{"id":"7e2ca84e-17eb-47f1-9f8a-9de62bca37ca","arxiv_id":"1908.09365","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves two theorems giving sufficient conditions for the two-term spectral asymptotics to be unchanged under a compact metric perturbation.","lead":"This math paper finds conditions that protect the second term in the eigenvalue asymptotics of a compact operator when the inner product is perturbed. It offers two sufficient criteria, one of which only requires a decay estimate on the perturbation's matrix elements.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1 reduces to sign-definite perturbations without showing that the matrix-element decay bound is inherited by B_+ and B_-, leaving the sandwich argument incomplete.","rationale":"Read in good faith, the paper gives plausible sufficient conditions for the stability of two-term spectral asymptotics under metric perturbation. The central claim is Theorem 1, and the reader accepted it with moderate confidence, identifying the matrix-element decay condition as the weakest assumption. My stress test found a more specific, load-bearing problem in how that assumption is used. The proof reduces arbitrary B to its positive and negative parts, but the decay condition is not shown to be invariant under the positive/negative part decomposition. Since B_+ and B_- are nonlinear functions of B, their matrix elements in the eigenbasis of K can behave very differently from those of B; the proof gives no argument that (B_+ h_n, h_m) or (B_- h_n, h_m) satisfies the same weighted bound. This is an internal gap in the proof, not a disagreement with external consensus. The proposed numerical test would settle whether the missing lemma is false in general. If the test confirms failure of inheritance, the theorem may still be true, but the current proof does not establish it. Therefore the verdict should be CONDITIONAL: accept only if the decomposition step is justified or replaced by a direct argument that avoids the positive/negative part reduction. My agreement with the reader is partial: I share the focus on the matrix-element decay assumption, but I locate the critical failure in the reduction step rather than in the hypothesis itself.","tokens_in":3480,"tokens_out":21104,"duration_ms":224267,"concrete_test":"Fix gamma = (1+delta)/2 > 1/2. Construct a family of self-adjoint compact operators B on l^2 with |(B h_n, h_m)| <= (mn)^{-gamma} but with |(B_+ h_n, h_m)| not bounded by C (mn)^{-gamma}. One concrete computational check: take N = 2000, K = diag(k^{-2}), choose a random orthogonal matrix U with entries |U_{kn}| ~ (1+|k-n|)^{-p} for large p, set D = diag((-1)^k k^{-q}), and B = U^* D U. Compute M_B = max_{n,m <= N} (nm)^gamma |(B h_n, h_m)| and M_+ = max_{n,m <= N} (nm)^gamma |(B_+ h_n, h_m)|. If M_B stays O(1) while M_+ grows with N, or fails to decay as n,m tend to infinity, then the reduction step in Theorem 1 is invalid. If instead M_+ also stays O(1) for all such U, the missing lemma may be true and the proof can be repaired.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"At the start of the proof of Theorem 1, B is decomposed as B = B_+ + B_- with B_+ >= 0 and B_- <= 0, and it is stated that by min-max it suffices to consider sign-definite perturbations. This eigenvalue sandwich is valid, but the subsequent estimates for the positive and negative cases rely on the matrix-element bound |(B h_n, h_m)_H| <= c (mn)^{-(1+delta)/2} for the sign-definite operator being analyzed (see the use of (B x, h_k) in equation (6) and the estimates leading to (7)-(8)). The hypothesis, however, is stated only for B. The operators B_+ and B_- are nonlinear spectral functions of B: B_+ = (|B| + B)/2 and B_- = (|B| - B)/2. In the fixed orthonormal basis {h_n}, the matrix elements of |B|, or of the positive and negative spectral parts, are not determined by those of B. Cancellations between positive and negative spectral subspaces that make (B h_n, h_m) small can disappear when passing to B_+ or B_-. Thus the claim 'it suffices to consider two cases' requires a missing lemma. Without such a lemma, the bounds (7) and (8), and consequently the conclusion J(x) >= lambda_n (1 - c n^{-(1+delta)}), do not follow for a general perturbation B. This gap is load-bearing because the theorem's conclusion for arbitrary (not sign-definite) B is not established by the proof as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies whether the two-term power-type spectral asymptotics lambda_n = (a n + b + O(n^{-delta}))^{-B} of a compact self-adjoint positive operator K are preserved when the Hilbert space scalar product is replaced by (h,g) -> ((I+B)h,g)_H, giving the generalized eigenproblem K h = lambda (I+B) h. Lemma 1 proves preservation under the norm decay condition ||B h_n|| <= c n^{-(1+delta)}. Theorem 1 claims the same conclusion under the weaker off-diagonal matrix-element decay |(B h_n, h_m)_H| <= c (mn)^{-(1+delta)/2}. Remark 2 extends the statement to two interlacing eigenvalue sequences. The proofs use spectral measure arguments in Lemma 1 and min-max/variational arguments in Theorem 1.","tokens_in":3788,"tokens_out":7350,"duration_ms":73294,"significance":"If the results are correct, they provide a concrete and checkable sufficient condition for the stability of two-term spectral asymptotics under metric perturbations, with potential applications to integro-differential operators arising in the theory of Gaussian processes. The paper is concise and fully self-contained, and Lemma 1 is cleanly proved. However, the main new result, Theorem 1, has a load-bearing gap in the reduction to sign-definite perturbations; the paper in its current form therefore does not establish the theorem for arbitrary (indefinite) B.","major_comments":[{"comment":"The reduction to the cases B >= 0 and B <= 0 is not justified. The proof applies the estimates in equations (6)-(8) to the sign-definite operators B_+ and B_- individually, and these estimates use the matrix-element bound |(B_+ h_n, h_m)| <= c (mn)^{-(1+delta)/2} (respectively for B_-). The hypothesis supplies this bound only for the original operator B. Since B_+ and B_- are obtained from B by the spectral calculus, their matrix elements in the fixed basis {h_n} are not determined by the matrix elements of B; cancellations that make (B h_n, h_m) small need not persist in B_+ or B_-. Thus the sandwich lambda_n^+ <= lambda_n <= lambda_n^- does not imply the conclusion from the two sign-definite cases without an additional inheritance lemma. This is a load-bearing gap: as written, Theorem 1 is not proven for an arbitrary (indefinite) B satisfying the stated hypothesis.","section":"Theorem 1, proof, first paragraph"}],"minor_comments":[{"comment":"Throughout Lemma 1 and Theorem 1 the letter B denotes both the perturbation operator in (2) and the positive exponent in the asymptotics (1), (3); this makes statements such as lambda_n = (a n + b + O(n^{-delta}))^{-B} ambiguous. The exponent should be given a different symbol, for example beta.","section":"Equations (1) and (3)"},{"comment":"The perturbed eigenvalues are not distinguished typographically from the unperturbed ones; for example, in part 1, \"evidently lambda_n <= lambda_n\" uses the same symbol on both sides, which is confusing and, taken literally, a tautology. Please introduce a distinct notation for the eigenvalues of the perturbed problem.","section":"Theorem 1, parts 1 and 2"},{"comment":"In the displayed formula for the derivative of the Rayleigh quotient there is an extra closing parenthesis after (B x, h_k)_H, making the quotient ambiguous as printed. The subsequent equation (6) shows the intended formula is correct, so this is a typographical issue rather than a mathematical one.","section":"Theorem 1, derivative computation"}],"recommendation":"major_revision","confidential_remarks":"The sole major comment concerns the central theorem and is not a presentation issue: without an inheritance lemma for the matrix-element decay of B_+ and B_-, the proof of Theorem 1 does not cover indefinite perturbations. If the authors can supply such a lemma or otherwise repair the argument, the paper may be suitable; if the gap cannot be fixed, the theorem as stated would need to be restricted or withdrawn."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is short and readable, and the question it asks is genuinely worth asking: the second term in spectral asymptotics is sensitive to metric perturbation, and sufficient conditions for its preservation are scarce. Lemma 1 is clean and, as far as I can tell, correct; the spectral-measure argument is standard and the norm decay condition is strong enough to do the job. Credit where due: the paper identifies a real gap in the literature and offers a natural pointwise decay condition that is globally weaker than the norm condition in Lemma 1. The Rayleigh-quotient technique with weighted sums is standard, but the combination is new and the two-term conclusion is not something I have seen stated before.\n\nThe problem is in the proof of Theorem 1. The reduction to sign-definite perturbations is invalid as written. The min-max sandwich gives λ_n^+ ≤ λ_n ≤ λ_n^-, but that only tells you that if both bounding problems have the desired asymptotics then the middle problem does too. It does not tell you that the matrix-element hypothesis for B transfers to B_+ and B_-. Those operators are spectral functions of B, and their matrix elements in the fixed basis {h_n} are not controlled by the matrix elements of B; cancellations that make (B h_n, h_m) small can vanish when you pass to the positive or negative spectral part. Without a lemma proving that B_+ and B_- inherit the decay bound, equations (7) and (8) are not justified, and the conclusion for non-sign-definite B does not follow. This is a load-bearing gap, not a typo. The minor presentation issues—the reused symbol B, the missing typographic distinction between unperturbed and perturbed eigenvalues, the derivative factor typo in Theorem 1—are real but secondary; the author should fix them anyway.","headline":"Theorem 1 has a real gap: the matrix-element decay condition on B does not pass to its positive and negative parts, so the main theorem is not proven as stated.","tokens_in":4277,"tokens_out":4293,"would_cite":false,"duration_ms":47333,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A75","47A55","47B06"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Hilbert-space metric perturbation whose matrix elements decay fast enough leaves the two-term power asymptotics of a compact operator's eigenvalues unchanged.","keywords":["spectral asymptotics","two-term asymptotics","compact operator","metric perturbation","generalized eigenproblem","Rayleigh quotient","min-max principle","matrix element decay"],"falsifier":"Take a diagonal $K$ with $\\lambda_n = n^{-2}$, choose a compact self-adjoint $B$ whose entries in the eigenbasis satisfy $|(B h_n,h_m)_H| = (mn)^{-(1+\\delta)/2}$ with alternating signs, and compute the generalized eigenvalues for large $n$. If the two-term expansion loses its $O(n^{-\\delta})$ remainder for any $\\delta>0$, the theorem's condition would not be sufficient.","tokens_in":3291,"feed_emoji":"🔢","tokens_out":8694,"duration_ms":79979,"temperature":0.7,"pith_summary":"This paper proves a stability theorem for two-term spectral asymptotics. If a compact self-adjoint operator $K$ has eigenvalues $\\lambda_n = (a n + b + O(n^{-\\delta}))^{-B}$ and the Hilbert-space metric is perturbed by a compact operator $B$ whose matrix elements in the eigenbasis of $K$ decay like $(mn)^{-(1+\\delta)/2}$, then the eigenvalues $\\tilde\\lambda_n$ of the generalized problem $Kh = \\tilde\\lambda(h + Bh)$ have exactly the same two-term asymptotics $\\tilde\\lambda_n = (a n + b + O(n^{-\\delta}))^{-B}$. This matters because one-term asymptotics are known to be stable under compact metric perturbations while the second term is delicate; the theorem supplies a checkable, pointwise condition under which the second term is preserved. The proof splits $B$ into positive and negative parts and uses min-max and Rayleigh-quotient estimates, rather than any special structure of the spectrum.","feed_headline":"Two-term spectrum survives metric perturbation","feed_subtitle":"A decay condition on off-diagonal matrix elements keeps the b-term in eigenvalue asymptotics intact.","key_machinery":"The argument is carried by the Rayleigh quotient $J(x) = (Kx,x)_H / ((x,x)_H + (Bx,x)_H)$ associated with the generalized eigenproblem. Restricting $J$ to the span of the first $n$ eigenfunctions of $K$ (or, for the negative part, to the tail subspace) reduces the problem to finite-dimensional comparisons. The pivotal identity (6) expresses the eigenfunction coefficients of the minimizer in terms of $(B\\hat x, h_k)_H$; the decay condition on the matrix elements then makes the sums $\\hat A$ and $\\check A$ in (7)\\textendash(8) small. This forces the Rayleigh minimizer value within $O(n^{-(1+\\delta)})$ of $\\lambda_n$, and the standard spectral-measure argument in Lemma 1 locks each generalized eigenvalue into a small interval free of other eigenvalues.","core_discovery":"The central discovery is Theorem 1: the preservation of the two-term asymptotics follows from the individual matrix-element bounds $|(B h_n, h_m)_H| \\le c (mn)^{-(1+\\delta)/2}$ together with the assumed $\\lambda_n = (a n + b + O(n^{-\\delta}))^{-B}$. Lemma 1 first settles the simpler case where $\\|B h_n\\|_H \\le c n^{-(1+\\delta)}$; Theorem 1 replaces that operator-level decay by a globally weaker, more pointwise decay condition. In both cases the conclusion is the same: the generalized eigenvalues $\\tilde\\lambda_n$ admit the expansion $(a n + b + O(n^{-\\delta}))^{-B}$. Remark 2 extends the same conclusion to spectra consisting of two interlacing sequences, such as even and odd indices with distinct constants.","pith_inferences":["The proof only uses the spacing and decay of the unperturbed eigenvalues, so the same trapping argument should survive for asymptotics with logarithmic corrections, provided consecutive $\\lambda_n$ stay separated by a power-like gap.","The threshold $(mn)^{-(1+\\delta)/2}$ suggests a natural boundary: perturbations with matrix elements of size $(mn)^{-1/2}$ might be able to shift the $b$-term, a question the paper does not address.","For concrete covariance kernels, the matrix-element condition can be checked from kernel smoothness, making the theorem a practical stability criterion for two-term Weyl asymptotics in Gaussian-process settings."],"forward_implications":["For every operator satisfying the hypothesis of Theorem 1, the generalized eigenproblem has two-term asymptotics with the same constants $a$, $b$, $B$, and the same $O(n^{-\\delta})$ remainder.","The positive and negative cases are handled separately, so for a general self-adjoint $B$ the conclusion is obtained by squeezing between the min-max bounds for $B_+$ and $B_-$.","By Remark 2, the result covers spectra with two interlacing eigenvalue sequences, so it applies to parity-class asymptotics such as $\\lambda_n^{(1)} = ((2n-1)a + b_1 + O(n^{-\\delta}))^{-B}$ and $\\lambda_n^{(2)} = (2 n a + b_2 + O(n^{-\\delta}))^{-B}$.","The sufficient condition is aimed at integro-differential operators arising in the theory of fractional Gaussian processes, where the metric perturbation corresponds to a concrete covariance term."],"supporting_citations":[{"why":"Supplies the min-max principle and the standard invariance of one-term power asymptotics under compact metric perturbations, which frames the problem.","marker":"[1]"},{"why":"Supplies the spectral-measure identities and the interval-free-spectrum inequality (4) used in Lemma 1.","marker":"[2]"},{"why":"Identifies the integro-differential operators from the theory of Gaussian processes to which the sufficient conditions are meant to apply.","marker":"[3]"}],"fun_headline_variants":["Pointwise decay preserves two-term spectrum","Weak matrix decay keeps eigenvalue expansion","Two-term spectral asymptotics survive perturbation","Off-diagonal decay locks the spectral b-term","Eigenvalue expansion robust to metric change"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the requirement that the perturbation's off-diagonal coefficients in the unperturbed eigenbasis decay like the product of the two indices raised to a power strictly above one-half; if that decay fails, the coefficient estimates (7) and (8) no longer close.","fun_headline_variants_meta":{"raw":{"variants":["Pointwise decay preserves two-term spectrum","Weak matrix decay keeps eigenvalue expansion","Two-term spectral asymptotics survive perturbation","Off-diagonal decay locks the spectral b-term","Eigenvalue expansion robust to metric change"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1553,"prompt_tokens":708,"completion_tokens":845,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":324,"completion_tokens_details":{"reasoning_tokens":782}},"tokens_in":324,"tokens_out":845,"duration_ms":9041,"temperature":1.0,"reasoning_tokens":782,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:14:24.787618+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a diagonal $K$ with $\\lambda_n = n^{-2}$, choose a compact self-adjoint $B$ whose entries in the eigenbasis satisfy $|(B h_n,h_m)_H| = (mn)^{-(1+\\delta)/2}$ with alternating signs, and compute the generalized eigenvalues for large $n$. If the two-term expansion loses its $O(n^{-\\delta})$ remainder for any $\\delta>0$, the theorem's condition would not be sufficient.","supporting_citations":[{"cited_title":"Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory","cited_arxiv_id":null,"evidence_quote":"Supplies the min-max principle and the standard invariance of one-term power asymptotics under compact metric perturbations, which frames the problem."},{"cited_title":"Spectral theory of self-adjoint operators in Hilbert space, 2nd ed., revised and extended","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral-measure identities and the interval-free-spectrum inequality (4) used in Lemma 1."},{"cited_title":"Spectral asymptotics for a class of integro-diffe rential equations arising in the theory of fractional Gaussian processes","cited_arxiv_id":null,"evidence_quote":"Identifies the integro-differential operators from the theory of Gaussian processes to which the sufficient conditions are meant to apply."}],"review_version":1}