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.ts

@@ -121,3 +121,61 @@ export function planHandoffReturn(opts: {
     shouldPlay: shouldResumeOnForeground(opts.wasPlaying, opts.nativeStateKind),
   };
 }
+
+/**
+ * Which of the two mutually exclusive background behaviours is armed.
+ *
+ * TRACES: UR-040, UR-041 | DR-266 | UT-246
+ *
+ * Backgrounding the app can either shrink the video into a picture-in-picture
+ * window (UR-041) or hand its audio off to the native player and drop the
+ * picture (UR-040). They are alternatives — the first keeps the video on
+ * screen, the second throws it away — so at most one may ever be armed.
+ */
+export interface BackgroundBehaviour {
+  /** The per-player background-audio toggle (UR-040). */
+  backgroundAudioArmed: boolean;
+  /** Whether leaving the app auto-enters PiP (UR-041). */
+  autoPipEnabled: boolean;
+}
+
+/** Arming/disarming the background-audio toggle flips auto-PiP the other way. */
+export function setBackgroundAudioArmed(armed: boolean): BackgroundBehaviour {
+  return { backgroundAudioArmed: armed, autoPipEnabled: !armed };
+}
+
+/**
+ * The user has asked for a PiP window, by pressing the button rather than by
+ * leaving the app.
+ *
+ * Exclusivity used to be enforced from one side only — arming the toggle
+ * suppressed auto-PiP — while the PiP button stayed live and ungated. Pressing
+ * it left both behaviours armed, and the video was then one stray background
+ * signal away from being handed off to audio-only while the user was watching
+ * it in the window. Pressing PiP is an unambiguous request to keep the picture,
+ * so it disarms the behaviour that throws the picture away.
+ */
+export function enteringPictureInPicture(_current: BackgroundBehaviour): BackgroundBehaviour {
+  return setBackgroundAudioArmed(false);
+}
+
+/**
+ * Whether the app is in a picture-in-picture window, for the purpose of
+ * deciding what backgrounding means.
+ *
+ * TRACES: UR-040, UR-041 | DR-266 | UT-246
+ *
+ * @param nativeFlag the Activity's `isInPictureInPictureMode`, sampled inside
+ *   `onStop()`
+ * @param sawPipEntered whether the frontend has seen `jellytau-pip-entered`
+ *   without a matching `jellytau-pip-exited`
+ */
+export function inPictureInPicture(nativeFlag: boolean, sawPipEntered: boolean): boolean {
+  // Either witness is enough. The native flag is a single sample taken inside
+  // onStop(); the frontend's is a latch, set by `jellytau-pip-entered` and
+  // cleared by `jellytau-pip-exited`. Both events reach the WebView through the
+  // same message queue in dispatch order, so a genuine exit is always known
+  // before the background signal that follows it — the latch can report a
+  // window that is still open, never one that has closed.
+  return nativeFlag || sawPipEntered;
+}