{"id":"7864f75b-6909-4400-b655-2655cbefdb89","arxiv_id":"2504.17012","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For nonlinear operator pencils that vary continuously in the gap metric, spectra and pseudospectra can be computed from finite-dimensional truncations with two or three successive limits, these limits are provably optimal, and Hermitian pencils are no easier.","lead":"This paper delivers the first general algorithms that compute spectra and pseudospectra of nonlinear eigenvalue problems without spurious or missing results, for any problem whose operators vary continuously in the gap metric. A generalist might read it because mechanical vibrations, photonic crystals, delayed biological systems, and fractional beam models all produce such problems, and until now none came with convergence guarantees.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence proof depends on a common-core basis condition that is not implied by gap continuity and is dropped without proof in the differential examples, so the 'universal' scope is narrower than claimed.","rationale":"I re-read the central arguments in good faith. Theorems 3.1 and 3.2 are internally coherent for the class Omega^U_NL as defined, the lower bounds are imported from established hardness results, and the algorithms' monotone convergence is plausible and correctly argued under the stated core assumptions. The load-bearing soft spot is precisely the common-core basis condition: it is essential for Lemma 3.2(ii), it is not implied by gap continuity alone, and it is dropped in the differential examples via an unproved assertion about z-dependent bases. The reader's weakest assumption identifies the same issue, and the appropriate response is to keep the CONDITIONAL verdict rather than accept or reject: the gap is concrete and addressable by adding a lemma for the relaxed setting or by narrowing the claims about the examples. I therefore recommend UNCHANGED.","tokens_in":27016,"tokens_out":20901,"duration_ms":221495,"concrete_test":"For Section 4.3, fix z = 1+i and analytically verify that finite linear combinations of the z-dependent Gram-Schmidt basis {f_n(z)} are dense in D(T(z)) and D(T(z)*) in the graph norm, i.e. prove that for every u in D(T(z)) there exist u_N in span{f_1,...,f_N} with ||u-u_N|| + ||T(z)(u-u_N)|| -> 0, and similarly for T(z)*. If this fails, recompute the example and qualify the no-invisibility claim; if it holds, add the missing lemma and state the relaxed convergence theorem so that the differential examples are covered by a proven result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main convergence guarantee rests on Lemma 3.2(ii), which uses Definition 2.4's assumption that span{e_n} is a core of T(z) and span{e_hat_n} is a core of T(z)* for every z. This is an additional hypothesis, not a consequence of gap continuity: for example, the family of multiplication operators T(z) = M_{1/(x-z)} on L^2(0,1) with U=(0,1) is gap-continuous, yet the intersection of the domains over z in U is {0}, so no fixed orthonormal basis can have finite spans forming cores for every z. Thus the abstract's 'only minimal continuity assumptions' overstates what Theorems 3.1 and 3.2 establish. The same assumption is dropped in Sections 4.3-4.5, where the authors use z-dependent Gram-Schmidt bases and assert without proof that 'the convergence results in Section 3.3 continue to hold under this relaxed assumption.' In particular, no core condition is verified for T(z)* in the wave-equation example. If the z-dependent basis is not a core of the adjoint, the second term in gamma_{n2}(z,T) can converge to a value above gamma(z,T), producing spectral invisibility. This is load-bearing because the differential examples are the primary evidence for broad applicability, and the central theorem does not literally cover them as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general computational framework for spectra and pseudospectra of nonlinear operator pencils T: U -> C(H1,H2) that are continuous in the gap topology. It introduces nonlinear injection moduli, proves their continuity, and uses monotone finite-dimensional truncations to construct two algorithms: Algorithm 1 computes pseudospectra with two (or one) limits, and Algorithm 2 computes spectra with three (or two) limits, depending on whether the evaluation set is Lambda_1 or Lambda_2. The main theoretical results are the SCI classifications in Theorems 3.1 and 3.2, including lower bounds showing optimality and the claim that Hermiticity does not reduce the number of limits for nonlinear problems. The paper also presents numerical examples for nonlinear shifts, Klein-Gordon equations, wave equations with acoustic boundary conditions, time-fractional beam equations, and a delayed reaction-diffusion system.","tokens_in":27199,"tokens_out":18350,"duration_ms":191087,"significance":"If the results stand, this is a substantial contribution to computational spectral theory. The central chain is sound: Lemma 3.1 characterizes the spectrum as the zero set of the injection modulus, Lemma 3.2 proves monotone convergence of the truncated moduli, Propositions 3.3 and 3.4 establish Attouch-Wets convergence of the algorithms, and Theorem 3.1 gives matching SCI upper and lower bounds. The constructions are parameter-free, do not use fitted constants, and the lower-bound reductions are explicit and machine-checkable in principle. The numerical examples are extensive and illustrate both the strengths of the method and the failure of naive truncation. However, the actual scope of the theorems is narrower than the abstract claims: the core-basis conditions in Definition 2.4 are extra hypotheses that are not implied by gap continuity, and the relaxed z-dependent basis used in Section 4.3 is asserted without proof. These issues are load-bearing because they affect the no-invisibility guarantee.","major_comments":[{"comment":"The abstract and the introduction state that the method requires only continuity with respect to the gap metric, but Definition 2.4 adds two further conditions: for every z in U, span{e_n} must be a core of T(z) and span{e_hat_n} must be a core of T(z)*. These conditions are not consequences of gap continuity. For example, T(z)=M_{1/(x-z)} on L^2(0,1) with U=(0,1) is gap-continuous, yet the intersection of the domains over all z is {0}, so no fixed orthonormal basis can have finite spans forming cores for every z. Lemma 3.2(ii) relies on both core conditions; if they fail, gamma_{n2}(z,T) can converge from above to a value strictly larger than gamma(z,T), and Algorithm 2 can miss true spectral points. Thus Theorem 3.1 classifies only the narrower class Omega_U_NL, and the claim of applicability to all gap-continuous pencils should be corrected, or the theorem extended to a properly stated weaker hypothesis.","section":"Abstract and §1; Definition 2.4 and Lemma 3.2(ii)"},{"comment":"Section 4.3 introduces a z-dependent orthonormal basis obtained by Gram-Schmidt and states without proof that 'the convergence results in Section 3.3 continue to hold under this relaxed assumption.' The theorems require a fixed basis whose spans are cores of both T(z) and T(z)*. In the wave-equation example the basis functions do not lie in the domain of T(z), the modified basis depends on z, and no core condition is verified for T(z)*. If the second term of gamma_{n2}(z,T) fails to converge down to gamma(z,T), spectral invisibility can occur. Sections 4.4 and 4.5 also do not verify the core condition for the bases used. Consequently, the differential-operator demonstrations are not consequences of Theorems 3.1 and 3.2 as written. The authors should either provide a proof for a relaxed-core theorem covering their z-dependent and Gram-Schmidt constructions, or explicitly present these examples as numerical demonstrations under an unverified heuristic assumption.","section":"§4.3–4.5"}],"minor_comments":[{"comment":"There are several typos, including 'Pseudosepctra' in the caption of Figure 1 and 'the prof if' in the proof of Proposition 3.4.","section":"Throughout"},{"comment":"The extension of the Attouch-Wets metric to the empty set is described by a convergence convention rather than by an explicit metric. For a fully rigorous SCI formulation, the empty set should be incorporated into the metric space with a defined distance, or the convention should be justified as a limiting metric.","section":"§2.3"},{"comment":"Several references are incomplete or lack publication details, including [9], [10], [11], [13], [24], [39], and [47]; for example, [9] lists only volume and pages without a journal or article title.","section":"References"},{"comment":"The notation for the truncated injection modulus alternates between gamma_{n2}(z,T) and gamma_n(z,T), and in Figure 4 the legend uses both; standardizing this notation would improve readability.","section":"§4.1–4.2"}],"recommendation":"major_revision","confidential_remarks":"I think the paper is within scope and the main theoretical machinery is correct. The central issue is the gap between the claimed scope and the core-basis assumptions, which the authors themselves flag in Section 4.3 but do not resolve. This is fixable by either extending the convergence theorem or revising the claims, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. It delivers the first SCI classifications and convergent algorithms for spectra and pseudospectra of nonlinear pencils that are continuous in the gap metric, and the main proof chain is sound. The catch is scope: the class in Definition 2.4 is not just gap-continuous pencils. It also requires a fixed orthonormal basis whose finite spans are cores of T(z) and T(z)* for every z. That is an extra hypothesis, and the differential examples in Sections 4.3–4.5 quietly drop it by using z-dependent bases with one sentence saying convergence still holds. That sentence is unproved. So the abstract's 'only minimal continuity assumptions' is stronger than what the theorems actually establish.\n\nCredit where it is due. Lemma 3.1 (spectrum as zero set of the minimum of injection moduli), Lemma 3.2 (monotone convergence of truncations), Propositions 3.3–3.4 (Attouch–Wets convergence), and the SCI lower bounds all check out on close reading. No fitted constants, no circularity. The embedding of the linear problem for lower bounds is legitimate. This genuinely generalizes the earlier holomorphic discrete-spectrum result and the Klein–Gordon classification, and the optimality results are a real step forward. The Riemannian metric to handle boundary effects is a nice touch.\n\nSoft spots, in proportion. First, the core-basis condition is load-bearing and not implied by gap continuity. The stress-test example, T(z) = M_{1/(x-z)} on L2(0,1) over U=(0,1), is gap-continuous but no fixed basis has finite spans forming cores for every z. So the word 'universal' overstates the class actually classified. Second, the z-dependent basis relaxation is unproved, and it matters for the wave, beam, and Lotka–Volterra examples. Without a verified core property for T(z)*, the adjoint term in gamma_{n2} can converge to a value above gamma(z,T), which would produce spectral invisibility. This is the one place where the demonstrations are not literally covered by Theorems 3.1–3.2. Third, minor: the displayed bound in Proposition 3.1 has an algebraic slip in the passage from the first expression to the second, and several spectral facts for examples are stated without derivation. Those are fixable.\n\nThis paper deserves a serious referee. My recommendation: send it to peer review, ask for a proof or precise sufficient conditions for the z-dependent basis extension, and require a rewritten abstract that states the core-compatible basis assumption honestly. With that fixed, this is a strong paper.","headline":"The SCI classification and convergent algorithms for gap-continuous nonlinear pencils are real and mostly correct, but the 'universal' scope is narrower than claimed: the core-basis condition is an extra hypothesis, and the differential examples rely on an unproved relaxation to z-dependent bases.","tokens_in":27827,"tokens_out":3298,"would_cite":true,"duration_ms":32429,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P30","46N40","47A10","47J10","65Hxx","65J10","65N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims the first general, convergent computational method for the spectra and pseudospectra of nonlinear spectral problems, applicable to any gap-continuous pencil.","keywords":["nonlinear spectral problems","spectral pollution","spectral invisibility","injection modulus","pseudospectra","solvability complexity index","gap topology","operator pencils"],"falsifier":"Run Algorithm 1 on any pencil in the stated class and compare its output to the true $Sp_\\epsilon(T)$ under the Attouch–Wets metric; any pencil where the limit differs would refute the convergence claim. A sharper falsifier targets the relaxed-basis assertion: for the acoustic wave pencil on the half-line with spectrum $\\{z:\\mathrm{Im}(z)\\ge 0\\}$, compute $\\gamma_{n_2}(z,T)$ using the paper's $z$-dependent orthonormal bases at a fixed $z$ with $\\mathrm{Im}(z)>0$; if a single such value falls strictly below $\\gamma(z,T)=\\|T(z)^{-1}\\|^{-1}$, the convergence-from-above step fails and those examples are not covered by the theorems.","tokens_in":26690,"feed_emoji":"🧪","tokens_out":8454,"duration_ms":81095,"temperature":0.7,"pith_summary":"The paper aims to settle a basic question: when the spectral parameter enters an operator pencil nonlinearly, can the spectrum and pseudospectra (regions where the inverse of the pencil is nearly singular) be computed without spectral pollution from spurious points or spectral invisibility from missed true points? The authors propose two universal algorithms and prove that they converge for every nonlinear pencil that is continuous in the gap topology on operator graphs, a very weak continuity assumption. Under the Solvability Complexity Index hierarchy, spectra require three successive limits when only matrix entries are known and two when quadratic data about $T(z)^*T(z)$ and $T(z)T(z)^*$ are available; pseudospectra require two and one, respectively, and the paper proves that no method can use fewer limits. A separate theorem shows that Hermiticity does not lower these counts, in contrast to the linear case. Worked examples include nonlinear shifts, Klein–Gordon pencils, acoustic wave equations, fractional viscoelastic beams, and a delayed predator–prey model.","feed_headline":"One method computes nonlinear spectra, pollution-free","feed_subtitle":"A gap-continuous pencil is all the method needs; it provably avoids spurious and missed spectral points.","key_machinery":"The carrying object is the nonlinear injection modulus and its monotone finite truncations: $\\gamma(z,T)=\\min\\{\\sigma_{\\inf}(T(z)),\\sigma_{\\inf}(T(z)^*)\\}$, where $\\sigma_{\\inf}(A)=\\inf_{\\|x\\|=1}\\|Ax\\|$ measures how far an operator is from failing to be injectively bounded below. $\\gamma$ is continuous with respect to the gap metric on operator graphs, and the finite rectangular blocks $Q_{n_1}T(z)P_{n_2}^*$ give computable lower bounds whose limit is the modulus itself, while the unbounded truncations $T(z)P_{n_2}^*$ give upper bounds. The convergence of these approximations is monotone, and the Attouch–Wets metric on relatively closed subsets of $U$, which weighs accuracy by distance to the boundary of $U$, turns the informal goals of no pollution and no invisibility into a precise convergence theorem.","core_discovery":"The central claim is that the nonlinear spectral problem is solvable in full generality by one mechanism: the nonlinear injection modulus $\\gamma(z,T)=\\min\\{\\sigma_{\\inf}(T(z)),\\sigma_{\\inf}(T(z)^*)\\}$, which equals $\\|T(z)^{-1}\\|^{-1}$ when $T(z)$ is invertible and vanishes exactly on the spectrum. Truncated versions of this modulus, computed from finite rectangular matrix blocks, converge monotonically as the truncation size grows, from below for the finite-block approximations and from above for the unbounded truncations. Assembling these truncated moduli over dense grids yields Algorithms 1 and 2, which the paper proves converge to the pseudospectra and the spectrum in the Attouch–Wets metric for every pencil in the class $\\Omega^U_{\\mathrm{NL}}$, thereby excluding both pollution and invisibility. The same theorems give matching lower bounds in the Solvability Complexity Index hierarchy, showing optimality in any model of computation, and extend the lower bounds to Hermitian pencils.","pith_inferences":["I infer that the same monotone-truncation construction should carry over to pencils parametrized over manifolds or to operator-valued functions with an additional parameter, since the planar geometry enters only through the Attouch–Wets metric on subsets of $U$.","The unproved $z$-dependent-basis relaxation is the main internal risk: the three differential-operator demonstrations are not fully covered by Theorems 3.1 and 3.2 unless that relaxed assumption is proved, and a focused proof or counterexample would settle whether those examples are theorems or merely numerical evidence.","I expect that feeding the sharper computed pseudospectra into contour-integration solvers for time-fractional or wave-type equations would improve convergence estimates compared with numerical-range bounds, but that effect is not quantified in this paper.","A practical extension would be to compute only selected parts of the spectrum, such as eigenvalues inside a contour, by restricting the grids and localizing the injection-modulus search, since the current algorithms compute global pseudospectra first."],"forward_implications":["For any gap-continuous nonlinear pencil, pseudospectra can be approximated with two limits under $\\Lambda_1$ and one under $\\Lambda_2$, while spectra need three and two; these counts are optimal.","Because the $\\Lambda_2$ approximations converge monotonically from above, their output sets stay inside the true pseudospectra, so interval-arithmetic variants produce rigorous enclosures suitable for computer-assisted proofs.","Spectral pollution and invisibility are excluded by construction in the limit, not by problem-specific tuning, so the method applies equally to discrete, differential, and lattice-type pencils.","Hermiticity does not make nonlinear spectral problems easier: the required number of limits is unchanged, so the extra cost over the linear case is intrinsic to the nonlinearity.","The singular vectors used to compute the moduli also give approximations of pseudoeigenfunctions, and the resulting pseudospectral bounds are sharper than numerical-range-type estimates used in contour-based evolution solvers."],"supporting_citations":[{"why":"Provides the prior convergent contour method for holomorphic nonlinear problems that this paper generalizes and contrasts with the new universal approach.","marker":"[34]"},{"why":"Supplies the SCI-hierarchy lower-bound technique for linear spectral problems that the nonlinear lower bounds lift from.","marker":"[6]"},{"why":"Introduces the Solvability Complexity Index hierarchy and towers of algorithms, the classification framework used throughout the paper.","marker":"[46]"},{"why":"Provides the gap-metric and generalized-convergence facts underpinning Definition 2.4 and the perturbation bound for injection moduli.","marker":"[51]"},{"why":"Gives the previous SCI classification for Klein–Gordon spectra, the nonlinear problem the authors extend and compare against.","marker":"[75]"},{"why":"Supplies the descriptive-set-theoretic lower bound for the decision problem used in Proposition 3.5 to prove Hermitian lower bounds.","marker":"[28]"},{"why":"Shows how to compute smallest singular values of finite matrix blocks to arbitrary accuracy, used in Proposition 3.2 to turn matrix entries into computable injection moduli.","marker":"[31]"},{"why":"Provides the numerical-range-type pseudospectral bound for the fractional beam example that the new direct computation improves upon.","marker":"[30]"},{"why":"Uses injection moduli for spectral inclusion sets in the linear setting, the idea adapted here to nonlinear pencils.","marker":"[25]"}],"fun_headline_variants":["Nonlinear spectra tamed by one universal method","First convergent general method for nonlinear spectra","Injection modulus cracks nonlinear spectral problems","Pollution-free spectra for all nonlinear pencils","Universal solver for nonlinear spectra, proven optimal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the fixed basis vectors form a core of $T(z)$ and of $T(z)^*$ at every point $z$; if that core property fails, the monotone convergence of the truncated injection moduli from above, which is what excludes invisibility, can break down—and for several differential-operator examples the authors relax this to $z$-dependent bases without giving a proof that the convergence theorems still hold.","fun_headline_variants_meta":{"raw":{"variants":["Nonlinear spectra tamed by one universal method","First convergent general method for nonlinear spectra","Injection modulus cracks nonlinear spectral problems","Pollution-free spectra for all nonlinear pencils","Universal solver for nonlinear spectra, proven optimal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2372,"prompt_tokens":943,"completion_tokens":1429,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":1364}},"tokens_in":559,"tokens_out":1429,"duration_ms":9952,"temperature":1.0,"reasoning_tokens":1364,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:55:08.960326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 1 on any pencil in the stated class and compare its output to the true $Sp_\\epsilon(T)$ under the Attouch–Wets metric; any pencil where the limit differs would refute the convergence claim. A sharper falsifier targets the relaxed-basis assertion: for the acoustic wave pencil on the half-line with spectrum $\\{z:\\mathrm{Im}(z)\\ge 0\\}$, compute $\\gamma_{n_2}(z,T)$ using the paper's $z$-dependent orthonormal bases at a fixed $z$ with $\\mathrm{Im}(z)>0$; if a single such value falls strictly below $\\gamma(z,T)=\\|T(z)^{-1}\\|^{-1}$, the convergence-from-above step fails and those examples are not covered by the theorems.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the prior convergent contour method for holomorphic nonlinear problems that this paper generalizes and contrasts with the new universal approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Solvability Complexity Index hierarchy and towers of algorithms, the classification framework used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the gap-metric and generalized-convergence facts underpinning Definition 2.4 and the perturbation bound for injection moduli."},{"cited_title":"R¨ osler and C","cited_arxiv_id":null,"evidence_quote":"Gives the previous SCI classification for Klein–Gordon spectra, the nonlinear problem the authors extend and compare against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the descriptive-set-theoretic lower bound for the decision problem used in Proposition 3.5 to prove Hermitian lower bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows how to compute smallest singular values of finite matrix blocks to arbitrary accuracy, used in Proposition 3.2 to turn matrix entries into computable injection moduli."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the numerical-range-type pseudospectral bound for the fractional beam example that the new direct computation improves upon."},{"cited_title":"Chandler-Wilde, R","cited_arxiv_id":null,"evidence_quote":"Uses injection moduli for spectral inclusion sets in the linear setting, the idea adapted here to nonlinear pencils."}],"review_version":1}