fix(player): stop PiP dropping the video and restarting the audio behind it

Watching in a picture-in-picture window would occasionally drop to audio-only,
and the audio would resume from wherever the video had been when PiP was
entered while the picture had carried on past it. Two independent faults, both
needed to produce that.

The position froze (DR-265). VideoPlayer tracks the absolute position in its
own `currentTime` rather than reading `videoElement.currentTime` at the point
of use, because transcoded HLS resets the element to 0 on every segment
rebuild. While playing, that variable had exactly one writer: a
requestAnimationFrame loop. RAF is driven by the document being rendered, and
an Android activity behind a PiP window is paused, so the loop stops while the
element plays on. The `timeupdate` handler that would have covered the gap was
written as a fallback "for when RAF isn't running" and gated itself on
`!isPlaying` -- switching itself off at precisely the moment it was the only
source left. Everything downstream froze with it: the seek bar, the ten-second
progress reports, the position mirrored into Rust, and the handoff. The gate is
now `shouldApplyTimeUpdate` and turns only on things that genuinely own the
position -- an in-flight seek, a seek-bar drag, an element below
HAVE_CURRENT_DATA. Both writers producing the same derived value costs nothing.

The handoff fired at all (DR-266). PiP and the background-audio handoff are
alternatives -- one keeps the picture, the other throws it away -- but
exclusivity was enforced from one side only: arming the toggle suppressed
*auto*-PiP, while the PiP button stayed ungated, so pressing it left both
armed. What then stood between them was `isInPictureInPictureMode`, sampled
once inside MainActivity.onStop(). That sample is not reliable: the keyguard
dismissing the window, the window being stashed, or OEM variance in when
onPictureInPictureModeChanged(false) lands can all leave the activity stopped
with a window still on screen and the flag reading false. Now entering PiP
disarms background audio, both directions go through one BackgroundBehaviour
pair, and the PiP question accepts either witness -- the native sample or the
frontend's latch over jellytau-pip-entered/exited. The latch cannot report a
window that has closed: both events reach the WebView through the same message
queue in dispatch order. The decision itself stays in Rust; the frontend only
supplies a fact it can establish more reliably than the activity can.

Red first, both: the existing behaviour was extracted into pure helpers, the
tests written against the correct behaviour, and both watched to fail before
either was changed.

src/lib/components/player/backgroundAudioHandoff.test.ts

@@ -7,6 +7,9 @@ import {
   shouldExitBackgroundAudio,
   shouldResumeOnForeground,
   type BackgroundAudioState,
+  setBackgroundAudioArmed,
+  enteringPictureInPicture,
+  inPictureInPicture,
 } from "./backgroundAudioHandoff";
 
 // TRACES: UR-040 | DR-052 | UT-060
@@ -137,3 +140,69 @@ describe("backgroundAudioHandoff", () => {
     });
   });
 });
+
+/**
+ * TRACES: UT-246 | DR-266
+ */
+describe("background behaviour exclusivity", () => {
+  describe("setBackgroundAudioArmed", () => {
+    it("disables auto-PiP when background audio is armed", () => {
+      expect(setBackgroundAudioArmed(true)).toEqual({
+        backgroundAudioArmed: true,
+        autoPipEnabled: false,
+      });
+    });
+
+    it("restores auto-PiP when background audio is disarmed", () => {
+      expect(setBackgroundAudioArmed(false)).toEqual({
+        backgroundAudioArmed: false,
+        autoPipEnabled: true,
+      });
+    });
+  });
+
+  describe("enteringPictureInPicture", () => {
+    it("disarms background audio when the user opens a PiP window", () => {
+      // THE REPORTED BUG, half one. Exclusivity was enforced in one direction
+      // only: arming the toggle suppressed auto-PiP, but the PiP *button* was
+      // still offered and still worked, leaving both behaviours live at once.
+      // A single stray background signal then handed a video the user was
+      // watching in a PiP window off to audio-only.
+      expect(
+        enteringPictureInPicture({ backgroundAudioArmed: true, autoPipEnabled: false }),
+      ).toEqual({ backgroundAudioArmed: false, autoPipEnabled: true });
+    });
+
+    it("leaves an already-exclusive state alone", () => {
+      const state = { backgroundAudioArmed: false, autoPipEnabled: true };
+      expect(enteringPictureInPicture(state)).toEqual(state);
+    });
+  });
+
+  describe("inPictureInPicture", () => {
+    it("trusts the native flag when the two agree", () => {
+      expect(inPictureInPicture(true, true)).toBe(true);
+      expect(inPictureInPicture(false, false)).toBe(false);
+    });
+
+    it("treats a live PiP window as PiP even when the native flag says otherwise", () => {
+      // THE REPORTED BUG, half two. `isInPictureInPictureMode` is sampled once,
+      // inside onStop(). There are orderings -- the keyguard dismissing the
+      // window, the window being stashed, OEM variance in whether
+      // onPictureInPictureModeChanged(false) lands first -- where the activity
+      // is stopped with a PiP window still on screen and that single boolean
+      // reads false. Backgrounding then means "the app is gone" and the video
+      // the user is watching is handed off to audio.
+      expect(inPictureInPicture(false, true)).toBe(true);
+    });
+
+    it("does not resurrect a window the frontend has already seen close", () => {
+      // jellytau-pip-exited and jellytau-background are both posted to the same
+      // WebView message queue, in that order, so a genuine exit is always known
+      // by the time the background signal is handled. Leaving playback running
+      // here would be the opposite defect: audio continuing after the user
+      // closed the window and left the app.
+      expect(inPictureInPicture(false, false)).toBe(false);
+    });
+  });
+});