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/core/tests/suite/client.rs

@@ -921,7 +921,7 @@ async fn includes_base_instructions_override_in_request() {
     let mut builder = test_codex()
         .with_auth(CodexAuth::from_api_key("Test API Key"))
         .with_config(|config| {
-            config.base_instructions = Some("test instructions".to_string());
+            config.base_instructions = Some(Some("test instructions".to_string()));
         });
     let codex = builder
         .build(&server)
@@ -953,6 +953,47 @@ async fn includes_base_instructions_override_in_request() {
     );
 }
 
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn omits_explicit_null_base_instructions_from_request() {
+    skip_if_no_network!();
+    // Mock server
+    let server = MockServer::start().await;
+    let resp_mock = mount_sse_once(
+        &server,
+        sse(vec![ev_response_created("resp1"), ev_completed("resp1")]),
+    )
+    .await;
+
+    let mut builder = test_codex()
+        .with_auth(CodexAuth::from_api_key("Test API Key"))
+        .with_config(|config| {
+            config.base_instructions = Some(None);
+        });
+    let codex = builder
+        .build(&server)
+        .await
+        .expect("create new conversation")
+        .codex;
+
+    codex
+        .submit(Op::UserInput {
+            items: vec![UserInput::Text {
+                text: "hello".into(),
+                text_elements: Vec::new(),
+            }],
+            final_output_json_schema: None,
+        })
+        .await
+        .unwrap();
+
+    wait_for_event(&codex, |ev| matches!(ev, EventMsg::TurnComplete(_))).await;
+
+    let request = resp_mock.single_request();
+    let request_body = request.body_json();
+
+    assert_eq!(request_body.get("instructions"), None);
+}
+
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
 async fn chatgpt_auth_sends_correct_request() {
     skip_if_no_network!();