feat(finale): server-stamped phase timer helper

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

src/lib/live-timer.ts

@@ -0,0 +1,36 @@
+/**
+ * Server-stamped phase timer math for the grand-finale ceremony.
+ *
+ * The cursor stores `phaseStartedAt` + `phaseDurationSeconds` plus a pause
+ * accumulator; every client derives the countdown locally from those stamps,
+ * so all screens agree and overtime is just a negative remainder.
+ */
+
+export type PhaseTimerState = {
+  phaseStartedAt: Date | string | null
+  phaseDurationSeconds: number | null
+  phasePausedAt: Date | string | null
+  phasePausedAccumMs: number
+}
+
+export function elapsedMs(t: PhaseTimerState, now: Date = new Date()): number {
+  if (!t.phaseStartedAt) return 0
+  const start = new Date(t.phaseStartedAt).getTime()
+  const end = t.phasePausedAt ? new Date(t.phasePausedAt).getTime() : now.getTime()
+  return Math.max(0, end - start - t.phasePausedAccumMs)
+}
+
+/** Seconds left on the phase timer; negative = overtime; null = no timer running. */
+export function remainingSeconds(t: PhaseTimerState, now: Date = new Date()): number | null {
+  if (!t.phaseStartedAt || t.phaseDurationSeconds == null) return null
+  return t.phaseDurationSeconds - Math.floor(elapsedMs(t, now) / 1000)
+}
+
+/** `5:05` for positive seconds, `+1:23` for overtime. */
+export function formatClock(seconds: number): string {
+  const over = seconds < 0
+  const abs = Math.abs(seconds)
+  const m = Math.floor(abs / 60)
+  const s = abs % 60
+  return `${over ? '+' : ''}${m}:${s.toString().padStart(2, '0')}`
+}