balance: smooth prestige income cliff, quadratic milestones, exponential combat scaling (#170, #171)

Reduce income_10 cost 30k→22.5k and income_11 80k→60k (25% cut each)
to ease the late-prestige runestone cliff without collapsing the timeline.

Change prestige milestone bonus from linear (n×25) to quadratic (n²×25)
so high-prestige milestones feel meaningful (P100 = 10k stones).

Replace linear prestige combat multiplier (1 + count×0.1) with exponential
(4^count) in both the tick engine and server-side boss route. Without this
the final boss (2×10^145 HP) was unreachable by ~112 orders of magnitude;
base-4 makes it achievable around P190, consistent with the 6-month target.

apps/api/src/data/prestigeUpgrades.ts

@@ -96,7 +96,7 @@ export const defaultPrestigeUpgrades: Array<PrestigeUpgrade> = [
     id:             "income_10",
     multiplier:     200,
     name:           "Eternal Rune I",
-    runestonesCost: 30_000,
+    runestonesCost: 22_500,
   },
   {
     category: "income",
@@ -105,7 +105,7 @@ export const defaultPrestigeUpgrades: Array<PrestigeUpgrade> = [
     id:             "income_11",
     multiplier:     500,
     name:           "Eternal Rune II",
-    runestonesCost: 80_000,
+    runestonesCost: 60_000,
   },
   // ── Click Power ───────────────────────────────────────────────────────────
   {

apps/api/src/routes/boss.ts

@@ -24,6 +24,13 @@ import { updateChallengeProgress } from "../services/dailyChallenges.js";
 import { logger } from "../services/logger.js";
 import type { HonoEnvironment } from "../types/hono.js";
 
+/**
+ * Exponential base for the prestige combat multiplier: Math.pow(base, prestigeCount).
+ * Replaces the former linear formula (1 + count * 0.1) to enable late-game zone progression.
+ * Must be kept in sync with prestigeCombatBase in apps/web/src/engine/tick.ts.
+ */
+const prestigeCombatBase = 4;
+
 const bossRouter = new Hono<HonoEnvironment>();
 
 bossRouter.use("*", authMiddleware);
@@ -38,8 +45,7 @@ const calculatePartyStats = (
     }
   }
 
-  // eslint-disable-next-line stylistic/no-mixed-operators -- prestige count * factor is clear
-  const prestigeMultiplier = 1 + state.prestige.count * 0.1;
+  const prestigeMultiplier = Math.pow(prestigeCombatBase, state.prestige.count);
 
   // Apply equipped weapon's combat bonus
   // eslint-disable-next-line capitalized-comments -- v8 ignore

apps/api/src/services/prestige.ts

@@ -159,7 +159,7 @@ const calculateProductionMultiplier = (
 
 /**
  * Returns the milestone runestone bonus for the given prestige count.
- * Every MILESTONE_INTERVAL prestiges awards milestone_number * MILESTONE_RUNESTONES_PER_INTERVAL stones.
+ * Every MILESTONE_INTERVAL prestiges awards milestone_number² * MILESTONE_RUNESTONES_PER_INTERVAL stones.
  * @param prestigeCount - The prestige count after the current prestige.
  * @returns The milestone runestone bonus, or 0 if not a milestone prestige.
  */
@@ -168,7 +168,7 @@ const calculateMilestoneBonus = (prestigeCount: number): number => {
     return 0;
   }
   const milestoneNumber = prestigeCount / milestoneInterval;
-  return milestoneNumber * milestoneRunestonesPerInterval;
+  return milestoneNumber * milestoneNumber * milestoneRunestonesPerInterval;
 };
 
 /**

apps/api/test/services/prestige.spec.ts

@@ -151,12 +151,12 @@ describe("calculateMilestoneBonus", () => {
     expect(calculateMilestoneBonus(5)).toBe(25);
   });
 
-  it("returns 50 at prestige 10", () => {
-    expect(calculateMilestoneBonus(10)).toBe(50);
+  it("returns 100 at prestige 10", () => {
+    expect(calculateMilestoneBonus(10)).toBe(100);
   });
 
-  it("returns 75 at prestige 15", () => {
-    expect(calculateMilestoneBonus(15)).toBe(75);
+  it("returns 225 at prestige 15", () => {
+    expect(calculateMilestoneBonus(15)).toBe(225);
   });
 });