{"id":"26337320-014f-4bc3-8e75-fdf3215be4bb","arxiv_id":"2607.26830","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Piecewise-linear FEM approximates the fractional Hardy constant at the sharp rate λ_{N,s,h}−λ_{N,s}∼1/|log h|² on bounded convex smooth domains.","lead":"The paper proves that the piecewise-linear Galerkin approximation of the fractional Hardy constant converges to the continuous constant at the sharp rate 1/|log h|². This extends the known classical Hardy (s=1) rate to the fractional setting and shows the same logarithmic bottleneck persists.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's strongest claim (Thm 1.1) is the two-sided asymptotic \\lambda_{N,s,h}(\\Omega)-\\lambda_{N,s}\\sim1/|log h|^2. The lower bound is indeed imported from Tzirakis, but once that continuous logarithmic improvement is granted the discrete argument is elementary and local (cut-off at scale h, inverse inequality on elements inside B_{2h}, Lemmas A.1–A.2). The upper bound is proved from scratch by constructing a cut-off competitor u_\\epsilon, applying the ground-state representation, controlling the three pieces J_1–J_3, and transferring the estimate to the FE space via standard interpolation and Gagliardo–Nirenberg; the case distinctions are lengthy but routine. No step requires an unstated restriction on s or N beyond those already listed, and the mesh assumptions (quasi-uniform, regular, convex smooth domain) are the usual ones that make the inverse and interpolation estimates hold. Consequently the reader's ACCEPT / low-correctness-risk assessment does not need adjustment. The suggested concrete test simply reconfirms the only external analytic input; a positive outcome leaves the verdict untouched.","tokens_in":22148,"tokens_out":573,"duration_ms":9988,"concrete_test":"Independently verify that the constant C_{N,s} in Tzirakis [16, Thm 5] remains strictly positive for every s\\in(0,min{1,N/2}) and every bounded domain containing the origin (e.g., by checking the proof for the endpoint s\\to(N/2)^- or by a direct numerical check of the deficit integral for the pseudo-minimizer truncated at scale h); if C_{N,s}=0 for some admissible s the discrete lower bound would collapse, otherwise the claim is unaffected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption call (black-box use of Tzirakis [16, Thm 5] for the lower bound) is accurate but not load-bearing against the paper's claim: the citation is standard, the remainder is used only qualitatively to extract the |log h|^{-2} rate after elementary cut-offs and inverse estimates (Lemmas A.1–A.2), and the matching upper bound is self-contained via the Frank–Lieb–Seiringer ground-state identity plus an explicit competitor. No internal inconsistency, hidden regime failure, or gap in the discrete analysis appears on a careful read of §§3–4 and the appendix. The central asymptotic therefore stands on the same footing as the classical (s=1) results it extends.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that the discrete fractional Hardy constant obtained by piecewise-linear Galerkin approximation on a quasi-uniform regular mesh of a bounded convex smooth domain Ω containing the origin satisfies λ_{N,s,h}(Ω)−λ_{N,s}∼1/|log h|² for s∈(0,min{1,N/2}) and all sufficiently small h. The lower bound is deduced from Tzirakis’s logarithmic improvement of the fractional Hardy inequality on domains, combined with elementary cut-offs near the origin, inverse estimates, and two Hardy-type lemmas in the appendix. The matching upper bound is obtained by constructing an explicit near-minimizer (a cut-off logarithmic perturbation of the pseudo-minimizer |x|^{-(N−2s)/2}), applying the Frank–Lieb–Seiringer ground-state representation, and controlling the piecewise-linear interpolant error via Gagliardo–Nirenberg and standard FEM estimates.","tokens_in":22272,"tokens_out":904,"duration_ms":27662,"significance":"The result cleanly extends the classical (s=1) Galerkin approximation theory for the Hardy constant developed by Ignat–Zuazua and Della Pietra et al. to the fractional setting, and shows that the same logarithmic rate persists. The upper-bound construction is self-contained and quantitative; the lower bound rests on a standard continuous deficit estimate used only qualitatively. The work fits a coherent program (Sobolev, fractional Sobolev, Hardy, fractional Hardy) and supplies a complete asymptotic rather than a one-sided bound. The detailed competitor analysis and the appendix lemmas are reusable technical contributions.","major_comments":[],"minor_comments":[{"comment":"The reduction of the upper bound from a general convex smooth Ω to the unit ball is stated in one sentence (“by comparison”). A short remark clarifying that λ_{N,s} is domain-independent and that the competitor can be transplanted (or that the discrete constant on a subdomain is at least as large) would make the argument self-contained for readers who have not seen the classical papers.","section":"Section 4, opening paragraph"},{"comment":"The free parameters α≥1, γ∈(0,1) and μ∈(0,1/2) are introduced without a brief summary of the admissible range that makes all estimates close simultaneously. A single sentence collecting the constraints (e.g. after Lemma 4.1 or before the choice ε=h^{γ/2}) would help.","section":"Section 4"},{"comment":"In (3.3)–(3.5) the passage from the weighted L² mass on B_h to the unweighted mass on B_{2h} via Lemmas A.1–A.2 and the inverse estimate is correct but dense. Signposting that the only role of the inverse estimate is to absorb the gradient term would improve readability.","section":"Section 3"},{"comment":"Several references appear as arXiv preprints with 2025–2026 identifiers (including the authors’ own related works). If any have since been accepted or published, the bibliographic data should be updated at proof stage.","section":"References"},{"comment":"Typographical consistency: the manuscript mixes “log” and “|log|” and occasionally writes log²(e/|x|) without parentheses; a uniform convention would be preferable. Also, the constant D=sup_{x∈Ω}|x| is introduced in Prop. 3.1 and immediately specialized to D≤1; stating the normalization 0∈Ω⊂B once at the beginning of Section 3 would avoid repetition.","section":"Sections 3–4"}],"recommendation":"accept","confidential_remarks":"The manuscript is technically solid and belongs in a strong numerical-analysis or applied-analysis journal. The dependence of the lower bound on Tzirakis [16] is standard and not a defect. No novelty or citation concerns; the self-citations to the classical-Hardy and fractional-Sobolev companions are appropriate and clearly distinguished from the new fractional-Hardy analysis."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: they get the sharp discrete rate λ_{N,s,h}−λ_{N,s}∼1/|log h|² for the fractional Hardy constant (p=2, s small) under piecewise-linear Galerkin on quasi-uniform meshes in a smooth convex domain containing the origin. Same rate as the classical s=1 papers they extend.\n\nWhat is actually new is the fractional statement itself. The strategy is not new—it is the Ignat–Zuazua / Della Pietra playbook (log improvement + ground-state identity + cut-off competitor + standard interpolation)—but carrying it through for the nonlocal seminorm is real work. The lower bound is short and clean once you grant Tzirakis’s continuous log improvement; the upper bound is long (lots of case splits on the double integral for the competitor) but structured, and the appendix lemmas on interpolation error and elementary Hardy cut-offs look correct. They do not hide the free parameters (α, γ, μ); they just choose them in ranges that close the estimates.\n\nSoft spots, in proportion: the lower bound is a black-box citation of Tzirakis [16, Thm 5]. That is standard and not a hidden flaw—the matching upper bound is self-contained via Frank–Lieb–Seiringer plus an explicit u_ε—so the ∼ relation stands on the same footing as the classical results. Novelty and significance are moderate: expected next case in a narrow program, not a reorganization of the field. Section 5 on L^p and Bregman variants is honest about what is missing. Citation pattern is appropriate; self-cites are to the direct predecessors.\n\nThis is for people already working on FEM approximation of sharp constants or fractional Hardy. A serious referee in math.NA or nonlocal analysis should see it. I would send it to peer review without hesitation; it is careful, reproducible on the page, and free of load-bearing gaps. Engage if that line is on your desk; otherwise file as the fractional counterpart you expected.","headline":"Solid, expected fractional extension of the classical Hardy FEM-rate program; the ~1/|log h|^2 asymptotic is real and carefully proved.","tokens_in":22971,"tokens_out":518,"would_cite":false,"duration_ms":17525,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N30","46E35"],"pacs":[],"model":"grok-4.5","headline":"Piecewise-linear Galerkin schemes recover the fractional Hardy constant at the same slow logarithmic rate as the classical case.","keywords":["Fractional Hardy Inequality","Galerkin approximation","Finite Element Method","Logarithmic improvement","Discrete optimal constant","Ground-state representation"],"falsifier":"Compute the discrete fractional Hardy constant on the unit ball for a sequence of successively refined quasi-uniform meshes and check whether the difference from the known continuous constant decays like 1/|log h|²; any other asymptotic would refute the theorem.","tokens_in":22976,"feed_emoji":"📐","tokens_out":837,"duration_ms":13077,"temperature":0.7,"pith_summary":"The paper asks how well a standard finite-element method can capture the sharp constant in the fractional Hardy inequality. On any bounded convex smooth domain that contains the origin, and for piecewise-linear elements on a quasi-uniform mesh of size h, the discrete constant lies above the true constant by a quantity that is asymptotically comparable to 1 over the square of the logarithm of h. The same rate was already known for the ordinary (local) Hardy inequality; the authors show that the non-local fractional version behaves identically. A sympathetic reader cares because the continuous constant is never attained, so the rate at which a numerical scheme approaches it is a concrete probe of how the deficit concentrates near the origin. The result also supplies a benchmark against which more exotic approximation spaces can later be tested.","feed_headline":"Fractional Hardy constant converges like 1 over log-squared h","feed_subtitle":"Piecewise-linear finite elements recover the sharp constant at the same slow rate known for the classical Hardy inequality.","key_machinery":"A logarithmic improvement of the fractional Hardy inequality on bounded domains (used for the matching lower bound) together with an explicit cut-off competitor built from the pseudo-minimizer |x|^{-(N-2s)/2} and a ground-state representation of the deficit (used for the upper bound).","core_discovery":"For every dimension N ≥ 1 and every fractional order s in (0, min{1, N/2}), the Galerkin constant λ_{N,s,h}(Ω) computed with continuous piecewise-linear elements on a regular quasi-uniform triangulation of a bounded convex smooth domain Ω containing the origin satisfies λ_{N,s,h}(Ω) − λ_{N,s} ∼ 1/|log h|² as h → 0.","pith_inferences":["The logarithmic rate is likely intrinsic to the fractional Hardy deficit itself rather than an artifact of piecewise-linear elements, so nonlinear approximation classes may still be limited by the same barrier.","Matching upper and lower bounds for the L^p fractional Hardy deficit would immediately yield analogous discrete rates once a logarithmic improvement on domains is available.","The same cut-off competitor technique should transfer, with only technical changes, to other non-local inequalities whose extremals concentrate at a point."],"forward_implications":["Standard linear finite-element codes cannot beat a logarithmic rate when approximating the fractional Hardy constant.","The same rate already known for the local Hardy inequality persists under non-local fractional diffusion.","Any future scheme that claims a faster rate must exploit structure beyond piecewise-linear approximation on quasi-uniform meshes.","The explicit near-minimizing sequence constructed in the paper can be reused to test other discrete spaces."],"fun_headline_variants":["Galerkin fractional Hardy gap shrinks as 1/log² h","P1 elements hit fractional Hardy constant at 1/|log h|²","Discrete fractional Hardy constant lags by 1/log-squared h","Sharp 1/|log h|² rate for FEM fractional Hardy constants","Piecewise-linear Galerkin matches known slow Hardy rate"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The lower bound stands or falls with a continuous logarithmic-improvement inequality that the paper imports as a black box; if that remainder is weaker than claimed, the discrete lower bound of order 1/|log h|² disappears.","fun_headline_variants_meta":{"raw":{"variants":["Galerkin fractional Hardy gap shrinks as 1/log² h","P1 elements hit fractional Hardy constant at 1/|log h|²","Discrete fractional Hardy constant lags by 1/log-squared h","Sharp 1/|log h|² rate for FEM fractional Hardy constants","Piecewise-linear Galerkin matches known slow Hardy rate"]},"model":"grok-4.5","effort":"low","cost_usd":0.002875,"raw_usage":{"total_tokens":979,"prompt_tokens":641,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":28748000,"prompt_tokens_details":{"text_tokens":641,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":259,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":641,"tokens_out":79,"duration_ms":5151,"temperature":1.0,"reasoning_tokens":259,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T19:47:48.617142+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the discrete fractional Hardy constant on the unit ball for a sequence of successively refined quasi-uniform meshes and check whether the difference from the known continuous constant decays like 1/|log h|²; any other asymptotic would refute the theorem.","supporting_citations":[],"review_version":1}