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/components/appChrome.tsx

@@ -341,6 +341,21 @@ function SessionDuration({ startedAt }: { startedAt: number }) {
   return fmtDuration(now - startedAt)
 }
 
+function IdleSince({ endedAt }: { endedAt: number }) {
+  // Time since the last final agent response. Re-ticks every second like
+  // SessionDuration so the read-out stays live while the session idles.
+  const [now, setNow] = useState(() => Date.now())
+
+  useEffect(() => {
+    setNow(Date.now())
+    const id = setInterval(() => setNow(Date.now()), 1000)
+
+    return () => clearInterval(id)
+  }, [endedAt])
+
+  return `✓ ${fmtDuration(now - endedAt)}`
+}
+
 const effortLabel = (effort?: string) => {
   const value = String(effort ?? '')
     .trim()
@@ -400,6 +415,7 @@ export function StatusRule({
   notice,
   usage,
   bgCount,
+  lastTurnEndedAt,
   liveSessionCount,
   sessionStartedAt,
   showCost,
@@ -488,6 +504,10 @@ export function StatusRule({
 
   const showBar = !!bar && fits(SEP + stringWidth(`[${bar}] ${pct != null ? `${pct}%` : ''}`))
   const showDuration = segs.duration && !!sessionStartedAt && fits(SEP + MAX_DURATION_WIDTH)
+  // Idle clock — time since the last final agent response. Hidden while busy
+  // (the FaceTicker's elapsed tail covers the live turn) and before the first
+  // turn completes. Shares the duration breakpoint and width reservation.
+  const showIdle = segs.duration && !busy && lastTurnEndedAt != null && fits(SEP + stringWidth('✓ ') + MAX_DURATION_WIDTH)
   const showCompressions = segs.compressions && compressions > 0 && fits(SEP + stringWidth(`cmp ${compressions}`))
   const showVoice = segs.voice && !!voiceLabel && fits(SEP + stringWidth(voiceLabel))
   const showSessionCount = !!sessionCountText && fits(SEP + stringWidth(sessionCountText))
@@ -567,6 +587,12 @@ export function StatusRule({
             <SessionDuration startedAt={sessionStartedAt!} />
           </Text>
         ) : null}
+        {showIdle ? (
+          <Text color={t.color.muted} wrap="truncate-end">
+            {' │ '}
+            <IdleSince endedAt={lastTurnEndedAt!} />
+          </Text>
+        ) : null}
         {showCompressions ? (
           <Text color={t.color.muted} wrap="truncate-end">
             {' │ '}
@@ -725,6 +751,7 @@ export function TranscriptScrollbar({ scrollRef, t }: TranscriptScrollbarProps)
 
 interface StatusRuleProps {
   bgCount: number
+  lastTurnEndedAt?: null | number
   liveSessionCount: number
   busy: boolean
   cols: number

ui-tui/src/components/appLayout.tsx

@@ -366,6 +366,7 @@ const StatusRulePane = memo(function StatusRulePane({
         cols={composer.cols}
         cwdLabel={status.cwdLabel}
         indicatorStyle={ui.indicatorStyle}
+        lastTurnEndedAt={status.lastTurnEndedAt}
         liveSessionCount={ui.liveSessionCount}
         model={ui.info?.model ?? ''}
         modelFast={ui.info?.fast || ui.info?.service_tier === 'priority'}