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Step 1: fixed scales and exact maxima. Throughout the proof the original potential v , the scales, and both templates are fixed. Constants used only to pass limits may depend on all of them. Choose t_k1 and interior maxima (z_k^0,b_k) supplied by Lemma 19 ([Regularized interior maxima]), for F_k(z)=Q_ t_k g_ b_k (z), G_k(z)=Q_ 1-t_k w_ c,b_k (z). The functions are C^ 1,1 in space: Proposition 7 ([Intermediate regularity and pole control]) applies to the first, and Lemma 4 ([Short time for a semiconvex datum]) to the second. At an exact maximum the velocities align. If x_k^0 and y_k^0 are its two minimizers and p_k^0=t_k^ -1 _ x_k^0 z_k^0 , then y_k^0=_ x_k^0 p_k^0 along the concatenated minimizing ray. Every finite positive active representation of p_k^0 satisfies the parameter inequality of Lemma 20 ([The parameter derivative]). The level relations and common section bounds of Lemma 21 ([Levels at a regularized maximum]) hold for these representations. Step 2: null sets are removed before Jensen sampling. Let N be the null set for the original potential in . For this fixed k , the final minimizer map Y_k(z) is locally Lipschitz and hence differentiable almost everywhere . In each coordinate chart the locally Lipschitz area formula implies that Z_k= z:DY_k(z) exists, DY_k(z)≠0,\ Y_k(z)∈N is a null set. Indeed the integral of | DY_k| on Y_k^ -1 (N) is zero. This argument does not require us to know in advance where DY_k is nonsingular. Apply the semiconvex maximum Lemma to F_k+G_k at z_k^0 , using a small fourth-order strictness penalty if needed. Its positive-measure conclusion allows us to choose a sequence z_ k,ν →z_k^0 off Z_k and off all the null sets where F_k,G_k , the prefix pole map, or Y_k lack the derivatives used below. We have (F_k+G_k)(z_ k,ν )⟶0, D^2(F_k+G_k)(z_ k,ν )≤η_ k,ν , η_ k,ν ⟶0 in the fixed chart. At every such point choose a representation with at most n+1 active logs, padding by zero weights only for compactness. Lemma 25 ([The pole matrix and endpoint lower jets]) applies at each point. Its matrices L_ k,ν are positive semidefinite and annihilate the active differences, exactly; its endpoint lower jets are m_ i,k,ν L_ k,ν . The matrices H_ k,ν satisfy . Step 3: take the first, fixed-prefix limits. For each fixed k , the local C^ 1,1 bounds are finite. After taking a subsequence in ν , all configuration parameters, weights, and the two matrices H_ k,ν ,L_ k,ν converge. Denote the limits by p_k,p_ i,k ,m_ i,k ,H_k,L_k and the corresponding base points. Discard zero limiting weights. The limiting endpoints are active at the exact maximum, and their weighted sum is p_k . In particular the universal parameter inequality applies to this limiting representation; we have not asserted forward persistence of any prescribed active vertex. We retain L_k≥0, L_kE_k=0, H_k≤L_k, E_k= p_ i,k -p_k:m_ i,k >0 . The quantitative proof of Lemma 26 ([A lower semibound before the center Jacobian]) has an error bounded by a fixed-scale constant times 1-t_k+|(F_k+G_k)(z_ k,ν )| . Passing ν→∞ therefore gives H_k≥c(b_k)/(D) -C(b_k)/(D^2)p_kp_k^* -C_ fixed (1-t_k). Since b_k≥b_->0 , for sufficiently large k this implies, after reducing c , H_k≥c(b_k)/(D) -C(b_k)/(D^2)p_kp_k^*. If E_k≥2 , a nonzero vector in E_k∩p_k^⊥ contradicts and . If E_k=0 , all active logs in the representation equal p_k and all endpoints equal the center. But B_c≤B_o at every level, contradicting the strict parameter inequality. Thus E_k= e_k . Orient e_k so q_k=p_k·e_k>0 ; the same two inequalities give q_k^2≥cD . Lemma 22 ([Weight, cap, and norm increment]) now supplies an index with m_ i,k ≥m_* , a strict outward displacement, and a bounded template value. We choose an index carrying a fixed share of the outward moment, as permitted by that Lemma, so its displacement has a positive lower bound at the fixed scales. Relabel it as index 1 . Step 4: fix the outer branch before bounding any center Hessian. Pass to a subsequence in k so that the selected configurations and their weights converge. Their centers and selected endpoints remain in the compact level ranges of Lemma 22 ([Weight, cap, and norm increment]). The selected limiting weight is at least m_* , its displacement is strictly outward, and its outer template value is below the prescribed cap. By imposing the template conditions under a slightly larger fixed cap, the limiting slope and curvature conditions have strict margins: B_o'>0 and B_o''<-K . The limiting selected pair is active. Lemma 27 ([Conjugacy is excluded by an active increasing modification]) excludes conjugacy of its modified branch, using only this activity and the original contact-volume bound. Its shortened true log is before cut. Lemma 28 ([Undoing the outer modification]) consequently supplies a neighborhood of this configuration with constants δ>0 and M<∞ such that its divided-action gain is at least δ and its determinant upper bound is at most M . These constants may depend on the fixed scales and templates. For every remaining k , return to its already chosen Jensen subsequence and choose ν=ν(k) sufficiently large. We require the configuration and matrices to be within 1/k of their fixed-prefix limits, the selected weight to be at least m_*/2 , and the selected endpoint to lie in that neighborhood. We also require both errors in , including their transformed versions in, to be at most (1-t_k)^3/k . All these requirements are compatible because the limit in ν is taken with k fixed. For the rest of the proof, a subscript k denotes this diagonal sequence of actual Jensen points. Step 5: the outer test bounds the exact pole matrix. At each diagonal point, Lemma 25 ([The pole matrix and endpoint lower jets]) gives the actual lower Taylor Hessian H_ o,k =m_ 1,k L_k≥0 at the selected endpoint. Use Lemma 28 ([Undoing the outer modification]) to undo the modification and call the resulting form V_ 1,k . The positive outer curvature term and the uniform divided-action baseline give V_ 1,k ≥m_ 1,k L_k+δ, V_ 1,k ≤M. Every eigenvalue of V_ 1,k is at least δ , so its largest eigenvalue is at most Mδ^ 1-n . Since m_ 1,k ≥m_*/2 , this bounds L_k from above. The comparison H_k≤L_k+o(1) and Lemma 26 ([A lower semibound before the center Jacobian]) now bound H_k from both sides. There has still been no use of a center density equation. The center extension gives uniformly nonsingular fixed-scale prefix-log coordinate maps, as in the proof of Lemma 26 ([A lower semibound before the center Jacobian]). Their derivatives and second derivatives are bounded; the gradients of G_k are bounded by the Lipschitz bound of the datum. Thus the coordinate change in converts the bound on H_k into \| G_k(z_k)\|≤C_ fixed . Step 6: the chosen centers are good original endpoints. Write τ_k=1-t_k , y_k=Y_k(z_k) , and K_k(z,y)=c(z,y)/τ_k . The envelope identity G_k=(K_k)_z(z,Y_k(z)) and differentiability of Y_k give G_k=(K_k)_ zz +(K_k)_ zy DY_k. In parallel orthonormal frames, the distance between z_k and y_k is O(τ_k) , and the smooth short-cost expansions are (K_k)_ zz =τ_k^ -1 +O(τ_k), (K_k)_ yy =τ_k^ -1 +O(τ_k), (K_k)_ zy =-τ_k^ -1 Π+O(τ_k), where Π is the appropriate parallel isometry. Solving the preceding identity, using, gives DY_k=Π^ -1 -τ_k G_k +O(τ_k^2). It is nonsingular for large k . The Jensen points were chosen off the set; hence y_k∉N . This is the promised justification for using the original Alexandrov and Jacobian statements at the centers. In the following formulas the scalar modifications are evaluated with b=b_k . At these points w_c=φ_c v has an Alexandrov Hessian. The minimizing property supplies a proximal subgradient ξ(z)=-(K_k)_y(z,Y_k(z))∈∂w_c(Y_k(z)) at every nearby point, even if w_c is not differentiable there. Apply Lemma 29 ([Subgradients at an Alexandrov point]) at the Alexandrov point y_k=Y_k(z_k) . Since Y_k is Lipschitz and differentiable at z_k , it gives, in normal coordinates at z_k and y_k , ξ(z_k+h)= w_c(y_k) + w_c(y_k)DY_k(z_k)h+o(|h|). Expanding the smooth function (K_k)_y in its two variables therefore gives w_c(y_k)+(K_k)_ yy DY_k+(K_k)_ yz =0. Thus stationary differentiation uses the subgradient expansion even when neighboring endpoints are nonsmooth. Solving it with gives, in parallel frames, w_c(y_k)= G_k(z_k)+O(τ_k). In particular these endpoint Hessians are uniformly bounded. The error statement follows directly from the displayed short-cost expansions and the bounded inverse in ; it does not assume a bound for w_c in advance. Step 7: insert the true center form and take matrix limits. Set r_k=τ_k^ -1 _ y_k z_k, l_k=φ_c'(v(y_k)), x_k=_ y_k r_k, p_k=_ x_k y_k. The true center endpoint is ζ_k=_ y_k (r_k/l_k) , and it is before cut because y_k∉N . Identify T_ x_k M with T_ x_k M by parallel transport over their short joining segment, and let B_k:T_ x_k M→T_ y_k M be the exponential differential at p_k after that identification. Define V_k=l_k B_k^* v(y_k)+A_ y_k (r_k/l_k) B_k>0. The exact chain rule for the scalar modification gives B_k^* w_c(y_k)+A_ y_k (r_k) B_k = , V_k +B_k^* A_ y_k (r_k)-l_kA_ y_k (r_k/l_k) B_k ; , +(φ_c''(v(y_k)))/(l_k^2) p_kp_k^* . All vectors on the last line are understood after the same isometric identification. Lemma 11 ([Isotropic gain after pullback]) bounds the middle term below by c(1-l_k) . Equation and the vanishing pole mismatch show that the left side differs from H_k by o(1) . The verification is the same smooth coordinate comparison as in Lemma 26 ([A lower semibound before the center Jacobian]), now using the bounded unregularized Hessians in place of the smooth support Hessians. This proves H_k≥V_k+c(b_k)/(D) -C(b_k)/(D^2)p_kp_k^*-o(1). It also bounds V_k from above, because V_k>0 and H_k is bounded. Taking the determinant in and using gives the exact factorization V_k =l_k^nσ_ x_k (p_k)^2 | D(_ · v)(y_k)| σ_ y_k (r_k/l_k) ≥c\, σ_ x_k (p_k)^2 σ_ y_k (r_k/l_k) . The squared factor is the determinant of the Hessian pullback. All scalar derivatives lie between 1/2 and 2 , so the constants in this density inequality are independent of K . There is no assertion that the limiting endpoint is an Alexandrov point of v . We take limits of matrices at the good endpoints just selected. The center modified rays have a uniform extension at the fixed scales: 1-l_k≥c b_-/D>0 , and the ray through x_k continues to the true endpoint through duration 1/l_k . Compactness of this family of proper minimizing subsegments therefore gives σ_ x_k (p_k)≥c_ fixed >0. Since 0<V_k≤C_ fixed , equation now implies σ_ y_k (r_k/l_k) ≥ c\,σ_ x_k (p_k)^2 V_k ≥c'_ fixed >0. Continuity of the exponential differential excludes a conjugate limit of the true center rays. An ordinary cut limit may remain; the specified nonsingular branch suffices. Finally take a subsequence so that b_k,H_k,L_k,V_k,V_ 1,k and all configuration parameters converge, in a smooth orthonormal trivialization. Discard any additional zero limiting weights. The selected weight remains at least m_* , and its outward displacement retains the positive fixed-scale lower bound from Step~3. The barycenter identity then gives another surviving vertex, so the surviving difference span is still the same nonzero line. The selected modified and true outer branches were already uniformly nonsingular in Step~4; the center modified branch has its fixed-scale extension. Equation gives a positive lower determinant in the limit, so V>0 . Equations ,, and give . At each selected outer endpoint in the diagonal sequence, formula , with H_ o,k =m_ 1,k L_k , has a positive divided-action term. Its scalar curvature coefficient satisfies - φ_o''(v(y_ 1,k )) l_ 1,k ^2 =- b_kB_o''((v(y_ 1,k )-a)/D) D^2l_ 1,k ^2 ≥cK b_k/D^2. Passing this inequality and along V_ 1,k →V_i gives and the other inequality in. No lower Taylor jet is asserted at the limiting endpoint. The weights, directions, and section-level estimates pass through the same chosen subsequence and give --. Their constants and the cap were fixed before K . Only δ,M , the rate of the prefix limit, and the Jensen radii depended on the fixed outer template.
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