{"id":"30b6fb0e-09f8-4e1c-89ed-c0e180d3d19c","arxiv_id":"2511.17780","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In corank-2 fat distributions, infinitely many prelegendrian tori share one formal class but are pairwise non-isotopic, detected by Legendrian contact homology of their canonical lifts.","lead":"This paper proves that the h-principle fails for prelegendrians, a new class of submanifolds of fat distributions — the higher-corank generalization of contact structures. It constructs infinitely many formally equivalent but genuinely non-isotopic tori in all dimensions, the first rigidity result for maximally non-integrable distributions outside contact topology.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 9.1's formal-class preservation is asserted by analogy; if the complex-tangent rotation cannot be made while preserving the formal prelegendrian condition, the Λ_s may not share a formal class.","rationale":"The reader's CONDITIONAL verdict is appropriate. The paper's architecture is coherent, the local computations I checked are consistent, and the cited external machinery (EES05b, Murphy, Gromov) is appropriate. However, the proof of Proposition 9.1 — the assertion that prelegendrian N-pushing preserves the formal prelegendrian class — is the place where the central claim is least secure. Unlike the contact N-pushing in Proposition 6.2, the prelegendrian formal condition is not just a homotopy of monomorphisms with a symplectic isotropy condition; it also requires the corank condition and isotropy with respect to the fat structure at every time. For a fixed front, the complex tangency is uniquely determined, so the claimed rotation must be non-holonomic in the fibre directions, and the paper does not demonstrate that the formal prelegendrian inequalities can be maintained throughout. This is a presentation gap rather than a demonstrated error, and it is likely fixable by a careful explicit construction or by appealing to a parametric h-principle for formal prelegendrian embeddings over a fixed embedding. The reader's secondary concern about intermediate immersed fronts is not as strong, because co-real front self-intersections lift to embedded prelegendrians. Thus I keep the verdict at CONDITIONAL/UNCHANGED, with the concrete test being a full explicit verification of Proposition 9.1 in the local model.","tokens_in":44672,"tokens_out":19987,"duration_ms":185792,"concrete_test":"Write out the missing formal prelegendrian homotopy in the local model of Proposition 9.1, for the lowest nontrivial case n=1 (so Λ has dimension 3, front a hypersurface in C^2), with an explicit bump function f and the vector field Z supplied by Proposition 3.9. At each time t in the interpolation, compute the image F_t(T_pΛ), project to T_{x(t)}X, and symbolically verify: (1) rank F_t = 3, (2) dim(F_t(T_pΛ) ∩ D) = 2, (3) isotropy of this intersection with respect to the curvature form. If no such explicit family exists, or if the rank drops at some t, then Proposition 9.1 fails and Theorem 1.10's formal-class claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.10 requires that the infinitely many prelegendrian tori Λ_s are all formally prelegendrian isotopic. This is achieved by combining Lemma 7.5 (formal Legendrian spinning isotopy via Reidemeister moves and K-pushings) with Theorem 9.2 / Proposition 9.1 (prelegendrian N-pushing preserves the formal prelegendrian class). The load-bearing step is the last claim of Proposition 9.1: the proof says it 'will follow by an argument similar to the one in Proposition 6.2' and that rotating a non-vanishing complex-tangent vector field Z towards ∂_z 'provides a formal rotation of the complex tangencies', after which one applies the contact argument. This interpolation is not written out. In the contact case, the formal isotopy is a homotopy of Legendrian monomorphisms satisfying a purely symplectic condition. In the prelegendrian case, Definition 2.17(1) imposes two additional algebraic conditions at every time: F_t(TΛ)+D has corank 1, and F_t(TΛ)∩D is isotropic in D with respect to the fat curvature form. For a genuine prelegendrian front, the complex-tangency plane W is forced to be T_xF∩J T_xF (Theorem 8.1 uniqueness), so any nontrivial rotation must use non-holonomic vertical components in the CP^n-fibre. The paper does not show that such a rotation can be performed continuously without the intersection dimension dim(F_t(TΛ)∩D) dropping below 2n or without isotropy failing. Since Theorem 1.10's same-formal-class assertion depends directly on this step, this is the most load-bearing gap. The reader's second concern — that Reidemeister II fronts force immersed intermediate prelegendrians — is less serious: front self-intersections do not make the prelegendrian lift non-embedded when the self-intersection locus is co-real (Proposition 8.4).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces prelegendrian submanifolds in corank-2 fat distributions, studies their fronts and formal classes, and proves that the h-principle fails for prelegendrian embeddings in the standard fat space (C^{2n+1},D_std). The main theorem constructs infinitely many prelegendrian (2n+1)-tori that are formally prelegendrian isotopic but not prelegendrian isotopic, because their canonical Legendrian lifts to the contactisation are distinguished by Legendrian contact homology. The paper also develops a stabilization operation for prelegendrians, proves robustness of the invariants under perturbations of the fat structure, and extends the constructions to general corank-2 fat manifolds in dimension six and to a subclass in higher dimensions.","tokens_in":44880,"tokens_out":14971,"duration_ms":139068,"significance":"If the proof is completed, this is the first rigidity result for submanifolds in maximally non-integrable distributions beyond the contact case, and it opens a promising new direction in the study of fat distributions. The framework developed here — prelegendrian fronts, co-real singular loci, prelegendrian spinning, and stabilization — is natural and well motivated. The paper makes good use of existing tools (front spinning, Legendrian contact homology, Murphy's h-principle) and includes explicit constructions and concrete models. The structural theorems, in particular the characterisation of simple prelegendrian fronts and the identification of contactisations with spaces of contact elements, are carefully presented and are substantial contributions in their own right.","major_comments":[{"comment":"The last claim of Proposition 9.1 — that p_N(Λ) is formally prelegendrian isotopic to Λ — is the load-bearing step for the same-formal-class assertion in Theorem 1.10, but it is only sketched. The proof says the argument is 'similar' to Proposition 6.2 and that rotating a non-vanishing complex-tangent vector field Z towards ∂z 'provides a formal rotation of the complex tangencies', after which the contact argument applies. This does not exhibit the required 1-parameter family of monomorphisms F_t covering the interpolating embeddings and does not verify the two algebraic conditions of Definition 2.17(1)(a)–(b) at every time. In particular, condition (b) — isotropy of F_t(TΛ)∩D with respect to the fat curvature form — is not automatic when the vertical (CP^n-fibre) component is rotated; the contact case of Proposition 6.2 only enforces the Legendrian condition, not the additional prelegen","section":"§9.1, Proposition 9.1"},{"comment":"The proof asserts that the prelegendrians Λ_s are formally isotopic because their fronts are connected by prelegendrian Reidemeister moves and N-pushings. The paper does not define 'formal prelegendrian isotopy' explicitly, and the proof in §11.2 does not specify the formal data along each segment of the concatenated homotopy. In particular, it is not shown how the formal isotopy produced by the N-pushing in Proposition 9.1 is glued to the genuine prelegendrian isotopies coming from Reidemeister moves, nor that the resulting path is a path of formal prelegendrian embeddings in the sense required by Definition 2.17(3). A complete proof should either prove that the three prelegendrian Reidemeister moves preserve the formal prelegendrian class (with an explicit formal homotopy) or state and prove a composition lemma for formal prelegendrian isotopies. Without this, the conclusion that all Λ","section":"§11.2, proof of Theorem 1.10"}],"minor_comments":[{"comment":"In the proof, 'open coition' should be 'open condition'.","section":"§8.1, Proposition 8.5"},{"comment":"The sentence 'the Legendrian L(Λ_s) are pairwise not Legendrian non isotopic' contains a double negation; it should read 'pairwise not Legendrian isotopic'.","section":"§11.2, proof of Theorem 1.10"},{"comment":"In item (4), 'contact contact submanifold' is a typo; it should be 'contact submanifold'.","section":"§2.2, Theorem 2.6"},{"comment":"The phrase 'an alternative characterisation of of fatness' has an extra 'of'.","section":"§4.1"},{"comment":"The word 'functioriality' should be 'functoriality'.","section":"§11.1, Lemma 11.1"},{"comment":"The caption says 'The first and fourth arrow indicate prelegendrian RII moves, and the last one several prelegendrian RI moves', but 'the last one' is the fourth arrow. The correspondence between arrows and moves should be clarified.","section":"Figure 5 caption"},{"comment":"The definition of a formal preisotropic embedding uses a family F_s with F_0 = df and (f,F_1) formal, but it is not stated whether intermediate F_s are required to satisfy the formal conditions. For clarity and for the notion of formal isotopy used later, this should be stated explicitly.","section":"Definition 2.17"}],"recommendation":"major_revision","confidential_remarks":"The central gap is the formal-class preservation in Proposition 9.1. The reader's stress-test concern about this step is legitimate and should be addressed directly. I do not see circularity, and the structural and computational parts of the paper appear sound. If the missing formal-homotopy construction is supplied, the paper would be a strong contribution to symplectic and contact topology. I would not require new invariants or a different strategy; the issue is local and likely repairable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the first genuine rigidity statement for submanifolds in maximally non-integrable distributions outside the contact case, and the architecture is convincing. But the proof that prelegendrian N-pushing preserves the formal class (Prop. 9.1) is sketched by analogy with the contact case, and that is the one load-bearing step I would want fully written out before trusting Theorem 1.10 as stated.\n\nWhat's actually new: the paper defines prelegendrians, proves a clean front criterion (Theorem 8.1: simple fronts whose singular loci are co-real), and builds a prelegendrian spinning and stabilization machine. The exotic (2n+1)-tori are distinguished by Legendrian contact homology of their canonical lifts, using the EES05b non-isotopic Legendrians. The extension to non-standard fat structures via nilpotentisation and Gray stability is a nice bonus. Several local computations I spot-checked are correct—the wedge-power calculation in Theorem 2.6 and the cusp-immersion computation in Proposition 8.5 both hold up. The paper is also honest about open questions, including whether rigidity holds for the distributions themselves.\n\nThe soft spot is exactly what the reader flagged. Proposition 9.1's proof says the formal isotopy 'will follow by an argument similar to Proposition 6.2' and that rotating a non-vanishing complex-tangent vector field Z 'provides a formal rotation of the complex tangencies.' Those are the lines that carry the same-formal-class claim for all Λ_s. In the contact case you only need to keep a symplectic condition; here you also need to preserve the corank-one condition and isotropy in D with respect to the fat curvature at every time. The paper does not walk through that check. I think it's very likely fixable—the construction of Z is given and the geometry is plausible—but as written it's a genuine gap.\n\nThe second concern—that Reidemeister II fronts might force the formal isotopy through immersions—looks like a non-issue. When the self-intersection locus is co-real, Proposition 8.4 says the lift is embedded, so those fronts give genuine prelegendrian isotopies.\n\nWho this is for: anyone working on h-principles for distributions or on higher-dimensional Legendrian rigidity. It deserves a serious referee; the main theorem is important if correct, and the exposition is good enough that a referee can focus on the one sketchy proof. My recommendation: send it to referees, and explicitly ask them to verify or refute Prop. 9.1's formal-class preservation.","headline":"First rigidity for prelegendrians in fat distributions, likely correct, but the formal-class preservation step in Prop. 9.1 needs a full proof.","tokens_in":45617,"tokens_out":4127,"would_cite":true,"duration_ms":37751,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D10","53D12","53D35","57R17","58A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For corank-2 fat distributions, formal prelegendrian classes split: infinitely many tori are formally equivalent yet pairwise non-isotopic.","keywords":["h-principle","fat distributions","prelegendrians","corank 2","Legendrian contact homology","co-real submanifolds","front spinning","contactisation"],"falsifier":"Exhibit a prelegendrian isotopy between Λ_s and Λ_t for two distinct indices, or a Legendrian isotopy between their canonical lifts in the contactisation C(C^{2n+1}, D_std); either would contradict Theorem 1.10. A more computational check is to compute the Legendrian contact homology of the lifts L(Λ_s) and test whether the resulting algebras for distinct s are quasi-isomorphic—a quasi-isomorphism between any pair would break the detection argument.","tokens_in":44377,"feed_emoji":"🌀","tokens_out":7124,"duration_ms":62511,"temperature":0.7,"pith_summary":"The paper introduces prelegendrian submanifolds in corank-2 fat distributions—maximally non-integrable even-rank distributions that generalize contact structures—and proves that their classification does not reduce to formal, bundle-theoretic data. In the standard fat space C^{2n+1}, for every n≥1, there are infinitely many prelegendrian (2n+1)-tori that all represent the same formal prelegendrian class but are pairwise not prelegendrian isotopic. The obstructions are detected by Legendrian contact homology of their canonical Legendrian lifts to the contactisation, so the failure of the h-principle is genuinely geometric. The same front-based machinery yields a prelegendrian stabilization whose lift is loose, and extends the constructions to arbitrary corank-2 fat distributions in dimension 6. This would be the first example of rigidity for maximally non-integrable distributions beyond contact topology.","feed_headline":"h-principle fails for prelegendrians in all dimensions","feed_subtitle":"Infinitely many tori lie in one formal class yet are genuinely distinct; Legendrian invariants see the difference.","key_machinery":"The load-bearing mechanism is the prelegendrian front calculus. Theorem 8.1 characterizes simple prelegendrian fronts: a simple front F ⊆ (X, J) is a prelegendrian front if and only if its singularity loci are co-real in the almost complex manifold (X, J). Prelegendrian spinning along a co-real embedding T^{2n} → C^{n+1} converts 1-dimensional Legendrian knots into prelegendrian (2n+1)-tori, and the canonical Legendrian lift of the result is exactly the Legendrian spinning of the input knot. Legendrian contact homology of that lift therefore becomes a prelegendrian invariant. Formal equivalence among the spun tori is produced by prelegendrian Reidemeister moves and prelegendrian N-pushing al","core_discovery":"The central discovery is Theorem 1.10: there exist infinitely many embeddings Λ_s : T^{2n+1} → (C^{2n+1}, D_std), s ∈ Z_{>0}, that are formally prelegendrian isotopic but pairwise not prelegendrian isotopic. Their canonical Legendrian lifts L(Λ_s) to the contactisation C(C^{2n+1}, D_std) are pairwise not Legendrian isotopic, distinguished by Legendrian contact homology. Because formal prelegendrian isotopy is the bundle-theoretic/homotopy notion and genuine prelegendrian isotopy is geometric, this is a failure of the h-principle for prelegendrian embeddings in all dimensions. The proof establishes a front criterion—simple prelegendrian fronts are exactly those whose singularity locus is co-r","pith_inferences":["Editorial inference: the front-and-canonical-lift strategy is a natural template for probing rigidity in corank k > 2 fat distributions; the missing ingredient there is a usable local model, not the invariant itself.","Editorial inference: the formal isotopy in the construction passes through fronts with transverse self-intersections, so the proof as written may only eliminate embedded isotopies while leaving the status of prelegendrian immersions open; smoothing those intermediate fronts would strengthen the conclusion.","Editorial inference: the stabilization construction suggests a provisional 'loose prelegendrian' class; if such a class obeys an h-principle, the exotic tori would be exactly the non-loose part, mirroring the tight/overtwisted split in contact topology."],"forward_implications":["In the standard corank-2 fat space, formal prelegendrian isotopy is strictly weaker than genuine prelegendrian isotopy, in every dimension n ≥ 1.","Legendrian contact homology of the canonical lift is an effective prelegendrian invariant: it can distinguish prelegendrians that are formally identical.","Prelegendrian stabilization produces, from any prelegendrian, another in the same formal class whose Legendrian lift is loose, yielding non-loose examples and exotic pairs including co-normal lifts of hypersurfaces.","The rigidity is stable: no compactly supported path of fat structures connects the exotic tori, since such a path would induce a contactomorphism of contactisations and preserve the Legendrian invariants.","Every corank-2 fat distribution in dimension 6 contains compact prelegendrians and admits stabilizations; in higher dimensions an analogous statement holds when the local nilpotentised model is the complex Heisenberg algebra."],"fun_headline_variants":["Infinite tori in one formal class, prelegendrians resist isotopy","h-principle fails: prelegendrians rigid in fat distributions","Formal twins parted by Legendrian invariants","Prelegendrian h-principle fails in all dimensions","Infinitely many tori, same formal class, distinct geometry"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the local prelegendrian N-pushing move and the intervening prelegendrian Reidemeister moves preserve the formal prelegendrian embedding class; the proof of this rotation/interpolation step is asserted by analogy with the contact case rather than written out in detail.","fun_headline_variants_meta":{"raw":{"variants":["Infinite tori in one formal class, prelegendrians resist isotopy","h-principle fails: prelegendrians rigid in fat distributions","Formal twins parted by Legendrian invariants","Prelegendrian h-principle fails in all dimensions","Infinitely many tori, same formal class, distinct geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000235,"raw_usage":{"total_tokens":1428,"prompt_tokens":925,"completion_tokens":503,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":669,"tokens_out":503,"duration_ms":4763,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T20:58:00.261279+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a prelegendrian isotopy between Λ_s and Λ_t for two distinct indices, or a Legendrian isotopy between their canonical lifts in the contactisation C(C^{2n+1}, D_std); either would contradict Theorem 1.10. A more computational check is to compute the Legendrian contact homology of the lifts L(Λ_s) and test whether the resulting algebras for distinct s are quasi-isomorphic—a quasi-isomorphism between any pair would break the detection argument.","supporting_citations":[],"review_version":1}