feat(v0.2): tab title = current folder (from OSC title)

- title-util.ts: folderFromTitle() parses 'user@host:~/path' → last folder
  segment (home stays ~); pure, dependency-free, 7 unit tests
- terminal-session.ts: onTitleChange → folderFromTitle so the tab auto-names
  to the cwd folder and updates live on cd
- Browser-verified: ~ → cd project → 'web-terminal' → cd docs → 'docs' → cd ~ → '~'.
  198 tests green, typecheck + build ok.

public/terminal-session.ts

@@ -11,6 +11,7 @@
 import { Terminal } from '@xterm/xterm'
 import { FitAddon } from '@xterm/addon-fit'
 import type { ClientMessage, ServerMessage } from '../src/types.js'
+import { folderFromTitle } from './title-util.js'
 
 const RESET = '\x1b[0m'
 const BOLD = '\x1b[1m'
@@ -94,7 +95,8 @@ export class TerminalSession {
     this.term.open(this.el)
 
     this.term.onData((data) => this.send(data))
-    this.term.onTitleChange((title) => this.onTitle?.(title))
+    // Tab title = current folder, derived from the shell's OSC title.
+    this.term.onTitleChange((title) => this.onTitle?.(folderFromTitle(title) ?? title))
 
     this.resizeObserver = new ResizeObserver(() => this.scheduleResize())
     this.resizeObserver.observe(this.el)

public/title-util.ts

@@ -0,0 +1,25 @@
+/**
+ * public/title-util.ts — derive a short tab title from a terminal OSC title.
+ *
+ * Pure, dependency-free (no xterm/DOM) so it's unit-testable in node.
+ */
+
+/**
+ * Derive the **current folder** from the terminal's OSC title.
+ *
+ * Most shells emit titles like `user@host:~/path/to/dir` (zsh/bash default).
+ * Take the part after the last ':' (the path), then its last segment (the
+ * folder name). `~` (home) stays `~`. Returns null for an empty title; falls
+ * back to the raw-ish basename when there's no `user@host:` shape (e.g. a
+ * running command title).
+ */
+export function folderFromTitle(title: string): string | null {
+  let s = title.trim()
+  if (!s) return null
+  const colon = s.lastIndexOf(':')
+  if (colon !== -1) s = s.slice(colon + 1).trim()
+  s = s.replace(/\/+$/, '') // strip trailing slash
+  const slash = s.lastIndexOf('/')
+  if (slash !== -1) s = s.slice(slash + 1)
+  return s || null
+}

test/title-util.test.ts

@@ -0,0 +1,33 @@
+import { describe, it, expect } from 'vitest'
+import { folderFromTitle } from '../public/title-util.js'
+
+describe('folderFromTitle', () => {
+  it('extracts the folder from user@host:~/path', () => {
+    expect(folderFromTitle('yiukai@Mac:~/Documents/git/web-terminal')).toBe('web-terminal')
+  })
+
+  it('keeps ~ for the home directory', () => {
+    expect(folderFromTitle('yiukai@Yiukais-MacBook-Pro:~')).toBe('~')
+  })
+
+  it('takes the basename of an absolute path', () => {
+    expect(folderFromTitle('user@host:/Users/yiukai/web')).toBe('web')
+  })
+
+  it('strips a trailing slash', () => {
+    expect(folderFromTitle('user@host:~/web/')).toBe('web')
+  })
+
+  it('handles a title with no user@host: prefix', () => {
+    expect(folderFromTitle('web-terminal')).toBe('web-terminal')
+  })
+
+  it('returns the last path segment when the whole title is a path', () => {
+    expect(folderFromTitle('/var/log/nginx')).toBe('nginx')
+  })
+
+  it('returns null for an empty/whitespace title', () => {
+    expect(folderFromTitle('   ')).toBeNull()
+    expect(folderFromTitle('')).toBeNull()
+  })
+})