{"id":"14bd2155-5a8c-498c-b6f0-45df9f878aee","arxiv_id":"2607.03152","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Almost-calibrated cohomogeneity-one Lagrangian mean curvature flows in C^n develop Type II singularities with precise curvature rate (T-t)^{-K/2} for admissible K,n, with cone pair tangent flow and smooth desingularization Type II limit.","lead":"The paper constructs Lagrangian mean curvature flows in complex space that form finite-time singularities with an exact, tunable curvature blow-up rate of order (T-t) to a negative power. This gives the first quantitative description of Type II singularities for this fully nonlinear geometric flow, with explicit tangent and blow-up models.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s strongest claim is an existence theorem for quantitatively precise Type II singularities of cohomogeneity-one LMCF. The construction is self-contained once the companion spectral theory is granted: the approximate solution (3), the modulation equations (Theorem 3.1), the L^2 evolution of the remainder (Lemma 3.2), the weighted C^{2,α} estimates in the three regions (Theorems 4.1, 4.18, Lemma 4.9), and the topological non-retraction argument (Corollary 5.5) form a closed chain. The reader correctly isolates the companion spectral gap as the sole external load-bearing assumption; within the present text that assumption is used cleanly and the numerology for admissible (n,K) is checked. No further soft spot (e.g., failure of the Liouville theorems 4.12–4.14, breakdown of the barriers, or gap in the Inverse Function Theorem of Proposition 2.10) is visible. Consequently the ACCEPT verdict stands and no adjustment is warranted. The concrete test simply reconfirms the black-box input that both the reader and the present analysis already flag as essential.","tokens_in":102712,"tokens_out":740,"duration_ms":6953,"concrete_test":"Independently verify the spectral gap (28) and the identification ϕ_{0,a}∼a^{-2}β_a of [SS26, Theorem 1.1 & Proposition 1.2] for the lowest two modes (K=2) in the model case n=3 (or n=4) by a direct numerical computation of the weighted Sturm–Liouville operator H_a on a truncated interval with the Gaussian weight dμ_a; if the computed gap is strictly less than 1-ε for small a, the modulation control of Section 3 fails and the box argument cannot close.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 is a rigorous existence construction of G-invariant LMCF solutions with the stated Type II rate (T-t)^{-K/2}, unique tangent flow a transverse pair of SL cones, and Type II limit a smooth SL desingularization aL. The argument is a Ważewski box (Section 5) around the approximate solution (3) built from the scale-dependent eigenfunctions of the companion [SS26]. The reader's weakest assumption correctly flags that the entire modulation analysis (Theorem 3.1, Lemma 3.2), energy estimates (Section 3), weighted Hölder estimates (Section 4, especially the inner-region Liouville argument of Proposition 4.15), and the topological exit map all rest on the spectral gap (28) and quantitative eigenfunction estimates of [SS26, Theorem 1.1]. Within the present manuscript, however, those ingredients are used as a black box in a logically consistent way: the geometric decomposition (Proposition 2.10) is justified by the Inverse Function Theorem once the L^2_a gap is granted, the modulation ODEs close under the smallness hypotheses of the box B, and the numerology of Propositions 5.3–5.4 is verified for the admissible (n,K). No internal inconsistency, circularity, or missing estimate appears in the 100+ pages of analysis. The only genuine external dependence is the companion paper itself; that is already acknowledged and does not constitute a flaw in the present argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs, for admissible pairs (n,K) (n≥4 and K≥2, or n=3 and K=2,3,4), a smooth almost-calibrated G-invariant Lagrangian mean curvature flow L_K(t) in C^n that develops a finite-time Type II singularity at the origin with the precise curvature rate sup |A| ∼ (T-t)^{-K/2}. The tangent flow is a transverse pair of cohomogeneity-one special Lagrangian cones, the Type II blow-up is a smooth cohomogeneity-one SL desingularization aL, and the initial Lagrangian-angle oscillation can be taken arbitrarily small. The argument reduces the G-invariant flow to a fully nonlinear scalar PDE for a profile function, expands the solution in the scale-dependent eigenfunctions of the companion paper [SS26], derives modulation equations and energy/Hölder estimates that keep the remainder small, and closes existence by a Ważewski box argument whose exit map would otherwise retract a K-ball onto its boundary.","tokens_in":103073,"tokens_out":825,"duration_ms":14272,"significance":"If correct, this is the first construction of finite-time LMCF singularities with a quantitatively precise curvature blow-up rate, answering the rate question left open by earlier Type-II results under the zero-Maslov assumption. It also supplies the first application of modulation/spectral methods to a fully nonlinear parabolic PDE and handles a non-L^2-integrable deformation of the tangent cone by working in the scale-dependent spaces L^2_a. The explicit rate, uniqueness of the tangent flow, and arbitrarily small initial angle oscillation make the examples useful for testing conjectures on singularity formation and for future gluing or surgery constructions. The dependence on the companion spectral theory is cleanly modular and does not diminish the geometric contribution of the present work.","major_comments":[],"minor_comments":[{"comment":"The dependence on the companion paper [SS26] is correctly acknowledged, but a short self-contained statement of the precise spectral-gap constant and the C^1-dependence of the eigenfunctions on a (beyond the citations to Theorems 1.1 and 1.2 of [SS26]) would make the present manuscript easier to read in isolation.","section":null},{"comment":"In Definition 5.1 the box parameters η±, γ, κ_in/out, ε_in/par/out, A_out are introduced with a long list of inequalities; a brief table or remark collecting the admissible ranges already verified in Propositions 5.3–5.4 would improve readability.","section":null},{"comment":"Remark 9 sketches a compact immersed zero-Maslov version via figure-eight caps and pseudolocality. The sketch is plausible but informal; either expand it into a short appendix with the necessary cutoff estimates or flag it more clearly as a heuristic outline.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., “Ważewski” vs. “Wazewski”, occasional missing spaces around ∼). These are purely cosmetic.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical, but the logical skeleton is complete and the numerology for the admissible (n,K) is carefully checked. The only external dependence is the companion spectral paper, which is already on arXiv and is used as a black box in a consistent way. I see no reason to delay acceptance pending further review of [SS26]; the present contribution stands on its own once that spectral theory is granted."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first construction of Lagrangian mean curvature flow singularities with an explicit Type II curvature rate of order (T-t)^{-K/2} for a range of K, together with unique tangent flow (transverse pair of cohomogeneity-one SL cones) and Type II limit a smooth SL desingularization. It answers Joy15 Problem 3.12(a) for this class of cones and starts from data with arbitrarily small Lagrangian-angle oscillation, so it is directly useful for Thomas-Yau type questions.\n\nWhat is new is the combination: they reduce the G-invariant flow to a fully nonlinear scalar PDE for the profile, expand around the shrinking family a(τ)L using the scale-dependent eigenfunctions of the companion [SS26], derive modulation ODEs for a and the coefficients b_i, control the remainder by energy estimates plus weighted Hölder estimates (barriers outside, blow-up + Liouville inside), and close existence by a Ważewski box whose exit map would retract a ball onto its sphere. The numerology that makes the box work for n≥4 (any K≥2) and n=3 (K=2,3,4) is carefully checked. The logical skeleton is complete and standard for the method; the estimates are written with explicit parameter dependence.\n\nThe only real external dependence is the spectral gap and quantitative eigenfunction estimates of [SS26]. That is acknowledged, used as a black box in a consistent way (geometric decomposition via IFT, modulation equations close under the box smallness), and is not circular. Soft spots are the usual ones for this style of paper: 100+ pages of estimates leave room for calculation slips, and the construction is cohomogeneity-one, so it does not yet give generic singularities. Neither undercuts the central claim.\n\nThis is for people working on LMCF, special Lagrangians, or Type II singularities of geometric flows. The math looks solid, the citation pattern is appropriate, and a serious editor should send it to referees. I would cite it and bring it to reading group.","headline":"Solid, first quantitative Type II rates for LMCF via modulation + box; rests cleanly on the companion spectral theory and is worth engaging.","tokens_in":103696,"tokens_out":519,"would_cite":true,"duration_ms":11312,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53E10","53C38","35K55"],"pacs":[],"model":"grok-4.5","headline":"Lagrangian mean curvature flows can form Type II singularities with exact curvature blow-up rate (T-t)^{-K/2}, even when started arbitrarily close to special Lagrangians.","keywords":["Lagrangian mean curvature flow","Type II singularity","special Lagrangian","cohomogeneity-one","modulation analysis","spectral gap","curvature blow-up rate","almost-calibrated"],"falsifier":"Either exhibit a zero-Maslov almost-calibrated Lagrangian mean curvature flow whose curvature blows up at a rate strictly between two consecutive powers (T-t)^{-k/2}, or prove that no such spectral gap exists for the linearized operator on the cohomogeneity-one special Lagrangians used in the construction.","tokens_in":103606,"feed_emoji":"∞️","tokens_out":773,"duration_ms":9123,"temperature":0.7,"pith_summary":"The paper constructs almost-calibrated, group-invariant Lagrangian mean curvature flows in complex n-space that develop finite-time singularities whose curvature blows up at a precise rate of order (T-t)^{-K/2}, for integers K at least 2 (with a short list of allowed K when n=3). The flows begin from data whose Lagrangian angle can be made arbitrarily close to constant, so they sit arbitrarily near the special Lagrangian condition. At the singular time the parabolic-scale tangent flow is a pair of special Lagrangian cones, while a faster Type II rescaling converges to a smooth special Lagrangian that desingularizes those cones. The construction works by modulating around a one-parameter family of shrinking special Lagrangian desingularizations and controlling the error with spectral gap estimates, thereby giving the first quantitative blow-up rates for this fully nonlinear geometric flow.","feed_headline":"Exact blow-up rates for Type II Lagrangian singularities","feed_subtitle":"Flows near special Lagrangians form singularities with curvature ~ (T-t)^{-K/2}","key_machinery":"A modulation analysis that expands the rescaled profile as a linear combination of the first K scale-dependent eigenfunctions of the linearized operator H_a about a shrinking family of special Lagrangian desingularizations, plus an orthogonal remainder controlled by the spectral gap; a Ważewski box argument then produces a solution that stays inside a carefully chosen shrinking set of coefficient bounds for all future time.","core_discovery":"For each admissible pair (n,K) there exists a smooth, non-compact, properly embedded, exact, almost-calibrated, G-invariant Lagrangian mean curvature flow in C^n that forms a finite-time Type II singularity at the origin with curvature satisfying 0 < liminf (T-t)^{K/2} sup|A| ≤ limsup (T-t)^{K/2} sup|A| < ∞, whose tangent flow is a transverse pair of cohomogeneity-one special Lagrangian cones and whose Type II blow-up is a smooth cohomogeneity-one special Lagrangian desingularization; the oscillation of the Lagrangian angle of the initial data can be taken arbitrarily small.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Exact Type II blow-up rates in Lagrangian mean curvature flow","LMCF forms Type II singularities with rate (T-t)^{-K/2}","Precise curvature rates for almost-calibrated Type II singularities","Controlled Type II blow-ups from near-special Lagrangian initial data","Quantified finite-time Type II singularities in cohomogeneity-one LMCF"],"cache_read_input_tokens":96512,"weakest_assumption_plain":"The whole argument depends on the existence of a quantitative spectral theory for the linearized operator about every small desingularization, including a spectral gap for the remainder after the first K modes; that theory is taken from a companion paper and is not proved here.","fun_headline_variants_meta":{"raw":{"variants":["Exact Type II blow-up rates in Lagrangian mean curvature flow","LMCF forms Type II singularities with rate (T-t)^{-K/2}","Precise curvature rates for almost-calibrated Type II singularities","Controlled Type II blow-ups from near-special Lagrangian initial data","Quantified finite-time Type II singularities in cohomogeneity-one LMCF"]},"model":"grok-4.5","effort":"low","cost_usd":0.006886,"raw_usage":{"total_tokens":1773,"prompt_tokens":842,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":68860000,"prompt_tokens_details":{"text_tokens":842,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":833,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":842,"tokens_out":98,"duration_ms":7590,"temperature":1.0,"reasoning_tokens":833,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T04:30:00.565059+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Either exhibit a zero-Maslov almost-calibrated Lagrangian mean curvature flow whose curvature blows up at a rate strictly between two consecutive powers (T-t)^{-k/2}, or prove that no such spectral gap exists for the linearized operator on the cohomogeneity-one special Lagrangians used in the construction.","supporting_citations":[],"review_version":1}