{"id":"3755b5dc-87c2-4983-b004-e18f56ddc830","arxiv_id":"2412.00187","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Bootstrap constraints plus QCD sum rules give a lower bound on the hadronic muon anomaly and an emergent rho-meson peak.","lead":"The paper derives a lower bound on the leading hadronic contribution to the muon's anomalous magnetic moment using only unitarity, crossing, analyticity, and QCD finite-energy sum rules, without fitting to g-2 data. It also finds that the bound's extremal solution produces a rho-meson-like peak, a possible bootstrap prediction of the rho mass.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed lower bound rests on FESR inputs at s0=1.19 GeV^2 with unquantified OPE truncation error, acknowledged in Appendix C; a shift in F_n beyond quoted errors would move Min[a_mu] by more than the stated +/-3.","rationale":"The reader identified the same weakest assumption: the FESR inputs at s0=1.19 GeV^2, with errors that do not quantify OPE truncation. The paper itself, in Appendix C, concedes that the OPE may break down at such low s0 and that a quantitative estimate is left for future work. This is the single most load-bearing concern because the entire lower-bound argument converts FESR values into inequality constraints; if those values are uncertain beyond the quoted box, the extremal spectral density and the resulting Min[a_mu] can change by more than the methodological +/-3 error quoted in the main result. The paper does provide useful supporting evidence: convergence in the primal truncation parameters N, L, P is checked, and Step 1 reproduces and improves the positivity-only bound of [19]. The scan over tolerance eps is a reasonable way to display dependence on the quoted errors, but it does not cover OPE truncation uncertainty. No independent machine-checked proof or released code is provided, so the numerical claims are not formally verified. The correct disposition is therefore the same conditional one reached by the reader: the central lower bound should be accepted only after the OPE truncation uncertainty at s0=1.19 GeV^2 is quantified, either by adding higher-dimensional terms or by demonstrating stability under changes of s0 and renormalization scale.","tokens_in":19352,"tokens_out":6839,"duration_ms":66927,"concrete_test":"Recompute the FESR central values in Appendix C at s0=1.19 GeV^2 including the first omitted higher-dimensional OPE term (for example, a dimension-eight gluon-quark condensate) and also under a factor-of-two variation of the renormalization scale; then rerun the Step-2 primal bootstrap with the shifted F_n. If Min[a_mu^LO-HVP] moves by more than 3 x 10^-10, the quoted lower bound is not robust to OPE truncation and the uncertainty on the bound must be widened accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result Min[a_mu^LO-HVP] = 688.4(+3/-3) x 10^-10 is obtained by imposing three FESR inequalities F_n in [mean - eps*error, mean + eps*error] with eps=1. The quoted errors are propagated from QCD parameters such as <alpha G^2> and kappa in [19]; they do not include the error of truncating the OPE at dimension six. Since s0=1.19 GeV^2 is close to the rho mass, the OPE is being used in a regime where its convergence is questionable. Appendix C explicitly states that at sufficiently low s0 the OPE may break down or its truncation error may become significant, and it leaves a quantitative estimate to future work. The eps-scan only rescales the quoted parameter errors; it cannot probe shifts of the central values caused by omitted higher-dimensional operators or non-OPE effects. If, for example, dimension-eight condensate contributions change F1 or F2 by a few times 10^-7, the feasible set changes and the extremal spectral density, hence Min[a_mu], shifts. The paper quotes only numerical/methodological uncertainties of +/-3, so the advertised lower bound is not robust against the largest acknowledged systematic. This is load-bearing because the consistency with the SM prediction, the exclusion of condensate regions, and the rho-peak claim all depend on the FESR constraints being the true QCD constraints.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a bootstrap calculation of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment. The author imposes unitarity (via a positive semi-definite matrix involving the spectral density, pion form factor, and pion partial waves), analyticity and crossing symmetry of pion amplitudes, and QCD finite-energy sum rules (FESRs) imposed as inequalities with a tolerance epsilon. Minimizing a_mu^LO-HVP over truncated basis expansions and extrapolating in the truncation order N gives Min[a_mu^LO-HVP] = 688.4(+3/-3) x 10^-10 for the conservative epsilon=1 case, and a combined lower bound a_mu^bootstrap-min-conservative = 11659176.3(+3/-3) x 10^-10, which is consistent with the SM data-driven value. The extremal spectral density exhibits a rho-like peak near sqrt(s)=0.73 GeV, which the paper labels a bootstrap prediction for the rho mass. Scanning epsilon from 1 to 0, the author reports an 'average' bound that is incompatible with the SM prediction but compatible with lattice and measurement, and uses this scan to exclude some QCD condensate parameter regions and to propose benchmark values for <alpha G^2> and kappa.","tokens_in":19662,"tokens_out":8378,"duration_ms":75414,"significance":"The paper is a serious attempt to bring S-matrix/bootstrap methods to bear on a precision observable, and it improves on the positivity-only bound of Li et al. [19] by imposing the full unitarity matrix. The numerical setup is described in unusual detail: convergence in N, P, and spin L is documented in Figs. 3-6, extrapolation procedures are stated, and the Step-1 comparison with Ref. [19] gives a useful calibration. If the reported lower bound could be certified as a rigorous bound, this would be a valuable non-perturbative constraint on the hadronic contribution to g-2. At present, however, the central bound is an extrapolated primal result rather than a dual certificate, and the FESR inputs at s0=1.19 GeV^2 carry an unquantified OPE truncation error that the paper itself acknowledges in Appendix C. These two issues are load-bearing for the main claims.","major_comments":[{"comment":"The quantity reported as Min[a_mu^LO-HVP] is obtained by minimizing the objective over a truncated polynomial basis (N up to 95) and then extrapolating to large N with fitted models such as a + b/N^2 and a + b exp(-0.07N). This is a primal computation: each finite-N optimum is the minimum over a restricted function space, and is therefore an upper bound on the true infimum of a_mu^LO-HVP over the full admissible set, not a certificate that all physical spectral densities satisfy a_mu >= Min. The extrapolation and the +/-3 error bars reflect model spread, not a proof of exclusion. The abstract and conclusion state that the paper 'establish[es] a lower bound'; to support that claim the author should either provide a dual SDP bound (or an explicit rigorous relaxation) or rephrase the central result as an approximate lower bound whose systematic uncertainty is quantified.","section":"Sec. III A, Eq. (A13), Figs. 3-5"},{"comment":"The central result uses FESR inputs at s0 = 1.19 GeV^2 with central values and errors taken from Ref. [19]. The errors in Eq. (8) propagate uncertainties in QCD parameters such as <alpha G^2> and kappa, but do not include the error from truncating the OPE at dimension six. The paper explicitly states in Appendix C that at sufficiently low s0 the OPE may break down or its truncation error may become significant, and that a quantitative estimate is left to future work. Because s0 is only slightly above the rho mass, this is not a negligible effect: a shift in F1 or F2 by a few times 10^-7 from higher-dimensional operators or non-OPE contributions would move the minimized bound by more than the quoted +/-3. The epsilon-scan only rescales the quoted parameter errors and cannot probe shifts of the central values. The paper should either provide a quantitative estimate of the OPE truncation error, or show that the bound is stable under variations of s0 and under inclusion of dimension-eight terms, before presenting the result as a conservative lower bound.","section":"Appendix C and Eq. (8)"},{"comment":"The claim that the rho-like peak near sqrt(s)=0.73 GeV is a 'bootstrap prediction for the rho-meson mass' is overstated. The FESR moments F0, F1, F2 in Eq. (8) are themselves derived from QCD sum rules whose central values depend on quark masses and condensates, and their relative magnitudes effectively set the position of the resonance peak in the reconstructed spectral density. The minimization is therefore not independent of the rho scale; it is a consistency check at best. To make the prediction claim meaningful, the author should demonstrate that the peak position is stable when the F_i are varied within their assigned errors (or within the tolerance scan), and discuss how much of the peak location is inherited from the input sum rules rather than from the bootstrap constraints. The text should also address the fact that the peak appears at 0.73 GeV, about 6% below the physical rho mass.","section":"Sec. II, Fig. 2, and Conclusion"}],"minor_comments":[{"comment":"In Sec. III C, 'Once can be more precise' should be 'One can be more precise'.","section":"Sec. III C"},{"comment":"In Appendix B, 'CDM-3 reported' should be 'CMD-3 reported'.","section":"Appendix B"},{"comment":"The notation ahad_mu appears without a definition; please define it consistently as aLO-HVP_mu or introduce it explicitly.","section":"Sec. II"},{"comment":"The phrase 'saturates our conservative lower bound' is ambiguous; the plot shows the SM value lying within error bars of the bound, not saturating it in the usual sense. Please rephrase.","section":"Fig. 1 caption"},{"comment":"The relation between rho^1_1(s) and ImPi(s), namely rho^1_1(s) x (2pi)^4 / s = ImPi(s), is stated only once; spell out the normalization when the FESR integrals are defined in Eq. (A13), since the powers of s and (2pi)^4 appear confusing at first reading.","section":"Eq. (2) and Eq. (A13)"},{"comment":"In Table II, the bootstrap row '63 335 284 682' sums to 682, but the conservative bound in the main text is 688.4 after adding charmonium and bottomonium contributions; clarify which total is being compared and why the table's total differs.","section":"Table II"},{"comment":"Reference [6] contains a second citation (F. Bhat et al.) inside the same bracket; please split or format it consistently.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is on the border between hep-th bootstrap methodology and precision QCD phenomenology; if the journal welcomes bootstrap applications to observables, the topic fits. The two load-bearing concerns are the absence of a rigorous dual certificate for the claimed lower bound and the unquantified OPE truncation error at the FESR scale. Both can potentially be addressed in revision by rephrasing claims and adding a sensitivity analysis. I do not see a circularity problem: the FESR inputs are external, but the paper does not fit a parameter to the target observable. One unusual feature is that the caption of Fig. 7 thanks anonymous referees; this should be removed or attributed properly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version first. The genuinely new thing here is that the paper runs a full S-matrix bootstrap—unitarity, crossing, analyticity, plus the FESRs—to bound a_mu^LO-HVP, and it gets a stronger lower bound than the positivity-only result from Li et al. The rho-like peak that shows up without any resonance ansatz is a nice demonstration, and the numerics look reasonably converged. The conservative bound at eps=1 is consistent with the data-driven SM value, and the paper is transparent about the method being primal rather than a rigorous dual bound.\n\nWhat it does well: the step-by-step implementation is clear, convergence in N, P, L is shown, comparisons with [19] are fair, and they don't oversell the average-over-tolerance result—they explicitly call it a heuristic. The window observable comparison is a nice check.\n\nSoft spots, in order. The one that matters most: the FESR central values at s0=1.19 GeV^2 carry an unquantified OPE truncation error, and the paper says so in Appendix C. The epsilon scan only rescales the quoted parameter errors; it cannot probe a shift of the central values themselves from omitted higher-dimensional condensates or non-OPE effects. If those shift F1 or F2 by a few times 10^-7, the feasible set moves and the lower bound moves by more than the quoted +/-3. Since the consistency with the SM prediction and the rho-peak claim both depend on these FESR constraints being the true QCD constraints, this is load-bearing. Second, the bound itself is from a truncated primal optimization with extrapolation, so calling it a 'lower bound' is conditional on the extrapolation being correct. Third, the rho mass has no error estimate, and the peak position is essentially set by the ratio of the FESR moments, so 'prediction' is a bit strong. None of these are fatal—the paper is honest about the first and third—but they are real.\n\nVerdict: this is a worthwhile paper for the bootstrap and g-2 communities. It deserves a serious referee and probably publication after the OPE uncertainty is quantified or at least clearly caveated in the abstract. My own view: the central claim 'bootstrap establishes a lower bound' is not yet fully established, but the method and the numerics are solid enough to engage with. Bring it to reading group if you want a good discussion.","headline":"A solid bootstrap application to the hadronic muon anomaly, with a real caveat about unquantified OPE truncation at the chosen sum rule scale.","tokens_in":20200,"tokens_out":2854,"would_cite":false,"duration_ms":23484,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A bootstrap of unitarity and QCD sum rules puts the muon anomaly's hadronic floor at 688.4 x 10^-10.","keywords":["muon anomalous magnetic moment","hadronic vacuum polarization","bootstrap constraints","finite energy sum rules","QCD condensates","rho resonance","unitarity","crossing symmetry"],"falsifier":"Recompute the three finite-energy sum rule moments $F_0,F_1,F_2$ at $s_0=1.19\\,\\mathrm{GeV}^2$ with a higher-order or resummed operator-product expansion and check whether they remain inside the quoted error intervals; if any moment moves outside, the bootstrap's feasible set changes and the lower bound must shift off $688.4\\times10^{-10}$. A precise experimental $R(s)$ scan around $\\sqrt{s}=0.73\\,\\mathrm{GeV}$ would also show whether the extremal single-peak spectral density is the true minimizer or whether additional near-peak strength is being missed.","tokens_in":19138,"feed_emoji":"🧲","tokens_out":13220,"duration_ms":105627,"temperature":0.7,"pith_summary":"The paper tries to establish that the leading-order hadronic contribution to the muon's anomalous magnetic moment can be bounded from below by a bootstrap: unitarity expressed as a matrix-positivity condition on the hadronic spectral density, pion partial wave, and vector form factor, together with analyticity, crossing symmetry, and QCD finite-energy sum rules. A sympathetic reader should care because this term carries the largest theoretical uncertainty in the muon g-2 discrepancy, and a rigorous lower bound would clarify whether the measured value points to new physics or to underestimated hadronic effects. The conservative result is a lower bound of $688.4^{+3}_{-3}\\times10^{-10}$ on the leading hadronic contribution and $11659176.3^{+3}_{-3}\\times10^{-10}$ on $a_\\mu$ itself, consistent with the data-driven Standard Model value within errors. The extremal spectral density that saturates the bound develops a rho-meson-like peak near $\\sqrt{s}=0.73\\,\\mathrm{GeV}$ without any resonance input, and varying the sum-rule tolerance moves the bound upward, with average-tolerance values matching lattice and experiment rather than the data-driven Standard Model.","feed_headline":"Bootstrap puts muon anomaly's hadronic floor at 688.4 x 10^-10","feed_subtitle":"Unitarity, analyticity, crossing, and QCD sum rules alone also produce a rho-like peak.","key_machinery":"The central object is the $3\\times3$ matrix $B(s)$ whose entries are the pion $P$-wave $S_1^1(s)$, the vector form factor $F_1^1(s)$, and the hadronic spectral density $\\rho_1^1(s)$; the bootstrap demands $B(s)\\succeq0$ for $s>4$, which encodes unitarity and implies the simple condition $\\rho_1^1(s)\\ge|F_1^1(s)|^2$ as one of its principal-minor constraints. Around this matrix the argument builds three further ingredients: the QCD finite-energy sum rules $F_n=\\frac{1}{s_0^{1+n}}\\int_4^{s_0}t^n\\frac{\\mathrm{Im}\\Pi(t)}{(2\\pi)^4}dt$ for $n=0,1,2$ at $s_0=1.19\\,\\mathrm{GeV}^2$, imposed as inequalities with tolerance $\\epsilon$ times the quoted error; partial-wave unitarity $|S_\\ell^I(s)|\\le1$ for isospin $I=0,1,2$ and spins up to $\\ell=9$; and tree-level chiral perturbation theory as a low-energy constraint. All functions are expanded in analytic basis sets suited to their crossing or dispersion properties, with truncation levels $P$ and $N$, and the minimization of $a_\\mu^{\\mathrm{LO\\text{-}HVP}}$ over the expansion coefficients is carried out with a semidefinite-program solver, with convergence in $N,P,\\ell$ checked numerically.","core_discovery":"The paper claims that the bootstrap constraints are strong enough to fix a genuine lower bound on $a_\\mu^{\\mathrm{LO\\text{-}HVP}}$. Imposing the full unitarity condition $B(s)\\succeq0$, partial-wave unitarity up to spin 9, analytic and crossing-symmetric pion amplitudes, and the three FESRs at $s_0=1.19\\,\\mathrm{GeV}^2$ with a conservative tolerance equal to their quoted errors gives Min$[a_\\mu^{\\mathrm{LO\\text{-}HVP}}]=688.4^{+3}_{-3}\\times10^{-10}$; adding the charmonium and bottomonium contributions and the precisely computed QED and electroweak terms yields $a_\\mu^{\\mathrm{bootstrap\\text{-}min\\text{-}conservative}}=11659176.3^{+3}_{-3}\\times10^{-10}$, which the data-driven Standard Model value saturates within errors. The extremal spectral density realizing this minimum is not fitted to data: it has essentially one peak near $\\sqrt{s}=0.73\\,\\mathrm{GeV}$, matching the rho resonance, which the paper presents as a bootstrap prediction of the rho mass. When the tolerance parameter $\\epsilon$ is scanned from 1 down to 0, the lower bound rises through values that disagree with the data-driven Standard Model prediction but remain consistent with the measured anomaly, until $\\epsilon\\lesssim0.2$ is excluded by measurement. The average over tolerance, $11659204.3^{+1.6}_{-1.6}\\times10^{-10}$, is saturated by the lattice evaluation and lies within the measured value's error bars, though the paper notes this average is a heuristic benchmark rather than a statistically rigorous result.","pith_inferences":["Editorial extension: applying the same positivity matrix to the electron or tau anomalies would produce analogous lower bounds that could be checked against independent lattice and dispersion evaluations, giving a low-cost cross-check of the method.","Editorial extension: the extremal one-peak spectral density implies a sharp qualitative prediction that the low-energy hadronic spectral function is dominated by the rho; a bootstrap extended with coupled channels or more data would either preserve this peak or force additional structure, which is a testable difference.","Editorial extension: the excluded region in the ($\\langle\\alpha G^2\\rangle,\\kappa$) plane suggests that an independent determination of either condensate would convert the experimental constraint into a two-sided bound on the other, something the paper leaves open.","Editorial extension: comparing the bootstrap's Euclidean window observables with lattice window data, where the present one-peak solution overshoots the intermediate window, could provide a sharper lattice-versus-bootstrap test than the total lower bound alone."],"forward_implications":["If the conservative bound is correct, no admissible spectral density satisfying unitarity, analyticity, crossing, and the FESRs can lower the leading hadronic contribution below $688.4\\times10^{-10}$; the data-driven Standard Model value already sits at that floor.","Full unitarity strengthens the light-quark bound from $630.7\\times10^{-10}$ (positivity plus FESRs) to $680.0\\times10^{-10}$, so the matrix positivity constraint is doing real work beyond simple spectral positivity.","The rho-like peak emerges dynamically from minimization, so the same bootstrap machinery can serve as a first-principles estimator of the rho mass ($\\sqrt{s}\\approx0.73\\,\\mathrm{GeV}$) without assuming any resonance shape.","Tightening the FESR tolerance moves the lower bound upward; tolerances below about $\\epsilon=0.2$ are ruled out by the measured muon anomaly, which translates into constraints on the gluon condensate and vacuum-saturation parameter.","The average-tolerance bound is saturated by the lattice value and compatible with the measured anomaly, so within the FESR uncertainties the bootstrap can accommodate both the measured value and the lattice, but not simultaneously the data-driven Standard Model value."],"supporting_citations":[{"why":"Supplies the three finite-energy sum rule moments at $s_0=1.19\\,\\mathrm{GeV}^2$ with central values and errors, and the positivity-only lower bound that the bootstrap improves on.","marker":"[19]"},{"why":"Introduces the positive-semidefinite unitarity matrix and the analytic basis expansions for the spectral density and form factor.","marker":"[8]"},{"why":"Provides the generalized unitarity relation among spectral density, pion partial wave, and form factor, and the chiral perturbation theory tolerance used at low energy.","marker":"[9]"},{"why":"Supplies the crossing-symmetric analytic ansatz for pion scattering amplitudes from which the partial waves are built.","marker":"[10]"},{"why":"Provides the semidefinite program solver used to impose the matrix positivity condition numerically.","marker":"[24]"},{"why":"Gives the Standard Model data-driven value of $a_\\mu$ and the precisely calculated electroweak and QED contributions added to the bootstrap bound.","marker":"[3]"},{"why":"Supplies the charmonium and bottomonium resonance contributions added to the light-quark bootstrap result.","marker":"[20]"},{"why":"Gives the lattice value that saturates the average-tolerance lower bound, used for comparison in the tolerance scan.","marker":"[5]"},{"why":"Provides the measured muon anomaly used to decide which tolerance choices are ruled out by experiment.","marker":"[1]"}],"fun_headline_variants":["Bootstrap pins hadronic muon floor at 688.4e-10","Bootstrap predicts rho peak, muon anomaly floor at 688.4e-10","QCD sum rules alone enforce muon anomaly lower bound: 688.4e-10","Muon anomaly floor from bootstrap: 688.4e-10, rho emerges","Bootstrap gives rho mass and hadronic muon floor at 688.4e-10"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the QCD finite-energy sum rules evaluated at $s_0=1.19\\,\\mathrm{GeV}^2$, with their published central values and error bars, genuinely constrain the hadronic spectral density; the paper itself notes in Appendix C that the operator-product-expansion truncation error at this scale is not quantified, so if that error shifts the sum rules beyond the quoted uncertainties, the derived lower bound changes.","fun_headline_variants_meta":{"raw":{"variants":["Bootstrap pins hadronic muon floor at 688.4e-10","Bootstrap predicts rho peak, muon anomaly floor at 688.4e-10","QCD sum rules alone enforce muon anomaly lower bound: 688.4e-10","Muon anomaly floor from bootstrap: 688.4e-10, rho emerges","Bootstrap gives rho mass and hadronic muon floor at 688.4e-10"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001047,"raw_usage":{"total_tokens":4469,"prompt_tokens":1086,"completion_tokens":3383,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":702,"completion_tokens_details":{"reasoning_tokens":3275}},"tokens_in":702,"tokens_out":3383,"duration_ms":21585,"temperature":1.0,"reasoning_tokens":3275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:38:31.958826+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the three finite-energy sum rule moments $F_0,F_1,F_2$ at $s_0=1.19\\,\\mathrm{GeV}^2$ with a higher-order or resummed operator-product expansion and check whether they remain inside the quoted error intervals; if any moment moves outside, the bootstrap's feasible set changes and the lower bound must shift off $688.4\\times10^{-10}$. A precise experimental $R(s)$ scan around $\\sqrt{s}=0.73\\,\\mathrm{GeV}$ would also show whether the extremal single-peak spectral density is the true minimizer or whether additional near-peak strength is being missed.","supporting_citations":[{"cited_title":"QCD bounds on leading-order hadronic vacuum polarization contributions to the muon anomalous magnetic moment","cited_arxiv_id":"2404.08591","evidence_quote":"Supplies the three finite-energy sum rule moments at $s_0=1.19\\,\\mathrm{GeV}^2$ with central values and errors, and the positivity-only lower bound that the bootstrap improves on."},{"cited_title":"Following [8] we use the ansatz ρ1 1(s) = − NX n=1 dn sin n arccos 8 s − 1 , F (s) = NX n=0 bn √ 4 − √4 − sp 4 + √4 − s !n","cited_arxiv_id":null,"evidence_quote":"Introduces the positive-semidefinite unitarity matrix and the analytic basis expansions for the spectral density and form factor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generalized unitarity relation among spectral density, pion partial wave, and form factor, and the chiral perturbation theory tolerance used at low energy."},{"cited_title":"Hence |SI ℓ [s, anm, bnm, P]| ≤1, for spins ℓ = 0, 1, 2, 3","cited_arxiv_id":null,"evidence_quote":"Supplies the crossing-symmetric analytic ansatz for pion scattering amplitudes from which the partial waves are built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Standard Model data-driven value of $a_\\mu$ and the precisely calculated electroweak and QED contributions added to the bootstrap bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the charmonium and bottomonium resonance contributions added to the light-quark bootstrap result."},{"cited_title":"and iso-spins I = 0 , 1, 2","cited_arxiv_id":null,"evidence_quote":"Gives the lattice value that saturates the average-tolerance lower bound, used for comparison in the tolerance scan."},{"cited_title":"This positive semi- definite matrix implies all the minors should be positive including determinant","cited_arxiv_id":null,"evidence_quote":"Provides the measured muon anomaly used to decide which tolerance choices are ruled out by experiment."}],"review_version":1}