{"id":"b89f38ff-3256-42f9-aa3f-b27d75c1c94c","arxiv_id":"2606.16270","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Products of spheres yield n-invertible cotangent bundles, so every compact Legendrian in ST*M has a Reeb chord of length at most the n-invertibility capacity.","lead":"The paper shows that if a manifold M is a product of spheres (or admits a submersion from one), every compact Legendrian in its unit cotangent bundle has a Reeb chord for any contact form. It does so by introducing an n-invertibility capacity and combining exact embeddings with Viterbo restriction maps.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript supplies a complete, self-contained geometric argument that converts the non-existence of short Reeb chords into an exact embedding, then uses Viterbo restriction and a filtered Künneth isomorphism to force the unit of SH(T*Λ) to vanish—an impossibility. The only non-elementary algebraic input is the string-topology calculation that realises n-invertibility for products of spheres; that calculation is classical in spirit and is isolated from the new geometric work. The reader’s weakest-assumption diagnosis is therefore accurate, yet it does not rise to a load-bearing flaw: the morphism is a theorem of the literature and the explicit classes are constructed by hand in §2.4. Consequently the ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":40883,"tokens_out":588,"duration_ms":6255,"concrete_test":"Independently verify the single-sphere identity A * Δ(B * [pt]) = [S^n] by reconstructing the completing-manifold class B = γs from the generalized section s : RP^n \to ST S^n of §2.4.2 and checking that the Chas–Sullivan product and BV operator reproduce the claimed relation in the bordism model of [BC25]; if the identity fails for even one n, the n-invertibility of T*(S^{n1} \times \tau \times S^{nk}) collapses and Theorem 10 no longer applies.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central chain (exact embedding of D_εT*Λ \times D(a) when no short chords exist \to Viterbo restriction killing PSS(H^n) \to unit vanishes in SH(T*Λ) by filtered Künneth \to contradiction with the unit in string topology) is internally consistent. The reader correctly flags the Viterbo–Abbondandolo–Schwarz–Abouzaid BV-algebra morphism H*(ΛM) \to SH(T*M) and the explicit string-topology calculation A * Δ(B * [pt]) = [S^n] (and its product extension) as the place where the argument is most dependent on prior work. Those ingredients are standard and are isolated in [BC25] and classical references; the new geometric constructions (Theorem 6, the filtered Künneth of Theorem 8, and the truncated restriction of Theorem 11) are developed carefully in the appendices and do not introduce circularity. No hidden assumption that would make c_ni infinite for the product-of-spheres case appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces n-invertibility of a Liouville manifold W (Definition 1) and the associated capacity c_ni(Ω) (Definition 2), then proves that every Legendrian Λ ⊂ ∂Ω bounds a non-constant Reeb chord of period at most c_ni(Ω) whenever the capacity is finite (Theorem 3). The argument proceeds by constructing an exact embedding D_εT*Λ × D(a) when no short chords exist (Theorem 6), applying the Viterbo restriction map (Theorem 7) and a filtered Künneth isomorphism (Theorem 8) to force the unit to vanish in SH(T*Λ), and obtaining a contradiction with string topology. As applications, the authors show that T*(product of spheres) is n-invertible via explicit string-topology classes (Theorem 4), deduce the Arnold chord conjecture for ST*M whenever M admits a submersion from such a product (Theorem 10), and bound rationality constants of aspherical Lagrangians in unit disk cotangent bundles of tori (Theorem 5) via truncated Viterbo restriction (Theorem 11).","tokens_in":41148,"tokens_out":746,"duration_ms":6167,"significance":"The work supplies a uniform, capacity-based upper bound on Reeb-chord lengths that recovers Zhou’s vanishing-SH results and extends them to aspherical bases such as T^n, where higher dilations and orientation tricks are unavailable. The geometric embedding of stabilized codisk bundles, the filtered Künneth map, and the truncated restriction for non-exact embeddings are carefully developed and of independent interest. The string-topology calculations that establish n-invertibility for products of spheres are topological and isolated from Floer theory, giving a clean, falsifiable criterion. Together these results settle the Arnold chord conjecture for a natural class of unit cotangent bundles and give the first rationality-constant bounds inside T*T^n.","major_comments":[],"minor_comments":[{"comment":"In Definition 1 and the subsequent persistence refinement, the notation for the ideal �_c and its colimit � is dense; a short clarifying sentence after Lemma 1.1 would help the reader track the filtration.","section":null},{"comment":"Theorem 6 is stated for a Legendrian isotopy satisfying a strict inequality y*α > a dt; the subsequent reduction to the Reeb flow (Theorem 9) uses the non-strict version. A one-line remark on the limiting argument would remove any ambiguity.","section":null},{"comment":"In §2.4.3 the homotopy between the classes P and D is continuous but not smooth; while the bordism-class model absorbs this, a parenthetical note that smoothing is possible would be reassuring.","section":null},{"comment":"Appendix A.1.2 introduces the constants 7 and 3 without immediate motivation; a forward reference to the action-window estimates that follow would improve readability.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., “n-invertible domains always have dimension 2n” versus the later use of half-dimension convention; “�P^n” for RP^n). These are easily corrected.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is ready for publication essentially as is. The dependence on the Viterbo–Abbondandolo–Schwarz–Abouzaid BV morphism and on the authors’ earlier work [BC25] is standard and cleanly isolated; no circularity or hidden assumption threatens the main theorems. The contemporaneous work of Guo–Zhou is properly acknowledged. I see no reason for further delay."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives a clean upper bound on the shortest Reeb chord of a Legendrian in the boundary of a Liouville domain: the n-invertibility capacity c_ni. When that capacity is finite, every Legendrian has a chord of length at most c_ni. The payoff is Theorem 10: if M is the image of a submersion from a product of spheres (so T^n is included), every compact Legendrian in ST*M has a Reeb chord for every contact form. That fills a genuine gap; higher-dilation and orientation tricks do not reach aspherical bases.\n\nWhat is new is the capacity itself (Definitions 1–2) and the explicit string-topology calculation that products of spheres are n-invertible (Theorem 4). The geometric step (Theorem 6) is a mild variant of known 1-jet neighbourhood ideas, and Viterbo restriction (exact and truncated) is taken from Zhou and the classical sources; the authors develop the filtered Künneth and the no-escape arguments carefully in the appendices so the chain is self-contained. The logical path is transparent: no short chords give an exact embedding of a stabilized codisk bundle, restriction kills the unit after Künneth, and string topology supplies the contradiction. The rationality-constant bound for aspherical Lagrangians in T*T^n is a nice extra application of the truncated map.\n\nThe soft spot is the dependence on the BV-algebra morphism H*(ΛM) → SH(T*M) and the explicit classes A, B that invert the top class. Those are isolated in the authors’ earlier framework and classical references; if the morphism fails to preserve the relevant products the capacity becomes infinite and the contradiction vanishes. That is a real but standard reliance, not circularity. The rest of the Floer package looks solid.\n\nThis is for people working on Reeb chords, symplectic cohomology capacities, or the Arnold conjecture on aspherical manifolds. It deserves a serious referee. I would accept it for peer review and would cite the capacity and the T^n case.","headline":"Solid quantitative chord bound via a new capacity that settles Arnold for ST*T^n and submersion images of products of spheres.","tokens_in":41745,"tokens_out":523,"would_cite":true,"duration_ms":5929,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D42","53D12","57R17"],"pacs":[],"model":"grok-4.5","headline":"If a Liouville domain is n-invertible, every Legendrian on its boundary has a Reeb chord of length at most the n-invertibility capacity.","keywords":["Reeb chords","Viterbo restriction","n-invertibility","Legendrian","Arnold chord conjecture","symplectic cohomology","string topology","rationality constant"],"falsifier":"An explicit compact Legendrian in the unit cotangent bundle of the n-torus that admits no Reeb chords for some contact form, or a computation showing that the unit never enters the ideal generated by the top PSS classes.","tokens_in":41785,"feed_emoji":"🔀","tokens_out":548,"duration_ms":4726,"temperature":0.7,"pith_summary":"The paper proves that the non-existence of short Reeb chords on a Legendrian forces an exact embedding of a stabilized disk cotangent bundle into the ambient Liouville domain. Viterbo restriction then transfers symplectic-cohomology structures, and when the domain is n-invertible the unit is forced to vanish after restriction, which is impossible for cotangent bundles. Consequently every compact Legendrian in the unit cotangent bundle of a manifold that admits a submersion from a product of spheres (for example the n-torus) admits a Reeb chord for every contact form. The same circle of ideas yields an upper bound on the rationality constants of aspherical non-exact Lagrangians inside the unit codisk bundle of the torus.","feed_headline":"Every Legendrian on an n-invertible boundary has a short Reeb chord","feed_subtitle":"n-invertibility forces chords on products of spheres and bounds rationality constants on the torus","key_machinery":"n-invertibility capacity: the infimal action at which the unit of symplectic cohomology enters the smallest Δ-invariant ideal generated by the image of the top-degree PSS classes; non-vanishing of this capacity produces the contradiction after Viterbo restriction.","core_discovery":"Every Legendrian in the contact boundary of a Liouville domain Ω bounds a non-constant Reeb chord of period at most the n-invertibility capacity of Ω, whenever that capacity is finite. In particular the Arnold chord conjecture holds for the unit cotangent bundle of any manifold that is the base of a submersion from a product of spheres.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["n-invertible domains force short Reeb chords on every boundary Legendrian","Legendrians in ST*M always admit Reeb chords if M is sphere-product base","Finite n-invertibility capacity bounds Reeb periods for all compact Legendrians","Arnold chord conjecture holds for unit cotangents of n-invertible manifolds","Inverting degree-n classes obstructs chordless Legendrians via Viterbo maps"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The isomorphism that identifies string topology of the free loop space with symplectic cohomology of the cotangent bundle must preserve the products and BV operators that invert the top-degree classes.","fun_headline_variants_meta":{"raw":{"variants":["n-invertible domains force short Reeb chords on every boundary Legendrian","Legendrians in ST*M always admit Reeb chords if M is sphere-product base","Finite n-invertibility capacity bounds Reeb periods for all compact Legendrians","Arnold chord conjecture holds for unit cotangents of n-invertible manifolds","Inverting degree-n classes obstructs chordless Legendrians via Viterbo maps"]},"model":"grok-4.5","effort":"low","cost_usd":0.005388,"raw_usage":{"total_tokens":1440,"prompt_tokens":717,"num_sources_used":0,"completion_tokens":108,"cost_in_usd_ticks":53880000,"prompt_tokens_details":{"text_tokens":717,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":615,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":717,"tokens_out":108,"duration_ms":5511,"temperature":1.0,"reasoning_tokens":615,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T13:51:45.824496+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit compact Legendrian in the unit cotangent bundle of the n-torus that admits no Reeb chords for some contact form, or a computation showing that the unit never enters the ideal generated by the top PSS classes.","supporting_citations":[],"review_version":1}