{"id":"7f7a6cd5-11e3-4136-b621-8efbcb20aae5","arxiv_id":"2411.13132","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the cubic nonlinear Klein-Gordon equation in the non-relativistic limit, the solution is approximated up to L² error ε⁴+(ε²t)^{α/4} by a first Schrödinger profile plus an ε² correction, for H^α data with α∈[4,8].","lead":"This paper derives a high-order asymptotic expansion for the 2D and 3D nonlinear Klein-Gordon equation in the non-relativistic limit, proving an error bound of order ε⁴ plus time-growth terms for H^α initial data. The result gives the first rigorous confirmation of high-order numerical observations and shows the bound is sharp.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.2 for α<8 hinges on an omitted justification in §5.2.3: uε−Φ1N−ε²Φ2N is not the remainder for a filtered NLKG solution, and its equation, initial data, and forcing terms are never written down.","rationale":"I read the paper in good faith. The central claim is the first analytic high-order expansion for the NLKG non-relativistic limit, with rate ε⁴+(ε²t)^{α/4} for α∈[4,8]. The regular case (Proposition 5.1) is structured consistently; the smallness-condition mismatch flagged by the reader appears to be a scaling notation issue that is harmless because the short-time estimate already produces the (ε²T)^{α/4} growth and the long-time conservation bound is below the target in the remaining range. The external Lei-Wu estimate (4.39) is a genuine dependence and worth checking, but I did not find an internal inconsistency in how it is quoted. The most load-bearing concern is the high-low frequency reduction in §5.2.3. The proof of Theorem 1.2 for α<8 rests on estimating uε−Φ1N−ε²Φ2N without deriving the equation that this quantity satisfies or verifying the size of its initial data and residual forcing in the norms required by the bootstrap. This is an omitted proof, not a known contradiction, so it does not force rejection; it does mean the theorem is not fully established as written for the stated range. A concrete derivation of the residual equation, as proposed in the test, would settle whether the N-balancing in Subcases A1–B3 is correct. Since the reader already recommended conditional acceptance, my read does not change that verdict, but it identifies a different and more internal weakness to be addressed. The paper deserves credit for the structure of the expansion and the careful Strichartz machinery in the regular case; the gap is localized to the reduction from H^α to H^8 data.","tokens_in":42291,"tokens_out":30200,"duration_ms":281166,"concrete_test":"Derive the exact equation for w = uε−Φ1N−ε²Φ2N by substituting into (1.1) and using the NLS and w-equations for v_N, w_N. Write the full residual R_N = ε²∂ttw−∆w+ε^{-2}w+|w+Φ1N+ε²Φ2N|²(w+Φ1N+ε²Φ2N)−|Φ1N+ε²Φ2N|²(Φ1N+ε²Φ2N) plus the profile-defect terms, and decompose R_N into the three types of Lemmas 3.1–3.3. Then check: (i) w(0) and ∂tw(0) are O(N^{-α}) in the scaled H^{1/2}×L² norm; (ii) the non-bootstrap part of R_N is bounded in Z_sc by C N^{-α}(1+ε²N²+ε²TN⁴) after all cancellations involving v_N, w_N; (iii) with N chosen as in Subcases A1–B3, the bootstrap for w closes and yields the target ε⁴+(ε²T)^{α/4}. If any term carries an unexpected N-power or lands in a stronger norm, the assertion 'we apply a similar argument to the proof of Proposition 5.1' is invalid and the theorem is unproven for α<8.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 5.2.3 the authors split v0 = P≤N v0 + P>N v0, define Φ1N, Φ2N through v_N and w_N, and then bound ‖r1‖ by ‖uε−Φ1N−ε²Φ2N‖ plus profile-difference terms. The latter are estimated by N^{-α}(1+ε²N²+ε²TN⁴) in (5.20), but the first term is dismissed with 'we apply a similar argument to the proof of Proposition 5.1'. Proposition 5.1, however, applies to the remainder uε′−Φ1−ε²Φ2 of a solution uε′ of (1.1) whose initial data match the profiles. Here uε is the original solution, and its initial data differ from Φ1N(0)+ε²Φ2N(0) by the high‑frequency part P>N u0 (and similarly for ∂t). No auxiliary solution uε,N is introduced, and the equation for w = uε−Φ1N−ε²Φ2N is not written down. The linearized bootstrap sees coefficients Φ1N+ε²Φ2N, whereas the true nonlinearity contains the full profile Φ1; the difference Φ1−Φ1N is O(N^{-α}) in L² but must be controlled in the Z_sc and S_sc norms used in Lemmas 3.1–3.3 after multiplication by w. Whether these products are small enough depends on the N-balances in Subcases A1–B3, but the displayed balancing never verifies this. If the residual forcing contains a term of order N^{-α} in a norm with a different N-power, the reduction to the regular case fails for α∈[4,8). This is an omitted step in the proof of the main theorem, not merely a notational issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the cubic nonlinear Klein-Gordon equation (1.1) in the non-relativistic limit regime for d=2,3 and proves an analytic high-order asymptotic expansion. The proposed expansion is u_ε ≈ Φ_1 + ε^2 Φ_2, where Φ_1 = e^{it/ε^2}v + c.c., Φ_2 = (1/8)e^{3it/ε^2}v^3 + e^{it/ε^2}w + c.c., v solves the cubic NLS (1.2), and w solves the Schrödinger-type equation (1.3). Theorem 1.2 claims the L^2 error is O(ε^4 + (ε^2t)^{α/4}) for H^α initial data with α∈[4,8]. Theorem 1.5 gives matching lower bounds by counterexamples. The proof strategy is to derive a residual equation for r_1, use Strichartz estimates and bootstrap arguments in scaled variables, prove the H^8 regular case, and then reduce to lower regularity by a high-low frequency decomposition of the initial data.","tokens_in":42660,"tokens_out":18472,"duration_ms":169687,"significance":"If the main theorem is correct, this is the first analytic second-order asymptotic expansion for this non-relativistic limit, improving the first-order result of Lei and Wu from ε^2 + (ε^2t)^{α/4} with α∈[1,4] to ε^4 + (ε^2t)^{α/4} with α∈[4,8]. The expansion coefficients are not fitted: they are determined by exact cancellation of the O(ε^2) residuals, and the lower bounds in Theorem 1.5 are constructed independently, so the claimed rate is genuinely falsifiable. The paper makes heavy but clearly identified use of external inputs: global scattering for the cubic NLS and the Lei–Wu first-order estimate. The overall architecture of the proof is credible, but two load-bearing points need correction before the main claim is fully established.","major_comments":[{"comment":"The reduction from Theorem 1.2 for α<8 to the regular case is incomplete. After defining Φ_1N and Φ_2N in (5.18), the proof bounds ‖u_ε−Φ_1N−ε^2Φ_2N‖ by 'a similar argument to the proof of Proposition 5.1'. Proposition 5.1, however, applies to a remainder with zero initial data and with profiles generated by the same initial data as the NLKG solution. Here u_ε has the full initial data (u_0,u_1), while the profiles are generated by P_≤N v_0. The difference u_ε(0)−Φ_1N(0)−ε^2Φ_2N(0) contains the P_>N part of the initial data, of size N^{−α}, and the time derivative has a similar P_>N contribution of size ε^{−2}N^{−α}. The equation, initial data, and forcing terms for w := u_ε−Φ_1N−ε^2Φ_2N are never written down, so the bootstrap lemmas of §3 are not demonstrably available for this function; in particular, the nonlinearity contains differences such as |Φ_1|^2Φ_1−|Φ_1N|^2Φ_1N multiplied by w, and the displayed N-balances in Subcases A1–B3 never verify the smallness of such products in the Z_sc and S_sc norms. This is not a notational issue: for α=4 with the chosen N=(ε^2T)^{−1/4}, the initial mismatch is N^{−α}=(ε^2T)^{α/4}, which is exactly the target rate and cannot be discarded without an argument.","section":"§5.2.3"},{"comment":"There is an inconsistency in the time normalization between Proposition 5.1 and Lemma 4.2. Lemma 4.2 is stated for the scaled variable R_1 satisfying the ε-independent equation (4.6), with a smallness condition ε^4T‖v_0‖_{Ḣ^{4+sc}}≤δ0. The final scaling formula (5.4) maps R_1 on [0,ε^{−2}T] back to r_1 on [0,T]. Therefore, when the proof of Proposition 5.1 is carried out for an original time interval of length T, the correct smallness condition is ε^2T‖v_0‖_{Ḣ^{4+sc}}≤δ0, not ε^4T‖v_0‖_{Ḣ^{4+sc}}≤δ0. The regular-case proof nevertheless splits the time axis using ε^4T, while the analogous non-regular condition (5.21) correctly uses ε^2T. As written, Lemma 4.2 is applied on a longer interval than its smallness condition allows. The threshold and the original/scaled interval notation should be made consistent throughout §5.1.","section":"§5.1 and Lemma 4.2"}],"minor_comments":[{"comment":"The symbol v_N is used both for the low-frequency solution of (5.6) and for the high-frequency difference v−v_N in (5.7); this makes (5.8)–(5.12) ambiguous. A separate notation such as v^N for the difference would clarify the estimates.","section":"§5.2.1"},{"comment":"The symbol ∂ttth appears to be a typo: the displayed expansion describes the third time derivative of h, so it should be ∂ttth or ∂_t^3h.","section":"§4.2.2, equation (4.16)"},{"comment":"References [15] and [16] appear to cite the same Faou–Schratz paper with identical title and journal data; if so, one duplicate should be removed.","section":"References"},{"comment":"The phrase 'There are extensive numerical and analytic results concerning that the solution...' should read 'concerning the fact that the solution...' or 'showing that...'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains a substantial and likely correct main result, and the two major issues identified above appear fixable within the paper's scope: the non-regular reduction needs a genuine proof of the remainder equation and initial-data control, and the time normalization in §5.1 needs correction. I recommend major revision rather than rejection, because the central expansion and rate are plausible and the gaps are local to the proof structure rather than contradictions in the stated theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first analytic high-order expansion in this regime, and the formal part is genuinely good. But the proof of Theorem 1.2 for α<8 has a gap in Section 5.2.3 that I would want fixed before trusting the stated range.\n\nWhat's new: Lei-Wu gave the first-order error ε²+(ε²t)^{α/4}. The authors add the ε²Φ₂ profile (the explicit ⅛e^{3it/ε²}v³ plus a corrector w), prove the error is ε⁴+(ε²t)^{α/4} for H^α data with α∈[4,8], and give lower-bound counterexamples showing the rates are sharp. If true, this is the right next step and it explains the numerical observations of Schratz-Zhao and Faou-Schratz.\n\nWhat's good: the derivation of Φ₂ via modulated Fourier expansion is clear, and the equation for w is natural. The bootstrap structure with Lemmas 3.1–3.3 is a sensible extension of Lei-Wu. The proof is long but mostly structured: Strichartz estimates for the scaled KG equation, scattering theory for cubic NLS, and a two-regime bootstrap. The counterexamples in Section 6 are serious work; the lower-bound construction using e^{it(-⟨∇⟩+1-Δ/2)}=1+⅛itΔ²+... is a believable route. I did not find a fatal flaw in the regular case (H^8 data).\n\nThe soft spot: Section 5.2.3, where the paper reduces from H^8 to H^α. The authors define v_N and w_N with low-frequency initial data and then claim ||uε-Φ₁N-ε²Φ₂N|| is handled by 'a similar argument to Proposition 5.1'. But Proposition 5.1 applies to the original uε with profiles matching the full initial data. Here uε is the original solution, so uε-Φ₁N-ε²Φ₂N has initial data mismatch from the high-frequency part P>N u₀, and the nonlinearity contains Φ₁-Φ₁N times the remainder. That residual forcing is never written down, and the bootstrap in Lemmas 3.1–3.3 would require controlling terms like (Φ₁-Φ₁N)·r₁ in the S_sc/Z_sc norms. The N-balances in Subcases A1–B3 do not display these terms; they only balance the profile differences and the smooth-profile error. It is plausible the gap is fixable by introducing an auxiliary solution with filtered initial data and adding a separate estimate for the difference, but that argument is absent. For α=8 the issue is harmless because the regular case applies, but for α∈[4,8) this is an omitted step in the main theorem.\n\nThere is also a smaller point: the smallness condition (5.21) uses ˙H^{4+sc}, but (5.22)–(5.23) are then stated with N-powers without spelling out how the bootstrap interval size and the eventual ε²t terms interact. This is cosmetic compared with the gap above.\n\nBottom line: this is a worthwhile paper with a novel result and a mostly credible proof, but the non-regular case needs repair. It deserves a serious referee; I would send it out, asking the authors to close the Section 5.2.3 gap or to state clearly that Theorem 1.2 is proved only for regular data.","headline":"First analytic high-order expansion for NLKG non-relativistic limit with sharp rates, but the non-regular range α∈[4,8) has a genuine missing step in §5.2.3.","tokens_in":43189,"tokens_out":2951,"would_cite":false,"duration_ms":29504,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B40","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the first analytic high-order asymptotic expansion for the cubic nonlinear Klein-Gordon equation in the non-relativistic limit, with an $L^2$ error of order $\\varepsilon^4+(\\varepsilon^2t)^{\\alpha/4}$ for $H^\\alpha$…","keywords":["nonlinear Klein-Gordon equation","non-relativistic limit","asymptotic expansion","Schrödinger profiles","convergence rate","Strichartz estimates","modulated Fourier expansion","sharpness"],"falsifier":"Take the Gaussian initial data $g(x)=A_0^{d/2}e^{-A_0^2|x|^2}$ used in Section 6.3.1 and compute the $L^2$ difference between the exact nonlinear Klein-Gordon solution and the two-term profile for small $\\varepsilon$ and $1\\le t\\le\\varepsilon^{-2}\\delta_0$; the paper's Lemma 6.1 predicts a lower bound growing like $\\varepsilon^4t^2\\|g\\|_{\\dot H^8}$. A direct numerical or analytic evaluation showing the error stays smaller than $c\\varepsilon^4t^2$, or grows with a different power of $t$, would refute the sharpness claim.","tokens_in":42075,"feed_emoji":"⚛️","tokens_out":13602,"duration_ms":109922,"temperature":0.7,"pith_summary":"This paper proves the first analytic high-order asymptotic expansion for the cubic nonlinear Klein-Gordon equation in the non-relativistic limit. Previous analytic results controlled the error of the leading Schrödinger profile by $\\varepsilon^2+(\\varepsilon^2t)^{\\alpha/4}$ for $H^\\alpha$ data with $\\alpha\\in[1,4]$; here a second profile is added and the leftover error is shown to be $\\varepsilon^4+(\\varepsilon^2t)^{\\alpha/4}$ for $H^\\alpha$ data with $\\alpha\\in[4,8]$. The correction consists of the third harmonic $\\tfrac18 e^{3it/\\varepsilon^2}v^3$ together with $e^{it/\\varepsilon^2}w$, where $v$ solves the cubic nonlinear Schrödinger equation and $w$ solves an explicit Schrödinger-type equation chosen to cancel all order-$\\varepsilon^2$ terms in the error equation. Counterexamples are constructed showing that this rate is optimal, up to a logarithmic factor for less regular data.","feed_headline":"Two-term profile cuts Klein-Gordon error to ε⁴","feed_subtitle":"The cubic correction makes the L² error ε⁴ + (ε²t)^{α/4} for H^α data, and the rate is sharp.","key_machinery":"The engine is the modulated Fourier (WKB) expansion: write the solution as a sum of terms $e^{imt/\\varepsilon^2}$ times profiles with $\\varepsilon$-independent time derivatives, and choose the profiles so that resonant terms cancel. The paper records three general lemmas: a forcing term of size $\\varepsilon^a O(1)$, an oscillatory forcing $\\varepsilon^{a-2}e^{imt/\\varepsilon^2}G(t,x)$ with $m\\neq1$, and a bootstrap-type nonlinearity each produce an order-$\\varepsilon^a$ remainder. Applying these to the first-order error equation, the authors split the known order-$\\varepsilon^2$ remainder into $\\varepsilon^2\\Phi_2+r_1$ and choose $\\Phi_2$ so that all order-$\\varepsilon^2$ and resonant terms vanish. The remaining error equation is controlled with Strichartz estimates, the fractional Leibniz rule, scattering bounds for the cubic nonlinear Schrödinger equation, a scaling transformation that removes $\\varepsilon$ from the equation, and a bootstrap on the high-order remainder; a high-low frequency decomposition of the initial data reduces the regularity from $H^8$ to $H^\\alpha$ with $\\alpha\\in[4,8]$.","core_discovery":"The central result is Theorem 1.2: for $d=2,3$ and $u_0,u_1\\in H^\\alpha(\\mathbb{R}^d)$, $\\alpha\\in[4,8]$, the solution $u^\\varepsilon$ of the nonlinear Klein-Gordon equation satisfies $\\|u^\\varepsilon-\\Phi_1-\\varepsilon^2\\Phi_2\\|_{L^2_x}\\le C(\\varepsilon^4+(\\varepsilon^2t)^{\\alpha/4})$ for all $t\\ge0$. Here $\\Phi_1=e^{it/\\varepsilon^2}v+c.c.$ is the usual first-order Schrödinger profile and $\\Phi_2=\\tfrac18 e^{3it/\\varepsilon^2}v^3+e^{it/\\varepsilon^2}w+c.c.$ is the new second-order profile, with $v$ solving $2i\\partial_t v-\\Delta v+3|v|^2v=0$ and $w$ solving the Schrödinger-type equation (1.3) whose initial value is fixed by matching the initial conditions of the remainder. The proof isolates the order-$\\varepsilon^2$ part of the first-order error, absorbs it into $\\Phi_2$, and shows the remaining profile $r_1$ satisfies an equation whose nonlinearities all fall into classes known to produce order-$\\varepsilon^4$ errors. Theorem 1.5 supplies matching lower bounds, so the $\\varepsilon^4$ term and the $(\\varepsilon^2t)^{\\alpha/4}$ growth are optimal.","pith_inferences":["Inference: the same cancellation mechanism should extend to still higher orders, with each additional $\\varepsilon^2$ level adding odd harmonics $e^{(2j+1)it/\\varepsilon^2}$ and requiring roughly four more derivatives of the data, following the regularity ladder $H^{4k}$ for a $k$-th order expansion.","Inference: the lower-bound analysis isolates the operator $1+\\tfrac{i}{8}t\\Delta^2$ as the dominant transport of the remainder, suggesting that the next correction beyond the nonlinear Schrödinger dynamics is a fourth-order (biharmonic) term; this could be tested by comparing the Klein-Gordon solution with the NLS evolution plus a biharmonic correction.","Inference: numerical integrators built on this two-term profile should show $\\varepsilon^4$ accuracy for $H^8$ data, and the sharp time growth gives a concrete benchmark for when the second correction is necessary."],"forward_implications":["For $H^8$ data the error is $O(\\varepsilon^4(1+t+t^2))$: two terms of the expansion are enough to push the small-$\\varepsilon$ error from order $\\varepsilon^2$ to order $\\varepsilon^4$.","For less regular data the error bound degrades as $(\\varepsilon^2t)^{\\alpha/4}$, so the expansion remains quantitatively useful over times up to roughly $\\varepsilon^{-2}\\delta_0$ before the time factor dominates.","The third harmonic $\\tfrac18 e^{3it/\\varepsilon^2}v^3$ is forced by the cubic nonlinearity; any correct second-order expansion must contain it, up to the stated freedom of absorbing terms into $w$.","The counterexamples show the bound $\\varepsilon^4+(\\varepsilon^2t)^{\\alpha/4}$ cannot be improved, up to a logarithmic factor for $H^\\alpha$ data.","The choice of $w$ is not unique, but all admissible decompositions lead to the same convergence rate."],"supporting_citations":[{"why":"Provides the first-order remainder estimate and bootstrap principle that the high-order construction is built on; its bound is quoted in estimate (4.39).","marker":"[21]"},{"why":"Supplies the numerical high-order expansions and observed $O(\\varepsilon^4)$ error that the paper now proves analytically.","marker":"[25]"},{"why":"Gives the explicit first- and second-order terms of the asymptotic expansion which motivated the choice of $\\Phi_2$.","marker":"[15]"},{"why":"Supplies the Strichartz estimates for the Schrödinger and Klein-Gordon evolutions used throughout the proof.","marker":"[19]"},{"why":"Provides the fractional Leibniz rule used to bound products of profiles and remainders.","marker":"[10]"},{"why":"Establishes scattering for the nonradial 2D cubic NLS, giving the global bounds on the profile $v$ needed in the bootstrap.","marker":"[12]"},{"why":"Gives the scattering result for the 3D cubic NLS used for the same purpose in dimension three.","marker":"[22]"},{"why":"Supplies the 3D cubic NLS smoothing and space-time bounds used in Lemma 2.6.","marker":"[17]"}],"fun_headline_variants":["First analytic error ε⁴ for nonlinear Klein-Gordon limit","Two-term expansion sharpens NLKG error to ε⁴ + (ε²t)^α/4","Klein-Gordon profile with cubic correction achieves ε⁴ accuracy","High-order expansion: Klein-Gordon error drops to ε⁴ for H^α data","Non-relativistic Klein-Gordon: first ε⁴ asymptotic error bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on a previously established first-order estimate: the leading error is at most $\\varepsilon^2$ plus $\\varepsilon^4T$ times a fixed Sobolev norm of the initial data, under the smallness condition that $\\varepsilon^4T$ times that norm is below a fixed constant. If that estimate fails or holds only on shorter times, the new $\\varepsilon^4$ convergence rate and its $(\\varepsilon^2t)^{\\alpha/4}$ growth would have to be modified.","fun_headline_variants_meta":{"raw":{"variants":["First analytic error ε⁴ for nonlinear Klein-Gordon limit","Two-term expansion sharpens NLKG error to ε⁴ + (ε²t)^α/4","Klein-Gordon profile with cubic correction achieves ε⁴ accuracy","High-order expansion: Klein-Gordon error drops to ε⁴ for H^α data","Non-relativistic Klein-Gordon: first ε⁴ asymptotic error bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000347,"raw_usage":{"total_tokens":2047,"prompt_tokens":1242,"completion_tokens":805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":858,"completion_tokens_details":{"reasoning_tokens":700}},"tokens_in":858,"tokens_out":805,"duration_ms":8182,"temperature":1.0,"reasoning_tokens":700,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:50:50.261948+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Gaussian initial data $g(x)=A_0^{d/2}e^{-A_0^2|x|^2}$ used in Section 6.3.1 and compute the $L^2$ difference between the exact nonlinear Klein-Gordon solution and the two-term profile for small $\\varepsilon$ and $1\\le t\\le\\varepsilon^{-2}\\delta_0$; the paper's Lemma 6.1 predicts a lower bound growing like $\\varepsilon^4t^2\\|g\\|_{\\dot H^8}$. A direct numerical or analytic evaluation showing the error stays smaller than $c\\varepsilon^4t^2$, or grows with a different power of $t$, would refute the sharpness claim.","supporting_citations":[{"cited_title":"Schratz and X","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical high-order expansions and observed $O(\\varepsilon^4)$ error that the paper now proves analytically."},{"cited_title":"Faou and K","cited_arxiv_id":null,"evidence_quote":"Gives the explicit first- and second-order terms of the asymptotic expansion which motivated the choice of $\\Phi_2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Strichartz estimates for the Schrödinger and Klein-Gordon evolutions used throughout the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fractional Leibniz rule used to bound products of profiles and remainders."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes scattering for the nonradial 2D cubic NLS, giving the global bounds on the profile $v$ needed in the bootstrap."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the scattering result for the 3D cubic NLS used for the same purpose in dimension three."},{"cited_title":"Ginibre and G","cited_arxiv_id":null,"evidence_quote":"Supplies the 3D cubic NLS smoothing and space-time bounds used in Lemma 2.6."}],"review_version":1}