{"id":"9b02bf78-b4b0-409b-9c4c-dcc29228ce73","arxiv_id":"2412.09555","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Ekeland-Hofer capacities equal the Gutt-Hutchings capacities for every star-shaped domain in R^{2n} and every index k.","lead":"This paper proves that two different sequences of symplectic capacities, the Ekeland-Hofer capacities and the Gutt-Hutchings capacities, always agree on star-shaped regions in even-dimensional space. The result settles a 35-year-old question about whether a classical variational invariant and a modern Floer-homology invariant measure the same geometric quantity.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The chain map Φ in Theorem 1.8 is unproved: the index-1 hybrid moduli space is shown to be compact in the sense of Theorem 5.5, but the gluing direction needed to identify the boundary as broken trajectories is absent, so ∂Φ = Φ∂ is not established.","rationale":"","tokens_in":24395,"tokens_out":12779,"duration_ms":122168,"concrete_test":"Write out the gluing theorem for the hybrid moduli space in §5. Concretely, take a broken trajectory (V,U) of index 1, where V is a Morse trajectory from (x,p) to (z,r) and U is a hybrid curve from (z,r) to (y,q) (or the symmetric order), preglue them to form an approximate solution, and apply the implicit function theorem in the weighted Sobolev spaces used in §5.1 to produce a smooth family of hybrid curves parametrized by the gluing length R ∈ (R0, ∞), converging to (V,U) as R → ∞. Then verify that the signed count of the two boundary strata equates to ∂Φ = Φ∂. If the gluing construction fails or requires additional hypotheses not stated in the paper, then Theorem 1.8 is unproved; if it succeeds, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests entirely on Theorem 1.8, whose proof in §5 constructs a chain map Φ by counting hybrid curves. For Φ to satisfy ∂Φ = Φ∂, the closure of the index-1 moduli space M̄^hyb((x,p),(y,q)) must be a compact 1-manifold whose boundary consists precisely of the two types of broken trajectories: a Morse segment followed by a hybrid curve (contributing to Φ∂) and a hybrid curve followed by a Floer segment (contributing to ∂Φ). The paper cites Theorem 5.5 for this statement, but Theorem 5.5 only proves the compactness half: via estimates (5.2)–(5.3) and citations to [Sch95] and [AM05], every sequence in M^hyb subconverges to some broken object. It does not provide a gluing theorem showing that every broken object of index 1 is the limit of a one-parameter family of smooth hybrid curves, nor does it establish that these limits form a collar neighborhood making M̄^hyb a manifold with boundary. Without gluing, the algebraic count of boundary points—the input for the chain map equation—is not defined, and the assertion in §5.3 that 'Φ is a homomorphism' does not follow from the stated results. This is the most load-bearing gap because Theorem 1.8 is the only bridge between the Morse-theoretic Ekeland–Hofer index and the Floer-theoretic Gutt–Hutchings capacities, and Lemmas 1.9–1.11 all depend on it. The uncited 'Crossing Energy Theorem' used for the diagonal is a standard action-energy argument and is not the principal obstacle; the missing gluing argument is a genuine unproved ingredient of the paper's central construction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the Ekeland–Hofer capacities of a star-shaped domain in R^{2n} agree with the Gutt–Hutchings capacities defined from positive S^1-equivariant symplectic homology, for every level k. The proof reduces this to Theorem 1.8, which asserts a chain-complex isomorphism between the S^1-equivariant Morse homology of the Hamiltonian action functional and the S^1-equivariant Floer homology of the same Hamiltonian, compatible with the U-map. The isomorphism is constructed in Section 5 by counting hybrid curves, where the Floer part is coupled to a gradient flow on the auxiliary S^{2N+1} factor.","tokens_in":24715,"tokens_out":3973,"duration_ms":73719,"significance":"If the main theorem is correct, it answers a question raised by Floer in 1988 and gives a rigorous bridge between variational capacities and Floer-theoretic capacities. The paper has real strengths: the reduction in Lemmas 1.9–1.11 is clean; the reformulations in Propositions 1.6 and 1.7 are explicit; the compactness and transversality sections contain substantial and credible arguments, including the estimates (5.2) and (5.3) and the use of Lemma 5.6; and the two sides of the desired equality are defined from independent inputs, so there is no circularity. However, the proof of the load-bearing isomorphism Theorem 1.8 is incomplete as written: the chain map property of the hybrid-curve count is asserted rather than proved, and the U-map compatibility is deferred. These gaps are local in the sense that they concern the proof of Theorem 1.8, but they are central to the paper's main claim.","major_comments":[{"comment":"The proof that Φ is a chain map is not complete. The text asserts that ∂Φ = Φ∂ follows from the fact that the closure of the index-1 hybrid moduli space is a 1-manifold whose boundary consists of broken trajectories, and it cites Theorem 5.5 for this. Theorem 5.5 proves H^1_loc-precompactness and subconvergence of every sequence to some broken object; it does not prove a gluing theorem showing that every index-1 broken object is the limit of a one-parameter family of smooth hybrid curves, nor that these limits form collar neighborhoods. Without gluing, the algebraic boundary count that defines the chain map equation is not shown to be well-defined. Since Theorem 1.8 is the bridge between the Morse-theoretic Ekeland–Hofer index and the Floer-theoretic Gutt–Hutchings capacities, this gap is load-bearing for Theorems 1.5 and 1.2.","section":"§5.3, Theorem 5.5"},{"comment":"The proof that Φ is an isomorphism uses the assertion that M^hyb((x,p),(x,p)) consists only of the constant solution, attributed to an unstated 'Crossing Energy Theorem'. This theorem is not cited or proved, and the constant-solution claim is what makes the diagonal entries of the matrix for Φ equal to 1. If the assertion fails, Φ need not be invertible, so this is another load-bearing point in the proof of Theorem 1.8.","section":"§5.3, Crossing Energy Theorem"},{"comment":"The compatibility of Φ with the U-map, required by Theorem 1.8 and used in Lemma 1.9, is deferred with the phrase 'mutatis mutandis from [GH18]' at the end of §5.3. In the parametrized S^1-equivariant setting, the U-map is not a formal consequence of the non-equivariant statement in [GH18]; the hybrid curve count must be shown to interact correctly with the additional S^1-family or CP^N structure. A proof or a precise citation of a statement covering the present chain model is needed.","section":"§5.3, U-map compatibility"},{"comment":"The proof of transversality for M^hyb is only sketched. The linearized operator DΘ is declared Fredholm with the stated index by a 'slight modification' of [Hec13, Theorem 4.4], and the implicit function theorem is invoked without proving surjectivity after the generic perturbation X. Since the moduli space has a non-compact domain and an infinite-dimensional unstable-manifold boundary condition, the surjectivity and regularity of the evaluation at zero are not automatic. This affects the well-definedness of the count defining Φ and needs a complete proof.","section":"§5.1, Theorem 5.3"}],"minor_comments":[{"comment":"The notation c^CH_k appears in Theorem 1.2 and several later statements; this should be c^GH_k, matching the Gutt–Hutchings capacities and the title.","section":"Theorem 1.2 and throughout"},{"comment":"The abstract says 'on all star-shaped domain in R^{2n}'; this should be 'on all star-shaped domains'. There is also a typo 'Inpired' in the first paragraph of Section 1.","section":"Abstract and Section 1"},{"comment":"The decomposition H = H+ ⊕ H− is written as a product with S^{2N+1}, but S^{2N+1} is not a vector space. This is a minor abuse of notation and should be clarified, for instance by working with the tangent space at a point or with local coordinates on S^{2N+1}.","section":"Section 4.2.1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about gluing is on point and is the main obstacle. The paper is well organized and the reduction is clean, but Section 5 is a proof sketch in the critical part. I recommend that the editor send the manuscript to an expert in Floer gluing and S^1-equivariant transversality. If the authors can supply the missing gluing theorem, the U-map compatibility proof, and full details for the transversality step, the main result is likely correct and the paper would be publishable. No concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Mike,\n\nQuick read of Gutt–Ramos. The headline is real: they claim c^EH_k = c^GH_k for all star-shaped domains, answering Ekeland–Hofer's 1989 extension question and Floer's S1-equivariant intuition. That's a significant result if it holds. The paper's architecture is clean: reformulate EH capacities in Fadell–Rabinowitz index terms and GH capacities via the U-map, then compare them through a chain isomorphism between equivariant Morse and Floer complexes of the action functional. The structural lemmas (1.6, 1.7, 1.9–1.11) are checkable, and I found no circularity; the two sides are defined independently. Credit where due: the paper is honest about what it needs, and the analytic sections show real work on compactness and transversality.\n\nThe soft spot is exactly the one the stress-test flags. Theorem 1.8/5.1 is the load-bearing bridge, and the chain map Φ is defined by counting hybrid curves. For ∂Φ = Φ∂, you need the index-1 moduli space to be a compact 1-manifold whose boundary is precisely the broken trajectories. Theorem 5.5 proves the compactness half—every sequence subconverges to some broken object—but it does not prove gluing: that every broken index-1 object appears as the limit of a one-parameter family, with the right collar structure. Without that, the boundary count is undefined and the claim that Φ is a chain map is asserted, not proved. The U-map compatibility is also deferred mutatis mutandis to GH18, and the 'Crossing Energy Theorem' used for the diagonal is uncited—though that last one is probably standard action-energy. These are genuine gaps, not nits, but they look fillable with standard techniques if the authors supply them.\n\nMy bottom line: this deserves a serious referee. The result is important, the framework is coherent, and the gaps are specific. But I wouldn't take the theorem as established from this version. Send it to review; the referee report should ask for the gluing argument and the U-map details. If they come through, this is a strong paper.","headline":"The main theorem is important and the architecture is coherent, but the written proof of the load-bearing chain isomorphism has a real gap: the gluing half of the moduli-space boundary argument is missing.","tokens_in":162,"tokens_out":2574,"would_cite":true,"duration_ms":31017,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","53D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that on every star-shaped domain in $\\mathbb{R}^{2n}$, the Ekeland–Hofer capacities coincide with the Gutt–Hutchings equivariant symplectic homology capacities at every level $k$.","keywords":["Ekeland–Hofer capacities","equivariant symplectic homology","Gutt–Hutchings capacities","star-shaped domains","Hamiltonian action functional","Fadell–Rabinowitz index","hybrid curves","U-map"],"falsifier":"A concrete check: compute the filtered equivariant Morse and Floer complexes for a simple admissible Hamiltonian with known Reeb orbits, such as an irrational ellipsoid, and compare the rank of the image of each $U^j$ map; any discrepancy would falsify Theorem 1.8 and hence Theorem 1.2. Alternatively, exhibit a nonconstant hybrid curve from a critical point to itself, which would violate the Crossing Energy Theorem used in Section 5.3.","tokens_in":24050,"feed_emoji":"📏","tokens_out":9642,"duration_ms":85588,"temperature":0.7,"pith_summary":"This paper claims that two seemingly unrelated sequences of symplectic capacities — the classical Ekeland–Hofer capacities, built by minimax arguments over loops in $\\mathbb{R}^{2n}$, and the Gutt–Hutchings capacities, built from positive $S^1$-equivariant symplectic homology — are actually the same invariant. The main theorem states that for every star-shaped domain $X \\subset \\mathbb{R}^{2n}$ and every $k \\in \\mathbb{N}_0$, $c^{\\mathrm{EH}}_k(X) = c^{\\mathrm{GH}}_k(X)$. If true, this answers a question posed in 1989 as well as Floer's suggestion from 1988 that Ekeland–Hofer capacities should be understood through $S^1$-equivariant homology. A sympathetic reader would care because it unifies variational and Floer-theoretic approaches to symplectic rigidity and extends the reach of the classical capacities.","feed_headline":"35-year-old question answered: two capacity sequences coincide","feed_subtitle":"On every star-shaped domain in R^{2n}, the classic Ekeland–Hofer capacities equal the Gutt–Hutchings capacities at every level.","key_machinery":"The carrier of the proof is a count of parametrized hybrid curves, denoted $\\Phi$. A hybrid curve consists of a negative gradient trajectory of the Hamiltonian action functional in $E = H^{1/2}(S^1, \\mathbb{R}^{2n})$ followed by a Floer trajectory, with the base point moving along a gradient flow of the Morse function $\\tilde f_N$ on $S^{2N+1}$; the map $\\Phi$ counts zero-dimensional such moduli spaces and is shown to be a chain isomorphism between the equivariant Morse and Floer complexes. It must additionally commute with the $U$-map, which is the algebraic shadow of the $S^1$ action. The Fadell–Rabinowitz index is reformulated as the maximal power of $U$ acting nontrivially, and this is what converts the chain isomorphism into an equality of capacities.","core_discovery":"On the paper's own terms, the discovery is a chain-level structural identity that forces the capacity identity. For every admissible Hamiltonian $H$ and every action level $c>0$, there is an isomorphism of chain complexes $CM^{S^1,c}(H) \\cong CF^{S^1,c}(H)$ between the equivariant Morse homology of the Hamiltonian action functional and the $S^1$-equivariant Floer homology of $H$, and this isomorphism commutes with the $U$-map. From this, the paper derives that the Fadell–Rabinowitz index of the sublevel sets $\\{A_H \\le c\\}$ equals the algebraic quantity $\\kappa_c(H)$ extracted from filtered Floer homology, which in turn is exactly the comparison needed to equate the two capacity sequences on every star-shaped domain.","pith_inferences":["One implication the paper leaves implicit is that the isomorphism should identify not just the capacities but the filtered morphism groups themselves, so one could compute the full $S^1$-equivariant symplectic homology of a star-shaped domain from Morse data alone.","A testable extension would be to check whether the same equality holds for domains that are not star-shaped but are still Liouville, or for the higher capacities defined by iterating the $U$-map more than once.","The reliance on a Crossing Energy Theorem suggests that a clean proof of that statement would put the isomorphism on firmer ground; constructing the gluing direction in the parametrized setting is the natural next target."],"forward_implications":["The two capacity sequences are interchangeable on star-shaped domains: any lower bound, computation, or existence result proved for one transfers to the other.","Ekeland–Hofer capacities, originally restricted to $\\mathbb{R}^{2n}$, inherit the Liouville-domain framework of Gutt–Hutchings capacities, so they can be extended beyond Euclidean domains.","The chain-level isomorphism means that higher structure beyond the capacities — the action filtration and the $U$-module structure — matches between the Morse and Floer sides.","The equality validates Floer's 1988 remark that Ekeland–Hofer capacities must come from $S^1$-equivariant homology, giving that program a precise theorem."],"supporting_citations":[{"why":"defines the Ekeland–Hofer capacities by variational minimax and the Fadell–Rabinowitz index, the object being compared.","marker":"[EH90]"},{"why":"defines the Gutt–Hutchings capacities from positive $S^1$-equivariant symplectic homology and supplies the $U$-map framework that the isomorphism must respect.","marker":"[GH18]"},{"why":"provides the infinite-dimensional Morse homology machinery used to build the equivariant Morse complex of the action functional.","marker":"[AM05]"},{"why":"supplies the transversality and compactness estimates for hybrid curves that the proof of Theorems 5.3 and 5.5 adapt.","marker":"[Hec13]"},{"why":"introduces the hybrid-curve chain map between Morse and Floer complexes that the map $\\Phi$ generalizes to the parametrized setting.","marker":"[AK22]"},{"why":"gives the family Floer homology construction of $S^1$-equivariant symplectic homology used in Section 3.","marker":"[BO16]"},{"why":"originates positive and $S^1$-equivariant symplectic homology and the action-filtration properties that the capacities rely on.","marker":"[Vit99]"},{"why":"establishes that Morse homology of the action functional agrees with ordinary homology for the sublevel sets used in Lemma 1.9.","marker":"[Abb99]"},{"why":"provides the $H^1_{\\mathrm{loc}}$-compactness criterion used to prove that hybrid moduli spaces precompactify into broken trajectories.","marker":"[Sch95]"}],"fun_headline_variants":["Ekeland-Hofer equals Gutt-Hutchings capacities: 35-year-old question settled","Proof: Ekeland–Hofer capacities match equivariant symplectic ones","Chain-level isomorphism equates two symplectic capacity sequences","35-year-old question settled: Ekeland-Hofer capacities = equivariant capacities","Star-shaped domains: Ekeland-Hofer and equivariant capacities proven equal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is the compactness and gluing behavior of the hybrid moduli spaces stated in Theorem 5.5: the closure of an index-1 family must be a compact one-manifold whose boundary is exactly the broken trajectories, and the $U$-map must commute with the count; in the paper the gluing direction and the $U$-map commutation are deferred rather than proved.","fun_headline_variants_meta":{"raw":{"variants":["Ekeland-Hofer equals Gutt-Hutchings capacities: 35-year-old question settled","Proof: Ekeland–Hofer capacities match equivariant symplectic ones","Chain-level isomorphism equates two symplectic capacity sequences","35-year-old question settled: Ekeland-Hofer capacities = equivariant capacities","Star-shaped domains: Ekeland-Hofer and equivariant capacities proven equal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00061,"raw_usage":{"total_tokens":2762,"prompt_tokens":787,"completion_tokens":1975,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":403,"completion_tokens_details":{"reasoning_tokens":1876}},"tokens_in":403,"tokens_out":1975,"duration_ms":15224,"temperature":1.0,"reasoning_tokens":1876,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:57:55.041744+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: compute the filtered equivariant Morse and Floer complexes for a simple admissible Hamiltonian with known Reeb orbits, such as an irrational ellipsoid, and compare the rank of the image of each $U^j$ map; any discrepancy would falsify Theorem 1.8 and hence Theorem 1.2. Alternatively, exhibit a nonconstant hybrid curve from a critical point to itself, which would violate the Crossing Energy Theorem used in Section 5.3.","supporting_citations":[],"review_version":1}