Proof. The product construction below is related to the diagonal approximation in Marino and Spagnolo [8]; we use its elementary coordinate-join arithmetic directly. First we show that each scalar triple strictly between the endpoints is interior. Given a nearby positive triple \((\alpha_1,\alpha_2, \alpha_3)\), choose \(M>\max_i\alpha_i\) and set \[d_i=\sqrt{1-\alpha_i/M}.\] Let \(t_i\) independently take the two values \(1-d_i,1+d_i\) with equal weights. Its arithmetic mean is one and its harmonic mean is \(1-d_i^2=\alpha_i/M\). Start with the eight scalar leaves \[ M t_1t_2t_3I. \tag{10}\] Joining the \(t_1\) pairs in direction 1 gives the diagonal triple \[(\alpha_1t_2t_3,Mt_2t_3,Mt_2t_3).\] Joining the \(t_2\) pairs in direction 2 gives \((\alpha_1t_3,\alpha_2t_3,Mt_3)\), and the direction-3 joins give \((\alpha_1,\alpha_2,\alpha_3)\). If the target tends to \((s,s,s)\), choose \(M\) tending to \(s\) from above. Every leaf then tends to \(s\), so all leaves lie in \((a,b)\) for sufficiently close targets.

Interior propagates through any nontrivial join. With the other child and a weight \(0<\theta<1\) fixed, the derivative of the join in its first child is diagonal, with entries \(\theta\) in the tangential positions and \(\theta[J_i]_i^2/\alpha_i^2>0\) in the normal position. It is invertible. The inverse function theorem, applied successively up a finite tree, proves openness in diagonal variables. Zero-weight joins can be removed. The diagonal class is permutation invariant; continuity of ordered eigenvalues now proves that its spectral extension is open, including at repeated eigenvalues. Varying the weight of an existing join simply supplies another finite tree with the same children, so its entire path, including its endpoints, stays in \(\mathcal G\).

For uniform coverage, take \(M=2C_0\) in Equation (10). For targets in \([c_0,C_0]^3\), all factors lie between \[\ell=1-\sqrt{1-c_0/(2C_0)}>0 \quad\text{and}\quad 2.\] All leaves therefore lie in \([M\ell^3,8M]\). For example, \(a=M\ell^3/2\) and \(b=16M\) put them strictly inside the leaf interval, uniformly for the entire required spectral range. ◻

Proof. For \(m=1\), surjectivity is ordinary nonzero contraction. For \(m=2\) write \[G=\begin{pmatrix}a&b\\b&c\\d&e\end{pmatrix},\qquad D(\xi)G=(\xi_1a+\xi_2b+\xi_3d, \xi_1b+\xi_2c+\xi_3e).\] If \(\lambda\in\mathbb R^2\) annihilates its range, then \[\xi_1\lambda_1=\xi_2\lambda_2 =\xi_1\lambda_2+\xi_2\lambda_1 =\xi_3\lambda_1=\xi_3\lambda_2=0.\] When \(\xi_3\ne0\) the assertion follows immediately; otherwise either \(\xi_1\) or \(\xi_2\) is nonzero and the first three equalities give the same conclusion. Thus \(D(\xi)D(\xi)^t\) is positive definite at every nonzero \(\xi\), where the transpose uses the specified inner products.

Set \(Q(\xi)=D(\xi)D(\xi)^t\) and define \[ P(\xi)=\det Q(\xi)I -D(\xi)^t\operatorname{adj}(Q(\xi))D(\xi). \tag{15}\] The identity \(Q\operatorname{adj}Q=(\det Q)I\) gives \(DP=0\) as a polynomial identity. On \(\ker D(\xi)\), \(P(\xi)\) is multiplication by the nonzero scalar \(\det Q(\xi)\), proving the image assertion. Both terms have degree \(2m\).

For completeness, let \(R\in\ker D(\xi)\) and choose \(R_0\in V_m\) with \(P(\xi)R_0=R\). Let \(h\) be a smooth one-periodic function of mean zero and let \(H\) be a one-periodic \(2m\)-fold primitive, so \(H^{(2m)}=h\). Such primitives are obtained by successive integration and subtraction of the mean. Given a smooth compactly supported cutoff \(\chi\), apply the constant coefficient differential operator \(P(\partial)\) to \[ k^{-2m}\chi(z)H(k\xi\cdot z)R_0. \tag{16}\] Its output lies in \(V_m\) and has zero column divergence exactly, by the polynomial identity. Terms with every derivative on \(H\) give \(\chi(z)h(k\xi\cdot z)R\). Each remaining term has at least one derivative on \(\chi\) and is \(O(k^{-1})\) uniformly for fixed \(\chi,h\). This proves the localization claim. ◻

Proof. Symmetry is immediate. Multiply by \(E\), use \(E^\dagger E=I\), and cancel the last two terms using \(R^tE=E^tR\). The norm bound follows from the same formula. ◻

Proof. Choose an initial compactly contained neighborhood of \(y_0\) on which the potential chart \(\Phi\) is defined and the selected gradients have a positive lower singular-value bound. We will choose a smaller neighborhood \(V\) from the frozen-error estimates below; the box \(Q\) will then have closure in \(V\). Until the constitutive correction is complete, the local matrices \(E,F,E_0,R\) and their primed versions contain only these \(m\) independent columns. We restore the full pair by the fixed affine relations at the end of the local calculation. Freeze \(A_0=A(y_0)\), \(E_0=E(y_0)\), and \(J_0=D\Phi(y_0)\). For a split as above, the physical polarization \(\Delta B E_0\) has zero normal component since the normal row of \(\Delta B\) vanishes. Its gradient pairing is symmetric. Hence \[ \xi=J_0^{-t}d,\qquad R=\frac{J_0\Delta B E_0}{\left\lvert \det J_0\right\rvert} \quad\text{satisfy}\quad R\in V_m,\quad D(\xi)R=0. \tag{18}\] For the last assertion, contract the displayed matrix with \(\xi\). For the first, the leading rows of \(J_0\) are the gradients of the selected potentials, so their entries are precisely the symmetric pairings with \(\Delta B\).

The construction below applies on any box \(Q\) with \(\overline Q\subset V\). After \(V\) has been chosen from the estimates below, fix such a box and a smooth cutoff \(0\leq\chi\leq1\), supported in \(Q\) and equal to one except on a thin margin. Choose a smooth periodic function \(h\) by convolving the two-state step function that equals \(\eta_-\) on a fraction \(\eta_+\) of the period and \(-\eta_+\) on the remaining fraction \(\eta_-\). Then \[ \int_0^1h=0,\qquad -\eta_+\leq h\leq\eta_-, \tag{19}\] and \(h\) equals the two pure values outside an arbitrarily small part of its period. The margin and transition lengths will be chosen to meet the prescribed volume error \(\delta\left\lvert Q\right\rvert\). Let \(H_1\) be a periodic primitive of \(h\). In Equation (16), use the cutoff \(\chi\circ\Phi^{-1}\). Transform the resulting flux perturbation back to the original coordinates and denote the perturbed flux by \(\widehat F_k\). It satisfies \[ \mathop{\mathrm{div}}\widehat F_k=0,\qquad E^t\widehat F_k=\widehat F_k^tE \tag{20}\] exactly, and equals the original flux outside a compact subset of \(Q\). The selected potentials have not yet changed.

Write \(\varphi(y)=\xi\cdot\Phi(y)\); thus \(\nabla\varphi(y_0)=d\). At the same time define on \(Q\), extending by the identity outside \(Q\), \[ X_k(y)=y+\frac{\Delta q}{k}\, d\,\chi(y)H_1(k\varphi(y)). \tag{21}\] If \(z_k(y)=\chi(y)h(k\varphi(y))\), differentiation gives \[ DX_k=I+\Delta q\,z_k\,d\otimes\nabla\varphi +O(k^{-1}). \tag{22}\] Throughout \(Q\), this is uniformly close to \(L(z_k)\) when \(V\) is sufficiently small and then \(k\) is large. The scalar \(z_k\) stays in the segment in Equation (13), because zero lies in that segment and \(0\leq\chi\leq1\). All \(q(z_k)\) are bounded away from zero. More explicitly, the rank-one determinant formula gives \[\det DX_k=q(z_k)+\Delta q\,z_k(\partial_d\varphi-1) +\frac{\Delta q}{k}H_1(k\varphi)\partial_d\chi.\] First shrink \(V\) so that \(\partial_d\varphi\) is uniformly close to one; after the box and profiles are fixed, take \(k\) large. The displayed determinant, which is the derivative of \(X_k\) along each line parallel to \(d\) in that direction, is then strictly positive. The displacement is parallel to \(d\) and compactly supported. Each such line is consequently mapped monotonically onto itself. This proves that \(X_k\) is a global smooth diffeomorphism, equal to the identity outside \(Q\), and mapping \(Q\) onto itself. Its derivative and inverse derivative have bounds independent of sufficiently large \(k\).

Push the potentials and preliminary fluxes forward by this map: \[ u_{j,k}'=u_j\circ X_k^{-1},\qquad E_k'\circ X_k=DX_k^{-t}E,\qquad F_k'\circ X_k=\frac{DX_k\widehat F_k}{\det DX_k}. \tag{23}\] The determinant is positive here. These fields satisfy exact column divergence and pairing symmetry, since \[(E_k'^tF_k')\circ X_k =\frac{E^t\widehat F_k}{\det DX_k}.\] The selected gradients remain independent. The discarded potentials are recomposed by the same map, so their original constant affine relations with the selected potentials remain valid. Restore their preliminary fluxes by the corresponding constant linear combinations of the selected fluxes, and push them by the same formula. We continue the constitutive estimates on the selected columns.

The errors in the frozen description are only errors in the constitutive approximation, not in these equations. More precisely, let \(\omega(V)\) bound the oscillations from their values at \(y_0\) of the fixed smooth fields and chart derivatives on \(V\), with \(\omega(V)\to0\) as \(V\) shrinks to \(y_0\). The frozen flux error can be seen directly. Put \(\Gamma(y)=\left\lvert \det J(y)\right\rvert J(y)^{-1}\), so that \(\Gamma(y_0)R=\Delta B E_0\). The localized leading wave gives \[\widehat F_k-B(z_k)E =(A-A_0)E+z_k\bigl[(\Gamma-\Gamma(y_0))R +\Delta B(E_0-E)\bigr]+O(k^{-1}).\] Consequently, for the fixed plan, \[\begin{align*} \widehat F_k-B(z_k)E&=O(\omega(V))+O(k^{-1}), \tag{24}\\ DX_k-L(z_k)&=O(\omega(V))+O(k^{-1}). \tag{25}\end{align*}\] All estimates here are absolute uniform bounds on the box. The constants multiplying \(\omega(V)\) depend only on the fixed plan and bounds on the initial neighborhood; \(z_k\) remains in its fixed segment. The constants in \(O(k^{-1})\) may also depend on the chosen box, cutoff, and waveform. This is why those choices can be fixed before the frequency is increased.

Define the constitutive defect \[\mathcal R_k=F_k'-C(z_k\circ X_k^{-1})E_k'.\] Writing \(K=DX_k\) and evaluating \(B,L\) at \(z_k(y)\), the pushforward formulas give the exact identity \[\mathcal R_k\circ X_k =\frac K{\det K}(\widehat F_k-BE) +\left(\frac{KBK^t}{\det K}-\frac{LBL^t}{\det L}\right)K^{-t}E.\] The map bounds and Equations (24)– (25) therefore give \[ \mathcal R_k=O(\omega(V))+O(k^{-1}). \tag{26}\]

Apply Lemma 7 to \(\mathcal R_k\), using the independent columns. Its symmetry condition is exact: both \(E_k'^tF_k'\) and \(E_k'^tC(z_k\circ X_k^{-1})E_k'\) are symmetric. The selected gradients retain a positive lower singular-value bound, because \(E_k'\circ X_k=DX_k^{-t}E\) and the map derivatives are bounded. Thus the correction \(H_k\) has a uniform bound by a local constant times \(\left\lvert \mathcal R_k\right\rvert\), and \[A_k'=C(z_k\circ X_k^{-1})+H_k\] maps the selected new gradients to their new fluxes. Restore every discarded flux by the same constant linear combination as its gradient. The common recomposition then gives the full two-column relation \(F_k'=A_k'E_k'\) and preserves the original fields outside the box. From this point onward, \(E,F,E_k',F_k'\) and \(\widehat F_k\) again denote the full two-column matrices; the bound for \(H_k\) is the one just obtained from the selected columns.

Choose \(\tau>0\) so that the symmetric \(\tau\)-neighborhood of the compact path \(C\) lies in \(\mathcal G\) and the \(\tau\)-balls about \(A^s\) lie in \(\mathcal O_s\) for \(s\in\{+,-\}\). This uses distance inside \(\mathrm{Sym}_3\). The coefficient of \(\omega(V)\) in the correction bound is fixed on the initial neighborhood, independently of the later box and profiles. Choose \(V\) small enough that this contribution is less than \(\tau/2\) and that the preceding map determinant is positive after a sufficiently small frequency error. For each box with closure in this \(V\), fix its cutoff and waveform and take \(k\) large enough that the remaining contribution is less than \(\tau/2\). Then \(A_k'\in\mathcal G\) throughout the box. The correction needs no bounds on derivatives of the defect. It is smooth inside the box; outside \(Q\) keep the original matrix. A jump of the matrix at the box boundary is harmless: the potentials and fluxes agree with the old ones near that boundary, and the matrix is only required to be smooth on an open cover up to null sets.

Where \(\chi=1\) and \(h\) is at the pure value for child \(s\), the matrix \(C\) is exactly \(A^s\), so the correction bound puts \(A_k'\) in \(\mathcal O_s\). Take \(Q_{k,s}\) to be the images under \(X_k\) of the open pure regions, omitting the endpoints of the waveform plateaus. The omitted plateau endpoints form a null set. The cutoff margin can have arbitrarily small volume, and the transition intervals can have arbitrarily small total length. The phase is nondegenerate on the box; changing to the chart and integrating in its linear phase direction shows that the preimages of the transition intervals have the corresponding small limiting volume. For the image measure, the determinant formula above gives \[0<\det DX_k\le \sup_{z\in[-\eta_+,\eta_-]}q(z) +\left\lvert \Delta q\right\rvert\sup_{z\in[-\eta_+,\eta_-]}\left\lvert z\right\rvert \sup_V\left\lvert \partial_d\varphi-1\right\rvert+1=:J_*,\] after the profiles are fixed and \(k\) is large enough to make the \(k^{-1}\) term in the determinant formula at most one in absolute value. Thus \(J_*\) is fixed by the preceding local data and \(V\), independently of the cutoff margin and transition lengths, and bounds the volume increase under \(X_k\). Choose the margin and transition lengths so that, for all sufficiently large \(k\), the complement of \(Q_{k,+}\cup Q_{k,-}\) has volume less than \(\delta\left\lvert Q\right\rvert\). The new smooth box pieces and the old pieces away from its faces retain the full-measure local rank cover.

Proof. For clarity we make the finite-depth argument explicit. In diagonal variables let \(T_0\) be the triples in \(\mathcal G^{\mathrm{diag}}\) whose distance from scalar triples is strictly less than \(\varepsilon\). Inductively, \(T_k\) contains \(T_{k-1}\) and the outputs of joins of two \(T_{k-1}\) children whose entire joining path lies in \(\mathcal G^{\mathrm{diag}}\). Equivalently these are targets with a representing tree of depth at most \(k\), ending in \(T_0\), with the indicated path condition. Take their spectral extensions when discussing matrices.

Each \(T_k\) is open and permutation invariant. This is immediate for \(T_0\). For a nontrivial join with children in \(T_{k-1}\), its invertible derivative in one child, computed in Lemma 4, absorbs any sufficiently small change of output into a change of that child. The child remains in the open set \(T_{k-1}\), and the whole path remains in the open class by compactness. This proves the induction; permutation invariance and spectral openness follow as before. Every member of \(\mathcal G\) lies in some \(T_k\), since its defining tree ends at scalar leaves.

Intersect the smooth rank patches with the set where \(A\in\mathcal G\). This intersection is open in each patch and still has full measure. Its subsets where \(A\in T_k\) are open and increase to the whole intersection. By finite measure, choose a finite \(K\) so that the portion not covered by these rank-regular \(T_K\) patches has volume less than \(\varepsilon/3\). If \(K=0\), the original subsolution already suffices. Otherwise fix \(\delta>0\) with \(\delta\left\lvert U\right\rvert<\varepsilon/(3K)\). At each of the \(K\) reductions, take a compact subset of the current good region losing volume less than \(\varepsilon/(3K)\), and use this fixed relative tolerance \(\delta\) in the local lemma before choosing its neighborhoods. Consider one reduction from \(T_k\) to \(T_{k-1}\), with \(1\leq k\leq K\). Choose every admissible neighborhood below inside its current open \(T_k\) rank patch.

At a rank-positive point, fix its smooth potential chart. If its matrix already belongs to \(T_{k-1}\), choose an admissible neighborhood on which \(A\in T_{k-1}\); every contained box then needs no change. Otherwise choose its frozen two-child plan, with children in \(T_{k-1}\). Apply Lemma 8 with both child neighborhoods equal to the spectral extension of \(T_{k-1}\) and with the already fixed relative tolerance \(\delta\). It supplies an admissible neighborhood for every contained grid box; the large-frequency family then permits any prescribed weak error. The local independent gradients have a positive smallest singular value on the initial compact neighborhood used by that lemma. At a rank-zero point, choose an admissible neighborhood on which the potentials are constant and their fluxes vanish. On any box with closure inside it, replace the matrix by any scalar strictly in \((a,b)\), without changing the fields.

These choices give admissible neighborhoods of every point in the chosen compact subset. A finite subcover of that compact set and a fine enough grid give finitely many disjoint box interiors, with closures inside admissible neighborhoods, covering the compact set except for grid faces. The frozen point need not lie in the grid box: Lemma 8 applies to every box whose closure lies in its admissible neighborhood. One can equivalently refine a grid successively. Each box is treated using its own chart, fixed frame, rank bound, and frozen plan; no continuous eigenframe on the whole domain is required.

For this reduction, construct every boxwise output from the same incoming subsolution, with the frozen data just selected. In each rank-positive box requiring a join, use the local lemma with the fixed relative tolerance \(\delta\). Paste the individual potential and flux differences and the matrix choices over the disjoint box interiors. The potential and flux differences have compact support in those boxes, so the pasted gradients and fluxes have each box’s constitutive relation, and the exact flux pairings add. Since the box interiors are disjoint, the sum of the exceptional volumes is less than \(\delta\sum_Q\left\lvert Q\right\rvert\leq\delta\left\lvert U\right\rvert<\varepsilon/(3K)\). The unchanged \(T_{k-1}\) boxes and the rank-zero scalar boxes create no exceptional volume. Together with them, the open pure regions give smooth patches with the local rank property on which the new matrix belongs to \(T_{k-1}\). The output is again a subsolution. In joined boxes this is part of the lemma; in the other boxes the fields are unchanged, and either the matrix is unchanged or it maps the zero gradients to the zero fluxes. The scalar replacement lies in \(\mathcal G\) and retains the local rank property. Matrix jumps at the finitely many box faces add only null seams, and the original cover remains where nothing changed.

Perform the next step only inside these new good patches and continue down to \(T_0\). Untreated exceptional portions can be left untouched. Every map is supported in a selected box and maps that box onto itself. Thus later operations can be confined to the current good set. The initial loss is less than \(\varepsilon/3\); the \(K\) compact-subset losses and the \(K\) local exceptional losses are each less than \(\varepsilon/(3K)\) per stage. Their total is less than \(\varepsilon\). Fixed global rank bounds or fixed tolerances over infinitely many plans are not needed.

Finally, because the box outputs are pasted from one incoming pair, the finite sum of their field differences is arbitrarily weakly small by taking their frequencies sufficiently large. Use a metric for the weak topology on the bounded set of all subsolution pairs, whose boundedness is shown below, and allot a sufficiently small part of the prescribed neighborhood to each of the finitely many stages. The final pair lies in the prescribed neighborhood, and its matrix is in \(T_0\) off a set of measure less than \(\varepsilon\), as asserted. ◻

Proof. Let \(P_N\) be the finite-rank orthogonal projections onto the first \(N\) vectors of a fixed orthonormal basis. Their restrictions to \(Z\) are weak-to-norm continuous and \(P_Nz\to z\) in norm for every \(z\). Here is the relevant Baire argument explicitly. On any nonempty closed subset \(K\) of \(Z\) and for \(\delta>0\), the closed sets \[K_N=\{z\in K:\left\lVert P_pz-P_qz\right\rVert\leq\delta \text{ for all }p,q\geq N\}\] cover \(K\). Compact metric spaces are Baire, so some \(K_N\) has nonempty relative interior. On that patch, letting \(p\) tend to infinity gives \(\left\lVert z-P_Nz\right\rVert\leq\delta\). Shrinking the patch using continuity of \(P_N\), its strong diameter is at most \(3\delta\).

Apply this inside the closure of an open set whose closure lies in any prescribed nonempty open subset of \(Z\). The relative patch meets the original open set, because that set is dense in its closure. Their intersection contains a nonempty open subset of \(Z\) of strong diameter at most \(3\delta\). Consequently, for each \(r>0\), the union of all open subsets of \(Z\) of strong diameter less than \(r\) is open and dense. The intersection of these sets for \(r=1,1/2,1/3,\ldots\) is dense by Baire, and every point of the intersection is a continuity point of the identity. ◻

Proof of Theorem 3. Choose a continuity point \((E,F)\in Z\) from Lemma 10. By the definition of \(Z\), choose subsolution pairs converging weakly to it, and apply Proposition 9 to each, with weak errors tending to zero and with \(\varepsilon_j=2^{-j}\). Denote the resulting subsolutions by \((A_j,E_j,F_j)\). Then \[ (E_j,F_j)\longrightarrow(E,F)\quad\text{strongly in }\mathcal H, \tag{30}\] and \(\mathop{\mathrm{dist}}(A_j,\mathbb RI)<2^{-j}\) except on a set of measure less than \(2^{-j}\). Put \(\lambda_j=\mathop{\mathrm{tr}}(A_j)/3\). Since \(A_j\in \mathcal G\), one has \(a<\lambda_j<b\), and the Frobenius-distance projection onto scalar matrices is exactly \(\lambda_jI\).

The exceptional measures are summable. After discarding their limsup, which has measure zero, we have \(A_j-\lambda_jI\to0\) pointwise. Passing to a subsequence in Equation (30) also gives pointwise convergence of \(E_j,F_j\) almost everywhere. At such a point, \[F_j-\lambda_jE_j=(A_j-\lambda_jI)E_j\longrightarrow0.\] If \(E\ne0\), it follows that \[ \lambda_j\longrightarrow\gamma= \frac{E:F}{\left\lvert E\right\rvert^2}\in[a,b],\qquad F=\gamma E, \tag{31}\] where \(E:F\) is the Frobenius pairing. If \(E=0\), the bound \(\left\lvert F_j\right\rvert\leq b\left\lvert E_j\right\rvert\) forces \(F=0\); define \(\gamma=a\) there. This defines one measurable scalar, common to both columns, with \(a\leq\gamma\leq b\) almost everywhere. No pointwise subsequence of the bounded coefficient sequence itself was assumed.

The strong convergence is essential for excluding concentration on small exceptional sets. Equivalently, it makes \(\left\lvert E_j\right\rvert^2\) uniformly integrable, so the coefficient defect times \(E_j\) also tends to zero in \(L^2\). A mere uniform energy bound would not suffice for that conclusion.

Finally the gradient and flux characterization of \(Z\) supplies \(v_1,v_2\in H^1(U)\) with the original traces, gradients \(E\), and the original normal-flux functionals. Since \(F=\gamma E\) and its columns have zero divergence, these are precisely Equations (7) and (8). ◻

Proof. Use local coordinates \(y_1=u,y_2=v,y_3\) and cover the source support by finitely many such coordinate boxes joined by overlaps in the regular region. A partition of unity splits the data. In any nonempty open overlap, smooth test pairs generate arbitrary vectors of the three moments in Equation (39). For example, take two bumps in the first component with different averages of \(v\), and one bump in the second component. Their moment vectors are \((1,0,\bar v_1)\), \((1,0,\bar v_2)\), and \((0,1,-\bar u_3)\), which are independent when \(\bar v_1\ne\bar v_2\). Such bumps exist because \(v\) is a coordinate. Transfer the moment vector of each piece along a spanning tree to one final box; the total vector there is zero. We have reduced to a pair satisfying all three conditions in a single box.

Solve \(\mathop{\mathrm{div}}F^i=r^i\) compactly by the box construction. Put \(q=F^1_2-F^2_1\). Integration by parts gives \[\int(y_2r^1-y_1r^2)=-\int q=0.\] Choose a compactly supported vector \(h\) with \(\mathop{\mathrm{div}}h=q\). Add to the first column the vector \[(\partial_2h_1,\ -\partial_1h_1-\partial_3h_3,\ \partial_2h_3)^t\] and to the second column the vector \[(\partial_2h_2,\ -\partial_1h_2,\ 0)^t.\] Both additions have zero divergence, and their change to \(F^1_2-F^2_1\) is \(-\mathop{\mathrm{div}}h=-q\). The corrected columns can therefore be completed to a symmetric matrix: set its third-row counterparts by symmetry and its unused \(33\) entry to zero. Since \(\nabla y_1=e_1\) and \(\nabla y_2=e_2\), its first two columns give Equation (40). Transform back to the original coordinates.

All operations involve fixed compact supports, fixed integrations, smooth multipliers, and finitely many derivatives. The overlap moment matrices have nonzero determinants; they and the coordinate Jacobians remain invertible under small smooth perturbations. This proves the stated uniform bounds, allowing a fixed finite loss of derivatives. ◻

Proof of Lemma 12. We first treat the nonzero case. Write the translated band as \(0<t<4\), and normalize the input fields there by \[\widetilde u_T(t,\theta)=u_{\rm in}(T+t,\theta)-T, \qquad \widetilde v_T(t,\theta)=\frac{v_{\rm in}(T+t,\theta)}{b e^{-\lambda T}}.\] They tend in every fixed smooth norm on the band to \[ U=t,\qquad V=e^{-\lambda t}\cos(kx). \tag{41}\] Indeed, the rate spectrum is discrete, so the first omitted rate in \(v_{\rm in}\), if present, is strictly larger than \(\lambda\); if there is no omitted mode, the normalized second field is already \(V\). For an \(H^{1/2}\) trace, Cauchy–Schwarz bounds every differentiated Fourier series on a translated band by the trace norm times a convergent sum of polynomial weights against the exponential factors. The positive rate gap then controls the normalized remainder; the decaying part of \(\widetilde u_T\) is handled in the same way.

Choose a fixed smooth cutoff which is zero near \(t=0\) and one near \(t=4\). Interpolate from \((\widetilde u_T,\widetilde v_T)\) to \((U,V)\) with this cutoff, and call the resulting pair \(u,v\). Both endpoint pairs are \(A_0\)-harmonic. Hence the source densities \(r^1=-\mathop{\mathrm{div}}(A_0\nabla u)\), \(r^2=-\mathop{\mathrm{div}}(A_0\nabla v)\) for the flat tensor are supported in a fixed interior subband and tend to zero in every fixed smooth norm.

These sources satisfy Equation (39) exactly. The first two equalities express equality of the mean axial slopes before and after interpolation. For the third, integrate the counterpart of Equation (38) for \(A_0\). The boundary expression is the difference of the cross-flux concomitants \[\int_{\mathbb T^2}(v\partial_tu-u\partial_tv)\,\,\mathrm d\theta.\] For the normalized input pair, continued harmonically along the translated end, this is independent of \(t\) and tends to zero at infinity: its first field is linear plus decaying modes, and its second has zero offset and zero slope. For the pure pair it vanishes by angular integration. Hence the difference is zero.

The three global moments now vanish, but the gradients may still fail to be independent near the walls of the limiting pair. For sufficiently large \(T\), use coordinates \((s=u,x,y)\), since \(u_t\) is close to one. Here \(v_x\) denotes differentiation with \(s,y\) held fixed. Near each limiting wall, the equation \(v_x=0\) has a unique nearby smooth graph \(x=x_j(s,y)\) by the implicit function theorem. Recenter by \(z=x-x_j(s,y)\). Then \[ v_z(s,0,y)=0,\qquad \left\lvert v_{zz}(s,0,y)\right\rvert\ge c_*>0 \tag{42}\] on a fixed narrow strip. After narrowing it if necessary, \(v_z/z\) extends smoothly and is bounded away from zero on the strip. The graphs and coordinates are periodic in \(y\) and converge smoothly to their limiting counterparts. Choose all \(2k\) strips disjoint.

We now remove the source near each wall. In its chart, \(r^1,r^2\) and \(H\) denote the transformed densities under the convention above, and derivatives hold the other recentered coordinates fixed. Set \[\begin{align*} \alpha(s,z,y)&=\int_0^z r^1(s,q,y)\,\,\mathrm dq,\\ G(s,z,y)&=\int_0^z \bigl(r^2-\partial_s(\alpha v_z)\bigr)(s,q,y)\,\,\mathrm dq -\alpha(s,z,y)v_s(s,z,y). \tag{43}\end{align*}\] Use only the \(s,z\) block of \(H\), taking \[H_{ss}=0,\qquad H_{sz}=H_{zs}=\alpha,\qquad H_{zz}=G/v_z.\] All entries involving \(y\) are zero. The quotient extends smoothly across \(z=0\): the numerator vanishes there, so \(G/z\) is smooth, while \(v_z/z\) is smooth and bounded away from zero. In particular \[H_{zz}(s,0,y)= \frac{r^2(s,0,y)-r^1(s,0,y)v_s(s,0,y)}{v_{zz}(s,0,y)}.\] Differentiating Equation (43) gives \[\partial_z\alpha=r^1,\qquad \partial_zG=r^2-\partial_s(\alpha v_z)-\partial_z(\alpha v_s).\] Since \(u=s\), the two flux columns are \((0,\alpha,0)^t\) and \((\alpha v_z,\alpha v_s+G,0)^t\); the displayed identities prove both equations in Equation (40). The fixed denominator bound and the smooth convergence of the wall charts show that this correction is small in every fixed smooth norm. Integration only in \(z\) preserves its support away from the axial ends. Multiply it by a cutoff in \(z\) which equals one in a narrower wall strip and vanishes outside the chosen strip. After subtracting the divergences of these compact corrections, the remaining sources are supported away from all walls. They still have three zero total moments by Equation (38).

The remaining source pair may nevertheless have nonzero moments in an individual regular strip between successive walls. We next transfer these imbalances across the walls without recreating a source there. To do so, choose any smooth \(a(s,y)\) supported in a fixed middle axial interval, put \(H_{sz}=H_{zs}=a\), \(H_{ss}=0\), and set \[ H_{zz}v_z= -2a\bigl(v_s(s,z,y)-v_s(s,0,y)\bigr) -a_s\bigl(v(s,z,y)-v(s,0,y)\bigr). \tag{44}\] The first divergence is zero because \(a\) is independent of \(z\). For the second, the \(z\) derivative of the right side is \(-2a v_{sz}-a_s v_z\), which cancels \(\partial_z(a v_s)+\partial_s(a v_z)\). Differentiating \(v_z(s,0,y)=0\) with the recentered \(z\) fixed gives \(v_{sz}(s,0,y)=0\). Hence both differences on the right of Equation (44) are of order \(z^2\). Thus division by \(v_z\) is again smooth and \(H_{zz}v_z\) vanishes on the wall. Cut this matrix off away from the wall. Its new sources are confined to the adjacent regular strips. Their three moments on one side, up to a common orientation sign, are \[ \int a(s,y)\bigl(1,v_s,v-sv_s\bigr)(s,0,y)\,\,\mathrm ds\,\,\mathrm dy; \tag{45}\] on the other side they are their negatives. This follows by integrating the two fluxes and Equation (38); their normal values on the wall are \(a\), \(av_s\), and \(a(v-sv_s)\). They are flux densities in this chart, so the wall integral uses \(\,\mathrm ds\,\,\mathrm dy\).

The map in Equation (45) has a bounded right inverse uniformly for large \(T\). On a limiting wall write \(V(s,0,y)=\sigma e^{-\lambda s}\), where \(\sigma\in\{1,-1\}\), and set \[w(s)=\bigl(1,-\sigma\lambda e^{-\lambda s}, \sigma(1+\lambda s)e^{-\lambda s}\bigr).\] Its three component functions are linearly independent on every open interval: after multiplying a linear relation by \(e^{\lambda s}\), two derivatives force its exponential coefficient to vanish, and the remaining affine coefficients then vanish. Choose a nonnegative smooth middle-supported function \(\chi\), positive on an interval, and use \(a_j(s,y)=\chi(s)w_j(s)\), \(j=1,2,3\). The limiting moment matrix is \(2\pi\int\chi(s)w(s)w(s)^t\,\,\mathrm ds\), which is positive definite. Its determinant remains nonzero for the smoothly perturbed wall coordinates and fields. This provides a fixed finite-dimensional right inverse with the asserted support and bounds.

The regular strips form a cyclic adjacency graph with \(2k\) vertices. Choose a spanning tree and transfer each strip’s three-vector imbalance towards a root. The root imbalance is zero because the total is zero. There are only finitely many transfers, so all corrections remain small with the sources. Their cutoff regions lie in fixed cores of the regular strips. Each regular strip is connected; away from the narrow wall neighborhoods \(v_x\) has a fixed nonzero sign and \((u,v,y)\) are regular local coordinates. The remaining source pair in each strip now has all three moments zero and support in a fixed compact subset of that strip. Lemma 13 removes it there.

Choose the wall strips and their transverse cutoffs first for the limiting pair in Equation (41), with the axial supports a positive distance from the band ends. In the complementary regular strips choose the source cores a positive distance from their bounding walls and the band ends, and fix the associated finite chart covers, overlap bumps, and right inverses. These choices persist for the smoothly close fields. Transforming each local matrix density back to the product coordinates and summing gives a correction \(H\) with \(\mathop{\mathrm{div}}(H\nabla u)=r^1\) and \(\mathop{\mathrm{div}}(H\nabla v)=r^2\). Thus both fields solve the equations for \(A_0+H\). The fixed bounds, including the finite derivative loss in Lemma 13, and the convergence of the sources in every smooth norm give \(\left\lVert H\right\rVert_{L^\infty}\to0\) as \(T\to\infty\). These constants are for the fixed end; no estimate uniform in \(\lambda\) or \(k\) is used. For large enough \(T\), \(A_0+H\) remains uniformly positive definite.

Return to the original band by setting \[u_T(T+t,\theta)=T+u(t,\theta),\qquad v_T(T+t,\theta)=b e^{-\lambda T}v(t,\theta),\] and translating \(H\) to \(H_T\). Linearity preserves both equations. Near the left and right edges, respectively, the pair is the original input and the exact target, and \(H_T=0\). The normal row of \(A_0\) is \((1,0,0)\), so the stated positive-\(t\) fluxes follow there. Finally, \(\partial_tu_T>0\) by smooth closeness to \(U=t\); away from the \(2k\) wall graphs, \(v_x\ne0\) in the \((s,x,y)\) coordinates, so the two gradients are independent. This proves the nonzero case.

If \(v_{\rm in}\) is identically zero, normalize and interpolate only the first field to \(t\). Its residual is small, compactly supported, and has mean zero because the two mean slopes agree. The compact divergence construction on the connected band gives a small compactly supported flux correction \(f\) with \(\mathop{\mathrm{div}}f\) equal to this residual. Since \(e=\nabla u\) stays nonzero, the symmetric matrix \[H=\frac{fe^t+ef^t}{\left\lvert e\right\rvert^2} -\frac{(f\cdot e)ee^t}{\left\lvert e\right\rvert^4}\] satisfies \(He=f\). It is smooth, compactly supported, and small because \(e\) stays bounded away from zero on this fixed band. Returning to the original band gives the stated first field, flux, and ellipticity, with \(\partial_tu_T>0\) and the zero field unchanged. ◻

Proof. The outer boxes of the children are, respectively, \[[-1.38,-0.24]\times[-0.912,0.912]\times[-0.57,0.57] \quad\hbox{and}\quad [0.24,1.38]\times[-0.912,0.912]\times[-0.57,0.57].\] They lie strictly inside the parent outer box. Both avoid the parent axial hole, whose first-coordinate half-width is \(Lh=0.048\), and the two boxes have separation at least \(0.48\).

For explicit collars, let \(q(\theta)=(q_1(\theta),q_2(\theta))\) parametrize the unit square perimeter by rays, and let \(\zeta(\psi)\) similarly parametrize the perimeter of \([h,1]\times[-1,1]\) about its center \(c_0=((1+h)/2,0)\). Both parameters have period \(2\pi\). Writing \[(r,z)=c_0+(1+\tau)\bigl(\zeta(\psi)-c_0\bigr),\] the map \[ (\theta,\psi,\tau)\longmapsto \bigl(Lr q_1(\theta),\,r q_2(\theta),\,z\bigr) \tag{54}\] is a product collar of \(\Sigma\) for sufficiently small \(\left\lvert \tau\right\rvert\). Indeed \(r\) stays strictly positive, and on each pair of perimeter faces the map and its inverse have bounded derivatives and nonzero Jacobian. The finitely many seams have measure zero. The child collars are its images under \(S_-\) and \(S_+\). Choose their widths small enough that they are disjoint.

For completeness, the central complement is connected. Above the child boxes, for example at height \(x_3=0.8\), its horizontal section is the connected parent rectangular annulus. A point between heights \(-0.57\) and \(0.57\) in the complement of the children can move vertically to that level: its horizontal position is outside the child annular footprints or inside one of their axial holes. A point below the child boxes first moves horizontally in the parent annulus to a position outside both boxes and then moves upward. The same argument applies after sufficiently thin collars are removed. Such removal slightly shrinks the parent cross-sectional rectangle in \((r,x_3)\) and slightly expands the two child rectangles; all the containment margins above remain positive.

We also check the local boundary regularity for these particular regions. Positive separation places each boundary point on just one boundary of a rectangular annular prism. A face is locally a half-space, and the convex edge and corner types are graphs of maxima or minima of finitely many affine functions after choosing a diagonal direction. For an inner top corner, the reentrant local model, after changes of coordinates and signs, is \(\{\max(x,y)>0,\ z<0\}\). On the line \((x,y,z)=(x_0,y_0,z_0)+t(1,1,-1)\), membership is equivalent to \[t>\max\bigl\{z_0,-\max(x_0,y_0)\bigr\}.\] The threshold is a Lipschitz function on the transverse plane \(x_0+y_0-z_0=0\). Inner edges give the corresponding two-dimensional max graph, and reversing the side of any of these boundaries reverses the inequality without changing the graph. The remaining edge and corner types give the same max/min description. Thus every boundary point has a local Lipschitz graph. The small collar adjustment keeps this rectangular form and the positive separation, so the central region is Lipschitz as claimed. ◻

Proof. On the fixed block, let \(V=(V_0,V_-,V_+)\) be the constant voltage vector and let \(p\) be the integrated outward conormal flux vector of its harmonic extension \(v\). The map \(V\mapsto p\) is linear, \(\sum_i p_i=0\), and \[\int_B b\left\lvert \nabla v\right\rvert^2=\sum_i V_i p_i.\] Thus its kernel consists exactly of common constants: zero flux implies zero energy, and connectedness of \(B\) then makes \(v\) constant. The induced map from voltage vectors modulo constants to zero-sum flux vectors is an isomorphism of two-dimensional spaces. Normalize its inverse by \(V_0=0\).

Extend \(v\) by \(V_\pm\) in \(D_\pm\) and by zero outside \(\overline D\). Matching traces give an \(H^1\) function \(W_p\) with compact support in \(\overline D\). The exact block flux property gives Equation (57) at the root. This identity holds for every \(H^1\) test: equivalently, one may first multiply the test by a smooth cutoff equal to one near \(\overline D\). The boundary trace functionals are continuous, since the measures have bounded surface densities. Invertibility of the finite-dimensional map, the maximum principle for the bounded boundary voltages, and the energy identity give \[\left\lVert W_p\right\rVert_\infty\le K\left\lvert p\right\rvert, \qquad \int\left\lvert \nabla W_p\right\rvert^2\le K\left\lvert p\right\rvert^2.\]

For \(j=\left\lvert \nu\right\rvert\), set \[W_{\nu,p}(x)=s^{-j}W_p(S_\nu^{-1}x),\] where \(W_p\) is understood to be zero outside \(\overline D\). A similarity of ratio \(r\) multiplies energy and integrated flux by \(r\) in dimension three when the scalar coefficient is copied without changing its values. The extra voltage factor \(r^{-1}\), here \(r=s^j\), therefore preserves the prescribed integrated charges and gives the bounds in Equation (58). The coefficient inside the children is irrelevant because this function has zero gradient there. Finally, \((S_\nu)_*\mu_\pm=\mu_{\nu\pm}\); hence the measure on each shared boundary is exactly the same from its two adjacent blocks. ◻

Proof. Test the weak equation for \(u\) with \(W_{\nu,p}\), and use Equation (57) with \(\psi=u\). This gives \[p_0m_\nu(u)+p_-m_{\nu-}(u)+p_+m_{\nu+}(u)=0 \qquad\text{whenever }p_0+p_-+p_+=0.\] The triple of averages is orthogonal to the zero-sum plane and is therefore constant. Iterating proves the assertion. If \(u\) is bounded, its traces have the same essential bounds: apply the trace map to the vanishing positive-part truncations beyond those bounds. Since every \(\mu_\nu\) is a probability measure, the last claim follows. ◻

Proof. At a fixed depth the cell interiors are disjoint. Combining Equations (58) and (59), with a larger constant \(K\) if necessary, gives \[ \left\lVert Z_j\right\rVert_\infty\le K(2s)^{-j},\qquad \left\lVert \nabla Z_j\right\rVert_2^2\le K(2s)^{-j}. \tag{61}\] Since \(2s=1.14>1\), both \(\sum_j\left\lVert Z_j\right\rVert_\infty\) and \(\sum_j\left\lVert \nabla Z_j\right\rVert_2\) converge. The first controls the \(L^2\) norms as well because \(\Omega\) is bounded. This proves the asserted convergence. Every partial sum is in \(H^1_0(\Omega)\) and vanishes off \(\overline D\); the same is true of its limit.

To show that \(w\ne0\), fix \(\eta\in C_c^\infty(\Omega)\) equal to one on a neighborhood of \(\overline{D_-}\) and zero on a neighborhood of \(\overline{D_+}\). The positive separation of these compact sets makes this possible. At every depth the signed total in the first left cell is \(1\) and that in the first right cell is \(-1\). Equation (60) therefore gives \[\int_\Omega\gamma\nabla w^{(N)}\cdot\nabla\eta=1.\] Strong \(H^1\) convergence yields \[ \int_\Omega\gamma\nabla w\cdot\nabla\eta=1, \tag{62}\] which proves nontriviality. No trace or pointwise value on the limiting nested set has been used. ◻

Proof. The product \((u-u_*)\phi\) belongs to \(H^1_0(\Omega)\), so it is an admissible test in Equation (60). Put \(M=\left\lVert u\right\rVert_\infty\). For each depth-\(N\) cell choose \(x_\nu\in\overline{D_\nu}\). The zero-average identity on its surface gives \[\int_{\Sigma_\nu}(\operatorname{Tr}u-u_*)\phi\,\,\mathrm d\mu_\nu =\int_{\Sigma_\nu}(\operatorname{Tr}u-u_*) \bigl(\phi-\phi(x_\nu)\bigr)\,\,\mathrm d\mu_\nu.\] Since \(\left\lvert \operatorname{Tr}u-u_*\right\rvert\le2M\) almost everywhere on this surface, the absolute value of the sum on the right-hand side of Equation (60) is at most \[ 4M\left\lVert \nabla\phi\right\rVert_\infty\,\operatorname{diam}(D)\,s^N. \tag{66}\] Here we used Equation (59) and \(\operatorname{diam}(D_\nu)=s^N\operatorname{diam}(D)\). This tends to zero. The fixed product test has square-integrable gradient, so strong \(H^1\) convergence of \(w^{(N)}\), together with boundedness of \(\gamma\), justifies passage to the limit on the left. Multiplication by \(\delta\) proves Equation (65). ◻

Proof. First take \(f\in C^\infty(\partial\Omega)\), and let \(u\) be its weak \(\gamma\)-harmonic extension. The weak maximum principle, obtained by testing truncations beyond the boundary bounds, gives \(\left\lVert u\right\rVert_\infty\le\left\lVert f\right\rVert_\infty\). Define \[ \widetilde u=u_*+\frac{u-u_*}{\rho}. \tag{67}\] The Sobolev chain rule applies to \(1/\rho\) on the interval \([1/2,3/2]\), and the product rule applies because both factors are bounded \(H^1\) functions. Hence \(\widetilde u\in H^1(\Omega)\), and it equals \(u\) off \(\overline D\), in particular near the outer boundary. Its flux is \[ \widetilde\gamma\nabla\widetilde u =\rho\gamma\nabla u-(u-u_*)\gamma\nabla\rho. \tag{68}\] This is an \(L^2\) vector field.

For \(\phi\in C_c^\infty(\Omega)\), use \(\rho\phi\in H^1_0(\Omega)\) in the original weak equation to obtain \[\int_\Omega\rho\gamma\nabla u\cdot\nabla\phi =-\int_\Omega\phi\gamma\nabla u\cdot\nabla\rho.\] Equation (65), expanded by the product rule, also gives \[\int_\Omega(u-u_*)\gamma\nabla\rho\cdot\nabla\phi =-\int_\Omega\phi\gamma\nabla\rho\cdot\nabla u.\] Subtracting proves that the divergence of the flux in Equation (68) is zero. All cross-gradient products are integrable by Cauchy–Schwarz; boundedness of the other factors also gives the stated \(L^2\) flux bound. Density of smooth tests in \(H^1_0(\Omega)\) proves the full weak equation. By uniqueness, \(\widetilde u\) is the \(\widetilde\gamma\)-harmonic extension of \(f\).

Now fix arbitrary \(g\in H^{1/2}(\partial\Omega)\) and choose an \(H^1\) extension \(V\) of \(g\). Multiply \(V\) by a smooth cutoff that equals one near \(\partial\Omega\) and is supported in an outer collar disjoint from \(\overline D\). The resulting \(V_0\) still has trace \(g\), and its gradient is supported where the two solution fluxes agree. Consequently the defining weak pairings satisfy \[\langle\Lambda_{\widetilde\gamma}f,g\rangle =\int_\Omega\widetilde\gamma\nabla\widetilde u\cdot\nabla V_0 =\int_\Omega\gamma\nabla u\cdot\nabla V_0 =\langle\Lambda_\gamma f,g\rangle.\] Both operators are bounded from \(H^{1/2}\) to \(H^{-1/2}\) by the elliptic energy estimate. Since smooth boundary functions are dense in \(H^{1/2}(\partial\Omega)\) on the ball, equality extends from smooth \(f\) to all boundary traces. This last step does not require extending the transformation in Equation (67) to unbounded solutions. ◻

Proof of Theorem 1. Lemma 14 permits the use of Proposition 11, and Equation (55) constructs a uniformly positive bounded scalar conductivity on the ball \(\Omega=B(0,3)\subset\mathbb R^3\). Lemmas 15–17 produce the bounded nonzero function used in Equation (63). The second scalar conductivity has the positive bounds in Equation (64), agrees with the first near the boundary, and differs from it on a set of positive measure. Set \(\gamma_0=\gamma\) and \(\gamma_1=\widetilde\gamma\), with common positive finite bounds obtained by taking the smaller lower bound and the larger upper bound. Proposition 19 proves equality of their full weak Dirichlet-to-Neumann operators. Thus these coefficients give the claimed counterexample in dimension three, disproving the assertion quantified over all \(n\ge3\). ◻

Proof of Corollary 2. Return to the notation of Corollary 2: let \(\Omega\) be its bounded connected Lipschitz domain, let \(c,C\) be the common bounds from Theorem 1, and let \(B_1,\ldots,B_m\) be the prescribed balls. Write \(B_i=B(x_i,r_i)\) and set \(U_i=B(x_i,r_i/2)\). The closed balls \(\overline U_i\) are separated from one another and from \(\partial\Omega\), even if some prescribed balls have tangent boundaries. Let \(a_0,a_1\) be the pair of Theorem 1, and put \(\Phi_i(y)=x_i+s_i y\), where \(s_i=r_i/6\). This maps \(B(0,3)\) onto \(U_i\). Copy the scalar coefficients by composition: \(a_{i,j}=a_j\circ\Phi_i^{-1}\) on \(U_i\). If \(q_{i,j}(h)\) is the minimum energy with trace \(h\in H^{1/2}(\partial U_i)\), change of variables gives \[q_{i,j}(h)=s_i \left\langle \Lambda_{a_j}(h\circ\Phi_i),h\circ\Phi_i\right\rangle.\] Thus \(q_{i,0}=q_{i,1}\). The coefficient values and their unit boundary collars are preserved; no amplitude rescaling is needed.

Define \(\gamma_\varepsilon=a_{i,\varepsilon_i}\) on \(U_i\) and \(\gamma_\varepsilon=1\) elsewhere. The stated bounds follow because the original unit collars imply \(c\leq1\leq C\). Let \(E=\Omega\setminus\bigcup_i\overline U_i\), and denote by \(E_{\rm ext}(f,h_1,\ldots,h_m)\) the minimum unit-conductivity energy on \(E\) with outer trace \(f\) and inner traces \(h_i\). The trace theorem and matching-trace \(H^1\) gluing, together with Equation (3), give \[\left\langle \Lambda_{\gamma_\varepsilon}f,f\right\rangle =\min_{h_i\in H^{1/2}(\partial U_i)} \left\{E_{\rm ext}(f,h_1,\ldots,h_m) +\sum_{i=1}^m q_{i,\varepsilon_i}(h_i)\right\} \qquad(f\in H^{1/2}(\partial\Omega)).\] Indeed, restriction of a global competitor gives one inequality, and gluing the subdomain minimizers gives the other; a global minimizer supplies an attaining tuple of interface traces. The right side is independent of \(\varepsilon\). Polarization therefore proves equality of the full weak operators.

If two indices differ at position \(i\), their conductivities differ inside \(U_i\) on a set of measure \(s_i^3\left\lvert \{a_0\ne a_1\}\right\rvert>0\). This proves all \(2^m\) choices are distinct. Their common boundary response is not asserted to be that of the constant conductivity one. ◻