{"id":"fb795c97-75ef-41ef-a76f-fe4c2e9d8e5f","arxiv_id":"2505.21176","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A generic two-band kinetic theory produces hydrodynamic equations with a density-dependent mass m⋆, an anomalous u(∇·u) term for band exponent a≠2, and corresponding viscosities, Lorenz numbers, and plasmons.","lead":"The paper derives Euler and Navier-Stokes equations for electrons in two-band materials with generic power-law dispersion and any dimension, finding an extra flow term that breaks boost invariance unless the band exponent is 2. It also computes viscosities, Lorenz and Prandtl numbers, and long-wavelength plasmons.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (18a), the Dirac-limit Euler equation, appears to have an incorrect coefficient and pressure term, contradicting the paper's own boost-symmetry analysis and the a=2 Galilean limit.","rationale":"The reader's verdict is CONDITIONAL, with the stated weakest assumption being the relaxation-time collision model (25) and the Onsager condition (37). Those are legitimate concerns about quantitative transport predictions, but they do not threaten the structural Euler-level claim. The Dirac-limit equation (18a), however, is a structural part of the central derivation: it is the paper's claimed exact closure at charge neutrality, and it is used in the symmetry discussion. The algebraic check above shows that (18a) is inconsistent with the moment equations unless some unstated closure is intended. In particular, the coefficient mismatch would make a=2, d=2 lose Galilean invariance at the Dirac point and would make a=1, d=2 admit a Galilean boost according to Appendix C, directly contradicting the paper's main narrative. This is an internal inconsistency, not a disagreement with external consensus, and it is concrete enough to be settled by re-derivation. If confirmed, the Dirac-limit hydrodynamic equations and all quantities derived from them (e.g., sound speed, viscosity limits, transport near the Dirac point) require correction. The Fermi-limit equation (22) and its boost-breaking analysis may still be correct, so the appropriate verdict remains CONDITIONAL rather than REJECT. The reader's rationale already noted that (18a) shows signs of an algebraic error, so this concern is partially aligned with the reader, even though the reader's formal weakest_assumption field focused on the relaxation-time model.","tokens_in":20208,"tokens_out":20074,"duration_ms":237455,"concrete_test":"Independently re-derive Eq. (18a) from Eqs. (14), (15), and (17): substitute n⟨p⟩ = m⋆ n u and P = (a/d)n⟨E⟩ into (14b), eliminate ∂t n via (14a), eliminate ∂t m⋆ via the energy equation (18b) or Eq. (23), and retain all leading-order terms in μ/kBT. Check whether the coefficient of (∇·u)u is (1-a)/d or (a-2)/d, and whether the pressure term is a/(d m⋆ n)∇[n⟨E⟩] or (1/m⋆)∇[n⟨E⟩]. If the derivation yields the latter coefficient and pressure term, Eq. (18a) must be corrected and the Dirac-limit conclusions revisited.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Dirac-limit Euler equation (18a) is load-bearing because it underpins the claim of exact closure at charge neutrality and the statement that only a=1 and a=2 allow a natural hydrodynamic closure. It is internally inconsistent with the moment equations (14) and the paper's own definitions. Using n⟨p⟩ = m⋆ n u, P = (a/d)n⟨E⟩, and eliminating ∂t n via (14a) and ∂t m⋆ via the energy equation (18b), the momentum equation (14b) reduces to ∂t u + (u·∇)u + (a-2)/d (∇·u)u + a/(d m⋆ n)∇[n⟨E⟩] = 0. Eq. (18a) instead gives +(1-a)/d (∇·u)u and (1/m⋆)∇[n⟨E⟩]. The discrepancy is not cosmetic: for a=2, Eq. (18a) predicts a spurious -(1/d)u∇·u term in a Galilean-invariant parabolic system; for a=1, d=2, the reduced 1D system has α=1, which Appendix C explicitly identifies as the Galilean-boost case. This directly contradicts the central claim that graphene-like linear bands do not inherit any boost-like symmetry. Since the derivation of (18a) is not shown, and the stated result is algebraically suspect, the Dirac-limit analysis and the associated sound speed, viscosity limits, and transport coefficients built on this closure are not reliable as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives Euler and Navier-Stokes-type hydrodynamic equations for two-band fermionic systems with power-law dispersion ω± = ±B k^a in d dimensions, using moment expansions of the Boltzmann equation and a generalized hydrodynamic mass m⋆. It analyzes boost symmetry, computes shear viscosity, thermoelectric transport coefficients, Lorenz and Prandtl numbers, and long-wavelength plasmons, with special attention to the Dirac (a=1) and parabolic (a=2) limits. The central result is Eq. (22), the Fermi-limit velocity equation containing a non-Galilean u(∇·u) term for a≠2, from which the subsequent transport and collective-mode results are built.","tokens_in":20597,"tokens_out":22031,"duration_ms":226704,"significance":"If the derivation is corrected, the paper provides a useful unified framework: it recovers the known shear-viscosity results for graphene (a=1, d=2) and for the dilute parabolic gas (a=2, d=3), it reproduces the Wiedemann-Franz law in the Fermi-liquid limit, and it gives concrete falsifiable predictions for η/s, Lorenz and Prandtl numbers, and plasmon dispersion as functions of the band exponent a and dimension d. The thermodynamic and transport coefficients are computed in closed form from the stated model without fitted parameters, apart from the explicitly phenomenological relaxation times. The main limitation, acknowledged in the text, is that the quantitative transport predictions inherit the relaxation-time ansatz (25) and the Onsager reciprocity condition (37), as the reader's report also notes.","major_comments":[{"comment":"The stated Dirac-limit Euler equation is not the correct reduction of the moment equations. Starting from Eqs. (14a)-(14c) with n⟨p⟩ = m⋆nu and Π = (a/d)n⟨E⟩I, and using ∂tT + u·∇T = -(a/d)T∇·u (which follows from Eq. (18b) and m⋆ ∝ T^{2/a-1}), the momentum equation reduces to ∂tu + (u·∇)u - (2-a)/d (∇·u)u + a/(d m⋆ n)∇[n⟨E⟩] = 0. Eq. (18a) instead has +(1-a)/d (∇·u)u and (1/m⋆)∇[n⟨E⟩]. The two differ at a=1, where the correct coefficient is -1/d rather than 0, and at a=2, where Eq. (18a) predicts a spurious -1/d term in a Galilean-invariant parabolic system. The pressure term also lacks the 1/n factor needed to be dimensionally consistent at charge neutrality. Because Eq. (18a) is used to discuss closure and boost symmetry in the Dirac limit, it must be corrected; the corrected equation is consistent with the general Fermi-limit equation (22).","section":"Sec. III A, Eq. (18a)"},{"comment":"As written, the temperature equation ∂tT + u·∇T + (a/d)∇·u = 0 is dimensionally inconsistent unless T is measured with k_BT = 1 throughout. The last term should be (a/d)T∇·u, i.e. ∂tT + u·∇T + (a/d)T∇·u = 0. This relation is used in deriving the sound speed (24), so the authors should verify which form was used and adjust Eq. (24) if necessary.","section":"Sec. III A, Eq. (23)"}],"minor_comments":[{"comment":"Eq. (B5) reads ∂tnc - ∇·(ncu) = 0, which differs by a sign from the main-text continuity equation (14a); the '+' sign should be used.","section":"Appendix B, Eq. (B5)"},{"comment":"The symbol α is used both for the band exponent (for example, 'if, and only if, α = 2' after Eq. (12)) and for the shallow-water coefficient in Appendix C. Please use a consistently for the band exponent to avoid confusion.","section":"Sec. II"},{"comment":"There are minor typographical issues: 'entalphy' should be 'enthalpy', and in Sec. IV 'well-defined ain the absence of disorder' should read 'well-defined in the absence of disorder'.","section":"Sec. III B"},{"comment":"The figure captions state τ0 = 0.6T exp(-|µ|/T) and τdis = 1, but because σ, κ, Pr and L are all proportional to these relaxation scales, the plotted curves should be explicitly labeled as illustrative model outputs rather than parameter-free predictions.","section":"Sec. V, Eq. (44)"}],"recommendation":"major_revision","confidential_remarks":"The Dirac-limit error in Eq. (18a) appears to be a genuine algebraic slip rather than a conceptual flaw, since the corrected equation is consistent with the central Fermi-limit result. A major revision is therefore appropriate; the paper should not be accepted before this equation and the related sound-speed discussion are fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper has a genuine new piece: for a two-band system with dispersion ω±=±Bk^a in d dimensions, the Euler equation in the Fermi limit acquires a u∇·u term with coefficient (2-a)/d, explicitly breaking boost invariance for a≠2. That generalization beyond graphene (a=1) and parabolic (a=2) is real, and the Lie-symmetry analysis in Appendix C is a solid piece of work. They also recover the known viscosity limits for a=1,2 and get a clean plasmon dispersion. Credit where due: the derivation is self-contained, no fitted parameters are used for the structure, and the paper is honest about the relaxation-time assumptions.\n\nNow the soft spots. The stress-test concern about Eq. (18a) is correct. Using their own definitions—n⟨p⟩=m⋆nu, P=(a/d)n⟨E⟩, continuity (14a), and energy equation (18b)—the momentum equation reduces to ∂tu+(u·∇)u+(a-2)/d (∇·u)u + a/(d m⋆ n)∇[n⟨E⟩]=0. Eq. (18a) instead has (1-a)/d and (1/m⋆)∇[n⟨E⟩]. The discrepancy is not cosmetic: for a=2 it leaves a spurious u∇·u term in a Galilean-invariant system, and for a=1,d=2 it makes the 1D reduction have α=1, which Appendix C identifies as the Galilean-boost case. That directly contradicts the paper's central claim about graphene. The Dirac-limit closure is stated without derivation, and as written it's wrong.\n\nThe viscosity appendix also disagrees with the main text: Eq. (D3) gives η = -2τ G_d d(d+a)P, which for d=2 is -(a+2)/2 τP, opposite sign and double the main text's (a+2)/4 τ(P++P−). This looks like a sign/normalization slip in the appendix, not a conceptual error, but it needs to be fixed.\n\nThe relaxation-time ansatz and the Onsager relation (37) are ad hoc, but the paper labels them as such; the plots are illustrative. That's acceptable if the limits are clearly stated.\n\nBottom line: the Fermi-limit framework and the boost-breaking claim are worth refereeing seriously. The Dirac-limit analysis and the viscosity appendix need major corrections. I'd send it to review with the expectation of a substantial revision.","headline":"A genuine new framework for two-band hydrodynamics with a broken-boost term, but the Dirac-limit Euler equation (18a) is algebraically inconsistent and the viscosity in Appendix D conflicts with the main text.","tokens_in":21076,"tokens_out":9824,"would_cite":false,"duration_ms":90182,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Hydrodynamics derived for two-band power-law electron systems is neither Galilean nor Lorentz invariant except for a=2, and yields viscosity, Lorenz and Prandtl numbers, and plasmons for every band exponent a.","keywords":["electronic hydrodynamics","two-band systems","power-law dispersion","hydrodynamic mass element","boost invariance","shear viscosity","Wiedemann-Franz law","plasmons"],"falsifier":"Measure $\\eta/s$ and the Lorenz number in a clean sample with known non-quadratic bands (for example rhombohedral trilayer graphene, $a = 3$, or a Weyl semimetal, $a = 1$) while sweeping temperature and chemical potential across the Dirac point: the paper predicts $\\eta/s$ independent of $T$ and $\\mu$ near charge neutrality and a diverging $L$ as disorder is removed, so observing $\\eta/s$ to grow with $T$, or $L$ to saturate at a finite value with fixed small disorder, would falsify the transport sector. Alternatively, in a current-biased two-band sample, the asymmetry of counter-propagating hydrodynamic plasmon modes predicted by Eq. (44) as $(a-2)/(2d)\\,u_0 k$ directly tests the boost-breaking term of Eq. (22).","tokens_in":19974,"feed_emoji":"🌊","tokens_out":17345,"duration_ms":156076,"temperature":0.7,"pith_summary":"This paper derives the Euler and Navier-Stokes equations for a two-band electron fluid whose bands disperse as $\\omega_\\pm = \\pm B k^a$ in arbitrary dimension $d$, starting from the Boltzmann equation and its collision moments. The central result is that the momentum equation contains a term $-\\frac{2-a}{d}\\,u(\\nabla\\cdot u)$ with no counterpart in ordinary fluids, and that this term removes boost symmetry: the symmetry analysis of the equations shows that the Galilean boost is a generator only for $a = 2$ (parabolic bands), so for generic power-law bands, including Dirac bands with $a = 1$, the macroscopic theory is neither Galilean nor Lorentz invariant. From the same framework the paper computes the shear viscosity (with exactly vanishing bulk viscosity), the thermoelectric conductivities, the Lorenz and Prandtl numbers, and the long-wavelength plasmon dispersion. Real materials, such as bilayer graphene, rhombohedral trilayer graphene, and Weyl semimetals, realize different values of $a$, so the analysis places them in one hydrodynamic description and identifies which transport anomalies are generic to two-band power-law systems.","feed_headline":"Two-band electron fluids break boost symmetry except for a=2","feed_subtitle":"Kinetic derivation yields viscosity, Lorenz and Prandtl numbers, and plasmons for any power-law band exponent a.","key_machinery":"The load-bearing object is the hydrodynamic mass element $m_\\star$ (Eq. 12), the proportionality constant between the average momentum density $n\\langle p\\rangle$ and the particle current $n u$; for power-law bands it scales as $m_\\star \\propto n^{(2-a)/d}$, so its density dependence generates the $-\\frac{2-a}{d}\\,u(\\nabla\\cdot u)$ term in Eq. (22) that breaks boost symmetry. The viscous sector rests on two additional pieces: the relaxation-time collision integral $C_\\pm = \\mp (f^{(+)}/\\tau_+ - f^{(-)}/\\tau_-)$ of Eq. (25), used inside a Chapman-Enskog expansion, and the Onsager reciprocity condition $\\alpha = \\bar\\alpha$ of Eq. (37), which ties the electron and hole relaxation times $\\tau_+$ and $\\tau_-$ to a common scale $\\tau_0$. The Lie-algebra computation on the reduced shallow-water-type system in Appendix C is what converts the presence of that extra term into the claim that no boost generator exists for $a \\neq 2$.","core_discovery":"The paper's central claim is that all macroscopic transport in a two-band power-law system is governed by a single density-dependent object, the hydrodynamic mass element $m_\\star$ defined by $n\\langle p\\rangle = m_\\star n u$ in Eq. (12). Because $m_\\star \\propto n^{(2-a)/d}$ in the Fermi liquid limit, the momentum equation (22) acquires the anomalous convective term $-\\frac{2-a}{d}\\,u(\\nabla\\cdot u)$. A Lie-algebra symmetry analysis of this equation (Appendix C) shows that for $a \\neq 2$ the symmetry algebra contains only translations and dilations, and that the boost generator appears only for $a = 2$; the paper concludes that the coarse-grained hydrodynamics is neither Galilean nor Lorentz invariant, and that even linear (relativistic-looking) dispersion does not produce a Lorentz-invariant fluid description. The theory nonetheless closes at the Dirac point ($\\mu/k_B T \\to 0$) and in the Fermi liquid limit ($\\mu/k_B T \\to \\infty$), and it yields a shear viscosity $\\eta = G_d(a)(\\tau_+ P_+ + \\tau_- P_-)$ with zero bulk viscosity, a Lorenz number that violates the Wiedemann-Franz law and diverges at charge neutrality in the clean limit, and plasmons that are gapped in three dimensions and scale as $k^{1/2}$ in two dimensions for any exponent $a$.","pith_inferences":["Beyond the paper: if the broken boost symmetry is real, the momentum current should fail a center-of-mass conservation test; measuring the streamline pattern around a constriction in a clean two-band sample with $a \\neq 2$ and comparing with the standard Navier-Stokes profile would expose the extra $u(\\nabla\\cdot u)$ term directly.","Beyond the paper: the $a$-dependent Doppler prefactor $(a-2)/(2d)$ turns the plasmon dispersion into a dynamical probe of the band exponent, so an asymmetric splitting of counter-propagating plasmon modes in a biased two-band material would be a quantitative, setup-specific signature of Eq. (44) that the paper itself does not propose.","Beyond the paper: the construction assumes isotropic power-law bands; extending the hydrodynamic mass element to tilted, anisotropic, or multi-pocket dispersions should produce a tensorial $m_\\star$ and further symmetry-breaking terms, and the same Lie-algebra test could classify which of those bands retain any boost-like symmetry.","Beyond the paper: because every quantitative number is proportional to $\\tau_0$, a first-principles computation of $\\tau_+$ and $\\tau_-$ from the Coulomb collision integral for a concrete material (say rhombohedral trilayer graphene) would convert the present framework into falsifiable magnitudes."],"forward_implications":["For any material whose low-energy bands disperse as $|k|^a$ with $a \\neq 2$, including Dirac systems with $a = 1$, the momentum equation contains $-\\frac{2-a}{d}\\,u(\\nabla\\cdot u)$, so the fluid is not Galilean even at the Euler level and momentum transport differs from ordinary Navier-Stokes flow.","The shear viscosity is fixed by band geometry through $\\eta = G_d(a)(\\tau_+ P_+ + \\tau_- P_-)$ with exactly zero bulk viscosity, and near the Dirac point the ratio $\\eta/s$ is independent of temperature and chemical potential, so clean low-temperature samples should behave as near-perfect fluids.","Near charge neutrality the Lorenz number $L = \\kappa/(\\sigma T)$ is not universal and diverges in the clean limit ($\\tau_{\\rm dis} \\to \\infty$), a direct breakdown of the Wiedemann-Franz law; in the Fermi liquid limit $L \\to L_0$ is restored.","The Prandtl number at the Dirac point, $\\Pr = (2d/a)(\\tau/\\tau_{\\rm dis})\\, k_B P/(S_+ + S_-)$, is much smaller than one when electron-electron scattering dominates and much larger than one when disorder dominates, making the ratio of momentum to heat diffusivity a tunable sample-quality diagnostic.","Hydrodynamic plasmons are gapped in 3D and gapless with $\\omega \\sim k^{1/2}$ in 2D for every $a$, but the Doppler shift carries the symmetry-breaking prefactor $(1 + (a-2)/(2d))\\,u_0 k$, so counter-propagating plasmon modes split asymmetrically when $a \\neq 2$."],"supporting_citations":[{"why":"Supplies the graphene Dirac band structure and the two-band platform that motivates the a=1 case.","marker":"[10]"},{"why":"Provides the bilayer-graphene parabolic two-band dispersion that serves as the a=2 reference case.","marker":"[13]"},{"why":"Supplies the kinetic-theory gradient expansion and the entropy-vortex-wave concept used to close the equations and identify collective modes.","marker":"[16]"},{"why":"Gives the graphene shear-viscosity result that the paper's formula reproduces for d=2, a=1, anchoring the viscosity derivation.","marker":"[18]"},{"why":"The electron-gas parabolic result (d=3, a=2) that the viscosity formula reduces to, and the moment-integration method used for the conservation equations.","marker":"[21]"},{"why":"The Chapman-Enskog procedure by which the viscous corrections to the stress tensor and heat current are computed from the collision operator.","marker":"[26]"},{"why":"Provides the Onsager transport-matrix framework and the reciprocity condition (Eq. 37) that ties the relaxation times to a common scale for all conductivity results.","marker":"[31]"},{"why":"Gives the collective-mode (plasmon) treatment for 2D Dirac systems that the paper's plasmon dispersion extends to general a.","marker":"[35]"},{"why":"The Lie-symmetry method used in Appendix C to show the boost generator exists only for a=2.","marker":"[37]"}],"fun_headline_variants":["Two-band fluids lose boost symmetry for a≠2","Anomalous convective term breaks boost in two-band fluids","No Lorentz boost for two-band electron hydrodynamics","Hydrodynamics of two-band electrons: boost only when a=2","Power-law two-band fluids yield novel hydrodynamic equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every quantitative prediction inherits the relaxation-time collision model $C_\\pm = \\mp(f^{(+)}/\\tau_+ - f^{(-)}/\\tau_-)$ of Eq. (25) and the Onsager reciprocity condition $\\alpha = \\bar\\alpha$ of Eq. (37) that merges $\\tau_+$ and $\\tau_-$ into a single scale $\\tau_0$, so if the real electron-electron scattering of a two-band system is not captured by those two assumptions, the computed viscosities, conductivities, Lorenz and Prandtl numbers would all shift even though the ideal Euler structure, including the broken boost symmetry, could still be correct.","fun_headline_variants_meta":{"raw":{"variants":["Two-band fluids lose boost symmetry for a≠2","Anomalous convective term breaks boost in two-band fluids","No Lorentz boost for two-band electron hydrodynamics","Hydrodynamics of two-band electrons: boost only when a=2","Power-law two-band fluids yield novel hydrodynamic equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1961,"prompt_tokens":965,"completion_tokens":996,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":918}},"tokens_in":581,"tokens_out":996,"duration_ms":11206,"temperature":1.0,"reasoning_tokens":918,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:34:55.400577+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $\\eta/s$ and the Lorenz number in a clean sample with known non-quadratic bands (for example rhombohedral trilayer graphene, $a = 3$, or a Weyl semimetal, $a = 1$) while sweeping temperature and chemical potential across the Dirac point: the paper predicts $\\eta/s$ independent of $T$ and $\\mu$ near charge neutrality and a diverging $L$ as disorder is removed, so observing $\\eta/s$ to grow with $T$, or $L$ to saturate at a finite value with fixed small disorder, would falsify the transport sector. Alternatively, in a current-biased two-band sample, the asymmetry of counter-propagating hydrodynamic plasmon modes predicted by Eq. (44) as $(a-2)/(2d)\\,u_0 k$ directly tests the boost-breaking term of Eq. (22).","supporting_citations":[{"cited_title":"Keimer and J","cited_arxiv_id":null,"evidence_quote":"Supplies the graphene Dirac band structure and the two-band platform that motivates the a=1 case."},{"cited_title":"Crossno, J","cited_arxiv_id":null,"evidence_quote":"Provides the bilayer-graphene parabolic two-band dispersion that serves as the a=2 reference case."},{"cited_title":"Levitov and G","cited_arxiv_id":null,"evidence_quote":"Supplies the kinetic-theory gradient expansion and the entropy-vortex-wave concept used to close the equations and identify collective modes."},{"cited_title":"Abramowitz and I","cited_arxiv_id":null,"evidence_quote":"Gives the graphene shear-viscosity result that the paper's formula reproduces for d=2, a=1, anchoring the viscosity derivation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The electron-gas parabolic result (d=3, a=2) that the viscosity formula reduces to, and the moment-integration method used for the conservation equations."},{"cited_title":"Esposito, J","cited_arxiv_id":null,"evidence_quote":"The Chapman-Enskog procedure by which the viscous corrections to the stress tensor and heat current are computed from the collision operator."},{"cited_title":"Pongsangangan, S","cited_arxiv_id":null,"evidence_quote":"Provides the Onsager transport-matrix framework and the reciprocity condition (Eq. 37) that ties the relaxation times to a common scale for all conductivity results."},{"cited_title":"Bohm and D","cited_arxiv_id":null,"evidence_quote":"Gives the collective-mode (plasmon) treatment for 2D Dirac systems that the paper's plasmon dispersion extends to general a."},{"cited_title":"Bernabeu, K","cited_arxiv_id":null,"evidence_quote":"The Lie-symmetry method used in Appendix C to show the boost generator exists only for a=2."}],"review_version":1}