feat: playable arcade — rooms, identity, client, deployment

Round length is now bounded: the multiplier follows a hyperbolic curve
diverging at 60s, replacing an exponential one where a 275x crash point
produced a two-and-a-half minute round.

Fixes seed reveal, which silently failed every round because pgx cannot
encode a fixed-size byte array as bytea.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

pkg/sim/crash_test.go

@@ -95,3 +95,33 @@ func TestTicksToMultiplierRoundTrips(t *testing.T) {
 		}
 	}
 }
+
+// No round may outlast the ceiling, however extreme the crash point.
+func TestRoundLengthIsBounded(t *testing.T) {
+	if got := MultiplierAt(RoundTicks); got != MaxMultiplier() {
+		t.Fatalf("curve past the ceiling = %v, want %v", got, MaxMultiplier())
+	}
+	// Even the most extreme crash point settles within the ceiling.
+	worst := fixed.FromInt(4_000_000_000)
+	if tick := TicksToMultiplier(worst); tick > RoundTicks {
+		t.Fatalf("extreme crash point needs %d ticks, ceiling is %d", tick, RoundTicks)
+	}
+}
+
+// Timings that matter for how the game feels.
+func TestCurveTimings(t *testing.T) {
+	for _, c := range []struct {
+		multiplier int64
+		maxSeconds float64
+	}{
+		{2, 20},   // the common case should arrive quickly
+		{10, 45},
+		{100, 56},
+	} {
+		tick := TicksToMultiplier(fixed.FromInt(c.multiplier))
+		secs := float64(tick) / TickHz
+		if secs > c.maxSeconds {
+			t.Errorf("%dx takes %.1fs, want under %.0fs", c.multiplier, secs, c.maxSeconds)
+		}
+	}
+}