feat(web): add privacy-friendly Umami custom events

Instrument the play flow with 9 content-free custom events (game_start,
art_style_select, style_image_upload, scene_reached, choice_select,
vision_click, tts_toggle, fullscreen_toggle, play_heartbeat) to measure
retention, engagement depth and session duration.

Privacy is enforced by construction, not convention:
- lib/analytics.ts types each event with a discriminated union, so a
  payload has no slot for free text — prompts, world guides, uploaded
  images and vision output can never reach analytics (compile-time
  guarantee, not a comment).
- track() no-ops without window.umami and never throws into the app.
- coarse 30s heartbeat fires only while the tab is visible.
- script stays gated on NEXT_PUBLIC_UMAMI_* env (blank → no script),
  honours Do-Not-Track, and locks to an exact data-domains allowlist.
- one-line on-site disclosure with a link, shown only when tracking is on.

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

app/play/page.tsx

@@ -26,6 +26,7 @@ import type {
   StartResponse,
   VisionResponse,
 } from "@infiplot/types";
+import { track } from "@/lib/analytics";
 
 const MUTED_STORAGE_KEY = "infiplot:muted";
 
@@ -373,6 +374,17 @@ function PlayInner() {
     mutedRef.current = muted;
   }, [muted]);
 
+  // Coarse liveness ping for active-time analytics. /play is a single SPA
+  // route, so page views alone read as ~0 duration; a 30s heartbeat (only
+  // while the tab is visible) gives Umami the timestamps to derive real
+  // engaged time. Content-free — no payload, and a no-op without the tracker.
+  useEffect(() => {
+    const id = window.setInterval(() => {
+      if (document.visibilityState === "visible") track("play_heartbeat");
+    }, 30_000);
+    return () => window.clearInterval(id);
+  }, []);
+
   // Whenever currentBeatId changes, append it to visited (skip consecutive dups)
   useEffect(() => {
     if (!currentBeatId) return;
@@ -472,6 +484,7 @@ function PlayInner() {
 
   // ── Mute persistence (read is via the useState lazy initializer above) ─
   const toggleMuted = useCallback(() => {
+    track("tts_toggle", { muted: !mutedRef.current });
     setMuted((prev) => {
       const next = !prev;
       try {
@@ -507,6 +520,7 @@ function PlayInner() {
   // ── Presentation mode toggle ─────────────────────────────────────────
   const togglePresentation = useCallback(async () => {
     const entering = !presentation;
+    track("fullscreen_toggle", { on: entering });
     if (entering) {
       try {
         if (!document.fullscreenElement) {
@@ -671,6 +685,7 @@ function PlayInner() {
         // beatAudioMap is populated lazily by the per-beat fetch effect once
         // currentScene becomes non-null (see fetchBeatAudio).
         setPhase("ready");
+        track("scene_reached", { scene_index: initial.history.length });
       })
       .catch((e) => setError(String(e)));
   }, [params, router]);
@@ -782,6 +797,7 @@ function PlayInner() {
       // beatAudioMap reset + per-beat fetches kicked off by the scene effect.
       setLastExitLabel(exitLabel);
       setPhase("ready");
+      track("scene_reached", { scene_index: newSession.history.length });
     } catch (e) {
       if ((e as { name?: string }).name === "AbortError") {
         setPhase("ready");
@@ -795,6 +811,17 @@ function PlayInner() {
   function onSelectChoice(choice: BeatChoice) {
     if (phase !== "ready" || !session || !currentScene) return;
 
+    const beatNext = currentBeatRef.current?.next;
+    const choiceIndex =
+      beatNext?.type === "choice"
+        ? beatNext.choices.findIndex((c) => c.id === choice.id)
+        : -1;
+    track("choice_select", {
+      scene_index: session.history.length,
+      choice_index: choiceIndex,
+      kind: choice.effect.kind,
+    });
+
     if (choice.effect.kind === "advance-beat") {
       // Pure local jump. No network. No pool changes.
       setCurrentBeatId(choice.effect.targetBeatId);
@@ -863,6 +890,7 @@ function PlayInner() {
         throw new Error(j.error ?? visionRes.statusText);
       }
       const decision = (await visionRes.json()) as VisionResponse;
+      track("vision_click", { result: decision.classify });
 
       if (decision.classify === "insert-beat") {
         setPhase("inserting-beat");