feat(ui): finish the workspace-sessions plan — tool filter, session search, explicit Resume

- Sidebar: a tool filter (shown once workspaces of two tools exist, kept in
  agenthub.sidebar.projects → tool; reordering under it keeps the hidden
  workspaces' places), and search that also finds a workspace by one of its
  session names, showing only the matching sessions.
- Workspace home: the repository it belongs to (root, main branch, worktree
  count) under the title, and each recent's cost.
- Transcript page: an explicit Resume for a Claude session started in a
  workspace's own folder (SessionResumeButton). It warns that a resume re-reads
  the whole transcript, re-reads the live panes, then starts in the first free
  pane and opens it; a live session offers Open pane instead.

client/src/components/lanes/WorkspaceHome.tsx

@@ -22,8 +22,8 @@ import {
   type PermissionMode,
   type RunHandle,
 } from "../../lib/api";
-import type { Lane, Session } from "../../lib/types";
-import { timeAgo } from "../../lib/format";
+import type { Lane, Project, Session } from "../../lib/types";
+import { fmtCost, timeAgo } from "../../lib/format";
 import { useAiTools } from "../../hooks/useAiTools";
 import { useProjectTreePrefs } from "../../lib/projectTreePrefs";
 import { ModelPicker, PromptEditor } from "../run/RunSetup";
@@ -137,6 +137,22 @@ function HomeBody({
     };
   }, [lane.id]);
 
+  // The repository this workspace belongs to (Settings → Projects used to
+  // show this): read once per visit, from the server's cached project list.
+  const [project, setProject] = useState<Project | null>(null);
+  useEffect(() => {
+    let live = true;
+    api.projects
+      .list()
+      .then(({ projects }) => {
+        if (live) setProject(projects.find((p) => p.workspaces.includes(lane.id)) ?? null);
+      })
+      .catch(() => undefined);
+    return () => {
+      live = false;
+    };
+  }, [lane.id]);
+
   // Suggestions are read when the folder form opens, not on every visit.
   useEffect(() => {
     if (!folderOpen) return;
@@ -268,6 +284,20 @@ function HomeBody({
             <MoreHorizontal className="h-5 w-5" />
           </button>
         </div>
+        {project?.root && (
+          <p
+            data-testid="workspace-home-facts"
+            className="mt-1 flex flex-wrap gap-x-3 font-mono text-[11px] text-fg-muted"
+          >
+            <span title={t("home.repoRoot")}>{project.root}</span>
+            {project.worktrees.find((w) => w.main)?.branch && (
+              <span title={t("home.mainBranch")}>
+                {project.worktrees.find((w) => w.main)?.branch}
+              </span>
+            )}
+            <span>{t("home.worktreeCount", { count: project.worktrees.length })}</span>
+          </p>
+        )}
 
         <div className="card mt-5 p-3">
           <PromptEditor
@@ -340,6 +370,11 @@ function HomeBody({
                   <span className="min-w-0 flex-1 truncate text-sm text-fg-primary">
                     {s.name || s.id.slice(0, 8)}
                   </span>
+                  {!!s.cost && s.cost > 0 && (
+                    <span className="shrink-0 text-xs tabular-nums text-fg-muted">
+                      {fmtCost(s.cost)}
+                    </span>
+                  )}
                   <span className="shrink-0 text-xs text-fg-muted">
                     {timeAgo(s.last_activity ?? s.started_at)}
                   </span>