Compatible smooth simplex faces have a neighbourhood extension

If $n=0$, take $O=\mathrm{\varnothing }$ and the empty map. Otherwise the face maps glue continuously to $g:\mathrm{\partial }D\to N$, because the boundary is a finite closed union and the maps agree on all intersections. Fix the data in [F2]. The only infinite-choice use in this proof is the countable embedding construction inherited there.

At $p\in \mathrm{\partial }D$ write $J=\{i:{\lambda }_i(p)=0\}$ for its incident faces, where ${\lambda }_i$ are barycentric coordinates. Choose one $k\mathrm{\notin }J$ and use the other $n$ coordinates as affine coordinates on $E$. Shrink around $p$ so every nonincident ${\lambda }_i$ remains positive. For a nonempty $I\subseteq J$, define $P_I$ by setting the coordinates in $I$ to zero and adding their sum to ${\lambda }_k$. Then $P_I{P}_L=P_{I\cup L}$, $P_{I}p=p$, and $P_I(D)\subseteq D$ by nonnegativity of the retained and added coordinates.

On $E_I=\{{\lambda }_i=0:i\in I\}$ take a smooth Euclidean extension $a_I$ of $jg$ on its actual face intersection near $p$: restrict an extension of $j{g}_i$ for one $i\in I$ to $E_I$. The intersections of the finitely many inverse extension domains give an open $V_p$ where every $a_I{P}_I$ is defined. Set $A_p={\sum }_{\mathrm{\varnothing }\mathrm{\ne }I\subseteq J}(-1{)}^{\mathrm{\mid }I\mathrm{\mid }+1}{a}_I{P}_I$. For $x\in V_p\cap \mathrm{\partial }D$, choose $j_0\in J$ with ${\lambda }_{j_0}(x)=0$. Pair each nonempty $I$ omitting $j_0$ with $I\cup \{j_0\}$. Their projected points coincide and lie in the actual intersection face, by step 2.1, so compatibility gives equal values with opposite signs. Only $I=\{j_0\}$ remains, giving $A_p(x)=jg(x)$. No agreement of the arbitrary extensions outside the actual simplex intersections was assumed.

Consider all local data from steps 2.1–3.1 together with balls $B(p,r)$ whose closed doubled balls lie in $V_p$. They form a cover of compact $\mathrm{\partial }D$. Extract finitely many, indexed by $a$, with extensions $A_a$ on neighbourhoods of their closed doubled balls. Define ${\eta }_a(x)=1-s((\mathrm{\parallel }x-p_a{\mathrm{\parallel }}^2-r_a^2)\mathrm{/}(3r_a^2))$. It equals one on the radius-$r_a$ ball and zero outside the radius-$2r_a$ ball. The product ${\eta }_a{A}_a$, extended by zero, is globally smooth: its support is contained in a closed ball strictly inside the domain of $A_a$. This uses a finite subcover of all eligible data, not a simultaneous point-indexed choice.

Put $w={\sum }_a{\eta }_a$ and $O_0=\{w>0\}$, an open neighbourhood of the boundary. On $O_0$, $A=({\sum }_a{\eta }_a{A}_a)\mathrm{/}w$ is smooth. At a boundary point every active $A_a$ equals $jg$, hence $A=jg$. Thus $O=O_0\cap A^{-1}(U)$ is open and contains the boundary, and $h=j^{-1}RA:O\to N$ is smooth and restricts to $g$. This proves the asserted neighbourhood extension.

In dimension one the boundary is two points and the same finite construction applies, regardless of their images; a filling is not inferred. Empty target with $n>0$ cannot satisfy the supplied-face hypothesis. Repeated or constant face values create no exception to the cancellation in step 3.1. The $n=0$ endpoint was treated in step 1.1; all other selections were finite. The boundaryless hypothesis is essential: compatibility alone does not guarantee nonnegative smooth extensions for a half-space target.