{"id":"de7225f8-7a6e-4f8a-84c0-71afb28c4de2","arxiv_id":"2509.00750","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"First Laplacian eigenstates on flat 2-tori of any shape are orbitally stable for 2D Euler dynamics up to translations, including new stable sinusoidal flows on hexagonal tori.","lead":"This paper proves that every first nontrivial sine-cosine flow on any flat two-dimensional torus, including the hexagonal torus, is stable under the incompressible Euler equations up to translations. It gives the first known family of such stable flows on a hexagonal torus, where the proof requires counting the possible shapes of equally distributed vorticity fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved Burton criterion (Prop 2.7) is the load-bearing external step; the main theorem is only as solid as that citation.","rationale":"The reader identified Proposition 2.7 as the weakest point, and I agree. The rest of the proof—the variational characterization, finite-orbit analysis, and the standard continuity argument from an isolated invariant set—is internally sound. I verified the main algebraic steps of Lemma C.1 and the orbit characterizations, and the moment equations in (3.9) appear consistent with direct integration for m=2,3,4. The only part that is not independently substantiated from the text is Proposition 2.7. Since this is a citation to an external paper rather than a demonstrated flaw, I do not think it changes the verdict: the claim may well be correct, but the paper would be stronger if Prop 2.7 were proved or its hypotheses checked. I kept the verdict UNCHANGED because the reader's moderate confidence already reflects this gap, and the stress-test does not uncover a specific counterexample or internal inconsistency.","tokens_in":15294,"tokens_out":27252,"duration_ms":300681,"concrete_test":"Write out a full proof of Proposition 2.7 for the flat torus setting, following the argument in [18, Section 5]. In particular, check whether the stability conclusion for Lp-admissible maps (defined only by energy and rearrangement conservation) holds for all 1<p<∞, without assuming the rearrangement class is bounded or that the domain has a boundary. If the proof from [18] requires p≥2 or a boundary/compactness argument that fails on the torus, then the theorem as stated is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.6 relies on Proposition 2.7, a Burton-type stability criterion, whose proof is omitted: it is only stated to follow 'from a similar argument as in [18, Section 5].' This is a critical step because it converts the variational characterization (Prop 3.1) and the finiteness of orbits (Props 3.2–3.4) into the orbital stability of O_bar_omega. If the criterion in [18] requires hypotheses that are not satisfied on a flat 2-torus—for example, a boundary, a specific smoothness class, or p≥2—then the proof of Theorem 1.6 does not work. The paper also does not state those hypotheses explicitly, so the reader cannot verify applicability from the text. This is not a demonstrated error, but it is a genuine gap in self-containedness: the main theorem is conditional on an unproved external proposition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that every first nonzero Laplacian eigenstate on an arbitrary flat 2-torus is orbitally stable under the 2D Euler dynamics modulo translations. The proof follows the Burton stability framework: it first characterizes the set C_ωbar = R_ωbar ∩ E1 as the set of maximizers of kinetic energy within the rearrangement class (Prop. 3.1), then proves that C_ωbar contains only finitely many translational orbits (Props. 3.2–3.4), so the orbit O_ωbar is isolated in C_ωbar. A Burton-type stability criterion (Prop. 2.7) is invoked to pass from the stability of the maximizer set to the stability of the isolated orbit. The main novelty is the hexagonal case, where the finiteness argument reduces to a polynomial system in the squared amplitudes.","tokens_in":15539,"tokens_out":13086,"duration_ms":150094,"significance":"If the main theorem is correct, it is a meaningful advance: it extends the known orbital-stability results for rectangular and square tori to arbitrary flat tori, and it provides the first family of orbitally stable sinusoidal Euler flows on a hexagonal torus. The proof is largely self-contained, uses no fitted parameters, and the finite-orbit counting for the six-dimensional eigenspace is an interesting piece of algebraic analysis. The main risks are the unproved external stability criterion (Prop. 2.7) and the unchecked Maple computation in Eq. (3.9); neither is currently presented in verifiable form, but both appear fixable within the manuscript's scope.","major_comments":[{"comment":"This Burton-type stability criterion is the unique external input that converts the variational characterization and finite-orbit isolation into orbital stability. Its proof is omitted (\"follows from a similar argument as in [18, Section 5]\") and the hypotheses are only implicit. Since Theorem 1.6 inherits all of its stability conclusion from this proposition, the author should either prove it in the paper or state the exact theorem from [18] with its full hypotheses and verify that they hold for a boundaryless flat torus and for all 1<p<∞.","section":"Section 2.3, Proposition 2.7"},{"comment":"The four moment equations are asserted to have been computed with Maple, but neither the computation nor the code/output is shown. These equations are load-bearing: they bound the number of triples (A1,A2,A3) to at most six via system (3.10) and hence bound the number of translational orbits in the 6D case. Without a verifiable derivation, Proposition 3.4 is not checkable. Please include an explicit integration or an appendix with the computer-algebra transcript.","section":"Section 3.2.3, Eq. (3.9)"}],"minor_comments":[{"comment":"In the sentence after the eigenvalue formula, \"4π|k|^2\" should be \"4π^2|k|^2\".","section":"Section 2.1, proof of Lemma 2.1"},{"comment":"There are unmatched parentheses after \"(0,0)\" in both displays; they should read \"\\{(0,0)\\}\".","section":"Equations (2.1)–(2.2)"},{"comment":"Typo: \"and and\" appears in the sentence introducing the orthonormal basis.","section":"Section 2.1, after Lemma 2.1"},{"comment":"The displayed equivalence is not a valid reformulation of Theorem 1.6. The reverse implication is generally false: a trajectory starting far from O_ωbar could later be within ε of it. This should be a single implication (or the wording should be corrected to avoid claiming equivalence).","section":"Remark 1.8, Eq. (1.16)"},{"comment":"The zero-mean condition on the vorticity is inherited from the mean-zero velocity assumption via Lemma A.3; it would help to state this explicitly when defining Lp-admissible maps so that the rearrangements are understood to be in the mean-zero subspace.","section":"Definition 1.5 and Section 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on the author's own prior results, particularly [18] for Proposition 2.7 and [19] for the rectangular/square cases. The missing proof of Prop. 2.7 is the main risk to the central claim. The Maple computation in Eq. (3.9) should also be made available, since it is essential to the hexagonal case. The false equivalence in Remark 1.8 should be corrected in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [name],\n\nThe hexagonal torus case is the actual news here, not the arbitrary-shape label. The paper gives the first orbitally stable sinusoidal flows on a hexagonal torus, and the proof is a clean chain: variational characterization via the energy-enstrophy inequality, finite count of translational orbits in the equimeasurable class, then a Burton-type stability criterion. The 2D and 4D cases are natural extensions of Wang–Zuo [19], but the 6D case is genuinely new, and the polynomial-system argument in Appendix C is a nice piece of elementary algebra. Lemma B.1, the characterization of translational orbits in the 6D eigenspace, is exactly the kind of detail that makes the whole thing work. The rigidity result in Section 4 is a bonus and it is fine.\n\nThe main soft spot is Proposition 2.7. It is the step that turns 'maximizers of kinetic energy in a rearrangement class are stable' into the final orbital stability of O_bar_omega, but the proof is just 'similar argument as in [18, Section 5]'. That is a citation, not a proof. If Burton's criterion requires hypotheses that are not met on a flat torus, or a version of the criterion that is not stated precisely, the main theorem does not go through. I don't think it is wrong—the paper's own Proposition 3.1 produces the right maximizer set—but a referee should ask for either a self-contained proof or an exact statement of the theorem in [18] that covers this case. The Maple computation in (3.9) is a smaller version of the same issue: the integrals are asserted, not shown. That is likely fine, but the author should make the computation available or at least include the key intermediate forms.\n\nNone of this undercuts the core result. The variational step is correct, the orbit counts are correct under the stated assumptions, and the proof of Theorem 1.6 is a standard isolation argument once Prop 2.7 is granted. The paper is honest about what is new: it explicitly says rectangular/square is in [19] and the contribution is the general case and in particular the hexagonal one. That is a localized but real advance.\n\nI would send this to a serious referee. It is a solid, focused paper in the mathematics of 2D Euler stability, and a referee can sort out the Prop 2.7 question without redoing the whole paper. If the author patches that citation gap and ships the Maple details, it is publishable as is.","headline":"The hexagonal torus case is the real news; the argument is clean but one imported Burton criterion is under-specified.","tokens_in":15953,"tokens_out":4285,"would_cite":true,"duration_ms":41157,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B35","35Q31","35P15","76B47"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every first Laplacian eigenstate on a flat 2-torus is orbitally stable up to translation for the incompressible Euler equation.","keywords":["2D Euler equation","orbital stability","Laplacian eigenstates","flat torus","hexagonal torus","rearrangement class","energy-enstrophy inequality","translational orbits"],"falsifier":"On the hexagonal torus with the eigenstate (1.13), a spectral Euler simulation starting from that state plus a small, localized perturbation should keep the L^2 distance to every translation of the eigenstate bounded by a small constant for all computed times; if a simulation that conserves energy and preserves the level-set distribution shows the flow drifting to another first-eigenstate orbit, Theorem 1.6 is false. Algebraically, one can evaluate the moment system (3.8)–(3.9) symbolically for a concrete choice of amplitudes and check whether the asserted bound of at most twelve translation o","tokens_in":15234,"feed_emoji":"🌀","tokens_out":11686,"duration_ms":139631,"temperature":0.7,"pith_summary":"On any flat 2-torus, the lowest-frequency sinusoidal vorticity fields—the first Laplacian eigenstates—are stable up to translation under the 2D Euler dynamics. The paper proves that if a vorticity distribution starts close to such a steady state in any L^p norm, it stays close, for all time, to some translation of that state. This extends earlier results on rectangular and square tori to every lattice shape, including the hexagonal torus, where the first eigenspace is six-dimensional and the streamlines have a richer vortex structure. The proof works by identifying the equimeasurable part of the first eigenspace with the kinetic-energy maximizers over the rearrangement class, then showing this class consists of finitely many translation orbits. A stability criterion then turns that finiteness into orbital stability of a single orbit.","feed_headline":"First Laplacian eigenstates on any flat torus are stable","feed_subtitle":"Hexagonal-torus sinusoidal flows gain their first known stable examples via a finite-orbit proof.","key_machinery":"The load-bearing mechanism is the equality M_ω̄ = C_ω̄: the maximizers of the kinetic energy functional E(f) = (1/2)∫ f G f dx over the rearrangement class R_ω̄ are exactly the first-eigenstate fields equimeasurable with ω̄. This is derived from the energy–enstrophy inequality, whose equality case is exactly E_1, and it converts the two conserved Euler invariants into a complete characterization of C_ω̄. The second mechanism is the finiteness of translation orbits inside C_ω̄, proved case by case: in dimension two the class is a single orbit; in dimension four equality of L^∞ and L^2 norms leaves at most two amplitude pairs; in dimension six the moment equations (3.8)–(3.9) reduce, via Lemma","core_discovery":"The paper's central claim is Theorem 1.6: for any first Laplacian eigenstate ω̄ on an arbitrary flat 2-torus and any 1<p<∞, for every ε>0 there is δ>0 such that every L^p-admissible map—any time evolution preserving kinetic energy and the vorticity distribution—starting within δ of ω̄ in L^p remains within ε of some translation of ω̄ at every later time. The proof proceeds in three steps. First, the energy–enstrophy inequality identifies the class C_ω̄ = R_ω̄ ∩ E_1, the first-eigenstate fields with the same vorticity distribution as ω̄, with the set of maximizers of kinetic energy over that distribution class. Second, a case analysis shows C_ω̄ is a finite union of translation orbits: one or","pith_inferences":["The six-dimensional count of at most twelve translation orbits is probably crude; an explicit symbolic evaluation of the moment integrals could sharpen the bound or expose hidden identifications among the six polynomial roots.","The same three-step scheme—variational characterization, finite-orbit analysis, then a stability criterion—is stated to work for other symmetric domains, so the result is a template rather than a one-off argument; a general theorem awaits a unified formulation.","The dynamics inside C_ω̄ is left unspecified, so a natural next test is whether actual Euler flow on a hexagonal torus can visit distinct translation orbits; the present theorem only says each orbit is isolated and stable, not which orbit the flow chooses."],"forward_implications":["Every first Laplacian eigenstate on a flat 2-torus—rectangular, square, hexagonal, or any other lattice shape—is stable modulo translations for 2D Euler in every L^p norm with p>1.","The hexagonal torus now has a family of orbitally stable sinusoidal steady states, the first such known examples in that setting.","The stability conclusion holds for every L^p-admissible map, not only for genuine Euler solutions, so the argument is insensitive to the finer details of how the flow map is constructed.","The hexagonal examples include stable states with saddle points, making them suitable base flows for constructing solutions with superlinear vorticity-gradient growth; Corollary 1.9 supplies such states.","The auxiliary rigidity result shows that on any flat 2-torus, a steady solution of the semilinear problem with φ′≤λ_1 must itself be a first eigenstate."],"supporting_citations":[{"why":"Proved the same theorem on rectangular and square tori; the present proof follows its three-step strategy and its L^p-admissible formulation.","marker":"[19]"},{"why":"Contains the proof of the stability criterion restated here as Proposition 2.7, on which the final step of the argument relies.","marker":"[18]"},{"why":"Original source for the principle that kinetic-energy maximizers over a rearrangement class are nonlinearly stable; Proposition 2.7 is the variant used here.","marker":"[5]"}],"fun_headline_variants":["Euler stability proven for all first eigenstates on flat tori","Hexagonal torus gets first stable sinusoidal flows","Orbital stability extends to every flat 2-torus","First eigenstates stable under Euler on any flat torus"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the stability criterion stated as Proposition 2.7 holds for arbitrary L^p-admissible maps under exactly the hypotheses listed; the paper does not prove it and refers to a similar argument in [18, Section 5], so if the criterion needs extra assumptions the final step from stability of the maximizer set to stability of a translation orbit fails; a secondary asserted input is the computer-algebra evaluation of the moment integrals behind (3.9), w","fun_headline_variants_meta":{"raw":{"variants":["Euler stability proven for all first eigenstates on flat tori","Hexagonal torus gets first stable sinusoidal flows","Orbital stability extends to every flat 2-torus","First eigenstates stable under Euler on any flat torus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1363,"prompt_tokens":718,"completion_tokens":645,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":577}},"tokens_in":462,"tokens_out":645,"duration_ms":7636,"temperature":1.0,"reasoning_tokens":577,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:16:33.048952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the hexagonal torus with the eigenstate (1.13), a spectral Euler simulation starting from that state plus a small, localized perturbation should keep the L^2 distance to every translation of the eigenstate bounded by a small constant for all computed times; if a simulation that conserves energy and preserves the level-set distribution shows the flow drifting to another first-eigenstate orbit, Theorem 1.6 is false. Algebraically, one can evaluate the moment system (3.8)–(3.9) symbolically for a concrete choice of amplitudes and check whether the asserted bound of at most twelve translation o","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved the same theorem on rectangular and square tori; the present proof follows its three-step strategy and its L^p-admissible formulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the proof of the stability criterion restated here as Proposition 2.7, on which the final step of the argument relies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Original source for the principle that kinetic-energy maximizers over a rearrangement class are nonlinearly stable; Proposition 2.7 is the variant used here."}],"review_version":1}