{"id":"e829fdaf-11c7-4326-9f15-47fda4bb55a8","arxiv_id":"2507.18724","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For an orderable Legendrian class with a positive Legendrian loop that extends to a contact loop, an unbounded invariant distance exists even when no positive contact loop exists.","lead":"New infinite distance functions are constructed on the space of Legendrian submanifolds inside a contact manifold, under weaker assumptions than were previously needed. This gives the first known case where such a distance exists without a positive loop of contactomorphisms, and it shows invariant distances on Legendrian isotopy classes are discrete.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof's main external dependence is [2, Prop 3.1], whose hypotheses should be checked against the Legendrian-loop family.","rationale":"The reader's weakest-assumption analysis correctly identifies the extension to a loop of contactomorphisms as the load-bearing mechanism: without it, ~φ^N would not be central in ^Cont0 and Proposition 4.2 would collapse. I agree that this is the most delicate point. However, it is an explicit hypothesis of Theorem A, not a hidden gap, and the corollary supplies a genuine example satisfying it. My reading of the proof found no internal error in the use of orderability, the integer-shift construction, or the unboundedness argument. The only step I would want verified before full acceptance is the applicability of [2, Prop 3.1] to the particular family ~Λ^T_*; the paper imports this finiteness statement without restating its hypotheses. That is a verification task rather than a demonstrated flaw, so the reader's ACCEPT verdict should stand unchanged.","tokens_in":17611,"tokens_out":44148,"duration_ms":456833,"concrete_test":"Locate [2, Prop 3.1] and check whether its hypotheses allow the family ~Λ^T_* to be generated by a positive Legendrian loop extended by an arbitrary loop of contactomorphisms based at the identity. If the proposition requires the extending loop itself to be positive (e.g. a Reeb flow), verify whether the proof of Proposition 4.2 can be adapted to the Legendrian-loop family directly, or restrict Theorem A accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The internal argument for Theorem A is coherent: orderability forces the positive Legendrian loop to be non-contractible, so the integer powers ~φ^N are distinct, strictly increasing order-automorphisms; centrality of ~φ^N in ^Cont0 follows because N∈Z makes it a deck transformation. The symmetry and triangle arguments for ℓ± are standard and the unboundedness computation is correct. The genuinely unproved external input is [2, Prop 3.1], which is quoted verbatim to ensure that the real-valued selectors ℓ^Λ*_± are finite for the family ~Λ^T_* defined by the Legendrian loop and its contact extension. The paper does not restate the hypotheses of that proposition. If [2, Prop 3.1] requires the reference family to be a positive loop of contactomorphisms, rather than a positive Legendrian loop with an arbitrary (possibly non-positive) contact extension, then the finiteness step in Proposition 4.2 would need a separate proof. I found no internal inconsistency, and the corollary provides a legitimate instance where the extension loop exists but is not globally positive, so the central claim appears sound conditional on the quoted result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs new unbounded invariant distances on the universal cover ~L(Λ*) of a Legendrian isotopy class, under the assumptions that ~L(Λ*) is orderable and that there exists a positive loop of Legendrians extending to a loop of contactomorphisms based at the identity. The key novelty is that the loop of contactomorphisms is not required to be positive. The distance is defined via integer-valued spectral selectors ℓ± that count turns of the positive Legendrian loop in the partial order. A corollary gives the first example of an orderable contact manifold for which ~L admits an unbounded invariant distance despite the absence of positive loops of contactomorphisms. The paper also proves Theorem B: every invariant distance on L(Λ*) is strongly discrete, and on ~L(Λ*) distinct underlying Legendrians are uniformly separated, via contact flexibility techniques. Additional results compare the new distance with the spectral distance and the Colin–Sandon oscillation distance, and an appendix identifies the set-theoretic universal cover with the topological one.","tokens_in":17845,"tokens_out":24797,"duration_ms":221953,"significance":"If the main theorem holds, it is a meaningful advance: it removes the standard assumption of a positive loop of contactomorphisms, replacing it with the weaker condition of a positive Legendrian loop admitting an arbitrary contact extension. The corollary on the unit cotangent bundle of the torus is striking, as it exhibits an orderable contact manifold with an unbounded invariant distance on ~L. Theorem B is a clean rigidity result, and its proof via contact flexibility is elegant. The paper also provides a full proof of the Weinstein neighborhood theorem (Lemma 3.5) and of the identification of the universal cover (Appendix), which are useful contributions in their own right. The writing is generally clear, and the main constructions are explicit and checkable.","major_comments":[{"comment":"The proof of finiteness of ℓ± depends on [2, Proposition 3.1], quoted to assert that the real-valued selectors ℓ^{Λ*}_± take finite values for the family ~Λ^T_* generated by the loop (φ^t). However, the hypotheses of that proposition are not stated. It is not clear whether [2, Prop 3.1] applies to a family generated by a loop of contactomorphisms that is not positive; if it requires a positive loop of contactomorphisms, the finiteness step in Proposition 4.2 lacks support. Since finiteness of ℓ± is load-bearing for Theorem A, the authors must either state the exact hypotheses of [2, Prop 3.1] and verify them for the family ~Λ^T_*, or provide a direct proof of the finiteness.","section":"Section 4, Proposition 4.2"},{"comment":"The proof of Proposition 4.2 contains reversed inequalities: from ~φ^{-N}·~Λ0 ⪯ ~Λ1 ⪯ ~φ^N·~Λ0 the correct conclusions are ℓ+(~Λ1,~Λ0) ≤ N and ℓ-(~Λ1,~Λ0) ≥ -N, not the reverse as printed. A further argument using monotonicity of the family ~φ^N·~Λ0 and antisymmetry is needed to rule out ℓ+ = -∞ and ℓ- = +∞. The proof of Proposition 4.3 also has an incorrect first equality: it should read ℓ+(~Λ1,~Λ0) = inf{N | ~Λ1 ⪯ ~φ^N·~Λ0}, with the subsequent equivalences adjusted accordingly. These are not mere typographical slips in peripheral material; they appear in the proof of the central distance properties, and the main theorem is not fully established until they are corrected.","section":"Section 4, Propositions 4.2 and 4.3"}],"minor_comments":[{"comment":"The notation [t] is used ambiguously, both for the class of t in S^1 = R/Z and for the corresponding coordinate in T^n = R^n/Z^n. Please clarify the notation to avoid confusion in the definition of the loop φ^t.","section":"Proof of the Corollary, Section 1.1"},{"comment":"The inequality ~φ^{Tm}_α·~Λ* ⪯ ~φ^T·~Λ* for T ≥ 0 (and its reverse for T ≤ 0) is stated without proof or reference. Since this is used to compare the spectral distance with the new distance, a short justification or a citation to a known comparison principle would be helpful.","section":"Section 5.1, proof of Theorem 5.1"},{"comment":"There are several typographical errors: 'Riemaniann metric' in Lemma 3.5, 'concanated' in the Appendix, and the running title appears as 'INV ARIANT' in the arXiv source. These should be corrected in the final version.","section":"Throughout"},{"comment":"In the proof of Proposition 5.5, the choice k0 := d(~Λ,~Λ') is valid because d is integer-valued, but the step k0A could be made more explicit by noting that dCS,osc(~Λ, ~φ^k·~Λ) ≤ kA for k = 2k0 via the triangle inequality; the current text is terse but correct.","section":"Section 5.2, Proposition 5.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the ideas are promising. The main risk is the unverified applicability of [2, Prop 3.1] to the family generated by a non-positive contact loop; since one of the authors is a coauthor of [2], this should be easy to resolve. The proof typos in Propositions 4.2 and 4.3 are fixable but currently render the main proof incomplete. If the authors confirm the hypotheses of [2, Prop 3.1] (or supply a direct finiteness proof) and correct the proof of Propositions 4.2–4.3, I would be willing to accept a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. It delivers a genuine improvement: prior unbounded invariant distances on the universal cover of a Legendrian isotopy class needed a positive loop of contactomorphisms; Theorem A only needs a positive Legendrian loop that happens to extend to some loop of contactomorphisms, not necessarily positive. That is not a reparametrization trick—the example in the corollary lives in a manifold where Cont0 is orderable, so no positive contact loop exists.\n\nWhat the paper does well: the distance d is explicit, integer-valued, and the metric axioms are checked directly (Propositions 4.2–4.5). The example in the corollary is concrete and instructive. Theorem B—discreteness of invariant distances—is a nice flexibility result, and the proof via Weinstein neighborhoods and Liouville squeezing is standard but clean. The appendix pinning down the C^k universal cover is a useful service.\n\nThe soft spot is the one the stress-test flags. Proposition 4.2 imports finiteness of the selectors ℓ^Λ*_± from [2, Prop 3.1], but the hypotheses of that proposition are not restated. The family ~Λ^T_* there is generated by a positive Legendrian loop with an arbitrary contact extension; [2] might have stated the result only for globally positive contact loops. My own reading of the positivity argument suggests the weaker hypothesis should suffice—the Hamiltonian is positive along the Legendrian track, which is all the order arguments use—but the paper should say so and either restate the proposition or give a short proof. That is a checkable, fixable issue, not a crack in the foundation.\n\nAlso minor: a typo in the proof of Proposition 4.2 (the line showing ~ψ·~φ^{N1}·~Λ_* ≤ ~ψ·~φ^{N1}·~Λ_* should have N2 on the right), and the notation for ~φ^T is compressed enough that a reader may trip on first pass.\n\nVerdict: the central construction is sound, conditional on the quoted external result. I would send this to a serious referee, with the explicit instruction to verify [2, Prop 3.1] against the Legendrian-loop setup. If that checks out, it is a solid accept.","headline":"A genuinely useful paper: it removes the positive-contact-loop assumption for unbounded invariant Legendrian distances, with the main caveat being a black-boxed finiteness result from the author's earlier work.","tokens_in":18341,"tokens_out":9132,"would_cite":true,"duration_ms":92066,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D10","53D35","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs unbounded invariant distances on universal covers of Legendrian isotopy classes using only a positive Legendrian loop, not a positive contactomorphism loop, and proves all invariant distances on Legendrian classes…","keywords":["Legendrian isotopy","invariant distance","universal cover","orderability","positive loop","contactomorphism","spectral distance","contact rigidity"],"falsifier":"In the torus example, take the Reeb-flow loop and check directly whether $\\tilde\\Lambda^{1/2}_* \\preceq \\tilde\\Lambda^1_*$ and $\\tilde\\Lambda^{-1}_* \\preceq \\tilde\\Lambda^{1/2}_*$ in the non-negative Legendrian order. The proof of Proposition 4.2 forces these order relations to hold and forces $\\ell_+(\\tilde\\Lambda^{1/2}_*,\\tilde\\Lambda_*)=1$; if any such order inequality fails for some real $T$, the integer-valued finiteness of $\\ell_\\pm$, and with it the unbounded distance of Theorem 4.1, would be false.","tokens_in":17425,"feed_emoji":"📏","tokens_out":16770,"duration_ms":145490,"temperature":0.7,"pith_summary":"The paper establishes that an unbounded invariant distance can exist on the universal cover of a Legendrian isotopy class without requiring a positive loop of contactomorphisms. It is enough to have an orderable space $\\tilde L(\\Lambda_*)$ and a positive loop of Legendrians that extends to a (not necessarily positive) loop of contactomorphisms. The distance is built by counting how many whole turns of this Legendrian loop separate two lifts in the order relation, and the proof shows the counting functions are finite integers. The same paper proves a complementary rigidity statement: every invariant distance on a Legendrian isotopy class is strongly discrete, so invariant distances on these spaces cannot have accumulating values. The concrete example of the unitary cotangent of the torus gives the first contact manifold whose contactomorphism group is orderable yet whose Legendrian space carries an unbounded invariant distance.","feed_headline":"Positive Legendrian loops alone yield unbounded invariant distances","feed_subtitle":"A positive Legendrian loop alone builds an unbounded metric; every invariant Legendrian metric is discrete.","key_machinery":"The load-bearing object is the pair of counting functions $\\ell_\\pm$ defined through the non-negative Legendrian order $\\preceq$ and the integer shifts $\\tilde\\varphi^N$ of the extended positive Legendrian loop. These functions measure, in whole turns of the loop, how far two lifts of Legendrian paths can be separated in the order; orderability guarantees the numbers are finite, and the extension to a contactomorphism loop makes $\\tilde\\varphi^N$ central, which gives the distance its invariance under the full universal cover of the contactomorphism group. The distance is then the max of $\\ell_+$ and $-\\ell_-$, and the unboundedness comes from the fact that $\\ell_+(\\tilde\\Lambda^T_*,\\tilde\\Lambda_*) = \\lceil T\\rceil$ along the loop.","core_discovery":"The central discovery is a construction of an unbounded invariant distance on $\\tilde L(\\Lambda_*)$ from order data alone. Under the assumptions that $\\tilde L(\\Lambda_*)$ is orderable and that a positive Legendrian loop $(\\Lambda^t_*)$ can be extended to a loop of contactomorphisms $(\\varphi^t)$ based at the identity, the paper defines $\\ell_+(\\tilde\\Lambda_1,\\tilde\\Lambda_0) = \\inf\\{N\\in\\mathbb Z\\mid \\tilde\\Lambda_1\\preceq \\tilde\\varphi^N\\cdot\\tilde\\Lambda_0\\}$ and $\\ell_-$ by the corresponding supremum, and proves these are integers with $\\ell_-\\le \\ell_+$. Then $d(\\tilde\\Lambda_0,\\tilde\\Lambda_1)=\\max\\{\\ell_+(\\tilde\\Lambda_1,\\tilde\\Lambda_0),-\\ell_-(\\tilde\\Lambda_1,\\tilde\\Lambda_0)\\}$ is a genuine invariant distance, compatible with the order, and unbounded because $d(\\tilde\\Lambda^T_*,\\tilde\\Lambda_*)\\ge T$ for all $T>0$. The proof relies on the centrality of the integer shifts $\\tilde\\varphi^N$ in the universal cover of the contactomorphism group, which is exactly where the extension condition is used. As a corollary, the unitary cotangent bundle of the $n$-torus yields an orderable contactomorphism group with an unbounded invariant distance on $\\tilde L$, the first such example.","pith_inferences":["An extension beyond the paper: the extension condition might be relaxable; if integer shifts could be defined from a Legendrian loop alone, the same formula would likely produce unbounded distances in orderable classes without any contactomorphism extension.","An extension beyond the paper: Theorem B's strong discreteness is probably a general contact-flexibility phenomenon, so one would expect any invariant distance on homogeneous spaces built from local Weinstein neighborhoods to be strongly discrete by the same squeezing argument.","An extension beyond the paper: the quasi-isometry with the spectral pseudo-distance suggests that the new distance is a discretized displacement spectrum; computing it on lifts separated by Reeb flow times could yield a numerical invariant of the contact form.","An extension beyond the paper: the paper conjectures that smooth orbit spaces of submanifolds of high codimension admit no unbounded invariant distance; a testable route would be to construct explicit bounded bi-invariant metrics on diffeomorphism groups."],"forward_implications":["Any orderable Legendrian class containing a positive Legendrian loop that extends to a contactomorphism loop admits an unbounded invariant distance given by the explicit max formula.","The unitary cotangent of the $n$-torus is the first example where the contactomorphism group has no positive loop but the Legendrian universal cover still carries an unbounded invariant distance.","Every invariant distance on a Legendrian isotopy class is strongly discrete, and on the universal cover any two lifts over different Legendrians are uniformly separated, so such distances cannot have values accumulating toward zero.","The constructed distance is compatible with the order relation and is quasi-isometric to the invariantized spectral pseudo-distance, so it encodes the same Reeb-dynamical information as spectral selectors up to bilipschitz equivalence.","The examples suggest a route to unbounded conjugation-invariant norms on the universal cover of the contactomorphism group even when that group is orderable, provided certain spectral bounds hold."],"supporting_citations":[{"why":"Supplies the spectral selectors and the finiteness result invoked to prove that the counting functions take integer values.","marker":"[2]"},{"why":"Introduces positive loops and the order relation, and the criterion that orderability equals absence of positive loops.","marker":"[17]"},{"why":"Provides the Sandon-type distance that the new distance generalizes and the discreteness result for bi-invariant metrics on contactomorphism groups.","marker":"[18]"},{"why":"Proves orderability of the universal cover for hypertight Legendrians, used to apply the main theorem to the torus example.","marker":"[9]"},{"why":"Shows the contactomorphism group of the unitary cotangent of the torus is orderable, yielding the first example with an orderable group but unbounded Legendrian space.","marker":"[12]"}],"fun_headline_variants":["Order data alone builds unbounded Legendrian distance","First unbounded invariant distance without positive loops","Legendrian metrics: always discrete, sometimes unbounded","New invariant distance on Legendrians, unbounded and integer","Unbounded invariant distance from order, no contact loop needed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the assumption that the positive Legendrian loop can be extended to a loop of contactomorphisms based at the identity; without that extension the integer shifts $\\tilde\\varphi^N$ that the distance counts are not available, and the finiteness and unboundedness arguments collapse.","fun_headline_variants_meta":{"raw":{"variants":["Order data alone builds unbounded Legendrian distance","First unbounded invariant distance without positive loops","Legendrian metrics: always discrete, sometimes unbounded","New invariant distance on Legendrians, unbounded and integer","Unbounded invariant distance from order, no contact loop needed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1490,"prompt_tokens":893,"completion_tokens":597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":521}},"tokens_in":509,"tokens_out":597,"duration_ms":6383,"temperature":1.0,"reasoning_tokens":521,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:12:12.891623+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the torus example, take the Reeb-flow loop and check directly whether $\\tilde\\Lambda^{1/2}_* \\preceq \\tilde\\Lambda^1_*$ and $\\tilde\\Lambda^{-1}_* \\preceq \\tilde\\Lambda^{1/2}_*$ in the non-negative Legendrian order. The proof of Proposition 4.2 forces these order relations to hold and forces $\\ell_+(\\tilde\\Lambda^{1/2}_*,\\tilde\\Lambda_*)=1$; if any such order inequality fails for some real $T$, the integer-valued finiteness of $\\ell_\\pm$, and with it the unbounded distance of Theorem 4.1, would be false.","supporting_citations":[{"cited_title":"0, 2550071","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral selectors and the finiteness result invoked to prove that the counting functions take integer values."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces positive loops and the order relation, and the criterion that orderability equals absence of positive loops."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Sandon-type distance that the new distance generalizes and the discreteness result for bi-invariant metrics on contactomorphism groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves orderability of the universal cover for hypertight Legendrians, used to apply the main theorem to the torus example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the contactomorphism group of the unitary cotangent of the torus is orderable, yielding the first example with an orderable group but unbounded Legendrian space."}],"review_version":2}