{"id":"b228fc60-e5e3-4840-b13b-90df43454669","arxiv_id":"2508.04209","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For any simplicial complex, the sum of the k largest upper-Laplacian eigenvalues is at most the sum of the (r+1)k largest r-degrees of (r-1)-faces, with applications to graphs and partite complexes.","lead":"A mathematician proved a sharp new inequality relating the largest Laplacian eigenvalues of any simplicial complex to sums of face degrees. This extends a classical graph result, yields the best known general bound for graphs, and proves a conjecture case for multipartite complexes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.8 is false as printed: it drops the complement in Lemma 2.7, so the proof of Corollary 4.2 and hence the derivation of Theorem 1.8 are invalid until corrected.","rationale":"The main argument for Theorem 1.6 is essentially sound; the undefined ratio in the definition of w(σ) when a face has zero r-degree is a genuine but easily patched gap, exactly as the reader noted. However, the most load-bearing concern I find is in the supporting chain for Theorem 1.8: Lemma 2.8 is false as printed because the complement operation is dropped when applying Lemma 2.7. This is not merely a sign typo in one displayed equation; it is a false lemma with a concrete counterexample, and it is used in the proof of Corollary 4.2 for k>(n−1)/2. The bound itself can still be rescued by restoring the complement and applying the first case to the complement graph, so the correct verdict remains CONDITIONAL rather than REJECT. My concern differs from the reader's weakest assumption, so I mark partial agreement, but the recommended verdict is unchanged: the paper should be accepted only after these corrections are made.","tokens_in":21862,"tokens_out":24780,"duration_ms":293992,"concrete_test":"Compute ε_1(K_4) and ε_2(K_4) from the definition ε_k(G)=Σ_{i=1}^k λ_i(L(G))−|E|. For K_4, this gives ε_1=−2 and ε_2=2, while nk−C(4,2)=−2. Plugging into Lemma 2.8 gives −2 = 2−2 = 0, which is false, settling that the printed lemma is incorrect. Then verify the corrected identity ε_1(K_4)=ε_2(\\overline{K_4})+4−6=0−2=−2 holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.8 states ε_k(G) = ε_{n−k−1}(G) + nk − C(n,2). This is false as written. In the proof, Lemma 2.7 is applied without carrying the complement: the term should be λ_{n−i}(L(\\bar G)), not λ_{n−i}(L(G)). The omission is not cosmetic; it makes the displayed lemma false. Counterexample: G=K4, n=4, k=1. The Laplacian eigenvalues are 4,4,4,0, so ε_1(K4)=4−6=−2, while ε_2(K4)=8−6=2 and nk−C(4,2)=4−6=−2. The printed Lemma 2.8 then asserts −2 = 2−2 = 0, a contradiction. The correct identity is ε_k(G)=ε_{n−k−1}(\\bar G)+nk−C(n,2). Corollary 4.2's second case k>(n−1)/2 relies on Lemma 2.8 to reduce to a smaller k; with the false statement the displayed equality is invalid. The final bound can be recovered by applying the first case to \\bar G (the first-case bound depends only on n and k, not on the graph), but a stated lemma in the chain to the headline graph bound is false as printed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.6: for any simplicial complex X, 1 ≤ r ≤ dim X and 1 ≤ k ≤ f_{r-1}(X)/(r+1), the sum of the k largest eigenvalues of the upper Laplacian L^+_{r-1}(X) is bounded by the sum of the (r+1)k largest r-degrees of (r-1)-faces. This sharp inequality extends Anderson–Morley. Corollary 1.7 gives the general bound f_r(X)+binom((r+1)k,2); Theorem 1.8 gives the graph bound sum_{i=1}^k λ_i(L(G)) ≤ |E|+k^2, improving Theorem 1.3; Theorem 1.9 and Corollary 1.10 prove the Duval–Reiner conjecture for (r+1)-partite r-dimensional complexes. Section 5 gives applications to forests, bounded-degree, planar, square-free, girth, and path/cycle-free graphs; Section 7 discusses extensions to signless Laplacians. The proofs use elementary tools: Ky Fan's inequality, Gershgorin's theorem, boundary matrices, and Bai's theorem as an external input.","tokens_in":22223,"tokens_out":16763,"duration_ms":169623,"significance":"The main theorem, once corrected, is a substantial contribution: it unifies and generalizes Anderson–Morley, supplies the best-known general upper bound for sums of Laplacian eigenvalues, and the partite result resolves a natural special case of Duval–Reiner. The proof is self-contained and the extremal examples in Section 3.1 are informative. The paper is also honest about where it falls short of Brouwer's conjecture and about the signless analogue not following from Bai's theorem. However, the submitted text contains several load-bearing errors that must be fixed; they are local and repairable.","major_comments":[{"comment":"Lemma 2.8 is false as printed. The correct identity is ε_k(G) = ε_{n−k−1}(\\bar G) + nk − C(n,2), not with G on the right. The proof misapplies Lemma 2.7, which itself is misstated: it should read λ_i(L(\\bar G)) = n − λ_{n−i}(L(G)). A counterexample to the printed lemma is G=K_4, k=1: ε_1(K_4)=−2, while the printed right-hand side gives ε_2(K_4)+4−C(4,2)=2+4−6=0. This lemma is used in Corollary 4.2's second case. The final bound can be recovered by applying the first case to \\bar G, so the error is repairable, but the false statement must be corrected.","section":"Section 2.3, Lemma 2.8"},{"comment":"In the proof of Theorem 4.1, the last sum on the right of (4.2) is '−∑_{i=2k+1}^n min{0,d_i−k}' but it must be '−∑_{i=2k+1}^n max{0,d_i−k}'. With the printed sign, adding (4.2) and (4.4) does not yield the claimed inequality for ε_k(G). This is a load-bearing typo for the derivation of Theorem 1.8.","section":"Section 4, Eq. (4.2)"},{"comment":"The weight w(σ)=min{d/deg_X(σ),1} is undefined when d=d^{(r)}_{(r+1)k}(X)=0 and deg_X(σ)=0. This case can occur: for example, r=1, k=2, and G=K_{1,2} has d=0. The proof should either handle d=0 separately (then the right-hand side equals the total degree sum and the inequality follows from trace) or state the convention that the corresponding L_i term is zero. As printed, Claim 3.3 and Claim 3.4 refer to an undefined expression. Claim 3.2's 'multiplicity one' assertion also fails for d_i=0, although its conclusion remains true. This is a gap in the main proof.","section":"Section 3, proof of Theorem 1.6"}],"minor_comments":[{"comment":"The statement as printed is missing the complement on the first Laplacian. It should read λ_i(L(\\bar G)) = n − λ_{n−i}(L(G)) for 1 ≤ i ≤ n−1.","section":"Section 2.3, Lemma 2.7"},{"comment":"The sentence 'w(σ_i)=1 for (r+1)k ≤ i ≤ f_{r−1}(X)' should be 'for i > (r+1)k'; at i=(r+1)k the value is d/d_i, which is 1 only when d_i>0.","section":"Section 3, after Eq. (3.2)"},{"comment":"There are several typographical issues, for example the displayed theorem statements in the abstract and main text are missing binomial coefficients due to formatting, and equation (4.2) should use max rather than min. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central inequality appears correct and the flaws are repairable: the false Lemma 2.8 is easily corrected by placing the complement in the right place, the sign in Eq. (4.2) is a typo, and the d=0 case in Theorem 1.6 can be handled separately. I therefore recommend major revision rather than rejection. No concerns about novelty or attribution; the paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper deserves a careful read. Theorem 1.6 is genuinely new: for every simplicial complex, in every dimension, and every k up to f_{r-1}/(r+1), the sum of the k largest upper-Laplacian eigenvalues is bounded by the sum of the (r+1)k largest r-degrees. It generalizes the k=1 case of Fan–Wu–Wang and the classical Anderson–Morley bound, and it is tight. The graphical consequence, sum_{i=1}^k λ_i(L(G)) ≤ |E| + k^2, improves the previous general bounds and gets closer to Brouwer's conjecture. The partite result, Corollary 1.10, is a new special case of Duval–Reiner's conjecture. The proof strategy—building a weighted Laplacian and then using Ky Fan plus Gershgorin—is clean and, once the errors below are fixed, convincing.\n\nThe soft spots are real but fixable. The reader's sign-typo reading of equation (4.2) is correct: the last term should be max{0, d_i − k}, not min{0, d_i − k}. More importantly, Lemma 2.8 is false as printed: the proof silently drops the complement from Lemma 2.7. The correct identity is ε_k(G) = ε_{n−k−1}(\\bar G) + nk − binom(n,2), not with G on the right-hand side. As written, the second case of Corollary 4.2 relies on this false statement. The bound can still be recovered by applying the first case to the complement, so the main theorem survives, but the printed proof needs the fix. There is also a small undefined-ratio in the definition of w(σ) when a degree is zero; it is cosmetic because the corresponding L_i is the zero matrix, but as written it is undefined in the main proof.\n\nNone of these issues look load-bearing. They are the kind of slips a careful referee would catch and the author can repair in a revision. The core mathematics—Lemma 3.1, the Ky Fan chain, and the partite argument—checks out. Researchers working on Laplacian spectrum bounds, simplicial Laplacians, or Grone–Merris/Duval–Reiner questions will benefit from this paper. I would send it to a serious referee and ask for the corrections before publication.\n\nBest,\n[You]","headline":"New sharp degree-sum bound for Laplacian eigenvalues; solid and worth refereeing, but Lemma 2.8 is false as printed and needs correction.","tokens_in":22732,"tokens_out":9055,"would_cite":true,"duration_ms":96062,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05E45","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that in every simplicial complex, the sum of the $k$ largest eigenvalues of the $(r-1)$-th upper Laplacian is at most the sum of the $(r+1)k$ largest $r$-degrees of codimension-one faces, a sharp bound that generalizes Ande","keywords":["sums of Laplacian eigenvalues","simplicial complexes","upper Laplacian","degree sequences","Brouwer's conjecture","Duval-Reiner conjecture","graph Laplacian","Ky Fan inequality"],"falsifier":"Diagonalize $L^+_{r-1}(X)$ exactly for all simplicial complexes on a small vertex set and compare each $k$-sum with the corresponding $(r+1)k$ largest $r$-degree sum; the first violation would refute Theorem 1.6. A minimal check is a graph consisting of a perfect matching plus isolated vertices with $k$ chosen so $d_{2k}=0$, where the proof's weight $d/\\deg$ is $0/0$ and a consistent convention must be supplied.","tokens_in":21755,"feed_emoji":"📐","tokens_out":12688,"duration_ms":132644,"temperature":0.7,"pith_summary":"This paper proves a high-dimensional analogue of a classical graph inequality: for any simplicial complex $X$, the sum of the $k$ largest eigenvalues of the $(r-1)$-th upper Laplacian is bounded by the sum of the $(r+1)k$ largest $r$-degrees of $(r-1)$-faces. The bound is sharp, and in the graph case $r=1$ it gives $\\sum_{i=1}^k \\lambda_i(L(G)) \\le |E| + k^2$, improving earlier bounds for all $k\\ge 3$ and moving toward Brouwer's conjecture. The same method verifies the Duval--Reiner conjecture for $(r+1)$-partite $r$-dimensional simplicial complexes. The overarching point is that degree sums, not just individual degree maxima, control Laplacian eigenvalue sums in every dimension.","feed_headline":"Laplacian sums bounded by top (r+1)k degrees in all dimensions","feed_subtitle":"For graphs it recovers |E|+k^2, a step toward Brouwer's open conjecture.","key_machinery":"The proof's load-bearing object is a weighted $r$-th lower Laplacian $L' = L^-_r(X) - \\sum_{i=1}^{(r+1)k} (1 - d/d^{(r)}_i(X)) L_i$, where $d$ is the $(r+1)k$-th largest $r$-degree and $L_i$ is the matrix supported on the $i$-th heaviest $(r-1)$-face, with exactly one nonzero eigenvalue $d^{(r)}_i(X)$. The weight function $w(\\sigma)=\\min\\{d/\\deg_X^{(r)}(\\sigma),1\\}$ makes every row of $L'$ have $\\ell^1$-norm at most $(r+1)d$, so the Frobenius--Gershgorin bound gives $\\lambda_1(L')\\le (r+1)d$. Ky Fan's inequality then removes the $L_i$ terms one by one, leaving exactly the sum of the top $(r+1)k$ $r$-degrees. The fact that $L^+_{r-1}(X)$ and $L^-_r(X)$ share all nonzero eigenvalues (Corollary","core_discovery":"The central claim is Theorem 1.6: for any simplicial complex $X$, any $1\\le r\\le \\dim(X)$, and any $k$ with $1\\le k\\le f_{r-1}(X)/(r+1)$, the sum of the $k$ largest eigenvalues of the $(r-1)$-th upper Laplacian $L^+_{r-1}(X)$ is at most the sum of the $(r+1)k$ largest $r$-degrees among $(r-1)$-faces. The inequality is tight: equality holds for complexes in which every $(r-1)$-face is contained in exactly one $r$-face, the graph case being a perfect matching. Anderson and Morley's bound $\\lambda_1(L(G))\\le d_1(G)+d_2(G)$ is exactly the case $r=1,k=1$. Two consequences are highlighted: the general bound $\\sum_{i=1}^k \\lambda_i(L^+_{r-1}(X)) \\le f_r(X)+\\binom{(r+1)k}{2}$, and, for graphs, $\\sum","pith_inferences":["Extension: the same weighted-decomposition argument should transfer to signless Laplacians of arbitrary complexes; the paper states the analogue (Theorem 7.5) without proof, and the underlying row-sum and Ky Fan steps do not use signs.","Extension: Corollary 4.2 hints at an interpolation between Bai's conjugate-degree bound and the new degree-sum bound; optimizing the two averages may yield Brouwer's exact $|E|+\\binom{k+1}{2}$ for all $k$, not just for the hereditary families treated in Section 5.","Extension: the equality case analysis suggests a structural question the paper leaves open: whether connected graphs can attain equality in Theorem 1.6 for $k>1$; if none can, the $|E|+k^2$ graph bound is never tight for connected graphs, strengthening the case for Brouwer's conjecture."],"forward_implications":["For every graph $G$ with $|V|\\ge k$, $\\sum_{i=1}^k \\lambda_i(L(G))\\le |E|+k^2$; this improves the previous best bounds for $k\\ge 3$ and is the graph-level content of Theorem 1.8.","The general complex bound $\\sum_{i=1}^k \\lambda_i(L^+_{r-1}(X))\\le f_r(X)+\\binom{(r+1)k}{2}$ follows immediately, giving a weak form of the high-dimensional Brouwer bound proposed as Conjecture 1.5.","For $(r+1)$-partite $r$-dimensional complexes, the stronger partite decomposition verifies the Duval--Reiner conjecture for this class (Corollary 1.10).","The inequality is tight: complexes in which every $(r-1)$-face meets exactly one $r$-face achieve equality; for $r=1$ this is a perfect matching, and star forests are near-extremal.","Hereditary graph classes inherit the bound; in particular square-free graphs satisfy Brouwer's conjecture for all $k\\ge 7$ and girth-at-least-5 graphs for all $k\\ge 1$ (Proposition 5.3)."],"supporting_citations":[{"why":"Supplies the classical bound $\\lambda_1(L(G))\\le d_1(G)+d_2(G)$ that Theorem 1.6 generalizes, and the complement-eigenvalue relation used in deriving the graph corollaries.","marker":"[6]"},{"why":"Proves the Grone--Merris conjecture; Theorem 1.8 is obtained by combining this theorem with Theorem 1.6.","marker":"[8]"},{"why":"Gives the spectrum-matching between upper and lower Laplacians (Corollary 2.6) used throughout, and formulates the conjecture resolved for partite complexes in Corollary 1.10.","marker":"[22]"},{"why":"Proves the $k=1$ case for every $r$, the immediate predecessor of Theorem 1.6, and supplies the bound used in Proposition 7.4.","marker":"[24]"},{"why":"Supplies the Gershgorin row-sum theorem used to bound $\\lambda_1(L')$ in the proof of Theorem 1.6.","marker":"[36]"},{"why":"Supplies the Ky Fan eigenvalue-sum inequality and the $AB$/$BA$ nonzero-eigenvalue identity, the two matrix tools at the core of the proof.","marker":"[46]"}],"fun_headline_variants":["Laplacian sums topped by (r+1)k largest degrees","New tight bound on Laplacian eigenvalue sums","Graph Laplacian sum improves to |E|+k^2","Anderson–Morley theorem extended to complexes","Simplicial Laplacians: sum ≤ max degree sum"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The argument requires that every $(r-1)$-face can be assigned the weight $w(\\sigma)=\\min\\{d/\\deg_X^{(r)}(\\sigma),1\\}$; when $d=0$ and a face has $r$-degree $0$, the ratio is $0/0$, and the paper does not state the convention that such faces contribute zero, leaving a gap in the literal proof of Theorem 1.6.","fun_headline_variants_meta":{"raw":{"variants":["Laplacian sums topped by (r+1)k largest degrees","New tight bound on Laplacian eigenvalue sums","Graph Laplacian sum improves to |E|+k^2","Anderson–Morley theorem extended to complexes","Simplicial Laplacians: sum ≤ max degree sum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000278,"raw_usage":{"total_tokens":1768,"prompt_tokens":1298,"completion_tokens":470,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":1042,"completion_tokens_details":{"reasoning_tokens":388}},"tokens_in":1042,"tokens_out":470,"duration_ms":6315,"temperature":1.0,"reasoning_tokens":388,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:50:05.261069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Diagonalize $L^+_{r-1}(X)$ exactly for all simplicial complexes on a small vertex set and compare each $k$-sum with the corresponding $(r+1)k$ largest $r$-degree sum; the first violation would refute Theorem 1.6. A minimal check is a graph consisting of a perfect matching plus isolated vertices with $k$ chosen so $d_{2k}=0$, where the proof's weight $d/\\deg$ is $0/0$ and a consistent convention must be supplied.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical bound $\\lambda_1(L(G))\\le d_1(G)+d_2(G)$ that Theorem 1.6 generalizes, and the complement-eigenvalue relation used in deriving the graph corollaries."},{"cited_title":"Bai, The Grone-Merris conjecture , Trans","cited_arxiv_id":null,"evidence_quote":"Proves the Grone--Merris conjecture; Theorem 1.8 is obtained by combining this theorem with Theorem 1.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the spectrum-matching between upper and lower Laplacians (Corollary 2.6) used throughout, and formulates the conjecture resolved for partite complexes in Corollary 1.10."},{"cited_title":"Fan, H.-F","cited_arxiv_id":null,"evidence_quote":"Proves the $k=1$ case for every $r$, the immediate predecessor of Theorem 1.6, and supplies the bound used in Proposition 7.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Gershgorin row-sum theorem used to bound $\\lambda_1(L')$ in the proof of Theorem 1.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Ky Fan eigenvalue-sum inequality and the $AB$/$BA$ nonzero-eigenvalue identity, the two matrix tools at the core of the proof."}],"review_version":1}