{"id":"57bb4056-aa6d-4506-ba85-1f9bdaa9e3d6","arxiv_id":"2606.21541","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs eigenfunctions with eigenvalues converging to the conical operator and proves a spectral gap estimate for G-equivariant functions on small-scale desingularized special Lagrangian cones.","lead":"The paper constructs a finite number of eigenfunctions for the linearized self-shrinker operator on scaled desingularizations of G-invariant special Lagrangian cones and proves a spectral gap on the orthogonal complement in a Gaussian weighted space. This supplies the analytic foundation for constructing Type II blow-up solutions of Lagrangian mean curvature flow in a companion paper.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the sole external hypothesis. Because the manuscript treats this as given setup rather than a claim to be proved here, and because the spectral results are derived from it without further circularity, the argument is internally sound under the stated hypothesis. The low reader confidence stems from abstract-only access; with the full text the conditional nature remains the only soft spot, but it is explicitly declared rather than concealed.","tokens_in":1692,"tokens_out":292,"duration_ms":13862,"concrete_test":"Confirm that the companion paper (or a separate existence result) supplies at least one concrete example of a G-invariant cone C together with the family aL-bar satisfying the convergence; if the family fails to exist for the cones of interest, the spectral statements remain formally correct but inapplicable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper states its central results conditionally on the explicit setup assumption that a scaled family of G-invariant special Lagrangian desingularizations aL-bar exists and converges to the cone C as a↘0. All subsequent constructions (eigenfunction approximation, spectral gap on the orthogonal complement, identification of the scaling mode) are derived from the linearized operator on these surfaces in the Gaussian-weighted space. No internal inconsistency, unjustified step, or hidden assumption in the spectral analysis itself is apparent from the stated claims.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript assumes the existence of a scaled family of G-invariant special Lagrangian desingularizations a ¯L converging to a G-invariant special Lagrangian cone C as a ↓ 0. It studies the linearized self-shrinker operator on a ¯L in the Gaussian-weighted L^{2} space of G-equivariant functions. For 0 < a ≪ 1, it constructs any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, proves a spectral gap estimate on the orthogonal complement of these modes, and identifies the lowest eigenfunction with the scaling mode of the desingularization. These results supply the analytic foundation for Type II blow-up constructions of Lagrangian mean curvature flow in a companion paper.","tokens_in":1772,"tokens_out":417,"duration_ms":15307,"significance":"If the results hold, the work supplies key spectral tools for analyzing stability and finite-time singularities in Lagrangian mean curvature flow near conical singularities, directly enabling Type II blow-up constructions via desingularization. The finite-mode approximation, spectral gap on the complement, and explicit identification of the scaling mode are load-bearing for controlling the linearized dynamics. The paper's conditional framing on the desingularization family is clearly stated, permitting the spectral analysis to rest on standard theory for the linearized operator in the weighted space.","major_comments":[],"minor_comments":[{"comment":"§1 (Introduction): the precise definition of the Gaussian-weighted inner product and the precise domain of G-equivariant functions should be stated explicitly before the main theorems, rather than deferred to later sections.","section":"§1"},{"comment":"The statement that eigenvalues 'converge to those of the limiting conical operator' would benefit from a short reminder of the spectrum of the conical operator (e.g., reference to its explicit eigenvalues or prior computation) to make the approximation claim self-contained.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The report provides no specific major comments or points requiring detailed response.","responses":[],"tokens_in":1227,"tokens_out":50,"duration_ms":7482,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is the construction, for small a, of any finite number of eigenfunctions on a L-bar whose eigenvalues approach those of the limiting cone operator, together with a gap on the orthogonal complement in the G-equivariant Gaussian-weighted space. They also match the lowest mode to the scaling variation of the desingularization. This is presented as the analytic setup for the companion paper on singularities.\n\nThe work is new in its explicit handling of the convergence and gap for these particular surfaces rather than relying on a purely abstract spectral description. The approach follows standard linearized self-shrinker analysis but applies it directly to the family of G-invariant special Lagrangian desingularizations.\n\nThe central assumption is the existence of the scaled family a L-bar converging to the cone C; once that is granted, the spectral constructions appear to rest on ordinary perturbation and orthogonality arguments without obvious circularity. The abstract gives no derivations, so the quality of the error controls and the precise rate of convergence as a to 0 cannot be checked here, but nothing in the stated claims suggests a load-bearing flaw.\n\nThis is a technical note aimed at researchers already working on Lagrangian mean curvature flow and finite-time singularities. Readers who need the spectral basis for blow-up constructions will find the results directly usable. It is worth sending to referees because it supplies a verifiable tool rather than an existence claim alone.","headline":"This paper works out the eigenfunction approximations and spectral gap for the linearized operator on the scaled desingularizations, giving the concrete analytic input needed for their Type II blow-up construction.","tokens_in":2270,"tokens_out":358,"would_cite":false,"duration_ms":13056,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For small desingularizations of special Lagrangian cones, the linearized operator has eigenfunctions converging to the cone's spectrum with a gap on the complement.","keywords":["special Lagrangian cone","desingularization","Lagrangian mean curvature flow","linearized self-shrinker operator","eigenvalue convergence","spectral gap","G-equivariant functions","self-shrinkers"],"falsifier":"Finding a specific cone and desingularization family where the eigenvalues of the linearized operator on a L-bar do not converge to the conical eigenvalues as a goes to zero, or where no gap appears.","tokens_in":2566,"feed_emoji":"","tokens_out":662,"duration_ms":30091,"temperature":0.7,"pith_summary":"The paper proves that given a G-invariant special Lagrangian cone with a converging family of desingularizations, the linearized self-shrinker operator on the small desingularized surface admits eigenfunctions whose eigenvalues approach those of the limiting cone. It constructs any finite number of such modes and shows a spectral gap on the orthogonal complement in the space of G-equivariant functions. The lowest mode is shown to correspond to the scaling variation of the desingularization. This provides the spectral control necessary to study the stability of these surfaces under Lagrangian mean curvature flow near the singular cone.","feed_headline":"Eigenvalues converge on desingularized special Lagrangian cones","feed_subtitle":"For small scales, finite modes match the conical spectrum while a gap holds on the orthogonal complement and the scaling mode is identified.","key_machinery":"The linearized self-shrinker operator on the desingularized special Lagrangian in the Gaussian-weighted L2 space restricted to G-equivariant functions.","core_discovery":"Assuming the existence of G-invariant special Lagrangian desingularizations a L-bar converging to the cone C as a approaches 0, the analysis of the linearized self-shrinker operator in the Gaussian weighted L2 space shows that for small a, any prescribed finite set of eigenfunctions can be constructed with eigenvalues converging to the conical operator's eigenvalues, a spectral gap estimate holds on the orthogonal complement, and the lowest eigenfunction is identified with the scaling mode of the desingularization.","pith_inferences":["The result allows perturbation methods to produce solutions that blow up in finite time with Type II singularities.","Similar spectral analysis could be applied to other classes of cones or flows where desingularizations exist.","Explicit examples of such desingularizations would permit numerical checks of the eigenvalue convergence rates."],"forward_implications":["Any finite number of prescribed modes from the conical operator can be realized approximately on the desingularization for small a.","A spectral gap separates the approximated modes from the rest of the spectrum.","The lowest eigenfunction corresponds exactly to the scaling mode induced by the desingularization parameter.","This spectral structure forms the basis for constructing Type II blow-up solutions of the flow."],"fun_headline_variants":["Finite modes converge to conical eigenvalues in desingularizations","Spectral gap proven on complement of Lagrangian cone eigenfunctions","Scaling mode is the lowest eigenfunction for small desingularizations","Self-shrinker spectrum analyzed for G-invariant desingularizations"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The assumption that there exists a scaled family of G-invariant special Lagrangian desingularizations converging to the cone as the scale goes to zero.","fun_headline_variants_meta":{"raw":{"variants":["Finite modes converge to conical eigenvalues in desingularizations","Spectral gap proven on complement of Lagrangian cone eigenfunctions","Scaling mode is the lowest eigenfunction for small desingularizations","Self-shrinker spectrum analyzed for G-invariant desingularizations"]},"model":"grok-4.3","cost_usd":0.005585,"raw_usage":{"total_tokens":2649,"prompt_tokens":615,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":55849500,"prompt_tokens_details":{"text_tokens":615,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1969,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":615,"tokens_out":65,"duration_ms":18424,"temperature":1.0,"reasoning_tokens":1969,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:17:32.823115+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a specific cone and desingularization family where the eigenvalues of the linearized operator on a L-bar do not converge to the conical eigenvalues as a goes to zero, or where no gap appears.","supporting_citations":[],"review_version":1}