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/src/client_common.rs

@@ -23,7 +23,7 @@ pub const REVIEW_EXIT_INTERRUPTED_TMPL: &str =
     include_str!("../templates/review/exit_interrupted.xml");
 
 /// API request payload for a single model turn
-#[derive(Default, Debug, Clone)]
+#[derive(Debug, Clone)]
 pub struct Prompt {
     /// Conversation context input items.
     pub input: Vec<ResponseItem>,
@@ -35,7 +35,7 @@ pub struct Prompt {
     /// Whether parallel tool calls are permitted for this prompt.
     pub(crate) parallel_tool_calls: bool,
 
-    pub base_instructions: BaseInstructions,
+    pub base_instructions: Option<BaseInstructions>,
 
     /// Optionally specify the personality of the model.
     pub personality: Option<Personality>,
@@ -44,6 +44,19 @@ pub struct Prompt {
     pub output_schema: Option<Value>,
 }
 
+impl Default for Prompt {
+    fn default() -> Self {
+        Self {
+            input: Vec::new(),
+            tools: Vec::new(),
+            parallel_tool_calls: false,
+            base_instructions: Some(BaseInstructions::default()),
+            personality: None,
+            output_schema: None,
+        }
+    }
+}
+
 impl Prompt {
     pub(crate) fn get_formatted_input(&self) -> Vec<ResponseItem> {
         let mut input = self.input.clone();