{"id":"b5ad7b19-4a15-4f89-bf93-ae3e2b7e4ef3","arxiv_id":"2507.13924","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every non-monotone toric symplectic four-manifold admits a Hamiltonian diffeomorphism and a Lagrangian torus that never intersects its own image under iteration.","lead":"This note extends the author's earlier counterexamples to Lagrangian Poincaré recurrence to every non-monotone toric symplectic four-manifold. It constructs Hamiltonian diffeomorphisms whose iterates move a Lagrangian torus so that it never meets itself.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 2 proves the counterexample only under the assumption that the hat H has width w>0; the note never shows that every non-monotone toric four-manifold admits such a representative, so the 'all non-monotone' claim is not established.","rationale":"The reader's weakest_assumption already flags the unverified w>0 hypothesis, so there is partial agreement. I isolated the width assumption as the single most load-bearing concern because it directly threatens the paper's advertised scope ('all non-monotone toric symplectic four manifolds') even if the external [3, Theorem 2.26] is accepted. The proof is otherwise coherent: given a positive-width hat and the lifting theorem, the translation argument and the irrational ratio choice produce non-recurrent Lagrangian tori, and the finite symplectic mapping class group argument promotes the symplectomorphism to a Hamiltonian one after passing to a power. The concern is not that the construction is wrong, but that its hypotheses have not been shown to cover the full claimed class. A classification check would settle this immediately. Because this is a verifiable gap rather than a demonstrated falsehood, the appropriate verdict remains conditional: accept once the width condition is confirmed for all non-monotone manifolds or the statement is revised.","tokens_in":3895,"tokens_out":13887,"duration_ms":157882,"concrete_test":"Classify Delzant polygons of non-monotone toric symplectic four-manifolds and, for each, compute the hat width w of the canonical type from [3, Proposition 3.17], using the recipe in [3, Definition 3.20]. If every non-monotone case has w>0, a one-sentence citation of that classification closes the gap; if any non-monotone case has w=0, the Section 2 construction does not cover it and the abstract must be weakened or the case treated separately.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is the abstract's 'extending the family of counterexamples to include all non-monotone toric symplectic four manifolds.' Section 2 begins its construction with 'Suppose that the hat H has width w>0' and the nodal tangle in Figure 4 is described as transforming 'part of an ε-hat of positive width.' No statement in the note proves that every non-monotone toric symplectic four-manifold can be represented by a nodal integral affine surface of canonical type with positive hat width. Figure 3 explicitly lists hats with w=0 for each of the five types A–E, and the text does not explain why such zero-width hats cannot occur outside the five monotone cases. If any non-monotone manifold has only canonical representatives with w=0, then the Section 2 construction does not apply to it, and the earlier results in [2] were already stated to exclude cases 2 and 3, so the 'all non-monotone' assertion would be unsupported. This is an internal gap in the note's argument, independent of the external reliance on [3, Theorem 2.26].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This note claims to extend counterexamples to Lagrangian Poincaré recurrence in dimension four to all non-monotone toric symplectic four-manifolds. It constructs a Hamiltonian diffeomorphism ψ and a Lagrangian torus L such that ψ^n(L)∩L=∅ for all n≥0. The construction uses nodal integral affine bases and nodal tangles. Section 1 treats non-monotone S^2×S^2 explicitly: starting from the base B0 of the almost toric fibration, the author performs nodal slides to obtain B1, defines an integral affine isomorphism τ:B0→B1, and invokes Theorem 2.26 of [3] to lift τ to a symplectomorphism ψ. Section 2 generalizes this to a nodal integral affine surface (B0,𝔑0) of canonical type (H,M,α_1,…,α_n), assuming the hat H has positive width w>0. The author again constructs a nodal tangle and an integral affine isomorphism τ, lifts it via [3, Theorem 2.26], and selects Lagrangian tori over levels where the ratio g(h)/(2(M-h)) is irrational. The paper concludes by using finiteness of the symplectic mapping class group to pass from a symplectomorphism to a Hamiltonian diffeomorphism.","tokens_in":4106,"tokens_out":5644,"duration_ms":65254,"significance":"If correct, the result would close the remaining cases 2 and 3 from [2, Remark 3.2] and establish that Lagrangian Poincaré recurrence fails for every non-monotone toric symplectic four-manifold, a substantial strengthening of the earlier construction. The note is concise and well illustrated, with explicit formulas for level-set lengths and a clear geometric mechanism. However, two load-bearing points prevent the result from being fully established as written: the construction depends on an unproved theorem from the author's unpublished preprint [3], and the positive-width hat hypothesis is not shown to hold for all non-monotone manifolds. With those gaps filled, the result would be a significant contribution to the study of Lagrangian dynamics on rational toric manifolds.","major_comments":[{"comment":"The abstract promises counterexamples for all non-monotone toric symplectic four-manifolds, but the construction in Section 2 is carried out only under the hypothesis that the canonical hat H has positive width w>0. Figure 3 explicitly displays zero-width hats for each type A–E, and the text does not explain why a non-monotone manifold must have a canonical representative with w>0, or how to modify a zero-width representative without changing the symplectic manifold. If some non-monotone manifolds admit only zero-width canonical hats, the supplied construction does not apply to them, and the earlier construction in [2] was already stated to exclude cases 2 and 3. Thus the 'all non-monotone' claim is not established as written. The author should either prove that every non-monotone toric four-manifold admits a base with w>0, or restrict the stated theorem and explain how the remaining zero-width cases are handled.","section":"Section 2, 'Suppose that the hat H has width w>0' (page 3)"},{"comment":"The map that produces the desired symplectomorphism ψ is obtained entirely from [3, Theorem 2.26], which asserts that an integral affine isomorphism between nodal integral affine bases lifts to a symplectomorphism fitting the stated diagram. This theorem is not stated or proved in the note, and [3] is an unpublished preprint, so the note is conditional on an external result whose proof the reader cannot check. Since Theorem 2.26 is load-bearing for both the S^2×S^2 case (Section 1) and the general case (Section 2), the author should include the precise statement, a proof or sketch, or clearly restrict the note to depend on [3] only as a published reference; as it stands, the main theorem cannot be independently verified from this note alone.","section":"Section 1 and Section 2, use of [3, Theorem 2.26]"}],"minor_comments":[{"comment":"The text says 'than repeat the construction' and should say 'then repeat the construction'.","section":"Page 1, paragraph after abstract"},{"comment":"The notation t· for the ℝ-action on B0∖ℱ^{-1}(M) is used without definition; please specify that it denotes the flow along the level-set circle in the chosen clockwise direction.","section":"Section 1, paragraph after Figure 1"},{"comment":"The formula 'integral affine length 2w+8(M−h)' uses w, but w is introduced in this section only as a parameter in the symplectic form; please state explicitly that this w is also the hat width appearing in the length formula.","section":"Section 1, level-set length formula"},{"comment":"The sentence 'Let f:B0∖(ℓ∪Δ_M)→R^2 be the length of the straight line segment...' is linguistically garbled; it should say 'Let f(x) be the length of the straight line segment...' and specify that f takes values in R.","section":"Section 2, definition of f"},{"comment":"The caption labels hats with 'A , w=0' etc.; the spacing and comma placement should be cleaned up for readability.","section":"Figure 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The note states at the top that this work has been incorporated into [3, Section 3.3]. Since [3] is the author's own preprint and the present note is essentially an excerpt from it, the editor may wish to consider whether standalone publication is appropriate, or whether the note should explicitly present itself as supplementary material to [3]. The major technical concern beyond the w>0 gap is the heavy reliance on the unpublished preprint [3, Theorem 2.26]; if [3] is not accepted or not available in final form, the note's main theorem is not independently verifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Joel, here's my take. The note does what it says: it extends the counterexample construction from [2] to the two remaining families of non-monotone toric four-manifolds. The new ingredient is a slightly different tangle that works when the hat has positive width, and the S^2×S^2 case is worked out carefully. If the machinery in [3] is sound, the result is almost certainly correct, and completing the toric picture is a legitimate advance.\n\nWhat I like: the construction is concrete, the figures help, and the use of the shear matrix to get the integral affine isomorphism is neat. The paper is short but not sloppy in its main argument.\n\nThe main problem is the hat-width assumption. Section 2 begins with 'Suppose that the hat H has width w > 0' and then proceeds. The note never proves that every non-monotone toric four-manifold can be represented by a canonical nodal integral affine surface with positive width. Figure 3 shows that w=0 can occur for every type, so this is not vacuous. If some non-monotone manifold only admits w=0 representatives, the construction doesn't apply, and the abstract's 'all non-monotone' claim is not established. This is a real gap, not a minor omission. The author should either prove the width condition or point to a theorem in [3] that does it.\n\nThe other issue is the heavy reliance on [3]: Theorem 2.26 and Definition 2.30 are central, and they come from an unpublished preprint. That makes the note hard to evaluate independently. A referee would need to verify those parts. Not a fatal flaw, but a reason to be cautious.\n\nMinor slips: 'n ≥ 0' should be 'n ≥ 1' in the intersection statements, and at the end of Section 2 the power m should be positive. These are easily fixed.\n\nMy bottom line: this deserves a serious referee. The construction is plausible, the extension is new, and the gaps are checkable. I would recommend sending it to review with the request that the author clarify the positive-width condition and make the dependence on [3] precise. I wouldn't cite the full 'all non-monotone' claim until that's settled, but the specific new cases are worth knowing about.","headline":"Extends Lagrangian recurrence counterexamples to all non-monotone toric four-manifolds, but the proof depends on an unproved width condition and on the author's unpublished nodal-tangle preprint.","tokens_in":4685,"tokens_out":4160,"would_cite":false,"duration_ms":44857,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs, in every non-monotone toric symplectic four-manifold of dimension four, a Hamiltonian diffeomorphism and a Lagrangian torus whose forward orbit never intersects the torus.","keywords":["Lagrangian Poincaré recurrence","toric symplectic four-manifold","Hamiltonian diffeomorphism","Lagrangian submanifold","nodal integral affine base","nodal tangle","almost toric fibration","symplectic mapping class group"],"falsifier":"Pick a non-monotone toric symplectic four-manifold whose Delzant polygon is small enough to check by hand and test the two load-bearing inputs: verify whether the shear-plus-reglue map is a true integral affine isomorphism at the ridge $\\Delta_M$, and compute the hat width $w$; a single manifold where the map has a corner singularity or $w=0$ would show that the construction does not cover all non-monotone cases.","tokens_in":1926,"feed_emoji":"🔄","tokens_out":2315,"duration_ms":124717,"temperature":0.7,"pith_summary":"This note aims to prove that Lagrangian Poincar\\'e recurrence fails for every non-monotone toric symplectic four-manifold: for each such manifold there is a Lagrangian torus $L$ and a Hamiltonian diffeomorphism $\\psi$ with $\\psi^n(L) \\cap L = \\emptyset$ for every $n \\in \\mathbb{N}$. The construction is the previous one from the author's earlier paper, now extended to close the exceptional gaps, which were the non-monotone products $S^2 \\times S^2$ with unequal factor areas and the one- or two-fold blow-ups of such products at critical size. The proof encodes the manifold as an almost toric fibration over a nodal integral affine base, performs nodal slides to produce an integral affine isomorphism of the base, and lifts that isomorphism to a symplectomorphism of the total space. Because the induced map translates Lagrangian fibers along closed level sets by an amount whose ratio to the level-set length can be made irrational, the orbit of the fiber never returns. If the construction is correct, the known counterexamples to Lagrangian recurrence now cover all non-monotone toric four-manifolds.","feed_headline":"All non-monotone toric 4-manifolds break Lagrangian recurrence","feed_subtitle":"A Hamiltonian map keeps a Lagrangian torus and all its iterates disjoint, in every non-monotone case.","key_machinery":"The central object is the nodal integral affine base $(B_0, \\mathfrak{N}_0)$ of an almost toric fibration: the usual Delzant polygon data plus a choice of hat and parked nodes, with a height function $\\mathcal{F}$ measuring integral affine distance to the boundary. The load-bearing identity is the level-set length formula $g(h) = 2w - k(M-h) + \\sum_i \\min\\{h - \\alpha_i, 0\\}$, where $w$ is the width of the hat, $M$ is the maximum height, $k$ is a constant determined by the hat class, and $\\alpha_i$ are the heights of the parked nodes. A nodal tangle that slides nodes and modifies the hat produces a second base $(B_1, \\mathfrak{N}_1)$ that is integral affine isomorphic to the first through an explicit shear-and-reglue map, and the supporting theorem lifts this isomorphism to a genuine symplectomorphism of the total space. The induced map translates fibers along the closed level sets, and the irrationality of $g(h)/(2(M-h))$ is what forces the Lagrangian orbit never to close up.","core_discovery":"On the author's own terms, the central claim is: for every non-monotone toric symplectic four-manifold $X$ there exists a Lagrangian torus $L \\subset X$ and a Hamiltonian diffeomorphism $\\psi$ of $X$ such that $\\psi^n(L) \\cap L = \\emptyset$ for all $n \\in \\mathbb{N}$. The manifold is described by a nodal integral affine base $(B_0, \\mathfrak{N}_0)$ with height function $\\mathcal{F}$; after a nodal tangle that slides parked nodes and modifies the hat, the new base $(B_1, \\mathfrak{N}_1)$ is integral affine isomorphic to the original via a map composed of a shear and a triangle cut-and-reglue. Using a lift theorem from the author's companion preprint, this base isomorphism becomes a symplectomorphism $\\psi$, which acts on the fibers above a level set as translation by integral affine distance $2(M-h)$. Since the level set has integral affine length $g(h) = 2w - k(M-h) + \\sum_i \\min\\{h - \\alpha_i, 0\\}$, choosing $h$ with $g(h)/(2(M-h))$ irrational makes the iterates $\\psi^n(\\pi_0^{-1}(x))$ pairwise disjoint from the starting fiber; finiteness of the symplectic mapping class group then gives a Hamiltonian iterate, namely $\\psi^m$ for some $m$, which still satisfies $(\\psi^m)^n(L) \\cap L = \\emptyset$ for all $n$. This directly covers the previously missing non-monotone $S^2 \\times S^2$ and critical blow-up cases.","pith_inferences":["A natural extension would be to formulate the same height-function and irrational-rotation mechanism in higher-dimensional toric manifolds, provided corresponding integral affine bases with closed level sets and integral affine automorphisms exist; the paper does not address this.","The completeness of the all-non-monotone statement currently depends on the unpublished companion theorem [3, Theorem 2.26]; a self-contained proof of that lift theorem would remove the main external dependency.","A direct test of the construction is to compute the hat width $w$ for every Delzant polygon of the claimed class; if some non-monotone polygon forces $w=0$, the note's general assumption \\'suppose $w>0$\\' would miss that case."],"forward_implications":["The previous exceptional cases—non-monotone $S^2 \\times S^2$ with unequal factor areas, and the one- or two-fold blow-ups of such products at size $c = \\min\\{a,b\\}/2$—now also admit Hamiltonian counterexamples to Lagrangian Poincar\\'e recurrence.","In every non-monotone toric symplectic four-manifold, the constructed Lagrangian torus is a fibre of an almost toric fibration, so the non-recurrence occurs on a geometrically natural Lagrangian rather than an exotic one.","Because the symplectic mapping class group is finite in these manifolds, the power $\\psi^m$ actually lies in $\\mathrm{Ham}(X)$, so the counterexample can be taken Hamiltonian even though the raw map $\\psi$ is only a symplectomorphism.","The arithmetic condition $g(h)/(2(M-h))$ irrational is open, so each manifold covered by the construction yields many non-returning Lagrangian fibres, one for each admissible irrational ratio."],"supporting_citations":[{"why":"Supplies the almost toric fibration and height-function machinery, the nodal tangle lift theorem (Theorem 2.26), and the hat classification used in the general construction.","marker":"[3]"},{"why":"Gives the original counterexample construction whose exceptional cases this note is designed to close.","marker":"[2]"},{"why":"Provides the finiteness of the symplectic mapping class group, used to promote the constructed symplectomorphism to a Hamiltonian diffeomorphism by taking an iterate.","marker":"[1]"}],"fun_headline_variants":["Non-monotone toric 4-manifolds all break Lagrangian recurrence","Every non-monotone toric 4-manifold breaks Lagrangian recurrence","Lagrangian recurrence fails for all non-monotone toric 4-manifolds","All non-monotone toric 4-manifolds are Lagrangian recurrence counterexamples","Counterexamples to Lagrangian recurrence extended to all non-monotone toric 4-manifolds"],"cache_read_input_tokens":6784,"weakest_assumption_plain":"The construction leans on the unpublished theorem that an integral affine isomorphism between nodal bases always lifts to a symplectomorphism, together with the unverified assumption that every non-monotone toric four-manifold has a hat of positive width, so if either gives way the claimed all-non-monotone counterexamples are not established.","fun_headline_variants_meta":{"raw":{"variants":["Non-monotone toric 4-manifolds all break Lagrangian recurrence","Every non-monotone toric 4-manifold breaks Lagrangian recurrence","Lagrangian recurrence fails for all non-monotone toric 4-manifolds","All non-monotone toric 4-manifolds are Lagrangian recurrence counterexamples","Counterexamples to Lagrangian recurrence extended to all non-monotone toric 4-manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000727,"raw_usage":{"total_tokens":3259,"prompt_tokens":950,"completion_tokens":2309,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":2200}},"tokens_in":566,"tokens_out":2309,"duration_ms":17150,"temperature":1.0,"reasoning_tokens":2200,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:14:19.013060+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a non-monotone toric symplectic four-manifold whose Delzant polygon is small enough to check by hand and test the two load-bearing inputs: verify whether the shear-plus-reglue map is a true integral affine isomorphism at the ridge $\\Delta_M$, and compute the hat width $w$; a single manifold where the map has a corner singularity or $w=0$ would show that the construction does not cover all non-monotone cases.","supporting_citations":[{"cited_title":"Nodal Tangles","cited_arxiv_id":"2506.23754","evidence_quote":"Supplies the almost toric fibration and height-function machinery, the nodal tangle lift theorem (Theorem 2.26), and the hat classification used in the general construction."},{"cited_title":"A counterexample to Lagrangian Poincaré recurrence in dimension four","cited_arxiv_id":null,"evidence_quote":"Gives the original counterexample construction whose exceptional cases this note is designed to close."}],"review_version":1}