diff --git a/test_libs/pyspec/tests/test_finality.py b/test_libs/pyspec/tests/test_finality.py
index 4aee1c538..9992d2a78 100644
--- a/test_libs/pyspec/tests/test_finality.py
+++ b/test_libs/pyspec/tests/test_finality.py
@@ -73,7 +73,7 @@ def next_epoch_with_attestations(state,
     return state, blocks, post_state
 
 
-def test_finality_from_genesis_rule_4(state):
+def test_finality_rule_4(state):
     test_state = deepcopy(state)
 
     blocks = []
@@ -81,8 +81,10 @@ def test_finality_from_genesis_rule_4(state):
         prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, True, False)
         blocks += new_blocks
 
+        # justification/finalization skipped at GENESIS_EPOCH
         if epoch == 0:
             check_finality(test_state, prev_state, False, False, False)
+        # justification/finalization skipped at GENESIS_EPOCH + 1
         elif epoch == 1:
             check_finality(test_state, prev_state, False, False, False)
         elif epoch == 2:
@@ -132,8 +134,6 @@ def test_finality_rule_2(state):
 
     blocks = []
     for epoch in range(3):
-        old_previous_justified_epoch = test_state.previous_justified_epoch
-        old_previous_justified_root = test_state.previous_justified_root
         if epoch == 0:
             prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, True, False)
             check_finality(test_state, prev_state, True, False, False)
@@ -144,8 +144,8 @@ def test_finality_rule_2(state):
             prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, False, True)
             # finalized by rule 2
             check_finality(test_state, prev_state, True, False, True)
-            assert test_state.finalized_epoch == old_previous_justified_epoch
-            assert test_state.finalized_root == old_previous_justified_root
+            assert test_state.finalized_epoch == prev_state.previous_justified_epoch
+            assert test_state.finalized_root == prev_state.previous_justified_root
 
         blocks += new_blocks
 
@@ -170,19 +170,24 @@ def test_finality_rule_3(state):
     blocks += new_blocks
     check_finality(test_state, prev_state, True, False, False)
 
+    # In epoch N, JE is set to N, prev JE is set to N-1
     prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, True, False)
     blocks += new_blocks
     check_finality(test_state, prev_state, True, True, True)
 
+    # In epoch N+1, JE is N, prev JE is N-1, and not enough messages get in to do anything
     prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, False, False)
     blocks += new_blocks
     check_finality(test_state, prev_state, False, True, False)
 
+    # In epoch N+2, JE is N, prev JE is N, and enough messages from the previous epoch get in to justify N+1.
+    # N+1 now becomes the JE. Not enough messages from epoch N+2 itself get in to justify N+2
     prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, False, True)
     blocks += new_blocks
     # rule 2
     check_finality(test_state, prev_state, True, False, True)
 
+    # In epoch N+3, LJE is N+1, prev LJE is N, and enough messages get in to justify epochs N+2 and N+3.
     prev_state, new_blocks, test_state = next_epoch_with_attestations(test_state, True, True)
     blocks += new_blocks
     # rule 3