feat(tui): verbose turn-phase status line with elapsed timing and cache hit rate

Squashes feat/tui-cache-hit-rate and feat/tui-verbose-status: the footer
shows cache hit rate and the running status line reports turn phase with
elapsed timing.

packages/ui/tui/tests/tui.spec.ts

@@ -818,6 +818,37 @@ describe('pi-tui chat lifecycle and transcript', () => {
     await dispose(result)
   })
 
+  it('shows the session cache hit rate in the footer and updates it live', async () => {
+    // Empty session: no input billed yet, so the cache segment is hidden.
+    // A cwd without "cache" in it keeps the negative assertion unambiguous.
+    const empty = await setup({ cwd: '/opt' })
+    expect(empty.terminal.output).toContain('↑0 ↓0')
+    expect(empty.terminal.output).not.toContain('cache')
+    await dispose(empty)
+
+    const result = await setup({
+      beforeMount(session) {
+        // Cold call: 10 billed input tokens, none served from cache.
+        appendAssistant(session, [{ type: 'text', text: 'cold' }], { inputTokens: 10, outputTokens: 5 })
+      },
+    })
+    expect(result.terminal.output).toContain('cache 0%')
+
+    result.terminal.output = ''
+    // Warm call lands live: 5 uncached + 30 cache-read + 5 cache-write billed
+    // input, so 30 of the 50 total prompt tokens are hits → 60%.
+    appendAssistant(result.session, [{ type: 'text', text: 'warm' }], {
+      inputTokens: 5,
+      outputTokens: 5,
+      cacheReadTokens: 30,
+      cacheWriteTokens: 5,
+    })
+    await tick()
+    expect(result.terminal.output).toContain('cache 60%')
+    expect(result.terminal.output).not.toContain('cache 0%')
+    await dispose(result)
+  })
+
   it('sends, steers, handles commands, global keys, and disposed-agent input', async () => {
     const result = await setup()