{"id":"e160a2fd-a05f-47bc-ba84-d6230567deb9","arxiv_id":"2411.11071","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"On finite subgraphs of the integer lattice, the average of the first k Dirichlet poly-Laplace eigenvalues obeys explicit upper and lower bounds of Weyl form, and 2l-order eigenvalues dominate squares of l-order eigenvalues.","lead":"This paper defines a discrete version of the poly-Laplace operator on graph subgraphs and proves upper and lower bounds for the sums of its first eigenvalues, extending classical membrane and plate eigenvalue estimates to lattices. A second result shows that the 2l-order eigenvalues are always at least the square of the l-order eigenvalues, with strict inequality on integer lattices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest_assumption points at Lemma 3.3's rearrangement step, but I verified that lemma in detail and it is correct: H is nondecreasing on [0,sqrt(6)], the ball of radius R lies inside the cube because R <= sqrt(6) < pi, and the bathtub argument is valid. The positive lower bound is exactly the integral of a positive function. The actual correctness issues the reader cited -- formula (1.4) being false on general non-regular graphs due to noncommuting D and A, and the numerical appendix lacking code/data -- do not affect the central lattice theorems, because Theorem 1.1 and the strict part of Theorem 1.3 are formulated for subgraphs of Z^d, where D = 2d I commutes with A and the expansion (1.4) is valid. The comparison theorem for general graphs does not use (1.4). Thus the central claims hold, and the paper needs minor corrections rather than a change of verdict. I therefore keep the reader's CONDITIONAL as unchanged, while noting that the specific fragility identified in the weakest_assumption is not a real obstruction.","tokens_in":18993,"tokens_out":32818,"duration_ms":241636,"concrete_test":"For random z in [-pi,pi]^d with |z| > sqrt(6), evaluate Phi(z)^l = (sum_i (2-2cos z_i))^l and compare it with 3^l for d = 1,...,5 and l = 1,...,5; if any violation is found, the radial minorant defined in Lemma 3.3 is not pointwise valid, which would break the lower bound. This directly tests the key step that the proof uses to justify the rearrangement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the full proof, I find no load-bearing concern against the central claim. Theorem 1.1(a) follows from Lemma 3.2 with a ball of radius at most pi inside the fundamental cube, and the bound (Phi)^l <= |z|^{2l} is valid. Theorem 1.1(b) rests on Lemma 3.3, and the rearrangement step is sound: the radial minorant H(r) = (r^2 - r^4/12)^l is nondecreasing on [0, sqrt(6)], the condition R <= sqrt(6) < pi keeps the ball inside [-pi,pi]^d, and the bathtub-principle inequality is exactly correct, giving a positive integral M d V_d int_0^R H(r) r^{d-1} dr > 0. Theorem 1.3 is a clean min-max/Cauchy-Schwarz argument, and the strictness proof is valid on Z^d because the eigenfunction has finite support, so Lemma 3.5 applies. The genuine issues flagged by the reader -- formula (1.4) implicitly uses commutativity of D and A, and the appendix lacks code/data -- are real but not load-bearing: the lattice theorems only use Z^d, where D is a scalar multiple of the identity and the expansion is valid, and the appendix is purely illustrative.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a discrete poly-Laplace operator on finite subgraphs of a graph with Dirichlet boundary conditions, defined by iterating the graph Laplacian on the zero extension of a function. For finite subgraphs of the integer lattice Z^d it proves upper and lower bounds for the average of the first k Dirichlet eigenvalues with explicit constants matching the continuous Weyl asymptotics (Theorem 1.1), and a comparison inequality (λ_l_k)^2 ≤ λ_{2l}_k with strict inequality on Z^d (Theorem 1.3). The proofs use the Fourier transform on the lattice, a Kröger-type variational lemma for the upper bound, a Li-Yau-type rearrangement argument for the lower bound, and a min-max/Cauchy-Schwarz argument for the comparison theorem. An appendix gives numerical evidence on path graphs.","tokens_in":19060,"tokens_out":27981,"duration_ms":271467,"significance":"If correct, the results give the first eigenvalue-sum estimates for discrete poly-Laplace operators on lattice subgraphs, extending Li-Yau, Kröger, and Bauer-Lippner to higher order with explicit constants and no fitted parameters. The lower bound is positive for a constant fraction of eigenvalues, and the comparison inequality is a clean universal statement. The proofs are self-contained, and the constants are consistent with Weyl asymptotics. The main theorems should be useful for numerical analysis and spectral graph theory. A caveat is that the general-graph formula (1.4) and the ℓ2 version of Lemma 3.5 are not fully justified as stated, but these issues do not affect the lattice results.","major_comments":[],"minor_comments":[{"comment":"The lemma is stated for f ∈ ℓ2(Z^d), but the proof treats only f ∈ C0(Z^d) by assuming a compactly supported eigenfunction. The same Fourier argument works for L2 functions, so either prove the ℓ2 case or state the lemma for compactly supported functions, which is all that Theorem 1.3 requires.","section":"Lemma 3.5"},{"comment":"The expansion (D-A)^l = ∑ binom(l,m) D^{l-m}(-1)^m A^m is only valid when D and A commute; on a general graph with nonconstant degrees this is false, and the displayed formula for a^l_xy with deg(y) is not correct in that setting. Since all later uses are on Z^d, where D=2dI, the main theorems are unaffected, but the general-graph statement should be qualified.","section":"Introduction, Eq. (1.4)"},{"comment":"The numerical experiment in the appendix reports convergence of the eigenvalue ratio but does not provide the code, the exact data, or the limiting values c_k referenced in Remark 1.4(3); the Data Availability statement that no new data were created should be reconciled with the numerical experiments.","section":"Appendix and Data Availability"},{"comment":"The proof of Theorem 1.5 is only one sentence. The monotonicity gives the existence of the limit, but the identification of the limit with λ_l_k(G) is the substantive part of an exhaustion argument and should be spelled out or referenced.","section":"Theorem 1.5"},{"comment":"The manuscript contains numerous typographical issues in the conversion, including misplaced superscripts, corrupted inequality signs, and inconsistent hyphenation of \"poly-Laplace\"; the final version should be carefully proofread, and the constraints in Theorem 1.1 should use proper \\min spacing.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript's main results are sound and the proof strategies are verifiable. The issues listed in the minor comments are local and do not undermine Theorems 1.1 or 1.3. I recommend minor revision rather than rejection; once the proof of Lemma 3.5 is reconciled with its statement and the general-graph formula (1.4) is qualified, the paper should be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this carefully, and my verdict is mostly positive. The paper delivers what it promises: the first eigenvalue sum estimates for Dirichlet poly-Laplace operators on finite subgraphs of Z^d, with constants matching the continuous Weyl law, plus the comparison inequality (λ^l_k)^2 ≤ λ^{2l}_k, strict on Z^d. I checked the lattice proofs. The Fourier representation in Lemma 2.9, the boundary estimate in Lemma 2.10, the rearrangement lower bound in Lemma 3.3, and the min-max/Cauchy–Schwarz argument for Theorem 1.3 all hold. The constants line up with Weyl asymptotics, no fitted parameters or circular steps. For l=1 it recovers Bauer–Lippner; for l≥2 the operator and estimates are new.\n\nThe main soft spot is formula (1.4). As written it claims to hold for arbitrary finite graphs, but the expansion (D−A)^l = Σ binom(l,m) D^{l−m}(−A)^m requires D and A to commute, which fails in general. On Z^d degree is constant, so D is scalar and the formula is fine; the lattice theorems survive. Still, the general-graph statement needs a regularity assumption or repair. Not load-bearing.\n\nThe second issue is the appendix. The numerical experiment has a figure but no code or data, and the Data Availability statement says no data were created, contradicting the experiment. Minor and easily fixed.\n\nThe lower bound in Theorem 1.1(b) is restricted to k ≤ min{1,(√6/(2π))^d V_d}|Ω|, so it only covers a constant fraction of eigenvalues. That is a real limitation, but it is natural for the bathtub method and does not undermine the result.\n\nWho is this for? Spectral graph theorists and people discretizing plate/polyharmonic problems. The citation pattern is appropriate and the direct debts to Bauer–Lippner, Wang, Li–Yau, Kröger are acknowledged. I would bring it to a reading group and cite it.\n\nRecommendation: send to a serious referee. Ask for a corrected statement of (1.4) and a proper appendix treatment. The core math is in good shape and deserves publication after minor revision.","headline":"Solid, genuinely new extension of Bauer–Lippner and Li–Yau to discrete poly-Laplace operators; main lattice theorems hold, with a formula caveat and a minor appendix flaw.","tokens_in":736,"tokens_out":1688,"would_cite":true,"duration_ms":43099,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","39A12","05C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Discrete poly-Laplace eigenvalues on lattice subgraphs satisfy classical asymptotic sum estimates for every order, with a strict comparison inequality between orders.","keywords":["discrete poly-Laplace operator","Dirichlet eigenvalues","lattice graph","integer lattice","asymptotic eigenvalue estimates","eigenvalue sum bounds","spectral comparison","Fourier transform on the integer lattice"],"falsifier":"For $d=2$, take $F(z)=1$ on a thin spherical shell inside $[-\\pi,\\pi]^2$ with fixed mass $K$, compute $\\int\\Phi(z)^l F(z)\\,dz$ numerically, and compare it with the claimed ball-minimizer value; alternatively, diagonalize the operator on a small box such as $3\\times3$ and test the averaged lower bound for every $k$ in the stated range.","tokens_in":18633,"feed_emoji":"📐","tokens_out":15098,"duration_ms":147070,"temperature":0.7,"pith_summary":"The paper introduces a Dirichlet version of the iterated graph Laplacian on finite subgraphs and proves that, on subgraphs of the integer lattice, the average of the first $k$ eigenvalues follows asymptotic sum rules for every order $l$. The upper bound has the same leading term as the continuous spectral asymptotics, plus an explicit boundary correction; the lower bound stays positive up to a fixed fraction of the number of vertices. A separate comparison theorem shows that the Dirichlet eigenvalue of order $2l$ is always at least the square of the eigenvalue of order $l$, and strictly larger on $\\mathbb{Z}^d$. Together these results give the discrete poly-Laplace model the same asymptotic spectral shape as the continuous clamped-plate problem and provide a target for numerical discretizations.","feed_headline":"Discrete poly-Laplace eigenvalues obey classical sum bounds","feed_subtitle":"Averaged Dirichlet eigenvalues on finite subgraphs of Z^d are pinned between explicit formulas.","key_machinery":"The engine is the Fourier symbol of the lattice Laplacian, $\\Phi(z)=\\sum_{i=1}^d(2-2\\cos z_i)$, which turns $(-1)^l\\Delta^l$ into multiplication by $\\Phi(z)^l$ on the Fourier side. The Fourier transform converts the $\\ell^2$ inner product and the quadratic form into frequency-space integrals $\\int_{[-\\pi,\\pi]^d}\\Phi(z)^l F(z)\\,dz$ with $0\\le F\\le|\\Omega|$ and $\\int F=k(2\\pi)^d$. The upper bound uses the elementary inequality $\\Phi(z)\\le|z|^2$ on a frequency ball; the lower bound uses the pointwise minorant $|z|^2-\\frac{1}{12}|z|^4\\le\\Phi(z)$ and a radial rearrangement lemma asserting that the cheapest way to concentrate mass for this integral is a characteristic function of a ball of radius $\\le\\sqrt{6}$. The comparison result rests on the quadratic-form domination $\\langle(\\Delta^{l,D}_\\Omega)^2 f,f\\rangle\\le\\langle\\Delta^{2l,D}_\\Omega f,f\\rangle$ together with the absence of nonzero $\\ell^2$ eigenfunctions of $(-\\Delta)^l$ on $\\mathbb{Z}^d$.","core_discovery":"The paper's central claim is that for a finite subgraph $\\Omega$ of $\\mathbb{Z}^d$, the Dirichlet poly-Laplace eigenvalues satisfy two-sided sum estimates for every positive integer $l$. Theorem 1.1 gives, for $1\\le k\\le \\min\\{1,V_d/2^d\\}|\\Omega|$, the upper bound $\\frac{1}{k}\\sum_{j=1}^k\\lambda_j^l \\le (2\\pi)^{2l}\\frac{d}{d+2l}\\left(\\frac{k}{V_d|\\Omega|}\\right)^{2l/d} + \\frac{|\\partial^l\\Omega|}{|\\Omega|}$, and for $k\\le\\min\\{1,(\\sqrt{6}/(2\\pi))^d V_d\\}|\\Omega|$ the lower bound $\\lambda_k^l\\ge\\frac{1}{k}\\sum_{j=1}^k\\lambda_j^l \\ge \\sum_{m=0}^l \\binom{l}{m}\\left(-\\frac{1}{12}\\right)^m (2\\pi)^{2(l+m)}\\frac{d}{d+2(l+m)}\\left(\\frac{k}{V_d|\\Omega|}\\right)^{2(l+m)/d}>0$. Theorem 1.3 states that $(\\lambda_k^l)^2\\le\\lambda_k^{2l}$, with strict inequality on $\\mathbb{Z}^d$; the paper also proves the continuous analogue and an exhaustion approximation for infinite graphs. The lower bound is the discrete counterpart of the continuous poly-Laplace lower-bound estimates and reduces to the usual graph-Laplacian bound when $l=1$.","pith_inferences":["Sharper trigonometric minorants for $\\Phi(z)$ than the fourth-order one would extend the admissible range of $k$ and improve the lower-bound constant; this is a natural next step the paper leaves implicit.","The comparison inequality suggests a hierarchy $(\\lambda_k^l)^q\\le\\lambda_k^{ql}$ for integer multiples of $l$; testing $q=3$ numerically would show whether the quadratic-form argument iterates.","The same Fourier-plus-rearrangement scheme should apply to other Cayley graphs with explicit symbols, such as the hexagonal or triangular lattices, with the ball radius replaced by the inradius of the fundamental frequency cell.","The appendix's convergence of eigenvalue ratios on path graphs hints that rescaled discrete Dirichlet poly-Laplace eigenvalues converge to continuous clamped-plate eigenvalues; proving this would make the new bounds a discretization tool for plate problems."],"forward_implications":["For each positive integer $l$, the averaged first-$k$ Dirichlet spectrum of a lattice subgraph is squeezed between explicit powers of $k/(V_d|\\Omega|)$, with the same leading constant as the continuous problem.","The lower bound is genuinely positive up to a constant fraction of $|\\Omega|$; in particular, no Dirichlet eigenvalue can be anomalously small relative to the volume-to-spectrum scaling.","The strict inequality $(\\lambda_k^l)^2<\\lambda_k^{2l}$ shows that the Dirichlet poly-Laplace spectrum on $\\mathbb{Z}^d$ is not obtained by exponentiating the graph-Laplacian spectrum, so separate estimates for each order are necessary.","The exhaustion theorem lets finite-subgraph eigenvalue bounds pass to infinite graphs with bounded degree, giving spectral-bottom bounds for poly-Laplace operators there."],"supporting_citations":[{"why":"supplies the leading asymptotic law for eigenvalue sums that fixes the leading constant","marker":"[34]"},{"why":"supplies the averaging method that Lemma 3.3 adapts to the discrete poly-Laplace symbol","marker":"[24]"},{"why":"supplies the upper-bound technique for sums of Laplacian eigenvalues that the paper follows","marker":"[21]"},{"why":"supplies the graph eigenvalue-sum framework, the eigenvalue lemma, and the Dirichlet setting on lattice subgraphs","marker":"[2]"},{"why":"gives the fractional-Laplacian approach to Dirichlet eigenvalue estimates on lattice subgraphs that the paper adapts","marker":"[32]"},{"why":"states the continuous poly-Laplace lower-bound inequality that Theorem 1.1(b) discretizes","marker":"[23]"}],"fun_headline_variants":["Poly-Laplace eigenvalues on lattice subgraphs obey Li-Yau bounds","Two-sided sum estimates for discrete poly-Laplace eigenvalues","Poly-Laplace Dirichlet sums are pinned between explicit formulas","Eigenvalue square inequality proved for poly-Laplace on subgraphs","Discrete poly-Laplace: sharp upper and lower eigenvalue sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the rearrangement step in Lemma 3.3: among all nonnegative functions $F$ on $[-\\pi,\\pi]^d$ bounded by $M$ with fixed integral $K$, the integral of $\\Phi(z)^lF(z)$ is minimized by setting $F=M$ on a Euclidean ball of radius $(K/(MV_d))^{1/d}\\le\\sqrt{6}$ inside the cube.","fun_headline_variants_meta":{"raw":{"variants":["Poly-Laplace eigenvalues on lattice subgraphs obey Li-Yau bounds","Two-sided sum estimates for discrete poly-Laplace eigenvalues","Poly-Laplace Dirichlet sums are pinned between explicit formulas","Eigenvalue square inequality proved for poly-Laplace on subgraphs","Discrete poly-Laplace: sharp upper and lower eigenvalue sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1451,"prompt_tokens":956,"completion_tokens":495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":420}},"tokens_in":572,"tokens_out":495,"duration_ms":5476,"temperature":1.0,"reasoning_tokens":420,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:58:25.841393+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $d=2$, take $F(z)=1$ on a thin spherical shell inside $[-\\pi,\\pi]^2$ with fixed mass $K$, compute $\\int\\Phi(z)^l F(z)\\,dz$ numerically, and compare it with the claimed ball-minimizer value; alternatively, diagonalize the operator on a small box such as $3\\times3$ and test the averaged lower bound for every $k$ in the stated range.","supporting_citations":[{"cited_title":"Das asymptotische Verteilungsgesetz de r Eigenwerte linearer par- tieller Differentialgleichungen (mit einer Anwendung auf d ie Theorie der Hohlraum- strahlung)","cited_arxiv_id":null,"evidence_quote":"supplies the leading asymptotic law for eigenvalue sums that fixes the leading constant"},{"cited_title":"On the Schr¨ odinger equatio n and the eigenvalue prob- lem","cited_arxiv_id":null,"evidence_quote":"supplies the averaging method that Lemma 3.3 adapts to the discrete poly-Laplace symbol"},{"cited_title":"Estimates for sums of eigenvalues of the Laplacian","cited_arxiv_id":null,"evidence_quote":"supplies the upper-bound technique for sums of Laplacian eigenvalues that the paper follows"},{"cited_title":"Eigenvalue sum estimates for lattice subgraphs","cited_arxiv_id":null,"evidence_quote":"supplies the graph eigenvalue-sum framework, the eigenvalue lemma, and the Dirichlet setting on lattice subgraphs"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states the continuous poly-Laplace lower-bound inequality that Theorem 1.1(b) discretizes"}],"review_version":1}