{"id":"630cfddc-fc06-4aa1-ba68-38a8215411ab","arxiv_id":"2506.06807","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symplectic fillings of the standard codimension-2 contact sphere in a symplectic ball are smoothly isotopic to the standard linear disk.","lead":"This paper proves that any symplectic filling of the standard contact sphere inside a larger standard contact sphere is smoothly unknotted, meaning it is diffeomorphic to the standard linear subspace. The result is the first step toward answering a question by Roger Casals about whether such fillings are symplectically standard.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem is conditional on an unproved almost-complex-structure existence/regularity claim: the corrupted passage after Prop. 4.9, on which Theorem 5.3 rests, must be supplied.","rationale":"The reader identified the weakest point exactly: the existence of a compatible almost complex structure making \\hat W holomorphic while preserving the transversality and homology-control needed in Propositions 4.8 and 4.9. The corrupted sentence after Proposition 4.9 is the only place this is asserted, and the subsequent suppression of the requirement means Theorem 5.3 is not fully proved as written. I do not see a fatal flaw or a counterexample: the gap is plausibly fillable by a careful perturbation argument, and the overall strategy is coherent. Therefore the verdict should remain CONDITIONAL rather than being upgraded to unconditional acceptance or downgraded to rejection. The concrete test proposed would settle whether the missing perturbation argument can actually be carried out within the constrained class of almost complex structures.","tokens_in":31383,"tokens_out":14941,"duration_ms":163463,"concrete_test":"Repair the passage by writing a complete proof that there exists J satisfying: (a) \\hat W is J-holomorphic; (b) on the positive end J is in the class of Proposition 4.6 and agrees with J_F near γ_pmax; (c) M_J(γ_pmax,q) is regular with signed count 1 and the quotient in Proposition 4.9 admits a degree-one evaluation map. Concretely, take the standard case U=D^{n-1}⊂D^n and an arbitrary filling W, and check whether a generic perturbation of J_F supported away from \\hat W can achieve transversality without moving J on the positive end or breaking the W-holomorphicity condition. If the only perturbations that yield Propositions 4.8–4.9 force J outside the class of Proposition 4.6, then Theorem 5.3 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.3, and hence Theorem 1.4/1.2, requires an almost complex structure J on the completion of V×D that simultaneously: (i) makes the arbitrary filling \\hat W a J-holomorphic hypersurface; (ii) is of the class controlled by Proposition 4.6 so that all relevant curves have trivial homology in the V-direction and satisfy the energy bounds; and (iii) is generic enough for Propositions 4.8 and 4.9, i.e. M_J(γ_pmax,q) has signed count 1 and the evaluation map on M_J(γ_pmax,ν)/∼ has degree 1. After Proposition 4.9 the text reads 'such that ?? 4.8?? 4.9 hold ... as we can perturb J near p and ν, which are outside of ˆW' and then says the requirement will be suppressed. This is not a cosmetic issue: Proposition 4.8 is proved by starting from the foliated almost complex structure J_F and then perturbing to achieve transversality. If the perturbation is constrained to keep \\hat W holomorphic, it is not automatic that the count #M_J(γ_pmax,q)=1, the degree-one property of ev, or the uniform bounds of Proposition 4.6 survive. The loop-pushing argument in Proposition 5.3 depends on these facts, so the unproved existence of such a J is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies symplectic fillings W of a codimension-two contact submanifold H = ∂(U × D) inside ∂(V × D), where V is a Liouville domain, with the main application being the standard contact submanifold (S^{2n−1}, ξ_std) in (S^{2n+1}, ξ_std) inside the standard ball. The central claims are that, for n ≥ 2, every such filling of the standard contact submanifold is smoothly unknotted in the ball, and more generally that the complement inclusion V × {(0,1)} \\ U × {(0,1)} → V × D \\ W induces a surjection on π1 and, when π1(V \\ U) is abelian, a homotopy equivalence. The method is pseudoholomorphic: the paper constructs L-simple almost complex structures and a holomorphic foliation, develops a Siefring-type intersection formula for punctured curves and holomorphic hypersurfaces in the L-simple setting, and uses holomorphic planes asymptotic to a maximal Reeb orbit to push loops in the complement to the boundary. A separate argument using holomorphic spheres proves an analogous rigidity statement for fillings of the binding of a trivial open book (Theorem 1.5).","tokens_in":31748,"tokens_out":5879,"duration_ms":60638,"significance":"If the proof is completed, the main theorem is a strong relative filling-rigidity result: it shows that symplectic fillings of the standard codimension-two contact submanifold are smoothly standard, a first concrete step toward Casals' question about symplectic unknottedness. The paper also contains useful technical contributions: the explicit holomorphic foliation and uniform homology bound of Proposition 4.6, the filtered Bourgeois–Oancea comparison in Theorem 4.3, and the pseudocycle arguments in Section 6. These are well-motivated and appear to be on the right track. However, the manuscript as written has load-bearing gaps, most importantly an asserted but unproved simultaneous existence of an almost complex structure that makes the arbitrary filling holomorphic and keeps the relevant moduli spaces regular with the required counts; this is used in the proof of Proposition 5.3 and hence in Theorems 1.4 and 1.2.","major_comments":[{"comment":"The passage after Proposition 4.9 asserts that for a symplectic filling W of H in V × D there exists an almost complex structure J on the completion such that W is J-holomorphic and Propositions 4.8 and 4.9 hold, with the explanation that one can perturb J near p and ν, which are outside W. This is not justified. Propositions 4.8 and 4.9 require J, at least on the positive end, to be a generic perturbation of the class considered in Proposition 4.6 so that M_J(γ_pmax, q) is regular with signed count 1 and the evaluation map on M_J(γ_pmax, ν)/∼ has degree 1. The condition that the arbitrary filling W be J-holomorphic is a closed constraint on J, and perturbing J only near the marked points p and ν does not automatically preserve transversality of the moduli spaces or the stated counts and degree. The proof of Proposition 5.3, and hence of Theorems 1.4 and 1.2, depends on this simultaneous-existence statement. A complete proof of this compatibility assertion is required.","section":"§4, passage after Proposition 4.9"},{"comment":"The loop-pushing argument in Proposition 5.3 is too compressed and is not sufficient as written. After defining the intervals I_i and the reparameterizations φ_i ∈ Aut(C,0), the text states that the C∞_loc limits from the two sides of a glued point may be different, and then defines paths p_{i,s} connecting s and φ_i^{-1}(s) with lim_{s→∞} |p_{i,s}| = ∞. It is not established that these paths can be chosen continuously in i and s, nor that the concatenated maps u_i(t)(s) converge on the glued boundary points to an actual holomorphic curve. The conclusion that ν_s for s ≫ 0 lies in R_+ × (Y \\ H) requires uniform or at least controlled convergence of the parametrized curves, which is not proved. Since this is the mechanism that proves π1-surjectivity, the argument needs to be written out in full.","section":"§5, Proposition 5.3"},{"comment":"The proof of Proposition 4.1 cites 'Theorem 2.10', but no such theorem exists in the manuscript; the closest statement is Example 2.10, which records that for f = −ε(x² + y²) the relevant eigenvalue has winding number 0. The conclusion u • W = 0 relies on the hidden intersection δ∞ being zero for this special f. The intended lemma should be stated and proved explicitly rather than referencing a nonexistent theorem. As written, the proof of Proposition 4.1, which is essential for ensuring that the probing curves do not intersect the filling W, is not complete.","section":"§4.1, Proposition 4.1"},{"comment":"In Section 6.2 the paper assumes an almost complex structure on V × CP^1 that is compatible with λ_V ⊕ ω_CP1, is product-type on the end, makes the arbitrary filling W holomorphic, and makes V × {(2,0)} and V × {∞} holomorphic hypersurfaces. The simultaneous existence of such a J is asserted without proof, and Proposition 6.5 further requires generic J for pseudocycle transversality. This is the same type of compatibility issue as the gap after Proposition 4.9 and underpins Theorem 1.5. The existence of such a J with all the required properties needs to be established.","section":"§6.2, list of almost complex structure conditions"}],"minor_comments":[{"comment":"The abstract and introduction promise a self-contained proof of the Siefring intersection formula, but Theorem 2.6 is stated without proof and the text refers to Wendl's book for the chain of arguments; Remark 2.8 further defers a needed variant to the in-preparation work [ABDRZ]. The self-containedness claim should be either fulfilled by a proof of Theorem 2.6 or softened.","section":"§2, abstract/introduction"},{"comment":"There are many cross-reference errors: for example 'Theorem 2.10' in Proposition 4.1, 'Theorem 3.1' for Proposition 3.1, 'Theorem 3.4' and 'Theorem 3.5' for Propositions 3.4 and 3.5, and 'Theorem 4.6'/'Theorem 4.8' in the proof of Proposition 4.8 for Propositions 4.6 and 4.8.","section":"Throughout"},{"comment":"The sentence after Proposition 4.9 contains corrupted text 'such that ?? 4.8?? 4.9 hold' and must be repaired; as printed it does not state a precise assertion.","section":"§4, after Proposition 4.9"},{"comment":"The proofs rely on the in-preparation works [ABDRZ] and [Sie] for essential ingredients (Remark 2.8 and the higher-dimensional intersection theory in Section 2). For a journal submission these dependencies should be disclosed explicitly and, ideally, the relevant statements should be proved or stated as assumptions.","section":"References"},{"comment":"Remark 5.4 discusses extensions to subcritical surgeries and explicitly says 'To rigorously prove those claims is non-trivial.' This is fine as a remark, but it should be separated more clearly from the main proof so that the reader does not confuse a stated extension with a proved theorem.","section":"§5, Remark 5.4"}],"recommendation":"major_revision","confidential_remarks":"The central idea is promising and the main theorem would be significant if the gaps are filled. However, the manuscript relies on several in-preparation papers and contains numerous corrupted cross-references and a load-bearing unproved almost-complex-structure existence claim. I recommend major revision rather than rejection, because the gaps appear fixable within the scope of the paper, but the current version is not yet suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zhengyi Zhou has a new relative filling rigidity theorem: any symplectic filling of the standard contact (S^{2n-1},ξ_std) inside (S^{2n+1},ξ_std) in the ball is smoothly unknotted for n≥2. That is a clean, substantial answer to the smooth part of Casals' question, and the proof strategy via Siefring intersection theory with the L-simple setup is coherent. The paper also gives a self-contained proof of the Siefring intersection formula in the L-simple case, which is a useful service—the original is an announced result.\n\nStrengths: the L-simple setup is carefully written; the foliation argument in §3.3 to control homology classes of holomorphic curves is a nice piece of work, and the application of Bourgeois-Oancea to get the required counts is executed with care. Section 6's pseudocycle argument is also clean.\n\nSoft spots, in order of seriousness. (1) The passage after Prop 4.9 asserts that we can find compatible almost complex structures on the completion such that W is holomorphic and Props 4.8/4.9 hold, and then says 'we will suppress the requirement.' This is load-bearing: Prop 5.3 uses the degree-one evaluation map and the counts from 4.8/4.9, and the asserted perturbation near p and ν does not obviously preserve those counts—the moduli spaces involve curves that can wander far from p and ν, and making W holomorphic imposes constraints on J that generally break the genericity used in the proofs of 4.8/4.9. The author needs to supply an actual argument. (2) The proof of Prop 5.3, the loop-pushing argument, is only sketched. The parametrization of the curves and the behavior at the gluing points need more detail to be convincing. This is a gap but likely fillable. (3) Minor: the paper cites two in-preparation works, but the main theorem doesn't reduce to them; they're building blocks. That's acceptable.\n\nOverall: the central claim is plausible and no fatal flaw jumps out, but the written proof is not complete as it stands. This is a paper for specialists in symplectic filling rigidity and intersection theory. It deserves a serious referee—with the gaps fixed it would be a solid publication. I'd send it to review with a request for major revision, focusing on the missing almost complex structure argument and a detailed proof of Prop 5.3.","headline":"Plausible and important result, but the proof has a load-bearing gap: the existence of an almost complex structure making W holomorphic while preserving the moduli-space counts is asserted in a corrupted passage, and the pi_1-argument is sketched.","tokens_in":32189,"tokens_out":3262,"would_cite":false,"duration_ms":33514,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D35","53D40","53D42","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Any symplectic filling of the standard contact sphere lying linearly inside a standard ball is smoothly unknotted.","keywords":["symplectic fillings","contact submanifolds","unknottedness","punctured holomorphic curves","intersection theory","L-simple almost complex structures","linearized contact homology","S1-equivariant symplectic cohomology"],"falsifier":"Find, for some $n\\ge 2$, a symplectic filling $W$ of the standard $(S^{2n-1},\\xi_{\\mathrm{std}})$ inside $(D^{n+1},\\omega_{\\mathrm{std}})$ whose complement $D^{n+1}\\setminus W$ has fundamental group not isomorphic to $\\mathbb{Z}$; the paper proves the complement is homotopy equivalent to $S^1$, so such an example would refute the unknottedness theorem.","tokens_in":31200,"feed_emoji":"🪢","tokens_out":18982,"duration_ms":170118,"temperature":0.7,"pith_summary":"This paper proves a relative version of filling rigidity: if $W$ is a symplectic filling of the standard contact sphere $(S^{2n-1},\\xi_{\\mathrm{std}})$ sitting linearly inside $(S^{2n+1},\\xi_{\\mathrm{std}})$ in the standard ball $(D^{n+1},\\omega_{\\mathrm{std}})$, then for every $n\\ge 2$ the pair $(D^{n+1},W)$ is smoothly unknotted, meaning diffeomorphic relative to the boundary to the standard linear disk. The result rules out many knotted codimension-two balls: those smooth models cannot be symplectic fillings of the simplest contact submanifold. The proof turns the filling into a holomorphic hypersurface for a carefully chosen almost complex structure and then probes the complement with punctured holomorphic curves, using intersection theory to show those curves stay away from the filling. A degree-one evaluation map from a moduli space of holomorphic planes forces the complement to have the fundamental group of a circle; in the sphere case the complement is homotopy equivalent to $S^1$, and a codimension-two unknotting theorem concludes smooth unknottedness. The same machinery gives general statements about the complements of symplectic submanifold fillings in any Liouville domain.","feed_headline":"Symplectic fillings of the standard contact sphere are unknotted","feed_subtitle":"For n≥2, each such filling is diffeomorphic, rel boundary, to the standard linear disk.","key_machinery":"The engine of the proof is a higher-dimensional intersection theory for punctured holomorphic curves against holomorphic hypersurfaces, set up in the L-simple framework: near the relevant Reeb orbits the almost complex structure is chosen so that the Cauchy–Riemann equation splits into a hypersurface direction and a normal direction, and the normal part has a linear asymptotic expansion in eigenfunctions of a self-adjoint operator. Winding numbers of those eigenfunctions define hidden intersections at punctures, and the intersection formula $$u\\bullet H = \\sum_{p:u(p)\\in H}\\delta(p,u,H) + \\sum_{p}\\delta_\\infty(p,u,H)$$ expresses the total intersection number as a sum of positive local terms; in particular $u\\bullet H=0$ forces the curve either to lie in $H$ or to avoid it. A holomorphic foliation of $\\hat V\\times \\mathbb{C}$ confines every holomorphic plane asymptotic to the maximal Reeb orbit $\\gamma_{p_{\\max}}$ to a fixed homology class, and a filtered isomorphism between linearized contact homology and $S^1$-equivariant positive symplectic cohomology for $V\\times D$ yields the one-point count $\\#\\mathcal{M}_J(\\gamma_{p_{\\max}},q)=1$ for generic $q$. That degree-one evaluation, applied to a loop in the complement, is what turns homology information into the fundamental-group surjection.","core_discovery":"The central claim is that relative topological rigidity holds for the simplest contact submanifold: for $n\\ge 2$, every symplectic filling $W$ of $(S^{2n-1},\\xi_{\\mathrm{std}})$ inside $(S^{2n+1},\\xi_{\\mathrm{std}})$ in $(D^{n+1},\\omega_{\\mathrm{std}})$ is smoothly unknotted, so the pair $(D^{n+1},W)$ is diffeomorphic, relative to the boundary, to the standard linear disk. More generally, for a Liouville domain $V$ and a codimension-two symplectic submanifold $U\\subset V$, any symplectic filling $W$ of $\\partial(U\\times D)$ in $V\\times D$ has complement homology isomorphic to that of $V\\setminus U$, and the inclusion induces a surjection on fundamental groups; when $\\pi_1(V\\setminus U)$ is abelian, $V\\times D\\setminus W$ is homotopy equivalent to $V\\setminus U$. The route taken is to choose almost complex structures making $\\hat W$ a holomorphic hypersurface in the completion, and to let intersection theory for punctured holomorphic curves show that the curves probing the complement never meet $\\hat W$, so their evaluation maps survive and control the topology of the complement.","pith_inferences":["The paper proves smooth unknottedness only; symplectic unknottedness remains open, and a natural conjecture is that the same intersection-theoretic control can be upgraded to a symplectomorphism statement.","The self-contained proof of the intersection formula in the L-simple setup is likely to be reusable in other constructions of holomorphic hypersurfaces, such as contact-homology computations with intersection information, which the paper mentions but does not develop.","The fundamental-group surjection is established only for the specific contact pairs $\\partial(U\\times D)$ inside $\\partial(V\\times D)$; the paper expects this rigidity to be special, and a test in the lowest case $n=2$, where the filling is a 4-manifold inside a 6-ball, would show how far the regularity assumption reaches."],"forward_implications":["For $n\\ge 2$, every symplectic filling of the standard $(S^{2n-1},\\xi_{\\mathrm{std}})$ inside $(D^{n+1},\\omega_{\\mathrm{std}})$ is smoothly the standard linear disk, so the smooth knot type of such a filling is unique.","For any Liouville domain $V$ and codimension-two symplectic submanifold $U\\subset V$, any symplectic filling of $\\partial(U\\times D)$ in $V\\times D$ has complement with the same homology as $V\\setminus U$, and $\\pi_1(V\\setminus U)$ maps onto $\\pi_1(V\\times D\\setminus W)$; if $\\pi_1(V\\setminus U)$ is abelian, the complement is homotopy equivalent to $V\\setminus U$.","A symplectic filling of the binding $\\partial V\\times\\{0\\}$ of the trivial open book in $\\partial(V\\times D)$ induces an isomorphism on homology and a surjection on fundamental groups; if $\\pi_1(V)$ is abelian, the inclusion is a homotopy equivalence.","Since the classical exact-filling theorem already forces $W$ to be a disk, the unknottedness statement is about the pair, and the proof works with the standard filling of $(S^{2n+1},\\xi_{\\mathrm{std}})$ replaced by any exact filling.","Smoothly knotted codimension-two balls are abundant, so the theorem implies that none of those knotted models can be symplectic fillings of the standard contact submanifold."],"supporting_citations":[{"why":"Gives the theorem that every exact filling of the standard contact sphere is diffeomorphic to a ball, so the filling $W$ in the main theorem is already a disk and only the pair needs to be unknotted.","marker":"[McD91]"},{"why":"Supplies the codimension-two unknotting theorem used to pass from the homotopy type of the complement to smooth unknottedness of the pair.","marker":"[Lev65]"},{"why":"Provides the L-simple contact forms and almost complex structures whose linear asymptotic expansions make the higher-dimensional intersection formula tractable.","marker":"[BH23]"},{"why":"Originates the intersection theory for punctured holomorphic curves of which the paper's formula is the higher-dimensional analog.","marker":"[Sie11]"},{"why":"Supplies the dimension-four intersection-theoretic chain of arguments and asymptotic winding statements the paper follows in the self-contained proof.","marker":"[Wen20]"},{"why":"Computes the filtered positive symplectic cohomology of $V\\times D$ and the homology isomorphism for fillings of $\\partial(V\\times D)$, used to identify the class of $\\gamma_{p_{\\max}}$ and the complement homology.","marker":"[Zho23]"},{"why":"Establishes the filtered isomorphism between linearized contact homology and $S^1$-equivariant symplectic cohomology that converts the algebraic computation into a statement about holomorphic curves through points.","marker":"[BO09a]"},{"why":"Supplies the index formula for holomorphic planes asymptotic to $\\gamma_p$, used to identify the unique local maximum contribution and the dimension of the moduli spaces.","marker":"[Zho21]"}],"fun_headline_variants":["Every symplectic filling of the standard contact sphere is unknotted","Symplectic fillings of contact sphere are smoothly unknotted for n≥2","Unknotted symplectic fillings: rigidity inside standard contact","Topological rigidity: symplectic fillings of S^{2n-1} are unknotted","Standard contact sphere fillings: all unknotted"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the existence of one geometric structure that both turns the filling into a holomorphic hypersurface and keeps the holomorphic-plane counting spaces regular, with degree-one evaluation maps; the paper asserts that such a structure can be arranged and then suppresses the requirement, and this assertion carries the main theorem.","fun_headline_variants_meta":{"raw":{"variants":["Every symplectic filling of the standard contact sphere is unknotted","Symplectic fillings of contact sphere are smoothly unknotted for n≥2","Unknotted symplectic fillings: rigidity inside standard contact","Topological rigidity: symplectic fillings of S^{2n-1} are unknotted","Standard contact sphere fillings: all unknotted"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000809,"raw_usage":{"total_tokens":3536,"prompt_tokens":918,"completion_tokens":2618,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":2517}},"tokens_in":534,"tokens_out":2618,"duration_ms":19907,"temperature":1.0,"reasoning_tokens":2517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:50:07.153423+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find, for some $n\\ge 2$, a symplectic filling $W$ of the standard $(S^{2n-1},\\xi_{\\mathrm{std}})$ inside $(D^{n+1},\\omega_{\\mathrm{std}})$ whose complement $D^{n+1}\\setminus W$ has fundamental group not isomorphic to $\\mathbb{Z}$; the paper proves the complement is homotopy equivalent to $S^1$, so such an example would refute the unknottedness theorem.","supporting_citations":[],"review_version":1}