fix: rewrite prestige/transcendence formula and rebalance progression

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", () => {