{"id":"b32484bd-f78b-4ca4-a687-9f235952ad97","arxiv_id":"2607.21159","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A symplectic T^{n−1} action on a closed 2n-manifold is Hamiltonian precisely when its orbits are contractible.","lead":"A group of rotations with one dimension less than half the space's dimension must be the Hamiltonian kind whenever its orbits can all shrink to points. This settles the last open case of a long-standing question in symplectic geometry and blocks certain circle actions from extending to larger torus actions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.2's local-freeness proof depends on an unproved transfer of Graham's strict log-concavity to cylinder-valued non-Hamiltonian momentum maps; if the transfer fails, the Splitting Lemma and Theorems 1.1/1.3 collapse.","rationale":"The paper's strongest claim (Theorem 1.1) is precisely stated, and the proof structure is credible: the Splitting Lemma, vanishing Chern class, and the non-torsion lemma are linked coherently. The single most load-bearing point is the unproved extension of Graham's strict log-concavity to the cylinder-valued non-Hamiltonian setting in Lemma 4.2. The reader identified exactly this as the weakest assumption. I do not see a demonstrated mathematical error, and the local-lift route seems likely to succeed; but because Lemma 4.2 underpins both Theorem 1.3 and Proposition 5.5, and the transfer is asserted rather than derived, I would not accept the paper as-is without requiring that derivation. Hence I move the verdict from ACCEPT to CONDITIONAL, pending the one-page proof (or a precise reference) establishing the local Hamiltonian lift and the strict inequality in the cylinder-valued case.","tokens_in":33096,"tokens_out":18037,"duration_ms":166508,"concrete_test":"Prove the local-lift lemma: for each y∈t*/P, choose a neighbourhood W and a section s:W→t* with ds=identity under the flat identification, and show (µ^{-1}(W), ω, s∘µ, s(W)) is a Hamiltonian complexity-one space whose Duistermaat–Heckman density equals f|_W. Then apply Graham's theorem to this local Hamiltonian space and verify that the strict inequality (4.3) holds for every codimension-one interior wall in W. If the verification goes through with no extra hypotheses, the gap is closed; if it requires an additional condition, Lemma 4.2 and the main theorems need revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem rests on Lemma 4.2. In its proof, after choosing an integral form and a cylinder-valued momentum map µ:M→t*/P, the authors assert (Section 4, after Eq. (4.3)) that Graham's strict inequality ∂_ξ f(y+) < ∂_ξ f(y−) at codimension-one interior walls, and the constancy of the derivative of f on chambers, \"are still true\" for the non-Hamiltonian cylinder-valued setup. This is not proved. Graham's theorem is stated for Hamiltonian complexity-one spaces. The natural justification is to take a small contractible neighbourhood W of y, choose a lift s:W→t*, and observe that on µ^{-1}(W) the action is Hamiltonian with momentum map s∘µ; but the paper does not supply this local-lift argument, and it is not immediate because µ is only defined modulo the period lattice P and the section must be chosen so that the derivative of the momentum map is the original one-form. Since this inequality is exactly what forces every codimension-one wall to contain t*_c and hence gives local freeness of T_c, this gap is load-bearing. Proposition 5.5 and Theorem 1.3 inherit the dependence. The missing step is likely fillable, but as written it is an assertion, not a proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that if a torus T acts symplectically on a closed connected symplectic manifold (M,ω) with dim M = 2n and dim T ≥ n−1, then the action is Hamiltonian if and only if its orbit maps are null-homotopic (Theorem 1.1). The main case dim T = n−1 is established through three structural results: the Splitting Lemma (Lemmas 1.5 and 4.2), which says that with isotropic orbits the non-Hamiltonian part acts locally freely; Proposition 5.5, which shows the real Chern class vanishes on the level sets of the cylinder-valued momentum map for the non-Hamiltonan part; and Theorem 1.3, which asserts that non-Hamiltonian subcircle actions represent non-torsion elements of π1(M). The paper also derives non-extendability results for symplectic circle actions and gives a topological characterization of reduced spaces of free circle actions with contractible orbits (Theorem 7.3).","tokens_in":33381,"tokens_out":12896,"duration_ms":111186,"significance":"If correct, Theorem 1.1 closes the remaining dimension range for the question of when Hamiltonianity of a symplectic torus action is detected by the topology of its orbits, extending the four-dimensional theorem of Lalonde–McDuff–Polterovich and giving new restrictions on non-Hamiltonian actions. The proof is well organized and uses external results — Benoist, Graham, Karshon–Tolman, Duistermaat–Pelayo — without circularity. The paper also provides independent, checkable consequences, such as non-extendability of known six-dimensional circle actions and cohomological obstructions for reduced spaces. The main concern is a specific unproved local-lift step in the proof of Lemma 4.2; it is likely fillable, but it is load-bearing for the central theorem.","major_comments":[{"comment":"The sentence 'Even though our action is not Hamiltonian, the above facts are still true...' is load-bearing and unproved. Graham's strict log-concavity is a theorem for Hamiltonian complexity-one spaces; to apply it to the cylinder-valued momentum map of a non-Hamiltonian T^{n-1}-action one must choose a contractible neighbourhood W of y in t*/P, lift it to t*, and check that on µ^{-1}(W) the lifted map is a genuine Hamiltonian momentum map. The manuscript supplies none of this. If the transfer failed, the conclusion that t*_c lies in every codimension-one wall — and hence Lemma 4.2, the Splitting Lemma, Proposition 5.5, Theorem 1.3, and Theorem 1.1 — would no longer follow. The gap is likely fillable, but it must be written out.","section":"§4, proof of Lemma 4.2 (after Eq. (4.3))"},{"comment":"The same local-lift issue recurs when the paper asserts that the T-action on µ^{-1}(W) is Hamiltonian and hence defines a complexity-one space. One needs to prove that a local section of the covering t* -> t*/P over the contractible set W yields a momentum map satisfying Hamilton's equation and proper over W; this also underlies Lemma 5.2's use of Hamiltonian T-models for a non-Hamiltonian action. As written, this is a second unsupported transfer of Hamiltonian facts to the cylinder-valued setup. Add the local-lift argument, and state explicitly the compactness hypothesis used in Lemma 5.2.","section":"§5, proof of Proposition 5.5 (paragraph after Eq. (5.10), beginning 'Since W is convex')"}],"minor_comments":[{"comment":"The statements say 'connected symplectic manifold' but the proofs use compactness (finite invariant cover). All applications are to closed manifolds, so add 'closed' to the hypotheses or justify the finiteness otherwise.","section":"§5, Lemma 5.2 and Proposition 5.5"},{"comment":"The sentence 'Because (R·P) \\ {0} is dense in the dual t*_c' is imprecise: when P is a lattice in t*_c, the set R·P equals t*_c. This does not affect the argument.","section":"§4, proof of Lemma 4.2"},{"comment":"The assertion 'The T-action is Hamiltonian if and only if P = {0}' is used repeatedly; a one-sentence justification or reference would help the reader.","section":"§2.3, Definition 2.13"},{"comment":"The notation '⟨c,π2(N)⟩ = Z' should be glossed as 'the image of the evaluation map equals Z', since c is a cohomology class and π2(N) is a group. This is clear from context but worth stating.","section":"§7, Lemma 7.1 and Theorem 7.3"},{"comment":"In Lemma C.2, the expression γ^{-b_j/a_j} should be interpreted as γ^{-b_j a_j^{-1}} in the finite cyclic group; the current notation is slightly abusive.","section":"Appendix C"}],"recommendation":"major_revision","confidential_remarks":"The gap in §4 is real but local. If the authors add the local-lift argument needed to transfer Graham's strict inequality to the cylinder-valued non-Hamiltonian setup, I would support acceptance. The self-citations are not problematic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Verdict: this is the right paper and the right result. It closes the last open dimension regime: for a symplectic T^{n-1} action on M^{2n}, Hamiltonian iff orbit maps are null-homotopic. That genuinely extends Lalonde–McDuff–Polterovich from four-manifolds to all dimensions, and the splitting lemma plus the non-extendability corollary are new. I went in expecting a hidden counterexample; I didn't find one. The architecture is coherent: DH log-concavity gives local freeness of the non-Hamiltonian part, vanishing of the real Chern class on level sets is the next step, and then the orbit topology follows from Lemma 3.4. The appendices are honest and useful.\n\nThe soft spot is exactly the one flagged in the stress test. In Lemma 4.2, once you replace a Hamiltonian momentum map by a cylinder-valued one, the proof asserts that Graham's strict log-concavity and the constancy of derivatives on chambers 'are still true' around the preimage of a small neighbourhood in t*/P. That is not proved. Graham's theorem is for Hamiltonian complexity-one spaces; the transfer to a non-Hamiltonian cylinder-valued setup needs a local section argument. It is very likely fillable—take a contractible neighbourhood in t*/P, lift to t*, and the lifted action is Hamiltonian—but the paper doesn't write it. Since Lemma 4.2 is what forces every codimension-one wall to contain t*_c and gives local freeness, this is a single load-bearing point. Proposition 5.5 and Theorem 1.3 inherit the dependence. I would not call it a fatal flaw; I would call it a required revision.\n\nThe citation pattern is fine. The self-citations are to prior published tools (Karshon–Tolman, Henigman), not to the target result. No circularity.\n\nWho is this for? Anyone working on symplectic torus actions, Hamiltonian vs non-Hamiltonian detection, or complexity-one spaces. It deserves a serious referee. I would send it out, and I would make acceptance contingent on the authors supplying the missing proof of the transfer (or an explicit local normal-form argument) in Lemma 4.2.","headline":"Resolves the last open dimension regime for Hamiltonian-vs-contractible-orbits, and the proof is mostly solid; the one load-bearing assertion in Lemma 4.2 needs to be proved before I'd sign off.","tokens_in":33935,"tokens_out":4604,"would_cite":true,"duration_ms":45833,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D20","53D35","57R17","57S15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A symplectic torus action of maximal size is Hamiltonian exactly when its orbits are contractible.","keywords":["symplectic torus actions","Hamiltonian actions","contractible orbits","cylinder-valued momentum maps","Duistermaat-Heckman measure","complexity one spaces","splitting lemma","non-extendability"],"falsifier":"A concrete counterexample would be a faithful symplectic T^{n-1} action on a closed connected 2n-dimensional symplectic manifold that is non-Hamiltonian and has at least one fixed point; Theorem 1.1 would then fail because the orbit maps of a fixed point are constant. At the proof level, one could look for a non-Hamiltonian complexity-one action with isotropic orbits whose cylinder-valued Duistermaat–Heckman density violates the strict one-sided derivative inequality ∂ξ f(y+) < ∂ξ f(y−) across a codimension-one wall; that failure would invalidate the Splitting Lemma.","tokens_in":32934,"feed_emoji":"🌀","tokens_out":6538,"duration_ms":51959,"temperature":0.7,"pith_summary":"This paper proves that for the largest torus symmetries that can act on a symplectic manifold without being Hamiltonian—a T^{n-1} action on a closed 2n-dimensional symplectic manifold—the action is Hamiltonian if and only if every orbit map is null-homotopic, equivalently, if orbits are contractible. This turns an analytic condition, existence of a momentum map, into a purely topological one, and it extends a four-dimensional theorem to all dimensions. The proof isolates the non-Hamiltonian part of the action and shows, using strict log-concavity of Duistermaat–Heckman densities, that it must act locally freely; a further vanishing of a real Chern class and a Moser-type isotopy argument then force its orbits to be essential. As a consequence, several a priori different properties—fixed points, Hamiltonianity, contractible orbits, null-homotopic orbit maps, and equivariant formality—coincide in this setting, and several known non-Hamiltonian circle actions can be shown not to extend to larger torus actions.","feed_headline":"Largest torus actions: Hamiltonian iff orbits are contractible","feed_subtitle":"In 2n dimensions, a symplectic T^{n-1} action is Hamiltonian precisely when its orbits are contractible—orbit topology detects dynamics.","key_machinery":"The load-bearing device is the Splitting Lemma: under isotropic orbits and dim T ≥ n−1, the torus decomposes as T = T_ham × T_c, where T_ham acts Hamiltonially and T_c acts locally freely. The lemma is obtained from strict log-concavity of the Duistermaat–Heckman density for complexity-one actions, applied to a cylinder-valued momentum map for the non-Hamiltonian part; this forces every one-dimensional non-Hamiltonian circle to be locally free. A second ingredient is the vanishing of the real Chern class on level sets of the cylinder-valued momentum map, proved by a Moser-type isotopy theorem for complexity-one spaces; combined with a long exact sequence of homotopy groups and a lemma on loc","core_discovery":"On a closed connected symplectic manifold (M^{2n}, ω), let T be a torus of dimension at least n−1 acting faithfully and symplectically. The paper's central claim is that the action is Hamiltonian if and only if its orbit maps are null-homotopic. Because Hamiltonian actions always have fixed points, the nontrivial direction is the converse: a non-Hamiltonian T^{n-1} action must have essential orbits. When the orbits are isotropic, the paper proves more: every non-Hamiltonian subcircle represents a non-torsion element of π₁(M), and the action splits as a product of a maximal Hamiltonian subtorus and a locally-free non-Hamiltonian part. These results imply that known six-dimensional non-Hamilto","pith_inferences":["If the strict log-concavity assumption for non-Hamiltonian cylinder-valued Duistermaat–Heckman densities can be verified directly, the proof would also give a construction guide: non-Hamiltonian T^{n-1} actions are essentially built from a Hamiltonian complexity-one piece and a locally-free torus bundle, suggesting a classification parallel to the known four-dimensional case.","Conjecture 8.5 points to a testable refinement: for non-isotropic orbits of rank 2r, the same conclusions should hold when dim T ≥ n+r−1; quotienting by the symplectic part of the action would reduce to the isotropic case, but only if the Moser-type isotopy theorem survives for orbifolds.","The reduced-space criterion of Theorem 7.3 could be used computationally to search for new free circle actions with contractible orbits: one should look for symplectic manifolds N admitting an integral class c with ⟨c, π₂(N)⟩ = Z and a diffeomorphism Ψ that reverses a straight line of symplectic forms in the direction c; candidates that are not K3 surfaces would answer the paper's Problem 8.3.","The non-extendability corollary suggests that the boundary of symplectic extendability in 6-manifolds is drawn by π₁: a circle action whose orbits are torsion in π₁ cannot sit inside a T² action."],"forward_implications":["For T^{n-1} actions on 2n-manifolds, Hamiltonianity is detected by orbit topology: contractible orbits, null-homotopic orbit maps, fixed points, and equivariant formality all become equivalent.","In dimension 6, non-Hamiltonian circle actions with contractible orbits, including free ones and ones with fixed points, cannot be extended to symplectic T² actions; in higher dimensions they cannot be extended to isotropic T^{n-1} actions.","Any non-Hamiltonian subcircle of an isotropic T^{n-1} action generates a non-torsion class in π₁(M), so the fundamental group records all non-Hamiltonian directions.","Reduced spaces of free symplectic circle actions with contractible orbits must satisfy ⟨c, π₂(N)⟩ = Z for some integral class c; aspherical symplectic manifolds and manifolds with second Betti number one are ruled out as reduced spaces.","Non-Hamiltonian T^{n-1} actions with isotropic orbits are constrained to local models where the Hamiltonian part is separated from a locally-free part, a structural step toward classifying such actions."],"fun_headline_variants":["Torus action Hamiltonian iff orbits contract","Contractible orbits imply Hamiltonian torus actions","Orbit contractibility distinguishes Hamiltonian actions","Non-Hamiltonian T^{n-1} actions have noncontractible orbits"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that the strict log-concavity inequality for Duistermaat–Heckman densities, known for Hamiltonian complexity-one actions, continues to hold around every interior wall after replacing the Hamiltonian momentum map by a cylinder-valued momentum map of a non-Hamiltonian T^{n-1} action; this transfer is asserted without proof and the Splitting Lemma, and hence the main theorems, depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Torus action Hamiltonian iff orbits contract","Contractible orbits imply Hamiltonian torus actions","Orbit contractibility distinguishes Hamiltonian actions","Non-Hamiltonian T^{n-1} actions have noncontractible orbits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001145,"raw_usage":{"total_tokens":4561,"prompt_tokens":696,"completion_tokens":3865,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":440,"completion_tokens_details":{"reasoning_tokens":3804}},"tokens_in":440,"tokens_out":3865,"duration_ms":25988,"temperature":1.0,"reasoning_tokens":3804,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:17:43.012426+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be a faithful symplectic T^{n-1} action on a closed connected 2n-dimensional symplectic manifold that is non-Hamiltonian and has at least one fixed point; Theorem 1.1 would then fail because the orbit maps of a fixed point are constant. At the proof level, one could look for a non-Hamiltonian complexity-one action with isotropic orbits whose cylinder-valued Duistermaat–Heckman density violates the strict one-sided derivative inequality ∂ξ f(y+) < ∂ξ f(y−) across a codimension-one wall; that failure would invalidate the Splitting Lemma.","supporting_citations":[],"review_version":1}