refactor: split prompt async gate modules

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

src/shared/prompt-async-gate/timing.ts

@@ -0,0 +1,46 @@
+export const DEFAULT_PROMPT_ASYNC_POST_DISPATCH_HOLD_MS = 2_000
+export const DEFAULT_PROMPT_DISPATCH_TIMEOUT_MS = 30_000
+export const DEFAULT_PROMPT_GATE_MESSAGES_FETCH_TIMEOUT_MS = 5_000
+export const DEFAULT_PROMPT_QUEUE_RETRY_MS = 250
+
+declare function setTimeout(callback: () => void, delay?: number): unknown
+declare function clearTimeout(timeout: unknown): void
+
+let promptGateMessagesFetchTimeoutMsForTesting: number | undefined
+
+export function _setPromptGateMessagesFetchTimeoutMsForTesting(value: number | undefined): void {
+  promptGateMessagesFetchTimeoutMsForTesting = value
+}
+
+export function getPromptGateMessagesFetchTimeoutMs(): number {
+  return promptGateMessagesFetchTimeoutMsForTesting ?? DEFAULT_PROMPT_GATE_MESSAGES_FETCH_TIMEOUT_MS
+}
+
+export function resetPromptGateTimingForTesting(): void {
+  promptGateMessagesFetchTimeoutMsForTesting = undefined
+}
+
+export async function withDispatchTimeout<T>(
+  operation: Promise<T>,
+  dispatchTimeoutMs: number,
+  operationName: string,
+): Promise<T> {
+  if (dispatchTimeoutMs <= 0) {
+    return operation
+  }
+
+  let timeoutID: unknown
+  const timeoutPromise = new Promise<never>((_, reject) => {
+    timeoutID = setTimeout(() => {
+      reject(new Error(`${operationName} timed out after ${dispatchTimeoutMs}ms`))
+    }, dispatchTimeoutMs)
+  })
+
+  try {
+    return await Promise.race([operation, timeoutPromise])
+  } finally {
+    if (timeoutID !== undefined) {
+      clearTimeout(timeoutID)
+    }
+  }
+}