{"id":"a391579a-c0ab-4c96-80ce-dd0685f4d2a1","arxiv_id":"2412.18620","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonlinear plasma oscillations and Landau damping are established for the Vlasov-Klein-Gordon system near radial equilibria, with electric field decay of order t^{-3/2} below the survival threshold.","lead":"This paper proves that, for the Vlasov-Klein-Gordon model of a plasma, small perturbations of stable radial equilibria produce electric fields that decay like t^{-3/2} through plasma oscillations, with an additional faster-decaying regular part. It is the first nonlinear Landau damping result in the regime where plasma oscillations, rather than phase mixing, drive the dynamics.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The spectral condition τ0^2>0 in (1.10) is not the survival threshold κ0 defined in (1.6); Theorem 2.4 yields undamped oscillatory modes at every wavenumber, so the advertised 'below threshold' regime is not characterized by the stated hypotheses.","rationale":"The reader's weakest_assumption correctly pointed to τ0^2>0 as the load-bearing spectral condition, and I agree that without it the linear argument collapses. However, the more precise formulation is that this condition is not the survival threshold κ0 defined in (1.6); the two quantities are conflated in the text. The reader's conditional verdict is based on the κ0 inconsistency and the forward reference in Section 6.7. I confirm both are real but presentation-level: the forward reference is not circular because Proposition 7.1's proof does not depend on Proposition 6.10's high-derivative source bounds, and the κ0 mismatch does not affect the internal validity of Theorem 1.1, which stands on the explicit hypotheses (1.10)–(1.11). The main theorem's proof is dense but the structural estimates—Green function decomposition, characteristic expansions, and the quadratic oscillation cancellation of Lemma 6.8—are consistent on inspection. I did not find a demonstrated fatal error, so the verdict should remain conditional rather than accept or reject. The proposed concrete test would settle whether the paper's advertised 'below survival threshold' regime is actually characterized by its hypotheses.","tokens_in":90184,"tokens_out":27814,"duration_ms":257646,"concrete_test":"Take μ(v) = C(1−|v|^2)_+ normalized to n_ions, and choose m0 so that τ0^2 = m0^2 + ∫φ'(〈v〉)dv > 0. First compute κ0 from (1.6) and from (2.12); they will differ, demonstrating the notational conflation. Then choose any |k| > κ0(1.6) and solve the dispersion relation M(iτ,k) = 0 in the form Ψ(x) = 0 from (2.17)–(2.18). Theorem 2.4 predicts a solution x* > |k|^2, giving a purely imaginary zero λ = ±i√x*. If such a solution exists, the introduction's claim that modes above the survival threshold are damped is false for this system, confirming that the theorem's hypothesis does not correspond to 'below survival threshold' as defined in (1.6).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise for the main theorem is the spectral stability condition τ0^2 = m0^2 + ∫φ'(〈v〉)dv > 0, stated in (1.10) and used in Propositions 2.3 and 2.5. This condition is genuinely needed: if τ0^2 ≤ 0, the linearized operator has a growing mode (Proposition 2.3's proof fails and M(0,0) = τ0^2 ≤ 0). The reader correctly identified this as the core hypothesis. However, the paper frames the result as 'below survival threshold' using a different quantity: κ0 in (1.6) is a wavenumber, while κ0 in (2.12) is defined as κ0^2 = −∫φ'(〈v〉)dv. These are not equivalent; for the explicit equilibrium μ(v) = C(1−|v|^2)_+ the two definitions differ by a factor of 2. More seriously, Theorem 2.4 constructs purely imaginary zeros λ± = ±iν*(|k|) for every k ∈ R^3, with ν*(|k|) > |k|, contradicting the introduction's claim that modes leave the imaginary axis for |k| > κ0. Thus the proof does not establish that the dynamics is below the survival threshold (1.6); it establishes decay for a Klein-Gordon model under an independent mass condition. This does not invalidate Theorem 1.1 as a theorem about (1.7)–(1.9), but it means the central claim's advertised regime is not what is proved. A secondary issue is that Section 6.7 invokes Proposition 7.1 before its proof; this is a forward reference that can be reordered, not a circularity, since Proposition 7.1's proof uses only the bootstrap assumptions and low-order source bounds.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies nonlinear Landau damping for the relativistic Vlasov–Klein–Gordon system (1.7)–(1.9) on R^3_x × R^3_v, near nonnegative, smooth, compactly supported radial equilibria µ(v)=φ(⟨v⟩), under the spectral stability condition τ0² = m0² + ∫φ′(⟨v⟩)dv > 0 stated in (1.10). Theorem 1.1 asserts that sufficiently small initial data satisfying (1.11) produce global solutions, and that the electric field decomposes as E = Σ± E_osc^± + E_r, with ||E_osc^±(t)||_{W^{1,p}_x} ≲ ǫ0⟨t⟩^{-3(1/2−1/p)} for p ∈ [2,8] and ||E_r(t)||_{W^{1,p}_x} ≲ ǫ0⟨t⟩^{-3(1−1/p)} for p ∈ [1,8]. A corollary claims scattering of particle trajectories. The proof proceeds through a linear spectral and Green-function analysis (Section 2), a Lagrangian formulation via nonlinear characteristics (Sections 3–5), a bootstrap for the nonlinear source density S(t,x) that separates particle–particle, particle–wave, and wave–wave interactions (Section 6), and decay estimates for the oscillatory convolution G_osc ⋆ ∇S (Section 7). The paper advertises the result as the first nonlinear Landau damping result in the plasma-oscillation, below-survival-threshold regime.","tokens_in":90444,"tokens_out":18288,"duration_ms":169680,"significance":"If the main theorem is correct, this is a substantial advance: it is the first nonlinear stability and damping result for a mean-field Vlasov model in the regime where plasma oscillations, rather than phase mixing, drive the dynamics, and it identifies the sharp Klein–Gordon decay rate t^{-3/2} for the oscillatory part. The proof is a genuine first-principles bootstrap with no fitted parameters; the dispersion analysis of Theorem 2.4, the non-stationary phase computations of Propositions 7.3–7.8, and the algebraic resonance cancellation in Lemma 6.8 giving the improved symbol bound (6.34) are concrete, checkable steps, and the claimed decay rates are falsifiable statements. The characteristic decomposition in Propositions 5.3–5.4 and 5.6 is an original and potentially reusable tool. The main caveats are that the advertised 'below survival threshold' regime is not actually characterized by the stated hypotheses (major comment 1), and that the internal organization of the proof has a circular dependency between Sections 6 and 7 (major comment 2).","major_comments":[{"comment":"The introduction and the linear theory present inconsistent phase diagrams. Section 1.4 claims that oscillatory modes exist only for 0 ≤ |k| ≤ κ0 and that 'as |k| increases past the critical wave number κ0, the phase velocity enters the range of admissible particle velocities... the dispersion functions λ±(k) leave the imaginary axis.' In contrast, Theorem 2.4, proved under the hypotheses used in Theorem 1.1, constructs exactly two imaginary-axis zeros λ±(k) = ±iν*(|k|) for every k ∈ R^3, with ν*(|k|) > |k| and hence phase velocity above the maximal relativistic particle speed; the proof of Theorem 2.4 explicitly rules out zeros in the region |τ| < |k| and finds exactly one mode with |τ| > |k| for each k. Moreover, the symbol κ0 denotes two different quantities: the survival threshold (1.6) in Section 1.4 and κ0² = −∫φ′(⟨v⟩)dv in (2.12) of Proposition 2.3, which are not equal in general (for instance they differ numerically for the explicit equilibrium discussed in the paper). The condition actually used throughout the bootstrap is (1.10), which is neither the formula (1.6) nor shown to be equivalent to the survival-threshold condition. Thus the title and abstract claim that the paper settles Landau damping 'below survival threshold' is not supported as stated; the paper proves a stability theorem for a mass-stable Vlasov–Klein–Gordon model at all wavenumbers. This should be fixed by either proving the phase diagram for (1.7)–(1.9) (including the above-threshold damped regime) or by explicitly rephrasing the advertised regime as the subcritical case τ0² > 0, and by using distinct notation for the quantities in (1.6) and (2.12).","section":"§1.4 vs. §2.4 (Theorem 2.4)"},{"comment":"The proof of Proposition 6.10 uses the estimate (6.67), namely the sharp bounds |B^β_x E_osc(t)| ≲ ǫ⟨t⟩^{-3/2} for |β| = n0 and |B^{α0−1}_x E_osc(t)| ≲ ǫ for |α0| = N0, which is asserted with 'see Proposition 7.1'. Proposition 7.1 is proved in Section 7, and its proof (via Lemma 7.2 and Propositions 7.3–7.8) uses the source-density bounds of Proposition 6.1 that Section 6 is supposed to establish. As the paper is organized, with Section 6 presenting Proposition 4.1 and Section 7 presenting Proposition 4.2, this is a genuine circular dependency, not merely a benign forward reference: the critical subcase |β| = N0 − 1 in the derivation of (6.65) cannot be closed from the bootstrap assumptions (4.6)–(4.9) alone. The fix is to state and prove Propositions 4.1 and 4.2 as a joint induction, to move the needed part of Proposition 7.1 before Section 6.7, or to replace the appeal to (6.67) in that corner case with an estimate obtained directly from the bootstrap assumptions.","section":"§6.7, proof of Proposition 6.10"},{"comment":"The linear theory that supports the entire nonlinear analysis is heavily delegated to prior work. Proposition 2.5 is described as a 'minor modification' of [23, Proposition 3.2], and the Klein–Gordon dispersive bounds (2.13)–(2.15) in Theorem 2.4 are said to follow from the 'similar lines' of [23, Theorem 2.18] with details skipped. Since the later Sections 4–7 rely on the uniform lower bound (2.16), the symbol bounds (2.27), and the dispersive estimates (2.32)–(2.36), these imports are load-bearing for the main theorem. Section 2.1 states that the linear analysis is provided 'for the sake of completeness', but the completeness claim is not met. The manuscript should either give complete proofs of these linear statements or explicitly declare them as imported results from [23] with the precise statements needed here.","section":"§2.5, Proposition 2.5"}],"minor_comments":[{"comment":"The double use of the symbol κ0 for the survival threshold in (1.6) and for the integral −∫φ′(⟨v⟩)dv in (2.12) is confusing even after the main issue is resolved; please use distinct notation and explicitly remark on the relation between the two quantities.","section":"§1.4, §2.3"},{"comment":"The phase-diagram bullets are stated for equilibria with finite maximal speed Υ and describe the linearized Vlasov–Poisson picture of [34]; the text should make explicit that this diagram is not the one proved for the Vlasov–Klein–Gordon system (1.7)–(1.9), whose phase velocity satisfies ν*(|k|) > |k| and thus never enters the range |v̂| < 1.","section":"§1.4"},{"comment":"There are small editorial issues: 'Thsi ends the proof' in Section 5.9; the spelling of the Faà di Bruno formula varies between 'Fa`a di Bruno' and 'Faa di Bruno'; and the notation ω±(k,v) is introduced in (5.4) and used again in Lemma 6.8 without restating the convention.","section":"§5.9, §5.8, §6.7"},{"comment":"In the proof of Corollary 1.1, the limiting distribution is denoted f8 in the displayed formula (4.23), while the corollary statement and the surrounding text use f∞; please unify the notation.","section":"§4.3 and Corollary 1.1"},{"comment":"In the proof of (2.34) the L^1 estimate states a decay of order ⟨t⟩^{-n−3} after the dyadic summation, while (2.34) claims ⟨t⟩^{-4+3/p−n}; the interpolation argument between L^1 and L^8 is only sketched and should be made explicit, since (2.34) is used directly in the proof of Theorem 1.1.","section":"§2.6, (2.34)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a very long companion to the author's prior works [22, 23, 34], and several crucial linear propositions are 'minor modifications' of results there. That is legitimate, but the editor may wish to assess the incremental novelty relative to those papers. The 'below survival threshold' framing will likely draw criticism from specialists in the linear theory, since Theorem 2.4 shows oscillatory modes at every wavenumber under the theorem's hypotheses and the two uses of κ0 are not identified; the author should be asked to state precisely what is proved about the phase diagram. The circularity between Section 6.7 and Section 7 is a structural issue that a careful reader will notice immediately; it is fixable but should not be left in the published version. If the joint induction is written out and the threshold claim is rephrased, the paper would be a strong contribution to the subject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it is the first nonlinear Landau damping result in the plasma-oscillation regime, and the main theorem is real: global existence near radial equilibria for the Vlasov-Klein-Gordon system, with the electric field split into an oscillatory part decaying like t^{-3/2} and a regular part decaying like t^{-3}, for finite Sobolev data. That is a genuine advance, and the physical-space iterative scheme treating particle-particle, particle-wave, and wave-wave interactions is an impressive piece of work. The paper also re-derives the linear theory in Section 2, so it is not simply leaning on prior work. Credit is earned here.\n\nThe soft spot is the survival-threshold framing. Eq. (1.6) defines kappa_0 as a wavenumber, depending on the maximal particle speed. Eq. (2.12) defines kappa_0^2 differently, as -int phi'(<v>) dv. These are not the same object; for an explicit compactly supported profile they differ by a factor of 2. More importantly, Theorem 2.4 constructs two imaginary-axis zeros for every k, with nu_*(|k|) > |k|, which contradicts the introduction's claim that modes leave the imaginary axis for |k| > kappa_0. The actual condition used in the proof is tau_0^2 = m_0^2 - kappa_0^2(2.12) > 0, a spectral mass condition, not a threshold on wavenumbers. So the paper proves stability for a Klein-Gordon model with positive effective mass, and the 'below survival threshold' terminology is misleading. This does not invalidate Theorem 1.1 as a theorem about (1.7)-(1.9), but it means the advertised physical regime is not what is proved. The author should be asked to clarify the relation between tau_0, kappa_0(1.6), and kappa_0(2.12), and to temper the abstract and introduction accordingly.\n\nOther issues are minor: Section 6.7 invokes Proposition 7.1 before it is proved (a forward reference, easily reordered), and a few estimates are delegated to [23] with 'minor modification.' Given the length, I could not verify every inequality in one pass, and the proof is not machine-checked. But I did not see an obvious fatal gap.\n\nWho is this for: kinetic theorists and PDE analysts working on Landau damping. It deserves a serious referee; the main theorem, if correct, is a major within-field advance. My recommendation: engage with it, send it to peer review, but require the author to fix the survival-threshold framing before publication.","headline":"First nonlinear result in the plasma-oscillation regime, but the advertised survival-threshold framing does not match the actual hypotheses; the theorem itself is substantial and deserves serious referee time.","tokens_in":91097,"tokens_out":3782,"would_cite":true,"duration_ms":37814,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q83","35B40","82D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonlinear Landau damping proven below the survival threshold.","keywords":["Landau damping","plasma oscillations","Vlasov-Klein-Gordon system","survival threshold","phase mixing","Klein-Gordon dispersion","nonlinear stability","plasma echoes"],"falsifier":"One concrete test is to solve the linearized Vlasov-Klein-Gordon system for a smooth compactly supported radial equilibrium with $\\tau_0^2>0$ and measure $\\|E(t)\\|_{L^\\infty}$ at low wavenumbers: the paper predicts oscillatory decay at rate $t^{-3/2}$, with the phase-mixing part negligible in comparison. If the observed decay were instead $t^{-1}$ or absent, the Green-function decomposition of the linear theory would be false. At the nonlinear level, a numerical check that the source density $S(t,x)$ satisfies the claimed weighted $L^p$ bounds for a single Sobolev-initial-data run would also falsify the bootstrap.","tokens_in":89835,"feed_emoji":"⚡","tokens_out":8569,"duration_ms":78033,"temperature":0.7,"pith_summary":"This paper proves global existence and nonlinear Landau damping for the relativistic Vlasov–Klein-Gordon system in three space dimensions, near general radial equilibria with compactly supported velocity profiles. The electric field is shown to split into Langmuir oscillatory waves that disperse like Klein-Gordon waves at rate $t^{-3/2}$, plus a regular part decaying at $t^{-3}$; particles scatter in the large time. This is the first nonlinear Landau damping result that operates below the survival threshold, where the dynamics is driven by collective oscillations rather than by phase mixing alone. If correct, it confirms the classical plasma-physics picture of Langmuir waves persisting and being damped by spatial dispersion even after nonlinear effects, plasma echoes, and finite-regularity perturbations are included.","feed_headline":"Proven: nonlinear Landau damping below survival threshold","feed_subtitle":"Electric field splits into Klein-Gordon waves decaying at $t^{-3/2}$ and a $t^{-3}$ remainder; particles scatter.","key_machinery":"The load-bearing object is the spacetime symbol $M(\\lambda,k)=\\lambda^2+|k|^2+m_0^2+\\int \\frac{ik\\cdot\\hat v}{\\lambda+ik\\cdot\\hat v}\\,\\varphi'(\\langle v\\rangle)\\,dv$, the Vlasov-Klein-Gordon analogue of the plasma dielectric function. Under the spectral condition $\\tau_0^2>0$, it has exactly two imaginary-axis zeros $\\lambda_\\pm(k)=\\pm i\\nu_*(|k|)$ whose dispersion is Klein-Gordon-like; the corresponding oscillatory Green kernels $G^{\\mathrm{osc}}_{\\pm}$ decay at $t^{-3/2}$, while the regular kernel $G^r$ decays faster and is governed by free transport. The nonlinear machinery is Lagrangian: characteristics are shown to be superpositions of a Klein-Gordon oscillation of amplitude $s^{-3/2}$ and a transport part of amplitude $s^{-1}$, and every quadratic interaction (particle-particle, particle-wave, wave-wave) is handled by time integrations by parts that decouple oscillations from phase mixing. A bootstrap scheme propagates sharp decay in low norms while allowing controlled growth in high norms, with the cascade parameter $\\delta_\\alpha=|\\alpha|/(N_0-1)$ entering through the asserted derivative bounds.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.1: for equilibria $\\mu(v)=\\varphi(\\langle v\\rangle)$ that are nonnegative, smooth, radial, and compactly supported, with $\\int \\varphi(\\langle v\\rangle)\\,dv=n_{\\mathrm{ions}}$ and $\\tau_0^2=m_0^2+\\int \\varphi'(\\langle v\\rangle)\\,dv>0$, any sufficiently small initial perturbation satisfying (1.11) yields a global solution whose electric field decomposes as $E=\\sum_{\\pm}E^{\\mathrm{osc}}_{\\pm}+E^r$, with the oscillatory part obeying $\\|E^{\\mathrm{osc}}_{\\pm}(t)\\|_{W^{1,p}_x}\\lesssim\\epsilon_0\\langle t\\rangle^{-3(1/2-1/p)}$ for $p\\in[2,8]$ and the regular part obeying $\\|E^r(t)\\|_{W^{1,p}_x}\\lesssim\\epsilon_0\\langle t\\rangle^{-3(1-1/p)}$ for $p\\in[1,8]$. The corollary is scattering of particle trajectories: final velocities exist, and the distribution function converges along free-transport characteristics at rate $t^{-1/2}$. In physical terms, Landau damping below the survival threshold is not exponential relaxation to a phase-mixed state; it is the slow dispersive decay of plasma oscillations.","pith_inferences":["A neighbouring problem the paper leaves implicit is the same below-threshold theorem for the relativistic Vlasov-Maxwell system, whose magnetic component already shows Klein-Gordon dispersion over the full wavenumber range in the linear analysis cited by the paper; the characteristic and resonant-integration framework here supplies the electric-field template for such a proof.","The condition $\\tau_0^2>0$ excludes both the critical case $\\tau_0^2=0$ and equilibria with infinite maximal velocity; near-threshold resonances and Gaussian-tail equilibria would presumably require a genuinely different mechanism, not a minor adaptation.","The stated Sobolev index $N_0\\ge 14$ is described by the author as non-optimal; a plausible refinement is to use paraproduct cancellations to reduce the number of derivatives, since the cascade parameter already allows growth in all but the top derivative and could likely be sharpened."],"forward_implications":["Below the survival threshold, the long-time electric field has two separated components: Klein-Gordon oscillatory waves decaying as $t^{-3/2}$ and a faster $t^{-3}$ phase-mixing/transport remainder, so oscillatory tails are generic rather than exponential in this regime.","Particles scatter: each trajectory has a well-defined final velocity, with $|V(t;x,v)-V_\\infty(x,v)|\\lesssim\\epsilon_0 t^{-3/2}$ and $|X(t;x,v)-x+tV_\\infty(x,v)|\\lesssim\\epsilon_0 t^{-1/2}$; consequently $f(t,x+t\\hat v,v)$ converges to a $C^1$ profile at rate $t^{-1/2}$.","Nonlinear Langmuir waves are stable objects: plasma echoes and particle-wave, wave-wave interactions do not destroy the linear Klein-Gordon dispersion, at least for small amplitude, finite-Sobolev, compactly supported data.","All previous nonlinear damping results lived in the phase-mixing regime; this result closes a gap by treating the regime in which the electric field itself drives the dynamics."],"supporting_citations":[{"why":"Supplies the physical identification of Langmuir oscillatory waves and the dielectric-function formalism whose zeros define the modes.","marker":"[39]"},{"why":"Provides the original Landau damping calculation, whose below-threshold regime this paper makes nonlinear.","marker":"[30]"},{"why":"Establishes the survival threshold $\\kappa_0$ and the linear Klein-Gordon dispersion of Langmuir modes that the proof starts from.","marker":"[34]"},{"why":"Gives the resolvent and Green-function construction for Vlasov-Maxwell that is adapted here to Vlasov-Klein-Gordon.","marker":"[23]"},{"why":"Confirms that linearized plasma oscillations disperse like Klein-Gordon waves; Theorem 1.1 extends this to the nonlinear level.","marker":"[22, 9]"},{"why":"Provides the benchmark nonlinear Landau damping result in the phase-mixing regime, which this paper complements below the threshold.","marker":"[32]"},{"why":"Supplies the screened Vlasov-Poisson Lagrangian characteristic framework that the nonlinear analysis borrows and refines.","marker":"[21]"},{"why":"Recent scattering theory for Vlasov-Klein-Gordon that underlies the treatment of particle-wave interaction used in the proof.","marker":"[35]"}],"fun_headline_variants":["Plasma oscillations beat exponential damping below survival threshold","Nonlinear Landau damping via Klein-Gordon decay at t^{-3/2}","Beyond phase mixing: Landau damping with plasma oscillations","Slow dispersive decay t^{-3/2} in nonlinear Landau damping"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the spectral-stability condition $\\tau_0^2=m_0^2+\\int\\varphi'(\\langle v\\rangle)\\,dv>0$; if this effective Klein-Gordon mass squared were zero or negative, the linearized oscillatory modes could be unstable and the whole Green-function decomposition and bootstrap would fail.","fun_headline_variants_meta":{"raw":{"variants":["Plasma oscillations beat exponential damping below survival threshold","Nonlinear Landau damping via Klein-Gordon decay at t^{-3/2}","Beyond phase mixing: Landau damping with plasma oscillations","Slow dispersive decay t^{-3/2} in nonlinear Landau damping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000376,"raw_usage":{"total_tokens":2108,"prompt_tokens":1156,"completion_tokens":952,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":878}},"tokens_in":772,"tokens_out":952,"duration_ms":7748,"temperature":1.0,"reasoning_tokens":878,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:39:27.304525+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete test is to solve the linearized Vlasov-Klein-Gordon system for a smooth compactly supported radial equilibrium with $\\tau_0^2>0$ and measure $\\|E(t)\\|_{L^\\infty}$ at low wavenumbers: the paper predicts oscillatory decay at rate $t^{-3/2}$, with the phase-mixing part negligible in comparison. If the observed decay were instead $t^{-1}$ or absent, the Green-function decomposition of the linear theory would be false. At the nonlinear level, a numerical check that the source density $S(t,x)$ satisfies the claimed weighted $L^p$ bounds for a single Sobolev-initial-data run would also falsify the bootstrap.","supporting_citations":[{"cited_title":"Klainerman and G","cited_arxiv_id":null,"evidence_quote":"Provides the original Landau damping calculation, whose below-threshold regime this paper makes nonlinear."},{"cited_title":"Masmoudi and W","cited_arxiv_id":null,"evidence_quote":"Establishes the survival threshold $\\kappa_0$ and the linear Klein-Gordon dispersion of Langmuir modes that the proof starts from."},{"cited_title":"Grenier, T","cited_arxiv_id":null,"evidence_quote":"Gives the resolvent and Green-function construction for Vlasov-Maxwell that is adapted here to Vlasov-Klein-Gordon."},{"cited_title":"Kunzinger, G","cited_arxiv_id":null,"evidence_quote":"Provides the benchmark nonlinear Landau damping result in the phase-mixing regime, which this paper complements below the threshold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the screened Vlasov-Poisson Lagrangian characteristic framework that the nonlinear analysis borrows and refines."},{"cited_title":"Mouhot and C","cited_arxiv_id":null,"evidence_quote":"Recent scattering theory for Vlasov-Klein-Gordon that underlies the treatment of particle-wave interaction used in the proof."}],"review_version":1}