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/components/ui/resourceBar.tsx

@@ -10,7 +10,11 @@
 /* eslint-disable complexity -- Many conditional resource and badge render paths */
 import { useState, type FocusEvent, type JSX } from "react";
 import { useGame } from "../../context/gameContext.js";
-import { RESOURCE_CAP, computeGoldPerSecond } from "../../engine/tick.js";
+import {
+  RESOURCE_CAP,
+  computeEssencePerSecond,
+  computeGoldPerSecond,
+} from "../../engine/tick.js";
 import type { Resource } from "@elysium/types";
 
 interface ResourceBarProperties {
@@ -83,12 +87,14 @@ const ResourceBar = ({
   const { gold, essence, crystals } = resources;
   let partyCombatPower = 0;
   let goldPerSecond = 0;
+  let essencePerSecond = 0;
   if (state !== null) {
     for (const adventurer of state.adventurers) {
       const contribution = adventurer.combatPower * adventurer.count;
       partyCombatPower = partyCombatPower + contribution;
     }
     goldPerSecond = computeGoldPerSecond(state);
+    essencePerSecond = computeEssencePerSecond(state);
   }
 
   let avatarUrl: string | null = null;
@@ -182,6 +188,13 @@ const ResourceBar = ({
                 </span>
                 <span className="resource-label">{"Gold/s"}</span>
               </div>
+              <div className="resource">
+                <span className="resource-icon">{"⚡"}</span>
+                <span className="resource-value">
+                  {formatNumber(essencePerSecond)}
+                </span>
+                <span className="resource-label">{"Essence/s"}</span>
+              </div>
               <div className={`resource${essenceFull
                 ? " resource-full"
                 : ""}`}>

apps/web/src/context/gameContext.tsx

@@ -1127,7 +1127,7 @@ export const GameProvider = ({
               return adventurer.unlocked && next.resources.gold >= cost;
             }).
             sort((adventurerA, adventurerB) => {
-              return adventurerB.combatPower - adventurerA.combatPower;
+              return adventurerB.level - adventurerA.level;
             });
           if (bestAdventurer !== undefined) {
             const purchaseCost

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.