Corollary 13 ([The complete quadratic kernel]). The kernel of c_2 is exactly K , a four-dimensional real vector space. In particular, the nonzero degree-one directions are genuine quadratic null directions.

First suppose q has finite angular type and c_2[q]=0 . Equation and the strict positivity just proved imply q=Q(v), U=0, F=0 s>0. Smoothness extends these equalities to the boundary. Coordinate spheres are umbilic under conformal changes of g_0 . At the base minimal sphere, the tracefree second-form variation due to a normal height h_1 is -(_Sh_1)^ : the radial curvature term is pure trace. Since h_1=-v|_S , the vanishing of U|_S implies (_σ(v|_S))^ =0 . For completeness, the divergence of this equation on the unit sphere gives (Δ_σv+2v)=0 . Subtracting the mean reduces it to _σ(v- v)=-(v- v)σ . This equation determines its solution along every geodesic from its value and differential at one point, and its solutions are precisely the linear coordinate functions. Thus v|_S is a constant plus a linear harmonic. Uniqueness of the decaying Dirichlet problem for L then excludes every other angular mode throughout M . Each degree-one radial coefficient is a multiple of the same decaying solution of V''+2 NrV'-(3+2 r^2 )V=0. Hence q∈ K . Conversely, every element of K is null, as we now check; the conditions U=F=0 do not impose a further restriction on its degree-one part. Take q=Q(v)∈ K . The first conformal coefficient u also has only constant and linear modes, by its Neumann equation and decaying uniqueness. Let h(s,x) be the first height coefficient of the CMC spheres, and put p=N/r . Conformal variation of mean curvature and first variation of the lapse give [H]_1=J_s^0h+4(u_s-pu), [f]_1=h_s+2u. Here the second identity follows by taking the normal component of the velocity of the graph s+ h(s,x) in g_0+4 u g_0 . On the degree-one part, the CMC equation therefore fixes h=-(2r^3)/(3m_0)(u_s-pu). All formulas in the next calculation are componentwise on that part. The background equations and give p'=(3m_0)/(r^3)-1 r^2 , p^2=1+1 r^2 -(2m_0)/(r^3), u''=-2pu'+(3+2 r^2 )u. Consequently, 1 r^3 ∂_s[r^3(u'-pu)] , =3p(u'-pu)+u''-p'u-pu'; , =(3+2 r^2 -p'-3p^2)u =(3m_0)/(r^3)u. It follows that h_s=-2u on the degree-one part, so its lapse fluctuation vanishes. The freely chosen radial mean of h changes [f]_1 only by a radial function, which disappears upon subtracting the mean. Thus F=0 independently of the radial labeling. At every background sphere, the tracefree second-form variation of these graphs is -(_ r^2σ h)^ ; the conformal variation and radial curvature terms are pure trace. The constant and linear modes of h have vanishing tracefree spherical Hessian, so U=0 as well. At S there is also exact agreement with the prescribed first minimal height: on degree one, D_1=(6m_0)/(a^3), h(0)=-(4u_s(0))/(D_1)=-v(0). We already have q=Q(v) , and hence proves c_2[q]=0 . Existence and uniqueness for each of the four boundary modes show that K has dimension at most four; it has dimension exactly four because and invertibility of D make the map from these boundary data to Q(v) injective. It remains to rule out additional null directions of arbitrary angular type. Let Z= q:c_2[q]=0 in the full smooth TT space. A nonnegative quadratic form has a linear kernel: if c_2[q]=0 , nonnegativity of c_2[q+th] for all real t forces its polarized bilinear form to vanish on (q,h) for every h . By Proposition 12 ([The quadratic deficit]), Z is also closed in \|·\|_β and invariant under rotations. For q∈ Z , let q_j be its polynomial rotational averages from Lemma 6 ([Rotation approximation and coefficient continuity]). Each such average is a norm limit of finite linear combinations of rotations of q ; therefore q_j∈ Z . Each q_j has finite angular type, so the preceding argument puts it in K . Since q_j→q in norm and the fixed finite-dimensional space K is closed in that norm, q∈ K . This proves the full assertion without applying a pointwise geometric limit to the CMC fields U and F .