Add experimental prediction forks that inherit parent context (#51884)

## Why

Prediction forks need to preserve the parent's effective context and request settings to maximize prompt-cache reuse.

## What changed

- Add `experimentalPredictionMode` to `thread/fork` for ephemeral forks of loaded, persisted parents. Reject configuration overrides, `path`, `lastTurnId`, and `beforeTurnId` in this mode.
- Inherit the parent's effective configuration, tools, environments, collaboration settings, disabled plugins, and persisted history.
- Enable `features.reasoning_effort_override` on the fork while preserving the parent's request-effort baseline, including when overrides were disabled on the parent.
- Report the fork's effective model, service tier, and reasoning effort in the response.

## Testing

Add coverage for matching parent and fork requests and prompt-cache keys across legacy and paginated history, invalid fork parameters, and reasoning-effort inheritance. Add a context snapshot for identical parent and fork continuations.

GitOrigin-RevId: 46b41f2ac331562b441dadff1a0a9f926c94880d

codex-rs/app-server-protocol/src/protocol/v2/thread.rs

@@ -544,6 +544,15 @@ impl From<ThreadTurnsListResponse> for TurnsPage {
 pub struct ThreadForkParams {
     pub thread_id: String,
 
+    /// Start from the loaded parent's effective config, tools, environments, and
+    /// persisted history to maximize prompt-cache reuse. Requires a loaded,
+    /// persisted parent and `ephemeral: true`. Enables `features.reasoning_effort_override`
+    /// on the fork. Cannot be combined with configuration overrides, `path`, `lastTurnId`,
+    /// or `beforeTurnId`.
+    #[experimental("thread/fork.experimentalPredictionMode")]
+    #[serde(default, skip_serializing_if = "std::ops::Not::not")]
+    pub experimental_prediction_mode: bool,
+
     /// Optional last turn id to fork through, inclusive.
     ///
     /// When specified, turns after `last_turn_id` are omitted from the fork.