feat: initial elysium idle game prototype

Sets up the full monorepo with pnpm workspaces. Includes shared types
package, Hono API with Discord OAuth/JWT auth, Prisma v6 + MongoDB
Atlas, and React + Vite frontend with game loop, five tabs, and
Discord-linked save/load.

apps/api/src/services/offlineProgress.ts

@@ -0,0 +1,45 @@
+import type { GameState } from "@elysium/types";
+
+const MAX_OFFLINE_SECONDS = 8 * 60 * 60; // 8 hours
+
+/**
+ * Calculates the gold earned whilst the player was offline.
+ * Capped at 8 hours to prevent exploit via system clock manipulation.
+ */
+export const calculateOfflineGold = (
+  state: GameState,
+  nowMs: number,
+): { offlineGold: number; offlineSeconds: number } => {
+  const elapsedSeconds = Math.min(
+    (nowMs - state.lastTickAt) / 1000,
+    MAX_OFFLINE_SECONDS,
+  );
+
+  const goldPerSecond = state.adventurers.reduce((total, adventurer) => {
+    if (!adventurer.unlocked || adventurer.count === 0) {
+      return total;
+    }
+
+    const upgradeMultiplier = state.upgrades
+      .filter(
+        (u) =>
+          u.purchased &&
+          (u.target === "global" ||
+            (u.target === "adventurer" && u.adventurerId === adventurer.id)),
+      )
+      .reduce((mult, u) => mult * u.multiplier, 1);
+
+    return (
+      total +
+      adventurer.goldPerSecond *
+        adventurer.count *
+        upgradeMultiplier *
+        state.prestige.productionMultiplier
+    );
+  }, 0);
+
+  return {
+    offlineGold: goldPerSecond * elapsedSeconds,
+    offlineSeconds: elapsedSeconds,
+  };
+};