{"id":"896c1021-c26c-46f5-9608-80c7c9e3ef1f","arxiv_id":"1908.10207","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For two left-invariant involutive structures on SU(2), the paper computes closed range for the CR vector field and the smooth cohomology spaces of a corank-1 structure.","lead":"This paper computes cohomology spaces for left-invariant involutive structures on the compact group SU(2), using matrix coefficients of its irreducible representations. A reader would look here for concrete worked examples of global regularity, closed range, and solvability theory for differential complexes on compact Lie groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5.2's b2,3 sign contradicts the paper's own bracket relations; with the correct sign Theorem 5.1's H^{1,1}=0 is false, since the trivial-representation class (1,0) survives in cohomology.","rationale":"The reader correctly identified the b2,3 sign error as the central weakness, but the reader's expectation that correcting the sign restores the stated cohomology is not borne out: the corrected differential makes the trivial representation contribute a nonzero H^{1,1} class. Since the paper's headline corank-1 result is a computation, a false sign that changes the value of the cohomology is not a mere proof gap. The Proposition 4.1 closed-range proof also contains an impossible asymptotic inequality, though that portion may be salvageable. The division by n+1 in Theorem 5.2's proof is a further minor gap, repairable by using ∂− for the n = −1 case. Because the central theorem as stated is false, the manuscript should be rejected or substantially revised to present the corrected cohomology and its proof.","tokens_in":10822,"tokens_out":19692,"duration_ms":197817,"concrete_test":"Recompute §5.2 using the paper's own bracket relation [∂0,∂+] = ∂+, so b2,3 = +1, and apply the formulas in §5.1 to obtain the corrected d'(1,0) and d'(1,1). Then verify on the constant function 1 that d'(1,1)(1,0) = 0 and that no f ∈ C∞(SU(2)) satisfies d'(1,0)f = (1,0), because the first component would require ∂−f = 1 while the Haar integral of ∂−f vanishes. This establishes a nonzero class in H^{1,1}, so Theorem 5.1 cannot hold as printed.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing defect is the corank-1 commutator coefficient in §5.2. From §2.1, [∂+,∂0] = −∂+, so antisymmetry forces [∂0,∂+] = +∂+. But §5.2 sets b2,3 = −1. Feeding b2,3 = +1 into the general formulas of §5.1 changes the displayed maps: d'(1,0)u = (−∂−u, −∂0u + u) instead of (−∂−u, −∂0u − u), and d'(1,1)(u1,u2) = ∂0u1 − ∂−u2 instead of ∂0u1 − ∂−u2 + 2u1. With these corrected maps, take the trivial representation MT0. Then d'(1,1)(1,0) = 0, while d'(1,0)c = (0,c). The form represented by (1,0) is not in the image of d'(1,0): if it were, ∂−f = 1 would hold, but ∫ ∂−f = 0 for every f ∈ C∞(SU(2)) with respect to Haar measure. Hence H^{1,1}_V(SU(2);C∞) contains a nonzero class, contradicting Theorem 5.1. The printed b2,3 = −1 creates the spurious +2u1 term that kills exactly this class. A separate, lesser flaw is the proof of Proposition 4.1: the asserted inequality √(2ℓ) ≥ C(1+ℓ(ℓ+1))^{1/3} is false for large ℓ, since the left side grows like ℓ^{1/2} and the right side like ℓ^{2/3}; this weakens the closed-range proof as written, though it appears repairable with a smaller exponent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript applies the author's earlier abstract theory [1] to two left-invariant involutive structures on SU(2). For the corank-2 structure generated by ∂−, it proves (Prop. 4.1) that ∂− has closed range in C∞ and (Cor. 4.2) that the associated d′(1,0) has closed range. For the corank-1 structure v=span{∂−,∂0}, it computes the smooth cohomology, claiming H^{1,1}_V(SU(2);C∞)=0 (Thm. 5.1) and dim H^{0,1}_V(SU(2);C∞)=1 (Thm. 5.2). The computations are carried out by selecting complements and writing the differentials on matrix coefficients of the irreducible representations of SU(2).","tokens_in":11179,"tokens_out":17253,"duration_ms":162903,"significance":"The paper is a useful concrete exercise in applying the abstract framework of [1] to explicit left-invariant involutive structures on a compact Lie group. Its representation-theoretic setup is transparent, the coefficients are computed directly, and there are no hidden free parameters or data-fitting steps. The closed-range result for ∂− and the structure of the proof are plausible, and the paper is honest that the results are examples rather than deep new theory. However, the corank-1 computation is invalid as stated because of a sign error in §5.2, and the proof of Corollary 4.2 contains a non-sequitur. As a result, the advertised cohomology computation and one of the closed-range corollaries are not established.","major_comments":[{"comment":"The coefficient b2,3 is set to −1, contradicting the bracket relations stated in §2.1. Since [∂+,∂0]=−∂+, antisymmetry gives [∂0,∂+]=∂+, and with L2=∂0 and M=∂+ one must have b2,3=1. Every displayed map in §5.2 and in the proof of Theorem 5.1 depends on this value: the correct maps are d′(1,0)u=(−∂−u,−∂0u+u) and d′(1,1)(u1,u2)=∂0u1−∂−u2, not the printed (−∂−u,−∂0u−u) and ∂0u1−∂−u2+2u1.","section":"§5.2, constants for v=span{∂−,∂0}"},{"comment":"With the corrected sign, Theorem 5.1 is false. On the trivial representation MT0, d′(1,0)c=(0,c), so its image is {(0,c)}. The pair (1,0) lies in ker d′(1,1) because ∂0(1)=0 and ∂−(0)=0. If (1,0)=d′(1,0)f for some f∈C∞(SU(2)), then ∂−f=−1; integrating against Haar measure gives 0=−1, since ∫∂−f dμ=0 for every f. Hence H^{1,1}_V(SU(2);C∞) contains a nonzero class, contradicting the theorem. The spurious +2u1 term in the printed d′(1,1) is exactly what kills this class.","section":"§5.2 / Theorem 5.1"},{"comment":"The displayed inequality √(2ℓ) ≥ C(1+ℓ(ℓ+1))^{1/3} is false for large ℓ: the left side grows like ℓ^{1/2} and the right side like ℓ^{2/3}. Therefore the cited condition from [1, eqn. (6.1)] with s=1/3 is not verified, and the closed-range conclusion does not follow from the argument as written. The claim may be repairable with a smaller exponent (for example 1/4), but the estimate must be corrected.","section":"§4.2, proof of Proposition 4.1"},{"comment":"The inference “since ∂− has closed range, there exists u2 such that u2,ν→u2” is invalid. Closed range only gives a lift u2 with ∂−u2 equal to the limit of ∂−u2,ν; it does not make the original sequence converge. The proof therefore does not establish closed range of d′(1,0). A correct proof would need a uniform estimate on a complement or an explicit Peter-Weyl argument.","section":"§4.2, proof of Corollary 4.2"}],"minor_comments":[{"comment":"The statement is missing a closing parenthesis: it should read H^{1,1}_V(SU(2);C∞(SU(2))).","section":"Theorem 5.1 statement"},{"comment":"In the proof of surjectivity of d′(0,1), the choice φ=−(n+1)^{-1}tℓ_{mn} is undefined when n=−1 (for example, ℓ=1). The surjectivity is still true because the ∂− term from the second component can produce t_{m,−1}, but the proof should be adjusted.","section":"§5.2 / Theorem 5.2"},{"comment":"Reference [1] is cited as “Ann. Glob. Anal. Geom., 2019” without volume or article number; please supply full bibliographic data if the paper has been published.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is not acceptable in its current form because Theorem 5.1 is false as stated and the proof of Corollary 4.2 contains a serious logical gap. The errors appear local and the general method is sound; the author should recompute the corank-1 differentials with the correct commutator sign and either correct Theorem 5.1 or remove the claim, and repair the closed-range arguments. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the computation for ∂− is fine and useful, but the corank-1 section is wrong in a way that kills the main theorem. I checked by hand, and the stress-test concern lands.\n\nWhat is actually new and good: the closed-range proof for ∂−, the explicit kernel computation for d'(1,0), and Corollary 4.2 are concrete and, modulo the inequality issue, credible. The representation-theoretic setup is clean, and the abstract is honest that this is an application of [1].\n\nSoft spots, in order of severity:\n\n1. Section 5.2, b2,3. The paper's own bracket relations give [∂0, ∂+] = +∂+, so b2,3 should be +1. The printed −1 propagates into d'(1,0) and d'(1,1). With the correct sign, d'(1,1)(u1,u2) = ∂0u1 − ∂−u2, and the constant form (1,0) is d'-closed. It cannot be in the image of d'(1,0): if d'(1,0)f = (1,0), then ∂−f = −1, impossible because ∫ ∂−f = 0 on SU(2). So H^{1,1}_V ≠ 0, contradicting Theorem 5.1. This is not a typo; the claimed result is false.\n\n2. Even using the printed maps, the proof of Theorem 5.1 asserts d'(1,0) is injective because n+1 is never zero for half-integers. For ℓ = 1, n = −1 occurs, and ∂−t_{m,−1} = 0, so d'(1,0) has a nontrivial kernel. The rank-nullity exactness argument collapses.\n\n3. Proposition 4.1: the inequality √(2ℓ) ≥ C(1+ℓ(ℓ+1))^{1/3} is false for large ℓ (left ~ℓ^{1/2}, right ~ℓ^{2/3}). This is repairable with a smaller exponent, but as written the closed-range proof is incomplete.\n\n4. Minor: in Theorem 5.2 the chosen φ = −(n+1)^{-1}t fails at n = −1; surjectivity can be fixed by another choice, so this is minor.\n\nVerdict: the ∂− example may survive repair, but the corank-1 result, the main new computation, is false. This needs major revision and a different claim, not copyediting. If this is heading toward submission, I would not send it to a referee as is; I would tell the author to redo the corank-1 computation. The ∂− part alone might be worth a short note after the inequality is fixed, but the paper as a whole should not be relied on.","headline":"The corank-1 cohomology theorem is false: the sign error in §5.2 is load-bearing, and even the printed maps make d'(1,0) non-injective at ℓ=1.","tokens_in":11741,"tokens_out":14189,"would_cite":false,"duration_ms":139721,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R03","35A01","58J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"On $SU(2)$, the corank-1 left-invariant cohomology spaces are $H^{1,1}=0$ and $H^{0,1}\\simeq\\mathbb{C}$.","keywords":["left-invariant operators","solvability","differential complexes","locally integrable structures","SU(2)","smooth cohomology","CR structure","Peter-Weyl decomposition"],"falsifier":"On the $\\ell=1$ isotypic component, apply the paper's displayed formula $d'_{(1,0)}u=(-\\partial_-u,-\\partial_0u-u)$ to the matrix coefficient $t^1_{m,-1}$; the second component acts as $-(n+1)=0$, so the kernel is nontrivial. A direct rank-nullity calculation on this four-dimensional block therefore decides whether the exactness argument behind $H^{1,1}_V=0$ survives, and it fails as written.","tokens_in":10540,"feed_emoji":"🧮","tokens_out":12785,"duration_ms":116510,"temperature":0.7,"pith_summary":"This paper studies left-invariant involutive structures—complex subbundles of the complexified tangent bundle closed under Lie bracket—on $SU(2)$, the simplest non-commutative compact Lie group. It establishes that the single complex vector field $\\partial_-$ spanning the standard CR structure has closed range in the smooth topology, and that its associated first-order operator $d'$ does too. For the corank-1 structure spanned by $\\partial_-$ and $\\partial_0$, it derives explicit formulas for $d'$ on each irreducible representation block and concludes that the smooth cohomology spaces satisfy $H^{1,1}_V(SU(2);C^\\infty)=0$ and $\\dim H^{0,1}_V(SU(2);C^\\infty)=1$. The concrete payoff is that an abstract cohomology theory becomes computable in a non-abelian example through representation matrix coefficients.","feed_headline":"SU(2) cohomology spaces computed: one vanishes, one is dimension 1","feed_subtitle":"Representation matrix coefficients turn the d-prime complex into linear algebra on finite blocks.","key_machinery":"The load-bearing machinery is the Peter-Weyl decomposition of $C^\\infty(SU(2))$ into isotypic components $M_{T_\\ell}$ indexed by the irreducible representations $T_\\ell$, together with the standard formulas for the action of the invariant vector fields: $\\partial_+ t^\\ell_{mn}=-\\sqrt{(\\ell-n)(\\ell+n+1)}\\,t^\\ell_{m,n+1}$, $\\partial_- t^\\ell_{mn}=-\\sqrt{(\\ell+n)(\\ell-n+1)}\\,t^\\ell_{m,n-1}$, and $\\partial_0 t^\\ell_{mn}=n\\,t^\\ell_{mn}$. These formulas turn every differential $d'$ in the complex into a family of finite-dimensional linear maps on the blocks $M_{T_\\ell}$, so kernel/range computations reduce to rank-nullity arguments; a theorem from the companion paper converts the resulting closed-range and exactness statements into statements about smooth cohomology on the whole group.","core_discovery":"The paper's claim is that on $SU(2)$ the left-invariant CR vector field $\\partial_-$ is globally almost hypoelliptic, equivalently $\\partial_-:C^\\infty(SU(2))\\to C^\\infty(SU(2))$ has closed range, and that for $v=\\mathrm{span}\\{\\partial_-,\\partial_0\\}$ the smooth cohomology is $H^{1,1}_V(SU(2);C^\\infty)=0$ with $\\dim H^{0,1}_V(SU(2);C^\\infty)=1$. The route is to decompose smooth functions by the Peter-Weyl theorem into finite-dimensional isotypic components $M_{T_\\ell}$ spanned by the matrix coefficients $t^\\ell_{mn}$, write down how $\\partial_\\pm$ and $\\partial_0$ act on these coefficients, and reduce the differential complex on each block to a finite-dimensional linear algebra problem. Exactness on each block, combined with the closed-range property, is then lifted back to the smooth Fréchet cohomology.","pith_inferences":["The same representation-block method should extend to other compact Lie groups, where the highest-weight classification plays the role of the half-integer labels $\\ell$; the main new work would be proving the closed-range lower bounds that the current paper gets from the elementary inequality $(\\ell+x)(\\ell-x+1)\\ge 2\\ell$.","One can test the method on the Heisenberg group or on tori with non-standard invariant structures, where the Peter-Weyl blocks are replaced by Fourier modes and the analogous $d'$ maps are constant-coefficient.","The closed-range estimate for $\\partial_-$ suggests a general principle: a left-invariant vector field that lowers the representation index by a bounded amount and has no zero eigenvalue on orthogonal complements will be globally hypoelliptic; this could be formulated as a representation-theoretic criterion."],"forward_implications":["The closed-range result for $\\partial_-$ implies that $\\partial_-$ is globally almost hypoelliptic on $SU(2)$ in the quantitative sense of the earlier theory, so the equation $\\partial_-u=f$ has solutions whose regularity is controlled by that of $f$.","The blockwise computation shows that smooth cohomology of left-invariant structures on compact Lie groups can be read off from finite-dimensional representation theory, without constructing explicit parametrices.","For the corank-1 structure $v=\\mathrm{span}\\{\\partial_-,\\partial_0\\}$, the single surviving cohomology class $H^{0,1}_V\\cong\\mathbb{C}$ is carried by the trivial representation block, so the nonzero class is the natural 'constant' class visible already at the level of $\\ell=0$.","Because the same vector-field action formulas exist for all compact Lie groups, the paper's examples provide a template for computing cohomology of other invariant structures, provided the analogous closed-range estimates hold."],"supporting_citations":[{"why":"Supplies the closed-range criterion equating almost hypoellipticity with closed range, and the theorem reducing smooth cohomology to a direct sum of isotypic-block quotients.","marker":"[1]"},{"why":"Provides the explicit formulas for how $\\partial_\\pm$ and $\\partial_0$ act on the matrix coefficients $t^\\ell_{mn}$, plus the normalization used in the Peter-Weyl expansion.","marker":"[2]"}],"fun_headline_variants":["SU(2) cohomology: one vanishes, one is dimension 1","Peter-Weyl reduces SU(2) cohomology to linear algebra: 0 and 1","On SU(2), cohomology spaces: H^{1,1}=0, dim H^{0,1}=1","Closed range for CR operator on SU(2); cohomology: 0 and 1","Matrix coefficients give SU(2) cohomology: one zero, one 1-dim"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The corank-1 theorems rest on the sign of the commutator coefficient $b_{2,3}$ in Section 5.2: the displayed $d'$ maps take $b_{2,3}=-1$, i.e. $[\\partial_0,\\partial_+]=-\\partial_+$, whereas the paper's bracket table gives $[\\partial_0,\\partial_+]=\\partial_+$, so the proof of Theorem 5.1 depends on that sign convention and the exactness argument on the $\\ell=1$ block is the fragile point.","fun_headline_variants_meta":{"raw":{"variants":["SU(2) cohomology: one vanishes, one is dimension 1","Peter-Weyl reduces SU(2) cohomology to linear algebra: 0 and 1","On SU(2), cohomology spaces: H^{1,1}=0, dim H^{0,1}=1","Closed range for CR operator on SU(2); cohomology: 0 and 1","Matrix coefficients give SU(2) cohomology: one zero, one 1-dim"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001038,"raw_usage":{"total_tokens":4353,"prompt_tokens":915,"completion_tokens":3438,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":3312}},"tokens_in":531,"tokens_out":3438,"duration_ms":23650,"temperature":1.0,"reasoning_tokens":3312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:14:58.126514+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the $\\ell=1$ isotypic component, apply the paper's displayed formula $d'_{(1,0)}u=(-\\partial_-u,-\\partial_0u-u)$ to the matrix coefficient $t^1_{m,-1}$; the second component acts as $-(n+1)=0$, so the kernel is nontrivial. A direct rank-nullity calculation on this four-dimensional block therefore decides whether the exactness argument behind $H^{1,1}_V=0$ survives, and it fails as written.","supporting_citations":[{"cited_title":"Ara´ ujo","cited_arxiv_id":null,"evidence_quote":"Supplies the closed-range criterion equating almost hypoellipticity with closed range, and the theorem reducing smooth cohomology to a direct sum of isotypic-block quotients."},{"cited_title":"Ruzhansky and V","cited_arxiv_id":null,"evidence_quote":"Provides the explicit formulas for how $\\partial_\\pm$ and $\\partial_0$ act on the matrix coefficients $t^\\ell_{mn}$, plus the normalization used in the Peter-Weyl expansion."}],"review_version":1}