fix: rewrite prestige/transcendence formula and rebalance progression

apps/api/src/services/prestige.ts

@@ -15,7 +15,6 @@ import type {
 } from "@elysium/types";
 
 const basePrestigeGoldThreshold = 1_000_000;
-const thresholdScaleFactor = 5;
 const runestonesPerPrestigeLevel = 10;
 const milestoneInterval = 5;
 const milestoneRunestonesPerInterval = 25;
@@ -29,7 +28,8 @@ const maxBaseRunestones = 200;
 
 /**
  * Calculates the gold threshold required for the next prestige.
- * Formula: BASE * SCALE_FACTOR^prestigeCount — each prestige makes the next threshold harder.
+ * Formula: BASE * (count + 1)^2 — polynomial growth that peaks around prestige 8–10
+ * then gets easier as the production multiplier overtakes it.
  * @param prestigeCount - The current number of prestiges completed.
  * @param thresholdMultiplier - An optional echo-upgrade multiplier applied to the threshold.
  * @returns The gold amount required to prestige.
@@ -40,7 +40,7 @@ const calculatePrestigeThreshold = (
 ): number => {
   return (
     basePrestigeGoldThreshold
-    * Math.pow(thresholdScaleFactor, prestigeCount)
+    * Math.pow(prestigeCount + 1, 2)
     * thresholdMultiplier
   );
 };
@@ -146,14 +146,15 @@ const calculateRunestones = (parameters: RunestoneParameters): number => {
 
 /**
  * Calculates the new prestige production multiplier.
- * Formula: 1.15^prestigeCount — exponential scaling per prestige.
+ * Formula: 1.25^prestigeCount — exponential scaling per prestige that eventually
+ * overtakes the polynomial threshold growth, making late prestiges progressively easier.
  * @param prestigeCount - The new prestige count.
  * @returns The production multiplier for the new prestige level.
  */
 const calculateProductionMultiplier = (
   prestigeCount: number,
 ): number => {
-  return Math.pow(1.15, prestigeCount);
+  return Math.pow(1.25, prestigeCount);
 };
 
 /**