{"id":"37df2325-24ed-4d1c-a776-f5e83fc94cb4","arxiv_id":"2607.03649","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Any closed orientable hypersurface in a contact manifold of dimension ≥5 is isotopic via a C^{0}-small isotopy to a C^{2}-robustly non-convex hypersurface.","lead":"Any closed orientable hypersurface in a contact manifold of dimension five or higher can be moved by an arbitrarily small continuous isotopy so that it becomes robustly non-convex. This shows convexity fails generically in a strong local sense and ties contact topology to robust chaotic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Thm 3) rests on three pillars: (i) the new Liouville obstruction for basic sets of the wrong index (Thm 10 / Cor. 11), (ii) C^0-local creation of a simple coindex-one cycle (Thm 14 / 4.1), and (iii) C^8-robustification of that cycle via a proper unfolding (Thm 15 / 6.21). Pillars (i) and (ii) are self-contained and elementary. Pillar (iii) invokes Li-Turaev Thm B, but only after the paper verifies every hypothesis of a proper unfolding inside the contact category (non-degeneracy Conditions 5.19–5.25, real center multipliers by Cor. 6.8, and the two-parameter family of Construction 6.23 whose splitting function and multiplier ratio satisfy Def. 5.30). The verification steps are concrete and use only contact Darboux charts, Legendrian/pre-Lagrangian transversality, and the standard neighborhood theorem for Reeb chords. Consequently the external black-box is applied correctly, the logical chain is complete, and the reader's ACCEPT verdict stands without adjustment.","tokens_in":45312,"tokens_out":600,"duration_ms":5130,"concrete_test":"Independently re-derive the signed-distance identity σ_g(L_{r,s})=s in Lemma 6.24 from Prop. 6.22(d) and the definition of the splitting function (Def. 5.29), confirming that the Reeb-chord length equals the metric distance under the weakly-compatible metric of Prop. 6.14; if the identity fails for the chosen Darboux ball B, the family is not a proper unfolding.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly flags the Li-Turaev application (Thm 6.21 / Construction 6.23) as the only non-local step, but the paper's own work closes the gap carefully: non-degeneracy of the coindex-one cycle is obtained by C^8-small plugs that enforce the four Conditions 5.19–5.25 via contact transversality (Lemmas 6.10, 6.18–6.20) and the contact-specific fact that center multipliers are real (Cor. 6.8); the subsequent 2-parameter family is built so that the splitting function equals the second parameter s while the multiplier ratio varies non-trivially with the first parameter r (Prop. 6.22 + Lem. 6.24). These verifications are local, explicit, and use only standard contact geometry. No hidden assumption or circularity appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that any closed orientable hypersurface in a contact manifold of dimension five or greater is isotopic, via an arbitrarily C^0-small isotopy, to a C^2-robustly non-convex hypersurface (Theorem 3). The argument proceeds in three steps: (i) hyperbolic basic sets of the characteristic foliation are Liouville, with sign determined by index (Theorem 3.15 / Theorem 10), yielding a heteroclinic obstruction to convexity (Corollary 3.16 / Corollary 11); (ii) simple coindex-one heterodimensional cycles of index (n-1,n) can be created by C^0-small contact Hamiltonian plugs (Theorem 4.1 / Theorem 14); (iii) any such cycle can be made non-degenerate by C^8-small plugs (Theorem 6.16) and then extended to a proper unfolding of characteristic foliations (Theorem 6.21 / Construction 6.23), so that Li-Turaev's theorem produces robust cycles and hence robust non-convexity (Theorem 15). The constructions are local and produce a robust deconvexifying plug.","tokens_in":45522,"tokens_out":1190,"duration_ms":8608,"significance":"If correct, the result is a strong counterpart to the Honda-Huang/Giroux C^0-density of convex hypersurfaces: non-convexity is not merely dense but can be forced robustly by arbitrarily small C^0 isotopies. It resolves a conjecture of the first author and places the convex/non-convex dichotomy squarely inside Bonatti's C^1-generic dynamics program via positive-negative heterodimensional cycles. Strengths include an independent ergodic divergence criterion for Liouville sets (Proposition 3.6, Lemma 3.7), explicit plug constructions that stay inside contact Hamiltonian manifolds, and a careful verification that the constructed 2-parameter family is a proper unfolding (Proposition 6.22 and Lemma 6.24). The work cleanly separates the new contact-geometric input from the external dynamical black box of Li-Turaev.","major_comments":[{"comment":"No load-bearing technical gaps were found. The application of Li-Turaev (Theorem 5.31) is the only non-local step, but the paper closes it carefully: non-degeneracy Conditions 5.19-5.25 are achieved by C^8-small contact Hamiltonian plugs using contact transversality (Lemmas 6.10, 6.18-6.20) and the contact fact that center multipliers are real (Corollary 6.8); the subsequent 2-parameter family is built so that the splitting function equals the second parameter s while the multiplier ratio varies non-trivially with the first parameter r (Proposition 6.22 + Lemma 6.24). These verifications are local and use only standard contact geometry.","section":null}],"minor_comments":[{"comment":"Throughout: many words are concatenated without spaces (e.g., 'Weprovethatanyclosedorientablehypersurface', 'arbitrarilyC0-smallisotopy', 'robustdeconvexifyingplug'). This appears to be a systematic typesetting artifact and should be corrected before publication.","section":null},{"comment":"Section 1, Definition 17 and Conjecture 18: the terminology 'positive-negative heterodimensional cycle' is introduced cleanly, but a short forward reference to the coindex-one restriction used in the body would help the reader see why higher-coindex examples remain open (Question 19).","section":null},{"comment":"Section 3.2, Proposition 3.6 (Div Criterion): the Hahn-Banach separation argument is standard but terse; a one-sentence reminder that the dual of the cohomology is precisely the space of signed invariant measures would improve readability for contact geometers.","section":null},{"comment":"Section 4.1, Lemma 4.3 (Orbit Creation): the construction via a Darboux chart and a momentum Hamiltonian is clear, but the size estimate |F+G|<=epsilon is only sketched; an explicit bound in terms of the support of the cutoff would make the C^0-smallness fully quantitative.","section":null},{"comment":"Section 5.3, Setup 5.18 and Conditions 5.19-5.25: the non-degeneracy package is long; a short summary table or diagram listing which condition controls which geometric feature (fragile transversality, strong foliations, saddle/focus ratio) would help navigation.","section":null},{"comment":"Section 6.4, Construction 6.23: the choice of two disjoint plugging domains U_G and U_H is essential for independence of the return map of C+ and the transition map of Gamma+; a one-line remark that the supports can be chosen arbitrarily small along the orbit and the fragile heteroclinic would make this transparent.","section":null},{"comment":"References: the arXiv numbers for Honda-Huang and for the first author's earlier blender paper are given; adding the published versions (if available) would be useful for the final version.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is ready for acceptance. The only external black box is Li-Turaev, which is correctly cited and carefully applied; the contact-geometric work is self-contained and of high quality. Fit for a top geometry/topology journal is excellent. No citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper delivers exactly what the abstract claims. The main theorem (any closed orientable hypersurface in a contact 5-or-higher manifold is C0-isotopic to a C2-robustly non-convex one) is a genuine upgrade of Chaidez's earlier existence result, and it sits as a clean counterpart to Honda-Huang density. The new pieces are the heteroclinic convexity obstruction (basic sets are Liouville by an ergodic divergence criterion, so a positive-negative cycle kills convexity) and the local cycle-creation plug that produces a simple coindex-one cycle; those are then fed into a carefully adapted proper unfolding so that Li-Turaev supplies the robust cycle.\n\nThe logical chain is transparent and local. Sections 3-4 give the obstruction and the explicit plug; Section 6 verifies non-degeneracy by contact Hamiltonian perturbations that enforce the four Li-Turaev conditions (center multipliers are automatically real by the contact linear algebra, and transversality is obtained by standard Legendrian/pre-Lagrangian moves). The 2-parameter family is built so the splitting function is literally the second parameter while the multiplier ratio moves with the first. No circularity, no free parameters, and the external input (Li-Turaev Thm B) is used exactly as stated.\n\nSoft spots are minor. The constructions are long and technical, and the paper invents a couple of convenient labels (\"robust deconvexifying plug,\" \"positive-negative cycle\"), but both are just packaging for objects already defined. The dependence on Li-Turaev is real, yet the contact-side verification that the family is a proper unfolding is self-contained and uses only standard tools. Citation pattern is appropriate: earlier blender work, Honda-Huang, and the dynamics literature are all in place.\n\nThis is for people working on higher-dimensional contact topology or on the interface with partially hyperbolic dynamics. It deserves a serious referee and will be cited. I would accept it for peer review without hesitation.","headline":"Clean strengthening of Chaidez's earlier non-convexity result: every hypersurface is C0-close to a C2-robustly non-convex one via a local deconvexifying plug built from Li-Turaev unfoldings.","tokens_in":46110,"tokens_out":527,"would_cite":true,"duration_ms":6223,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D10","37D30","57R17"],"pacs":[],"model":"grok-4.5","headline":"Any closed orientable hypersurface in a contact 5-or-higher manifold can be C0-small isotoped to one that is C2-robustly non-convex.","keywords":["convex hypersurface","characteristic foliation","heterodimensional cycle","contact Hamiltonian manifold","robust non-convexity","deconvexifying plug","Liouville set"],"falsifier":"Exhibit a closed orientable hypersurface in a contact five-manifold that remains convex under every C0-small isotopy, or verify that the constructed two-parameter family fails one of the multiplier-ratio or signed-distance regularity conditions of a proper unfolding.","tokens_in":46214,"feed_emoji":"🌀","tokens_out":730,"duration_ms":6075,"temperature":0.7,"pith_summary":"In contact manifolds of dimension five and higher, convex hypersurfaces are already known to be dense in the C0 topology. This paper shows that the complementary phenomenon is equally dense: every closed orientable hypersurface can be moved by an arbitrarily small continuous isotopy so that the resulting surface remains non-convex under every sufficiently small C2 perturbation. The authors obtain the result by building a local “deconvexifying plug.” First they prove that hyperbolic basic sets of the characteristic foliation are automatically Liouville, with sign controlled by index; a positive-negative heterodimensional cycle is therefore an immediate convexity obstruction. They then construct, by C0-small ambient deformations, a simple coindex-one cycle of the correct indices, make it non-degenerate, and unfold it into a two-parameter family to which a recent persistence theorem of Li-Turaev applies. The resulting robust cycle sits inside a plug that can be inserted anywhere, converting any hypersurface into a robustly non-convex one. The theorem therefore supplies a strong counterpart to the Honda-Huang-Giroux density theorem and frames the convex/non-convex dichotomy as a Palis-type alternative in contact topology.","feed_headline":"Every hypersurface can be C0-moved to a robustly non-convex one","feed_subtitle":"In contact dimension five and up, non-convexity becomes as dense as convexity itself","key_machinery":"The robust deconvexifying plug: a C0-small ambient deformation of a standard model region that inserts a non-degenerate coindex-one heterodimensional cycle and then unfolds it so that Li-Turaev persistence produces a C1-open set of nearby characteristic foliations still carrying a positive-negative cycle, thereby obstructing convexity.","core_discovery":"Any closed orientable hypersurface in a contact manifold of dimension five or greater is isotopic, by an arbitrarily C0-small isotopy, to a hypersurface that is C2-robustly non-convex. The obstruction is a robust positive-negative heterodimensional cycle of indices (n-1,n) in the characteristic foliation; such a cycle can be created and then robustified by a local deconvexifying plug.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Any hypersurface C0-moves to robustly non-convex in dim ≥5","Robust non-convexity dense for contact hypersurfaces via C0 isotopy","C0-small isotopy yields robustly non-convex hypersurface in high dim","Deconvexifying plug makes non-convexity robust for hypersurfaces","Heterodimensional cycles obstruct convexity robustly after C0 move"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"That the two-parameter family of contact Hamiltonian structures built by successive plug insertions really is a proper unfolding in the sense required by the Li-Turaev persistence theorem.","fun_headline_variants_meta":{"raw":{"variants":["Any hypersurface C0-moves to robustly non-convex in dim ≥5","Robust non-convexity dense for contact hypersurfaces via C0 isotopy","C0-small isotopy yields robustly non-convex hypersurface in high dim","Deconvexifying plug makes non-convexity robust for hypersurfaces","Heterodimensional cycles obstruct convexity robustly after C0 move"]},"model":"grok-4.5","effort":"low","cost_usd":0.003834,"raw_usage":{"total_tokens":1105,"prompt_tokens":651,"num_sources_used":0,"completion_tokens":105,"cost_in_usd_ticks":38340000,"prompt_tokens_details":{"text_tokens":651,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":349,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":651,"tokens_out":105,"duration_ms":3280,"temperature":1.0,"reasoning_tokens":349,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T00:56:01.939486+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a closed orientable hypersurface in a contact five-manifold that remains convex under every C0-small isotopy, or verify that the constructed two-parameter family fails one of the multiplier-ratio or signed-distance regularity conditions of a proper unfolding.","supporting_citations":[],"review_version":1}