{"id":"079de4d4-cffd-401d-b23a-d6faa40f289b","arxiv_id":"2608.10493","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Arnol'd's chord conjecture is proved for weakly non-orderable contact manifolds satisfying a sharp Lagrangian displacement-energy bound, including many prequantization and Brieskorn manifolds.","lead":"This paper proves Arnol'd's chord conjecture for a new class of contact manifolds: whenever the contact manifold is weakly non-orderable and has rigid symplectizations, every closed Legendrian submanifold has a Reeb chord. The result covers Brieskorn manifolds, cosphere bundles of rank-one symmetric spaces, and most prequantization spaces, with a uniform bound on the minimal chord length.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Short-window RGW proof of Proposition 8 for arbitrary prequantizations is the fragile step; if d^2=0 fails, Theorem 14's contradiction collapses.","rationale":"I read the paper in good faith. The main proof of Theorem 14 is clean conditional on the displacement-type hypothesis, and the Liouville-fillable, hypertyght, and symplectically aspherical filling arguments are plausible and largely standard. The genuinely load-bearing step is the sharp lower bound e(L) ≥ ρ_L for totally rational Lagrangians in the symplectization; without it, the contradiction at the end of §2.1 disappears. For arbitrary prequantizations, this bound is established only by a short-window RGW argument that is sketched and that invokes the unpublished Daemi–Fukaya trilogy. The reader's weakest_assumption identifies exactly this point, and I agree with that assessment. My concern is not that the authors are dishonest or that the result is false; it is that a complete verification of the RGW d^2=0 statement in the present non-monotone, divisor-exterior setting is necessary before the strongest claims are accepted. Consequently, the reader's CONDITIONAL verdict remains appropriate: no change is needed.","tokens_in":10957,"tokens_out":14508,"duration_ms":149109,"concrete_test":"Re-derive the short-window RGW complex for the prequantization of S^2×S^2 with an integral non-monotone area form, so the base is neither symplectically aspherical nor monotone. Following [16, Thm 2.8 and 2.16] together with Lemma 17, verify explicitly that the components of the RGW boundary in Items (2) and (3) of [16, Thm 2.8] all have energy at least ρ_L and therefore vanish in an action window (−ρ_L, ε), and that no sphere bubble contained in the zero section contributes to d^2 in that window. If any non-zero count survives, Proposition 8's general case is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive premise is displacement-type (Definition 7): the sharp lower bound e(L) ≥ ρ_L for every totally rational (relatively spin) Lagrangian in SY. For arbitrary prequantizations, Proposition 8's proof of this bound is only a sketch (end of §2.3): it asserts that short-window RGW complexes are well-defined, with d^2=0 and non-trivial continuation maps, because contributions of disk-like RGW configurations have energy at least ρ_L by Lemma 17 and so vanish in an action window shorter than ρ_L. This is the exact step the contradiction in Theorem 14 needs: the lower bound is used to derive ℓ(h) ≥ T, contradicting the positive-loop construction with ℓ(h) < T. The argument depends on the unpublished Daemi–Fukaya trilogy [14–16] and on a non-monotone adaptation of [16, Thm 2.16] that is not written out. Lemma 17 only controls disk-like configurations with zero intersection with the zero section; sphere bubbles in the zero section are deferred to [16]. A gap here—for example, a sub-ρ_L sphere bubble contributing to d^2, or a failure of the compactification to apply outside their monotone hypotheses—would invalidate Theorem 1's arbitrary-prequantization relatively spin case. The Corollary 4 Hamiltonian formula also has an apparent T/S typo, but that is secondary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a new general form of Arnol'd's chord conjecture: for every contact form on a weakly non-orderable 'displacement-type' contact manifold, every closed Legendrian has a non-constant Reeb chord (Theorem 14), and the proof yields a uniform upper bound on the chord length. The main applications (Theorem 1 and Theorem 3) use Proposition 8 to identify large classes of displacement-type manifolds, including many prequantizations, Liouville-fillable manifolds, and links of isolated weighted homogeneous hypersurface singularities. The proof strategy relates Mohnke's Lagrangian construction to the Hofer displacement energy in the symplectization and to contact Hofer geometry of non-negative paths.","tokens_in":11111,"tokens_out":19131,"duration_ms":167079,"significance":"If the technical steps in Proposition 8 can be completed, this is a substantial breakthrough: it unifies and extends Mohnke's and Hutchings-Taubes results, establishes the chord conjecture for arbitrary prequantization spaces in the relatively spin case, and provides the first uniform bounds on chord length in this generality. The conceptual reduction of the chord conjecture to a displacement-energy lower bound plus weak non-orderability is elegant and likely to be influential. The proof of Theorem 14 is coherent and self-contained conditional on Definition 7, and Lemma 17 is a clean, useful computation. The main risk is the unfinished proof of the displacement-type property in the most general cases, where the argument relies on unpublished work and sketches.","major_comments":[{"comment":"The proof that short-window RGW Floer complexes satisfy d^2=0 in the general prequantization case is not written out and contains a specific gap. The text asserts that all RGW configurations contributing to Items (2) and (3) of [16, Theorem 2.8] have energy at least rho_L by Lemma 17, but Lemma 17 only applies to disk-like relative homology classes with zero intersection with the zero section; it says nothing about sphere bubbles contained in the zero section, whose positive symplectic area in M can be smaller than rho_L. Since d^2=0 and the non-triviality of the continuation maps are the justification for the displacement-type inequality e(L) >= rho_L, a failure here would invalidate the contradiction in Theorem 14 for the relatively spin case of arbitrary prequantizations. The appeal to a non-monotone adaptation of [16, Theorem 2.16] without presenting the adaptation is not sufficient for a proof in a research article.","section":"§2.3 (Proposition 8, general prequantization case, final two paragraphs)"},{"comment":"The displacement-type claim for prequantizations of monotone symplectic manifolds with minimal Chern number at least two is not proven in the manuscript. The argument presented for the symplectically aspherical case concludes that bubble configurations must be constant because the zero section is symplectically aspherical; in the monotone case the zero section contains holomorphic spheres, so this argument does not apply. The text defers to [40, Remark 3.11] for index estimates without stating or proving them, and it is unclear how the short-window argument excludes positive-area sphere bubbles in the zero section. Since Theorem 1 explicitly includes this case, the paper's statement of Proposition 8 goes beyond what is demonstrated.","section":"§2.3 (Proposition 8, monotone prequantization case)"}],"minor_comments":[{"comment":"In the formula h(t,x)=S−f(ϕ^{(1−t)T}_R x), the time parameter in the exponent should be S, not T, to generate ϕ^S; this appears to be a typo.","section":"§2.2 (proof of Corollary 4)"},{"comment":"The word 'mononotone' appears twice in place of 'monotone'; please fix these typos.","section":"§2.3 (Proposition 8)"},{"comment":"The notation 'minsupp_F ρ' is undefined; please define it explicitly when introducing the perturbation support.","section":"§2.3 (hypertight case)"},{"comment":"The notation H^D_2(E,i_ε(L);Z) is not defined; please clarify that it denotes the relative homology group of disk classes, or introduce a definition.","section":"Lemma 17"},{"comment":"The assertion that Mohnke's Lagrangian L is totally rational with rationality constant arbitrarily close to T is stated without a proof or reference; a brief justification would improve readability.","section":"§2.1 (proof of Theorem 14)"}],"recommendation":"major_revision","confidential_remarks":"The key step of Proposition 8 rests on the unpublished Daemi–Fukaya trilogy [14,15,16], and the non-monotone adaptation is only sketched. The editor may wish to consider whether acceptance is appropriate while the central technical tool remains a preprint. There is also some overlap with the forthcoming work of Z. Zhou mentioned in Remark 12, which may raise disclosure concerns. If the author can supply the missing RGW details, or restrict Theorem 1 to the cases where the proof is complete, the paper would be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: it ties Mohnke's Lagrangian construction to contact Hofer geometry through the notion of displacement-type, and proves the chord conjecture for every weakly non-orderable displacement-type contact manifold. That theorem (14) is clean and the applications are substantial—Brieskorn manifolds, many prequantizations, Liouville-fillable manifolds, and a uniform bound on chord length. I read the core argument carefully and it holds: the totally rational Lagrangian L built from K and the Reeb flow has rationality constant arbitrarily close to T, and if there is no chord, the canonical lift of phi^T displaces L. Then a positive loop gives a non-negative Hamiltonian of Hofer length < T generating phi^T, contradicting the sharp lower bound e(L) >= rho_L. The logic is coherent.\n\nWhere I agree with the stress-test note is the proof of Proposition 8 in the general prequantization case. The short-window RGW complex is load-bearing, and the paper only sketches why d^2=0: it asserts that disk-like RGW configurations have energy at least rho_L by Lemma 17, so they don't affect the window. But Lemma 17 only controls configurations with zero intersection with the zero section, and the non-monotone adaptation of [16, Thm 2.16] is not written out. The sphere bubbles in the zero section are deferred to [16], which is unpublished. This is a real gap in the exposition, not a manufactured one. It may be fillable, but a referee cannot verify it from the text. The monotone case with minimal Chern number at least two also relies on a sketch via [40]. For the aspherical and Liouville-fillable cases, the argument is complete modulo standard tools.\n\nThe Corollary 4 proof has a T/S typo in the displayed Hamiltonian formula, but that's minor.\n\nOverall: the central idea is good, the paper is honest about its dependence on the Daemi-Fukaya machinery, and the main theorem is a genuine step forward. I would send it to a strong journal and insist the referee gets a complete treatment of the RGW short-window argument, or the author explicitly marks the general prequantization case as conditional on [14-16]. The rest of the paper stands alone. This deserves serious refereeing.","headline":"A serious advance on the chord conjecture, with a clean main theorem and a fragile technical core in the general prequantization case; worth refereeing, but the sketch of Proposition 8 needs to be either completed or explicitly marked conditional.","tokens_in":11730,"tokens_out":3268,"would_cite":true,"duration_ms":29078,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D10","53D35","53D40","53D42"],"pacs":[],"model":"deepseek-v4-flash","headline":"The chord conjecture holds for every weakly non-orderable contact manifold of displacement type, yielding Reeb chords for all Legendrians in many prequantizations and Brieskorn manifolds.","keywords":["chord conjecture","Reeb chords","Legendrian submanifolds","contact orderability","Hofer displacement energy","prequantization","Brieskorn manifolds","RGW compactifications"],"falsifier":"Find a totally rational Lagrangian $L$ in the symplectization of a weakly non-orderable contact manifold whose Hofer displacement energy is strictly smaller than its rationality constant $\\rho_L$; the displacement-type premise would fail and the contradiction in Theorem 14 would collapse. Concretely, in a general prequantization one can search for an RGW configuration with energy below $\\rho_L$ and nonzero intersection with the zero section, which would invalidate the $d^2=0$ step in the short-window argument.","tokens_in":10654,"feed_emoji":"🔗","tokens_out":18032,"duration_ms":133315,"temperature":0.7,"pith_summary":"The paper aims to prove the chord conjecture—that every closed Legendrian submanifold has a non-constant Reeb chord for every contact form—for a large class of closed contact manifolds, characterized by two geometric properties. A manifold is weakly non-orderable when it admits a positive loop of contactomorphisms—a one-parameter loop of contact transformations with strictly positive contact Hamiltonian; it is of displacement type when every totally rational Lagrangian in its symplectization has Hofer displacement energy at least its rationality constant $\\rho_L$. The main theorem states that every weakly non-orderable displacement-type manifold satisfies the conjecture, and relatively spin Legendrians do so under the weaker Q displacement-type assumption. The applications cover prequantizations over symplectically aspherical and suitable monotone base manifolds, all relatively spin Legendrians in arbitrary prequantizations, and links of weighted homogeneous hypersurface singularities including Brieskorn manifolds. The same argument gives a uniform upper bound on the length of the shortest chord for forms proportional to a standard period-one form.","feed_headline":"Proved: chord conjecture holds on weakly non-orderable manifolds","feed_subtitle":"It yields Reeb chords for all Legendrians in many prequantizations and Brieskorn manifolds, with bounded length.","key_machinery":"The engine of the proof is a contradiction between two estimates on the Hofer displacement energy of one Lagrangian. Starting from a closed Legendrian $K$ with no Reeb chord of length up to $T$, the paper builds a Lagrangian $L$ in the symplectization from the images of $K$ under the Reeb flow for times in $[0,T-\\delta]$, with the endpoints pushed outward by the Liouville flow; $L$ is totally rational with rationality constant $\\rho_L$ arbitrarily close to $T$, and if no chord exists then the time-$T$ Reeb flow displaces $L$. The displacement-type property gives $e(L) \\ge \\rho_L \\approx T$. Weak non-orderability, however, supplies a positive loop of contactomorphisms whose non-negative contact Hamiltonian generates the time-$T$ Reeb flow with contact Hofer length strictly less than $T$; the canonical lift of this Hamiltonian displaces $L$ with Hofer energy $< T$, contradicting the lower bound. The technical work is in establishing the displacement-type inequality, especially in prequantizations, where a lemma comparing intersection number with the zero section to symplectic area (Lemma 17) and short action-window counts of RGW configurations—a specific holomorphic-curve compactification—are used.","core_discovery":"The central claim, stated as Theorem 14, is that the chord conjecture holds for every weakly non-orderable displacement-type contact manifold, and for relatively spin Legendrians in weakly non-orderable Q displacement-type manifolds. Weak non-orderability means a positive loop of contactomorphisms exists; displacement type means that for every totally rational Lagrangian $L$ in the symplectization, the Hofer displacement energy satisfies $e(L) \\ge \\rho_L$, where $\\rho_L$ is the rationality constant of $L$. The theorem is applied to prequantizations of closed symplectically aspherical manifolds, to prequantizations of monotone manifolds with minimal Chern number at least two, to Liouville-fillable prequantizations, to all relatively spin Legendrians in arbitrary prequantizations, and to links of isolated weighted homogeneous hypersurface singularities such as Brieskorn manifolds. It also yields Corollary 4: for $\\alpha = f \\alpha_0$ with $\\alpha_0$ the standard period-one form, a chord of length at most $2\\max f$ exists.","pith_inferences":["The same contradiction mechanism should extend the conjecture to all contact manifolds with periodic Reeb flow: they are weakly non-orderable, so proving the displacement-type inequality is the remaining step.","The uniform bound of twice the maximum of the conformal multiplier suggests that a quantitative version of the chord conjecture, with one chord whose length is controlled only by the contact form, may hold in all dimensions for this class.","The short-window configuration argument for general prequantizations looks replaceable: any Floer-theoretic framework that can rule out low-energy contributions in an action window shorter than the rationality constant would recover the same theorem, which is why alternative transversality approaches are mentioned as future work."],"forward_implications":["Every prequantization of a closed symplectically aspherical manifold satisfies the chord conjecture for every Legendrian.","Every prequantization of a closed monotone symplectic manifold with minimal Chern number at least two satisfies the chord conjecture.","In any prequantization, the chord conjecture holds for every relatively spin Legendrian, which includes all Legendrians diffeomorphic to spheres.","The chord conjecture holds for all links of isolated weighted homogeneous hypersurface singularities, in particular Brieskorn manifolds with all exponents at least two.","For a contact form that is a positive function multiple of the standard period-one form, a Reeb chord exists with length at most twice the maximum of the multiplier, uniformly in the Legendrian."],"supporting_citations":[{"why":"Supplies the construction of a Lagrangian from a Legendrian and its Reeb-flow images, with rationality constant close to the flow time.","marker":"[32]"},{"why":"Establishes the sharp lower bound on Hofer displacement energy of Lagrangians in geometrically bounded manifolds, the displacement-type inequality.","marker":"[11]"},{"why":"Provides the contact Hofer norm and canonical-lift estimates converting short non-negative contact paths into Hamiltonian displacements of small Hofer energy.","marker":"[38]"},{"why":"Gives the equivalence between orderability and the existence of positive or non-negative contact loops used to define weak non-orderability.","marker":"[22]"},{"why":"Supplies the result that a contactomorphism whose contact Hofer norm equals its Reeb-flow time forces strong orderability, the contradiction target.","marker":"[25, 26]"},{"why":"Provides the RGW compactifications and virtual perturbation counts used to define Floer complexes and continuation maps in short action windows for general prequantizations.","marker":"[14, 15, 16]"}],"fun_headline_variants":["Weakly non-orderable manifolds have Reeb chords","Reeb chords proven for weakly non-orderable contact manifolds","Chord conjecture solved on a new class of manifolds","Uniform bound on Reeb chord length for new class","Brieskorn manifolds now covered by chord conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every totally rational Lagrangian in the symplectization has Hofer displacement energy at least its rationality constant $\\rho_L$; in the most general prequantization case this is only sketched and depends on a short action-window counting argument whose foundations are supplied by a trilogy of preprints.","fun_headline_variants_meta":{"raw":{"variants":["Weakly non-orderable manifolds have Reeb chords","Reeb chords proven for weakly non-orderable contact manifolds","Chord conjecture solved on a new class of manifolds","Uniform bound on Reeb chord length for new class","Brieskorn manifolds now covered by chord conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000278,"raw_usage":{"total_tokens":1617,"prompt_tokens":871,"completion_tokens":746,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":666}},"tokens_in":487,"tokens_out":746,"duration_ms":6348,"temperature":1.0,"reasoning_tokens":666,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:19:47.003416+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a totally rational Lagrangian $L$ in the symplectization of a weakly non-orderable contact manifold whose Hofer displacement energy is strictly smaller than its rationality constant $\\rho_L$; the displacement-type premise would fail and the contradiction in Theorem 14 would collapse. Concretely, in a general prequantization one can search for an RGW configuration with energy below $\\rho_L$ and nonzero intersection with the zero section, which would invalidate the $d^2=0$ step in the short-window argument.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the construction of a Lagrangian from a Legendrian and its Reeb-flow images, with rationality constant close to the flow time."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the sharp lower bound on Hofer displacement energy of Lagrangians in geometrically bounded manifolds, the displacement-type inequality."},{"cited_title":"Shelukhin","cited_arxiv_id":null,"evidence_quote":"Provides the contact Hofer norm and canonical-lift estimates converting short non-negative contact paths into Hamiltonian displacements of small Hofer energy."},{"cited_title":"Eliashberg and L","cited_arxiv_id":null,"evidence_quote":"Gives the equivalence between orderability and the existence of positive or non-negative contact loops used to define weak non-orderability."}],"review_version":1}