Lemma 31 ([The rough axial estimate]). Let g range over metrics with uniform ellipticity and bounded local C² norms on a three-dimensional ball. Suppose that z∈C² , |z|+|dz|≤M , and A:∇^2z=G, A=I-(1-χ)e⊗e, e=∇z|∇z|, 0<c≤χ≤1, |G|≤M, …

We give the compactness and improvement arguments, including the treatment of the face. The identities involving third derivatives are distributional. For a C^2 function and metric their trace-reversed form follows by C^2 smoothing: the fluxes converge locally uniformly, the Hessian quadratics converge in L^1 , and the Ricci terms converge locally uniformly. The original axial equation is substituted only after this identity has been established; neither χ nor G is differentiated or mollified. The same argument on each side of a face retains the normal flux jump. Thus the proof applies in particular to the C^ 2,α functions arising in continuation below. A Hessian estimate above exponent two. Normalize the metric at the center of a small coordinate ball. The coordinate principal matrix a in Equation satisfies |I-a|_ F ≤1-c/2 after shrinking the ball by an amount determined by the metric bounds. For a compactly supported function Y on Euclidean space, the Fourier transform gives \|D^2Y\|_2=\|ΔY\|_2 , where the matrix norm is Frobenius. The Calder\'on--Zygmund estimate at any fixed exponent larger than two , interpolated with this identity, implies that the norm of D^2Δ^ -1 :L^s⟶L^s tends to one as s2 . Choose s_0>2 sufficiently close to two that this norm times 1-c/2 is less than one. Writing ΔY=(I-a):D^2Y+a:D^2Y and absorbing gives the global estimate. Apply it to ζY , and use \|DY\|_ L^s(B_r) ≤εr\|D^2Y\|_ L^s(B_r) +C_ ε r^ -1 \|Y\|_ L^s(B_r) . Using intermediate balls and absorbing successively gives, for 2≤s≤s_0 , \|D^2Y\|_ L^s(B_ r/2 ) ≤C(\|a:D^2Y\|_ L^s(B_r) +r^ -2 \|Y\|_ L^s(B_r) ). For clarity, the successive absorption uses radii tending geometrically from r/2 to r : choose the interpolation coefficient smaller than the inverse of the geometric loss to the power 2s . The resulting series converges. This proves Equation without a derivative of a . The trace-reversed identity. In Euclidean coordinates put H=D^2z , q=1-χ , and W=|Dz|^2/2 . Since He= D W/|Dz| wherever Dz≠0 , direct multiplication gives the identity, also valid when Dz=0 , A DW-GDz=(H-(trH)I)Dz, div((H-(trH)I)Dz) =|H|^2-(trH)^2. Moreover trH=qH(e,e)+G . Choose (1-c)^2<θ<1 . Young's inequality then gives (trH)^2≤θ|H|^2+C_cG^2. If |Dz|≤1 and S=1-|Dz|^2 , we consequently have div(A DS+2GDz) ≤-κ|D^2z|^2+C_cG^2, κ>0. This is a divergence inequality with measurable coefficients; it has no Dχ term. Intrinsically the last member of Equation gains Ric_g(∇z,∇z) . Passing to coordinates with the volume density and replacing covariant by ordinary derivatives adds bounded flux and scalar errors. On rescaled balls whose metric tends to the Euclidean metric in C^1 , whose smooth-side curvature tends to zero, and whose normalized forcing tends to zero, these errors have the form div(a_1 DS+B) ≤-κ_1|∇^2z|^2+ε, |B|≤ε, with fixed ellipticity constants and ε→0 . A gradient drop or entry into an affine regime. Fix 0<r_*<1/32 . For every b_0>0 there exist k_*∈(r_*,1) and an error tolerance ε_*>0 such that a normalized solution on B_1 , with |∇z|≤1 and errors at most ε_* , satisfies one of _ B_ r_* |∇z|≤k_*; _ B_ r_* |z-L|≤b_0r_*, 12≤|DL|≤2 for an affine L . To prove this, suppose the assertion fails, let the errors tend to zero, and let the gradient supremum on B_ r_* tend to one. The scalar weak Harnack inequality , applied to the nonnegative supersolution S in Equation, gives (1 |B_ 1/3 | ∫_ B_ 1/3 S^p)^ 1/p ≤C(_ B_ r_* S+ε)⟶0 for a fixed p>0 . Equation gives a uniform local L^2 bound for DS . Here is the extra truncation step needed to retain the negative Hessian term. Testing the supersolution inequality by ζ^2(b-S)_+ yields ∫_ S<b ζ^2|DS|^2 ≤Cb^2∫|Dζ|^2+C\|B\|_∞^2 +Cεb. Indeed the derivative of (b-S)_+ contributes the negative of the elliptic energy on S<b ; the cutoff and B terms are absorbed by Young's inequality. Since S→0 in measure, split DS into the sets S<b and S≥b . The first has limiting L^1 norm at most Cb by Equation; the second has vanishing L^1 norm by the uniform L^2 bound and its vanishing measure. Thus DS→0 locally in L^1 . Testing Equation with a fixed nonnegative cutoff now gives D^2z→0 locally in L^2 . After subtraction of a constant, the gradient bound gives uniform compactness, and the limit is affine. Its slope has length one because S→0 in measure. This contradicts failure of the second alternative in Equation. Improvement near a nonzero affine function. Suppose on B_1 that |z-L|≤b≤b_0 , 1/4≤|DL|≤4 , and the gradient is bounded by a fixed constant. Suppose also that metric oscillation, connection errors, and forcing are at most bδ . For Y=(z-L)/b , Equation gives a uniform W^ 2,s_0 bound on smaller balls; its source is bounded by Cδ . Put e_0=DL/|DL| . The directions satisfy |e⊗e-e_0⊗e_0| ≤C1,b|DY|+o(1). Hence they converge in measure to the fixed axis as b_0,δ→0 . H\"older's inequality, with exponent 2s_0/(s_0-2) for this bounded coefficient error, proves \|R\|_ L^2(B_ 3/4 ) ⟶0, R=(Δ_ e_0^⊥ +χ∂_ e_0 ^2)Y. Only the axis is frozen; χ remains an arbitrary measurable function of position. There is a zero-Dirichlet correction U∈W^ 2,2 (B_ 3/4 )∩ W^ 1,2 _0(B_ 3/4 ) with (Δ_ e_0^⊥ +χ∂_ e_0 ^2)U=R, \|U\|_ W^ 2,2 ≤C_c\|R\|_2. Indeed solve U=Δ_D^ -1 ((1-χ)U_ e_0e_0 +R) by contraction in the Hessian norm. On a convex ball integration by parts gives ∫|D^2U|^2≤∫(ΔU)^2; the difference is the boundary mean curvature times (∂_νU)^2 , which is nonnegative. Thus the contraction factor is at most 1-c . Poincar\'e's inequality controls the lower derivatives. In dimension three the Sobolev embedding W^ 2,2 ↪C^ 0,1/2 gives \|U\|_∞→0 . This is the dimension-specific uniform correction required here. Write Y_0=Y-U , and rotate so that e_0 is the third coordinate axis. Then v=∂_3Y_0∈W^ 1,2 satisfies ∂_1^2v+∂_2^2v+∂_3(χ∂_3v)=0 distributionally: it is exactly the distributional derivative of Δ_ e_0^⊥ Y_0+χ∂_3^2Y_0=0 . The scalar divergence-form H\"older estimate and its energy inequality therefore give, on smaller balls, [v]_ C^ α_1 ≤C, ∫_ B_r |Dv|^2≤Cr^ 1+2α_1 for some α_1∈(0,1) . Since ΔY_0=(1-χ)∂_3v=:F_0 , Cauchy--Schwarz gives ∫_ B_r |F_0|≤Cr^ 2+α_1 . This also controls all the first derivatives of Y_0 . To see the point explicitly, localize F_0 to a slightly larger ball and take its Newton potential P . In the difference DP(x)-DP(y) , with h=|x-y| , the two near regions have size bounded by C∑_ j≥0 (2^ -j h)^ α_1 ≤Ch^ α_1 , by Equation. On the annuli of radius r≥2h , the derivative of the gradient kernel is bounded by Cr^ -3 , giving Ch∑_ r=2^jh r^ α_1-1 ≤Ch^ α_1 . The remainder Y_0-P is harmonic. Its local norms are bounded by the already established W^ 2,2 bound and the potential bound. Consequently DY_0 is uniformly H\"older on an interior ball. Choose 0<α_2<α_1 , then a sufficiently small fixed λ∈(0,1/8) so that the Taylor error of Y_0 is at most 12λ^ 1+α_2 . Finally choose b_0 and δ so that the uniform correction is smaller than the other half. We obtain an affine L' with _ B_λ |z-L'|≤bλ^ 1+α_2 , |DL'-DL|≤Cb. Iteration and the boundary interface. In the improvement statement b is an allowed upper bound for the error, and need not equal the actual error. Fix b_0,δ,λ as above, decreasing b_0 so that Cb_0/(1-λ^ α_2 )<1/4 , and then choose k_*,ε_* in Equation. Initially shrink to a radius R_0 and normalize the gradient by a fixed bound Λ_0 so that all errors are at most (ε_*,b_0δ) . After j drops the physical radius and gradient normalization are R_j=R_0r_*^j and Λ_j=Λ_0k_*^j . In particular the normalized forcing is R_jG/Λ_j , and is multiplied by r_*/k_*<1 at each drop. Metric and connection errors decrease at least by r_* and curvature errors by r_*^2 . At the first affine entry the physical radius is R_e=r_*R_j , with the same Λ_j . Rescale that ball and apply Equation with the prescribed upper bound b=b_0 , even if the actual affine error is smaller. On iteration i use b_i=b_0λ^ iα_2 . The forcing divided by b_i is at most δλ^ i(1-α_2) ; the same comparison controls the connection errors. The slope increments are summable by the choice of b_0 , so the axis remains separated from zero. Only constants are subtracted in the rescaled original equation, so its axis is always the actual gradient axis. These two regimes give uniform affine approximation of order r^ 1+α , including the transition scale, for any sufficiently small 0<α≤ α_2, k_*/ r_*. The affine approximation criterion for H\"older gradients can also be seen directly: affine slopes at consecutive dyadic scales differ by Cr^α , hence converge; comparing overlapping balls gives the same modulus between two centers. Thus the first estimate in Equation follows. Subtract the first jet in Equation. Its error is O(r^ 1+α ) ; connection terms are bounded. Squaring the resulting L^2 estimate proves the second estimate. For a face use Gaussian coordinates (x',x_3) , subtract its constant value, reflect z oddly and g evenly. The reflected function is C^1∩W^ 2,s , the metric is Lipschitz, and each smooth side has bounded curvature. The coordinate Hessian estimate applies unchanged. In the trace-reversed identity the only additional term is the jump of the normal flux at x_3=0 . Since the tangential gradient vanishes there, ((∇^2z-(Δz)g)∇z)·ν =-(∂_νz)tr_ T ∇^2z. The constant boundary value gives |tr_ T ∇^2z|≤C|II|\,|dz| . Thus the jump of the density-weighted flux is a bounded plane density J(x') , of size C|II|\,|dz|^2 . It can be cancelled exactly by adding the coordinate vector -J(x')1_ x_3>0 ∂_3 to the flux; its divergence is -Jδ_ x_3=0 , and no tangential derivative of J is used. Under a spatial scale R_j and gradient normalization Λ_j , the normal flux jump becomes R_jJ/Λ_j^2 . Since the current gradient bound is |dz|≤Λ_j , its size is at most CR_j|II| , also after any number of gradient drops. The smooth-side Ricci error is at most CR_j^2|Ric| . Both therefore tend to zero independently of the shrinking gradient normalization. Hence Equation, the compactness alternative, and the affine improvement all remain valid across the face. This proves the boundary assertion.