docs: sync CLAUDE.md and READMEs with codex lanes and the tmux run surface

Add the codex-lane facts and doc-sync marker to CLAUDE.md, document the
per-lane tool picker in README/README.vi, and bring server/README.md and
client/README.md's API and routing tables up to date with the tmux-backed
run endpoints and the /workspace route.

client/README.md

@@ -449,11 +449,13 @@ graph TB
     Feed["/activity (redirect)"]
     Workflows["/workflows"]
     CcConfig["/cc-config (redirect)"]
-    Run["/run"]
+    Lanes["/lanes (redirect)"]
+    Run["/run (redirect)"]
+    Workspace["/workspace"]
     Settings["/settings (tabs: dashboard | claude-code | ...)"]
     NF["/* (NotFound)"]
 
-    Root --> Dashboard & Sessions & Detail & Feed & Workflows & CcConfig & Run & Settings & NF
+    Root --> Dashboard & Sessions & Detail & Feed & Workflows & CcConfig & Lanes & Run & Workspace & Settings & NF
 
     style Dashboard fill:#3B82F6
     style Detail fill:#3B82F6
@@ -479,7 +481,10 @@ function App() {
           <Route path="workflows" element={<Workflows />} />
           {/* legacy path - Claude config now lives in a Settings tab */}
           <Route path="cc-config" element={<Navigate to="/settings?tool=claude-code" replace />} />
-          <Route path="run" element={<Run />} />
+          {/* legacy paths - lanes and runs are one page now */}
+          <Route path="lanes" element={<Navigate to="/workspace" replace />} />
+          <Route path="run" element={<Navigate to="/workspace" replace />} />
+          <Route path="workspace" element={<Workspace />} />
           <Route path="settings" element={<SettingsHub />} />
           <Route path="*" element={<NotFound />} />
         </Route>