feat(cli,tui): show time since last final agent response on the status bar (#44265)

Adds an idle clock to the context/status bar in both the prompt_toolkit CLI
and the Ink TUI: once a turn completes, a dim '✓ <elapsed>' segment shows how
long the session has been idle since the last final agent response. Hidden
while a turn is live (the per-prompt elapsed timer covers that) and before
the first turn completes.

- cli.py: track _last_turn_finished_at when the agent thread exits, surface
  it via _format_idle_since() in the snapshot, render in both the wide
  fragments path and the plain-text fallback.
- ui-tui: stamp lastTurnEndedAt when busy flips false after a live turn,
  thread it through appStatus -> StatusRule, render via a ticking IdleSince
  segment sharing the duration breakpoint/width budget.

ui-tui/src/app/interfaces.ts

@@ -368,6 +368,7 @@ export interface AppLayoutProgressProps {
 export interface AppLayoutStatusProps {
   cwdLabel: string
   goodVibesTick: number
+  lastTurnEndedAt: null | number
   sessionStartedAt: null | number
   showStickyPrompt: boolean
   statusColor: string

ui-tui/src/app/useMainApp.ts

@@ -173,6 +173,7 @@ export function useMainApp(gw: GatewayClient) {
   const [voiceRecordKey, setVoiceRecordKey] = useState<ParsedVoiceRecordKey>(DEFAULT_VOICE_RECORD_KEY)
   const [sessionStartedAt, setSessionStartedAt] = useState(() => Date.now())
   const [turnStartedAt, setTurnStartedAt] = useState<null | number>(null)
+  const [lastTurnEndedAt, setLastTurnEndedAt] = useState<null | number>(null)
   const [goodVibesTick, setGoodVibesTick] = useState(0)
   const [bellOnComplete, setBellOnComplete] = useState(false)
 
@@ -500,10 +501,14 @@ export function useMainApp(gw: GatewayClient) {
   useEffect(() => {
     if (ui.busy) {
       setTurnStartedAt(prev => prev ?? Date.now())
-    } else {
+    } else if (turnStartedAt != null) {
+      // Only stamp the idle marker when a turn was actually live — busy is
+      // also false on mount and we don't want a phantom "done" timestamp
+      // before the first turn has completed.
+      setLastTurnEndedAt(Date.now())
       setTurnStartedAt(null)
     }
-  }, [ui.busy])
+  }, [ui.busy, turnStartedAt])
 
   useConfigSync({ gw, setBellOnComplete, setVoiceEnabled, setVoiceRecordKey, sid: ui.sid })
 
@@ -1090,6 +1095,7 @@ export function useMainApp(gw: GatewayClient) {
       // essentials and truncates this further on narrow terminals.
       cwdLabel: fmtCwdBranch(cwd, gitBranch, 28),
       goodVibesTick,
+      lastTurnEndedAt: ui.sid ? lastTurnEndedAt : null,
       sessionStartedAt: ui.sid ? sessionStartedAt : null,
       showStickyPrompt: !!stickyPrompt,
       statusColor: statusColorOf(ui.status, ui.theme.color),
@@ -1103,6 +1109,7 @@ export function useMainApp(gw: GatewayClient) {
       cwd,
       gitBranch,
       goodVibesTick,
+      lastTurnEndedAt,
       sessionStartedAt,
       stickyPrompt,
       turnStartedAt,