diff --git a/lucene/CHANGES.txt b/lucene/CHANGES.txt
index 12b058fe14e7..8fbde7a4616f 100644
--- a/lucene/CHANGES.txt
+++ b/lucene/CHANGES.txt
@@ -288,6 +288,8 @@ Bug Fixes
 
 * GITHUB#12736: Fix NullPointerException when Monitor.getQuery cannot find the requested queryId (Davis Cook)
 
+* GITHUB#12770: Stop exploring HNSW graph if scores are not getting better. (Ben Trent)
+
 Build
 ---------------------
 
diff --git a/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphSearcher.java b/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphSearcher.java
index c18c16c38f53..0135fc5a4119 100644
--- a/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphSearcher.java
+++ b/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphSearcher.java
@@ -174,8 +174,7 @@ private int[] findBestEntryPoint(RandomVectorScorer scorer, HnswGraph graph, lon
           }
           float friendSimilarity = scorer.score(friendOrd);
           visitedCount++;
-          if (friendSimilarity > currentScore
-              || (friendSimilarity == currentScore && friendOrd < currentEp)) {
+          if (friendSimilarity > currentScore) {
             currentScore = friendSimilarity;
             currentEp = friendOrd;
             foundBetter = true;
@@ -243,7 +242,7 @@ void searchLevel(
         }
         float friendSimilarity = scorer.score(friendOrd);
         results.incVisitedCount(1);
-        if (friendSimilarity >= minAcceptedSimilarity) {
+        if (friendSimilarity > minAcceptedSimilarity) {
           candidates.add(friendOrd, friendSimilarity);
           if (acceptOrds == null || acceptOrds.get(friendOrd)) {
             if (results.collect(friendOrd, friendSimilarity)) {
diff --git a/lucene/core/src/test/org/apache/lucene/search/BaseKnnVectorQueryTestCase.java b/lucene/core/src/test/org/apache/lucene/search/BaseKnnVectorQueryTestCase.java
index 03531253abcf..d45902d3bb3c 100644
--- a/lucene/core/src/test/org/apache/lucene/search/BaseKnnVectorQueryTestCase.java
+++ b/lucene/core/src/test/org/apache/lucene/search/BaseKnnVectorQueryTestCase.java
@@ -281,12 +281,12 @@ public void testScoreEuclidean() throws IOException {
 
       DocIdSetIterator it = scorer.iterator();
       assertEquals(3, it.cost());
-      assertEquals(1, it.nextDoc());
-      assertEquals(1 / 6f, scorer.score(), 0);
-      assertEquals(3, it.advance(3));
+      assertEquals(2, it.nextDoc());
       assertEquals(1 / 2f, scorer.score(), 0);
+      assertEquals(4, it.advance(4));
+      assertEquals(1 / 6f, scorer.score(), 0);
 
-      assertEquals(NO_MORE_DOCS, it.advance(4));
+      assertEquals(NO_MORE_DOCS, it.advance(5));
       expectThrows(ArrayIndexOutOfBoundsException.class, scorer::score);
     }
   }
@@ -384,7 +384,7 @@ public void testExplain() throws IOException {
         assertEquals(0, matched.getDetails().length);
         assertEquals("within top 3 docs", matched.getDescription());
 
-        Explanation nomatch = searcher.explain(query, 4);
+        Explanation nomatch = searcher.explain(query, 1);
         assertFalse(nomatch.isMatch());
         assertEquals(0f, nomatch.getValue());
         assertEquals(0, matched.getDetails().length);