feat: show per-stage timing (듣기/LLM/TTS) in the transcript channel

The transcript channel only showed STT and LLM seconds. Add wall-clock
start/end times and durations for listening, LLM and TTS so it's obvious
what takes long; STT surfaces as the gap between listening end and LLM start.

- bridge: emit llm_start_ms/llm_end_ms on meta and tts_*_ms on the end event
- bot: capture the listening window, assemble full timing after the stream,
  and render a per-stage breakdown in the transcript message

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

bot/src/userbot.test.ts

@@ -0,0 +1,70 @@
+import { test, expect } from "bun:test";
+
+// userbot.ts imports the runtime `config`, which requires DISCORD_GUILD_ID.
+process.env.DISCORD_GUILD_ID ||= "test-guild";
+const { formatTurnMessage } = await import("./userbot.ts");
+
+test("formatTurnMessage shows a per-stage timing breakdown with durations", () => {
+  const msg = formatTurnMessage({
+    transcript: "안녕하세요",
+    reply: "네, 안녕하세요",
+    listenStartMs: 10_000,
+    listenEndMs: 12_000, // 듣기 2.0s
+    llmStartMs: 12_500, // STT/전송 gap 0.5s
+    llmEndMs: 14_100, // LLM 1.6s
+    ttsStartMs: 14_100,
+    ttsEndMs: 15_000, // TTS 0.9s
+  });
+
+  expect(msg).toContain('🗣️ "안녕하세요"');
+  expect(msg).toContain("🤖 답변: 네, 안녕하세요");
+  expect(msg).toContain("👂 듣기 2.0초");
+  expect(msg).toContain("🧠 LLM 1.6초");
+  expect(msg).toContain("(STT/전송 0.5초)");
+  expect(msg).toContain("🔊 TTS 0.9초");
+  // Total spans listening start -> TTS end.
+  expect(msg).toContain("합계 5.0초");
+});
+
+test("formatTurnMessage omits the STT gap note when it is negligible", () => {
+  const msg = formatTurnMessage({
+    transcript: "테스트",
+    reply: "응답",
+    listenStartMs: 0,
+    listenEndMs: 1000,
+    llmStartMs: 1000, // no gap
+    llmEndMs: 2000,
+  });
+  expect(msg).not.toContain("STT/전송");
+  expect(msg).toContain("🧠 LLM 1.0초");
+});
+
+test("formatTurnMessage reports a dropped turn with only the listening stage", () => {
+  const msg = formatTurnMessage({
+    transcript: "",
+    reply: "",
+    note: "너무 짧음(<300ms)",
+    listenStartMs: 0,
+    listenEndMs: 200,
+  });
+  expect(msg).toContain("❌ 너무 짧음(<300ms)");
+  expect(msg).toContain("👂 듣기 0.2초");
+  expect(msg).not.toContain("🧠 LLM");
+  expect(msg).not.toContain("🔊 TTS");
+});
+
+test("formatTurnMessage falls back to the plain line when no timing is present", () => {
+  const msg = formatTurnMessage({ transcript: "안녕", reply: "하이" });
+  expect(msg).toBe('🎤 들음 → 🗣️ "안녕"\n🤖 답변: 하이');
+  expect(msg).not.toContain("⏱️");
+});
+
+test("formatTurnMessage drops out-of-order (negative) spans instead of showing junk", () => {
+  const msg = formatTurnMessage({
+    transcript: "안녕",
+    reply: "하이",
+    listenStartMs: 5000,
+    listenEndMs: 4000, // negative span -> omitted
+  });
+  expect(msg).not.toContain("👂 듣기");
+});