{"id":"3d855f6f-f903-40da-bcd9-bd72439ca059","arxiv_id":"2507.01416","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hamiltonian unlinked subsets of a symplectic ball satisfy the spectral capacity packing inequality c(K1) + c(K2) <= a, and violating that inequality forces Hamiltonian linking.","lead":"This note proves a dichotomy: two disjoint sets inside a symplectic ball that can be moved apart by a Hamiltonian motion must satisfy a capacity-packing inequality, while any pair that violates the inequality must be Hamiltonian linked. It gives a common framework for several rigidity results in symplectic topology, including boundary minimality, a closing lemma, and bounds on holomorphic disk areas.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's packing dichotomy hinges on Lemma 15 being an equality for arbitrary disjointly supported isotopies, but the cited references may only establish a max inequality under sign or convention hypotheses; if the equality fails in the paper's Floer convention, the proof collapses.","rationale":"The paper's main dichotomy is transparently reduced to two lemmas, and the surrounding argument is clean; this is a genuine strength. However, the proof of Theorem 1 does not itself prove Lemma 15, and the cited literature is invoked in a form that may not match the stated equality: the max formula for disjointly supported Hamiltonians is often a max inequality, and equality can require non-negativity or a specific convention for spectral invariants. Because the entire packing inequality in §2.4 depends on the lower bound c(φ1φ2^{-1}) ≥ c(φ1), a sign or convention mismatch here would invalidate Theorem 1, not merely a secondary corollary. Lemma 16 is similarly load-bearing and is cited from a self-authored preprint, so independent verification is warranted. The reader's weakest assumption was already Lemma 15; I agree with that identification but sharpen it to the specific sign/convention and equality-versus-inequality issue. No evidence of actual falsehood is available from the manuscript, so the appropriate recommendation is to keep the verdict conditional: Theorem 1 should be regarded as established only after Lemma 15 is checked against the cited sources and, ideally, recomputed in the paper's Floer convention, and after Lemma 16 from [AAC24] is independently confirmed.","tokens_in":9296,"tokens_out":33915,"duration_ms":401368,"concrete_test":"Independently compare Lemma 15 with the exact statements in [GT23, Theorem 1.1] and [Tan22]: check whether they prove equality or only c(H#K) ≤ max(c(H), c(K)) and whether they require non-negative Hamiltonians or a particular path convention. Then run a computational test in the paper's convention: take φ0 = id and φ1 supported in a small ball and generated by a strictly negative autonomous Hamiltonian, and compute c(φ0φ1) directly from the definition in §2. If c(φ0φ1) ≠ max(0, c(φ1)), Lemma 15 is false as stated and Theorem 1 needs a repaired argument; if equality holds for these examples, also test a pair with one positive and one negative generating Hamiltonian to confirm the sign-independence of the identity.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central reduction in §2.4 requires the exact identity c(φ0φ1) = max(c(φ0), c(φ1)) for compactly supported isotopies whose supports lie on opposite sides of a hyperplane. In the proof, this identity is applied as c(φ1φ2^{-1}) = max(c(φ1), c(φ2^{-1})) and c(φ2φ1^{-1}) = max(c(φ2), c(φ1^{-1})); only the lower bounds c(φ1φ2^{-1}) ≥ c(φ1) and c(φ2φ1^{-1}) ≥ c(φ2) are then needed to force c(φ1)+c(φ2) ≤ γ(Ω). Lemma 15 is stated without restriction on the signs of the generating Hamiltonians and without specifying the path-composition convention (simultaneous sum versus concatenation). The cited results [HLRS16, GT23, Tan22] are formulated as a max inequality for disjointly supported Hamiltonians, and it is not automatic that the equality, rather than merely ≤ max, holds in the present Floer-theoretic convention for isotopies of arbitrary sign. If only the inequality c(φ0φ1) ≤ max(c(φ0), c(φ1)) is available, the needed lower bound c(φ1φ2^{-1}) ≥ c(φ1) is not established and the packing inequality does not follow. Lemma 16, the second black box, is cited from the self-authored [AAC24] and is equally load-bearing, since the final line of the proof applies it to ψ = φ1φ2^{-1} supported in U1∪U2 ⊂ B(a). Thus the soundness of Theorem 1 currently rests on an unstated verification of Lemma 15's exact hypotheses and on an independent check of [AAC24].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines two compact sets in R^{2n} to be Hamiltonian unlinked if some compactly supported Hamiltonian diffeomorphism sends them to opposite sides of a hyperplane, and Hamiltonian linked otherwise. Its main result (Theorem 1) states that if two disjoint compact subsets of the standard ball B(a) are Hamiltonian unlinked, then the sum of their spectral capacities is at most a. The proof conjugates by an unlinking diffeomorphism, applies a maximum formula for spectral invariants of disjointly supported isotopies and the spectral-diameter bound for the ball, and concludes the capacity inequality. The paper then derives several corollaries, including boundary minimality of the ball, a closing lemma that leads to a characterization of uniformly convex Zoll domains, a statement about areas of J-holomorphic disks bounded by Lagrangians, and two further results (Theorems 9 and 12) about capacities of toric subsets of the boundary and about Reeb chords between unlinked Legendrians.","tokens_in":9642,"tokens_out":12253,"duration_ms":100157,"significance":"If Theorem 1 is established, it is a clean and potentially useful observation connecting symplectic packing inequalities with Hamiltonian linking. The proof is remarkably short and the applications are wide-ranging. The main obstruction to accepting the paper as it stands is that the central maximum-formula lemma is not proved, and the cited literature does not clearly support the exact equality used. The paper also depends essentially on the spectral-diameter theorem of the author's own [AAC24]. The manuscript is clearly written and the main idea is elegant, but the central claim requires a missing verification.","major_comments":[{"comment":"Lemma 15 is the only input that converts the conjugated isotopies into quantities controlled by the spectral diameter. In the proof of Theorem 1 (§2.4), the equality c(φ0φ1) = max{c(φ0), c(φ1)} is applied to isotopies supported on opposite sides of a hyperplane, and only the lower bounds c(φ1φ2^{-1}) ≥ c(φ1) and c(φ2φ1^{-1}) ≥ c(φ2) are actually needed. The lemma is stated without proof and with a citation to [HLRS16, GT23]. The discussion in §1.1 suggests those references prove a max inequality for generating-function capacities of subsets, or for disjointly supported Hamiltonians under additional hypotheses, and the author explicitly notes in §1.3.5 that a max inequality is insufficient for the argument. The exact equality for spectral invariants of arbitrary compactly supported isotopies with supports on opposite sides of a hyperplane, in the Floer convention of [AAC24, CZ24], needs a complete proof or a precise theorem-level reference with all hypotheses verified. This is load-bearing: without this equality, Theorem 1 does not follow.","section":"§2.2 (Lemma 15)"},{"comment":"Lemma 16 is stated for isotopies supported in B(a), but the proof of Theorem 1 is written for a domain Ω with a = c(Ω) = γ(Ω), and the final bound c(φ1φ2^{-1}) + c(φ2φ1^{-1}) ≤ a is applied to an isotopy supported in U1 ∪ U2 ⊂ Ω. Please state the version needed for arbitrary Ω with γ(Ω) = a, or restrict the proof (and statement) to Ω = B(a). Since Lemma 16 is cited from the author's own preprint [AAC24], the manuscript should also make the precise dependence clear, for example by giving a theorem statement and a reference with a theorem number.","section":"§2.3 and §2.4 (Lemma 16 and its use)"}],"minor_comments":[{"comment":"The passage from disjoint compact sets K1, K2 to disjoint open neighborhoods U1, U2 that are 'also unlinked' is true but should be justified explicitly by applying the unlinking diffeomorphism and taking preimages of separating neighborhoods.","section":"§2.4"},{"comment":"The notation E(...) is not defined, and the identities a = c(K ∪ K^c) and c(K ∪ K^c) ≤ c(K) + c(K^c) are used without proof; these steps need justification, as they are not standard consequences of spectral capacity alone.","section":"§2.8 (Theorem 9)"},{"comment":"The construction from [Moh01] is not described in enough detail to verify that the two Lagrangians are disjoint, unlinked, and have capacities arbitrarily close to the minimal Reeb chord length; please expand this argument.","section":"§2.9 (Theorem 12)"},{"comment":"The assertion that a small ball inside U ∩ Ω is unlinked with Ω \\ U is not immediate and should be proved or supported by a reference.","section":"§1.1 (Corollary 3)"},{"comment":"There are minor typographical issues, including 'proves the the maximum formula' in §1.1 and 'pertubations' in §1.3.3.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on two papers coauthored by the author, [AAC24] and [CZ24]. This is not a problem in itself, but the editor may wish to confirm that those results are publicly available and that the dependence is fully transparent. The paper's breadth of applications exceeds the detail of the proofs for several corollaries; the core of the paper is Theorem 1, and the referee report focuses on that."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result, Theorem 1, is genuinely new as far as I can tell: a packing-versus-linking dichotomy that packages several rigidity phenomena into one statement. The proof is short and clean: conjugate by the unlinking Hamiltonian, apply a max formula to split the two spectral invariants, then bound the sum by the spectral diameter. If the two black-box lemmas hold, the argument goes through, and the corollaries are a nice payoff. I also give the author credit for being transparent: Corollary 3 is explicitly flagged as recovering Ziltener's boundary minimality, and Theorem 10 is attributed to Ziltener rather than claimed as new. That is good scholarship.\n\nThe soft spots are real but not necessarily fatal. The proof of Lemma 15 is a citation to [HLRS16, GT23, Tan22], and the stress-test note raises a fair point: the known results in those papers are often stated as a max inequality, not an equality, and sometimes with sign restrictions. The proof of Theorem 1 needs the equality direction, specifically the lower bounds c(φ1φ2^{-1}) ≥ c(φ1) and c(φ2φ1^{-1}) ≥ c(φ2). If the cited references only give c(φ0φ1) ≤ max(c(φ0), c(φ1)), those lower bounds do not follow and the packing inequality collapses. This is the load-bearing spot. It might be sorted out by a careful check of conventions, and I would not be surprised if the equality is true in the right setting, but as it stands the central proof depends on an unverified hypothesis. The second black box, Lemma 16, comes from the author's own [AAC24]; that is not a problem per se, but an independent check would help, especially since the final step needs it for the product isotopy supported in U1∪U2.\n\nSome secondary results are sketched rather than fully proved (Theorem 9, Theorem 12, and parts of the closing lemma). For a short note this is acceptable if the main theorem holds, but a referee should ask for details or at least a clear roadmap.\n\nWho should read this? Anyone working on spectral capacities, Hamiltonian dynamics, or symplectic rigidity. It connects several known threads and gives a useful organizing principle. I would send it to a serious referee, not desk reject. The referee should focus on verifying Lemma 15's exact hypotheses in the conventions used, and on checking Lemma 16. If those hold, the paper is publishable as a short research note.","headline":"A short, honest note whose new packing-versus-linking dichotomy is elegant and likely correct, but the referee should check the exact hypotheses of the cited max formula and the spectral-diameter result.","tokens_in":10207,"tokens_out":4506,"would_cite":true,"duration_ms":52408,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","53D35","53D12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A packing inequality or a Hamiltonian link: the dichotomy inside a symplectic ball.","keywords":["Hamiltonian linking","symplectic packing","spectral capacity","spectral diameter","maximum formula","boundary minimality","Floer homology","J-holomorphic disks"],"falsifier":"Find two disjoint compact subsets $K_1,K_2$ of $B(a)$ that can be separated by a Hamiltonian diffeomorphism to opposite sides of a hyperplane but satisfy $c(K_1)+c(K_2)>a$; Theorem 1 asserts no such pair exists.","tokens_in":9041,"feed_emoji":"🔗","tokens_out":5568,"duration_ms":56464,"temperature":0.7,"pith_summary":"This paper establishes a dichotomy for two disjoint compact sets inside a standard symplectic ball: if the pair cannot be separated by a Hamiltonian diffeomorphism so that they lie on opposite sides of a hyperplane, then their spectral capacities must sum to at most the ball's capacity. Equivalently, any pair whose spectral capacities sum to more than the ball's capacity is necessarily Hamiltonian linked. The observation matters because it turns a packing obstruction into a linking statement and vice versa, giving a single inequality that controls both phenomena. From it the paper derives boundary minimality of the ball, a closing lemma for Reeb orbits, and bounds on areas of J-holomorphic disks with Lagrangian boundary.","feed_headline":"Two sets in a symplectic ball either link or obey a packing bound","feed_subtitle":"When two sets' capacities sum past the ball's size, they must be Hamiltonian linked; separable pairs always pack.","key_machinery":"The machinery is spectral capacity $c(K)$, the Floer-theoretic invariant that assigns a nonnegative size to a compact set, together with the spectral diameter $\\gamma(\\Omega)$ of the space of isotopies supported in $\\Omega$. The load-bearing identity is the maximum formula: for isotopies supported on opposite sides of a hyperplane, the spectral invariant of their product is the maximum of the two individual invariants. This converts the conjugated pair of isotopies into two product isotopies whose spectral invariants dominate the original capacities, and the spectral-diameter equality $\\gamma(B(a))=a$ then bounds their sum. The proof of Theorem 1 reduces the whole packing-versus-linking alternative to exactly this formula together with the ball's spectral-diameter computation.","core_discovery":"The central claim is Theorem 1: for compact disjoint $K_1,K_2\\subset B(a)$ in the standard symplectic ball of capacity $a$, either $K_1$ and $K_2$ are Hamiltonian linked, or $c(K_1)+c(K_2)\\le a$. Here Hamiltonian linked means the pair cannot be mapped by a compactly supported Hamiltonian diffeomorphism to opposite sides of a hyperplane. The proof conjugates by the diffeomorphism that separates the sets, applies the maximum formula to the conjugated isotopies, and uses the fact that the spectral diameter of the ball equals its capacity to bound the sum by $a$. A corollary is that any domain $\\Omega$ with $c(\\Omega)=\\gamma(\\Omega)$ is boundary minimal, so removing an open set touching the boundary strictly lowers its capacity.","pith_inferences":["If the maximum formula extends to a wider class of disjoint pairs, the same proof would give the packing inequality $c(K_1)+c(K_2)\\le a$ for those pairs; the paper explicitly notes exact and boundary-$\\pi_1$-injective Liouville embeddings as known cases.","An affirmative answer to the paper's Question 1 would settle the conjecture that every Lagrangian in the interior of $B(a)$ bounds $J$-holomorphic disks of area at most $a/2$, since a boundary set of capacity at least $a/2$ would force the alternative in Corollary 6 to fail.","Theorem 13 shows that sets of arbitrarily small positive spectral capacity can link the ball, so any future criterion for linking must use more than positivity of capacity; this sharpens the open question of whether zero-capacity sets can link.","The same dichotomy may be testable in low dimensions by computing spectral capacities of explicit toric subsets of the ball, since Theorem 9 already computes capacities of boundary subsets cut out by toric inequalities."],"forward_implications":["A compact set $K\\subset\\partial B(a)$ with $c(K)=b$ is Hamiltonian linked with every smaller ball $B(c)\\subset B(a)$ once $c>a-b$, giving a higher-dimensional camel-type obstruction (Corollary 2).","Any domain $\\Omega$ with $c(\\Omega)=\\gamma(\\Omega)$ is boundary minimal: deleting any open set that touches $\\partial\\Omega$ strictly lowers the capacity (Corollary 3).","Boundary minimality yields a closing-type lemma: for any nonempty open subset of the boundary there is a Reeb orbit through it whose period plus a cohomological correction is bounded by $c(\\Omega)$ (Lemma 4).","For any Lagrangian in the interior of $B(a)$ and any admissible almost complex structure, either the Lagrangian bounds a $J$-holomorphic disk of area at most $a-b$, or it is linked with a fixed boundary set $K$ of capacity $b$ (Corollary 6).","A uniformly convex domain is Zoll if and only if it is boundary minimal (Theorem 5)."],"supporting_citations":[{"why":"Proves the spectral diameter of the ball equals its capacity, $\\gamma(B(a))=c(B(a))=a$, supplying the upper bound in Theorem 1.","marker":"[AAC24]"},{"why":"Proves the maximum formula for spectral invariants of hyperplane-separated supports, the key identity in the proof of Theorem 1.","marker":"[HLRS16]"},{"why":"Provides a max inequality for spectral invariants of disjointly supported Hamiltonians that feeds into the maximum formula.","marker":"[Tan22]"},{"why":"Establishes the maximum formula in the Floer-theoretic setting used here and extends it to pairs of boundary-$\\pi_1$-injective Liouville embeddings.","marker":"[GT23]"},{"why":"Relates spectral capacity of a Lagrangian to areas of holomorphic disks, used in Corollary 6.","marker":"[Her04]"},{"why":"Supplies the convention for spectral capacity and the disk-area bound used in Corollary 6.","marker":"[CZ24]"},{"why":"Introduced boundary minimality and the chord argument that the paper recovers in the ball case.","marker":"[Zil16]"},{"why":"Identifies the spectral capacity of a convex domain with the shortest Reeb period, used in Lemma 4 and Theorem 5.","marker":"[AK22]"}],"fun_headline_variants":["Symplectic balls: either linked or packable","Link or pack: a symplectic dichotomy","Hamiltonian link or packing bound in balls","Two sets: linked or capacity-bound pack","In symplectic balls, disjoint sets either link or pack"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the maximum formula holding exactly: when two Hamiltonian isotopies have supports on opposite sides of a hyperplane, the spectral invariant of their product must equal the larger of the two individual invariants, not merely lie close to it or be bounded by it.","fun_headline_variants_meta":{"raw":{"variants":["Symplectic balls: either linked or packable","Link or pack: a symplectic dichotomy","Hamiltonian link or packing bound in balls","Two sets: linked or capacity-bound pack","In symplectic balls, disjoint sets either link or pack"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000501,"raw_usage":{"total_tokens":2334,"prompt_tokens":713,"completion_tokens":1621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":329,"completion_tokens_details":{"reasoning_tokens":1548}},"tokens_in":329,"tokens_out":1621,"duration_ms":13098,"temperature":1.0,"reasoning_tokens":1548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:52:26.329856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find two disjoint compact subsets $K_1,K_2$ of $B(a)$ that can be separated by a Hamiltonian diffeomorphism to opposite sides of a hyperplane but satisfy $c(K_1)+c(K_2)>a$; Theorem 1 asserts no such pair exists.","supporting_citations":[],"review_version":1}