturbed subordinated Bessel kernel p^{(α)}_{ζ,η}, whose sharp bounds were proved in the authors' earlier work [BJM24]. The identification is established for η>0 within the paper (§4.3), while the η<0 case (§4.4) proceeds through the limit (4.72) of Lemma 4.17, which is derived from the Duhamel estimate (4.71) (Lemma 4.16) and the bounds (4.69)–(4.70) (Lemma 4.15), both proved in Appendix B using 3G inequalities imported from [BM25] and [BJM24]. The α=2 case is handled by unitary equivalence with Bessel operators and the explicit results of [MNS18].","tokens_in":34157,"tokens_out":27269,"duration_ms":281026,"significance":"I found the main theorem credible and the proof strategy sound. The bounds are genuinely sharp (matching upper and lower) and contain no free parameters: the comparison kernel and the ground-state weights are explicit, and the constants depend only on d, ℓ, α, η. The identification of the Hardy operator's form with the generator of the perturbed Bessel semigroup is new and is the technically substantial part of the paper. The η>0 case is self-contained relative to the cited kernel bounds, and the η<0 case, while resting on a chain of estimates deferred to Appendix B, is internally consistent: I checked the scaling argument that yields p^{(1,D)}_t=O(t²) for fixed r≠s, so the stress-test concern about (4.71) does not land as a detected error. The paper is also admirably explicit about what is imported: Theorem 3.7 is quoted from [BJM24], Remark 1.2 states that the ℓ-dependence of constants is not tracked, and the main theorem nowhere assumes the target Hardy heat-kernel bounds, so there is no circularity. The main weaknesses are statement-accuracy problems in the central bridge theorem and its auxiliaries, and the concentration of verification burden in an unreviewed preprint.","major_comments":[{"comment":"Theorem 4.2(2) asserts the form equality I_{ζ,η}=E_{ζ,η} for all ζ∈(−1/2,∞) and η∈(−α,(2ζ+1−α)/2], but the proof for η<0 is delegated to Theorem 4.18, which is stated only for ζ∈[0,∞). The overclaimed range is not needed for Theorem 1.1, since ζ=(d_ℓ−1)/2≥0 for every d≥1 and ℓ≥0, so the main result survives; nevertheless, as printed, Theorem 4.2(2) claims more than is proved. Since the ingredients used in the η<0 proof (Lemmas 4.15–4.17) are all stated for ζ∈(−1/2,∞), an extension may be routine, but the authors must either supply it or restrict the statement of Theorem 4.2(2) accordingly.","section":"§4.2, Theorem 4.2(2); §4.4, Theorem 4.18"},{"comment":"As stated, Lemma 4.16 involves no η, but p^{(1,D)}_t is defined through (3.13) with q(z)=Ψ_ζ(η)z^{−α}, so the quantity being estimated depends on the coupling while the right-hand side of (4.71) does not. Since Ψ_ζ(η)→−∞ as η↓−α, (4.71) cannot hold with a constant independent of η; the statement needs the hypothesis η∈(−α,(2ζ+1−α)/2] (or the coupling constant), with the implicit constant allowed to depend on it. The application in Lemma 4.17 uses only a fixed η, so this is a statement-accuracy fix rather than a flaw in the main proof.","section":"§4.4.2, Lemma 4.16 and (4.71)"},{"comment":"The critical limit (4.72), which is the hinge of the η<0 identification, is justified in a single sentence. For verifiability, please expand the argument: bound the integral in (4.73) by p^{(1,D)}_t(t,s,r) using p^{(α)}_{ζ,η}≤p^{(α)}_ζ from Theorem 3.6(5), apply Lemma 4.16, use the scaling (4.68), and show from Lemma 4.15 that for fixed r≠s the quantities G₀(1,r/t^{1/α},s/t^{1/α})+G₀(1,s/t^{1/α},r/t^{1/α}) are O(t), that G̃(1,r/t^{1/α},s/t^{1/α}) and G̃(1,s/t^{1/α},r/t^{1/α}) are o(1), and that t/(r∧s)^α is O(t); since p^{(α)}_ζ(t,r,s)=O(t), this yields p^{(1,D)}_t(t,r,s)=O(t²) for ζ≥0 and O(t²)+o(t) for ζ<0, so (4.73)=o(t) as required. The exposition should also mark which term of (4.71) uses [BM25, Theorem 3.1] (the case ζ≥0) versus [BJM24, Lemma 2.5] (the case ζ<0).","section":"§4.4.2, proof of Lemma 4.17"}],"minor_comments":[{"comment":"The hypothesis reads \"η∈(−α, 2ζ+1−α/2]\", which is typeset ambiguously; the intended interval is (−α,(2ζ+1−α)/2], matching the range used in Theorem 3.7.","section":"§3.2, Theorem 3.6"},{"comment":"The symbol p^{(1,D)}_t(r,s) in (4.71) and p^{(1,D)}_t(t,r,s) in the proof of Lemma 4.17 mix two notational conventions; please adopt one convention with the time variable explicit.","section":"§4.4 and Appendix B"},{"comment":"The sentence \"E^{(α)}_{d_ℓ,η}[[u]_{ℓ,m}] = I_{ζ,η}[u], u∈D(I_{ζ,η}), holds for all u∈L²(R₊, r^{d_ℓ−1}dr)\" conflates the maximal domain with L²; please restate it so that the identity is claimed for u in the maximal domain, with both sides allowed to equal +∞.","section":"§4.2, Theorem 4.2(1)"},{"comment":"This theorem is the decisive quantitative input, and it is cited to the arXiv preprint [BJM24] (arXiv:2409.02853); please indicate the publication status of [BJM24] or reproduce the needed statements, since the η<0 part of Theorem 4.2 depends on (4.71) and on (3.18).","section":"§3.2, Theorem 3.7"},{"comment":"The match between (rs)^{−η}p^{(2)}_{ζ−η}(t,r,s), the claimed weight (1∧r/t^{1/2})^{−η}(1∧s/t^{1/2})^{−η}, and the lower bound (3.8b) would be easier to check with one additional line of algebra displaying the cancellation (rs)^{−η}(rs)^{−ζ+η}=(rs)^{−ζ}.","section":"§5, Eq. (1.19)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central quantitative inputs — the sharp bounds of Theorem 3.7 and the two 3G inequalities behind (4.71) — come from [BJM24], which is listed as an arXiv preprint (arXiv:2409.02853). I read the chain (4.73)→(4.71)→(4.69)–(4.70) carefully and found it internally consistent, so my recommendation rests on statement-accuracy and verifiability issues rather than on a detected mathematical error; still, the correctness risk of Theorem 1.1 is concentrated in an unreviewed companion paper by the same authors. It would be reasonable to ask the authors for the current status of [BJM24], or to make the few imported estimates (Lemma 2.5, Lemma 3.14, and Theorem 1.1 of that paper) self-contained in an appendix. The self-citation density (BM24, BM25, BJM24, FMS21, FM25) is high but reflects a coherent research program and is not inappropriate here. If the authors fix the range of Theorem 4.2(2), repair the statement of Lemma 4.16, and expand the proof of Lemma 4.17, I would expect the paper to be acceptable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what you should know: the bounds in Theorem 1.1 are not new in isolation—they have exactly the same shape as the sharp bounds for the Schrödinger-perturbed Bessel kernels p^{(α)}_{ζ,η} proved in the authors' earlier paper [BJM24]. The contribution here is the bridge: Theorem 4.2 identifies the quadratic form of the Hardy operator in a fixed angular momentum channel with the form generated by p^{(α)}_{ζ,η}. Once that identification is in hand, Theorem 1.1 drops out from known results.\n\nThe bridge is built carefully. The η>0 case is clean, and the η<0 case is genuinely hard: it needs the ground-state representation with compensation and delicate estimates for the first Duhamel term (Lemmas 4.15–4.17). I checked the scaling and the main line of the argument in the appendix and found no concrete error. The limit (4.72) follows because G_0 is O(t), making the Duhamel term o(t). This part is a verification burden rather than a detected flaw.\n\nTwo soft spots. First, Lemma 4.16's statement omits the coupling constant Ψ_ζ(η), which is finite in the range used; that is a presentation issue, not a mathematical one. Second, Theorem 4.2(2) claims ζ∈(−1/2,∞), but the η<0 case is proved in Theorem 4.18 only for ζ≥0. That is an unsupported overclaim as written. It does not affect Theorem 1.1, where ζ=(d_ℓ−1)/2≥0, but it should be fixed or qualified in revision. Remark 1.2 honestly states that the constants are not tracked in ℓ; that limits applications but does not invalidate the theorem.\n\nThere is heavy reliance on the authors' own prior work. That is legitimate here—the cited inequalities are the actual tools, not placeholders. I would not call the self-citation a flaw.\n\nWho will get value: analysts working on Hardy operators, heat kernels, and Dirichlet forms, and people using these estimates in the relativistic-atom program. The paper is a serious piece of mathematics and the main theorem is, as far as I can tell, correct. I'd send it to a good referee. My recommendation: engage, accept after the ζ-range overclaim is addressed.","headline":"The genuinely new step is the form identification in Theorem 4.2, not the bounds themselves; the η<0 case is a verification burden, not a detected error, and the paper deserves a serious referee.","tokens_in":34796,"tokens_out":4280,"would_cite":true,"duration_ms":46158,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47D08","60J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves sharp two-sided heat kernel bounds for the fractional Laplacian with Hardy potential in each angular momentum channel: the kernel is uniformly comparable to the free channel kernel times ground-state weights $(1\\wedge…","keywords":["Hardy inequality","heat kernel","fractional Laplacian","angular momentum channel","Bessel kernel","ground state representation","Dirichlet form","sharp estimates"],"falsifier":"Take a concrete channel with $\\alpha=1$, $d=3$, $\\ell=1$ and $\\eta\\in(-1,0)$, and a compactly supported smooth radial function $u$. The theorem predicts $I_{\\zeta,\\eta}[u]=E_{\\zeta,\\eta}[u]$ and $D(I_{\\zeta,\\eta})=D(E_{\\zeta,\\eta})$; finding any $u$ that makes one form finite and the other infinite, or for which the ratio of the two forms differs from 1, would disprove Theorem 4.2(2). A numerical check: evaluate $p^{(\\alpha)}_{\\zeta,\\eta}(t,r,s)$ at very small $t$ and $r\\neq s$, and compare with $\\nu_\\zeta(r,s)t$; the claimed limit (4.72) requires the ratio to tend to 1, so a persistent deviation would falsify the transfer.","tokens_in":33582,"feed_emoji":"⚛️","tokens_out":9583,"duration_ms":101981,"temperature":0.7,"pith_summary":"This paper establishes sharp, two-sided heat kernel bounds for the operator $(-\\Delta)^{\\alpha/2} - \\kappa|x|^{-\\alpha}$, the fractional Laplacian with an inverse-power Hardy potential, acting on functions with fixed angular momentum $\\ell$. The claim is that on each such channel the heat kernel is comparable, uniformly in space and time, to the free channel kernel multiplied by the ground-state weights $(1\\wedge r/t^{1/\\alpha})^{-\\eta}(1\\wedge s/t^{1/\\alpha})^{-\\eta}$, with explicit comparable formulas for the free kernel itself. Since Hardy operators model relativistic atoms and appear as scaling limits of more complicated operators, the bounds give quantitative control of the electron density near the nucleus channel by channel, and refine the known equivalence of Sobolev norms for such operators. The proof works by identifying the heat kernel of the Hardy operator with a Schr\\\"odinger perturbation of a subordinated Bessel heat kernel on the half-line, and proving equality of the corresponding quadratic forms.","feed_headline":"For the Hardy Laplacian, heat kernels factorize with ground-state weights","feed_subtitle":"On each angular momentum channel, the kernel is comparable to the free kernel weighted by $r^{-\\eta}$.","key_machinery":"The machinery has two layers. First, the angular momentum decomposition: writing functions as $u(|x|)|x|^\\ell Y_{\\ell,m}(x/|x|)$ reduces the fractional Laplacian with Hardy potential on $\\mathbb{R}^d$ to an operator $L_{\\kappa,\\ell}$ on the half-line with weighted measure $r^{d_\\ell-1}dr$, where $d_\\ell=d+2\\ell$ is the effective dimension. Second, the form-identification theorem (Theorem 4.2): the quadratic form $I_{\\zeta,\\eta}$ of the ground-state representation with $h(r)=r^{-\\eta}$ is shown to equal the Dirichlet form $E_{\\zeta,\\eta}$ of the Schr\\\"odinger-perturbed subordinated Bessel heat kernel $p^{(\\alpha)}_{\\zeta,\\eta}$; equality of forms implies equality of the self-adjoint operators and hence of heat kernels. The positive-$\\eta$ case is handled by form cores and density of $C_c^\\infty$; the negative-$\\eta$ case uses a ground-state representation and estimates of the first Duhamel term via 3G inequalities. Once the forms coincide, the sharp bounds for $p^{(\\alpha)}_{\\zeta,\\eta}$ from Theorem 3.7 transfer verbatim.","core_discovery":"Let $d_\\ell=d+2\\ell$ and let $\\kappa=\\Phi^{(\\alpha)}_{d_\\ell}(\\eta)$ parameterize the coupling constant through the monotone function defined in (1.11). For $\\alpha\\in(0,2]\\cap(0,d_\\ell)$ and $\\eta\\in(-S,(d_\\ell-\\alpha)/2]$, the heat kernel of $L_{\\kappa,\\ell}$---the radial part of $(-\\Delta)^{\\alpha/2}-\\kappa|x|^{-\\alpha}$ acting on angular momentum $\\ell$---satisfies the uniform comparability (1.18) for $\\alpha<2$ and the $\\asymp$-version (1.19) for $\\alpha=2$. The paper establishes this by proving that the semigroup generated by the ground-state quadratic form $I_{(d_\\ell-1)/2,\\eta}$ is exactly the Schr\\\"odinger perturbation of the subordinated Bessel semigroup $p^{(\\alpha)}_{(d_\\ell-1)/2,\\eta}$, and then invoking the sharp bounds for that kernel. In particular, for $\\eta=0$ the bare channel kernels have the explicit comparabilities (1.20) and (1.21).","pith_inferences":["An implicit consequence is that on each channel the Hardy semigroup is obtained from the free semigroup by a Doob $h$-transform with $h(r)=r^{-\\eta}$, up to a comparability factor; making this precise could give large-deviation or endpoint regularity information beyond the two-sided bounds.","The authors do not track the dependence of the constants on $\\ell$; a testable extension is to prove the same bounds with constants that grow only polynomially or exponentially in $\\ell$, which would permit summation over channels and strengthen the density results.","The $\\eta$ range extends to negative values, so the same machinery should also give sharp bounds for repulsive inverse-power potentials, a case not emphasized in the physical motivation.","The $\\alpha\\to 2$ limit is not uniform in $\\eta$; one could test whether the $\\sim$ bounds for $\\alpha<2$ converge to the $\\asymp$ bounds for $\\alpha=2$ with fixed $d$, $\\ell$, and $\\eta$."],"forward_implications":["On every angular momentum channel, the heat kernel of the Hardy operator is comparable to the unperturbed channel kernel multiplied by $(1\\wedge r/t^{1/\\alpha})^{-\\eta}(1\\wedge s/t^{1/\\alpha})^{-\\eta}$, uniformly in $r,s,t>0$.","For $\\alpha<2$ the free channel kernel has the explicit comparability $t/(|r-s|^{1+\\alpha}(r+s)^{d_\\ell-1}+t^{(1+\\alpha)/\\alpha}(t^{1/\\alpha}+r+s)^{d_\\ell-1})$, giving sharp near- and off-diagonal behavior.","For $\\alpha=2$ the bound is a Gaussian-type comparability $e^{-(r-s)^2/(ct)}/(\\sqrt{t}\\,(rs+t)^{(d_\\ell-1)/2})$, with possibly different constants in the exponential upper and lower bounds.","The kernel $\\exp(-tL_{\\Phi,\\ell})(r,s)$ is jointly continuous in $r,s,t>0$.","The stated applications include upper bounds for relativistic-atom ground-state densities in fixed angular momentum channels, $\\varrho^H_\\ell(r)\\lesssim r^{-2\\eta}$ for small $r$, and refined Sobolev-norm equivalences for Hardy operators."],"supporting_citations":[{"why":"Supplies the sharp two-sided bounds for the Schr\\\"odinger-perturbed subordinated Bessel kernel $p^{(\\alpha)}_{\\zeta,\\eta}$ quoted as Theorem 3.7, which Theorem 1.1 transfers to the Hardy operator.","marker":"[BJM24, Theorem 1.1]"},{"why":"Provides the 3G inequality for $\\zeta\\ge 0$ used in Lemma 4.16 to estimate the first Duhamel term.","marker":"[BM25, Theorem 3.1]"},{"why":"Provides the companion 3G inequality for $\\zeta\\in(-1/2,0)$ used in the same estimate.","marker":"[BJM24, Lemma 2.5]"},{"why":"Gives the ground-state representation, angular momentum decomposition, and L\\'evy kernel identity that define $I_{\\zeta,\\eta}$ and connect $E^{(\\alpha)}_{d_\\ell,\\eta}$ to $E_\\zeta$.","marker":"[BM24]"},{"why":"Cited in Lemma 4.5 to get the form equality $E_{\\zeta,\\eta}=I_{\\zeta,\\eta}$ on the smaller domain $D(E_\\zeta)$.","marker":"[BGJP19, Lemma 5.1]"},{"why":"Supplies the exact $\\alpha=2$ heat-kernel identity $\\exp(-tL_{\\Phi,\\ell})(r,s)=(rs)^{-\\eta}\\exp(-tL_{(d_\\ell-1)/2-\\eta})(r,s)$ used for (1.19).","marker":"[MNS18, Theorem 4.12, Proposition 4.14]"},{"why":"Sharp Hardy inequalities in each angular momentum channel justify non-negativity of $L_{\\kappa,\\ell}$ precisely up to $\\kappa_c^{(\\alpha)}(d_\\ell)$.","marker":"[Yaf99]"}],"fun_headline_variants":["Heat kernels for Hardy Laplacian factorize into free kernel times r^{-η}","Channel-wise heat kernel bounds: comparable to free kernel times r^{-η}","Sharp heat kernel comparability for Hardy-perturbed fractional Laplacian","Hardy Laplacian heat kernel: exact factorization with ground-state weight"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the quadratic form of the Hardy operator in a channel and the form generated by the Schr\\\"odinger-perturbed Bessel semigroup are equal, with the same domain; the proof for negative coupling relies on integral estimates from the authors' previous papers, and if that equality fails the heat kernel bounds do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Heat kernels for Hardy Laplacian factorize into free kernel times r^{-η}","Channel-wise heat kernel bounds: comparable to free kernel times r^{-η}","Sharp heat kernel comparability for Hardy-perturbed fractional Laplacian","Hardy Laplacian heat kernel: exact factorization with ground-state weight"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000651,"raw_usage":{"total_tokens":2944,"prompt_tokens":863,"completion_tokens":2081,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":1998}},"tokens_in":479,"tokens_out":2081,"duration_ms":18855,"temperature":1.0,"reasoning_tokens":1998,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:19:32.676675+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete channel with $\\alpha=1$, $d=3$, $\\ell=1$ and $\\eta\\in(-1,0)$, and a compactly supported smooth radial function $u$. The theorem predicts $I_{\\zeta,\\eta}[u]=E_{\\zeta,\\eta}[u]$ and $D(I_{\\zeta,\\eta})=D(E_{\\zeta,\\eta})$; finding any $u$ that makes one form finite and the other infinite, or for which the ratio of the two forms differs from 1, would disprove Theorem 4.2(2). A numerical check: evaluate $p^{(\\alpha)}_{\\zeta,\\eta}(t,r,s)$ at very small $t$ and $r\\neq s$, and compare with $\\nu_\\zeta(r,s)t$; the claimed limit (4.72) requires the ratio to tend to 1, so a persistent deviation would falsify the transfer.","supporting_citations":[],"review_version":1}