fix: resolve sync count inflation, add essence/s display, sort auto-buy by level

Closes #147: patch functions now detect actual changes before incrementing
the patched counter, preventing inflated sync reports.

Closes #149: computeEssencePerSecond exported from tick engine and shown
in the resource bar dropdown alongside Gold/s.

Closes #150: auto-buy now sorts adventurers by level descending for
semantic clarity, ensuring highest-tier units are purchased first.

apps/web/src/engine/tick.ts

@@ -195,6 +195,54 @@ export const computeGoldPerSecond = (state: GameState): number => {
   return goldPerSecond;
 };
 
+/**
+ * Computes the current essence per second for the given game state,
+ * applying all relevant multipliers (upgrades, prestige, echo, crafted, companion).
+ * @param state - The current game state.
+ * @returns The total essence per second.
+ */
+export const computeEssencePerSecond = (state: GameState): number => {
+  const runestonesEssence = state.prestige.runestonesEssenceMultiplier ?? 1;
+  const craftedEssenceMultiplier
+    = state.exploration?.craftedEssenceMultiplier ?? 1;
+  const companionBonus = getActiveCompanionBonus(
+    state.companions?.activeCompanionId,
+    state.companions?.unlockedCompanionIds ?? [],
+  );
+  const companionEssenceMult
+    = companionBonus?.type === "essenceIncome"
+      ? 1 + companionBonus.value
+      : 1;
+
+  let essencePerSecond = 0;
+  for (const adventurer of state.adventurers) {
+    if (!adventurer.unlocked || adventurer.count === 0) {
+      continue;
+    }
+    const upgradeMultiplier = state.upgrades.
+      filter((upgrade) => {
+        const isGlobal = upgrade.target === "global";
+        const isThisAdventurer
+          = upgrade.target === "adventurer"
+          && upgrade.adventurerId === adventurer.id;
+        return upgrade.purchased && (isGlobal || isThisAdventurer);
+      }).
+      reduce((mult, upgrade) => {
+        return mult * upgrade.multiplier;
+      }, 1);
+    const contribution
+      = adventurer.essencePerSecond
+      * adventurer.count
+      * upgradeMultiplier
+      * state.prestige.productionMultiplier
+      * runestonesEssence
+      * craftedEssenceMultiplier
+      * companionEssenceMult;
+    essencePerSecond = essencePerSecond + contribution;
+  }
+  return essencePerSecond;
+};
+
 /**
  * Pure function — applies one game tick to the state.
  * DeltaSeconds: time elapsed since last tick.