{"id":"523ddfd8-4668-4a21-b68e-d1670669ba57","arxiv_id":"2502.01470","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Existence of smooth 3D Euler solutions concentrating along N nearly parallel helical vortex filaments whose rotation speed matches the Klein-Majda-Damodaran model to leading order.","lead":"This paper proves that certain swirling fluid motions in three dimensions really do follow a long-used approximation for thin vortex filaments. The constructed solutions rotate at exactly the speed the approximation predicts, for families of N interlocking helical filaments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Speed-matching step (6.31)-(6.32) rests on sketched cancellations; if any hidden term of order εμ√|logε| survives, α_ε is not pinned to the KMD value and the model-matching claim collapses.","rationale":"The reader's weakest_assumption correctly identifies the speed-matching step as the point where the KMD match is decided. My own reading of the estimates suggests the asserted orders are plausible: the \\tilde E contribution is computed from (4.9) and yields the displayed term; the nonlinear terms are controlled by (εμ)^2|logε| which is o(εμ√|logε|); the outer-correction term is O(ε^{1+σ}), also negligible; the boundary term is O((εμ)^m|logε|^{m/2}) with m > 2, negligible. So the concern is not that the identities are false, but that they are not proved in the paper. Because the proof is a continuation argument with a final scalar equation for α, the entire justification of the KMD rotation speed rests on (6.31)-(6.32). A fully detailed verification is therefore a necessary condition for acceptance. The secondary issue of α_ε > 0 conflicting with the expansion for parameters with 1/h^2 < (N-1)/r^2 is a genuine inconsistency in the theorem statement, but it is easily repaired by dropping the positivity condition and does not affect the construction. Since the reader's conditional verdict already reflects these concerns, my stress-test does not move the verdict.","tokens_in":36817,"tokens_out":21043,"duration_ms":171432,"concrete_test":"Re-derive (6.31)-(6.32) by a complete expansion of ∫ H Z1 and the boundary term. Specifically: (i) substitute the constructed φ = φ1 + d0φ2o into the inner equation; (ii) list every term in H that is not explicitly \\tilde E; (iii) compute each integral against Z1 over Bρ using the known decay of φ1 and φ2o; and (iv) verify that the sum of all terms proportional to εμ√|logε| equals exactly 2εμ√|logε|(r/h^2 - (N-1)/r - αr/2)∫ U y1 Z1, with all others o(εμ√|logε|). If the computation exposes an additional term of order εμ√|logε|, Theorem 1's speed-matching fails; if it confirms the cancellation, the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1 is that the constructed solution rotates with α_ε = 2(1/h^2 - (N-1)/r^2) + O(log|logε|/|logε|), matching the KMD model. This value is obtained at the very end of the proof by solving d1[φ,ϕ,α] = 0. The key input is the pair of identities (6.31)-(6.32). Equation (6.31) asserts that the only contribution of order εμ√|logε| to ∫ H Z1 comes from the explicit term involving r/h^2 - (N-1)/r - αr/2; equation (6.32) asserts all remaining terms, including the boundary integral from integration by parts and the B_ε[φ] + b_ε φ terms, are of order (εμ)^{2-a} with a > 0. The proof of (6.31) explicitly computes only the \\tilde E part of H; the nonlinearity and outer-correction contributions are dismissed in one sentence as (εμ)^{2-a}Y(α), and the proof of (6.32) bounds the boundary term by (εμ√|logε|)^m and the B_ε/b_ε terms by (εμ)^2|logε|, which are indeed smaller than (εμ)^{2-a}, but the displayed order is then used as if it were the sharp size. The concern is not that a specific estimate is obviously false, but that the pivotal cancellation is asserted rather than demonstrated. If a hidden contribution of order εμ√|logε| survives, for instance from the interaction of the inner correction φ1 with the boundary or from a non-vanishing y2-odd component not killed by (3.28), then the equation d1 = 0 would determine α only up to an O(1) error, so α_ε would not agree with the KMD rotation speed at leading order and the theorem would not justify the model.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This article constructs exact smooth solutions of the 3D incompressible Euler equations whose vorticity is concentrated in thin tubular neighborhoods of N nearly parallel circular helices of radius r/sqrt(|log epsilon|) and pitch 2*pi*h, rotating rigidly with angular speed alpha_epsilon = 2(1/h^2 - (N-1)/r^2) + O(log|log epsilon|/|log epsilon|), exactly the speed predicted by the formal Klein-Majda-Damodaran (KMD) model for its N-helix central configuration. The proof reduces 3D Euler via helical symmetry to the 2D elliptic problem (1.8)-(1.11), assembles an approximate solution from epsilon-regularized Green functions of L = -div(K grad) (Sections 2-4), and produces an exact solution by an inner-outer gluing scheme (Sections 5-6). The speed is selected by the solvability condition d1[phi,phi,alpha] = 0, whose leading term is claimed in (6.31)-(6.32) to vanish exactly at the KMD value. Theorem 2 states an analogous result for N helices surrounding a central straight filament; its proof is omitted. The paper explicitly leaves the justification of the KMD model for general initial data open.","tokens_in":37157,"tokens_out":53969,"duration_ms":440279,"significance":"If fully demonstrated, the central claim is a significant step in the rigorous theory of vortex filaments: it constructs genuine Euler solutions realizing the KMD helical central configurations at the near-parallel scale 1/sqrt(|log epsilon|), going beyond the earlier fixed-distance helical constructions of [16,21]. Credit is due for the honest scope statement (the full KMD justification is explicitly left open), for the transparent use of the helical symmetry reduction, and for the fact that the KMD speed emerges from the solvability analysis as a computed quantity rather than an imposed one: the constant 2(1/h^2 - (N-1)/r^2) is traced back to the explicit interaction coefficients in (4.10)-(4.12) and (6.31), so the matching is not circular. The paper is nevertheless conditional in two respects: the pivotal verification (6.31)-(6.32) that pins alpha to the KMD value is currently asserted at the level of 'we claim' with one-line dismissals, and Theorem 2, advertised in the abstract as part of the rigorous justification, is not proved. Both points are, in my assessment, repairable within the scope of the manuscript.","major_comments":[{"comment":"The advertised KMD matching alpha_epsilon = 2(1/h^2 - (N-1)/r^2) + O(log|log epsilon|/|log epsilon|) rests entirely on the two assertions (6.31) and (6.32), and as written these are not proved. In the proof of (6.31) only the E_tilde = epsilon^2 mu^2 S(psi*) contribution is actually computed; the remaining contributions to I = integral_{B_rho} H Z_1 (the quadratic term epsilon^2 mu^2 N(sum eta_i Phi_i + phi) and the linearized coupling epsilon^2 mu^2 F'(psi* - alpha|log epsilon||x|^2/2) phi = (e^Gamma + b_epsilon) phi) are dismissed in one sentence. The displayed bound |epsilon^2 mu^2 F'(...) phi| <= C U(y)|phi| together with (6.9) (|phi| ~ epsilon^{1+sigma} in the inner region) yields only integral U|phi||Z_1| = O(epsilon^{1+sigma}), which is larger than the asserted remainder (epsilon mu)^{2-a}Y(alpha) for the allowed parameter ranges (a in (0,1/2) in (6.32), sigma < a); similarly, the quadratic term is controlled only at order (epsilon mu)^2|log epsilon|. These sizes are still o(epsilon mu sqrt(|log epsilon|)), so the conclusion of (6.31) is plausible, but the asserted order is not what the displayed estimates deliver. More consequential: the final 'by continuity' step requires the remainder in (6.31)-(6.32) to be uniform in alpha, and requires the root of d1[phi(alpha),phi(alpha),alpha] = 0 to be localizable with accuracy log|log epsilon|/|log epsilon|; the alpha-dependence of the fixed points phi_1(alpha) (from (6.19), Proposition 6.2) and phi(alpha) (from (6.7), Proposition 6.1) is never fed into the error budget, and the only Lipschitz-in-alpha estimate available, (6.11), is not used here. I checked the terms I initially suspected most - the boundary integral in (6.32) (order (epsilon mu sqrt(|log epsilon|))^m with m > 2) and the B_epsilon-hat[phi_1] contribution (order (epsilon mu)^2|log epsilon|) - and those do come out at the required higher orders provided (6.22) and (6.33) hold; the gap is in the statements above. Request: rewrite the proofs of (6.31)-(6.32) with explicit integrals for every term of H, explicit uniform-in-alpha remainder bounds, and a rigorous implicit-function/sign-change argument for the resulting equation in alpha; alternatively, soften the theorem to assert only existence of a rotation speed in a bounded range.","section":"Section 6, Proof of Theorem 1, Eqs. (6.31)-(6.32)"},{"comment":"Theorem 2 - one of the two configurations in the abstract's claim 'we rigorously justify this model for two configurations' - is stated and never proved; the text says the proof 'is a relatively straightforward adaptation of the proof of Theorem 1' and is omitted. This is not merely a presentation issue: the N+1 configuration requires structurally new ingredients. The central filament at x' = 0 sits at the center of the shrinking polygon, so the interaction sums in (3.21) and (3.22), the choice of mu, and the coefficient A(alpha) in (4.10) must be recomputed with the additional center contribution; that contribution is exactly what changes the rotation speed from -2(N-1)/r^2 to -2(N+1)/r^2. In addition, the dihedral-symmetry analysis of Section 3 and Appendix A must now handle a rotationally invariant contribution at the origin, and the regularity of the approximate stream function near the new singular point at 0 needs independent checking. A referee cannot verify the 'straightforward adaptation' from the manuscript. Please either (i) provide the full proof, (ii) provide a detailed outline identifying every lemma whose statement changes and how, or (iii) state the N+1 configuration only as a remark with the caveat that the proof is omitted, and align the abstract and introduction accordingly.","section":"Section 1, Theorem 2, and abstract"}],"minor_comments":[{"comment":"The statement 'There exist alpha_epsilon > 0' is incompatible with the displayed asymptotics for parameters with 2(1/h^2 - (N-1)/r^2) < 0 (e.g., N = 3, r = 1, h = 10), and the claimed stationary case r^2 = (N-1)h^2 + o(1) forces alpha_epsilon -> 0, contradicting strict positivity. Either add the hypothesis 2(1/h^2 - (N-1)/r^2) > 0, or allow real-valued alpha_epsilon and state the stationary case in a limiting sense.","section":"Section 1, statement of Theorem 1"},{"comment":"The displayed integral 'integral_{B_rho} E_tilde(y) y_1 Z_1 dy' in the proof of (6.31) is not the quantity defined in I = integral_{B_rho} H Z_1; presumably the factor y_1 refers to the leading term of E_tilde, but the display is misleading and should be corrected. Also, the first-order term in the statement of (4.9) is printed as c_1 Gamma(y) + A(alpha), whereas the derivation in the proof of Proposition 4.1 obtains c_1 Gamma(y) + R_epsilon(3h^2+R_epsilon^2)/(2h(h^2+R_epsilon^2)^{3/2}) + A_1(alpha); the definitions of A(alpha) in (4.10) and A_1(alpha) in (4.11)-(4.12) should be aligned.","section":"Section 6, proof of (6.31); Section 4, Proposition 4.1"},{"comment":"The letter a is used for different objects with incompatible ranges: a in (3/4,1) in Proposition 4.1 and (4.8), a in (0,1/2) in (6.32), and the same letter enters the condition 2 < m < 2 + a in Proposition 6.2. Use distinct letters, since the order assertions in (6.32) and the condition on m depend on which 'a' is meant.","section":"Section 4 (Proposition 4.1), Section 6 (Eq. (6.32) and Proposition 6.2)"},{"comment":"The estimate '|integral_{B_rho} B_epsilon[phi] Z_1 dy| <= C epsilon mu sqrt(|log epsilon|) [||phi_1||_* + |d_0|]' overshoots the natural bound by a factor |log epsilon|: with R_epsilon = r/sqrt(|log epsilon|), the coefficients displayed in (6.33) are of order epsilon mu / sqrt(|log epsilon|), which after using (6.22) gives order (epsilon mu)^2. The overestimate is harmless for the conclusion, but the displayed bound should be corrected.","section":"Section 6, proof of (6.32)"},{"comment":"The sentence 'Our result encompasses stationary solutions ...' should specify in which sense the stationary case is obtained, given that Theorem 1 requires alpha_epsilon > 0 while the stationary condition r^2 = (N-1)h^2 + o(1) forces alpha_epsilon -> 0.","section":"Section 1, paragraph after Theorem 1"}],"recommendation":"major_revision","confidential_remarks":"The technical core relies heavily on the authors' companion paper [21] (Guerra-Musso, AIHPC 2024): Proposition 2.2 is proved only by a sketch 'analogous to Proposition 2.2 in [21]', Lemma 6.1 is quoted from [21], and the overall inner-outer gluing architecture follows [21] with the new ingredients being the KMD scale, the interaction coefficient A(alpha), and the speed selection. My main worry, beyond the two points in the report, is that the verification of (6.31)-(6.32) may be routine in the authors' private notes but is not checkable by a reader in the present form; I would recommend that the editor ask for a genuinely complete treatment of that step, including the uniformity in alpha needed for the implicit-function conclusion. If the authors prefer to demote Theorem 2 to a remark with an omitted proof, that is acceptable to me, but the abstract must then be adjusted. The fit with the journal is good (math.AP, vortex filaments, singular perturbation gluing)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is the first rigorous Euler solution that realizes the KMD nearly-parallel filament model at its natural scale for the regular-polygon configurations. The vorticity is concentrated along N helices with radius and mutual separation of order 1/sqrt(|log eps|), and the rotation speed is pinned to the KMD value 2(1/h^2 - (N-1)/r^2) at leading order. That is a real step beyond [16], [5], and the authors' own [21]. The reduction to the 2D elliptic problem (1.11), the approximate solution, and the inner-outer gluing are coherent and in the standard del Pino-Musso-Wei style. The dihedral and evenness symmetries are used effectively.\n\nThe soft spots are real but not fatal. The main one is the speed-matching step, identities (6.31)-(6.32). These pin alpha to the KMD value: they say the only order-eps mu sqrt(|log eps|) contribution to the projected error is the explicit term linear in alpha, and everything else is lower order. The leading term from the error is computed, but the nonlinearity, outer correction, boundary term, and B_eps/b_eps contributions are dismissed with order bounds and symmetry arguments that are not written out. The orders look plausible, and I do not see an obvious counter-term, but for a step this central the paper should show the cancellations explicitly. A referee should demand this.\n\nTheorem 2 is stated without proof, although the abstract advertises both configurations. If it is a routine adaptation, say so and sketch the differences. As written, the abstract overclaims. Also, the strict positivity alpha_epsilon > 0 conflicts with the expansion when the leading KMD speed is negative; restrict the parameter range or drop positivity.\n\nThe reliance on [21] for technical lemmas is acceptable since [21] is published, but the paper is not self-contained.\n\nBottom line: the structural logic holds up and the result is significant. The load-bearing identities need a complete proof before acceptance. This deserves serious peer review.","headline":"First rigorous Euler solutions at the KMD scale for the N-helix configurations; construction credible, but the speed-pinning step (6.31)-(6.32) is sketched too briefly for the main claim to be considered fully checked.","tokens_in":37814,"tokens_out":3743,"would_cite":true,"duration_ms":34761,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","76B47","35J61"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any prescribed radius and pitch, the three-dimensional Euler equations admit smooth solutions whose vorticity concentrates along N helices rotating at the speed predicted by the Klein–Majda–Damodaran model.","keywords":["nearly parallel vortex filaments","helical vortex filaments","3D Euler equations","Klein–Majda–Damodaran model","gluing method","elliptic singular perturbation","vortex filament conjecture","central configurations"],"falsifier":"A direct computation of the projected integral $d_1=\\gamma_1\\int_{B_\\rho}(H+B_\\varepsilon\\varphi+b_\\varepsilon\\varphi)Z_1$ for a small case such as $N=2$ or $N=3$, using the fixed-point $\\varphi$ and $\\phi$, would settle the speed-matching claim: if a contribution of order $\\varepsilon\\mu\\sqrt{|\\log\\varepsilon|}$ survives beyond the leading term in (6.31), or if the $y_2$-odd sums in (3.28) fail to vanish at that order, then $\\alpha_\\varepsilon$ would acquire an undesired correction and the KMD matching in Theorem 1 would collapse.","tokens_in":36452,"feed_emoji":"🌀","tokens_out":5370,"duration_ms":52622,"temperature":0.7,"pith_summary":"This paper proves that a well-known formal model for how nearly parallel vortex filaments interact is actually realized, in special symmetric configurations, by exact solutions of the three-dimensional incompressible Euler equations. The authors construct smooth Euler flows whose vorticity is concentrated, in a small-core limit, along N circular helices arranged as a rotating regular polygon, and they show the rotation speed of this configuration matches the speed predicted by the Klein–Majda–Damodaran filament model at leading order. A second theorem does the same for N helices surrounding a central straight filament. If correct, this turns the model's rotating central configurations from an asymptotic guess into a proven existence statement for the underlying PDE.","feed_headline":"N nearly parallel helical vortices are exact Euler flows","feed_subtitle":"Rigorous construction confirms the Klein–Majda–Damodaran rotation speed for polygon and centered configurations.","key_machinery":"The argument is carried by the screw-symmetry reduction: a helical Euler solution is encoded in a scalar planar stream function $\\psi$ solving $\\nabla\\cdot(K\\nabla\\psi)+F(\\psi-\\frac{\\alpha}{2}|\\log\\varepsilon||x|^2)=0$, where $K$ is the explicit $2\\times 2$ matrix coming from the helical change of variables. Around each filament the operator $L=-\\nabla\\cdot(K\\nabla)$ is a small perturbation of the Laplacian, and the approximate stream function is assembled from scaled copies of the Liouville-profile Green's function $\\Gamma(y)=\\log(8/(1+|y|^2)^2)$, centered at the polygon vertices $P_j$, with the parameter $\\mu$ and the speed $\\alpha$ chosen to cancel the $\\log|\\log\\varepsilon|$ terms. The load-bearing identity is the projected inner equation: the obstruction $d_1$ computed against the kernel direction $Z_1=\\partial_{y_1}\\Gamma$ must vanish, and equations (6.31)-(6.32) pin $\\alpha$ to the KMD value; the $y_2$-odd components cancel by dihedral symmetry, so only the radial direction matters. This projected-kernel mechanism is what converts a merely approximate solution into one with the model's exact leading-order rotation speed.","core_discovery":"Theorem 1 asserts that for any radius $r>0$ and pitch parameter $h\\neq 0$ there is a rotation speed $\\alpha_\\varepsilon>0$ and a smooth Euler vorticity $\\vec\\omega_\\varepsilon$ such that $\\vec\\omega_\\varepsilon$ converges in distribution to $8\\pi$ times the sum of Dirac masses along the tangent directions of the N helices $\\gamma^\\varepsilon_j(s,t)$ of radius $r/\\sqrt{|\\log\\varepsilon|}$ and pitch $2\\pi h$, with $\\alpha_\\varepsilon = 2(1/h^2 - (N-1)/r^2)+O(\\log|\\log\\varepsilon|/|\\log\\varepsilon|)$. Theorem 2 gives the analogous $N+1$ configuration: N helices rotating around a straight central filament, with the adjusted speed $2(1/h^2 - (N+1)/r^2)$. The proof constructs the stream function as a superposition of regularized Green's functions for the helical-symmetry-reduced elliptic operator, then solves the resulting equation by an inner-outer gluing scheme; the predicted speed is selected by requiring the projected inner problem's obstruction $d_1$ to vanish. If the $\\alpha_\\varepsilon$ expansion holds, these are, to the authors' knowledge, the first rigorous Euler realizations of the KMD model's rotating polygon and centered configurations at the model's own scaling.","pith_inferences":["Editorial: the same gluing strategy is likely to apply to other symmetric solutions of the KMD equation, such as nested regular polygons, whenever the projected inner problem has the same kind of symmetry cancellation.","Editorial: the explicit $O(\\log|\\log\\varepsilon|/|\\log\\varepsilon|)$ correction to $\\alpha_\\varepsilon$ is a concrete prediction that numerical simulations of helical vortex lattices could, in principle, test at finite $\\varepsilon$.","Editorial: the paper's construction gives no information about stability or about how generic initial data evolve; it proves existence of special solutions, leaving the dynamical vortex-filament conjecture untouched."],"forward_implications":["The rotating regular-polygon and centered configurations of the Klein–Majda–Damodaran model are not merely formal: they are realized at leading order by genuine Euler solutions.","The stationary case $r^2=(N-1)h^2+O(\\log|\\log\\varepsilon|/|\\log\\varepsilon|)$ yields stationary vorticity concentrations along N helices.","The deformation of each filament and the separation between filaments both scale as $1/\\sqrt{|\\log\\varepsilon|}$, exactly the scaling regime assumed by the KMD model.","A general derivation of the KMD model from the Euler equations for arbitrary filament configurations remains open; the paper settles two symmetric configurations, not the full conjecture."],"supporting_citations":[{"why":"Supplies the formal Klein–Majda–Damodaran asymptotic model whose rotating central configurations are being rigorously realized.","marker":"[29]"},{"why":"Provides the elliptic singular-perturbation framework and the traveling-helix construction that the present gluing scheme adapts.","marker":"[16]"},{"why":"Gives the preceding cluster-of-helices construction, including the analysis of the operator $L$ and the linear solver used here.","marker":"[21]"},{"why":"Identifies the three-dimensional kernel of the linearized Liouville operator $\\Delta+e^\\Gamma$, which drives the projected inner problem.","marker":"[3]"},{"why":"Shows the analogous nearly-parallel filament model is rigorously derivable from Gross–Pitaevskii, the comparison that frames the open Euler problem.","marker":"[25]"},{"why":"Provides the rigorous vortex-filament-conjecture framework under a concentration assumption, against which this construction is positioned.","marker":"[24]"}],"fun_headline_variants":["Exact helical vortex polygons in Euler flow","KMD helical vortex model rigorously exact","Helical vortex filaments: exact Euler solutions","N-helix vortex configurations proven exact"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction fixes the rotation speed to the model's value only if, in the projected inner equation, the dominant term is exactly the one displayed in (6.31) and every other contribution is genuinely smaller; if those cancellations or order estimates fail, a solution still exists but its rotation speed is not pinned to the Klein–Majda–Damodaran value.","fun_headline_variants_meta":{"raw":{"variants":["Exact helical vortex polygons in Euler flow","KMD helical vortex model rigorously exact","Helical vortex filaments: exact Euler solutions","N-helix vortex configurations proven exact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000473,"raw_usage":{"total_tokens":2326,"prompt_tokens":898,"completion_tokens":1428,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1374}},"tokens_in":514,"tokens_out":1428,"duration_ms":14297,"temperature":1.0,"reasoning_tokens":1374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T15:13:11.456563+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computation of the projected integral $d_1=\\gamma_1\\int_{B_\\rho}(H+B_\\varepsilon\\varphi+b_\\varepsilon\\varphi)Z_1$ for a small case such as $N=2$ or $N=3$, using the fixed-point $\\varphi$ and $\\phi$, would settle the speed-matching claim: if a contribution of order $\\varepsilon\\mu\\sqrt{|\\log\\varepsilon|}$ survives beyond the leading term in (6.31), or if the $y_2$-odd sums in (3.28) fail to vanish at that order, then $\\alpha_\\varepsilon$ would acquire an undesired correction and the KMD matching in Theorem 1 would collapse.","supporting_citations":[{"cited_title":"Kenig, G","cited_arxiv_id":null,"evidence_quote":"Supplies the formal Klein–Majda–Damodaran asymptotic model whose rotating central configurations are being rigorously realized."},{"cited_title":"D´ avila, M","cited_arxiv_id":null,"evidence_quote":"Provides the elliptic singular-perturbation framework and the traveling-helix construction that the present gluing scheme adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the preceding cluster-of-helices construction, including the analysis of the operator $L$ and the linear solver used here."},{"cited_title":"Baraket, and F","cited_arxiv_id":null,"evidence_quote":"Identifies the three-dimensional kernel of the linearized Liouville operator $\\Delta+e^\\Gamma$, which drives the projected inner problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the analogous nearly-parallel filament model is rigorously derivable from Gross–Pitaevskii, the comparison that frames the open Euler problem."}],"review_version":1}