feat(gate): foreground caption fetches take priority over the background sweep (ADR-026)

A user waiting on a Summarize click shared the per-IP caption gate equally with
the background firehose, so on a busy IP the click was slow or 429'd. Add a
context-marked priority lane: the web path (engineProcessor.ProcessVideo) marks
its context foreground; the gate serves foreground immediately while background
fetches yield until no foreground is pending. Threaded via a context value (no
new signatures) + a process-wide foregroundPending counter. Clicks are rare, so
the background barely loses throughput; the waiting human gets the cleaner slot.

Drops the credentials probe: ADR-010 and captions.go already settle it — the
timedtext/InnerTube path rejects authenticated requests and the OAuth token does
not authenticate it anyway, so auth cannot help and can hurt. Documented in
ADR-026 rather than built.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

cmd/tapir/processor.go

@@ -113,6 +113,11 @@ type engineProcessor struct {
 }
 
 func (p *engineProcessor) ProcessVideo(ctx context.Context, userID, videoID string) error {
+	// This is the user-initiated (foreground) path — a click on "Summarize",
+	// "Try now", or a pasted URL. Mark the context so the caption gate gives it
+	// priority over the background sweep (ADR-026, Pillar A).
+	ctx = youtube.ForegroundContext(ctx)
+
 	row, err := p.store.GetVideoRow(ctx, userID, videoID)
 	if err != nil {
 		return fmt.Errorf("load video %q: %w", videoID, err)