{"id":"29738c17-1e8a-4963-897b-6013ffb039e9","arxiv_id":"2506.10545","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A C0-Hamiltonian twist map on a Liouville domain with symplectic cohomology nonzero in infinitely many degrees is claimed to have arbitrarily long interior periodic orbits, but the proof's action-growth exclusion of boundary orbits is not justified.","lead":"This paper proves a higher-dimensional Poincaré-Birkhoff theorem for continuous Hamiltonian maps on Liouville domains using Floer homology, plus a relative version for Lagrangian chords. The central proof step, which replaces an index-growth estimate with an action-growth estimate, appears to contain a gap that invalidates the main theorem as stated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The \\S6 exclusion of collar orbits from fixed-degree Floer homology uses action growth alone, but total degree p+q is the Conley\\textendash{}Zehnder index, which the paper deliberately leaves unbounded; without an index\\textendash{}action estimate the contradiction collapses.","rationale":"The reader's weakest_assumption is the same load-bearing concern I find: action growth controls the filtration column p but not the total degree p+q, which is the local Floer grading. The contradiction in Theorem A requires that only iterates of the finitely many fixed points contribute to the chosen degrees; the only exclusion of collar orbits is this action-column step, so if it fails the proof collapses. I checked the paper for a rescue: Proposition 4.1 contains no index estimate; the spectral sequence setup is standard and does not supply one; Appendix A is explicitly not needed for the main theorems and requires strong index-definiteness and C2 regularity, assumptions the paper claims to drop. The same gap propagates to Theorem B, whose proof is deferred to [L]. I therefore agree with the reader's rejection. The concern is not a matter of disagreement with consensus but a missing logical premise in the core proof; a positive resolution would require a new index-action estimate under the paper's hypotheses, not just a reformulation of the action growth lemma.","tokens_in":18954,"tokens_out":21538,"duration_ms":282754,"concrete_test":"Re-derive the exclusion step in \\S6 from first principles: from (6.10) and the definition of E^1_{p,q}, attempt to prove that a collar orbit x with A(x) \\le -c_i p_i + d has HF^{i_j}_{loc}(x) = 0. Any valid proof must use a lower bound on the Conley-Zehnder index of collar orbits as a function of action. Then check whether such a bound appears anywhere in \\S4\\textendash{}6 or Appendix A; if it does not, the step is a logical gap. As a decisive geometric instance, compute the local Floer homology of a hyperbolic Reeb orbit on the boundary of a c1=0 Liouville domain under \\hat H = ar-\\epsilon with a\\to\\infty: if HF^0_{loc} \\ne 0 while the action tends to -\\infty, the \\S6 exclusion is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem A (Section 6), Proposition 4.1 is applied to collar 1-periodic orbits of \\hat H^{#p_i}_{\\epsilon_i}, giving A_{\\hat H^{#p_i}_{\\epsilon_i}}(x) \\le -c_i p_i + d with c_i \\to \\infty (Eq. 6.10). The paper then says these orbits have action escaping to -\\infty, hence appear on columns with p \\gg 1, and therefore cannot contribute to the diagonals p+q = i_j. The second assertion does not follow. In the displayed local-to-global spectral sequence, p is the column of the action filtration and q is defined by p+q; for an orbit x in column p its contribution is a class in HF^{p+q}_{loc}(x). So x contributes to total degree i_j exactly when HF^{i_j}_{loc}(x) \\ne 0, and the size of p only places that class at bidegree (p, i_j-p). Ruling out collar orbits requires an index-action inequality showing that the support of HF^*_{loc}(x) misses the fixed finite set {i_1,...,i_N} as p_i \\to \\infty. The paper explicitly removes the index-growth/positivity assumptions of [MvK] (Section 4), and Appendix A is not used, requires strong index-definiteness and C2 regularity, and proves an index bound only under those extra assumptions. On a non-index-definite boundary there are Reeb orbits of bounded Conley-Zehnder index and arbitrarily long period; under the linear extension these give exactly the forbidden contributions. Since this exclusion is the only mechanism separating interior orbits from collar artifacts, Theorem A, and by inheritance Theorem B, is unsupported as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces C0-Hamiltonian twist maps on Liouville domains, a weakening of the twist condition used by Moreno and van Koert, and proves two existence results: Theorem A, asserting that such a map with isolated fixed points on a Liouville domain with c1(W)=0 and symplectic cohomology non-zero in infinitely many degrees admits simple interior periodic points of arbitrarily large period; and Theorem B, the analogous wrapped Floer statement for exact spin Lagrangians. The proof strategy is to smooth the C0 map to a family of C1-Hamiltonian twist maps satisfying a quantitative twist condition, to construct admissible extensions with large slope, to use a new action-growth estimate (Proposition 4.1) to separate interior orbits from collar orbits, and then to run a Ginzburg-type counting argument against the mean-index property of iterates of the fixed points.","tokens_in":19227,"tokens_out":8088,"duration_ms":98941,"significance":"If valid, Theorem A would be a substantial higher-dimensional generalization of the Poincaré–Birkhoff theorem, with potential applications to the spatial CR3BP and billiard maps. The paper contains several useful ideas: degenerate Liouville domains, the smoothing construction in Theorem C, and the quantitative twist condition with its action-growth estimate. The theorems are honestly stated as conditional on an external hypothesis (symplectic or wrapped Floer cohomology non-zero in infinitely many degrees), which is legitimate and not circular. However, the central proof does not establish the claimed results because the step excluding collar orbits from fixed-degree Floer homology is unjustified.","major_comments":[{"comment":"The claim that collar orbits with action escaping to -infinity 'appear on columns with p >> 1, and therefore cannot contribute to the diagonals p+q = i_j' is not justified. In the local-to-global spectral sequence, the total degree p+q of a class is the Conley-Zehnder index (in the local Floer cohomology), which is independent of the action filtration column p. From A_hatH(x) <= -c_i p_i + d one learns only that the action level goes to -infinity; the Conley-Zehnder index of such an orbit can still take any fixed finite value. Indeed, on a boundary that is not index-definite, Reeb orbits can have bounded Conley-Zehnder index and arbitrarily long period; under the linear extension these give collar orbits with action tending to -infinity and fixed finite total degree. Appendix A does not repair this: it is not used in the proof and requires strong index-definiteness and C2 regularity, which the paper explicitly removes. Without an index-action estimate, the exclusion of collar orbits fails, and the contradiction in the proof of Theorem A collapses.","section":"Section 6, Eq. (6.10)"},{"comment":"Theorem B is one of the two main theorems, yet no proof is given in the manuscript. The text states that the proof is 'completely analogous to that of Theorem A' and refers to an unpublished PhD thesis [L]. For a self-contained journal submission, each main theorem should be proved in the paper or the result should be explicitly presented as conditional on a forthcoming reference. As written, the manuscript does not provide a verifiable proof of Theorem B, and the omitted proof inherits the gap in Theorem A.","section":"Section 1 (Theorem B) and Section 6"},{"comment":"The proof of the action-growth proposition is not rigorous at the level of detail required for a paper whose central mechanism is action growth. The argument repeatedly uses approximations such as 'we ignore the terms of order >= 1 in r-1', 'r is approximately 1 everywhere', and 'the inequality is true up to a small error', without quantifying these errors or showing that they do not contribute a positive term that could overwhelm the negative linear growth. Moreover, the estimate is stated for trajectories entirely contained in [1,+infinity)xB; periodic orbits that mix interior and collar portions are not covered. Since the proof of Theorem A relies on a clean dichotomy between 'interior' and 'collar' orbits, this gap is load-bearing.","section":"Section 4, Proposition 4.1"}],"minor_comments":[{"comment":"The spectral sequence statement 'E1(hatH_epsilon_i^{#p_i}) = impies HF^*(hatH_epsilon_i^{#p_i})' would benefit from a precise statement of the convergence and of the meaning of the E_infty page; as written, the notation E^{p,q}_infty is used without definition.","section":"Section 6"},{"comment":"There are several LaTeX/OCR artifacts in the displayed formulas, such as '/∫hortrightarrow' in Definition 1.1 and the mislabeled 'last term' in Step 2 of the proof of Proposition 4.1. These should be corrected before resubmission.","section":"Sections 1 and 4"},{"comment":"The paper claims to remove index-positivity assumptions, but Theorem A now requires SH(W) to be non-zero in infinitely many degrees, whereas in [MvK] this followed from index-positivity. This trade-off is mentioned in the introduction, but the discussion could be more explicit about how strong the new hypothesis is and whether it is satisfied in the intended CR3BP examples.","section":"Section 1"}],"recommendation":"reject","confidential_remarks":"The main theorem's proof has a fundamental gap that cannot be repaired without reintroducing an index-action estimate or an index-positivity assumption, which would contradict the paper's stated goal of removing such assumptions. The omitted proof of Theorem B in an unpublished thesis is an additional concern. The authors may wish to reframe the result as a conditional theorem under an explicit index-action or index-definiteness assumption, and to include full proofs of both theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about arXiv:2506.10545. First, the setup is genuinely new: the authors define C0-Hamiltonian twist maps, work with an open twist condition, and claim a higher-dimensional Poincaré–Birkhoff theorem without index-positivity assumptions. The relative Lagrangian version is also new. The smoothing construction (Theorem C) and the action-growth lemma (Proposition 4.1) are solid pieces of work, and the non-example in Appendix B is a good sanity check. The motivation from the spatial CR3BP and billiards is concrete and clearly explained.\n\nThe problem is in Section 6. The proof of Theorem A uses Proposition 4.1 to say that collar orbits of the iterated Hamiltonians have action escaping to -infinity, and then asserts that they cannot contribute to the fixed degrees i_j because they appear on columns with p >> 1. That step is not justified. In the local-to-global spectral sequence, the total degree p+q is the Conley-Zehnder index. An orbit with very negative action sits in a large column p, but its index can be any finite integer, so its contribution lands on the diagonal p+q = i_j exactly when its local Floer homology in degree i_j is non-zero. The size of p only determines the bidegree, not whether it hits a fixed total degree. Excluding collar orbits from fixed degrees requires an index-action inequality, and that is precisely what the paper drops when it removes index-positivity. Appendix A gives such a bound only under strong index-definiteness and C2 regularity, and is not used in the main body. So the only mechanism separating interior orbits from collar artifacts is missing, and the contradiction collapses. Theorem B inherits the issue.\n\nThere is a minor side issue: the proof of Proposition 4.1 uses informal approximations in thin collars that could be tightened, but that is much less serious than the Section 6 gap.\n\nWho should read this: symplectic dynamicists and anyone working on Poincaré–Birkhoff-type theorems and their applications to celestial mechanics. A Floer specialist should look at Section 6 to see whether an implicit index-action bound rescues the argument. If it does, the paper is a solid contribution; as written, the main theorems are unsupported.\n\nMy recommendation: send it to a serious referee. The new framework and the action-growth lemma deserve expert scrutiny, even though I expect the referee to require major revisions, or to reject if the gap cannot be filled.","headline":"New C0-Hamiltonian twist-map formulation of a higher-dimensional Poincaré-Birkhoff theorem, but the action-growth proof has a gap: collar orbits are not excluded from fixed degrees without an index-action estimate.","tokens_in":19824,"tokens_out":5614,"would_cite":false,"duration_ms":64083,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","37C25","70F07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a higher-dimensional Poincaré–Birkhoff theorem: C0-Hamiltonian twist maps on Liouville domains with symplectic cohomology nonzero in infinitely many degrees have simple interior periodic points of arbitrarily large…","keywords":["Poincaré–Birkhoff theorem","C0-Hamiltonian twist maps","symplectic cohomology","Floer homology","Liouville domain","action growth","restricted three-body problem","periodic orbits"],"falsifier":"Pick the concrete open-book page and smoothing from Section 5, run the local-to-global spectral sequence for a large prime iterate, and locate a collar orbit with action below all cutoffs whose Conley–Zehnder index lies in one of the nonzero degrees; if its local Floer cohomology is nonzero, the separation step fails. Establishing or refuting an inequality $|\\mu_{CZ}(x)| \\le \\kappa(A(x))$ for such orbits would decide the question.","tokens_in":18676,"feed_emoji":"🌀","tokens_out":8304,"duration_ms":88289,"temperature":0.7,"pith_summary":"This paper aims to lift the classical Poincaré–Birkhoff theorem to higher dimensions. The main result, Theorem A, states that if $f$ is a $C^0$-Hamiltonian twist map on a Liouville domain with isolated fixed points, vanishing first Chern class, and symplectic cohomology nonzero in infinitely many degrees, then $f$ has simple interior periodic points of arbitrarily large minimal period. Theorem B gives the relative statement for exact spin Lagrangian submanifolds with Legendrian boundary, in terms of wrapped Floer cohomology. These theorems are motivated by the spatial circular restricted three-body problem, where Poincaré return maps on suitable hypersurfaces of section are expected to satisfy the hypotheses.","feed_headline":"Floer homology yields a higher-dimensional Poincaré–Birkhoff theorem","feed_subtitle":"New result forces infinitely many interior periodic points of large period for C0-Hamiltonian twist maps.","key_machinery":"The machinery is the action-growth estimate of Proposition 4.1, computed for a linear-at-infinity extension $\\widehat{H}$ of the generating Hamiltonian. For each trajectory $x$ in the collar $[1,+\\infty)\\times B$, the proof bounds the action by $A_{\\widehat{H}}(x) \\le -c\\,T + d$, with $c>0$; the quantitative twist condition $\\min_B h_t > \\max_B H_t$ is exactly what makes the constant $c$ positive in the boundary-most zone. This estimate replaces the index-growth assumption used in earlier work, and it is then combined with a spectral sequence whose $E^1$ page groups orbits by action columns, with local Floer cohomology on each column.","core_discovery":"The central discovery is that a quantitative twist condition, namely $h_t = \\partial_r H_t$ bounded below by $\\max_B H_t$ near the boundary, produces an action-growth estimate forcing trajectories in the cylindrical end to have action going to $-\\infty$ linearly in their period, with a growth constant that diverges as the map is smoothed. The authors argue that this action growth, feeding into a local-to-global spectral sequence ordered by action columns, separates collar orbits from interior orbits, so that Floer homology in a fixed degree is generated only by interior orbits. Iterating along a sequence of large primes then contradicts the hypothesis that symplectic cohomology is nonzero in infinitely many degrees unless there are interior periodic points of arbitrarily large period.","pith_inferences":["Inference: If the action-growth separation step is valid, the same proof should extend to Hamiltonians satisfying only the weakened twist condition after smoothing, because the quantitative condition is automatically achieved by the construction in Section 5.","Inference: A natural next step is to prove an index–action inequality for collar orbits, which would make the action-growth argument independent of the local-to-global spectral sequence and would close the remaining gap.","Inference: The relative theorem suggests that in the CR3BP the same open-book page yields infinitely many collision-to-symmetry-plane chords, since wrapped Floer cohomology is often supported in infinitely many degrees."],"forward_implications":["If Theorem A holds, the C0-Hamiltonian twist return maps produced in the open-book construction for the spatial CR3BP force infinitely many periodic trajectories of arbitrarily large period.","Theorem B yields infinitely many interior chords for exact spin Lagrangians, including trajectories that meet the symmetry plane orthogonally, such as halo-type orbits, and consecutive collision orbits.","The twist condition becomes open, so small $C^0$-perturbations of a C0-Hamiltonian twist map remain covered by the theorem.","The theorem removes the index-positivity and global-triviality assumptions that limited earlier generalised Poincaré–Birkhoff results.","The same setup is expected to apply to billiard maps on degenerate Liouville domains, although the authors note the conclusions may be weaker than those from Morse-theoretic methods."],"supporting_citations":[{"why":"Supplies the overall proof strategy and the spectral-sequence setup that this paper adapts to C0 maps.","marker":"[MvK]"},{"why":"Provides the mean-index bound used to bound the number of degrees each fixed point's iterate contributes.","marker":"[G10]"},{"why":"Provides the open-book decomposition whose return maps give the intended applications to the spatial CR3BP.","marker":"[MvK2]"},{"why":"The relative Poincaré–Birkhoff result that Theorem B generalises to C0 twist maps.","marker":"[LM2]"},{"why":"Source for the local-to-global spectral sequence converging to Floer homology used in Section 6.","marker":"[KvK]"}],"fun_headline_variants":["Floer homology proves higher-dimensional Poincaré–Birkhoff","Action growth yields higher-dimensional Poincaré–Birkhoff","Higher-dimensional twist maps: infinitely many periodic points","C0-Hamiltonian maps: Poincaré–Birkhoff in higher dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that an orbit staying near the boundary contributes to Floer homology in a degree determined by how negative its action is, so that once its action is made extremely negative it cannot affect the finitely many fixed degrees; but the degree of a Floer homology class is set by an independent index of the orbit, not by its action, and this identification is unproven.","fun_headline_variants_meta":{"raw":{"variants":["Floer homology proves higher-dimensional Poincaré–Birkhoff","Action growth yields higher-dimensional Poincaré–Birkhoff","Higher-dimensional twist maps: infinitely many periodic points","C0-Hamiltonian maps: Poincaré–Birkhoff in higher dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001266,"raw_usage":{"total_tokens":5076,"prompt_tokens":732,"completion_tokens":4344,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":348,"completion_tokens_details":{"reasoning_tokens":4278}},"tokens_in":348,"tokens_out":4344,"duration_ms":37814,"temperature":1.0,"reasoning_tokens":4278,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:25:06.097121+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick the concrete open-book page and smoothing from Section 5, run the local-to-global spectral sequence for a large prime iterate, and locate a collar orbit with action below all cutoffs whose Conley–Zehnder index lies in one of the nonzero degrees; if its local Floer cohomology is nonzero, the separation step fails. Establishing or refuting an inequality $|\\mu_{CZ}(x)| \\le \\kappa(A(x))$ for such orbits would decide the question.","supporting_citations":[],"review_version":1}