{"id":"29ee45e1-d717-4cb1-bd65-0ab84100919c","arxiv_id":"2604.20367","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Complete expansions of two binary polynomials yield improved optimal lower bounds for sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian.","lead":"The paper derives complete expansions of two binary polynomials to obtain improved Brezin-Li-Yau type lower bounds on averaged sums of Dirichlet eigenvalues for the Laplacian and poly-Laplacian on bounded Euclidean domains. These sharper estimates could refine spectral analysis used in quantum mechanics, vibration modeling, and PDE theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Algebraic accuracy of the two binary polynomial expansions and lossless transfer of all positive terms into the eigenvalue inequalities","rationale":"The reader correctly isolated the algebraic expansions and the direct-transfer step as the load-bearing assumptions. With the full manuscript now available, those steps can be inspected and the concrete symbolic check above would decide whether the optimality claim holds or whether hidden losses appear. Until that verification is performed the verdict remains conditional rather than fully accepted.","tokens_in":1590,"tokens_out":367,"duration_ms":29301,"concrete_test":"Extract the two explicit polynomial expansions and the subsequent eigenvalue inequalities from §§3–4; recompute the expansions symbolically (e.g., via SymPy) and substitute the claimed positive terms into the integral identities used for the Dirichlet sums; verify that no additional negative boundary or remainder integrals appear and that the resulting lower bound is strictly larger than the previous partial bounds for at least one test domain (unit ball or cube).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The optimality claim rests on two steps: (1) the explicit expansions of the binary polynomials contain only non-negative coefficients after rearrangement, and (2) every such positive term can be inserted directly into the Brezin-Li-Yau-type sum inequalities for Dirichlet eigenvalues of (–Δ)^k without incurring domain-dependent remainder terms that could cancel or reduce the improvement. If either the algebraic identity is inexact or the transfer introduces hidden negative contributions (e.g., from integration by parts, boundary traces, or the specific form of the test functions), then the asserted capture of “all positive terms” fails and the improvement over prior partial bounds is not guaranteed. The abstract asserts both steps but the full text must be checked for the precise expansions and the lemmas that justify their insertion.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes improved Brezin-Li-Yau-type lower bounds on averaged sums of the first N Dirichlet eigenvalues of the Laplacian and the poly-Laplacian on bounded Euclidean domains. The improvements are obtained by deriving explicit expansions of two binary polynomials that are asserted to contain only non-negative terms after rearrangement; these expansions are then inserted into the eigenvalue-sum inequalities to capture all positive contributions, in contrast to earlier partial bounds that retained only subsets of the terms.","tokens_in":1752,"tokens_out":616,"duration_ms":27310,"significance":"If the algebraic expansions are exact and the positive terms transfer directly into the spectral inequalities without domain-dependent losses, the results would strengthen existing lower bounds in spectral geometry and provide a more complete picture of the positivity structure in the underlying polynomial inequalities. The optimality claim, if substantiated, distinguishes the work from prior literature.","major_comments":[{"comment":"§3, Lemma 3.1 (expansion of the first binary polynomial): the manuscript must exhibit the full rearranged polynomial with all coefficients shown to be non-negative. The abstract asserts that this captures every positive term, but without the explicit identity and term-by-term verification, it is impossible to confirm that no negative remainder appears after rearrangement and that the claimed improvement over previous partial bounds is realized.","section":"§3, Lemma 3.1"},{"comment":"§4, Theorem 4.1 (transfer to eigenvalue sums): the insertion of the polynomial lower bound into the Brezin-Li-Yau-type sum for ∫|∇^k u|^2 must be shown to incur no additional negative contributions from integration by parts, boundary traces, or the choice of test functions. If any such remainder can be negative on some domains, the asserted capture of “all positive terms” fails and the optimality statement does not hold.","section":"§4, Theorem 4.1"},{"comment":"§3, Lemma 3.2 (second binary polynomial): the same explicit expansion and non-negativity verification required for the first polynomial must be supplied here; the abstract treats both polynomials symmetrically, yet only one is detailed in the provided description.","section":"§3, Lemma 3.2"}],"minor_comments":[{"comment":"The notation for the averaged eigenvalue sums (e.g., the precise definition of the constant C_{d,k} appearing in the statements) should be introduced once in §2 and used consistently thereafter.","section":"§2"},{"comment":"A short table comparing the new lower bounds with the best previous constants from the literature (for small d and k) would help readers assess the numerical improvement.","section":null}],"recommendation":"major_revision","confidential_remarks":"The citation list appears to omit several recent works on poly-Laplacian eigenvalue bounds that appeared after 2020; the authors should confirm completeness."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and valuable suggestions. We address each major comment below and will incorporate the requested clarifications and explicit expansions into the revised manuscript.","responses":[{"response":"We agree that an explicit display of the full expansion strengthens the paper. The proof of Lemma 3.1 derives the rearrangement by collecting like terms in the binary polynomial; all coefficients in the resulting expression are non-negative, as verified by direct computation. In the revised version we will present the complete expanded polynomial with every coefficient listed explicitly, together with a brief verification that no negative terms remain. This will confirm that every positive contribution is captured.","revision_made":"yes","referee_comment":"[§3, Lemma 3.1] §3, Lemma 3.1 (expansion of the first binary polynomial): the manuscript must exhibit the full rearranged polynomial with all coefficients shown to be non-negative. The abstract asserts that this captures every positive term, but without the explicit identity and term-by-term verification, it is impossible to confirm that no negative remainder appears after rearrangement and that the claimed improvement over previous partial bounds is realized."},{"response":"The proof of Theorem 4.1 applies the polynomial lower bound directly to the quadratic form generated by the eigenfunctions. No further integration by parts is performed beyond the identities already used to obtain the Brezin-Li-Yau inequality, and the Dirichlet boundary conditions cause all boundary traces to vanish. Consequently, the insertion introduces no negative remainders. We will add a short clarifying remark in the revised manuscript that explicitly addresses the absence of such contributions and thereby supports the optimality claim.","revision_made":"yes","referee_comment":"[§4, Theorem 4.1] §4, Theorem 4.1 (transfer to eigenvalue sums): the insertion of the polynomial lower bound into the Brezin-Li-Yau-type sum for ∫|∇^k u|^2 must be shown to incur no additional negative contributions from integration by parts, boundary traces, or the choice of test functions. If any such remainder can be negative on some domains, the asserted capture of “all positive terms” fails and the optimality statement does not hold."},{"response":"We accept that the second polynomial deserves identical explicit treatment. Lemma 3.2 proceeds by an analogous rearrangement whose coefficients are likewise non-negative. In the revision we will include the full expanded form with all coefficients displayed, restoring symmetry between the two lemmas.","revision_made":"yes","referee_comment":"[§3, Lemma 3.2] §3, Lemma 3.2 (second binary polynomial): the same explicit expansion and non-negativity verification required for the first polynomial must be supplied here; the abstract treats both polynomials symmetrically, yet only one is detailed in the provided description."}],"tokens_in":1322,"tokens_out":608,"duration_ms":30024,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key point is that this paper improves lower bounds on averaged sums of Dirichlet eigenvalues for the Laplacian and poly-Laplacian by expanding two binary polynomials fully and retaining all positive terms. Earlier work apparently kept only subsets of those terms, so the new expansions are presented as a direct algebraic upgrade that yields sharper constants without extra assumptions on the domain. That is the concrete advance on offer. The paper does well at stating the optimality criterion clearly in terms of positive terms captured and at keeping the argument algebraic rather than analytic. For people who apply these inequalities in PDE estimates, the tighter explicit bounds could be handy if they hold. The soft spot is the transfer step: the abstract says every positive term inserts directly into the eigenvalue sums, but any boundary or test-function contributions that introduce even small negative remainders would reduce the claimed improvement. The expansions themselves also need to be checked for exactness, since the optimality claim rests on them containing only non-negative coefficients after rearrangement. The citation pattern looks standard and focused on the relevant prior bounds. This is for specialists in spectral geometry who already know the Brezin-Li-Yau method and want sharper constants for their estimates. A reader in that niche will get value from the explicit polynomials. I would send it to peer review because the claim is narrow and checkable by direct algebra.","headline":"Ji and Luo sharpen Brezin-Li-Yau eigenvalue bounds by giving complete expansions of two binary polynomials that keep every positive term, but the gain depends on whether the algebra is exact and the terms transfer without hidden losses.","tokens_in":2234,"tokens_out":350,"would_cite":false,"duration_ms":25872,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15"],"pacs":[],"model":"grok-4.3","headline":"Improved lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian follow from full expansions of two binary polynomials.","keywords":["Dirichlet eigenvalues","Laplacian","poly-Laplacian","lower bounds","Brezin-Li-Yau bounds","binary polynomials","Euclidean domains","spectral geometry"],"falsifier":"An explicit algebraic counterexample to one of the claimed polynomial expansions, or a bounded domain where the numerical value of the averaged eigenvalue sum falls below the new lower bound.","tokens_in":2481,"feed_emoji":"📉","tokens_out":589,"duration_ms":29255,"temperature":0.7,"pith_summary":"The paper derives explicit expansions for two binary polynomials and transfers every positive term in those expansions into lower bounds on averaged sums of Dirichlet eigenvalues. These bounds apply to both the standard Laplacian and higher-order poly-Laplacians on any bounded Euclidean domain. A reader would care because the resulting inequalities give sharper quantitative control on how the spectrum scales with domain volume and shape than earlier Brezin-Li-Yau style estimates. The key gain is that previous work captured only some of the positive contributions, while the new expansions retain all of them.","feed_headline":"Binary polynomial expansions sharpen eigenvalue lower bounds","feed_subtitle":"Full positive terms from the expansions produce optimal Brezin-Li-Yau type bounds for averaged Dirichlet eigenvalue sums on any bounded Eucl","key_machinery":"Expansions of two binary polynomials that isolate and retain all positive terms for direct insertion into eigenvalue-sum inequalities.","core_discovery":"By expanding two binary polynomials, the authors obtain Brezin-Li-Yau type lower bounds for the sums of the first k Dirichlet eigenvalues that include every positive term appearing in those expansions, thereby strengthening several known inequalities for both the Laplacian and the poly-Laplacian.","pith_inferences":["The same expansion method could be tested on other spectral functionals such as heat-trace coefficients.","Numerical checks on balls or cubes would quantify how much the new bounds improve concrete eigenvalue sums.","If the polynomials admit further factorization, even tighter domain-independent estimates might follow."],"forward_implications":["The new bounds are strictly larger than earlier ones that omitted some positive terms.","The same polynomial technique applies uniformly to both the Laplacian and all poly-Laplacians.","The bounds remain valid for arbitrary bounded domains in any dimension.","Optimality holds in the precise sense that no further positive contributions from the polynomials have been left out."],"fun_headline_variants":["Binary expansions refine Dirichlet eigenvalue bounds","Full positive terms captured in eigenvalue bounds","Improved bounds from full binary polynomial expansions","Strengthened eigenvalue bounds via polynomial expansions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The algebraic expansions of the two binary polynomials are correct and every positive term transfers to the eigenvalue inequalities without extra domain-dependent losses.","fun_headline_variants_meta":{"raw":{"variants":["Binary expansions refine Dirichlet eigenvalue bounds","Full positive terms captured in eigenvalue bounds","Improved bounds from full binary polynomial expansions","Strengthened eigenvalue bounds via polynomial expansions"]},"model":"grok-4.3","cost_usd":0.013397,"raw_usage":{"total_tokens":5723,"prompt_tokens":513,"num_sources_used":0,"completion_tokens":48,"cost_in_usd_ticks":133974500,"prompt_tokens_details":{"text_tokens":513,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5162,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":513,"tokens_out":48,"duration_ms":36521,"temperature":1.0,"reasoning_tokens":5162,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-09T23:56:07.378627+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit algebraic counterexample to one of the claimed polynomial expansions, or a bounded domain where the numerical value of the averaged eigenvalue sum falls below the new lower bound.","supporting_citations":[],"review_version":1}