The full collared EIT class satisfies the centered range and source interfaces

The space $V$ is closed in the intrinsic Sobolev Hilbert space, since convergence there implies $L^2$ convergence and preserves vanishing on the collar. For its elements, zero extension is bounded into $H^{s_1}({\mathbb{R}}^3)$ by [F3]. Fourier Cauchy–Schwarz gives $\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\mathrm{\parallel }a{\mathrm{\parallel }}_V$ because $s_1>3\mathrm{/}2$. Thus $V$ embeds continuously into the bounded collared coefficient space. The set $\mathcal{C}-{\sigma }_0$ is convex and closed in $V$: convexity preserves the pointwise bounds, and uniform convergence from this embedding preserves them under a norm limit.

Choose a smooth scalar function $\vartheta$ with $0\le \vartheta \le 1$, equal to one on $[m,M]$ and zero outside $(m\mathrm{/}2,2M)$, and set $g(t)=1+{\int }_1^t\vartheta (u)du$. Then $g(t)=t$ on $[m,M]$, $g(1)=1$, and $m\mathrm{/}2\le g(t)\le 2M$ globally. All its first and second derivatives are bounded. Put $b(a):=g({\sigma }_0+a)-1$. Since ${\sigma }_0+a=1$ outside $B_{\rho }$ and $g(1)=1$, $b(a)\in E$ for every $a\in V$. Moreover, $a\mapsto b(a)$ is globally Lipschitz from $V$ into $E$ and continuously Fréchet differentiable, with derivative $b^{\mathrm{\prime }}(a)h=g^{\mathrm{\prime }}({\sigma }_0+a)h$ and globally Lipschitz derivative; its Taylor remainder is bounded by $\frac{1}{2}\mathrm{\parallel }{g}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}^2$. Use the centered map [F1] with fixed strict ellipticity bounds $m_{\ast }:=m\mathrm{/}4$ and $M_{\ast }:=4M$, and denote it by $G_{m_{\ast },M_{\ast }}$. Since $m_{\ast }<m\mathrm{/}2\le 1+b(a)\le 2M<M_{\ast }$, every $b(a)$ is admissible for that map. Define explicitly $\mathrm{\Gamma }(a):=G_{m_{\ast },M_{\ast }}(b(a))=G_{m_{\ast },M_{\ast }}(g({\sigma }_0+a)-1)(a\in V)\mathrm{.}$ The chain rule gives ${\mathrm{\Gamma }}^{\mathrm{\prime }}(a)h=G_{m_{\ast },M_{\ast }}^{\mathrm{\prime }}(b(a))[g^{\mathrm{\prime }}({\sigma }_0+a)h]$. The bounded and Lipschitz derivatives of $b$ and the Lipschitz derivative of the centered supplier give all the asserted global bounds for $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\mathrm{\prime }}$. For ${\sigma }_0+a\in \mathcal{C}$, $g({\sigma }_0+a)={\sigma }_0+a$, so this formula agrees with the centered physical observation.

The identities for $K,r,F$ and the inverse follow by direct substitution. The map $T$ is globally Lipschitz with $L_T\le \mathrm{Lip}(\mathrm{\Gamma })+\mathrm{\parallel }{\mathrm{\Gamma }}^{\mathrm{\prime }}(0)\mathrm{\parallel }$, so both $r$ and $r^{-1}$ have Lipschitz constant at most $1+L_T$ in the sum norm. In particular $\mathrm{\parallel }P{r}^{-1}(a,b)-P{r}^{-1}(a^{\mathrm{\prime }},b^{\mathrm{\prime }})\mathrm{\parallel }\le (1+L_T)\mathrm{\parallel }(a,b)-(a^{\mathrm{\prime }},b^{\mathrm{\prime }})\mathrm{\parallel }$. The restriction of a bijection to $U$ is a bijection onto $r(U)$; its projected inverse point has the same coefficient and zero artificial voltage, hence remains in $U\cap X_0$. The extension to all of $D=X$ supplies the global factorization domain required by the original generic hypotheses without changing the physical forward map on any admitted point.

The indicator of the convex set in step 1.1 is convex, and adding the quadratic functional makes $\check{R}$ proper and nonnegative. For any admitted $a$, expansion of its square gives $\check{R}(a)-\check{R}(a^{†})-{\xi }^{†}(a-a^{†})=\frac{1}{2}\mathrm{\parallel }a-a^{†}{\mathrm{\parallel }}_V^2$; outside the set the subgradient inequality is automatic because the left value is infinite. This proves the asserted subgradient and exact Bregman formula. On a fixed sublevel $\check{R}(a)\le R_0$, the coefficients have bounded intrinsic $H^{s_1}$ norm; subtracting the constant background and zero-extending gives a uniform whole-space bound by [F3]. Thus [F2] applies uniformly on that sublevel, including the truth. Applying [F3] yields the VSC source term $C[\log (1\mathrm{/}\mathrm{\parallel }\mathrm{\Gamma }(a)-\mathrm{\Gamma }(a^{†})\mathrm{\parallel }){]}^{-\nu }$ for small data differences. Replacing the norm argument by its square changes only a fixed constant in the logarithm. Choose $u_0$ small enough and then enlarge $C$ to dominate the source inequality also at larger data differences; on the fixed sublevel the absolute linear pairing is uniformly bounded. The truncated logarithmic index in [F4] is then a valid concave VSC index on the entire sublevel.

Concavity and $\psi (0)=0$ imply subadditivity: for positive $a+b$, $\psi (a)\ge a\psi (a+b)\mathrm{/}(a+b)$ and $\psi (b)\ge b\psi (a+b)\mathrm{/}(a+b)$; summing gives $\psi (a+b)\le \psi (a)+\psi (b)$. Hence $C_{\psi }=1$. The derivative remainder of $r$ has only the second component $\mathrm{\Gamma }(a_1)-\mathrm{\Gamma }(a_2)-{\mathrm{\Gamma }}^{\mathrm{\prime }}(a_2)(a_1-a_2)$. Integrating the Lipschitz derivative along the segment gives norm at most $L\mathrm{\parallel }{a}_1-a_2{\mathrm{\parallel }}_V^2\mathrm{/}2$. Since $\mathrm{\parallel }{a}_i\mathrm{\parallel }<R$ and the first component of $r(x_1)-r(x_2)$ is $a_1-a_2$, this is at most $LR\mathrm{\parallel }r(x_1)-r(x_2){\mathrm{\parallel }}_Z$. No differentiability of the boundary of $\mathcal{C}$ is needed, since the map was differentiated on the full ambient space.

If $\widehat{r}=r(a,z)\in r(U)$, then $\mathrm{\parallel }\widehat{r}-r(a^{†},0){\mathrm{\parallel }}_Z\le (1+L_T)\mathrm{\parallel }a-a^{†}{\mathrm{\parallel }}_V+\mathrm{\parallel }z{\mathrm{\parallel }}_Y$. Squaring gives a bound by $4(1+L_T{)}^2{D}_{\check{R}}^{{\xi }^{†}}(a,a^{†})+2\mathrm{\parallel }z{\mathrm{\parallel }}_Y^2$. Because $\mathrm{\parallel }z\mathrm{\parallel }<R$, concavity gives $\psi (C_P\mathrm{\parallel }z{\mathrm{\parallel }}^2)\ge (\mathrm{\parallel }z{\mathrm{\parallel }}^2\mathrm{/}{R}^2)\psi (C_P{R}^2)$. The displayed ${\tilde{C}}_Q$ therefore proves the full frozen compatibility inequality for every admitted range variable. Its denominator is positive since the logarithmic index is positive away from zero. The stated $R_P$ has the required lower bound with equality and vanishes at zero.

All projected coefficients in $U$ have $\mathrm{\parallel }{\sigma }_0+a{\mathrm{\parallel }}_{s_1,\mathrm{\Omega }}\le \mathrm{\parallel }{\sigma }_0{\mathrm{\parallel }}_{s_1,\mathrm{\Omega }}+R$, so one fixed $R_0$ covers their regularizer values and the truth. The preceding steps verify every structural range, domain, Lipschitz, projection, subgradient, VSC, compatibility and penalty hypothesis. With $p=2$ and $C_{\psi }=1$, the frozen constants are $t=1\mathrm{/}4-2c_{\mathrm{t}\mathrm{c}\mathrm{r}}^2$, $v=3c_{\mathrm{t}\mathrm{c}\mathrm{r}}^2$, $\lambda =1\mathrm{/}2$ and $\bar{C}=4$. The bound $c_{\mathrm{t}\mathrm{c}\mathrm{r}}<1\mathrm{/}\sqrt{20}$ makes $t>0$ and $q=v\mathrm{/}t<1$. The displayed choice of $c_{\beta \alpha }$ is exactly the required linked-parameter bound because $1-b=1\mathrm{/}4$ and $\gamma -C_{\psi }=\gamma -1$. Thus the original generic estimates apply on the admitted set with their remaining stated algorithmic hypotheses. The ambient extension and indicator do not change any admitted objective value.