{"id":"1c78e6d4-6999-4b7c-8e3f-71167a715711","arxiv_id":"2607.24169","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Supersolution construction via potentials with integrable Laplacian yields null-critical Hardy weights on general (possibly non-locally finite) graphs, including fractional Laplacians with the continuum constant on Z^d.","lead":"The paper gives a new criterion that produces optimal Hardy weights on general graphs from superharmonic potentials, without needing local finiteness. This yields optimal Hardy weights for fractional Laplacians on arbitrary graphs and recovers the expected continuous constant and asymptotics on the integer lattice.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The paper internally proves criticality, but the decisive implication from unequal boundary sums to null-criticality—and hence optimality near infinity—is imported from two propositions in a companion preprint.","rationale":"The reader identified the right general location—the null-criticality criterion in Theorem 8—but characterized the issue mainly as lack of sharpness. That does not by itself weaken the stated implication, especially because the authors explicitly present a sufficient rather than necessary criterion and give an example where the hypothesis fails. The stronger correctness risk is the outsourced proof of the decisive positive-criticality consequences. The internal argument for criticality is detailed and appears coherent, and the fractional-laplacian asymptotics are consistent with the stated Riesz-kernel expansion. I would therefore not reject the paper, but I would make acceptance of the null-critical/optimality conclusions conditional on checking the cited companion propositions in exactly the generality used here.","tokens_in":29227,"tokens_out":742,"duration_ms":315529,"concrete_test":"Independently re-derive [HKP26c, Propositions 21 and 22] under precisely the hypotheses of Theorem 8: a general connected graph with killing term, no local-finiteness or bounded-oscillation assumption, Lu∈ℓ¹(X,m), and u∈D₀∪P. In particular, prove that positive criticality of w implies ∑mLu=∑cu (and u^{1/2}∈D₀). If an additional hypothesis is needed, or a positive-critical example with unequal sums exists, Theorem 8 must be restricted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 8 is the abstract engine behind Theorems 1–3. Lemmas 7–14 and Proposition 14 in §4 give a detailed internal proof of the estimate Q_v(λ_T(u)) ≤ 4 log(T)||Lu||₁ and of criticality. The final paragraph, however, does not prove the decisive implication needed for null-criticality. It cites [HKP26c, Propositions 21 and 22] for the assertions that positive criticality forces u^{1/2} ∈ D_0 and ∑_X mLu = ∑_X cu. Only then does the assumed inequality of these sums imply that w cannot be positive critical. Theorem 1 additionally relies on [HKPS26, Theorem 1] to equate G(c/m)<1 with that sum condition for every superharmonic potential. If either companion result has narrower hypotheses—such as local finiteness, Lu≥0, transience, or restrictions on D_0∪P—then the present general null-criticality conclusion would need corresponding restrictions. This is more load-bearing than the criterion’s non-necessity: a sufficient condition need not be sharp, and the N₀ example does not contradict the theorem. I see no indication that the cited propositions are false, but their exact hypothesis match is not demonstrated here.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper develops a new criterion for optimal (null-critical) Hardy weights on general graphs via the supersolution construction w = L(u^{1/2})/u^{1/2}, replacing the properness/bounded-oscillation hypotheses of [KPP18b] by the integrability condition Lu ∈ ℓ¹(X,m) together with u ∈ D₀ ∪ P. The abstract engine is Theorem 8 (criticality, and null-criticality under ∑ mLu ≠ ∑ cu), proved internally via a truncation family λ_T, the ground state transform, and the estimate Q_v(λ_T(u)) ≤ 4 log T ‖Lu‖₁ (Lemmas 7–14, Proposition 14). Theorem 1 and Theorem 2 specialize to superharmonic potentials and to the Green function; Theorem 3 applies this to the fractional Laplacian L_σ on arbitrary (non-locally-finite) stochastically complete graphs, where the Green function is the Riesz kernel; Theorem 4 shows that on Z^d the resulting weight has leading term c_{d,σ}|x|^{-2σ} with the constant expected from the continuum. Section 6 extends the result to subcritical Schrödinger operators via the ground state transform (Theorem 19).","tokens_in":20487,"tokens_out":4260,"duration_ms":36479,"significance":"If the results hold, this is a solid advance in discrete criticality theory. It removes the local-finiteness restriction of earlier supersolution constructions and thereby yields the first optimal Hardy weights for fractional Laplacians on general graphs, with only standard structural hypotheses (transience, stochastic completeness). The criterion is essentially parameter-free: once the graph and the potential u are fixed, w is defined from L and u, not fitted, and the sufficient conditions (Lu ∈ ℓ¹, ∑ mLu ≠ ∑ cu) are checkable and are characterized in terms of stochastic completeness at infinity. On Z^d the weight matches the expected continuum constant c_{d,σ} with controlled error (Theorem 4), a falsifiable asymptotic statement. The internal proof of Theorem 8 is self-contained and carefully written, with explicit citation of the Green formula and Fatou/dominated-convergence passages. The authors are also candid about the non-sharpness of the criterion (remark after Theorem 1, the N₀ example after Theorem 2), which strengthens rather than weakens the paper.","major_comments":[{"comment":"The final step of the proof of Theorem 8 (last paragraph of §4) imports the decisive implication for null-criticality from the companion preprint [HKP26c, Propositions 21 and 22]: that positive criticality of w forces u^{1/2} ∈ D₀ and ∑_X mLu = ∑_X cu. The hypotheses of those propositions are not restated here, and there is a visible setting mismatch to check: Theorem 8 is formulated for Hardy *type* weights w : X → R that may take negative values, whereas [HKP26c] is framed around the fractional Laplacian and (in the cited criticality theory) non-negative weights. If Propositions 21–22 assume w ≥ 0, or impose restrictions on the class D₀ ∪ P or on m, then Theorem 8 as stated would need corresponding restrictions. Since null-criticality is the central conclusion of the paper, I ask the authors to quote the two propositions verbatim (or in a lemma) and verify explicitly that their hypothe","section":"§4, proof of Theorem 8"},{"comment":"The deduction of Theorem 1 from Theorem 8 rests on [HKPS26, Theorem 1]: for every superharmonic u ∈ P, G(c/m) < 1 if and only if ∑ mLu ≠ ∑ cu. Theorem 2 similarly routes through [HKPS26, Theorem 1/Theorem 26]. HKPS26 is a concurrent preprint by (a superset of) the same authors, and the equivalence is stated here only by citation. Given that Theorems 1–2 are the headline results, the manuscript should state the precise hypotheses of the cited theorems (e.g., any measure-finiteness, local-finiteness, or regularity assumptions) and confirm they apply to every transient connected graph over (X,m) and every superharmonic u ∈ P. This is related to Major Comment 1 and can presumably be handled in the same added remark.","section":"§4, proofs of Theorem 1 and Theorem 2"}],"minor_comments":[{"comment":"The citation reads '[KL W21, Theorem 6.26x]' — the trailing 'x' appears to be a stray character (presumably a part label such as 6.26 (a)/(b)).","section":"§4, proof of Corollary 9"},{"comment":"'Propostion 6' is misspelled (should be 'Proposition 6').","section":"§4, proof of Lemma 13"},{"comment":"Theorem 15(b) proves transience of b_σ for all σ ∈ (0,1] under a Sobolev inequality of dimension d > 2, but Theorem 3 states the conclusion only for σ ∈ (0,1). Presumably this is because L_σ is defined only for σ ∈ (0,1) in this paper, but the discrepancy should be flagged in one phrase to avoid confusion.","section":"§2 Theorem 3 vs. §5 Theorem 15"},{"comment":"Theorem 8 is stated for a 'connected graph' with no transience assumption, while the hypothesis u ∈ P only makes sense when the Green function exists (transient case), and Lemma 7(b) invokes the Green operator. For recurrent graphs the potential class is empty and the statement degenerates (e.g., constant u gives w = c). A one-line comment on the standing assumptions under which Theorem 8 is non-vacuous would help the reader.","section":"§4, Theorem 8"},{"comment":"In the first display of the proof, the left-hand side m( w̃−w)(vφ_n)^2(x) has x free while the right-hand side sums over X; it should be stated explicitly that the inequality holds for every fixed x ∈ X (using m > 0), which is what the subsequent T → ∞ argument uses.","section":"§4, proof of Proposition 14"},{"comment":"For σ = 1, d ≥ 3 the error term q = 3 is attributed to [KPP18b, Theorem 7.2]; please confirm that this reference indeed yields O(|x|^{-3}) (rather than only the leading term), since the announced improvement to q = 4 is deferred to [HKP26a].","section":"§2, Theorem 4"},{"comment":"The definitions of λ^±_T use the indicator 1_{[0,t]}(s) with s ranging over [1/T,1] resp. [1,T]; for t ≤ 0 this is empty and the notation is slightly compressed. A half-sentence clarifying λ^±_T(t) = 0 for t ≤ 0 (used later, e.g., λ_T(u(x)) for u(x) < 1/T in Proposition 14) would smooth the reading.","section":"§4, Lemma 10"},{"comment":"The citation pattern leans heavily on concurrent preprints of the authors ([HKP26a/b/c], [HKPS26], [Hak25]). This is natural for a research program, but for archival value the key imported statements (Green formula for potentials, Propositions 21–22 of HKP26c, Theorem 1 of HKPS26, the Riesz-kernel asymptotics of HKP26b) should be restated with hypotheses in the final version so the paper remains readable independently of the companions.","section":"References / throughout"}],"recommendation":"minor_revision","confidential_remarks":"The two load-bearing citations (HKP26c Props. 21–22; HKPS26 Thm. 1) are to concurrent preprints by the same authors. I have no reason to doubt those results, and the hypothesis match looks plausible given the shared setting, but it is not verifiable from the present manuscript alone; the editor may wish to ask the authors to include the precise statements, which should settle the point quickly. Subject to that, the internal arguments are clean and the paper is a good fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real advance is a usable supersolution criterion that no longer needs local finiteness, properness, or bounded oscillation. For any transient connected graph, a strictly positive superharmonic potential u with Lu in ℓ¹ gives a critical Hardy-type weight w = L(u^{1/2})/u^{1/2}; the Green-function case is then automatic when there is no killing, and the fractional Laplacian inherits optimal weights on general graphs. On Z^d they recover the expected leading constant c_{d,σ}|x|^{-2σ}. That is exactly the obstruction that blocked earlier discrete supersolution work from reaching non-locally finite operators.\n\nWhat they do well is internal and careful. Lemma 7 (Lipschitz images of potentials), the λ_T cut-offs, the Q_v estimates, and the reduction of Theorems 1–2 to the Hardy-type Theorem 8 are written with explicit Green formulae and Fatou passages; the ground-state transform extension in §6 is clean. The Z^d asymptotics are short and honest: they compare to the earlier weight from HKP26b and control the difference via the Riesz-kernel expansion. Self-citation of the concurrent notes is heavy but mostly for background characterizations (stochastic completeness, Riesz kernels), not for the new criterion itself.\n\nThe soft spot is real but limited. Criticality is proved in-house (Prop. 14). Null-criticality, and therefore “optimal near infinity,” is imported from Propositions 21–22 of HKP26c (and the G(c/m) equivalence from HKPS26). The paper flags that Lu ∈ ℓ¹ and u ∈ P are sufficient but not necessary, and gives the N_0 counter-example when G(c/m)=1. I see no sign the cited propositions are false, yet a referee will want the hypothesis match checked so the generality claim does not silently inherit extra assumptions. That is a documentation issue, not a crack in the argument.\n\nThis is for people who work on discrete Hardy inequalities, fractional graph Laplacians, or criticality theory. It deserves a serious referee. I would cite the criterion and the fractional application, and I would bring it to reading group.","headline":"Clean sufficient criterion that drops local finiteness and gives optimal fractional Hardy weights on general graphs, with continuum asymptotics on Z^d; null-criticality step leans on companion preprints.","tokens_in":21446,"tokens_out":559,"would_cite":true,"duration_ms":11173,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R02","31C20","47B39","39A12"],"pacs":[],"model":"grok-4.5","headline":"A new supersolution criterion produces optimal Hardy weights on general graphs, including fractional Laplacians, matching the continuous constant on the lattice.","keywords":["Hardy inequality","supersolution construction","null-critical weights","fractional Laplacian","Green function","graphs","Riesz kernel","stochastic completeness"],"falsifier":"Exhibit a transient graph with killing for which the Green function satisfies G(c/m) = 1, compute w = L(G_o^{1/2})/G_o^{1/2}, and check whether the ground state lies in L^2(w m); if it does, null-criticality fails exactly as the criterion predicts.","tokens_in":21389,"feed_emoji":"📐","tokens_out":900,"duration_ms":19362,"temperature":0.7,"pith_summary":"Hardy inequalities give quantitative control on functions by comparing energy to a weighted L2 norm. This paper supplies a new, weaker criterion that turns a positive superharmonic potential into an optimal Hardy weight via the supersolution construction. The criterion drops local finiteness and the older oscillation and properness assumptions, so it applies to non-locally finite graphs and in particular to fractional Laplacians. For ordinary Laplacians without killing, the Green function always yields a null-critical (hence optimal) weight. On the integer lattice the resulting weight for the fractional Laplacian has the same leading constant and decay as the continuum theory predicts.","feed_headline":"Optimal Hardy weights now work on general graphs","feed_subtitle":"A weaker supersolution test covers fractional Laplacians and recovers the continuum constant on the lattice","key_machinery":"The supersolution construction: form w = L(u^{1/2})/u^{1/2} from a positive superharmonic potential u. The main technical lemma controls the energy of truncated logarithms of u, producing a null sequence that proves criticality and, under the Green identity, null-criticality.","core_discovery":"On any transient connected graph, every strictly positive superharmonic potential u with Lu integrable produces the critical Hardy-type weight w = L(u^{1/2})/u^{1/2}; when an extra Green-potential identity holds (equivalently G(c/m) < 1), the weight is null-critical and therefore optimal near infinity. Specializing to the Green function and to the Riesz kernel of the fractional Laplacian yields optimal weights on general graphs, and on Z^d the weight asymptotics match the continuous constant c_{d,σ}|x|^{-2σ}.","pith_inferences":["The removal of local-finiteness barriers suggests the same criterion could produce optimal weights for other non-local discrete operators whose Green kernels are known only through spectral calculus.","When the lattice error term can be sharpened (as the paper notes is possible for σ = 1), the discrete–continuous comparison becomes quantitative enough for sharp spectral-gap or eigenvalue bounds.","Graphs that are stochastically complete at infinity yet retain residual heat at infinity remain the natural test bed for whether null-criticality can survive mild killing."],"forward_implications":["Every transient graph without killing admits an optimal Hardy weight built directly from its Green function.","Fractional Laplacians on general stochastically complete graphs that satisfy a Nash or Sobolev inequality receive explicit null-critical Hardy weights from the Riesz kernel.","On Z^d the fractional Hardy weight has leading term identical to the continuum constant c_{d,σ}|x|^{-2σ}.","The same construction extends, via ground-state transform, to positive Schrödinger operators on graphs."],"fun_headline_variants":["Supersolution test yields optimal Hardy weights on any graph","Optimal Hardy weights for fractional Laplacians on general graphs","Green's function gives optimal Hardy weight without local finiteness","Fractional Laplacian Hardy weights match continuum on the lattice","Critical Hardy weights from superharmonic potentials on graphs"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The weight is optimal only when the superharmonic function is an integrable potential and a Green-potential identity fails to hold; with a killing term that identity can hold and optimality is lost.","fun_headline_variants_meta":{"raw":{"variants":["Supersolution test yields optimal Hardy weights on any graph","Optimal Hardy weights for fractional Laplacians on general graphs","Green's function gives optimal Hardy weight without local finiteness","Fractional Laplacian Hardy weights match continuum on the lattice","Critical Hardy weights from superharmonic potentials on graphs"]},"model":"grok-4.5","effort":"low","cost_usd":0.004378,"raw_usage":{"total_tokens":1220,"prompt_tokens":681,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":43784000,"prompt_tokens_details":{"text_tokens":681,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":460,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":681,"tokens_out":79,"duration_ms":8051,"temperature":1.0,"reasoning_tokens":460,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T22:00:55.308298+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a transient graph with killing for which the Green function satisfies G(c/m) = 1, compute w = L(G_o^{1/2})/G_o^{1/2}, and check whether the ground state lies in L^2(w m); if it does, null-criticality fails exactly as the criterion predicts.","supporting_citations":[],"review_version":1}