Proof. Write \(g=ds^2+k(s,z)\), with \(k\) a tangential positive matrix, and put \(\rho=(k^{ab}\xi_a\xi_b)^{1/2}\). Green’s formula identifies the measured operator with the density \(\nabla^A_\nu u\,dS_g\). Relative to the coordinate density, its principal symbol is \[(\det k)^{1/2}\rho I.\] Its square determines \(C=(\det k)k^{-1}\). Since the tangential dimension is \(m=n-1\), \[\det C=(\det k)^{m-1},\qquad k=(\det C)^{1/(m-1)}C^{-1}.\] Thus the boundary metrics and densities agree. Division by their common density gives equality of the ordinary covariant unit-normal DN operators locally on \(\Gamma\). Localization on both sides by cutoffs supported in \(\Gamma\) gives equality of their full symbols.

We spell out the symbol calculation, extending the metric factorization of (Lee and Uhlmann 1989) to the present scalar-principal system. Recovery of a connection together with the metric by this even–odd symbol separation is also developed in (Gabdurakhmanov 2021, Theorem 1.1 and Section 3.3); the calculation below uses complex Hermitian fibers. In normal gauge, \[L_{g,A}=-\partial_s^2-\alpha\partial_s+P, \qquad \alpha=\tfrac12\operatorname{tr}(k^{-1}\partial_s k),\] where \(P\) is the connection Laplacian on each tangential slice. With \(D_z=-i\partial_z\), its degree-one symbol consists of the scalar metric term and \(-2ik^{ab}A_a\xi_b\). Its degree-zero symbol uses only slice coefficients and their tangential derivatives. The local Dirichlet factorization has the form \[L_{g,A}=(-\partial_s+B-\alpha)(\partial_s+B) \pmod{\Psi^{-\infty}}, \qquad b\mathbin{\#}b-\alpha b-\partial_s b\sim\sigma(P),\] where \(b\sim\rho I+b_0+b_{-1}+\cdots\) is a left symbol and \[b\mathbin{\#}c\sim \sum_{\mu}\frac{(1/i)^{|\mu|}}{\mu!} (\partial_\xi^\mu b)(\partial_z^\mu c).\] The positive root \(\rho I\) selects the decaying Dirichlet branch. The Poisson parametrix consequently identifies \(B(0)\), modulo a smoothing operator, with the outward normal derivative. This factorization is valid for smooth coefficients: its principal root is scalar and each subsequent matrix coefficient is solved by scalar division by \(2\rho\). The true inverse Dirichlet problem changes only the local smoothing remainder.

The part of \(b_0\) involving the new jets is \[ \frac14\left(\operatorname{tr}_k\partial_s k -\frac{(\partial_s k)(\xi^\sharp,\xi^\sharp)}{\rho^2}\right)I -\frac{ik^{ab}A_a\xi_b}{\rho}, \qquad \xi^\sharp=k^{-1}\xi. \tag{4}\] This follows from \(\partial_s\rho=-(\partial_s k)(\xi^\sharp,\xi^\sharp)/(2\rho)\) and the degree-one factorization equation. At degree \(1-j\), \(j\ge2\), the new normal derivatives arise only in \(\partial_s b_{2-j}/(2\rho)\). Inductively their contribution is \[ \frac{1}{(2\rho)^{j-1}} \left[ \frac14\left(\operatorname{tr}_k\partial_s^j k -\frac{(\partial_s^j k)(\xi^\sharp,\xi^\sharp)}{\rho^2}\right)I -\frac{ik^{ab}(\partial_s^{j-1}A_a)\xi_b}{\rho} \right]. \tag{5}\] Every omitted term involves lower normal jets, with tangential derivatives. Differentiating \(\rho\), a raised index, or a coefficient multiplying the preceding highest jet produces only such lower-jet expressions.

Subtract the two symbol recursions after their lower jets have been identified. The metric term in (5) is even in \(\xi\), whereas the connection term is odd. The odd term determines all components of the new connection jet. If the even term vanishes for a symmetric tensor \(T\), then \(T(v,v)=\operatorname{tr}_k(T)\) for every unit vector \(v\). Polarization gives \(T=\operatorname{tr}_k(T)k\), and taking trace gives \((m-1)\operatorname{tr}_k(T)=0\). As \(m\ge2\), \(T=0\). Beginning with the recovered boundary metric, this proves all normal jets by induction; their tangential derivatives recover all mixed jets. The remaining normal metric components and normal connection component are fixed by the coordinate and gauge choices. ◻

Proof. Choose a boundary chart compactly contained in \(\Gamma\), and a nonnegative smooth bump \(b\) whose support is compact in that chart and whose positivity set \(P=\{b>0\}\) is nonempty and connected. Using the separate normal collars, move the boundary from \(s=0\) to the graph \(s=-b(z)\), with \(b\) small enough to remain inside an extended collar. The common open cap is \[E=\{(z,s):-b(z)<s<0\}.\] Extend the first coefficient fields smoothly to negative \(s\), retaining positive definiteness of the metric and skew-Hermitian values of the connection. Use the same exterior extension for the second pair. Lemma 2 makes the resulting pasting smooth to all orders on a neighborhood of \(\operatorname{supp}b\). Restricting it to \(s\ge-b(z)\) gives smoothness also at the flat edges of the bump. The bundles remain trivial in the extended normal gauges.

For the energy comparison only, identify the extensions by the identity on the original \(M\) and by cap coordinates on \(E\). In collar charts this map and its inverse are continuous and piecewise smooth with bounded first derivatives, and hence bi-Lipschitz. They identify the relevant \(H^1_0\) spaces. Fix a vector source \(f\) compactly supported in \(E\). The solution \(w_j=L_{j,D}^{-1}f\) is the unique minimizer of \[\mathcal J_j(w)=\frac12\int_{N_j}|d_{A_j}w|_{g_j}^2\,dV_{g_j} -\operatorname{Re}\int_E\langle f,w\rangle\,dV_{g_j} \quad\text{on }H^1_0(N_j;\mathbb C^2).\] Its trace on the original boundary is smooth and supported in \(\operatorname{supp}b\Subset\Gamma\), because the original and extended boundaries coincide wherever \(b=0\). Transport \(w_1\) to \(N_2\), and inside the original \(M\) replace it by the second harmonic extension of this trace. The replacing difference has zero trace on \(\partial M\), so its zero extension is in \(H^1_0(N_2)\). The competitor is therefore admissible even at the narrowing edges of the cap.

On \(M\), the source is zero and \(w_1\) is the first harmonic extension of its trace. Equality of the measured quadratic forms makes its interior energy equal to that of the replacement. On \(E\), both coefficients and the source pairing are unchanged. Thus \(\min\mathcal J_2\le\min\mathcal J_1\). Reversing the argument gives equality, and uniqueness of the minimizer proves \(w_1=w_2\) on \(E\).

Normalize the Green matrix by \[L_{j,x}G_j(x,y)=\delta_y^{g_j}(x)I,\qquad G_j(\cdot,y)|_{\partial N_j}=0,\qquad L_{j,D}^{-1}f(x)=\int_{N_j}G_j(x,y)f(y)\,dV_{g_j}(y).\] The source densities on \(E\) coincide. Varying the source gives equality of the distribution kernels on \(E\times E\), hence pointwise equality off the diagonal. Self-adjointness of the Dirichlet inverse gives the adjoint symmetry in (6). ◻

Proof. The last assertion is smooth Dirichlet regularity. For the first two, a complex-linear functional on values or jets is represented by a dual-vector distribution \(T\), supported at one or two points. Suppose it annihilates all source solutions, and define \(v=(L_D^{-1})^tT\) by distributional transposition. Then \[\langle v,f\rangle=\langle T,L_D^{-1}f\rangle=0 \quad(f\in C_c^\infty(U;\mathbb C^2)), \qquad L^tv=T.\] Here the transpose and pairing are bilinear on the dual bundle; no complex conjugation convention is needed. Elliptic regularity makes \(v\) smooth off the support of \(T\). The interior with one or two points removed is connected: a path can be diverted around each point in a punctured coordinate ball when \(n\ge2\). Unique continuation from the nonempty set obtained by removing these points from \(U\) makes \(v\) supported at those points in the interior. If a point belongs to \(U\), the asserted initial vanishing is distributional there as well.

At each support point a distribution is a finite sum of derivatives of the point mass. If its highest derivative order is \(k\), its image under \(L^t\) has exact highest order \(k+2\). Indeed, the highest homogeneous coefficient vector polynomial is multiplied by the scalar elliptic quadratic polynomial; this multiplication is injective. Consequently no nonzero order-zero distribution can be the image of a point-supported distribution. There are therefore no value annihilators, proving both surjectivity claims.

For a two-jet annihilator at \(p\notin\overline U\), the image \(T\) has order at most two. Thus \(v=c\delta_p\), with \(c\) a dual-fiber vector. Its image is precisely the functional \(u\mapsto c(Lu(p))\). Conversely all such functionals annihilate the source solutions. Finite-dimensional duality gives the exact stated jet space. In particular, values and gradients may be prescribed freely, since the Hessian trace can be chosen to satisfy the equation. ◻

Proof. If two possible images of the same point were distinct, their evaluation identities would impose a fixed invertible linear relation between the evaluations at two distinct points of \(N_1^\circ\). This contradicts joint surjectivity in Lemma 4. Once the image agrees, surjectivity at the source point determines \(J\) uniquely. The same argument using the evaluations in \(N_2\) rules out two base points with the same image. Thus the maps glue on every overlap. Their union is connected because every domain contains \(E\), and retains all the properties in (8).

For fixed \(p\), the evaluation identity and the common source density give (9) when \(q\in U_0\), away from the pole. As a function of \(q\), the adjoint of each side solves the same pulled-back connection equation after the bundle identification. Injectivity of \(\Phi\) ensures that \(q=p\) is the only possible pole. Unique continuation on the connected set \(W\setminus\{p\}\) proves the identity everywhere claimed. ◻

Proof. Choose a small coordinate ball near an interior frontier and a point of \(W\) close enough to that frontier that its distance to the complement is attained inside the chart. The Euclidean ball with that center and radius lies in \(W\) and touches its complement at an interior point \(p\). Its boundary gives the required smooth one-sided contact hypersurface.

If \(p_k\in W\) tends to \(p\), compactness of \(N_1\) and \(U(2)\), using the fixed global trivializations, gives a subsequence for which \(\Phi(p_k)\to q\) and \(J(p_k)\to J_0\). Continuity of every source solution gives \[ J_0 I_2(p)=I_1(q). \tag{11}\] The point \(q\) cannot lie on the boundary, where the right side would be zero, since \(I_2(p)\) is onto. Two distinct possible limits \(q\) contradict joint surjectivity on \(N_1\); at a fixed \(q\), surjectivity of \(I_2(p)\) determines \(J_0\). The full limits therefore exist. Also \(q\notin E\): otherwise its identity match on \(E\) and (11) would relate the evaluations at the distinct points \(q,p\) of \(N_2\). Both \(p\) and \(q\) consequently avoid \(\overline{U_0}\).

Choose two source solutions whose columns give an invertible matrix \(F_1(q)\). Their corresponding second-side columns give \(F_2(p)\), invertible by (11). Shrink the charts so both matrices are invertible. In a harmonic frame, \(v=F_j^{-1}w\) satisfies an equation with scalar elliptic principal part and no zeroth-order term, since every column of \(F_j\) is harmonic. Lemma 4 is invariant under this smooth frame change. It therefore supplies matrices \(X_1^l\) with value zero and gradients equal to any chosen real coordinate covectors times \(I\). Choose those covectors independently, and let \(X_2^l\) be the corresponding source matrices expressed in \(F_2\).

Set \(x^l=\frac12\operatorname{Re}\operatorname{tr}X_1^l\) and \(y^l=\frac12\operatorname{Re}\operatorname{tr}X_2^l\). The first functions form coordinates at \(q\). On matched points, the source identity implies \(JF_2=F_1\circ\Phi\) and \(X_2^l=X_1^l\circ\Phi\). Thus the scalar coordinates and the metric Gram matrices of their differentials agree under the isometry. Passing to \(p,q\), their Gram matrices agree and are positive definite. Hence the \(y^l\) are coordinates at \(p\). Fix a target chart neighborhood \(U_1\) on which the \(x^l\) are coordinates. The full convergence of partner points permits a source chart neighborhood \(U_2\) so small that \(\Phi(W\cap U_2)\subset U_1\). Choose a small coordinate ball \(\Omega\) about zero contained in both \(x(U_1)\) and \(y(U_2)\), and restrict the two charts to their inverse images of \(\Omega\). For a matched point in the restricted source chart, \(x(\Phi(p'))=y(p')\in\Omega\); its partner therefore lies in the restricted target chart. Thus the old map is coordinate identity throughout its domain in these charts. The matrix gradient identity at \(q\) now reads \(\partial_iX_1^l(0)=\delta_i^lI\); equality on the one-sided region and smoothness give the same identity for \(X_2^l\).

Because the harmonic-frame equation has no zeroth-order term, a jet with zero value and gradient is constrained only by \(g_1^{ij}(0)H_{ij}=0\). The jet lemma supplies a finite basis of these Hessians and corresponding source pairs. They agree on the known region, so their complete jets agree at the contact point. The same conclusion holds for every smooth coefficient or paired function already identified there.

Finally polar-orthonormalize the original harmonic frames: write \(F_j=H_jS_j\), with \(H_j\) unitary and \(S_j\) positive Hermitian. On the matched set, \(JF_2=F_1\) implies \(S_2=S_1\) and \(JH_2=H_1\). In the unitary frames \(H_j\), the old bundle identification is therefore \(I\); connection matrices and Green matrices agree there. We continue to write \(F_j\) for the harmonic-frame matrices in these unitary frames. A common real linear coordinate change makes the metric at zero Euclidean and the contact hyperplane horizontal. Apply the same linear change to the list of \(X_j^l\); this preserves all identities in (10). Reverse the final normal coordinate if necessary to put the known side below its graph. In these coordinates the Hessian constraint is the one in the statement. ◻

Proof. Let \(\mathcal H_1(D_1)\) be the closed subspace of \(L^2(D_1;V_1)\) consisting of distributional \(\mathsf L_1\)-solutions. Restriction \(R:\mathcal H_1(D_1)\to L^2(S_1;V_1)\) is compact by interior elliptic estimates and is injective by unique continuation. The spectral theorem for \(R^*R\) gives an orthonormal basis \((w_j)\) of \(\mathcal H_1(D_1)\) such that \[ (w_i,w_j)_{L^2(S_1)}=\mu_j^2\delta_{ij},\qquad 0<\mu_j\le1. \tag{15}\] For every \(N\) there is a uniform constant \(C_N\) with \[ \mu_j\le C_N(1+j)^{-N}. \tag{16}\] Indeed, multiply a solution by a cutoff equal to one near \(\overline{S_1}\) and compactly supported in \(D_1\). Interior estimates bound its \(H^k\) norm by its \(L^2(D_1)\) norm. Fourier truncation in a containing cube, component by component, gives an approximation of rank \(O(A^n)\) and error \(O(A^{-k})\). The minimax characterization of singular values yields \(\mu_j\le C_kj^{-k/n}\). The fixed fiber dimension changes only the constants.

For \(y\in S_2\), expansion in the first variable gives \[F(x,y)=\sum_j w_j(x)\otimes b_j(y),\qquad \|b_j\|_{L^2(S_2;V_2)}\le M.\] Contracting the first tensor index against \(\overline{w_j}\) extends the coefficients to \(D_2\): \[ b_j(y)=\mu_j^{-2}\int_{S_1} \sum_\alpha\overline{w_{j,\alpha}(x)}F_{\alpha}(x,y)\,dx, \qquad \|b_j\|_{L^2(D_2;V_2)}\le M\mu_j^{-1}. \tag{17}\] Here \(F_\alpha(x,y)\in V_2\) are the first-index components in an orthonormal basis of \(V_1\); the displayed contraction leaves a vector in \(V_2\). Because the second system acts on that remaining index, \(\mathsf L_2b_j=0\). Restriction orthogonality proves that this formula agrees with the original coefficient on \(S_2\). The assumed interpolation gives \[\|w_j\|_{L^2(O_1)}\le C\mu_j^{\gamma_1},\qquad \|b_j\|_{L^2(O_2)}\le CM\mu_j^{\gamma_2-1}.\] Thus the tensor series converges absolutely in \(L^2(O_1\times O_2)\): the norm of its \(j\)th term is at most \(CM\mu_j^{\gamma_1+\gamma_2-1}\), and (16) makes these bounds summable. The limit solves both equations distributionally. Interior estimates for their elliptic sum give smoothness and the claimed bounds.

Agreement on the second arm requires a completeness check. Take \(z\in L^2(S_1;V_1)\) orthogonal to the closure of \(R\mathcal H_1(D_1)\). The \(V_2\)-valued function \[y\longmapsto\int_{S_1}\sum_\alpha\overline{z_\alpha(x)}F_\alpha(x,y)\,dx\] solves the second system and vanishes on \(S_2\), hence vanishes on connected \(D_2\). Every second-arm slice therefore belongs to the closed restricted range. Its expansion in the orthonormal basis \(w_j/\mu_j\) is complete, with coefficients \(\mu_jb_j(y)\). Consequently the series equals \(F\) on \(S_1\times O_2\); it converges there by the positive exponent \(\gamma_2\). On \(O_1\times S_2\) it is the original expansion, converging with exponent \(\gamma_1\). These establish both required agreements. Only upper bounds for \(\mu_j\) have been used, so uniformity does not require continuous choices of singular bases or lower bounds on singular values. ◻

Proof. We first split the product Hessian condition into a strict weight in each factor. Their sum will lie below a defining function for the given side, with a quadratic gap. That gap places two intersecting product sets in the given side; the weights then yield interpolation exponents greater than one half, allowing Lemma 7 to apply.

Proof. Starting from the top, let \(a_*\) be the infimum of levels down to which an extension agreeing with the seed has been obtained. In the uniform version include uniform bounds in this property. At a non-seed point of \(B\times\{a_*\}\), Proposition 8 uses compact data strictly above that level. Take a finite cover of the level by such output neighborhoods and seed neighborhoods. Their data are contained above \(a_*+\eta\) for one \(\eta>0\), so an already reached level supplies them with the needed bounds.

Shrink the outputs in \((z,a)\) coordinates to boxes \(B_i\times(a_*-d_i,a_*+d_i)\). Every connected component of an overlap contains vertical segments to a common height greater than \(a_*\). Both germs equal the preceding extension there. Unique continuation for \(\mathsf L_1^x+\mathsf L_2^y\) makes them identical on that overlap component. The same argument gives agreement with \(\mathcal S\), because a vertical segment moving upward from a seed point stays in the seed. Boundary neighborhoods use the given solution. A finite subcover has a positive minimum output width and therefore passes the putative last level, or includes the endpoint \(a_-\). This proves continuation on the whole cylinder. The same finite constructions retain all strict margins under small perturbations. Their Carleman, series and interior estimates prove the uniform assertion. Data bounded up to a limiting surface are never required: each local construction uses compact sets strictly above it. ◻

Proof. For the final comparison with the full original cross, fix at the outset \[\widehat B=B+\eta+4,\qquad \widehat d=\min(d/2,1/4).\] We construct the continuation for these enlarged location bounds and smaller separations, and then restrict to the stated target. We first cross the limiting plane, then deform variable-radius tubes to reach those separations. This follows the two-stage construction of (OpenAI 2026, sec. 3, “Fixed-separation initialization”). We first choose the Euclidean geometric ratios, including the opening of the region reached in the first stage. We next choose the entire tube cylinder and a bounded positive-separation range containing it. The first-stage constructions are then made on that range. Only after all these fixed sets have been chosen do we decrease \(\lambda\).

Proof. Write \(a_\delta=(\log\ell_\delta)'\). With \(s=t/\delta\), \[a_\delta(t)=\delta^{-1}A(s),\qquad A(s)=\frac{e^s}{(1+e^s)\log(1+e^s)}.\] The function \(A\) is positive and decreasing: setting \(u=e^s\) gives \[A'(s)=\frac{u\bigl(\log(1+u)-u\bigr)} {(1+u)^2\log(1+u)^2}<0.\] At the negative end \(A\) tends to one, and at the positive end it is comparable to \((1+s)^{-1}\). Differentiating its convergent expressions at the two ends gives \(|A^{(j)}|\le C_jA^{j+1}\); positivity gives the same bounds on the remaining compact interval. Consequently, on the fixed range of \(t\), \[a_\delta\ge c>0,\qquad a_\delta'\le0,\qquad |\partial_t^j\log\ell_\delta|\le C_j a_\delta^j, \qquad a_\delta\le\ell_\delta^{-1}.\] Since \(|dt|_\gamma\) is bounded above and below, \(\beta=p_*a_\delta|dt|_\gamma\). The identity \[\operatorname{Hess}_\gamma f_* =p_*a_\delta\operatorname{Hess}_\gamma t +p_*a_\delta'\,dt\otimes dt\] and (34) prove the first two bounds in (36). The chain rule gives the derivative bounds; their constants can be independent of \(p_*\) because \(p_*a_\delta^j\le p_*^j a_\delta^j\) for \(j\ge1\).

The estimate \(0<\ell_\delta'<1\) proves the Lipschitz assertion. Moreover, \[\partial_\delta\ell_\delta =\log(1+e^s)-\frac{s e^s}{1+e^s}>0.\] The displayed expression tends to zero at both ends and its derivative in \(s\) is \(-s e^s/(1+e^s)^2\), so it is strictly positive at finite \(s\). The limit of \(\ell_\delta\) is \(\max(t,0)\). Finally, \[r_\delta\beta^2\le C R_*p_*^2\ell_\delta^{p_*-2},\] and \(\ell_\delta\) is uniformly bounded on the fixed depth interval. This proves (37). ◻

Proof. Put \(b=\nabla^\gamma r\), so that \(|b|=r\beta\). Parallel transport along the radial \(\gamma\)-geodesic identifies the endpoint tangent spaces; let \(\omega\) be its unit direction from \(y\) to \(x\). Multiply the normalized vectors in Proposition 8 by the common factor \(r\). Their real and imaginary parts can be written as \[U_1=\omega+i v,\qquad v\perp\omega,\quad |v|=1, \qquad U_2=q+i w.\] All lengths in the following calculation use \(\gamma\). Set \(s=\omega+b\). In a \(\gamma\)-orthonormal frame at \(y\), the exact conditions on the second vector are \[ q=\frac{Qs}{s^TQs},\qquad s\cdot w=0,\qquad w^TQ^{-1}w=\frac1{s^TQs}. \tag{42}\] Here the matrix of \(a\) is the identity. Therefore \[ s\cdot U_2=1,\qquad q=\frac{s}{|s|^2}+O(|h|),\qquad |w|=|s|^{-1}+O(|h|). \tag{43}\] The assumptions make \(b\) and \(h\) small. Both partial gradients of \(\rho\) are consequently nonzero, and \(|U_2|\) is bounded above and below.

With the product \(\gamma\)-connection, the endpoint Hessian of half the squared distance has value \[ |U_1-U_2|^2+O(r^2)(|U_1|^2+|U_2|^2). \tag{44}\] To obtain the error, parametrize the radial geodesic on \([0,1]\). In a parallel frame its Jacobi field with prescribed endpoint values differs from the affine interpolant by \(O(r^2)\) in \(C^1\), because the curvature is bounded and the velocity has size \(r\). The second variation formula gives (44). Replacing the second connection by that of \(Q^{-1}\) contributes only \(O(r^2)\): its difference from the \(\gamma\)-connection is \(O(|h|+|\partial h|)=O(\varepsilon r)\), while \(|d\rho|=O(r)\).

The relation \(s\cdot U_2=1\) implies \(\omega\cdot(U_1-U_2)=b\cdot U_2\). Also \[\operatorname{Hess}_\gamma(r^2/2) =2\,dr\otimes dr+r^2\operatorname{Hess}_\gamma f_*.\] Thus the Hessian sum in the criterion, multiplied by \(r^2\), equals \[ |\pi_{\omega^\perp}(U_1-U_2)|^2-|b\cdot U_2|^2 -r^2\operatorname{Hess}_\gamma f_*(U_2,\overline{U_2}) +O(r^2). \tag{45}\] For a real bilinear form, evaluation on a complex vector and its conjugate means the sum of the evaluations on its real and imaginary parts, as in the criterion.

We record the cancellation quantitatively. The real projected difference is \(-\pi_{\omega^\perp}b+O(|b|^2+|h|)\). The imaginary part obeys \[|\pi_{\omega^\perp}w| =1-\omega\cdot b+O(|b|^2+|h|).\] Comparing this length with \(|v|=1\) and squaring gives \[|\pi_{\omega^\perp}(U_1-U_2)|^2 \ge |b|^2-C\bigl(|b|^3+|b||h|+|h|^2\bigr).\] The real and imaginary directions of \(U_2\) are almost orthonormal; hence \[|b\cdot U_2|^2\le \bigl(1+C(|b|+|h|)\bigr)|b|^2.\] Using \(|b|=r\beta\), \(|h|\le\varepsilon r\), and (36), expression (45) is bounded below by \[ c r^2\beta- C\bigl(r^2+r^3\beta^3+\varepsilon r^2\beta+ \varepsilon^2r^2\bigr). \tag{46}\] Divide the error terms by \(r^2\beta\). Their ratios are bounded by \(C/\beta\), \(Cr\beta^2\), \(C\varepsilon\), and \(C\varepsilon^2/\beta\), respectively. Increasing \(p_*\) controls the first, and then decreasing \(\varepsilon\) controls the others by (41). This proves strict positivity. ◻

Proof. Deform \(\eta\) from \(1\) down to \(\delta\), using \((y,\omega)\) in the unit sphere bundle of \(\gamma\) as base coordinates and \[x=\exp_y^\gamma(r_\eta(y)\omega).\] The base over the closed cylinder \(Y\) is compact. The radius has strictly positive derivative in \(\eta\), so these variables give cylinder coordinates on every fixed interval \(\delta\le\eta\le1\). The positive side of a level is the direction of increasing \(\eta\). Since \(r_\eta\ge r_\delta\), condition (40) holds at every intermediate radius. Lemma 13 applies there.

At \(\eta=1\) the separations have a positive minimum on \(Y\). The same is true throughout \(t>t_b\) by (39). Proposition 10 therefore supplies the top and positive-depth data, with open room and uniform bounds at any fixed order. In the lower strip \(y_n<-1/4\), use the original kernel \(G_1\); this strip lies in the dilated known side for small \(\lambda\). The initialization agrees with this kernel on their overlap.

These two base regions cover every lateral face of \(Y\): its bottom face is in the lower strip, and on a remaining lateral or upper face outside that strip one has \(t_0\ge3/4\). Both regions are independent of \(\eta\) and hence upward closed. Together with a neighborhood of the top level they supply exactly the seed required by Lemma 9. That lemma and the strict sphere test extend the kernel through the whole cylinder, preserving agreement on the seed. Normal-neighborhood room and positive radii on each fixed interval ensure that all surfaces remain off the diagonal. ◻

Proof. The Dirichlet problem on each ball is invertible. In the unitary frame this follows from the positive connection energy: a zero-boundary null solution is parallel and hence zero. Invertibility is preserved by the harmonic-frame change and by positive scalar multiplication of the equation. Its Dirichlet solution operator determines the boundary covariant normal derivative from the boundary value. Transpose this boundary operator onto the smooth matrix coefficient in the first term of (47), then multiply by the frame factors and pull the surface density back under \(\omega\mapsto\exp_y^\gamma(r(y)\omega)\). This gives the smooth matrix density in (49). Smooth dependence of the ball Dirichlet problem on its center and positive radius gives smooth dependence of \(\psi\).

To prove the local harmonicity assertion, fix \(y_0\) and \(u=F_1v\). The kernel is defined on an open neighborhood of the compact sphere over \(y_0\). Choose a fixed slightly larger surrounding surface inside that neighborhood and inside the domain of \(u\). After restricting the center neighborhood, this surface surrounds all the moving spheres and the intervening collars remain in the common domain. Green’s identity on each collar replaces (47) by its integral over the fixed surface. Both first-variable equations are homogeneous in the collar. Applying the second-variable operator under the fixed integral now proves that \(S(y;u)\) is second-system harmonic: the columns of \(\mathcal G(x,y)^*\) solve that system. Equation (48) proves the assertion in harmonic frames. This argument applies to arbitrary local harmonic inputs; they need not extend throughout the coordinate region.

If \(y_n<-1/4\), the kernel is \(G_1\) and the Green representation formula gives \(S(y;u)=u(y)\). On that region \(F_1=F_2\), so \(T_yv=v(y)\). For a fixed pair \((v_1,v_2)\), both \(T_yv_1\) and \(v_2(y)\) solve the second harmonic-frame equation on connected \(Y^\circ\) and agree in the lower strip. Unique continuation proves their equality. Finally constant columns are harmonic in both harmonic frames and agree there, so the same argument yields the first identity in (50). ◻

Proof. On the lower cutoff region, \(T_y\) is evaluation at the center in the harmonic frame. Normalize its equation to principal matrix \(a=\gamma^{-1}\), use a smooth \(\gamma\)-orthonormal frame to pull it back under \(z\mapsto\exp_y^\gamma(rz)\), and multiply by \(r^2\). With center and radius regarded as independent parameters, the resulting Dirichlet systems extend smoothly to \(r=0\), are uniformly elliptic, and have the Euclidean Laplacian as limit; their lower-order coefficients vanish at \(r=0\). The Dirichlet inverses are uniformly bounded, either by their limiting inverse and perturbation or by the positive energy before the change of frame. Elliptic boundary regularity, applied after differentiation in the parameters, shows that the evaluation density at the center has uniform angular smooth bounds and smooth center–radius dependence. The center remains a fixed positive distance from the unit sphere in these coordinates. Substituting the varying radius costs only \(|dr|_\gamma=r\beta\), which is uniformly bounded by (41) and the lower bound for \(\beta\).

This evaluation density is the density constructed in Proposition 15. Indeed both represent the same functional on traces of functions harmonic near the closed ball. These traces are dense in smooth boundary data. For a prescribed smooth trace \(\varphi(\omega)\), solve on the ball of radius \(r(1+\eta)\) with boundary value \(\varphi\) transported along the radial rays. On pulling this problem back to the closed unit ball, its solution \(z_\eta\) converges in every \(C^k\) norm to the solution \(z_0\) at radius \(r\), by smooth dependence of the Dirichlet inverse and boundary regularity. Its trace on the original sphere is \(z_\eta(\omega/(1+\eta))\), which converges in \(C^k(S_y)\) to \(z_0(\omega)=\varphi(\omega)\). The corresponding physical solutions are harmonic near the original closed ball. Thus the two smooth density matrices agree.

On the other cutoff region \(t>t_b\), the radii have the positive lower bound (39); all their required derivatives are uniformly bounded in \(\delta\). Proposition 10 gives uniform kernel bounds there. The smooth Dirichlet construction of the density therefore gives the same bounds for \(\psi_\delta\) and its first base derivatives. Compactness of the cutoff derivative support completes the proof. ◻

Proof. First regard the radius \(s\) as an independent parameter. Pull the first scalar equation back by \(z\mapsto\exp_y^\gamma(sz)\) and multiply by \(s^2\). Its principal coefficients are \(\delta^{ij}+O(s^2)\) and its first-order coefficients are \(O(s)\). The same statement holds for the system, with scalar principal coefficients times \(I\). Both Dirichlet operators extend smoothly to \(s=0\), where they equal the Euclidean Laplacian. On each fixed Sobolev scale their Dirichlet inverses are uniformly bounded and smooth for sufficiently small \(s\), by elliptic Dirichlet estimates and invertibility at zero.

For these smooth strongly elliptic Dirichlet families on the unit ball, the boundary radial derivative operator is classical of order one. Its principal symbol depends only on the principal coefficients and the differentiating vector. The boundary symbol construction, obtained by choosing the decaying normal root and solving the successive transport equations, is smooth in \((y,s)\); see, for example, Joshi and Lionheart (2005, sec. 3, Proposition 3.1). Their factorization is for the Hodge Laplacian; its construction applies to the present Dirichlet systems because their principal symbol is scalar times \(I\), while the transport equations allow matrix coefficients. The true Dirichlet inverse corrects the parametrix by a smoothing operator. Differentiating this inverse on arbitrarily high fixed Sobolev scales bounds every required smooth kernel seminorm of the correction. Thus these assertions hold in the full symbol topology, including smoothing remainders and parameter derivatives.

At \(s=0\) the radial operator is \(\mathsf B\), and the first \(s\)-derivative of its principal symbol vanishes because the principal coefficients have no linear term in \(s\). Taylor expansion in the symbol topology therefore gives, for the scalar equation, \[ \mathsf B+s\mathcal B_1(y)+s^2\mathcal B_2(y,s), \qquad \mathcal B_1\in\Psi^0,\quad\mathcal B_2\in\Psi^1. \tag{58}\] The system and scalar radial operators have identical principal symbols for every \(s\) and coincide at zero. Their difference is consequently \(s\) times a smooth family in \(\Psi^0\).

Differentiating the center of the exponential map while transporting \(\omega\) parallel gives a Jacobi field with initial derivative zero. In the unit-ball coordinates its velocity is \[s^{-1}(e_i+O(s^2));\] putting \(s=r(y)\) adds \(f_i\omega\). Tangential derivatives act directly on boundary data and are identical for the scalar and matrix equations. For the radial component use (58). The leading terms are \(r^{-1}\mathsf A_i+f_i\mathsf B\); the order-one errors have size \(O(r)\) and the order-zero errors are bounded, since \(r\beta\le\varepsilon/\beta\). This proves (56) after sphere adjunction. The radial difference \(r\Psi^0\), multiplied by the center velocity \(O(r^{-1}+\beta)\), proves (57). Base derivatives cost at most one factor \(\beta\) each by the profile bounds. Sphere transposition and commutation with the smooth angular connection fields preserve these orders and bounds. ◻

Proof. Both operators change angular degree by exactly one, so it suffices to prove the estimate at one input degree \(k\ge1\). Identify that component with a harmonic homogeneous polynomial \(v(z)\) of degree \(k\). Use the Fischer inner product, in which the monomials \(z^\alpha\) are orthogonal with squared norm \(\alpha!\). Polynomial differentiation \(a_i=\partial_{z_i}\) is adjoint to multiplication \(M_i=z_i\). On harmonic polynomials of degree \(k\) the Fischer squared norm is the normalized squared sphere norm multiplied by \[N_k=n(n+2)\cdots(n+2k-2),\qquad N_0=1.\] For completeness, integration against a standard real Gaussian identifies the Fischer norm on trace-free homogeneous polynomials with their Gaussian \(L^2\) norm; integrating radially then gives the factor \(N_k\). These classical identities are also recorded in (Render 2008, Equation (2.1) and Theorem 2.1).

Set \(d=k+n/2-1\) and use \(D_i^\pm=\pm\nabla_i+k f_i\) at this fixed degree. For harmonic polynomials \(q_i\) of degree \(k\), the Laplacian of \(\sum_iM_iq_i\) is \(2\sum_i a_iq_i\). Since \[\Delta_z(|z|^2s)=4d\,s\qquad(s\in\mathcal H_{k-1}),\] its harmonic projection is \(\sum_iM_iq_i-|z|^2\sum_i a_iq_i/(2d)\). Orthogonality of this projection and its complement shows that its squared norm is \[\sum_i\|q_i\|^2+\sum_{i,j}(a_jq_i,a_iq_j) -d^{-1}\Bigl\|\sum_i a_iq_i\Bigr\|^2.\] Changing from Fischer adjoints to sphere adjoints gives \[\mathsf A_i^*|_{\mathcal H_k} =\frac{N_{k+1}}{N_k}\Pi_{k+1}M_i,\] where \(\Pi_{k+1}\) is harmonic projection in homogeneous degree \(k+1\). Consequently \(N_k\) times the left side of (66) is \[ \begin{split} (2d+2)\left[\|D^+v\|^2 +\sum_{i,j}(a_jD_i^+v,a_iD_j^+v) -\frac1d\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2\right] -2d\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2. \end{split} \tag{67}\] All norms in this calculation include integration over the base and use the polynomial inner product in the fibers. The factors are \(N_{k+1}/N_k=2d+2\) and \(N_k/N_{k-1}=2d\).

Write \(C_{ij}=2k(\operatorname{Hess}_\gamma f_*)_{ij}\) and \[C[av]=\int_Y\sum_{i,j}C_{ij}\langle a_iv,a_jv\rangle\,dV_\gamma.\] Covariant integrations by parts give \[\begin{align*} \sum_{i,j}(a_jD_i^+v,a_iD_j^+v) &=\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2-C[av] +O(k^2)\|v\|^2,\tag{68}\\ \Bigl\|\sum_i a_iD_i^+v\Bigr\|^2 &\le k\|D^-v\|^2-C[av]+O(k^2)\|v\|^2,\tag{69}\\ \|D^-v\|^2 &=\|D^+v\|^2+\int_Y\mathop{\mathrm{tr}}C\,|v|^2\,dV_\gamma. \tag{70}\end{align*}\] Here and below a form denoted \(O(k^2)\|v\|^2\) has absolute value at most a fixed constant times that quantity. To verify the first two formulas, move a \(D_j^+\) across the pairing and commute \(D_j^-D_i^+\) to \(D_i^+D_j^--C_{ij}\). The reordered term in (69) is at most \(k\|D^-v\|^2\), because \(\sum_i|a_iw|^2=k|w|^2\) for \(w\in\mathcal H_k\). The extra commutator is curvature: it acts as a sum of rotations of norm \(O(k)\) at degree \(k\), and the two contractions supply one more factor \(k\). This proves the claimed error bounds. Expanding the two squares proves (70). For complex functions all the displayed forms are understood through their real parts.

Substituting (68) into (67) rewrites that expression as \[ \begin{split} &(2+2/d)\left[d\|D^+v\|^2 -\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2-dC[av]\right]\\ &\qquad+2\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2 +O(k^3)\|v\|^2. \end{split} \tag{71}\] We must control the remaining negative divergence square while retaining both an ordinary derivative estimate and an additional half-order angular derivative estimate. The latter requires a coefficient of size \(k/\beta\) in front of \(|D^+v|^2\), including when \(k\) is much larger than \(\beta\). Choose a small constant \(\vartheta>0\). On the fraction \(1-\vartheta\) of its negative divergence square use (69)–(70). On the remaining fraction use the pointwise convex combination \[ \begin{split} \Bigl|\sum_i a_iD_i^+v\Bigr|^2 &\le(1-\zeta)(k+n-1)|D^+v|^2\\ &\quad+\zeta\left(2\Bigl|\sum_i a_iD_i^-v\Bigr|^2 +8k^3\beta^2|v|^2\right), \qquad \zeta=c_1/\beta. \end{split} \tag{72}\] The fraction \(1-\vartheta\) in the integrated estimates is constant; the \(y\)-dependent \(\zeta\) is used only in this pointwise inequality. The first bound follows from the adjoint multiplication operators, since the contraction \((q_i)\mapsto\sum_i a_iq_i\) has squared norm at most \(k+n-1\) on polynomials of degree \(k\). The second follows from \(D_i^++D_i^-=2k f_i\) and \(|\sum_i f_i a_i v|^2\le k\beta^2|v|^2\).

Combining these bounds in (71), the coefficient retained for \(|D^+v|^2\) is \[ (2+2/d)\bigl[d-k-\vartheta(n-1) +\vartheta\zeta(k+n-1)\bigr]. \tag{73}\] Since \(d-k=(n-2)/2\), a sufficiently small fixed \(\vartheta\) makes this at least a fixed positive constant plus a positive multiple of \(c_1k/\beta\). The coefficient of \(|\sum_i a_iD_i^-v|^2\) is nonnegative when \(\beta_0\) is sufficiently large. The Hessian contribution is exactly \[ -(2+2/d)\left[(d-1+\vartheta)C[av] +(1-\vartheta)k\int_Y\mathop{\mathrm{tr}}C\,|v|^2\,dV_\gamma\right]. \tag{74}\] For \(k\ge1\) and \(n\ge3\), both coefficients inside this expression are positive. The Hessian hypothesis bounds it below by \(c k^3\int\beta|v|^2\). This absorbs the curvature error \(O(k^3)\|v\|^2\) by increasing \(\beta_0\), and absorbs the last term of (72) by choosing \(c_1\) sufficiently small. We have therefore retained positive multiples of \[\|D^+v\|^2,\qquad \int_Y\frac{k}{\beta}|D^+v|^2\,dV_\gamma, \qquad k^3\int_Y\beta|v|^2\,dV_\gamma.\] The first controls both \(\|\nabla v\|^2\) and \(k^2\int\beta^2|v|^2\), since its cross term is \(-k\int\Delta_\gamma f_*\,|v|^2\ge0\). For the weighted derivative use the pointwise inequality \[\frac{k}{\beta}|\nabla v|^2 \le2\frac{k}{\beta}|D^+v|^2+2k^3\beta|v|^2.\] No differentiation of \(\beta^{-1}\) is required. Dividing by \(N_k\), and then summing the orthogonal degree components, proves (66); the weights \((1+k)^s\) are comparable to \(k^s\) because the zero degree is excluded. ◻

Proof. Let a dagger denote the full adjoint for \(dV_\gamma d\sigma\). Covariant integration and the degree shift give \[ Z:=T^\flat-T^\dagger=\sum_i f_i\mathsf A_i. \tag{76}\] A direct expansion yields \[ \begin{split} &\|(2T+\mathcal U)\varphi\|^2- \|(2T^\flat+\mathcal U^\dagger)\varphi\|^2\\ &\qquad=4\bigl(\|T\varphi\|^2-\|T^\flat\varphi\|^2\bigr) +\langle\mathcal E\varphi,\varphi\rangle, \end{split} \tag{77}\] where \[ \mathcal E=2[T^\dagger,\mathcal U]+2[\mathcal U^\dagger,T]+[\mathcal U^\dagger,\mathcal U] -2\bigl(\mathcal U Z+(\mathcal U Z)^\dagger\bigr). \tag{78}\] We will prove \[ |\langle\mathcal E\varphi,\varphi\rangle| \le C\varepsilon\mathcal N(\varphi)^2. \tag{79}\]

Here are the complete order and weight counts needed for this claim. An operator has count \((m,s)\) if its coefficient at \(\partial_y^\alpha\) is angular order \(m-|\alpha|\), of size \(O(\beta^{s-|\alpha|})\), with the derivative allowances already specified. Adjoints preserve the count, and products add the counts. When a base derivative hits a coefficient it consumes one differential order and adds a factor \(\beta\), so it preserves the weight index. A commutator of scalar angular operators loses one angular order: the products of their principal symbols cancel. Hence a scalar commutator loses one in total order while preserving the sum of the weight indices. The counts of \(T,T^\dagger,\mathcal U,Z\) are, respectively, \[(2,1),\quad(2,1),\quad(2,0),\quad(1,1),\] and every occurrence of \(\mathcal U\) carries the factor \(\varepsilon\). Thus (78) has count at most \((3,1)\) and a factor \(O(\varepsilon)\); the quadratic \(\mathcal U\) term is smaller.

There is a further cancellation that the total count alone does not express: no third base derivative survives. To check it explicitly, use an orthonormal sphere trivialization and write the base measure as \(\mu(y)\,dy\). The sphere probability measure is then fixed. Write \[\mathcal U=c^{ij}\partial_i\partial_j+\mathcal B^i\partial_i+\mathcal C, \qquad c^{ij}=rQ^{ij}=c^{ji}\in\mathbb R.\] The coefficients \(\mathcal B^i\) and \(\mathcal C\) are angular operators. Replacing a horizontal derivative by a coordinate derivative plus its angular connection term changes only base orders at most one, so these are all the second-base coefficients. If a star denotes the sphere adjoint, direct integration by parts gives \[\begin{align*} \mathcal U^\dagger-\mathcal U ={}&\left(2\mu^{-1}\partial_j(\mu c^{ij}) -\mathcal B^i-(\mathcal B^i)^*\right)\partial_i\\ &+\mu^{-1}\partial_i\partial_j(\mu c^{ij}) -\mu^{-1}\partial_i\bigl(\mu(\mathcal B^i)^*\bigr) +\mathcal C^*-\mathcal C. \end{align*}\] In particular it has base order at most one. In \([T^\dagger,\mathcal U]\) and its partner, the possible third-base-derivative coefficient is a commutator of an angular coefficient with \(rQ^{ij}\), which is zero by (64). For \([\mathcal U^\dagger,\mathcal U]\), first write it as \([\mathcal U^\dagger-\mathcal U,\mathcal U]\). The operator \(\mathcal U^\dagger-\mathcal U\) has no second base derivative, since the coefficient in (64) is real. Taking adjoints against the smooth base density only produces lower base orders. Its remaining possible third-base-derivative coefficients commute for the same reason. Finally, \(\mathcal U Z\) already has at most two base derivatives. It follows that \(\mathcal E\) is a sum of terms of precisely the following three permitted types: \[ \varepsilon\beta^{-1}\Psi^1\partial_y^2, \qquad \varepsilon\Psi^2\partial_y, \qquad \varepsilon\beta\Psi^3. \tag{80}\] This argument also covers lower orders, since \(\beta\ge1\).

For the first type in (80), integrate once in the base. The principal form is bounded by \(C\varepsilon\int\beta^{-1}\|\partial_y\varphi\|_{1/2}^2\). A derivative of its coefficient costs one factor \(\beta\) and leaves an \(\varepsilon\Psi^1\partial_y\) term. This includes differentiation of the weight itself: \(|d\beta|\le C\beta^2\) implies \(|d(\beta^{-1})|\le C\). The second type has the form bound \[C\varepsilon\int_Y \|\partial_y\varphi\|_{1/2}\|\varphi\|_{3/2}\,dV_\gamma \le C\varepsilon\int_Y\left( \beta^{-1}\|\partial_y\varphi\|_{1/2}^2 +\beta\|\varphi\|_{3/2}^2\right)dV_\gamma.\] The third type is bounded by \(C\varepsilon\int\beta\|\varphi\|_{3/2}^2\). The lower-order term from differentiating a coefficient satisfies the same bound. Replacing coordinate derivatives by horizontal derivatives adds a bounded angular first derivative, again controlled by these norms. A fixed finite partition of the base region has the same harmless lower-order errors. This proves (79).

The constants in the preceding estimates are uniform for all \(\beta\ge\beta_0\) once a fixed lower threshold is met. Choose \(\varepsilon_0\) small relative to the constant in Lemma 18. Dropping the nonnegative second squared norm in (77) gives \(\mathcal N(\varphi)\le C\|(2T+\mathcal U)\varphi\|\). Finally, \[\|\mathcal J\varphi\|\le C\|\varphi\|_{L^2(Y;H^1(S_y))} \le C\beta_0^{-1}\mathcal N(\varphi),\] so a sufficiently large \(\beta_0\) absorbs \(\mathcal J\) and proves (75).

All these operations use angular symbol estimates pointwise in the base, and ordinary base differential operators of degree at most two. Formal adjoints, the commutator expansion and the one integration by parts use at most the five base coefficient derivatives specified in (63). Forming these coefficients uses at most one base derivative of a trace operator; six trace derivatives therefore suffice. Since its weight dependence is through \(r\) and \(f_i\), profile derivatives through order seven suffice, within the eight allowed in (52). Differentiating a factor such as \(h/r\) five times uses only five derivatives of \(h\), with the remaining factors bounded by the permitted powers of \(\beta\). Higher angular symbol seminorms depend on the fixed smooth first-metric family: the tensor \(h\) is evaluated at the center, so angular differentiation introduces no further center derivatives of it. The scalar drifts have fixed smooth background bounds. Thus the eight smallness derivatives in (53) suffice in every fixed dimension; no smallness of all derivatives of \(h\) is used. ◻

Proof. Put \(q^i=b_2^i-b_{2,0}^iI\) and use \(D_i=D_{i,0}+K_i\) from (57). Let \(L_M\) denote left multiplication by a matrix \(M\). Expansion, without interchanging any factors, gives the identity \[ \begin{split} P-P_0=r\bigl[&Q^{ij}(D_{i,0}K_j+K_iD_{j,0}+K_iK_j)\\ &+L_{q^i}D_{i,0}+L_{b_2^i}K_i\bigr]. \end{split} \tag{81}\] Thus \(D_{i,0}K_j\) denotes composition and includes differentiation of \(K_j\). There is no derivative of \(q^i\) or \(b_2^i\) in this formula. Left multiplication commutes with the row action of a sphere operator when its matrix depends only on the center, but no such commutation is needed in (81).

The trace expansion gives \(\mathcal C_{i,0}^*\in(r^{-1}+\beta+1)\Psi^1\). Together with \(K_i\in\Psi^0\), \(\nabla K_i\in\beta\Psi^0\), and bounded drifts, (81) consequently yields \[ \|(P-P_0)\varphi\|_{L^2} \le C\left(\|\varphi\|_{L^2(Y;H^1(S_y))} +\|r\nabla\varphi\|_{L^2}\right). \tag{82}\] To see the weights explicitly, every term with a derivative of \(\varphi\) in the base is \(r\Psi^0\nabla\varphi\). Products of \(K_i\) with the leading \(r^{-1}\mathsf A_j^*\) leave a bounded \(\Psi^1\) term after multiplication by \(r\); products with \(f_j\mathsf B\) have coefficient \(O(r\beta)\). Differentiating \(K_j\) gives \(r\beta\Psi^0\). The remaining products have smaller orders and bounded coefficients. Since \(r\beta\le \varepsilon/\beta\), all constants in (82) are uniform as the lower threshold \(\beta_0\) increases.

Apply Proposition 19 to the four entries of \(\varphi\). By the definition of \(\mathcal N\), \[\|\varphi\|_{L^2(Y;H^1)}\le\beta_0^{-1}\mathcal N(\varphi), \qquad \|r\nabla\varphi\|_{L^2} \le\varepsilon\beta_0^{-2}\mathcal N(\varphi).\] It follows that \[\mathcal N(\varphi) \le C_0\|P\varphi\|_{L^2} +C_0C(\beta_0^{-1}+\varepsilon\beta_0^{-2}) \mathcal N(\varphi).\] Increase the fixed lower threshold so that the last coefficient is at most \(1/2\). This proves the assertion. In the parameter choices for the construction, this increase is made when \(p_*\) is chosen, before \(\varepsilon\) and the radius amplitude are fixed. ◻

Proof. The reproduction identity \(\int\psi_\delta\,d\sigma_y=I\) makes \(\varphi=\chi(\psi_\delta-I)\) mean zero on each sphere. It is smooth and compactly supported in the base. Equation (59) gives \[P\varphi=-\chi PI+[P,\chi](\psi_\delta-I).\] The first term has the uniform bound just established. For the second, (83) requires only one angular derivative and \(r\nabla\psi_\delta\) on the cutoff derivative region. The uniform bounds of Lemma 16 supply these: in the lower region the density is the normalized ball evaluation kernel, and in the other region the radii have a fixed positive lower bound. System coercivity bounds \(\mathcal N(\varphi)\), hence \(\|\varphi\|_{L^2}\), uniformly. Adding the fixed matrix function \(\chi I\) proves (84). ◻

Proof. As \(\lambda\to0\), the coordinate columns converge smoothly to \(y^l e_\alpha\), \(1\le l\le n\), \(1\le\alpha\le2\), where \(e_\alpha\) is the standard basis of \(\mathbb C^2\). The additional columns converge to the trace-free quadratic polynomials specified by their Hessians at the origin. At any point \(y\), the latter give arbitrary trace-free Hessians, and the affine columns correct their gradients to any prescribed values. Thus the limiting jets span all \((p,H)\) with \(\sum_iH_{ii}=0\), uniformly for \(y\in Y\).

The spaces in (87) form a smooth vector bundle of constant rank: contraction with the positive definite tensor \(a\) is onto \(\mathbb C^2\). They can be trivialized by the gradient and the trace-free part of the Hessian, the remaining trace being determined by (87). In this trivialization the selected jets give a smooth matrix of constant full row rank at \(\lambda=0\). Compactness of \(Y\) preserves this rank, with a positive lower bound on its nonzero singular values, for small \(\lambda\). If this matrix is \(T\), the right inverse \(T^*(TT^*)^{-1}\) has the asserted smoothness and uniform bounds. Each selected column satisfies (87), so its range is precisely the stated space. ◻

Proof. Writing \(\rho(y)=\|\psi(y,\cdot)\|_{L^2(S_y)}\), uniform normal coordinates and Cauchy–Schwarz give \(|M(y)|\le Cr(y)\rho(y)\). Taylor expansion in the fixed chart, followed by (89), gives \[ v_{2,\nu}-v_{1,\nu} =M^i\partial_i v_{1,\nu}+E_\nu, \qquad |E_\nu(y)|\le Cr(y)^2\rho(y). \tag{93}\] This calculation uses only the uniform second derivatives of the selected columns and \(|x-y|\le Cr\); the coordinate displacement is a scalar, so no matrix factors have been interchanged.

Apply the same calculation to the coordinate matrices. Subtracting (90) shows that the uniformly invertible map defining \(D\) sends \(D-M\) to an error bounded by \(Cr^2\rho\). Therefore \[|D-M|\le Cr^2\rho, \qquad |D|\le Cr\rho, \qquad |R_\nu|\le Cr^2\rho.\] Multiplication by \(\chi\) and integration prove (92). ◻

Proof. By Lemma 11, \(r^{1/p_*}\) has a uniform Lipschitz constant. Set \(s=r(y)^\eta\). Since \(\eta>1/p_*\), for \(r(y)\) sufficiently small every \(z\in B(y,s)\) satisfies \[\big|r(z)^{1/p_*}-r(y)^{1/p_*}\big| \le C r(y)^\eta\le\tfrac12 r(y)^{1/p_*}.\] Consequently \(c r(y)\le r(z)\le C r(y)\) on this ball, with constants depending only on the fixed \(p_*\). Because \(\chi=1\) on a neighborhood of the compact set \(K\), the ball lies in that neighborhood after a further fixed reduction of \(\rho_0\). The weighted bounds yield \[\|D\|_{L^2(B(y,s))}\le Cr(y),\qquad \|R_\nu\|_{L^2(B(y,s))}\le Cr(y)^2.\]

For any smooth vector- or matrix-valued function \(U\) with a bounded \(C^m\) norm, Taylor expansion to degree \(m-1\) and equivalence of norms on the finite-dimensional polynomial space give \[ |\partial^\alpha U(y)| \le C s^{-|\alpha|-n/2}\|U\|_{L^2(B(y,s))} +C_m s^{m-|\alpha|},\qquad |\alpha|<m. \tag{95}\] Indeed the Taylor remainder has \(L^2\) norm at most \(C_m s^{m+n/2}\), and, after rescaling the ball to unit radius, each coefficient of its Taylor polynomial is bounded by that polynomial’s \(L^2\) norm. This proves the displayed inequality componentwise.

For \(|\alpha|\le10\), the first term of (95) applied to \(D\) is at most \(Cr(y)^{1-\eta d_*}=Cr(y)^{1/2}\); for \(R_\nu\) it is at most \(Cr(y)^{2-\eta d_*}=Cr(y)^{3/2}\). The second term is at most \(C_m r(y)^{\eta(m-10)}\le C_m r(y)^{3/2}\), taking \(\rho_0\le1\). This proves (94). ◻

Proof. Denote the right-hand side of (96) for the column \(v_{1,\nu}\) by \(Z_\nu\). Its derivatives through order eight are bounded by \(Cr^{3/2}\) on the stated set. For \(\mathcal L_2R_\nu\) this uses derivatives of \(R_\nu\) through order ten in (94). Every derivative of either remaining product contains a derivative of \(h\) of order at most eight and a derivative of \(D\) of order at most nine or eight, respectively. Thus (88) and (94) give the factor \(r\,r^{1/2}\). The other factors are fixed smooth derivatives of the selected columns.

At each point, Lemma 22 permits linear combinations of the identities (96) for which the combined gradient is zero and the combined Hessian is any prescribed \(a\)-trace-free symmetric covariant tensor with values in \(\mathbb C^2\). Such Hessians satisfy (87) with zero gradient. They annihilate the \(\mathcal Y\) term and test precisely the \(\gamma\)-trace-free part of the coefficient of the Hessian.

More explicitly, choose a smooth uniformly bounded basis \(H_s\) of scalar symmetric Hessians with \(a^{ij}(H_s)_{ij}=0\), and use the right inverse in Lemma 22 to realize \(H_se_\alpha\), for \(\alpha=1,2\), with zero gradient. The resulting combinations of \(Z_\nu\) give the two columns of the contraction of the projected coefficient with \(H_s\). Inverting this spatial duality defines \(\mathcal E\) and proves (99). The coefficients realizing these prescribed jets are smooth functions of \(y\) and give zero combined gradient identically. We form these combinations before differentiating, so the \(\mathcal Y\) term is identically zero. The resulting formula expresses \(\mathcal E\) using only the \(Z_\nu\) and coefficients with uniformly bounded derivatives. Differentiating that formula through order eight proves (100), without introducing derivatives of \(\mathcal Y\). ◻

Proof. Differentiate the equation twice. At a point, its highest-order part is a linear map from the tensor of third derivatives of \(W\) to the twice-differentiated trace-free symmetric gradient. We first show that this map is injective. A constant linear change of coordinates sets \(a^{ij}=\delta^{ij}\) at the point, and each real component of \(V\) can be considered separately.

A tensor in the kernel defines a homogeneous cubic vector field \(Z\) whose trace-free symmetric gradient vanishes: that gradient is homogeneous quadratic and all of its second derivatives are zero. Hence \[ \partial_iZ_j+\partial_jZ_i=2\delta_{ij}\sigma, \qquad \sigma=\frac1n\operatorname{div}Z. \tag{103}\] Differentiating and combining the three permutations of the indices gives \[\partial_i\partial_jZ_k =\delta_{jk}\partial_i\sigma +\delta_{ik}\partial_j\sigma -\delta_{ij}\partial_k\sigma.\] Taking divergence in \(k\) yields \[(n-2)\partial_i\partial_j\sigma=-\delta_{ij}\Delta\sigma.\] Its trace gives \((2n-2)\Delta\sigma=0\); since \(n\ge3\), the Hessian of \(\sigma\) is zero. The preceding formula therefore gives \(\partial_i\partial_j\partial_lZ_k=0\), so the homogeneous cubic \(Z\) vanishes. This proves injectivity.

Let \(T_a\) denote this injective map on third-derivative tensors, using fixed Euclidean inner products on the finite-dimensional tensor spaces. It has the smooth left inverse \((T_a^*T_a)^{-1}T_a^*\). Uniform ellipticity and compactness of bounded coefficient ranges give uniform bounds for this inverse. All other terms in the twice-differentiated equation contain at most two derivatives of \(W\) or of \(E\). Applying the left inverse proves the claim, including the smooth coefficient bounds after further fixed numbers of differentiations. ◻

Proof. Apply Lemma 26 to (102) with \(V=\mathcal V_0\). For each multi-index \(\alpha\) of order three, it gives an identity of the form \[ \partial^\alpha W =\sum_{|\zeta|\le2}C_{\alpha\zeta}\partial^\zeta W +\sum_{|\zeta|\le2}F_{\alpha\zeta} \partial^\zeta\mathcal E_0, \tag{105}\] with uniformly smooth coefficient maps.

Fix \(y\in K\) with \(r(y)<\rho_0\). By the core geometry of Section 4.2, the vertical segment \[z(s)=(y',s),\qquad -\tfrac12\le s\le y_n,\] stays in \(K\), and \(r(z(s))\le r(y)\). At its initial point, \(D\) and all its derivatives vanish: the paired coordinate matrices agree on an open neighborhood in the known lower region. The same holds for \(W\).

Let \(\mathcal W(s)\) consist of all coordinate derivatives of \(W\) through order two at \(z(s)\). Differentiating this vector with respect to \(s\) either gives another one of its components or a third derivative, to which (105) applies. Thus \[\mathcal W'(s)=A(s)\mathcal W(s)+F(s), \qquad \mathcal W(-\tfrac12)=0, \qquad |A(s)|\le C, \qquad |F(s)|\le Cr(y)^{3/2}.\] The bound for \(F\) follows from (100), since the entire segment has radius below \(\rho_0\). Its length is uniformly bounded, so the elementary integral form of Gronwall’s inequality gives \[\sum_{|\alpha|\le2}|\partial^\alpha W(y)| \le Cr(y)^{3/2}.\]

This argument holds at every point of the stated set. Differentiating (105) \(k-3\) times, for \(3\le k\le9\), expresses the derivatives of \(W\) of order \(k\) in terms of its derivatives through order \(k-1\) and derivatives of \(\mathcal E_0\) through order \(k-1\). Induction and (100) therefore give the same bound through order nine. Notice that the last step uses precisely eight derivatives of \(\mathcal E_0\).

Equation (101) now gives the same estimates for \(d\) and hence for \(D\). Finally, differentiate (99) through order eight. Its right-hand side uses derivatives of \(D\) through order nine and of \(\mathcal E\) through order eight, all bounded by \(Cr^{3/2}\). This proves (104). ◻

Proof. First fix the reference geometry and uniform smooth bounds for all sufficiently small dilations. Choose \(p_*\) large enough for the sphere geometry, the angular estimate, and Lemma 24. Next choose \(\varepsilon\) sufficiently small and \(R_*\) sufficiently small to satisfy those results, including \(r\beta^2\le\varepsilon\). As explained in Remark 12, every fixed-separation range needed for the continuation and cutoff bounds is then fixed before \(\lambda\). Proposition 21 and all estimates of this section have constants uniform for sufficiently small \(\lambda\).

Choose \(0<\rho_*<\rho_0\), with \(\rho_*\le1\), so small that \(C\rho_*^{1/2}\le\varepsilon/2\) in (104). Thus, wherever that estimate applies and \(r_\delta<\rho_*\), it improves (88) to \(H_8\le\varepsilon r_\delta/2\).

As \(\lambda\to0\), \(h\to0\) in \(C^8(Y)\). The numbers \[\rho_*,\qquad \min_Y r_1,\qquad R_*(2t_b)^{p_*}\] are positive and fixed independently of \(\lambda\). We can therefore choose one sufficiently small \(\lambda\) such that \[ \sup_Y H_8\le\frac{\varepsilon}{2} \min\big\{\rho_*,\min_Yr_1,R_*(2t_b)^{p_*}\big\}. \tag{106}\] We also make the known lower region contain \(\{y_n<-1/4\}\) on the fixed ranges, as in the sphere construction. Fix this \(\lambda\) for the rest of the proof.

Let \[\mathcal I=\{\delta\in(0,1]: H_8(y)\le\varepsilon r_\delta(y) \text{ for every }y\in Y\}.\] It contains \(1\) by (106). For any \(\delta\in\mathcal I\), the preceding estimates improve the inequality to \(H_8\le\varepsilon r_\delta/2\) throughout \(Y\). Indeed, Proposition 27 handles the points in \(K\) with \(r_\delta<\rho_*\). Equation (106) handles all points with \(r_\delta\ge\rho_*\). At a point outside \(K\), either \(y_n<-1/2\), where equality of the coefficients is already known, or \(t>2t_b\). In the latter case \(\ell_\delta(t)\ge t\), so \(r_\delta\ge R_*(2t_b)^{p_*}\) and (106) again applies.

The set \(\mathcal I\) is relatively closed in \((0,1]\) by continuity and compactness of \(Y\). It is also relatively open: at any positive \(\delta\), the radius has a positive minimum on \(Y\), so the strict factor \(1/2\) just proved persists for nearby parameter values in the weaker defining inequality. Since \((0,1]\) is connected, \(\mathcal I=(0,1]\).

On the fixed open set \(K^\circ\cap\{t<0\}\), which contains the origin, \(r_\delta\to0\) as \(\delta\to0\). The coefficients \(h,D,R_\nu\) are independent of \(\delta\). The provisional bound and (94) therefore imply \[h=0,\qquad D=0,\qquad R_\nu=0\] there. In particular all selected pairs agree. Subtracting their equations now gives \[(B_2^i-B_1^i)\partial_i v_{1,\nu}=0 \qquad(1\le\nu\le N).\] The gradient projection of the space (87) is all of \((\mathbb C^2)^n\): for any gradient one can choose the trace of the Hessian to satisfy that constraint. Lemma 22 therefore implies \(B_1=B_2\). Together with \(Q=a\), this proves equality of the normalized harmonic-frame operators. The definition of \(Q\) shows that the two inverse metrics are positive scalar multiples, so the metrics are smoothly conformal. Undoing the fixed dilation proves the stated conclusion in the original contact coordinates. ◻

Proof. Let \[C_j=F_j^{-1}(dF_j+A_jF_j).\] These are the connection matrices in the harmonic frames; they need not be skew-Hermitian in the standard matrix inner product. The negative of the actual connection Laplacian, expressed in such a frame, is \[\Delta_{g_j,C_j} =g_j^{il}\bigl((\partial_i+C_{j,i})(\partial_l+C_{j,l}) -\Gamma^k_{il}(g_j)(\partial_k+C_{j,k})\bigr).\] Its zeroth-order term is zero, because the columns of \(F_j\) are harmonic. The common-principal-part hypothesis therefore says \[ e^{2\omega}\Delta_{g_2,C_2}=\Delta_{g_1,C_1}. \tag{107}\] Indeed their second-order coefficients agree after the indicated conformal normalization, their first-order coefficients agree by assumption, and both zeroth-order coefficients vanish.

Put \(\phi=(n-2)\omega/2\). The conformal change of the Christoffel symbols gives \[e^{2\omega}\Delta_{g_2,C_2} =\Delta_{g_1,C_2} +2\langle d\phi,\partial+C_2\rangle_{g_1}.\] Comparison of first-order terms with (107) yields \(C_2=C_1-d\phi\,I\). Substitution, with matrix multiplication in its displayed order, gives \[\begin{align*} \Delta_{g_1,C_1-d\phi I} +2\langle d\phi,\partial+C_1-d\phi I\rangle_{g_1} &=\Delta_{g_1,C_1} -(\Delta_{g_1}\phi+|d\phi|_{g_1}^2)I. \end{align*}\] Here \(\Delta_{g_1}=\operatorname{div}_{g_1}\nabla_{g_1}\). The zeroth-order terms in (107) consequently imply \[\Delta_{g_1}\phi+|d\phi|_{g_1}^2=0, \qquad \Delta_{g_1}(e^\phi)=0.\] On the known open side the metrics already agree, so \(e^\phi=1\). Unique continuation on \(\Omega\) gives \(e^\phi=1\) everywhere. Since \(n\ge3\) and \(\omega\) is real, \(\omega=0\), and hence \(C_1=C_2\).

The last equality is equivalent to \(d_{A_1}(F_1F_2^{-1}u)=F_1F_2^{-1}d_{A_2}u\). Although the harmonic frames are not unitary, \(J_{\mathrm{loc}}=F_1F_2^{-1}\) is. To see this, parallel transport the intertwining identity along any path starting in the known side. Both connection transports are unitary, and \(J_{\mathrm{loc}}\) is the old unitary identification at the initial point. It remains unitary along the path; connectedness of \(\Omega\) proves the assertion everywhere.

For an arbitrary source \(f\in C_c^\infty(U_0;\mathbb C^2)\), both \(w_{1,f}\) and \(J_{\mathrm{loc}}w_{2,f}\), in the common coordinate domain, are solutions of the first homogeneous connection equation. The charts avoid \(\overline{U_0}\). Their difference vanishes on the known side, so unique continuation makes it zero on \(\Omega\). Thus all evaluation identities hold, not only those for the finite family used in the local argument. On an overlap with the old match, the two proposed first-side images of a point, together with their invertible fiber maps, would impose an invertible linear relation between the two first-side evaluation maps. Joint surjectivity in Lemma 4 forces the image points to coincide. Surjectivity at the second-side point then forces the fiber maps to coincide as well. Thus the local identification agrees with the old match on every overlap. ◻

Proof. If its domain had an interior frontier, Proposition 6 and the dilation in Section 2.5 would supply the local data used in Sections [sph:section]–[mat:section]. Proposition 28 gives conformally equal metrics and equal normalized harmonic-frame equations on a neighborhood of the contact point in the original charts. Apply Lemma 29. Coordinate identity and \(J_{\mathrm{loc}}\) extend the old match to a neighborhood containing the frontier point. The concluding assertion of Lemma 29 ensures agreement on every overlap. This contradicts maximality. The domain, already nonempty and open, has no relative frontier in the connected interior, and is therefore all of \(N_2^\circ\).

The image is open because \(\Phi\) is a local isometry. To see that it is relatively closed, suppose \(\Phi(p_k)\to q\in N_1^\circ\). Compactness gives a subsequence with \(p_k\to p\in N_2\) and \(J(p_k)\to J_0\in U(2)\). If \(p\in\partial N_2\), continuity of the evaluations and (8) would give \(I_1(q)=J_0I_2(p)=0\), contrary to Lemma 4. Thus \(p\) is interior, and continuity gives \(\Phi(p)=q\). Connectedness of \(N_1^\circ\) makes the image all of that interior. Injectivity was proved in Lemma 5; the inverse is smooth because the map is a bijective local diffeomorphism. ◻

Proof. An onto interior isometry preserves lengths of curves in both directions and therefore preserves intrinsic distance. The metric completion of the interior of a compact smooth Riemannian manifold with boundary is the manifold with its length metric. In particular, a curve touching the boundary can be pushed slightly into an inward collar with arbitrarily small change of length; uniform local metric comparisons identify the resulting completion with the original compact topology. Thus \(\Phi\) and its inverse extend to mutually inverse homeomorphisms of \(N_2,N_1\), mapping boundary to boundary.

The extensions are smooth. In a sufficiently thin uniform collar, the distance \(s_j\) to \(\partial N_j\) is smooth and its gradient generates inward unit normal flow. The isometry preserves distance to boundary, so \(s_1\circ\Phi=s_2\), and in the interior it intertwines these gradient fields. On a fixed small level \(s_2=s_0>0\), the map is already smooth. For \(0\le s\le s_0\), express it using that smooth level map and the smooth normal flows on the two sides. This gives a smooth extension to \(s=0\); the inverse has the same construction. Parallel transport for \(A_1,A_2\) along these normal curves similarly extends \(J\) smoothly, retaining unitarity and the connection intertwining identity.

The maps are identity on the open cap, hence on its closure by continuity. With closure taken in \(N_j\), the original interiors satisfy \[M_j^\circ=N_j^\circ\setminus\overline E.\] Bijectivity and identity on \(\overline E\) preserve these complements, and hence their closures, the original copies of \(M\). The restricted diffeomorphism fixes the attachment patch \(P=\{b>0\}\), and \(J=I\) there.

We must recover the boundary identity on every part of \(\Gamma\), which may have more than one component. For \(f\in C_c^\infty(\Gamma;\mathbb C^2)\), let \(u_j\) now denote the harmonic solutions on the original manifolds with boundary value \(f\). On the second manifold define \(\widetilde u_1=J^{-1}(u_1\circ\Phi)\). The metric and connection identities make \(\widetilde u_1\) second-harmonic. On \(P\), it has the same boundary value as \(u_2\). It also has the same covariant normal derivative there. Indeed equality of the measured forms, tested against arbitrary functions supported in \(P\), gives the equality of smooth densities \[(\nabla^{A_1}_{\nu_1}u_1)\,dS_{g_1} =(\nabla^{A_2}_{\nu_2}u_2)\,dS_{g_2} \quad\text{on }P.\] The densities agree by Lemma 2, the isometry transports the outward normal, and the bundle map equals \(I\) on \(P\). Thus \(\widetilde u_1-u_2\) has zero Cauchy data there. Boundary and interior unique continuation give \(\widetilde u_1=u_2\) throughout \(M\).

Taking boundary traces yields \[ J(p)f(p)=f(\Phi(p)) \quad\bigl(p\in\partial M,\ f\in C_c^\infty(\Gamma;\mathbb C^2)\bigr). \tag{108}\] If \(p\in\Gamma\) and \(\Phi(p)\ne p\), choose \(f\) nonzero at \(p\), supported in a small neighborhood of \(p\) avoiding \(\Phi(p)\). Equation (108) contradicts invertibility of \(J(p)\). Hence \(\Phi(p)=p\) on \(\Gamma\). Arbitrary vector values of \(f(p)\) then give \(J(p)=I\) there. ◻

Proof of Theorem 1. The boundary-identity gauges \(V_j\) of Section 2 put the connections in normal gauge. Lemma 2 and Proposition 3 produce common exterior Green data. Lemmas 4 and 5 define the maximal match, and Proposition 6 supplies the local contact data wherever an interior frontier is present. Proposition 28, followed by Lemma 29, removes that frontier. Propositions 30 and 31 give a smooth global metric and connection identification fixing the entire measured patch.

To distinguish the original connections in this final step, write \(A_j^{\mathrm{orig}}\) for them, so the connections used in the proof were \[A_j=V_j^{-1}A_j^{\mathrm{orig}}V_j+V_j^{-1}dV_j.\] In the original trivializations the bundle map from second to first fiber is \[U=(V_1\circ\Phi)J V_2^{-1}.\] It is smooth and unitary, and equals \(I\) on \(\Gamma\), since both original gauges equal \(I\) on the entire boundary. The intertwining identity is \(d_{\Phi^*A_1^{\mathrm{orig}}}(Uu)=U d_{A_2^{\mathrm{orig}}}u\). Expanding it gives \[A_2^{\mathrm{orig}} =U^{-1}(\Phi^*A_1^{\mathrm{orig}})U+U^{-1}dU, \qquad g_2=\Phi^*g_1.\] These are the asserted identities. The argument uses densities and a single nonempty cap; it requires neither orientability nor connectedness of the boundary or of \(\Gamma\). ◻