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/__tests__/paths.test.ts

@@ -0,0 +1,70 @@
+import { afterEach, beforeEach, describe, expect, it } from 'vitest'
+
+import { fmtCwdBranch, shortCwd } from '../domain/paths.js'
+
+describe('shortCwd', () => {
+  const origHome = process.env.HOME
+
+  beforeEach(() => {
+    process.env.HOME = '/Users/bb'
+  })
+
+  afterEach(() => {
+    process.env.HOME = origHome
+  })
+
+  it('collapses HOME to ~', () => {
+    expect(shortCwd('/Users/bb/proj/repo')).toBe('~/proj/repo')
+  })
+
+  it('leaves non-HOME paths alone', () => {
+    expect(shortCwd('/tmp/work')).toBe('/tmp/work')
+  })
+
+  it('truncates long paths from the left with ellipsis', () => {
+    const out = shortCwd('/var/long/deeply/nested/workspace/here', 10)
+    expect(out.startsWith('…')).toBe(true)
+    expect(out.length).toBe(10)
+    expect('/var/long/deeply/nested/workspace/here'.endsWith(out.slice(1))).toBe(true)
+  })
+
+  it('keeps paths shorter than max intact', () => {
+    expect(shortCwd('/a/b', 10)).toBe('/a/b')
+  })
+})
+
+describe('fmtCwdBranch', () => {
+  const origHome = process.env.HOME
+
+  beforeEach(() => {
+    process.env.HOME = '/Users/bb'
+  })
+
+  afterEach(() => {
+    process.env.HOME = origHome
+  })
+
+  it('returns bare cwd when branch is null', () => {
+    expect(fmtCwdBranch('/Users/bb/proj', null)).toBe('~/proj')
+  })
+
+  it('returns bare cwd when branch is empty', () => {
+    expect(fmtCwdBranch('/Users/bb/proj', '')).toBe('~/proj')
+  })
+
+  it('appends branch in parens', () => {
+    expect(fmtCwdBranch('/Users/bb/proj', 'main')).toBe('~/proj (main)')
+  })
+
+  it('truncates the path to keep the branch tag readable', () => {
+    const out = fmtCwdBranch('/Users/bb/very/deeply/nested/project/folder', 'feature-branch', 30)
+    expect(out).toMatch(/ \(feature-branch\)$/)
+    expect(out.length).toBeLessThanOrEqual(30)
+  })
+
+  it('truncates very long branch names from the right', () => {
+    const out = fmtCwdBranch('/Users/bb/p', 'a-very-long-feature-branch-name')
+    expect(out).toMatch(/^~\/p \(…/)
+    expect(out).toContain(')')
+  })
+})