fix: rewrite prestige/transcendence formula and rebalance progression

apps/api/test/routes/transcendence.spec.ts

@@ -158,7 +158,7 @@ describe("transcendence route", () => {
       const res = await post("/buy-upgrade", { upgradeId: "echo_income_1" });
       expect(res.status).toBe(200);
       const body = await res.json() as { echoesRemaining: number; purchasedUpgradeIds: string[] };
-      expect(body.echoesRemaining).toBe(95); // 100 - 5
+      expect(body.echoesRemaining).toBe(98); // 100 - 2
       expect(body.purchasedUpgradeIds).toContain("echo_income_1");
     });
 

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

@@ -55,15 +55,18 @@ const makeMinimalState = (overrides: Partial<GameState> = {}): GameState =>
 
 describe("calculatePrestigeThreshold", () => {
   it("returns base threshold at count 0", () => {
+    // base × (0+1)^2 = 1_000_000 × 1 = 1_000_000
     expect(calculatePrestigeThreshold(0)).toBe(1_000_000);
   });
 
-  it("returns 5× at count 1", () => {
-    expect(calculatePrestigeThreshold(1)).toBe(5_000_000);
+  it("returns 4× base at count 1", () => {
+    // base × (1+1)^2 = 1_000_000 × 4 = 4_000_000
+    expect(calculatePrestigeThreshold(1)).toBe(4_000_000);
   });
 
-  it("returns 25× at count 2", () => {
-    expect(calculatePrestigeThreshold(2)).toBe(25_000_000);
+  it("returns 9× base at count 2", () => {
+    // base × (2+1)^2 = 1_000_000 × 9 = 9_000_000
+    expect(calculatePrestigeThreshold(2)).toBe(9_000_000);
   });
 
   it("applies threshold multiplier correctly", () => {
@@ -128,12 +131,12 @@ describe("calculateProductionMultiplier", () => {
     expect(calculateProductionMultiplier(0)).toBe(1);
   });
 
-  it("returns 1.15 at count 1", () => {
-    expect(calculateProductionMultiplier(1)).toBeCloseTo(1.15);
+  it("returns 1.25 at count 1", () => {
+    expect(calculateProductionMultiplier(1)).toBeCloseTo(1.25);
   });
 
   it("scales exponentially", () => {
-    expect(calculateProductionMultiplier(10)).toBeCloseTo(Math.pow(1.15, 10));
+    expect(calculateProductionMultiplier(10)).toBeCloseTo(Math.pow(1.25, 10));
   });
 });
 

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

@@ -97,20 +97,21 @@ describe("isEligibleForTranscendence", () => {
 
 describe("calculateEchoes", () => {
   it("handles prestige count of 0 by treating it as 1", () => {
-    // safeCount = max(0, 1) = 1; floor(853 / sqrt(1)) = 853
-    expect(calculateEchoes(0, 1)).toBe(853);
+    // safeCount = max(0, 1) = 1; floor(224 / sqrt(1)) = 224
+    expect(calculateEchoes(0, 1)).toBe(224);
   });
 
   it("calculates echoes at count 1", () => {
-    expect(calculateEchoes(1, 1)).toBe(853);
+    // floor(224 / sqrt(1)) = 224
+    expect(calculateEchoes(1, 1)).toBe(224);
   });
 
   it("decreases echoes with higher prestige count", () => {
     const echoesAt1 = calculateEchoes(1, 1);
     const echoesAt4 = calculateEchoes(4, 1);
     expect(echoesAt4).toBeLessThan(echoesAt1);
-    // floor(853 / sqrt(4)) = floor(853 / 2) = 426
-    expect(echoesAt4).toBe(426);
+    // floor(224 / sqrt(4)) = floor(224 / 2) = 112
+    expect(echoesAt4).toBe(112);
   });
 
   it("applies echoMetaMultiplier", () => {
@@ -118,6 +119,11 @@ describe("calculateEchoes", () => {
     const withMult = calculateEchoes(1, 2);
     expect(withMult).toBe(base * 2);
   });
+
+  it("returns 50 echoes at the target prestige 20", () => {
+    // floor(224 / sqrt(20)) = floor(224 / 4.472) = floor(50.09) = 50
+    expect(calculateEchoes(20, 1)).toBe(50);
+  });
 });
 
 describe("buildPostTranscendenceState", () => {