feat(openai): Add verbosity parameter support for GPT-5 family models (#16660)

OpenAI's GPT-5 model family supports a verbosity parameter to control
the length and detail of responses. This parameter accepts three values:
'low', 'medium', or 'high'.

Changes:
- Added verbosity parameter to completion() and acompletion() signatures
- Added verbosity to DEFAULT_CHAT_COMPLETION_PARAM_VALUES in constants.py
- Added verbosity to get_optional_params() in utils.py
- Added verbosity to GPT-5 supported params list
- Updated OpenAI docs with verbosity usage examples
- Added comprehensive test for verbosity parameter

Supported models: gpt-5, gpt-5.1, gpt-5-mini, gpt-5-nano, gpt-5-codex, gpt-5-pro

tests/test_litellm/llms/openai/test_gpt5_transformation.py

@@ -155,6 +155,32 @@ def test_gpt5_codex_supports_function_calling(config: OpenAIConfig):
     assert "tools" in supported_params
 
 
+def test_gpt5_verbosity_parameter(config: OpenAIConfig):
+    """Test that verbosity parameter passes through correctly for GPT-5 models.
+
+    The verbosity parameter controls output length and detail for GPT-5 family models.
+    Supported values: 'low', 'medium', 'high'
+    Supported models: gpt-5, gpt-5.1, gpt-5-mini, gpt-5-nano, gpt-5-codex, gpt-5-pro
+    """
+    # Test all valid verbosity values
+    for verbosity_level in ["low", "medium", "high"]:
+        params = config.map_openai_params(
+            non_default_params={"verbosity": verbosity_level},
+            optional_params={},
+            model="gpt-5.1",
+            drop_params=False,
+        )
+        assert params["verbosity"] == verbosity_level
+
+    # Test with different GPT-5 models
+    for model in ["gpt-5", "gpt-5.1", "gpt-5-mini", "gpt-5-nano", "gpt-5-codex"]:
+        params = config.map_openai_params(
+            non_default_params={"verbosity": "low"},
+            optional_params={},
+            model=model,
+            drop_params=False,
+        )
+        assert params["verbosity"] == "low"
 def test_gpt5_1_reasoning_effort_none(config: OpenAIConfig):
     """Test that GPT-5.1 supports reasoning_effort='none' parameter.