diff --git a/python/cog/predictor.py b/python/cog/predictor.py
index f86753420f..27c65d616a 100644
--- a/python/cog/predictor.py
+++ b/python/cog/predictor.py
@@ -282,20 +282,30 @@ class Config:
                         parameter.default.default is PydanticUndefined
                         or parameter.default.default is ...
                     ):
-                        if PYDANTIC_V2:
-                            parameter.default.default = None
-                        else:
+                        parameter.default.default = None
+                        if not PYDANTIC_V2:
                             parameter.default.default_factory = None
-                            parameter.default.default = None
                 default = parameter.default
 
+        extra: Dict[str, Any] = {}
         if PYDANTIC_V2:
             # https://github.com/pydantic/pydantic/blob/2.7/pydantic/json_schema.py#L1436-L1446
             # json_schema_extra can be a callable, but we don't set that and users shouldn't set that
             if not default.json_schema_extra:  # type: ignore
-                default.json_schema_extra = {}  # type: ignore
+                default.json_schema_extra = {"x-order": order}  # type: ignore
             assert isinstance(default.json_schema_extra, dict)  # type: ignore
-            extra = default.json_schema_extra  # type: ignore
+            # In Pydantic 2.12.0 the json_schema_extra field is copied into a variable called "_attributes_set"
+            # that gets created in the constructor.
+            # This means that changes to that dictionary after the construction don't take effect during the render
+            # to openapi schema JSON.
+            # To get around this, we will reference the dictionary in the attributes_set variable and make changes to
+            # json_schema_extra take effect.
+            if hasattr(default, "_attributes_set"):
+                if "json_schema_extra" not in default._attributes_set:  # type: ignore
+                    default._attributes_set["json_schema_extra"] = {"x-order": order}
+                extra = default._attributes_set["json_schema_extra"]  # type: ignore
+            else:
+                extra = default.json_schema_extra  # type: ignore
         else:
             extra = default.extra  # type: ignore
         extra["x-order"] = order
diff --git a/python/cog/server/helpers.py b/python/cog/server/helpers.py
index c7af410aa2..9c86efa438 100644
--- a/python/cog/server/helpers.py
+++ b/python/cog/server/helpers.py
@@ -461,6 +461,7 @@ def update_openapi_schema_for_pydantic_2(
     _extract_enum_properties(openapi_schema)
     _set_default_enumeration_description(openapi_schema)
     _restore_allof_for_prediction_id_put(openapi_schema)
+    _ensure_nullable_properties_not_required(openapi_schema)
 
 
 def _remove_webhook_events_filter_title(
@@ -501,7 +502,7 @@ def _update_nullable_anyof(
             if len(non_null_items) < len(value) and not in_header:
                 openapi_schema["nullable"] = True
 
-    elif isinstance(openapi_schema, list):
+    elif isinstance(openapi_schema, list):  # type: ignore
         for item in openapi_schema:
             _update_nullable_anyof(item, in_header=in_header)
 
@@ -593,3 +594,13 @@ def _restore_allof_for_prediction_id_put(
             ref = value["$ref"]
             del value["$ref"]
             value["allOf"] = [{"$ref": ref}]
+
+
+def _ensure_nullable_properties_not_required(openapi_schema: Dict[str, Any]) -> None:
+    schemas = openapi_schema["components"]["schemas"]
+    for schema in schemas.values():
+        properties = schema.get("properties", {})
+        nullable = {k for k, v in properties.items() if v.get("nullable", False)}
+
+        if "required" in schema and nullable:
+            schema["required"] = [k for k in schema["required"] if k not in nullable]