{"id":"2fc41416-96d2-4fa5-9ace-9787d6f36042","arxiv_id":"2603.18808","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"There exists a global fat (4,6)-distribution on R^6 that has two Reeb directions but is nowhere diffeomorphic to a complex contact structure.","lead":"This paper constructs a fat rank-4 distribution on R^6 with two Reeb directions that cannot come from a complex contact structure, answering an open question in the negative. Specialists should read it for the first explicit separation between real fat distributions with Reeb directions and complex-contact structures.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central counterexample appears correct (Lemma 3.10 formula checks out); the load-bearing failure is Theorem 4.1/1.3: the κ-perturbations are all pullbacks of D by a z1-shift, so they cannot yield infinite codimension.","rationale":"After checking the central computation in good faith, I find Lemma 3.10 correct: decomposing ν=X2 with λ=1/2-ic/(2Δ) gives Jν=(c/Δ)X2-(2/Δ)X1; the -2/Δ coefficient is right. The subsequent S-nonvanishing argument also appears internally consistent: the reduction to 2[A,ν]-J[A,ν]-[A,Jν] is sign-correct, and A1=x2·4(1+x2^2)/Δ^3·(2-4/(cΔ)) cannot vanish on an open interval. So the paper's answer to Bhowmick's question is credible. The reader's second concern is the load-bearing one: κ(y1)dy1 is exact, so each H_κ is the pullback of D under a z1-shift; hence Theorem 4.1 supplies only one diffeomorphism class, not an infinite-dimensional family of non-equivalent germs, and Theorem 1.3 is unproved. Because the title/abstract make the infinite-codimension claim a main contribution, the manuscript as submitted is not acceptable; I would keep the reader's REJECT, though the reason should be the failed moduli argument, not the alleged algebra error.","tokens_in":20536,"tokens_out":20595,"duration_ms":188639,"concrete_test":"Compute Φ^*λ1 and Φ^*λ2 explicitly for Φ=(x1,x2,y1,y2,z1-∫_0^{y1}κ(s)ds,z2). If the pullback identities hold for arbitrary κ (they do), Theorem 4.1's family is a single diffeomorphism class, so the claimed infinite codimension is not established. Separately, to confirm the central computation, expand ν=X2 as λu1+conj(λu1) with λ=1/2-i(x2^2+1)/(2Δ) and verify the X1-coefficient of Jν is -2/Δ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Recomputing Lemma 3.10 removes the reader's stated algebraic objection. With c=x2^2+1, Δ=sqrt(3-2x2^2-x2^4), and λ=1/2-i c/(2Δ), ν=X2=λu1+conj(λu1); hence Jν=iλu1-i conj(λu1)=(c/Δ)X2-(2/Δ)X1. The printed -2/Δ coefficient is correct, and the sign algebra in Eqs. (18)-(19) checks out. The serious gap is Theorem 4.1/1.3. Since α1_κ=α1_0-κ(y1)dy1=α1_0-d(∫κ), the global diffeomorphism Φ(x1,x2,y1,y2,z1,z2)=(x1,x2,y1,y2,z1-∫_0^{y1}κ(s)ds,z2) satisfies Φ*λ1=α1_κ and Φ*λ2=α2. Thus every H_κ is diffeomorphic to D; the family lies in a single orbit and cannot demonstrate infinite codimension of complex-contact germs. The primary counterexample (Theorem 3.13) is not affected by this gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the converse of the known implication ``complex contact structure on a real 6-manifold gives a fat (4,6)-distribution with two Reeb directions.'' It constructs an explicit pair of 1-forms on R^6 whose kernel distribution is fat in |x_2|<1, admits two Reeb directions, and whose Čap--Eastwood tensor S is shown to be nonvanishing on every open set. By Theorem 2.13 this means no germ is induced by a complex contact structure. The construction is then globalized by a diffeomorphism to all of R^6 (Theorem 3.13). Section 4 claims an infinite-dimensional family of such germs and concludes infinite codimension of complex-contact germs among fat germs with Reeb directions (Theorems 4.1 and 1.3). The main counterexample is explicit and appears sound; the infinite-codimension part does not, as explained below.","tokens_in":20854,"tokens_out":14270,"duration_ms":121540,"significance":"If the main counterexample stands, it settles Bhowmick's Question 1.1 in the negative and shows that fat distributions with Reeb directions form a strictly larger class than those underlying complex contact structures, even at the germ level. The paper has real strengths: the construction is fully explicit; the fatness, Reeb, and nonvanishing-S computations are written out; and the globalization via tan((pi/2)x_2) is transparent. I verified that the alleged algebraic error in Lemma 3.10 is not an error: the coefficient -2/Δ is correct, and the sign algebra leading to Eq. (20) is consistent. The serious defect is confined to Section 4: the claimed infinite-dimensional family is actually a single diffeomorphism orbit, so Theorems 4.1 and 1.3 are not established as stated.","major_comments":[{"comment":"The perturbation term is exact: α1_κ = α1_0 - κ(y1)dy1 = α1_0 - d(∫_0^{y1} κ(s)ds). Therefore the explicit global diffeomorphism Φ(x1,x2,y1,y2,z1,z2) = (x1,x2,y1,y2,z1 - ∫_0^{y1} κ(s)ds, z2) satisfies Φ^*λ1 = α1_κ and Φ^*λ2 = α2. Hence every (R^6,H_κ) in the family is diffeomorphic to the original (R^6,D). Since non-existence of a complex contact structure is diffeomorphism-invariant, condition (iii) gives no new information. More importantly, a family contained in one Diff-orbit cannot establish that complex-contact germs have infinite codimension in the space of fat germs with Reeb directions. The proof's assertion that ``the differentials have identical expressions and therefore the same proof applies'' misses this exactness issue. This is load-bearing for Theorem 1.3, which is currently unsupported.","section":"Section 4, Theorem 4.1 and Theorem 1.3"},{"comment":"Even if the family were nontrivial, the manuscript does not define the relevant topology or notion of codimension on the space of germs, nor does it explain why a family parametrized by an infinite-dimensional function space implies infinite codimension rather than merely a large family of equivalence classes. The exactness defect makes this point moot for the present family, but the paper should either supply a rigorous definition and argument or state the weaker result that there exist infinitely many (non-equivalent) examples.","section":"Section 4 proof structure"}],"minor_comments":[{"comment":"The vector denoted η in the first paragraph and in Eq. (18)-(19) is the ν defined just before: ν = ∂_{x2} + y2∂_{z1} + y1∂_{z2}. Please use one symbol throughout.","section":"Proposition 3.12"},{"comment":"The proof of Lemma 3.11 is phrased slightly informally: ``A1 = x2 f(x2), and x2 only vanishes at the origin whereas f vanishes at most in a finite set, so A1 cannot vanish on an open set.'' Since a product of two nonzero functions can vanish at isolated points, the argument is valid, but it would be clearer to state explicitly that an open interval would have to be contained in the zero set of at least one factor.","section":"Lemma 3.11"},{"comment":"In the proof, after defining β1, β2, the displayed formulas for (23)-(24) are labelled as λ1, λ2; this is correct but the preceding sentence could confuse the reader. Also, the diffeomorphism ϕ maps {|x2|<1} to R^6, so the phrase ``global distribution'' is justified, but this should be stated explicitly as a pullback of the fat region.","section":"Theorem 3.13"},{"comment":"The statement says ``for every point p in M'' but the proof only transfers the origin germ by a local chart. This is fine, but the theorem assumes M is a smooth 6-manifold; no orientation issue is discussed. Since the Čap--Eastwood construction in the paper uses an oriented manifold, the hypotheses should be stated precisely (e.g., oriented M).","section":"Theorem 1.3"}],"recommendation":"major_revision","confidential_remarks":"The reader's report flagged Lemma 3.10 as algebraically wrong, but I checked it and the printed coefficient -2/Δ is correct. The real problem is Theorem 4.1: the κ(y1)-perturbation is exact and the whole family is pulled back from D by a z1-shift. This is not a minor typo; it invalidates the advertised infinite-codimension result. The central counterexample (Theorem 3.13) appears sound and may well be publishable. I recommend major revision: either repair the perturbation argument with a genuinely non-exact deformation and prove the codimension statement, or remove/rewrite Section 4 as a weaker remark. The paper's claim to have answered the question ``completely'' should be moderated if the infinite-codimension part cannot be substantiated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I've read the paper through twice. The thing to know: the primary construction is real. The fat (4,6)-distribution D in Theorem 3.1 has Reeb directions and its Čap–Eastwood tensor S is nonzero on every open set in |x2|<1, so it is nowhere locally equivalent to a complex contact structure. That answers Bhowmick's Question 1.1 in the negative, and it was not known even at germ level. The global version in Theorem 3.13 is a straightforward but legitimate pushforward under arctangent.\n\nThe reader's reported algebraic error in Lemma 3.10 is not an error. Recomputing with c=x2^2+1, Δ=sqrt(3-2x2^2-x2^4), the decomposition ν=(1/2 - ic/(2Δ))u1 + conjugate gives Jν=(c/Δ)X2-(2/Δ)X1, exactly as printed. The sign algebra in Proposition 3.12 checks out. So the central counterexample survives.\n\nThe soft spot is Section 4, and it is serious. The family α1_κ differs from α1_0 by κ(y1)dy1, which is d(∫_0^{y1}κ). The coordinate change z1→z1-∫κ pulls D back to H_κ while fixing α2. So every H_κ is diffeomorphic to D. The parametrization by κ is infinite-dimensional as a set of forms, but it gives one germ modulo diffeomorphism, not infinitely many. Theorem 4.1, and hence Theorem 1.3, is not established by this construction. The claim of infinite codimension may still be true, but it needs a different argument or should be dropped from the abstract.\n\nThe paper is otherwise honest and self-contained. It uses the Čap–Eastwood criterion as an external theorem, cites the relevant literature, and does not lean on the author's own h-principle papers for the load-bearing step. The main result is worth knowing.\n\nMy recommendation: send it to peer review. The referee should ask for the infinite-codimension claim to be corrected or removed; the primary counterexample is publishable as is. Anyone working on fat distributions or complex contact geometry should read at least Section 3.","headline":"The main counterexample is solid and answers Bhowmick's question; the advertised infinite-codimension conclusion is not proved, because the κ-family collapses under a z1-shift diffeomorphism.","tokens_in":21312,"tokens_out":3623,"would_cite":true,"duration_ms":34700,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C15","53D10","58A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A fat (4,6)-distribution with two Reeb directions need not support a complex contact structure on any open set, even up to diffeomorphism.","keywords":["fat distributions","complex contact structures","Reeb directions","corank-2 distributions","almost complex structure","obstruction tensor","infinite codimension","dimension six"],"falsifier":"Recompute the obstruction tensor directly for the distribution in Theorem 3.1, taking R=∂z2 and ν=∂x2+y2∂z1+y1∂z2, and inspect the ∂z1 coefficient A1(x2) of (2[JR,ν]−J[JR,ν]−[JR,Jν]) mod D. The paper's formula gives A1(x2)=x2·f(x2) with f zero only when x2^2=−1+√2; if a recomputation yields A1≡0 on an open interval inside (−1,1), the main theorem is false.","tokens_in":20424,"feed_emoji":"📐","tokens_out":11565,"duration_ms":107501,"temperature":0.7,"pith_summary":"Complex contact 3-manifolds, viewed as real 6-manifolds, always carry a fat corank-2 distribution with two distinguished Reeb directions. The paper shows the converse fails: it writes down explicit 1-forms defining a fat (4,6)-distribution on R^6 with two Reeb directions, then uses the canonical almost complex structure determined by any fat distribution and its obstruction tensor to prove the structure is not complex contact on any open set, even after any local diffeomorphism. This answers an open converse question at the level of germs. The same construction, with a smooth function inserted into one of the 1-forms, yields infinitely many such germs, so complex-contact germs form an infinite-codimension subset of fat germs with Reeb directions.","feed_headline":"Fat distributions with Reeb directions need not be complex contact","feed_subtitle":"The converse fails: such distributions are not complex contact even locally or up to diffeomorphism.","key_machinery":"The machinery is the canonical almost complex structure J assigned to a fat (4,6)-distribution on an oriented 6-manifold: J is uniquely characterized by preserving the distribution, matching the orientation, and making the Levi curvature complex-bilinear. Associated to J is the obstruction tensor S(u,v)=[u,v]+J[Ju,v] mod D, whose vanishing on all pairs is equivalent to the distribution germ being complex contact. The paper constructs J explicitly from a complex root of the Levi quadratic form of the model distribution, computes it on the quotient by the Reeb directions and on the distribution itself, and then evaluates S on the pair R=∂z2, ν=∂x2+y2∂z1+y1∂z2. The z1-component of the resulting","core_discovery":"The paper's central object is the rank-4 distribution (R^6,D) cut out by the 1-forms λ1 = dz1 − y1 dx1 − y2 dx2 − (x2^3/3 + x2 + 2x1) dy1 and λ2 = dz2 − y2 dx1 − y1 dx2. This distribution is fat on the open slab |x2|<1 and admits two commuting Reeb directions ∂z1, ∂z2. For any fat distribution on an oriented 6-manifold there is a canonical almost complex structure; the paper computes it for this example and evaluates the obstruction tensor S, whose vanishing is exactly the condition that the germ be induced by a complex contact structure. The relevant component of S(R,ν) is shown to be a nonzero function of x2 on every open interval, so no germ can be complex contact. A diffeomorphism from t","pith_inferences":["The explicit bracket calculation suggests the non-vanishing of S is stable: because A1(x2) only vanishes at isolated points, a small C∞ perturbation of the 1-forms that keeps the distribution fat will generically preserve the obstruction, matching the paper's infinite-codimension conclusion.","One could test the same method on distributions obtained by replacing the polynomial 3−2x2^2−x2^4 in the definition of t with nearby positive polynomials; if a similar coefficient function remains nonzero on every interval, the counterexample extends to a whole neighbourhood in the space of fat distributions.","The hyperbolic analogue is left open: whether a hyperbolic (4,6)-distribution with Reeb directions must be a product of two real contact germs. The same obstruction-tensor strategy, if adapted, could settle it."],"forward_implications":["Fat (4,6)-distributions with Reeb directions strictly contain the class that comes from complex contact structures, already at the level of germs.","Germs of horizontal immersions into such distributions need not be locally equivalent to 1-jet prolongations of holomorphic maps, since the complex-contact model is no longer forced.","At every point of every 6-manifold, the space of complex-contact distribution germs has infinite codimension inside the space of fat distribution germs with Reeb directions.","The canonical almost complex structure and the tensor S give an effective local certificate: to check whether a fat germ is complex contact, one computes whether S vanishes."],"fun_headline_variants":["Fat with Reeb directions doesn't imply complex contact","First fat distribution with Reeb directions not complex contact","Reeb directions are not enough: fat but not complex contact","Fat distribution with two Reeb directions escapes complex contact","No complex contact for this fat distribution despite Reeb directions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof rests on the correctness of the explicit formula for the canonical almost complex structure and on the resulting coefficient A1(x2) being nonzero on every interval; if those bracket computations contain an algebraic slip, the claim that S never vanishes is not established.","fun_headline_variants_meta":{"raw":{"variants":["Fat with Reeb directions doesn't imply complex contact","First fat distribution with Reeb directions not complex contact","Reeb directions are not enough: fat but not complex contact","Fat distribution with two Reeb directions escapes complex contact","No complex contact for this fat distribution despite Reeb directions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000422,"raw_usage":{"total_tokens":1969,"prompt_tokens":670,"completion_tokens":1299,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":1220}},"tokens_in":414,"tokens_out":1299,"duration_ms":9812,"temperature":1.0,"reasoning_tokens":1220,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T17:53:47.137068+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the obstruction tensor directly for the distribution in Theorem 3.1, taking R=∂z2 and ν=∂x2+y2∂z1+y1∂z2, and inspect the ∂z1 coefficient A1(x2) of (2[JR,ν]−J[JR,ν]−[JR,Jν]) mod D. The paper's formula gives A1(x2)=x2·f(x2) with f zero only when x2^2=−1+√2; if a recomputation yields A1≡0 on an open interval inside (−1,1), the main theorem is false.","supporting_citations":[],"review_version":1}