\section{Collars and all-jet compatibility} \label{app:collars-all-jets} This appendix records the differential-topological mechanisms used in \cref{sec:naturalization,sec:relative-extensions}. Its first purpose is to distinguish compatibility on an open product germ from equality on a central face. Its second is to retain a \emph{specified full collar} through the relative rigidity argument; replacing that collar by a smaller unspecified one would not suffice for the marked construction. Compatible inward germs produce one collar, and signed bicollars make the local and spectator formulas agree on an open neighborhood. The subsequent rigidity argument retains that specified collar by comparing the relevant path components and then choosing a jointly smooth isotopy. Thus the same protected neighborhood survives both gluing and contraction. \subsection{Germs and jets} \begin{definition}[Agreement of all tangential and normal derivatives] \label{def:all-jet-agreement} Let \(c_i:Z\times(-a,a)\to M_i\) be smooth product charts and let \(F,G:M_0\to M_1\) be smooth wherever the displayed expressions are defined. Put \[ \widetilde F=c_1^{-1}Fc_0,\qquad \widetilde G=c_1^{-1}Gc_0. \] We say that \(F\) and \(G\) agree with all jets along \(Z\) if, for every multi-index \(\alpha\) and \(m\ge0\), \[ \partial_z^\alpha\partial_t^m \bigl[\widetilde F-\widetilde G\bigr](z,0)=0. \] We say that they agree on an open product germ if there is an open neighborhood \(\mathcal U\subset Z\times(-a,a)\) of \(Z\times\{0\}\) on which \(\widetilde F=\widetilde G\). Open-germ equality implies all-jet agreement. The converse is false for general smooth maps, and is never used here: every gluing argument below proves the stronger open-germ equality. \end{definition} \begin{warning} Equality of values on \(Z\times\{0\}\), even with matching first normal derivative, does not imply smoothness of a piecewise-defined map across \(Z\). Likewise, equality on the artificial annulus \(\mathcal A_{-+}^D\) does not verify the disk-extension hypothesis on the whole protected annulus \(\mathcal A_{01}^D\). \end{warning} \subsection{Relative extension of compatible collar germs} \begin{lemma}[Relative collar extension] \label{lem:multigerm-relative-collar-extension} Let \(M\) be a compact smooth manifold with boundary and let \(O_0\Subset O_1\subset\partial M\). Suppose a finite collection of smooth collar embeddings is prescribed on open subsets of \(O_1\), all pointing strictly inward and agreeing on every overlap. Then there is \(b>0\) and a collar \[ d:\partial M\times[0,b)\longrightarrow M \] which agrees with every prescription on \(O_0\) through a positive depth. If the prescribed germs agree on open neighborhoods of the closures of their domains, the resulting agreement includes all jets at the domain boundaries. \end{lemma} \begin{proof} After shrinking the finitely many collar depths, push forward \(\partial_t\) on the complete prescribed product germs: \[ X_{\mathrm{fix}}\bigl(c_\nu(x,t)\bigr) =(c_\nu)_*(\partial_t). \] The overlap hypotheses make this a single smooth field on an open neighborhood of the prescribed product images, not merely a field on their central faces. On the boundary it points strictly inward. Choose any smooth field \(Y\), defined near the boundary and strictly inward there. Extend \(X_{\mathrm{fix}}\) slightly beyond the smaller prescribed images. With a cutoff \(\chi\), equal to one on those images over \(O_0\) and supported over \(O_1\), put \[ X=\chi X_{\mathrm{fix}}+(1-\chi)Y. \] The cone of inward-pointing vectors is convex, so \(X\) remains strictly inward. Extend it to a neighborhood of the boundary. Compactness and the inverse function theorem give a uniform \(b>0\) for which the positive-time flow of \(X\) is a collar embedding. On the region where \(\chi=1\), the new flow and each prescribed collar solve the same autonomous initial-value problem throughout the retained product germ. Uniqueness of integral curves makes them equal there. The last assertion follows because this is open-germ equality in the sense of \cref{def:all-jet-agreement}. \end{proof} In \cref{sec:relative-extensions}, this lemma is applied five times to the model spectator balls and ten times to the realized spectator balls. The compatibility checks precede those applications. Taking a minimum of the resulting fifteen positive widths is legitimate only afterward, which is why \(u_{\mathrm{cmp}}\) is defined at \eqref{eq:comparison-depth} rather than at the vertex stage. \subsection{Signed internal bicollars} Let \(E_{e,u}\) be the internal disk separating the local and spectator halves of \(K_u\). Orient its disk factor by the boundary orientation of the local half. The spectator boundary orientation is then the opposite one. A signed bicollar is written \[ c_{e,u}^{E_e}:E_e\times(-\eta,\eta)\longrightarrow K_u, \] with negative coordinate on the local side and positive coordinate on the spectator side. If \(s_{\mathrm{loc}},s_{\mathrm{sp}}\ge0\) are the two inward variables, the endpoint maps have the formulas \begin{align} \widehat\Theta_{e,u}(z,s_{\mathrm{loc}}) &=c_{e,u}^{E_e}(z,-s_{\mathrm{loc}}), \label{eq:appendix-local-bicollar}\\ \Xi^z_{e,u}(z,s_{\mathrm{sp}}) &=c_{e,u}^{E_e}(z,+s_{\mathrm{sp}}). \label{eq:appendix-spectator-bicollar} \end{align} Both are orientation preserving because the induced orientations on the two copies of \(E_e\) are opposite. \begin{lemma}[All-jet two-sided gluing] \label{lem:all-jet-two-sided-gluing} Let \[ K_u=K_{e,u}^{\mathrm{loc}}\cup_{E_{e,u}}L_{e,u}^{\mathrm{sp}} \] and suppose smooth positive diffeomorphisms on the two closed halves extend to one signed bicollar and both have coordinate expression \[ (z,t)\longmapsto(z,t). \] Then their union is a smooth orientation-preserving diffeomorphism. Its inverse is obtained by gluing the inverse half-maps. \end{lemma} \begin{proof} The two maps agree on an open neighborhood of the central disk in the signed coordinate, not only on the disk itself. Their union is therefore locally one smooth map there and is smooth away from the disk by hypothesis. The same argument applies to the inverse formulas. Positivity holds on either half and hence everywhere by connectedness. \end{proof} For the maps prepared in \cref{prop:fresh-naturalized-representatives}, \eqref{eq:internal-bicollar-germs} gives more explicitly, for every multi-index \(\alpha\) and \(m\ge0\), \begin{align*} \partial_z^\alpha\partial_t^m \left[ (c_{e,w}^{E_e})^{-1} \widehat\Theta_{e,w}\widehat\phi_e^{\mathrm{cut}} \widehat\Theta_{e,v}^{-1}c_{e,v}^{E_e}(z,t)-(z,t) \right]&=0,\\ \partial_z^\alpha\partial_t^m \left[ (c_{e,w}^{E_e})^{-1} \Xi^z_{e,w}(\Xi^z_{e,v})^{-1} c_{e,v}^{E_e}(z,t)-(z,t) \right]&=0. \end{align*} These equations are the differential content of the corrected local/spectator gluing. The disk \(E_e\) here is internal to the cut ball; it is not the global exterior disk \(E\). \subsection{Fixed collar versus fixed boundary} For \(M=D^2\) or \(D^3\), fix a specified smooth closed full collar \[ c:\partial M\times[0,a]\longrightarrow M,\qquad C=c(\partial M\times[0,a]). \] Define, with the compact-open \(C^\infty\)-topology, \begin{align*} G_C&=\Diff^\infty(M\mathbin{\mathrm{rel}}C),\\ G_J&=\{g\in\Diff^\infty(M): j_{\partial M}^\infty g=j_{\partial M}^\infty\id\},\\ G_\partial&=\Diff^\infty(M\mathbin{\mathrm{rel}}\partial M). \end{align*} \begin{lemma}[Retention of a specified full collar] \label{lem:specified-collar-components} The inclusion \(G_C\hookrightarrow G_\partial\) induces a bijection on path components. \end{lemma} \begin{proof} The specified collar is compared to pointwise boundary fixing in two stages: first fixed-tube data are compared with infinite-jet data, then infinite-jet data are compared with boundary data. Keeping both stages visible is what allows the original full collar, rather than an unspecified thinner one, to remain fixed in the relative mapping class. Equip a slightly larger collar with a product metric. The prescribed \(u\)-lines are normal geodesics, and a radial contraction supported in the outer buffer makes the specified \(C\), not a smaller replacement, an interior normal tube in Cerf's sense. Cerf's fixed-tube versus fixed-infinite-jet comparison gives \[ \pi_0(G_C)\xrightarrow{\cong}\pi_0(G_J), \] and his fixed-infinite-jet versus pointwise-boundary comparison gives \[ \pi_0(G_J)\xrightarrow{\cong}\pi_0(G_\partial) \] \cite[II.4.2.3, Corollary~3; II.5.2.1, Proposition~9 and Corollary~2]{Cerf1961}. These are two distinct steps; the second may not be omitted. Their composition proves the claim. \end{proof} \begin{lemma}[Jointly smooth representative of a path] \label{lem:jointly-smooth-path} Let \(A\subset M\) be fixed. Suppose \[ g_0,g_1\in\Diff^\infty(M\mathbin{\mathrm{rel}}A) \] lie in the same path component for the \(C^\infty\)-topology, then they are joined by a jointly smooth isotopy through maps fixed on \(A\). The isotopy can be made constant in time near both endpoints. \end{lemma} \begin{proof} Cerf's local smooth endpoint section supplies a neighborhood \(V\) of the identity and a smoothly parameterized isotopy from \(\id\) to every element of \(V\) \cite[Lemma~1.5.3]{Cerf1961}. Given a continuous path \(\gamma\) from \(g_0\) to \(g_1\), compactness of its parameter interval and the topological-group structure give a subdivision \(0=t_0<\cdots