feat: add titles system with unlock tracking and character sheet display

Titles are earned by reaching milestones (quests, bosses, gold, clicks,
adventurers, guild, prestige, transcendence, apotheosis, achievements,
longevity) and are permanent - never lost on prestige/transcendence/
apotheosis resets. 20 titles available at launch.

Also fixes a pre-existing P2034 write-conflict on the load backfill path
and the exactOptionalPropertyTypes violation in the quest failure handler.

apps/api/src/services/titles.ts

@@ -0,0 +1,51 @@
+import type { GameState } from "@elysium/types";
+import { TITLES } from "../data/titles.js";
+
+export const checkAndUnlockTitles = (
+  currentUnlocked: string[],
+  state: GameState,
+  guildName: string,
+  createdAt: number,
+): string[] => {
+  const metrics: Record<string, number | boolean> = {
+    totalClicks: state.player.totalClicks,
+    totalGoldEarned: state.player.totalGoldEarned,
+    bossesDefeated: state.bosses.filter((b) => b.status === "defeated").length,
+    questsCompleted: state.quests.filter((q) => q.status === "completed").length,
+    prestigeCount: state.prestige.count,
+    transcendenceCount: state.transcendence?.count ?? 0,
+    apotheosisCount: state.apotheosis?.count ?? 0,
+    adventurerTotal: state.adventurers.reduce((sum, a) => sum + a.count, 0),
+    achievementsUnlocked: state.achievements.filter((a) => a.unlockedAt !== null).length,
+    guildFounded: guildName.trim().length > 0,
+    playedDays: Math.floor((Date.now() - createdAt) / 86_400_000),
+  };
+
+  const newlyUnlocked: string[] = [];
+
+  for (const title of TITLES) {
+    if (currentUnlocked.includes(title.id)) continue;
+
+    const { type, amount } = title.condition;
+    let earned = false;
+
+    if (type === "guildFounded") {
+      earned = metrics.guildFounded === true;
+    } else if (amount !== undefined) {
+      earned = (metrics[type] as number) >= amount;
+    }
+
+    if (earned) {
+      newlyUnlocked.push(title.id);
+    }
+  }
+
+  return newlyUnlocked;
+};
+
+export const parseUnlockedTitles = (raw: unknown): string[] => {
+  if (Array.isArray(raw)) {
+    return raw.filter((item): item is string => typeof item === "string");
+  }
+  return [];
+};