{"id":"88a3ffd2-d001-49e9-b996-950ace07a1e1","arxiv_id":"2608.04976","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For Ginzburg-Landau vortices of degree 2 and 3, the only bounded zero modes are the three geometric symmetries, and the Morse indices are 2 and 6.","lead":"Vortices with winding number two or three in the Ginzburg-Landau equation are shown to have no hidden zero modes, and their unstable directions are exactly two and six. The proof uses new explicit bounds on the vortex profiles and a comparison argument that may extend to higher degrees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The (3,3)-block index bound relies on an unshown Bernstein-coefficient positivity check; this is the least independently verifiable step supporting the n=3 Morse index and kernel claims.","rationale":"The reader's stated weakest assumption is the unproved bound P(t)<(n+2)^2 before (3.11), and their rationale also flags the asserted Bernstein-coefficient positivity for (3,3). I regard the Bernstein check as the more load-bearing concern because it is the least independently verifiable step in the argument: it is a finite but nontrivial computation, the relevant coefficient tables are not included, and it is exactly the step that excludes an extra bounded or L2 kernel in the most delicate Fourier block. The P-bound is also unproved, but for the two explicit lower barriers it reduces to verifying a rational inequality by clearing denominators; it is an elementary fillable gap rather than a serious correctness risk. The rest of the proof has substantial independent support: the explicit upper and lower profile bounds are proven via Lemma 2.2, the m=0 and m=1 symmetry blocks are handled by exact ground-state identities, the high-m comparison (3.7) is clean, and the negative test functions for the lower index bounds are explicit and checkable. I found no fatal mathematical error. The verdict should remain CONDITIONAL: the central claim is plausible and mostly well-supported, but the (3,3) Bernstein positivity must be independently reproduced before the n=3 nondegeneracy and Morse-index statements can be fully accepted.","tokens_in":20684,"tokens_out":10567,"duration_ms":116876,"concrete_test":"Recompute the right-interval supersolution for (n,m)=(3,3) in exact rational arithmetic: from (3.24) and the displayed coefficients c0,...,c16, form J2(z), substitute w=z/(1+z), and compute the Bernstein coefficients of q1(w)=bJ2(w/4), q2(w)=bJ2((w+1)/4), q3(w)=bJ2((w+2)/4), q4(w)=bJ2((w+12)/16), q5(w)=bJ2((w+13)/16), and q6(w)=bJ2((w+7)/8) using formula (3.18). If every coefficient is nonnegative, the assertion in §3.2 is confirmed; if any coefficient is negative, the upper bound ν(Q_3^{(3)})≤1 does not follow from the written proof, and the proof would need a modified supersolution or an independent argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.2 and the n=3 index bound hinge on admissibility of the supersolutions in §3.2. For the (n,m)=(3,3) block, the paper asserts that after compactifying the right interval via w=z/(1+z) and subdividing at 0, 1/4, 1/2, 3/4, 13/16, 7/8, 1, all Bernstein coefficients of bJ2(w) are strictly positive, and that the analogous q1,...,q6 checks succeed. Only the coefficients c0,...,c16 of J2(z) are displayed; the resulting Bernstein coefficient tables are not given, and no code is included. This is not a cosmetic omission: Proposition 3.2 uses this positivity to conclude ν(Q_3^{(3)})≤1, while the explicit negative test function only proves ind≥1. A single negative Bernstein coefficient on any subinterval would invalidate the claimed upper bound and leave open the possibility of an extra L2 kernel or a second negative direction. The auxiliary assertion just before (3.11) that P(t)<(n+2)^2 is also stated without proof, but for the two explicit barriers it is an elementary polynomial inequality; the Bernstein claim is the more serious gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the standard degree-two and degree-three equivariant Ginzburg-Landau vortices are nondegenerate and computes their Morse indices: the C^2∩L^∞ kernel is exactly the span of the phase and two translation Jacobi fields, the L^2 kernel is trivial, and the full real Morse indices are 2 (n=2) and 6 (n=3). The proof combines new explicit algebraic upper and lower barriers for the radial profiles f_2 and f_3, a Fourier block decomposition of the linearized Hessian, ground-state identities for the symmetry blocks m=0 and m=1, a comparison argument that rules out all blocks with m≥2n−1, and one-dimensional Sturm-Liouville supersolution arguments for the remaining intermediate blocks. Explicit negative test functions provide matching lower bounds for the index.","tokens_in":20941,"tokens_out":18085,"duration_ms":203215,"significance":"If correct, Theorem 1.1 settles a natural open question for n=2,3: there are no hidden bounded Jacobi fields beyond the geometric symmetries, and the exact number of negative Hessian directions is known. The method is a substantial contribution to the spectral theory of Ginzburg-Landau vortices: the explicit global barriers for f_2 and f_3 are new, the m=0,1 ground-state identities are clean and fully analytic, and the comparison reducing high Fourier blocks to the translation block is elegant. The paper also gives explicit negative test functions, so the lower index bounds are concrete and checkable. The main caveat is that several decisive positivity checks for the (3,3) block are asserted rather than documented, so the upper index bound for n=3 is not currently independently verifiable from the text.","major_comments":[{"comment":"The assertion P(t)<(n+2)^2 is stated without proof. This inequality is load-bearing: it guarantees A_m(t)>0, which is used to prove that the projection (c,d)↦d is injective on every nonpositive subspace of Q^(n)_{m,g} and hence that ν(Q^(n)_m)≤k follows from the scalar supersolution argument. For the n=2 barrier the inequality is immediate, but for the n=3 barrier it is a nontrivial polynomial inequality. The authors should include a proof of P(t)<(n+2)^2 for both barriers used in the paper.","section":"Section 3, before (3.11); Proposition 3.2"},{"comment":"The positivity checks for the (3,3) block are not documented. The paper asserts that, after degree elevation to 10, all Bernstein coefficients of J_1 on [0,17/4] are nonnegative, and that after compactification w=z/(1+z) and subdivision at 0,1/4,1/2,3/4,13/16,7/8,1, all Bernstein coefficients of bJ_2(w) and of q_1,...,q_6 are strictly positive, but no coefficient tables or code are provided. These checks are load-bearing: Proposition 3.2 uses them to conclude ν(Q^(3)_3)≤1, and a single negative Bernstein coefficient would invalidate the claimed upper bound. The authors should include the complete verification, either as explicit tables of Bernstein coefficients or as reproducible code, or give an independent certificate that all required inequalities hold.","section":"Section 3.2, m=3 case"},{"comment":"For the lower bound f_2>ϕ_{C*}, the paper displays the formula for E_2(ϕ_C)/ϕ_C but does not state the needed sign conclusion. To apply Lemma 2.2 for the lower bound one must show E_2(ϕ_{C*})≥0; this follows from the fact that C*=30+6√21 makes the discriminant of the numerator vanish, but the proof as written leaves this step implicit. Please spell out this verification, since the lower barrier for n=2 underpins all subsequent estimates for the degree-two vortex.","section":"Section 2, Proposition 2.3"}],"minor_comments":[{"comment":"The coefficient list for p(s) is hard to read because some commas are missing: \"−547200000 1088506800\" and \"772436736 −449390592\" should be separated by commas.","section":"Section 3.2, m=2 case"},{"comment":"There are typographical errors: \"uniquness\" should be \"uniqueness\" and \"realted\" should be \"related\".","section":"Section 1"},{"comment":"The phrase \"the kevaluation maps\" should read \"the k evaluation maps\".","section":"Section 3, Proposition 3.2 proof"},{"comment":"The claim that multiplication by a bounded matrix-valued function converging to zero at infinity is form-compact relative to the radial operators is not correct in two dimensions for the 1/r^2 perturbation; the subsequent coercivity argument is sufficient for the finiteness of the negative spectrum, so the Weyl-theorem justification should be revised or removed.","section":"Section 2, Lemma 2.1"},{"comment":"The negative test-function estimates for m=2 and m=4 are asserted without the computational details that are shown for m=3; please include the corresponding integral bounds or refer to a supplementary file.","section":"Section 3.2, negative test functions"},{"comment":"The statement that the proof in [5] relies on a variational argument requiring additional justification is vague; if this criticism is retained, the authors should identify the precise defect in the admissibility of the test function in [5, Theorem 1.8].","section":"Section 1, discussion of [5]"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in substance, and the analytic architecture is strong. The main obstacle to acceptance is the undocumented Bernstein-coefficient verification for the (3,3) block; this is a finite, checkable computation, so the revision should be able to close the gap by supplying tables or code. The authors may also want to soften the criticism of [5] to avoid unnecessary controversy, since the present paper does not need that comparison for its main results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this paper carefully. The main result is new and significant: a complete proof of nondegeneracy for the degree-two and degree-three Ginzburg–Landau vortices, with exact Morse indices, filling a real gap left by Beaulieu's general claim. The proof is not a repackaging; the comparison barriers for f2 and f3 are explicit and proved, the Fourier block structure is handled cleanly, and the reduction to scalar Schrödinger forms via supersolutions is a genuinely useful technique. The m=0 and m=1 blocks are treated rigorously with exact ground-state representations, and the high-mode comparison m≥2n−1 is elegant. I found no fatal error in the analytic architecture, and the lower bounds on the index via explicit negative test functions are convincing.\n\nThat said, there is a load-bearing gap in the n=3 case. In §3.2, for the (3,3) block, the authors assert that after compactifying and subdividing, all Bernstein coefficients of the six polynomials q1,...,q6 are strictly positive. They display the coefficients of J2(z) but not the Bernstein coefficients on the subdivided intervals, and they do not include the Mathematica code. This is not cosmetic: the claim is exactly what drives the upper bound ν(Q_3^{(3)})≤1, and a single negative coefficient would break Proposition 3.2. The m=2 block, by contrast, lists the Bernstein coefficients explicitly; the contrast makes the omission for m=3 conspicuous. Also, the assertion just before (3.11) that P(t)<(n+2)^2 for the explicit barriers is stated without proof. It looks like an elementary polynomial check, so I treat it as minor but still something that should be written out.\n\nIf the Bernstein positivity check is correct, the paper is a solid contribution. If it is wrong, the n=3 Morse index claim collapses. The manuscript as posted does not let a reader verify this independently, which is the one thing standing between this and a clean accept.\n\nMy bottom line: this deserves a serious referee. Send it out, but require the authors to provide either the full Bernstein coefficient tables or a short script that reproduces the check, and expand the proof of P(t)<(n+2)^2. For the right reader—someone working on vortex gluing, stability, or spectral properties of Ginzburg–Landau—it is worth the time.","headline":"A serious proof of nondegeneracy and Morse indices for n=2,3 GL vortices, with a reproducible-computation gap in the n=3 case that should be fixed before publication.","tokens_in":21494,"tokens_out":3229,"would_cite":true,"duration_ms":38537,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q56","35J61","35B35","35P15"],"pacs":[],"model":"deepseek-v4-flash","headline":"At degrees two and three, the standard Ginzburg–Landau vortex is nondegenerate: its only bounded kernel directions are the three geometric symmetries, and its Morse index is exactly 2 (degree two) or 6 (degree three).","keywords":["Ginzburg-Landau vortices","nondegeneracy","Morse index","vortex linearization","Fourier block decomposition","explicit barriers","Jacobi fields","Schrödinger operators"],"falsifier":"A direct spectral computation would settle the theorem: numerically solve the radial eigenvalue problems of the linearized operator at $u_2$ and $u_3$ — especially the blocks $m=2$ (for $n=2$) and $m=2,3,4$ (for $n=3$) — and count negative eigenvalues and zero modes; the theorem requires exactly one negative eigenvalue and no $L^2$ kernel in each of those blocks, none in any other block, and the only bounded zero modes should be the three geometric ones. The asserted barrier inequality is even cheaper to test, since $P(t)<(n+2)^2$ is an explicit one-variable rational inequality for the two models; a single point where it fails would expose the gap in the reduction.","tokens_in":20440,"feed_emoji":"🌀","tokens_out":26255,"duration_ms":251144,"temperature":0.7,"pith_summary":"The paper sets out to settle an open spectral question for the two lowest higher-degree Ginzburg–Landau vortices: the standard solutions $u_n(r,\\theta)=f_n(r)e^{in\\theta}$ for $n=2,3$ have no explicit formula, so their linearized operators have resisted direct analysis. The main theorem proves that these vortices are nondegenerate — the only bounded kernel directions of the linearized operator are the three geometric symmetries (one phase rotation and two translations) — and that the Morse index, the number of negative directions of the energy's second variation, is exactly $2$ for $n=2$ and $6$ for $n=3$. The result matters because vortex-based constructions must invert the linearized operator on the complement of its kernel: a hidden kernel mode would introduce an extra modulation parameter, and the index fixes how many unstable directions a min-max or evolutionary argument has to handle. Since higher-degree vortices are not energy minimizers, the stability argument used for the degree-one vortex cannot be recycled; the proof instead sandwiches $f_n$ between explicit algebraic barriers, decomposes perturbations into Fourier blocks, and reduces each block to a one-dimensional Schrödinger form whose positivity is certified by explicit supersolutions. If the theorem is right, the paper delivers the first rigorous nondegeneracy results for higher-degree Ginzburg–Landau vortices, and the same method is expected to extend to all degrees.","feed_headline":"2 and 6: exact instability counts for degree-2 and -3 vortices","feed_subtitle":"Vortices of degree 2 and 3 have no hidden instability modes, making them safe building blocks for larger solutions.","key_machinery":"Three pieces of machinery carry the proof. (1) Explicit algebraic barriers replace the unknown profile: Proposition 2.3 gives $r^2/\\sqrt{r^4+4r^2+30+6\\sqrt{21}}<f_2(r)<r^2/\\sqrt{r^4+4r^2+24}$ and $g_{4000}(r)<f_3(r)<g_C(r)$ for $0<C<792$, where $g_C(r)=r^3/\\sqrt{r^6+9r^4+99r^2+C}$; Lemma 2.2 certifies each barrier by the sign of the profile operator $E_n(g)$, via a maximum-principle argument on the ratio $f_n/g$. (2) The linearized operator is split into radial Fourier blocks $Q_m^{(n)}$ (following the conjugate-form treatment in [35]), which are mutually orthogonal, so each block can be analysed separately. (3) For the intermediate blocks $2\\le m\\le 2n-2$ the paper replaces $f_n$ by a lower barrier $g$, giving $Q^{(n)}_m\\ge Q^{(n)}_{m,g}$, changes to logarithmic time $t=\\log r$, and completes the square in one component using $A_m(t)=(n+m)^2-P(t)>0$ (the asserted bound $P(t)<(n+2)^2$); discarding the nonnegative square leaves the scalar Schrödinger form $\\ell_m[d]=\\int_{\\mathbb R}(\\dot d^2+V_m d^2)\\,dt$ with $V_m=-P+A_mW/(A_m+W)$. Positivity is verified by explicit supersolutions on each side of a splitting point — for example $1-x/4$ and $1-4/x$ for the block $(2,2)$ — and Proposition 3.2 converts $k$ splitting points into the bound $\\nu(Q_m^{(n)})\\le k$; matching negative directions come from explicit test functions such as $a=(r^2+2)^{-1}$. All polynomial positivity checks reduce, via the Bernstein-basis criterion of Lemma 3.3, to nonnegativity of explicit coefficient lists.","core_discovery":"The central claim, Theorem 1.1, is that for $n=2,3$ the linearized operator $L_n v=-\\Delta v-(1-f_n^2)v+2u_n\\,\\mathrm{Re}(u_n\\bar v)$ at the vortex $u_n$ satisfies\n$$\\ker_{$C^{2}$(\\mathbb $R^{2}$;\\mathbb C)\\cap L^\\infty(\\mathbb $R^{2}$)}L_n=\\mathrm{span}_{\\mathbb R}\\{iu_n,\\partial_{x_1}u_n,\\partial_{x_2}u_n\\},$$\nso the $L^2$ kernel is trivial, and the Morse indices are $\\mathrm{ind}(L_2)=2$ and $\\mathrm{ind}(L_3)=6$. Theorem 1.2 locates every part of the spectrum at the block level: the $m=0$ Fourier block contributes only the phase mode; the $m=1$ block is nonnegative with kernel spanned by the pair $(q,-p)$, where $q=f_n'+n f_n/r$ and $p=n f_n/r-f_n'$, corresponding to the two translations; every block with $m\\ge 2n-1$ is strictly positive; and among the intermediate blocks only $m=2$ (for $n=2$) and $m=2,3,4$ (for $n=3$) carry exactly one negative eigenvalue each, with no kernel. Since each block has two real angular copies, the full real indices are $2\\cdot 1=2$ and $2\\cdot 3=6$.","pith_inferences":["A concrete next step the paper leaves implicit: before constructing analytic supersolutions for $n=4$, one can run the same Fourier-block and scalar-reduction scheme numerically for the blocks $m=2,\\dots,6$; the authors expect multiple negative eigenvalues per block at larger $n$, and a numerical scan would show exactly where the difficulty begins.","If the conjectured bound $\\mathrm{ind}\\le n(n-1)$ holds, the unstable dimension of the degree-$n$ vortex grows quadratically in $n$, which numerically coincides with the conjectured index of the degree-$n$ complex sine-Gordon II vortex and with the number of free parameters in its Adler–Moser polynomials; that coincidence suggests a hidden combinatorial law for the vortex spectrum.","A cheap testable consequence of the reduction: for any candidate barrier, the single inequality $P(t)<(n+m)^2$ controls whether the 'complete the square and discard' step is legitimate, so the screening of future barriers reduces to evaluating one rational function.","If the nondegeneracy transfers to construction problems on bounded domains or magnetic models that use a planar vortex as the inner solution, the kernel classification means the matching equations contain precisely the three geometric unknowns — exactly the input such constructions assume."],"forward_implications":["For $n=2,3$ the linearized operator is invertible on the complement of the three geometric modes, so any gluing or Lyapunov–Schmidt reduction built on a rescaled vortex needs exactly three modulation parameters (one phase, two positions) and gains no hidden solvability conditions.","The Morse indices $2$ and $6$ give the exact dimension of the descending space of the Ginzburg–Landau energy at these vortices, which is the number of unstable spectral channels an evolution has before the neutral symmetries are factored out.","Every Fourier block with $m\\ge 2n-1$ is strictly positive for the degrees treated, so any instability or kernel of a higher-degree vortex can only occur in the finite strip $2\\le m\\le 2n-2$.","The explicit barriers for $f_2$ and $f_3$ are global algebraic functions, usable in any later argument that needs a definite sign in the transition region rather than an asymptotic series.","The paper expects the same barrier-plus-comparison scheme to prove nondegeneracy for all higher degrees, with the number of intermediate blocks growing as $2n-3$."],"supporting_citations":[{"why":"The monograph's Chapter 3 supplies the conjugate quadratic form, the radial Fourier block decomposition used in (2.1), and the matched asymptotic analysis near the origin and infinity that the paper adapts to general degree.","marker":"[35]"},{"why":"Establishes existence and uniqueness of the symmetric vortex profile $f_n$, the object whose linearized spectrum the paper computes.","marker":"[6]"},{"why":"Provides the qualitative properties of the radial profile $f_n$ (positivity, monotonicity, asymptotics) that the barrier and supersolution arguments use.","marker":"[25]"},{"why":"Claims nondegeneracy for the degree-$n$ vortex for every $n$; the present paper argues that proof is incomplete and supplies the rigorous version for $n=2,3$.","marker":"[5]"},{"why":"Gives the numerical evidence that these vortex solutions are nondegenerate, which Theorem 1.1 turns into a proof.","marker":"[4]"},{"why":"Shows the degree-two vortex of the complex sine-Gordon II equation is degenerate and provides explicit solution formulas whose rational form inspires the algebraic barriers.","marker":"[14]"},{"why":"Proves positivity of the second variation at the degree-one vortex, the baseline argument that cannot be reused because higher-degree vortices are not minimizers.","marker":"[33]"},{"why":"Establishes minimality and nondegeneracy of the degree-one vortex through a Hardy-type inequality, the model nondegeneracy statement the paper extends.","marker":"[16]"}],"fun_headline_variants":["Vortex Morse indices: 2 for degree 2, 6 for degree 3","Exact instability counts: degree-2 vortices have 2, degree-3 have 6","Nondegenerate degree-2 and -3 vortices: Morse indices 2 and 6","Degree-2 vortex has exactly 2 unstable modes; degree-3 has 6","No hidden instabilities: degree-2 and -3 vortices have indices 2 and 6"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on an inequality stated without proof just before (3.11): for the two explicit comparison profiles used in the proof, the quantity $P(t)=x(1-g(x)^2)$ with $x=e^{2t}$ stays below $(n+2)^2$, which guarantees the coefficient $A_m(t)$ is positive; if that bound failed at any point, the reduction of each Fourier block to a one-dimensional Schrödinger form — the step that carries the whole Morse-index upper bound — would break down.","fun_headline_variants_meta":{"raw":{"variants":["Vortex Morse indices: 2 for degree 2, 6 for degree 3","Exact instability counts: degree-2 vortices have 2, degree-3 have 6","Nondegenerate degree-2 and -3 vortices: Morse indices 2 and 6","Degree-2 vortex has exactly 2 unstable modes; degree-3 has 6","No hidden instabilities: degree-2 and -3 vortices have indices 2 and 6"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000766,"raw_usage":{"total_tokens":3385,"prompt_tokens":922,"completion_tokens":2463,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":2343}},"tokens_in":538,"tokens_out":2463,"duration_ms":18229,"temperature":1.0,"reasoning_tokens":2343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:27:42.064800+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct spectral computation would settle the theorem: numerically solve the radial eigenvalue problems of the linearized operator at $u_2$ and $u_3$ — especially the blocks $m=2$ (for $n=2$) and $m=2,3,4$ (for $n=3$) — and count negative eigenvalues and zero modes; the theorem requires exactly one negative eigenvalue and no $L^2$ kernel in each of those blocks, none in any other block, and the only bounded zero modes should be the three geometric ones. The asserted barrier inequality is even cheaper to test, since $P(t)<(n+2)^2$ is an explicit one-variable rational inequality for the two models; a single point where it fails would expose the gap in the reduction.","supporting_citations":[{"cited_title":"Pacard and T","cited_arxiv_id":null,"evidence_quote":"The monograph's Chapter 3 supplies the conjugate quadratic form, the radial Fourier block decomposition used in (2.1), and the matched asymptotic analysis near the origin and infinity that the paper adapts to general degree."},{"cited_title":"Berger and Y .Y","cited_arxiv_id":null,"evidence_quote":"Establishes existence and uniqueness of the symmetric vortex profile $f_n$, the object whose linearized spectrum the paper computes."},{"cited_title":"Herv´e and M","cited_arxiv_id":null,"evidence_quote":"Provides the qualitative properties of the radial profile $f_n$ (positivity, monotonicity, asymptotics) that the barrier and supersolution arguments use."},{"cited_title":"Beaulieu, Bounded solutions for an ordinary differential system from the Ginzburg–Landau theory,Proc","cited_arxiv_id":null,"evidence_quote":"Claims nondegeneracy for the degree-$n$ vortex for every $n$; the present paper argues that proof is incomplete and supplies the rigorous version for $n=2,3$."},{"cited_title":"Barashenkov, V .S","cited_arxiv_id":null,"evidence_quote":"Gives the numerical evidence that these vortex solutions are nondegenerate, which Theorem 1.1 turns into a proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the degree-two vortex of the complex sine-Gordon II equation is degenerate and provides explicit solution formulas whose rational form inspires the algebraic barriers."},{"cited_title":"del Pino, P","cited_arxiv_id":null,"evidence_quote":"Establishes minimality and nondegeneracy of the degree-one vortex through a Hardy-type inequality, the model nondegeneracy statement the paper extends."}],"review_version":1}