feat(tui): append git branch to cwd label in status bar

Adds useGitBranch hook (async, cached, 15s TTL) and fmtCwdBranch
helper so the footer shows `~/repo (main)` instead of just `~/repo`.
Degrades silently when git is unavailable or cwd is outside a repo.

Partial fix for #12267 (TUI portion; #12277 covers the Python side).

ui-tui/src/hooks/useGitBranch.ts

@@ -0,0 +1,72 @@
+import { execFile } from 'node:child_process'
+import { promisify } from 'node:util'
+
+import { useEffect, useState } from 'react'
+
+const TTL_MS = 15_000
+const TIMEOUT_MS = 500
+
+const pexec = promisify(execFile)
+const cache = new Map<string, { at: number; branch: null | string }>()
+const inflight = new Map<string, Promise<null | string>>()
+
+const resolveBranch = async (cwd: string): Promise<null | string> => {
+  try {
+    const { stdout } = await pexec('git', ['-C', cwd, 'rev-parse', '--abbrev-ref', 'HEAD'], { timeout: TIMEOUT_MS })
+    const b = stdout.trim()
+
+    return !b || b === 'HEAD' ? null : b
+  } catch {
+    return null
+  }
+}
+
+const fetchBranch = (cwd: string): Promise<null | string> => {
+  const pending = inflight.get(cwd)
+
+  if (pending) {
+    return pending
+  }
+
+  const p = resolveBranch(cwd).finally(() => inflight.delete(cwd))
+  inflight.set(cwd, p)
+
+  return p
+}
+
+export function useGitBranch(cwd: string): null | string {
+  const [branch, setBranch] = useState<null | string>(() => cache.get(cwd)?.branch ?? null)
+
+  useEffect(() => {
+    let cancelled = false
+
+    const tick = async () => {
+      const hit = cache.get(cwd)
+
+      if (hit && Date.now() - hit.at < TTL_MS) {
+        if (!cancelled) {
+          setBranch(hit.branch)
+        }
+
+        return
+      }
+
+      const b = await fetchBranch(cwd)
+      cache.set(cwd, { at: Date.now(), branch: b })
+
+      if (!cancelled) {
+        setBranch(b)
+      }
+    }
+
+    void tick()
+    const id = setInterval(() => void tick(), TTL_MS)
+
+    return () => {
+      cancelled = true
+      clearInterval(id)
+    }
+  }, [cwd])
+
+  return branch
+}