refactor: load agent identity runtime eagerly (#19763)

## Summary

AgentIdentity auth previously registered the process task lazily behind
a `OnceCell`. That meant the auth object could be constructed before its
runtime task binding was known.

This PR makes AgentIdentity auth load the runtime task at auth load time
and stores the resulting process task id directly on the auth object.
The model-provider call path can then read a concrete task id instead of
handling a missing lazy value.

## Stack

1. [refactor: make auth loading
async](https://github.com/openai/codex/pull/19762) (merged)
2. **This PR:** [refactor: load AgentIdentity runtime
eagerly](https://github.com/openai/codex/pull/19763)
3. [fix: configure AgentIdentity AuthAPI base
URL](https://github.com/openai/codex/pull/19904)
4. [feat: verify AgentIdentity JWTs with
JWKS](https://github.com/openai/codex/pull/19764)

## Important call sites

| Area | Change |
| --- | --- |
| `AgentIdentityAuth::load` | Registers the process task during auth
loading and stores `process_task_id`. |
| `CodexAuth::from_agent_identity_jwt` | Awaits AgentIdentity auth
loading. |
| model-provider auth | Reads a concrete `process_task_id` instead of an
optional lazy value. |
| AgentIdentity auth tests | Mock task registration now covers eager
runtime allocation. |

## Design decisions

AgentIdentity auth now treats task registration as part of constructing
a usable auth object. That matches how callers use the value: once auth
is present, the model-provider path expects the task-scoped assertion
data to be ready.

## Testing

Tests: targeted Rust auth test compilation, formatter, scoped Clippy
fix, and Bazel lock check.

codex-rs/login/src/auth/manager.rs

@@ -212,7 +212,7 @@ impl CodexAuth {
                     "agent identity auth is missing an agent identity token.",
                 ));
             };
-            return Self::from_agent_identity_jwt(&agent_identity);
+            return Self::from_agent_identity_jwt(&agent_identity).await;
         }
 
         let storage_mode = auth_dot_json.storage_mode(auth_credentials_store_mode);
@@ -246,9 +246,9 @@ impl CodexAuth {
         .await
     }
 
-    pub fn from_agent_identity_jwt(jwt: &str) -> std::io::Result<Self> {
+    pub async fn from_agent_identity_jwt(jwt: &str) -> std::io::Result<Self> {
         let record = AgentIdentityAuthRecord::from_agent_identity_jwt(jwt)?;
-        Ok(Self::AgentIdentity(AgentIdentityAuth::new(record)))
+        Ok(Self::AgentIdentity(AgentIdentityAuth::load(record).await?))
     }
 
     pub fn auth_mode(&self) -> AuthMode {
@@ -322,16 +322,6 @@ impl CodexAuth {
         }
     }
 
-    pub async fn initialize_runtime(
-        &self,
-        _chatgpt_base_url: Option<String>,
-    ) -> std::io::Result<()> {
-        match self {
-            Self::AgentIdentity(auth) => auth.ensure_runtime().await,
-            Self::ApiKey(_) | Self::Chatgpt(_) | Self::ChatgptAuthTokens(_) => Ok(()),
-        }
-    }
-
     /// Returns `None` if Codex backend auth does not expose an account id.
     pub fn get_account_id(&self) -> Option<String> {
         match self {
@@ -749,7 +739,9 @@ async fn load_auth(
     }
 
     if let Some(agent_identity) = read_codex_agent_identity_from_env() {
-        return CodexAuth::from_agent_identity_jwt(&agent_identity).map(Some);
+        return CodexAuth::from_agent_identity_jwt(&agent_identity)
+            .await
+            .map(Some);
     }
 
     // Fall back to the configured persistent store (file/keyring/auto) for managed auth.
@@ -1400,12 +1392,7 @@ impl AuthManager {
             tracing::error!("Failed to refresh token: {}", err);
             return Some(auth);
         }
-        let auth = self.auth_cached()?;
-        if let Err(err) = auth.initialize_runtime(self.chatgpt_base_url.clone()).await {
-            tracing::error!("Failed to initialize auth runtime: {err}");
-            return None;
-        }
-        Some(auth)
+        self.auth_cached()
     }
 
     /// Force a reload of the auth information from auth.json. Returns