feat(frontend): live room view, beat connectivity test, session resume

- PodView now shows LIVE LiveKit participants (updates as people join/leave,
  active-speaker highlight) instead of the static roster — no refresh needed.
- "Play beat" button publishes a generated 4/4 beat into the room so every
  participant hears the same audio (speaker lights up) — a real connectivity test.
- "Share my screen" is now a deliberate button; joining only connects to the
  room (fast/reliable), so a denied/slow screen prompt no longer fails the join.
- Session persisted in sessionStorage with auto-reconnect, so a refresh keeps
  you in the room instead of dropping back to the pod list.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

frontend/src/lib/beat.ts

@@ -0,0 +1,77 @@
+export interface BeatHandle {
+  track: MediaStreamTrack;
+  stop: () => void;
+}
+
+/**
+ * Generate a simple 4-on-the-floor beat (kick + hi-hat) as an audio
+ * MediaStreamTrack to publish into a LiveKit room. Also routes to the local
+ * speakers so the publisher hears it too. Pure Web Audio — no asset/CORS.
+ */
+export function startBeat(): BeatHandle {
+  const ctx = new AudioContext();
+  void ctx.resume();
+  const dest = ctx.createMediaStreamDestination();
+  const master = ctx.createGain();
+  master.gain.value = 0.5;
+  master.connect(dest); // -> published track (remote listeners)
+  master.connect(ctx.destination); // -> local speakers (publisher)
+
+  const bpm = 120;
+  const spb = 60 / bpm;
+  let next = ctx.currentTime + 0.1;
+  let beat = 0;
+
+  function kick(time: number, accent: boolean) {
+    const osc = ctx.createOscillator();
+    const g = ctx.createGain();
+    osc.frequency.setValueAtTime(accent ? 180 : 150, time);
+    osc.frequency.exponentialRampToValueAtTime(50, time + 0.12);
+    g.gain.setValueAtTime(0.0001, time);
+    g.gain.exponentialRampToValueAtTime(accent ? 1 : 0.7, time + 0.005);
+    g.gain.exponentialRampToValueAtTime(0.0001, time + 0.18);
+    osc.connect(g);
+    g.connect(master);
+    osc.start(time);
+    osc.stop(time + 0.2);
+  }
+
+  function hat(time: number) {
+    const size = Math.floor(ctx.sampleRate * 0.05);
+    const buffer = ctx.createBuffer(1, size, ctx.sampleRate);
+    const data = buffer.getChannelData(0);
+    for (let i = 0; i < size; i++) data[i] = Math.random() * 2 - 1;
+    const noise = ctx.createBufferSource();
+    noise.buffer = buffer;
+    const hp = ctx.createBiquadFilter();
+    hp.type = 'highpass';
+    hp.frequency.value = 7000;
+    const g = ctx.createGain();
+    g.gain.setValueAtTime(0.12, time);
+    g.gain.exponentialRampToValueAtTime(0.0001, time + 0.05);
+    noise.connect(hp);
+    hp.connect(g);
+    g.connect(master);
+    noise.start(time);
+    noise.stop(time + 0.05);
+  }
+
+  const timer = window.setInterval(() => {
+    while (next < ctx.currentTime + 0.2) {
+      kick(next, beat % 4 === 0);
+      hat(next + spb / 2);
+      next += spb;
+      beat++;
+    }
+  }, 50);
+
+  const track = dest.stream.getAudioTracks()[0]!;
+  return {
+    track,
+    stop: () => {
+      window.clearInterval(timer);
+      track.stop();
+      void ctx.close();
+    },
+  };
+}