[proofplan] The proof uses the local Tumanov propagation theorem for wedge extendibility along short CR curves in $C^{2,\alpha}$ generic CR manifolds. Starting from the initial wedge extension at $\gamma(0)$, we define the set of times up to which wedge extendibility has propagated and prove that its supremum cannot be less than $1$. The key point is that local propagation is stable after shrinking the ambient neighbourhood and wedge cone, so the extension obtained at one time becomes valid initial data for the next short segment of the curve. [/proofplan]

[step:State the precise local propagation theorem used in the argument]Fix any auxiliary Riemannian distance $d_M$ on $M$ induced from a $C^1$ Riemannian metric. We use the following local uniform form of Tumanov's analytic-disc propagation theorem for wedges: let $M\subset\mathbb C^N$ be a generic $C^{2,\alpha}$ CR submanifold with $0<\alpha<1$, let $q\in M$, let $U_q\subset M$ be an open neighbourhood of $q$, and let \begin{align*} g:U_q\to\mathbb C \end{align*} be a continuous CR function. Then there are a relatively open neighbourhood $O_q\subset U_q$ of $q$ and a number $\delta_q>0$ with the following property. If $q'\in O_q$, if $g$ is the continuous boundary value on an edge neighbourhood in $U_q$ of a [holomorphic function](/page/Holomorphic%20Function) on a wedge with edge $M$ at $q'$, and if \begin{align*} \eta:[a,b]\to O_q \end{align*} is a continuous piecewise $C^1$ curve with $\eta(a)=q'$, tangent to $T^cM$ wherever its derivative exists, and satisfying $d_M(\eta(t),q')<\delta_q$ for every $t\in[a,b]$, then $g$ is holomorphically extendible to a wedge with edge $M$ at every point $\eta(t)$, $a\le t\le b$. The propagated wedge may be taken after shrinking the ambient neighbourhood and edge neighbourhood, and its normal cone may depend on the endpoint. Here a wedge with edge $M$ at a point $a\in M$ means an [open set](/page/Open%20Set) which, after choosing local $C^{2,\alpha}$ coordinates flattening $M$ and a real complementary normal space to $T_aM$, contains points obtained from an edge neighbourhood in $M$ by adding sufficiently small normal vectors lying in some non-empty open cone. The normal cone is the corresponding open cone in the normal quotient. Thus shrinking a wedge means replacing the edge neighbourhood, the cone, or the size parameter by smaller ones while preserving non-empty interior of the cone. This is the standard local propagation-along-CR-curves form of Tumanov's wedge propagation theorem for existing wedge extendibility, proved by analytic discs and the continuity principle. It should be distinguished from the separate finite-type wedge-[extension theorem](/theorems/59): here no finite-type or minimality hypothesis is used, because wedge extendibility is assumed at the starting point and the theorem propagates that already-existing extension along sufficiently short CR curves. Its hypotheses match the present theorem: $M$ is generic of class $C^{2,\alpha}$, $f$ is continuous and CR on the open set $U$, the initial datum at each use is a holomorphic wedge extension with continuous boundary value $f$, and every restricted curve used below is continuous piecewise $C^1$ and tangent to $T^cM$ wherever the derivative exists.[/step]

[guided]The global argument below has only one analytic input, so we state it with the exact hypotheses we need. After fixing an auxiliary Riemannian distance $d_M$ on $M$, Tumanov's local propagation theorem says: if $M\subset\mathbb C^N$ is a generic $C^{2,\alpha}$ CR submanifold, $q\in M$, $U_q\subset M$ is open, and \begin{align*} g:U_q\to\mathbb C \end{align*} is continuous and CR, then there are a neighbourhood $O_q\subset U_q$ of $q$ and a number $\delta_q>0$ such that wedge extendibility of $g$ at any starting point $q'\in O_q$ propagates along every continuous piecewise $C^1$ curve in $O_q$ that starts at $q'$, is tangent to $T^cM$ wherever its derivative exists, and stays within $d_M$-distance $\delta_q$ of $q'$. The conclusion is local: at each endpoint $q''$ it gives an open neighbourhood $V_{q''}\subset\mathbb C^N$, a wedge $W_{q''}\subset V_{q''}$ with edge $M\cap V_{q''}$ at $q''$, and a holomorphic function on $W_{q''}$ whose continuous boundary value agrees with $g$ on $M\cap V_{q''}\cap U_q$. Let us also make the wedge language explicit. A wedge with edge $M$ at $a\in M$ is an open set which, in local coordinates and after choosing a real normal complement to $T_aM$, opens from $M$ in the directions of a non-empty open cone in the normal quotient. This cone is the normal cone. Tumanov's theorem does not require the same cone at all points; when the theorem propagates the extension, the edge neighbourhood, the ambient neighbourhood, and the cone are all allowed to shrink and to vary with the endpoint. We verify the hypotheses in the present setting. The manifold $M$ is generic and has class $C^{2,\alpha}$ by assumption. The function \begin{align*} f:U\to\mathbb C \end{align*} is continuous and CR by assumption. The curve \begin{align*} \gamma:[0,1]\to U \end{align*} is continuous and piecewise $C^1$, and at every point where $\dot\gamma(t)$ exists it lies in $T^c_{\gamma(t)}M$. Finally, the initial hypothesis gives a holomorphic wedge extension at $p_0=\gamma(0)$. Therefore Tumanov's theorem can be applied to every sufficiently short restricted segment of $\gamma$ once we know that the segment remains in a small neighbourhood on which the local theorem is valid.[/guided]

[step:Define the reachable time set] For each $s\in[0,1]$, say that $s$ is reachable if for every $r\in[0,s]$ there exist an open neighbourhood $V_r\subset\mathbb C^N$ of $\gamma(r)$, a wedge $W_r\subset V_r$ with edge $M\cap V_r$ at $\gamma(r)$, and a holomorphic function \begin{align*} F_r:W_r\to\mathbb C \end{align*} whose continuous boundary value on $M\cap V_r\cap U$ is $f$. Define \begin{align*} A:=\{s\in[0,1]:s\text{ is reachable}\}. \end{align*} The initial hypothesis says exactly that $0\in A$, using the given neighbourhood $V_0$, wedge $W_0$, and holomorphic function $F_0$. Hence $A$ is non-empty. Since $A\subset[0,1]$, the number \begin{align*} \sigma:=\sup A \end{align*} is well-defined and satisfies $0\le \sigma\le 1$. [/step]

[step:Show that wedge extendibility holds at the supremum time] Let $(s_j)_{j=1}^{\infty}$ be a sequence in $A$ such that $s_j\to\sigma$ as $j\to\infty$ and $s_j\le \sigma$ for all $j\in\mathbb N$. Such a sequence exists by the definition of supremum. Since $\gamma$ is continuous, $\gamma(s_j)\to\gamma(\sigma)$ in $M$. Apply the local propagation theorem with $q=\gamma(\sigma)$ and $U_q=U$. It gives a relatively open neighbourhood $O_q\subset U$ of $\gamma(\sigma)$ and a number $\delta_q>0$ with a uniform propagation conclusion for all starting points $q'\in O_q$. Choose an index $j_0\in\mathbb N$ so large that $\gamma(s_{j_0})\in O_q$ and \begin{align*} \gamma([s_{j_0},\sigma])\subset O_q. \end{align*} After increasing $j_0$ if necessary, continuity of $\gamma$ also gives \begin{align*} d_M(\gamma(t),\gamma(s_{j_0}))<\delta_q \end{align*} for every $t\in[s_{j_0},\sigma]$. Since $s_{j_0}\in A$, the function $f$ is wedge-extendible at the starting point $\gamma(s_{j_0})$. The restricted curve \begin{align*} \gamma|_{[s_{j_0},\sigma]}:[s_{j_0},\sigma]\to U \end{align*} is continuous and piecewise $C^1$ after restricting the original finite partition to $[s_{j_0},\sigma]$, and it is tangent to $T^cM$ wherever its derivative exists. Applying the local propagation theorem with starting point $q'=\gamma(s_{j_0})$ gives wedge extendibility of $f$ at $\gamma(\sigma)$. It remains to check the full definition of reachability at $\sigma$. Let $r\in[0,\sigma]$. If $r=\sigma$, the preceding paragraph gives the required wedge extension at $\gamma(r)$. If $r<\sigma$, then the definition of $\sigma=\sup A$ gives an element $s\in A$ with $r\le s\le\sigma$. Since $s\in A$, the defining property of reachability for $s$ gives the required wedge extension at every point $\gamma(\tau)$ with $\tau\in[0,s]$, in particular at $\tau=r$. Hence the defining property of reachability holds for every $r\in[0,\sigma]$, and $\sigma\in A$. [/step]

[step:Propagate beyond the supremum unless it is the final time]Assume, for contradiction, that $\sigma<1$. Since $\gamma(\sigma)\in U$ and $U$ is open in $M$, continuity of $\gamma$ gives a number $\varepsilon_0>0$ such that \begin{align*} \gamma([\sigma,\min\{1,\sigma+\varepsilon_0\}])\subset U. \end{align*} By the previous step, $f$ is wedge-extendible at $\gamma(\sigma)$. Apply the local propagation theorem at the starting point $\gamma(\sigma)$ to the short CR curve \begin{align*} \gamma|_{[\sigma,\min\{1,\sigma+\varepsilon\}]}:[\sigma,\min\{1,\sigma+\varepsilon\}]\to U \end{align*} for some sufficiently small number $\varepsilon$ with $0<\varepsilon\le\varepsilon_0$. The theorem gives wedge extendibility of $f$ at every $\gamma(r)$ with \begin{align*} \sigma\le r\le \min\{1,\sigma+\varepsilon\}. \end{align*} Together with $\sigma\in A$, this implies that $\min\{1,\sigma+\varepsilon\}\in A$. Because $\sigma<1$ and $\varepsilon>0$, this contradicts the definition of $\sigma$ as the supremum of $A$.[/step]

[guided]We now explain the global continuation argument in full. The set $A$ records the times up to which wedge extendibility has already been established at every earlier point of the curve. The previous step proved the closed-endpoint property: if $\sigma=\sup A$, then $\sigma\in A$. We prove that this supremum cannot stop before time $1$. Assume $\sigma<1$. Because $\gamma(\sigma)\in U$ and $U$ is open in the relative topology of $M$, continuity of \begin{align*} \gamma:[0,1]\to U \end{align*} gives a number $\varepsilon_0>0$ such that the forward segment remains in $U$: \begin{align*} \gamma([\sigma,\min\{1,\sigma+\varepsilon_0\}])\subset U. \end{align*} Since $\sigma\in A$, the function $f$ has a holomorphic wedge extension at $\gamma(\sigma)$. We may therefore use this extension as the initial wedge extension in Tumanov's local propagation theorem. Choose $\varepsilon$ with $0<\varepsilon\le\varepsilon_0$ small enough that the restricted curve \begin{align*} \gamma|_{[\sigma,\min\{1,\sigma+\varepsilon\}]}:[\sigma,\min\{1,\sigma+\varepsilon\}]\to U \end{align*} lies in the local neighbourhood and stays within the $d_M$-distance threshold supplied by Tumanov's theorem at the starting point $\gamma(\sigma)$. This restricted curve is still continuous and piecewise $C^1$ after intersecting the original finite partition with the restricted interval, and it is tangent to $T^cM$ wherever its derivative exists, because these properties are inherited from $\gamma$. Tumanov's theorem therefore gives wedge extendibility of $f$ at every point $\gamma(r)$ with \begin{align*} \sigma\le r\le \min\{1,\sigma+\varepsilon\}. \end{align*} For times $r\le\sigma$, wedge extendibility is already known because $\sigma\in A$. For times between $\sigma$ and $\min\{1,\sigma+\varepsilon\}$, wedge extendibility is supplied by the local propagation theorem. Hence the defining property of reachability holds up to the later time $\min\{1,\sigma+\varepsilon\}$, so \begin{align*} \min\{1,\sigma+\varepsilon\}\in A. \end{align*} Since $\sigma<1$ and $\varepsilon>0$, this produces an element of $A$ strictly larger than $\sigma$, contradicting the definition of $\sigma$ as $\sup A$. Therefore the only possible value is $\sigma=1$.[/guided]

[step:Conclude wedge extendibility at every point of the CR curve] The contradiction in the preceding step proves that $\sigma=1$. Since $\sigma\in A$, the definition of $A$ gives the following for every $t\in[0,1]$: there exist an open neighbourhood $V_t\subset\mathbb C^N$ of $\gamma(t)$, a wedge $W_t\subset V_t$ with edge $M\cap V_t$ at $\gamma(t)$, and a holomorphic function \begin{align*} F_t:W_t\to\mathbb C \end{align*} whose continuous boundary value on $M\cap V_t\cap U$ is $f$. This is precisely the asserted propagation of wedge extendibility along $\gamma$. [/step]