Honor null thread instructions (#16964)

- Treat explicit null thread instructions as a blank-slate override
while preserving omitted-field fallback behavior.
- Preserve null through rollout resume/fork and keep explicit empty
strings distinct.
- Add app-server v2 start/fork coverage for the tri-state instruction
params.

codex-rs/codex-api/src/common.rs

@@ -153,7 +153,8 @@ impl From<VerbosityConfig> for OpenAiVerbosity {
 #[derive(Debug, Serialize, Clone, PartialEq)]
 pub struct ResponsesApiRequest {
     pub model: String,
-    pub instructions: String,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub instructions: Option<String>,
     pub input: Vec<ResponseItem>,
     pub tools: Vec<serde_json::Value>,
     pub tool_choice: String,
@@ -196,7 +197,8 @@ impl From<&ResponsesApiRequest> for ResponseCreateWsRequest {
 #[derive(Debug, Serialize)]
 pub struct ResponseCreateWsRequest {
     pub model: String,
-    pub instructions: String,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub instructions: Option<String>,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub previous_response_id: Option<String>,
     pub input: Vec<ResponseItem>,

codex-rs/codex-api/tests/clients.rs

@@ -266,7 +266,7 @@ async fn streaming_client_retries_on_transport_error() -> Result<()> {
 
     let request = ResponsesApiRequest {
         model: "gpt-test".into(),
-        instructions: "Say hi".into(),
+        instructions: Some("Say hi".into()),
         input: Vec::new(),
         tools: Vec::new(),
         tool_choice: "auto".into(),
@@ -302,7 +302,7 @@ async fn azure_default_store_attaches_ids_and_headers() -> Result<()> {
 
     let request = ResponsesApiRequest {
         model: "gpt-test".into(),
-        instructions: "Say hi".into(),
+        instructions: Some("Say hi".into()),
         input: vec![ResponseItem::Message {
             id: Some("msg_1".into()),
             role: "user".into(),