feat(memory): structured reflection + correction detection in MemoryMiddleware (#1620) (#1668)

* feat(memory): add structured reflection and correction detection

* fix(memory): align sourceError schema and prompt guidance

---------

Co-authored-by: Willem Jiang <willem.jiang@gmail.com>

backend/tests/test_memory_prompt_injection.py

@@ -119,3 +119,38 @@ def test_format_memory_skips_non_string_content_facts() -> None:
     # The formatted line for a list content would be "- [knowledge | 0.85] ['list']".
     assert "| 0.85]" not in result
     assert "Valid fact" in result
+
+
+def test_format_memory_renders_correction_source_error() -> None:
+    memory_data = {
+        "facts": [
+            {
+                "content": "Use make dev for local development.",
+                "category": "correction",
+                "confidence": 0.95,
+                "sourceError": "The agent previously suggested npm start.",
+            }
+        ]
+    }
+
+    result = format_memory_for_injection(memory_data, max_tokens=2000)
+
+    assert "Use make dev for local development." in result
+    assert "avoid: The agent previously suggested npm start." in result
+
+
+def test_format_memory_renders_correction_without_source_error_normally() -> None:
+    memory_data = {
+        "facts": [
+            {
+                "content": "Use make dev for local development.",
+                "category": "correction",
+                "confidence": 0.95,
+            }
+        ]
+    }
+
+    result = format_memory_for_injection(memory_data, max_tokens=2000)
+
+    assert "Use make dev for local development." in result
+    assert "avoid:" not in result