{"id":"fbfa4799-6d6e-4592-b742-fd7870e9e067","arxiv_id":"2411.11422","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Contactomorphism groups of R^{2n+1} admit a dense conjugacy class, those of R^{2n}×S^1 do not, and Sandon's spectral norm is C^0-locally bounded and extendable to the C^0-closure.","lead":"This paper proves that the C^0-closed group of compactly supported contactomorphisms of R^{2n+1} has the Rokhlin property (a single element whose conjugates are dense), while the same group on R^{2n}×S^1 does not. It also introduces a conjugation-invariant norm, a metric measuring conjugacy-class size, and shows Sandon's spectral norm is C^0-locally bounded and extends to contact homeomorphisms.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central dichotomy is supported by the paper's construction and a sound C0-extension of the cited non-squeezing theorem.","rationale":"The paper's central claims are internally coherent. Theorem 1.3's positive half constructs a concrete element whose conjugacy class is dense; the construction is standard but carefully executed, and the infinite product is indeed a C0-limit of smooth compactly supported contactomorphisms. The negative half depends on the Eliashberg-Kim-Polterovich non-squeezing theorem, which is a well-established external input; the paper explicitly supplies the C0-approximation argument needed to push non-squeezing from smooth contactomorphisms to their closure. I checked that this argument is valid: the image of the open ball under a C0-close smooth approximation is contained in the slightly enlarged ball, so the smooth non-squeezing theorem applies. The subsequent fixed-point argument is sound once one chooses the Hamiltonian perturbation to move the closed ball, which is possible with a bump function. The extension eγ of Sandon's norm is also supported by the local boundedness and lower semicontinuity arguments; the only notable omission is that conjugation-invariance of eγ is not explicitly proved, but it follows by approximating the conjugating homeomorphism by smooth contactomorphisms and using lower semicontinuity. The stated limitations in Remarks 1.15 and Question 1.6 are honest. No p-hacking, circular reasoning, or overclaiming was found; the reliance on external Floer/generating-function machinery is appropriate for the field.","tokens_in":25719,"tokens_out":58970,"duration_ms":605242,"concrete_test":"Independently verify the C0-extension step in §3.2: for f in the closure of Cont0,c(R2n×S1) with f(B(R)×S1)⊂B(r)×S1, prove that for every ε>0 a smooth contactomorphism fε with dC0(f,fε)<ε satisfies fε(B(R)×S1)⊂B(r+ε)×S1, and then replay the contradiction of Theorem 1.3 using the compact margin of supp(g) to handle the closed-ball boundary. If this step fails, the negative half of the dichotomy loses its non-squeezing input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reviewing the proof of Theorem 1.3, I do not find a load-bearing flaw. The positive half is a careful 'Rokhlin element' construction: a separable dense sequence is shrunk and translated into disjoint small supports, a cut-off Reeb flow separates one factor, and the remaining tail is shrunk away; the infinite product is a C0-limit of smooth compactly supported contactomorphisms, so it lies in the closure. The negative half rests on the cited EKP contact non-squeezing theorem (Theorem 2.7), and the only genuinely load-bearing step is its extension from Cont0,c to the C0-closure in §3.2. That step is correct: if f in the closure maps B(R)×S1 into B(r)×S1, any smooth contactomorphism within C0-distance ε maps into B(r+ε)×S1, and choosing ε<R−r violates Theorem 2.7. In the contradiction, compactness of supp(g) gives the needed margin when h_n^{-1} maps the closed ball B(r+1) into B(r). Minor expository omissions, such as a boundary point of supp(g) lying on ∂B(2) and the conjugation-invariance of eγ being asserted in Theorem 1.11 without an explicit proof, are easily repaired and do not affect the central claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the C^0 topology on the identity component of compactly supported contactomorphism groups and on its C^0 closure. The main result is Theorem 1.3: the C^0 closure on R^{2n+1} has the Rokhlin property, while the C^0 closure on R^{2n}×S^1 does not, and the dichotomy is tied to contact squeezing versus contact non-squeezing. The paper also proves that Sandon's spectral norm γ is C^0-locally bounded (Theorem 1.7), extends γ to a conjugation-invariant norm eγ on the C^0 closure (Theorem 1.11), introduces a conjugation norm bounded above by the fragmentation norm (Theorem 1.13), proves that the conjugation-connectedness distance d_cc is unbounded (Theorem 1.17), and obtains non-Rokhlin results for prequantization spaces (Theorems 1.19 and 1.20). A further result on the C^0-continuity of Albers-Merry spectral invariants is proved in Theorem 1.22.","tokens_in":25893,"tokens_out":32300,"duration_ms":320149,"significance":"The paper makes a substantial contribution to C^0-contact topology. The Rokhlin dichotomy is a new and natural rigidity/flexibility statement: the positive half is proved by an explicit ball-squeezing construction of a Rokhlin element, and the negative half is reduced, via a correct C^0-approximation argument, to the Eliashberg-Kim-Polterovich contact non-squeezing theorem. The spectral-norm results give a new quantitative rigidity phenomenon: γ is not continuous, but it is locally bounded with an explicit constant on the 1/2-ball, and it extends to the C^0 closure. The proofs are detailed and largely self-contained, and the main claims are derived from independent published results (contact non-squeezing, Sandon's invariants) rather than from circular reasoning. The paper is careful to state and prove the auxiliary continuity and local-boundedness facts it relies on, and its constructions are explicit and falsifiable.","major_comments":[],"minor_comments":[{"comment":"The proof does not explicitly verify the conjugation-invariance of eγ, although this is asserted in the theorem statement. The claim follows from the fact that conjugation by a fixed element is a τ_C0-homeomorphism of Cont0,c and from the conjugation-invariance of γ, but the argument should be written out.","section":"§5.2 (Theorem 1.11)"},{"comment":"The assertion that the inclusion W×S^1 → W×P induces an injection on free-loop homotopy classes is only sketched. Please spell out the reduction to conjugacy classes in π1 and the group-theoretic fact that in the free group powers of a commutator are conjugate only when the exponents agree; as written, this is too terse for a lemma on which Theorem 1.22 depends.","section":"§2.7 (Lemma 2.19)"},{"comment":"It should be stated explicitly why the infinite product g is a well-defined continuous homeomorphism: each factor moves points by at most the diameter of its support, the supports are disjoint with diameters tending to zero, and the supports accumulate only at a single boundary point where the displacement tends to zero.","section":"§3.1 (proof of Theorem 1.3, positive half)"},{"comment":"The symbol Conj(g) is used both for the conjugacy class in Definition 1.2 and for its closure in Section 1.3 and in the proof of Theorem 1.3. This should be made consistent, for example by writing \\overline{\\mathrm{Conj}(g)} for the closure.","section":"Notation"},{"comment":"In the observation that non-squeezing passes to the C^0 closure, the inequalities 1 ≤ πr^2 < πR^2 are assumed; later in the contradiction one should explicitly say that r can be enlarged so that πr^2 ≥ 1 before applying Theorem 2.7.","section":"§3.2 (proof of the second part of Theorem 1.3)"},{"comment":"The proof of lower semicontinuity of eγ is unnecessarily indirect and implicitly relies on the maximum in the definition of eγ being attained. A direct proof is shorter: if a < eγ(φ), some neighbourhood U of φ has min_{U'} γ > a, and then every ψ ∈ U satisfies eγ(ψ) > a, which gives lower semicontinuity. The current argument should be replaced or clarified.","section":"§5.2 (proof of Theorem 1.11)"},{"comment":"There are a few typos that should be corrected, including 'insted' in §2.6, 'mix-max' for 'min-max', and 'Be the definition' in the proof of Theorem 1.11.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"This is a strong paper and I have found no load-bearing mathematical error. My recommendation of minor revision reflects local expository gaps rather than any scientific concern. The omitted justifications in Lemma 2.19 and Theorem 1.11 are straightforward to supply and should not require substantial reworking."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a good paper and the reader's accept verdict is right. The main results are genuinely new — the Rokhlin property dichotomy between R^{2n+1} and R^{2n}×S^1, the C0-local boundedness of Sandon's spectral norm, the extension to eγ on the C0-closure, the conjugation norm inequality, and the d_cc metric. The positive half of the dichotomy is a careful explicit construction of a Rokhlin element; the negative half is a clean application of EKP non-squeezing plus a correct C0-approximation argument that extends non-squeezing to the closure. The proof of that extension is short and sound: if f in the closure maps B(R)×S1 into B(r)×S1, approximate f by smooth contactomorphisms within ε and use the margin R-r. I checked the part that worried me most and it holds.\n\nThe spectral norm part is also in good shape. The action-spectrum estimate in Proposition 4.1 is elegant, and it gives local boundedness of γ with the sharp-looking factor 2. The extension eγ via limit inferior is a natural construction and the paper proves non-degeneracy, triangle inequality, local boundedness, and lower semicontinuity. The new conjugation norm and d_cc are useful quantitative tools; Theorem 1.13 and the unboundedness of d_cc follow cleanly.\n\nThe soft spots are minor and mostly expository. Lemma 2.19's claim that Spec'_{α}(ϕ) ⊂ Spec_α(ϕ) is only sketched via a topological injectivity statement about free loop classes; that deserves a few more lines. Theorem 1.11 states that eγ is conjugation-invariant but the proof as written does not explicitly show conjugation-invariance — it proves symmetry and triangle inequality, and conjugation-invariance is presumably immediate from the definition and the conjugation-invariance of γ, but it should be stated. There is also a tiny boundary issue in the support argument in §3.2 when supp(g) touches ∂B(2); that is easily patched. None of these affect the central claims.\n\nThe dependence on EKP non-squeezing and Sandon's machinery is appropriate — these are cited, not hidden. No circularity, no fitted parameters.\n\nWho should read this: anyone working in C0 symplectic/contact topology or on conjugation-invariant norms. It deserves a serious referee. I'd send it out; the fixes I'd request are cosmetic and local.","headline":"A solid, genuinely new C0-contact topology paper; the Rokhlin dichotomy is real and the quantitative tools are reusable, with only minor expository gaps.","tokens_in":26498,"tokens_out":1628,"would_cite":true,"duration_ms":14987,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D10","53D35","53D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"On $\\mathbb{R}^{2n+1}$, contact squeezing produces a contact homeomorphism whose conjugacy class is $C^0$-dense, while on $\\mathbb{R}^{2n}\\times S^1$ contact non-squeezing prevents any such element; the paper also extends the spectral…","keywords":["C0 contact topology","Rokhlin property","contactomorphism group","conjugacy class","spectral norm","contact non-squeezing","fragmentation norm","prequantization spaces"],"falsifier":"Take the explicit map $g$ built in Section 3.1 and test whether its conjugacy class is $C^0$-dense in $\\operatorname{Cont}_{0,c}(\\mathbb{R}^{2n+1})$: a single open $C^0$ ball containing no conjugate of $g$ would disprove the Rokhlin claim. For the negative half, find a $C^0$-limit contact homeomorphism $f$ on $\\mathbb{R}^{2n}\\times S^1$ and radii with $\\pi r^2\\ge 1<\\pi R^2$ such that $f(B(R)\\times S^1)\\subset B(r)\\times S^1$; that would break the non-squeezing theorem and with it the no-Rokhlin conclusion.","tokens_in":25458,"feed_emoji":"🌀","tokens_out":13750,"duration_ms":117683,"temperature":0.7,"pith_summary":"At the level of $C^0$ limits, this paper finds that a single geometric dichotomy governs how large the conjugacy classes of contactomorphism groups can be. It proves that the $C^0$-closure of compactly supported contactomorphisms of standard contact $\\mathbb{R}^{2n+1}$ has the Rokhlin property — one element's conjugacy class is dense — while the same group on $\\mathbb{R}^{2n}\\times S^1$ does not. The split is exactly the split between contact squeezing (balls can be pushed into arbitrarily small balls, giving the dense conjugacy class) and contact non-squeezing (large balls resist being compressed, blocking any dense conjugacy class). The paper also shows that the integer-valued spectral norm is locally bounded in the $C^0$ topology and extends to a norm on the $C^0$-closure, which yields an unbounded distance measuring how far the group is from having the Rokhlin property. A reader should care because this turns smooth contact rigidity into a topological property of homeomorphism groups and provides quantitative $C^0$ invariants where none were available.","feed_headline":"Where balls squeeze, a contact group has a dense conjugacy class","feed_subtitle":"On $\\mathbb{R}^{2n}\\times S^1$ non-squeezing blocks it, and the spectral norm extends to $C^0$ limits.","key_machinery":"The pair that carries the argument is squeezing versus non-squeezing. Squeezing is provided by explicit compactly supported contactomorphisms $\\Psi_a$ that act as $(x,y,z)\\mapsto(ax,ay,a^2z)$ on a ball; these let the Rokhlin construction place scaled copies of a dense family into disjoint tiny balls. Non-squeezing is provided by the theorem that $B(R)\\times S^1$ cannot be mapped into $B(r)\\times S^1$ by a compactly supported contactomorphism when $1\\le\\pi r^2<\\pi R^2$, a statement that survives passage to the $C^0$-closure. The quantitative half is carried by the spectral norm $\\gamma(\\phi)=\\lceil c_+(\\phi)\\rceil-\\lfloor c_-(\\phi)\\rfloor$, defined from the integer parts of generating-function spectral invariants. The key new fact is that $\\gamma$ is $C^0$-locally bounded: maps within $C^0$-distance less than $1/2$ have $\\gamma$-values differing by at most $2$. This allows $\\gamma$ to be extended by a limit-inferior formula to a lower semicontinuous, conjugation-invariant norm on the $C^0$-closure, which is then the engine for the unboundedness of $d_{cc}$.","core_discovery":"The central claim is stated as Theorem 1.3: $(\\operatorname{Cont}_{0,c}(\\mathbb{R}^{2n+1}),\\tau_{C^0})$ has the Rokhlin property, while $(\\operatorname{Cont}_{0,c}(\\mathbb{R}^{2n}\\times S^1),\\tau_{C^0})$ does not. Here $\\operatorname{Cont}_{0,c}$ denotes the $C^0$-closure of the identity component of compactly supported contactomorphisms. For the positive half the paper constructs, from a countable $C^0$-dense sequence of contactomorphisms, a single contact homeomorphism $g$ whose conjugates approximate every element of the group; the construction works because any piece of support can be squeezed into a tiny ball, translated to a prescribed location, and later enlarged to approximate the target while the remaining pieces are collapsed. For the negative half, contact non-squeezing implies that some large ball must meet the support of every conjugate, and so no conjugate of a map supported in a small ball can approximate a half-rotation that moves every point. The same obstruction, upgraded by spectral invariants, proves that the conjugation metric $d_{cc}$ is unbounded on $\\mathbb{R}^{2n}\\times S^1$.","pith_inferences":["The explicit construction suggests a recipe: on any separable $C^0$-contact group where compactly supported squeezing is available for arbitrarily small balls, the same disjoint-copies argument should produce a Rokhlin element.","The metric $d_{cc}$ is a new invariant of the topological group structure; if it is unbounded on other manifolds, it can distinguish conjugation dynamics even where continuous conjugation-invariant observables are constant.","If a triangle inequality could be established for the spectral invariants used on prequantization spaces, the results on $\\mathbb{R}^{2n}\\times S^1$ would transfer to that setting; the paper's local boundedness result is the missing ingredient, and its Conjecture 1.23 predicts the same unboundedness of $d_{cc}$ there.","The positive half relies on separability of the $C^0$ group; on non-separable contact manifolds the construction yields only density in a separable subgroup, so whether the Rokhlin property holds in full generality is left open by this method."],"forward_implications":["If the dichotomy is correct, the two $C^0$-closed contactomorphism groups are topologically different: one has a dense conjugacy class and the other does not, so they cannot be isomorphic as topological groups.","The spectral norm $\\gamma$ is locally bounded in the $C^0$ topology, so its extension $\\widetilde{\\gamma}$ is a genuine conjugation-invariant norm on contact homeomorphisms of $\\mathbb{R}^{2n}\\times S^1$, giving a quantitative $C^0$ invariant.","Because $\\widetilde{\\gamma}$ is unbounded, the conjugation-connectedness metric $d_{cc}$ is an unbounded extended metric, which is a stronger obstruction than the mere failure of the Rokhlin property.","For prequantization spaces $W\\times S^1$ whose symplectic homology vanishes, and for $W\\times\\mathbb{R}^{2m}\\times S^1$ when the relevant symplectic capacity of the Liouville domain is finite, the same non-squeezing argument rules out the Rokhlin property.","A spectral invariant from Floer-theoretic methods is $C^0$-continuous near the identity, giving a route toward spectral norms on more general contact manifolds."],"supporting_citations":[{"why":"Supplies the contact non-squeezing theorem for $B(R)\\times S^1$, the engine of the negative Rokhlin results and their prequantization analogues.","marker":"[16]"},{"why":"Provides the sharp form of non-squeezing for all radii with $\\pi r^2\\ge 1$, which the paper cites as Theorem 2.7.","marker":"[8]"},{"why":"Gives the spectral invariants and non-squeezing results on prequantization spaces used in Theorems 1.19, 1.20 and 1.22.","marker":"[1]"},{"why":"Defines the integer-valued spectral norm $\\gamma$ and proves it is a conjugation-invariant norm; unboundedness of $\\gamma$ drives the quantitative obstructions.","marker":"[39]"},{"why":"Provides the generating-function spectral invariants and the lift formula used to compute $\\gamma$ in the auxiliary examples.","marker":"[38]"},{"why":"Supplies the background on conjugation-invariant norms and spectral norms, including the fact that no fine conjugation-invariant norm exists.","marker":"[40]"},{"why":"Gives the general bound between conjugation-invariant norms and the fragmentation norm that Theorem 1.13 refines.","marker":"[20]"},{"why":"Introduces weak conjugacy and the criterion that the metric $d_{cc}$ quantifies.","marker":"[29]"},{"why":"Shows that vanishing symplectic homology gives finiteness of the spectral capacity used to apply non-squeezing on $W\\times S^1$.","marker":"[4]"}],"fun_headline_variants":["Rokhlin property splits contact C^0 groups by manifold","Non-squeezing blocks dense conjugacy in contact C^0 group","Contact homeos: Rokhlin on R^{2n+1}, not on R^{2n}×S^1","Spectral norm extended to C^0 contact homeomorphisms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is contact non-squeezing in the $C^0$-closure: once a ball $B(R)\\times S^1$ has cross-sectional area $\\pi R^2$ exceeding $\\pi r^2\\ge 1$, no compactly supported contactomorphism — and no $C^0$ limit of such maps — can move it inside $B(r)\\times S^1$; if that fails, the obstructions on $\\mathbb{R}^{2n}\\times S^1$ and on prequantization spaces collapse.","fun_headline_variants_meta":{"raw":{"variants":["Rokhlin property splits contact C^0 groups by manifold","Non-squeezing blocks dense conjugacy in contact C^0 group","Contact homeos: Rokhlin on R^{2n+1}, not on R^{2n}×S^1","Spectral norm extended to C^0 contact homeomorphisms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1224,"prompt_tokens":905,"completion_tokens":319,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":231}},"tokens_in":521,"tokens_out":319,"duration_ms":3090,"temperature":1.0,"reasoning_tokens":231,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:33:00.603471+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit map $g$ built in Section 3.1 and test whether its conjugacy class is $C^0$-dense in $\\operatorname{Cont}_{0,c}(\\mathbb{R}^{2n+1})$: a single open $C^0$ ball containing no conjugate of $g$ would disprove the Rokhlin claim. For the negative half, find a $C^0$-limit contact homeomorphism $f$ on $\\mathbb{R}^{2n}\\times S^1$ and radii with $\\pi r^2\\ge 1<\\pi R^2$ such that $f(B(R)\\times S^1)\\subset B(r)\\times S^1$; that would break the non-squeezing theorem and with it the no-Rokhlin conclusion.","supporting_citations":[{"cited_title":"Eliashberg, S","cited_arxiv_id":null,"evidence_quote":"Supplies the contact non-squeezing theorem for $B(R)\\times S^1$, the engine of the negative Rokhlin results and their prequantization analogues."},{"cited_title":"Chiu, Nonsqueezing property of contact balls","cited_arxiv_id":null,"evidence_quote":"Provides the sharp form of non-squeezing for all radii with $\\pi r^2\\ge 1$, which the paper cites as Theorem 2.7."},{"cited_title":"Albers and W","cited_arxiv_id":null,"evidence_quote":"Gives the spectral invariants and non-squeezing results on prequantization spaces used in Theorems 1.19, 1.20 and 1.22."},{"cited_title":"Sandon, An integer-valued bi-invariant metric on the group of contactomorphisms of R2n× S1","cited_arxiv_id":null,"evidence_quote":"Defines the integer-valued spectral norm $\\gamma$ and proves it is a conjugation-invariant norm; unboundedness of $\\gamma$ drives the quantitative obstructions."},{"cited_title":"Sandon, Contact homology, capacity and non-squeezing in R2n × S1 via generating func- tions","cited_arxiv_id":null,"evidence_quote":"Provides the generating-function spectral invariants and the lift formula used to compute $\\gamma$ in the auxiliary examples."},{"cited_title":"Sandon, Bi-invariant metrics on the contactomorphism groups","cited_arxiv_id":null,"evidence_quote":"Supplies the background on conjugation-invariant norms and spectral norms, including the fact that no fine conjugation-invariant norm exists."},{"cited_title":"Fraser, L","cited_arxiv_id":null,"evidence_quote":"Gives the general bound between conjugation-invariant norms and the fragmentation norm that Theorem 1.13 refines."},{"cited_title":"Le Roux, S","cited_arxiv_id":null,"evidence_quote":"Introduces weak conjugacy and the criterion that the metric $d_{cc}$ quantifies."},{"cited_title":"Benedetti and J","cited_arxiv_id":null,"evidence_quote":"Shows that vanishing symplectic homology gives finiteness of the spectral capacity used to apply non-squeezing on $W\\times S^1$."}],"review_version":1}