{"id":"28c44c75-1de2-41ac-a53f-cdd590291aa9","arxiv_id":"2607.01809","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For non-resonant logarithmic simple co-rank-one foliations on threefolds, reduced tangential arcs live on the invariant separatrix divisor, so foliated discrepancies equal ordinary log discrepancies of the normalised branch–conductor adjunction pairs via an Ein–Mustaţă–Yasuda codimension formula.","lead":"This paper measures singularities of certain three-dimensional foliations by studying formal curves that stay tangent to the foliation. It reduces those measurements to ordinary singularity theory on the surfaces and curves that make up the foliation’s separatrix system, giving new tests for log canonicity and minimal discrepancies.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the paper's own non-resonance hypothesis.","rationale":"The central claim (tangential EMY formula via branch–conductor reduction) rests on reduced-arc confinement, which the paper both proves under non-resonance and shows fails without it. All subsequent steps (seminormal coequaliser, crepant adjunction, independence under common refinements, ordinary EMY/MJ on strata) are standard once confinement is granted, and the manuscript carefully tracks the restrictions (toroidal category, branch vs conductor data, relative MJ). The reader's CONDITIONAL verdict already encodes exactly these scope limits; no stronger objection (e.g., a gap in the coequaliser or a failure of crepancy) appears. The concrete test simply reconfirms the elementary residue calculation that the paper already uses for sharpness.","tokens_in":40203,"tokens_out":517,"duration_ms":5019,"concrete_test":"Independently recompute the leading coefficient of γ*ω in the pure logarithmic model of Thm 4.1 for a concrete resonant triple (λ1,λ2,λ3)=(1,1,-1) with a=(1,1,2) and a non-resonant triple (1,1,1) with a=(1,1,1); confirm that the t^{A-1} term vanishes exactly when a·λ=0 and is nonzero otherwise, matching Prop. 4.3 and the confinement claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (positive non-resonance of residues, Def. 2.1) is correctly identified as load-bearing for the confinement theorem (Thm 4.1 / 1.2) that underpins the whole branch–conductor reduction and thus Cor. 1.4 / Thm 12.3. The paper itself proves sharpness (Prop. 4.3) and repeatedly restricts statements to the non-resonant logarithmic simple adapted toroidal sector; the comparison a_tan = A_F is likewise restricted to branch data. No further internal inconsistency or hidden gap in the reduction chain (confinement \to coequaliser presentation of reduced arcs \to crepant adjunction \to ordinary EMY on strata) is visible on a careful reading of the local charts and functoriality arguments. The theory is deliberately narrow; the manuscript already flags the limits.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper develops a tangential arc-space approach to foliated discrepancies for logarithmic simple co-rank one foliations on threefolds, relative to a fixed invariant normal-crossing separatrix divisor. In the non-resonant logarithmic simple adapted setting, reduced tangential arcs centred on the tangential locus are confined to the invariant divisor (Theorem 4.1 / Theorem 1.2). The reduced tangential sector is then presented by the seminormal branch–conductor system (coequaliser of reduced arc functors). Foliated adjunction transfers the discrepancy calculus to ordinary log pairs on the normalised branches and conductors; the Ein–Mustaţă–Yasuda theorem on those strata yields a tangential codimension formula (Corollary 1.4 / Theorem 12.3) identifying logarithmic codimensions of adapted toroidal tangential divisorial cylinders with the tangential discrepancies a_tan. For adapted toroidal invariant divisors read on normalised branches one has a_tan = A_F (usual foliated log discrepancy). Applications include toroidal tangential inversion of adjunction, branch–conductor descriptions of non-lc/non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of tmld_tor, and a relative Mather–Jacobian refinement on the canonical image separatrix system.","tokens_in":40466,"tokens_out":1054,"duration_ms":9335,"significance":"If the reduction chain holds, the paper supplies a clean arc-space dictionary for a restricted but natural class of foliated singularities (logarithmic simple non-resonant, adapted toroidal). The confinement argument is elementary and explicit in formal power series; the subsequent transfer of ordinary EMY / Mather–Jacobian formulas to the branch–conductor strata is carefully functorial under adapted blow-ups and crepant under common refinements. The comparison a_tan = A_F for branch data, the lower-semicontinuity statement for tmld_tor, and the relative MJ refinement are concrete contributions that sit usefully between Carter’s jet schemes of foliations and the foliated MMP of Cascini–Spicer–Svaldi. The theory is deliberately narrow and the manuscript flags its own limits (non-resonance, reduced infinite arcs, image-system MJ); that honesty is a strength rather than a defect.","major_comments":[{"comment":"The central load-bearing hypothesis is positive non-resonance of the logarithmic residues (Definition 2.1). Theorem 4.1 / Theorem 1.2 and the whole branch–conductor presentation rest on it; Proposition 4.3 correctly shows sharpness. The manuscript already restricts every global statement to the non-resonant logarithmic simple adapted toroidal sector and never claims the resonant or second simple type. No further major technical gap is visible in the reduction chain (confinement → coequaliser of reduced arcs → crepant adjunction → ordinary EMY on strata). The comparison a_tan = A_F is likewise correctly restricted to branch data (Corollary 1.5 / Corollary 11.3). I therefore raise no load-bearing objection that would require a major revision of the argument.","section":null}],"minor_comments":[{"comment":"The introduction and abstract correctly emphasise that only reduced infinite arcs are used, yet several later passages still allude to Carter jet schemes without always repeating the reduced-infinite restriction. A single clarifying sentence at the start of §4 would help the reader.","section":null},{"comment":"Notation for the two discrepancy conventions (α(E;G,Δ_W) versus A_F) is introduced in Notation 3.2 but reappears with slight variations later; a short table or a consistent choice of symbol throughout would improve readability.","section":null},{"comment":"The Mather–Jacobian section (§16) is long and partly independent of the main EMY transfer. A brief roadmap at the beginning of that section, or a clearer separation of the relative versus intrinsic statements, would help.","section":null},{"comment":"Examples 19.1–19.7 are useful local models; a short pointer in the introduction to which example illustrates coefficient-one cancellation versus transverse defect would make them easier to locate.","section":null},{"comment":"Minor typographical inconsistencies appear in the arXiv version (e.g., spacing around “Ein–Mustaţă–Yasuda”, occasional missing accents). These are easily cleaned.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is carefully written and the technical core is sound within its stated hypotheses. The main risk for a general audience is that the hypotheses (non-resonance + fixed adapted toroidal category + reduced infinite arcs) make the results look narrower than the abstract might suggest; the authors already flag this, so I do not regard it as a defect. Fit for a solid specialised journal in algebraic geometry / singularities is good; for a very broad venue the restricted scope might be an issue, but that is an editorial rather than a mathematical question."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is that Corrêa gives a clean geometric reduction: under positive non-resonance, reduced tangential arcs stay on the invariant divisor, so the tangential sector is presented by the normalised branch–conductor system, and ordinary EMY (and MJ) on the adjunction pairs become a tangential codimension formula. That is new relative to Cascini–Spicer, Spicer–Svaldi, Carter, and the classical arc-space literature, and it yields the first arc-space lower-semicontinuity statement for a toroidal tangential foliated mld-type invariant in this sector, plus inversion of adjunction and non-lc/non-klt descriptions on the strata.\n\nWhat works well is the honesty of the setup. Theorem 4.1 is elementary formal power series: the leading coefficient of γ*ω is (a·λ)∏c_i, nonzero by non-resonance. The paper proves sharpness (Prop. 4.3) and the finite-jet obstruction (Ex. 4.5), so the choice of reduced infinite arcs is justified rather than hand-waved. Functoriality of the seminormal pushout, crepant adjunction compatibility, and the transfer of EMY to strata are written carefully in SNC charts. Comparison a_tan = A_F is correctly restricted to branch data; MJ is relative to the image system unless separatrices algebraise. Circularity is low: discrepancies come from ordinary formulas after the reduction.\n\nSoft spots are mostly the ones the paper already flags. Non-resonance is load-bearing; without it the whole branch–conductor presentation collapses. Scope is deliberately narrow (log simple non-resonant co-rank one on threefolds, fixed adapted toroidal category). The manuscript is long and the proofs are not machine-checked, so a referee will want to audit the local charts and the common-refinement arguments, but I do not see a hidden gap in the reduction chain.\n\nThis is for people working on foliated MMP or arc-space discrepancies who need a usable tangential calculus in the non-resonant logarithmic sector. It deserves a serious referee. I would engage with it and expect to cite the confinement and the tangential EMY formula when the setting matches.","headline":"Solid, carefully scoped arc-space reduction for non-resonant logarithmic foliations; the confinement theorem and tangential EMY transfer are real and usable inside the stated sector.","tokens_in":41089,"tokens_out":552,"would_cite":true,"duration_ms":6852,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","14B05","32S65","14E18"],"pacs":[],"model":"grok-4.5","headline":"Tangential arcs on non-resonant foliations reduce foliated discrepancies to ordinary log-pair calculations on branches and conductors.","keywords":["foliated discrepancies","tangential arcs","separatrix–conductor system","Ein–Mustaţă–Yasuda","Mather–Jacobian discrepancy","logarithmic foliations","minimal log discrepancy","adjunction"],"falsifier":"In a pure logarithmic model whose residues satisfy a positive resonance relation, exhibit an explicit reduced formal arc that is tangent yet whose generic point leaves the invariant divisor; if such arcs exist, confinement (and therefore the codimension formula) fails.","tokens_in":41022,"feed_emoji":"📐","tokens_out":774,"duration_ms":6201,"temperature":0.7,"pith_summary":"The paper shows that for logarithmic simple co-rank one foliations on threefolds, when residues satisfy a positive non-resonance condition, reduced infinite arcs that stay tangent to the foliation and centre on a prescribed invariant divisor cannot leave that divisor. Those arcs are therefore exactly the ordinary arcs on the divisor, which can be presented by a finite normalised system of branches and their pairwise conductors. Foliated adjunction turns the branches and conductors into ordinary log pairs; the classical Ein–Mustaţă–Yasuda codimension formula on those pairs then becomes a tangential codimension formula that recovers a new tangential discrepancy. On adapted toroidal invariant divisors read on branches, this tangential discrepancy coincides with the usual foliated log discrepancy of the foliated MMP. The same identification yields toroidal inversion of adjunction, a branch–conductor description of the tangential non-lc and non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of the toroidal tangential minimal log discrepancy, and a relative Mather–Jacobian refinement on the image separatrix system. A sympathetic reader cares because the construction supplies the first arc-space calculus that computes foliated discrepancies by ordinary discrepancy calculations on a finite collection of surfaces and curves.","feed_headline":"Tangential arcs turn foliated discrepancies into ordinary ones","feed_subtitle":"Non-resonant arcs stay on the separatrix, so branch–conductor pairs compute the discrepancies","key_machinery":"The reduced tangential arc-confinement theorem (and its coequaliser presentation by the normalised separatrix–conductor system). Non-resonance forces every reduced tangential formal arc centred on the invariant divisor to factor through that divisor; the resulting coequaliser of branch and conductor arc spaces carries all subsequent codimension and discrepancy calculations.","core_discovery":"In the non-resonant logarithmic simple adapted setting, the reduced tangential arc sector of a co-rank one foliation on a threefold is identified with the ordinary reduced arc space of a fixed invariant normal-crossing separatrix divisor, and is therefore presented by the coequaliser of arcs on the normalised branches and conductors. Foliated adjunction transfers the discrepancy calculus to ordinary log pairs on those strata; applying the Ein–Mustaţă–Yasuda theorem there yields that the tangential logarithmic codimension of every adapted toroidal tangential divisorial cylinder equals q times the corresponding tangential discrepancy, which for branch data agrees with the usual foliated discre","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Tangential arcs reduce foliated discrepancies to ordinary log pairs","Separatrix branches and conductors compute the tangential discrepancies","Non-resonant arcs confine to the fixed invariant separatrix divisor","Foliated adjunction transfers discrepancies to branch-conductor pairs","Arc codimensions of toroidal cylinders equal tangential discrepancies"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The residues of the logarithmic form never form a positive integer relation; if they do, reduced tangential arcs can escape the prescribed invariant divisor and the whole branch–conductor reduction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Tangential arcs reduce foliated discrepancies to ordinary log pairs","Separatrix branches and conductors compute the tangential discrepancies","Non-resonant arcs confine to the fixed invariant separatrix divisor","Foliated adjunction transfers discrepancies to branch-conductor pairs","Arc codimensions of toroidal cylinders equal tangential discrepancies"]},"model":"grok-4.5","effort":"low","cost_usd":0.005588,"raw_usage":{"total_tokens":1546,"prompt_tokens":826,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":55880000,"prompt_tokens_details":{"text_tokens":826,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":655,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":826,"tokens_out":65,"duration_ms":11406,"temperature":1.0,"reasoning_tokens":655,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T08:34:56.821145+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"In a pure logarithmic model whose residues satisfy a positive resonance relation, exhibit an explicit reduced formal arc that is tangent yet whose generic point leaves the invariant divisor; if such arcs exist, confinement (and therefore the codimension formula) fails.","supporting_citations":[],"review_version":2}