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.go

@@ -12,10 +12,17 @@ const HouseEdgeBP int64 = 200
 // TickHz is the simulation rate. Rounds advance in whole ticks only.
 const TickHz = 60
 
-// growthPerTickBP is multiplier growth per tick, in basis points of the current
-// value. At 6bp and 60Hz the multiplier reaches 2x in roughly 19 seconds, which
-// is long enough to feel the climb and short enough to keep rounds moving.
-const growthPerTickBP int64 = 6
+// RoundTicks is the hard ceiling on a round's length: 60 seconds at 60Hz.
+//
+// The multiplier follows a hyperbolic curve that diverges at exactly this
+// tick, so no round can run longer no matter how extreme the crash point.
+// An exponential curve has no such bound — a 275x round on one takes over two
+// and a half minutes, which is unplayable when a dozen people are waiting.
+const RoundTicks = 60 * TickHz
+
+// MaxMultiplier is the largest value the curve expresses, reached on the final
+// tick. Crash points at or above it settle when the round hits its ceiling.
+func MaxMultiplier() fixed.F { return MultiplierAt(RoundTicks - 1) }
 
 // CrashPoint derives the multiplier at which a round ends, as a pure function of
 // the seed.
@@ -48,31 +55,49 @@ func CrashPoint(seed [32]byte) fixed.F {
 	return cp
 }
 
-// step is the per-tick growth factor, 1 + growthPerTickBP/10000, in Q32.32.
-func step() fixed.F {
-	return fixed.One + fixed.F(growthPerTickBP<<32/10000)
-}
-
-// MultiplierAt returns the multiplier displayed at a given tick of the round,
-// compounding from 1.0.
+// MultiplierAt returns the multiplier displayed at a given tick.
+//
+//	m(t) = 1 / (1 - t/T)^2
+//
+// It starts at 1.0, rises slowly at first, and accelerates without bound as t
+// approaches T. That acceleration is the tension: the longer you hold, the
+// faster the number moves away from you, and the less time you have to react.
+// It is also O(1), so a long round costs no more per tick than a short one.
 func MultiplierAt(tick int) fixed.F {
-	m := fixed.One
-	s := step()
-	for i := 0; i < tick; i++ {
-		m = m.Mul(s)
+	if tick <= 0 {
+		return fixed.One
 	}
-	return m
+	// Clamp the tick, not the value: clamping the value would make the curve
+	// step backwards at the boundary if rounding put the last computed point
+	// above the nominal ceiling.
+	if tick >= RoundTicks {
+		tick = RoundTicks - 1
+	}
+	// remaining = 1 - tick/T, always in (0, 1].
+	remaining := fixed.One - fixed.FromInt(int64(tick)).Div(fixed.FromInt(RoundTicks))
+	return fixed.One.Div(remaining.Mul(remaining))
 }
 
-// TicksToMultiplier returns the first tick at which MultiplierAt reaches m.
+// TicksToMultiplier returns the first tick at which MultiplierAt reaches m,
+// inverting the curve: t = T * (1 - 1/sqrt(m)).
 func TicksToMultiplier(m fixed.F) int {
-	cur := fixed.One
-	s := step()
-	for tick := 0; tick < 1_000_000; tick++ {
-		if cur >= m {
-			return tick
-		}
-		cur = cur.Mul(s)
+	if m <= fixed.One {
+		return 0
 	}
-	return 1_000_000
+	if m >= MaxMultiplier() {
+		return RoundTicks
+	}
+	inv := fixed.One.Div(fixed.Sqrt(m))
+	t := fixed.FromInt(RoundTicks).Mul(fixed.One - inv).Int()
+
+	// Rounding in fixed point can land a tick early; step forward to the first
+	// tick that genuinely reaches the target.
+	tick := int(t)
+	for tick > 0 && MultiplierAt(tick-1) >= m {
+		tick--
+	}
+	for tick < RoundTicks && MultiplierAt(tick) < m {
+		tick++
+	}
+	return tick
 }