{"id":"f15c1e66-77fd-4c9c-87f5-2ebd213ee61f","arxiv_id":"2608.10608","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Certified spectral analysis and numerics prove the degree-one vortex linearized operator has one internal mode with eigenvalue in [0.777471875,0.77747375] and negative Fermi Golden Rule coefficients.","lead":"Rigorous analysis and interval-arithmetic computations determine the spectrum of the linearized operator around a stable magnetic vortex in a relativistic gauge theory. The result proves a single internal wobble mode and certifies the nonlinear damping that makes the vortex asymptotically stable over time.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1(i) and the effective-damping claim rest on Proposition 3.7 and Proposition 4.2, both deferred to not-yet-available companion papers; the standalone paper does not prove them.","rationale":"Good-faith reading: the paper is the first of a three-part series and explicitly punts the distorted Fourier transform and normal-form analysis to the companion papers. The reader's weakest assumption identifies exactly this. I see no internal contradiction in the analytic sections or the certified numerics as presented; Lemmas 6.2, 6.3, 7.3, and Proposition 4.3 have clear statements, and the archived CAPD certificate is a strong reproducibility artifact. The load-bearing risk is the unproved Proposition 3.7: it is the only basis for Theorem 1.1(i), and the Fermi Golden Rule computation additionally uses this kernel. Similarly, Proposition 4.2 is necessary for the phrase \"effective nonlinear damping\"; without it, knowing D1,D2,D12<0 is not enough for asymptotic stability. Since this gap is acknowledged by the authors and the companion papers may supply the missing proofs, a conditional verdict is appropriate. No new fatal flaw found; the verdict remains conditional.","tokens_in":71847,"tokens_out":15549,"duration_ms":242557,"concrete_test":"Inspect, once posted, [Luh+26b] and [Luh+26c]: verify that Proposition 3.7 is proved with the stated kernel E, connection-coefficient bounds, and unitarity eF eF^{-1}=I, and that Proposition 4.2 is proved with the remainder bounds that turn (4.8) into damping. If either companion paper is missing or the proof does not deliver these statements, the paper's main theorem and Fermi Golden Rule damping conclusion are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorem 1.1(i) and of the abstract's \"effective nonlinear damping\" are not proved in this paper. The purely absolutely-continuous spectrum statement is derived entirely from Proposition 3.7 (the distorted Fourier transform), which is introduced with \"We now state, without proof\" and whose construction is deferred to [Luh+26b, Theorem 1.2]. Likewise, the Fermi Golden Rule coefficients in (4.7) are evaluated using the kernel E from Proposition 3.7, and the step from strict negativity of D1,D2,D12 to damping is Proposition 4.2, deferred to [Luh+26c, Sections 9 and 10]. The paper explicitly flags both deferrals (Section 3.2, Remark 4.1, and the paragraph before Proposition 4.2). Therefore, if the companions are unavailable or contain a gap, Theorem 1.1(i) and the nonlinear-damping conclusion do not follow from anything in this manuscript. The in-paper spectral and numerical work is detailed and internally consistent, but it establishes Propositions 3.3 and 4.3 only conditional on the deferred distorted Fourier construction and normal-form analysis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the spectral theory of the linearized operator M around the degree-one vortex in the abelian Yang-Mills-Higgs model at self-dual coupling, in the context of equivariant perturbations. Using the Bogomolny factorization, the authors reduce the spectral analysis of the matrix operator M to two scalar strongly singular half-line operators L1 and L2. They combine analytic estimates (Frobenius expansions, comparison and Volterra arguments, Seto's bound) with CAPD-based interval arithmetic to prove several spectral statements: absence of discrete spectrum and threshold resonance for L1, absence of threshold resonance for L2, and existence of exactly one gap eigenvalue of L2 (and hence of M) with a two-dimensional eigenspace, with certified enclosure λ^2 in [0.777471875,0.77747375]. The paper also certifies strict negativity of the three Fermi Golden Rule coefficients D1,D2,D12, which is the non-degeneracy condition needed for nonlinear damping of the internal mode. A substantial part of the argument, however, is deferred: the distorted Fourier transform for M (Proposition 3.7), the asymptotic and symbol bounds for the generalized eigenfunctions (Proposition 3.5), and the normal-form ODE and remainder estimates connecting the signs of D1,D2,D12 to effective damping (Proposition 4.2) are all stated as results of the companion papers [Luh+26b] and [Luh+26c]. Thus the paper's in-house contributions are Propositions 3.3 and 4.3, conditional on those deferred inputs.","tokens_in":72004,"tokens_out":8831,"duration_ms":136202,"significance":"If the deferred companion papers are valid, this manuscript provides a carefully executed hybrid analytic-numerical proof of a nontrivial spectral statement for a gauge-theory soliton: the linearized vortex has a unique internal mode, no threshold resonance, and a negative definite Fermi Golden Rule form. The methodology is transparent: the interval enclosures are derived from the model equations rather than fitted to the desired output; the analytic-numerical split is clearly documented; and the repository with CAPD code and 220-bit transcripts is a strong reproducibility feature. The proof of the eigenvalue count via Seto's bound, the threshold-resonance exclusion via Wronskian matching, and the certified tail estimates are all substantial in-paper contributions. However, the headline claims in Theorem 1.1(i) and in the abstract's 'effective nonlinear damping' are not proved in this manuscript: they depend on the distorted Fourier construction and on the normal-form analysis that are explicitly deferred to the two companion papers.","major_comments":[{"comment":"Theorem 1.1(i), the claim that the restriction of M to the continuous spectral subspace has purely absolutely continuous spectrum [1,∞), is not established within this manuscript. The proof of Theorem 1.1 invokes Proposition 3.7, which is introduced with 'We now state, without proof' and is deferred to [Luh+26b, Theorem 1.2], together with Proposition 3.5, whose asymptotic and symbol bounds are also deferred. The purely a.c. statement is stronger than the absence of eigenvalues and threshold resonances proved in Sections 6 and 7; it requires the distorted Fourier basis and its completeness, which are not proved here. The authors should either include the construction and proof of Proposition 3.7 or explicitly restate Theorem 1.1(i) as a conditional theorem dependent on the companion paper.","section":"Section 3.2 and proof of Theorem 1.1"},{"comment":"The abstract's assertion that the certified signs yield 'effective nonlinear damping of the internal mode' is not proved in this paper. Proposition 4.2, which contains the normal-form ODE, the remainder estimates, and the step from D1,D2,D12<0 to the differential inequality (4.8), is explicitly deferred to [Luh+26c, Sections 9 and 10]; Remark 4.1 and the paragraph before Proposition 4.2 say exactly this. What the present manuscript proves is Proposition 4.3, namely the strict negativity of the dissipative coefficients, assuming the distorted Fourier representation of Proposition 3.7. The wording of the abstract and of Section 1.4 should be revised so that the nonlinear damping conclusion is presented as conditional on the normal-form analysis in the companion paper.","section":"Section 4.1, Proposition 4.2 and the abstract"},{"comment":"The central quantitative claims, including the eigenvalue enclosure in Lemma 7.3 and the Fermi Golden Rule lower bounds in (4.9), are certified by CAPD interval arithmetic at 220-bit precision and are reported through transcripts in an external repository. This is a standard and welcome practice in computer-assisted proofs, and the analytic-numerical division is clearly explained. Nevertheless, the correctness of these claims depends on the executing code and environment, which are not independently verified in the manuscript. The authors should state more prominently that the proofs are hybrid and that the numerical certificate is machine-checked only by running the provided code; if a simpler independent verifier for the final inequalities exists, it would reduce this risk.","section":"Sections 7-9, numerical certification"}],"minor_comments":[{"comment":"The symbol λ denotes the coupling constant in Section 1.1 and then is redefined as the internal frequency after Theorem 1.1. The redefinition is stated explicitly, but a separate symbol (e.g., ω) for the internal frequency would avoid confusion for readers scanning the paper.","section":"Throughout"},{"comment":"The notation H2 is used both for the scalar operator in (3.6) and for the cubic Hamiltonian in (4.2). The authors note this conflict, but renaming the Hamiltonian in (4.2) (e.g., H_cubic) would make the text clearer.","section":"Section 4.1"},{"comment":"The companion papers [Luh+26b], [Luh+26c], and the certificate repository [Luh+26a2] are cited as preprints without stable identifiers. Since the present paper's main claims depend on the first two, the authors should add arXiv numbers or other permanent identifiers once available, and archive the repository in a persistent service.","section":"References"},{"comment":"The sentence 'The interval J0 has width 9.0·10^{-13}, whereas that lemma reduces this to 1.263·10^{-16}' could be misleading: Lemma 5.9 actually provides an interval Icert of width 1.57·10^{-14} around the refined value, and the stated 1.263·10^{-16} is the error bound around the midpoint. Please reword to distinguish the two quantities.","section":"Paragraph before Lemma 5.9"},{"comment":"The sentences defining L1,L2,L12 appear twice, once in the completion of the proof of Proposition 4.3 and once at the beginning of Section 9.2. One of the repetitions should be removed.","section":"Section 9.2"}],"recommendation":"major_revision","confidential_remarks":"The in-paper analytic and numerical work appears careful and internally consistent; I do not see a clear mathematical error in Propositions 3.3 and 4.3 as conditional statements. The main issue is that Theorem 1.1(i) and the effective-damping conclusion are not proved in this manuscript because they depend on the distorted Fourier transform and normal-form analysis in two companion papers. If the journal's policy permits papers in a planned series to state results conditional on forthcoming companions, the revision should at minimum reframe the abstract and Theorem 1.1 accordingly. If not, acceptance should wait until the companion proofs are available for review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The reader's take is about right, and I largely agree with it. This paper proves, with a careful mix of analysis and interval arithmetic, the spectral reduction for the linearized operator around the degree-one vortex, the absence of threshold resonances, the uniqueness and sharp enclosure of the internal mode, and certified negativity of the Fermi Golden Rule coefficients. The in-paper work is thorough and internally consistent: the vortex profile estimates, the Seto-count argument, the Wronskian matching, and the tail bounds all hang together. The archived CAPD certificates and transcripts are a real reproducibility artifact, and the fact that no free parameters are fitted is a genuine strength.\n\nNow the soft spots, in proportion. Theorem 1.1(i) — the purely absolutely continuous spectrum on [1,∞) — and the effective nonlinear damping conclusion are not proved in this manuscript. They depend on Proposition 3.7 (the distorted Fourier transform for the matrix operator) and Proposition 4.2 (the normal-form ODE with remainder estimates), both deferred to the not-yet-available companion papers [Luh+26b, Luh+26c]. The paper flags these deferrals explicitly, so it is not hiding anything, but it does mean the headline claims are conditional. The in-paper results are solid, but they are statements about the spectral reduction and the FGR sign assuming those deferred constructions exist and have the stated properties.\n\nThe circularity burden is low: the certified enclosures are model-derived, not fitted. The main engineering assumption is that CAPD's 220-bit interval arithmetic and the supplied transcripts are correct; I did not independently re-run the code, and I would not expect a referee to do so either, but spot-checking the transcript format and a few of the stated enclosures is reasonable.\n\nWho is this for? Specialists in soliton stability, spectral theory, and computer-assisted proof. It deserves a serious referee, not a desk rejection. My recommendation: send it to review, with the expectation that the referee checks the in-paper proofs and samples the certificates. Final acceptance should hinge on the companion papers being public and correct, or on the authors agreeing to state Theorem 1.1(i) and the damping claim as conditional on those papers.","headline":"Solid certified spectral analysis for the degree-one vortex, but the headline spectral and damping claims rest on two companion papers; referee the in-paper work and the certificates.","tokens_in":72571,"tokens_out":1710,"would_cite":true,"duration_ms":19683,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P05","35P15","35Q51","35B35","47E05","65G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the linearized operator around the degree-one vortex has one internal mode at a certified frequency and that the Fermi Golden Rule coefficients are negative, so the mode is radiatively damped.","keywords":["abelian Yang-Mills-Higgs","degree-one vortex","internal mode","Fermi Golden Rule","distorted Fourier transform","threshold resonance","interval arithmetic","computer-assisted proof"],"falsifier":"Run the threshold Wronskian and the three Fermi Golden Rule radial integrals with an independent validated interval-arithmetic solver: if the Wronskian interval at $r=10$ contains $0$, or if any of $D_1, D_2, D_{12}$ is certified nonnegative, the paper's central claim is refuted.","tokens_in":1948,"feed_emoji":"🌀","tokens_out":3093,"duration_ms":120727,"temperature":0.7,"pith_summary":"This paper proves the spectral facts that a planned stability proof for the degree-one vortex in the abelian Yang-Mills-Higgs model needs. It shows that the linearized operator around the vortex has purely continuous spectrum from $1$ upward, with no threshold resonance, and exactly one discrete gap eigenvalue $\\lambda^2$, enclosed in $[0.777471875, 0.77747375]$, with a two-dimensional eigenspace called an internal mode. It also certifies by interval arithmetic that the three Fermi Golden Rule coefficients $D_1, D_2, D_{12}$ are strictly negative, the sign condition under which the internal mode is nonlinearly damped rather than persistent. These are the first of three papers; the companion papers supply the distorted Fourier basis and the normal-form estimates that complete the argument. If the conclusion holds, the internal mode is unique, nondegenerate, and radiatively damped, which is the configuration the companion papers need for asymptotic stability.","feed_headline":"Vortex's unique internal mode found — and it damps out","feed_subtitle":"Interval arithmetic fixes the mode at squared frequency 0.77747 and proves the Fermi-Golden-Rule sign that damps it.","key_machinery":"The central object is the supersymmetric factorization $\\mathbf L = B^*B$ and its partner $\\widetilde{\\mathbf L} = BB^* = \\operatorname{diag}(L_1, L_2)$, obtained from the Bogomolny equations at self-dual coupling. This identity uncouples the two-component linearized problem into two scalar radial Schrödinger operators, which after conjugation become half-line operators $H_1$ and $H_2$. The proof runs on a division of labor: analytic Frobenius series near $r=0$, certified outward-rounded interval arithmetic on compact intervals, and Volterra, barrier, and tail estimates at infinity, all producing enclosures of the vortex profile, the eigenpair, the distorted Fourier basis, and the Fermi Golden Rule integrals. The sign of those integrals is the mechanism: negative coupling coefficients route energy from the bound state into radiation.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.1 together with Proposition 4.3: the linearized operator $\\mathbf M$ has purely absolutely continuous spectrum $[1,\\infty)$ on its continuous subspace, the threshold is not resonant, and the unique positive gap eigenvalue $\\lambda^2$ lies in $[0.777471875, 0.77747375]$, with eigenspace spanned by $Y_1 = \\lambda^{-1}(U\\psi,-\\psi',0,0)^t$ and $Y_2 = \\lambda^{-1}(0,0,U\\psi,-\\psi')^t$, where $\\psi$ is the $L^2_{rdr}$-normalized ground state of $-\\Delta + U^2$. The Fermi Golden Rule coefficients $D_1, D_2, D_{12}$, formed from the distorted Fourier transform of the quadratic interaction $Q_s(Y_i,Y_j)$ at the resonant frequency $k_\\lambda = \\sqrt{4\\lambda^2-1}$, are certified negative; the code proves the equivalent positive lower bounds $b^{(0)}_1 > 0.017297889$, $b^{(0)}_2 > 0.047248801$, and $b^{(0)}_{12} > 0.022940050$. Because $C_{12}$ is controlled by Cauchy-Schwarz by $D_1 D_2$, the three negative signs imply positive damping constants $\\Gamma_0, \\Gamma_1$ in the differential inequality for $|Z_1|^2 + |Z_2|^2$.","pith_inferences":["If the companion nonlinear analysis succeeds, the decay rate of the internal-mode amplitude should be governed by the certified quantities $b^{(0)}_1, b^{(0)}_2, b^{(0)}_{12}$, so an independent time-dependent simulation of small equivariant perturbations could test the predicted slow radiative damping rate.","The same hybrid analytic-interval-arithmetic pipeline is portable to $n$-vortices and to non-self-dual couplings, where the internal mode count is not known a priori; the crossing dichotomy and Setô bound suggest the eigenvalue count is the key quantity to certify next.","The paper's theorem is exactly as strong as the underlying interval-arithmetic transcripts and the deferred companion estimates, so an independent implementation of the same validated computations would be a decisive check of the spectral and damping claims."],"forward_implications":["The linearized flow near the vortex has no trapped state at the bottom of the continuum: the continuous spectrum starts at $1$ and is purely absolutely continuous, so radiation disperses rather than accumulating.","There is exactly one internal frequency, so any equivariant perturbation has a two-dimensional bound-state component whose squared frequency is known to within $2\\cdot 10^{-6}$.","With $D_1, D_2, D_{12} < 0$, the leading-order cubic terms in the internal-mode ODE are dissipative, so $|Z_1|^2 + |Z_2|^2$ satisfies a differential inequality with positive damping rates, the mechanism that makes the internal mode decay instead of surviving as a periodic state.","No separate numerical certification of the mixed coefficient $C_{12}$ is needed, since $|C_{12}|^2 \\le D_1 D_2$ follows from the positive spectral measure.","These spectral and Fermi Golden Rule inputs are exactly what the companion papers use to construct the linear dispersive estimates and the nonlinear normal-form stability argument."],"supporting_citations":[{"why":"Deferred companion paper that supplies Proposition 3.5 and Proposition 3.7, the asymptotic and symbol bounds and the distorted Fourier basis for the linearized operator.","marker":"[L¨uh+26b]"},{"why":"Deferred companion paper providing Proposition 4.2, the normal-form ODE and remainder estimates from which the nonlinear damping conclusion follows.","marker":"[L¨uh+26c]"},{"why":"Source of the radiation-damping mechanism: shows that when the Fermi Golden Rule holds, linear bound states become damped for the nonlinear flow.","marker":"[SW99]"},{"why":"Introduced the Fermi Golden Rule as the condition under which periodic and quasiperiodic bound states are unstable for nonlinear equations.","marker":"[BP95; Sig93]"},{"why":"Provides the general procedure for constructing the distorted Fourier transform and spectral representation that the paper follows.","marker":"[GZ06; KMS; KST08]"},{"why":"Supplies the two-dimensional bound used to prove that the scalar operator $L_2$ has at most one eigenvalue in $(0,1)$.","marker":"[Set73]"},{"why":"Supplies the sharp modified Bessel function ratio bound used in the tail and derivative estimates for the internal eigenfunction and the Weyl solutions.","marker":"[YC17]"}],"fun_headline_variants":["Vortex's unique internal mode proven to damp out","Interval arithmetic nails vortex mode's damping sign","Single vortex mode, twin eigenspace, definite damping","Vortex's lone internal mode fades: rigorous proof"],"cache_read_input_tokens":74752,"weakest_assumption_plain":"The central claim collapses if the companion papers' distorted Fourier basis and normal-form remainder estimates fail, or if the 220-bit interval arithmetic certificate is wrong.","fun_headline_variants_meta":{"raw":{"variants":["Vortex's unique internal mode proven to damp out","Interval arithmetic nails vortex mode's damping sign","Single vortex mode, twin eigenspace, definite damping","Vortex's lone internal mode fades: rigorous proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1521,"prompt_tokens":1073,"completion_tokens":448,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":385}},"tokens_in":689,"tokens_out":448,"duration_ms":5433,"temperature":1.0,"reasoning_tokens":385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:00:11.929374+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the threshold Wronskian and the three Fermi Golden Rule radial integrals with an independent validated interval-arithmetic solver: if the Wronskian interval at $r=10$ contains $0$, or if any of $D_1, D_2, D_{12}$ is certified nonnegative, the paper's central claim is refuted.","supporting_citations":[],"review_version":1}