Background agents → sidebar section (owner): the retired floating dock's successor

The old activity dock (fixed bottom-right, every view) was a cross-session
widget parked inside the session view — the same boundary confusion that
got the plan window moved. The sidebar is the app's cross-session surface,
so the agents section lives at its bottom strip now (user-locked design):

- ALWAYS present: empty means empty — the header ('Background agents',
  matching the Sessions label language) sits alone; nothing hides and
  nothing decorates (the total-cost span only renders when populated).
- Populates and grows when processes run: same blobs (sub-agents +
  extraction runs, one poll, inferred cost, elapsed heartbeat), capped by
  max-h-[40dvh] + internal scroll so a pile never crowds the session list.
- Per-blob END button (unchanged abort contract); the detail popover
  becomes an INLINE expansion — the section scrolls, a floating popover
  would clip at its bounds.
- Sidebar-native skin: border-t separator matching the sidebar's edge,
  session-row hover language, no floating-card chrome.
- Sidebar closed = the learner opted out; no ambient signal for now
  (user decision — invisibility until a clearer observability story).
- Narrow-mode count badge/drawer machinery retired with the dock.

app/layout.tsx: ActivityDock mount removed, ActivityProvider stays (the
sidebar consumes the context). Live-verified: populated rows with inline
detail + session link, abort makes the row vanish, idle renders the
header alone, closed sidebar hides the section with nothing floating.
Gates: typecheck, lint, 423/423 tests.

components/sidebar.tsx

@@ -4,6 +4,7 @@ import { createContext, useCallback, useContext, useEffect, useMemo, useState }
 import Link from "next/link";
 import { usePathname, useRouter } from "next/navigation";
 import { PanelLeftCloseIcon, PanelLeftOpenIcon, PlusIcon, SearchIcon } from "lucide-react";
+import { ActivitySection } from "@/components/activity-dock";
 import { Button } from "@/components/ui/button";
 import { Input } from "@/components/ui/input";
 import { cn } from "@/lib/utils";
@@ -285,8 +286,15 @@ function SessionsSidebar() {
 						))
 					)}
 				</div>
-			</div>
-		</aside>
+
+					{/* Background agents (owner, sidebar round): the section is
+					    ALWAYS here — empty means empty (the header sits alone); it
+					    populates and grows when processes run. The session list is
+					    flex-1, so this pins to the bottom. Sidebar closed = the
+					    learner opted out (no ambient signal, user decision). */}
+					<ActivitySection />
+				</div>
+			</aside>
 	);
 }