fix(ai): handle statusless hosted search completions

packages/ai/src/providers/openai-shared.ts

@@ -3064,7 +3064,7 @@ export async function processResponsesStream<TApi extends Api>(
 				}
 				closeOpenItem(event.output_index, item.id, entry, item.call_id, prefixedFunctionCallItemKey(item.call_id));
 				stream.push({ type: "toolcall_end", contentIndex, toolCall, partial: output });
-			} else if (item.type === "web_search_call" && item.status === "completed") {
+			} else if (item.type === "web_search_call" && (item.status === undefined || item.status === "completed")) {
 				// A completed provider-hosted web search is progress evidence even when
 				// the model never surfaced an answer; the agent loop continues from it.
 				sawCompletedWebSearchCall = true;

packages/ai/test/openai-responses-stream-terminal.test.ts

@@ -673,6 +673,28 @@ describe("processResponsesStream: terminal events", () => {
 		expect(output.stopDetails).toEqual({ type: "pause_turn" });
 	});
 
+	test("pauses a status-less hosted web search done item that has no visible assistant output", async () => {
+		const output = makeOutput();
+		const stream = { push: () => {}, end: () => {} } as never;
+
+		await processResponsesStream(
+			makeStream([
+				{
+					type: "response.output_item.done",
+					output_index: 1,
+					item: { type: "web_search_call", id: "ws_1", action: { type: "search" } },
+				},
+				{ type: "response.completed", response: { id: "resp_search", status: "completed" } },
+			]),
+			output,
+			stream,
+			makeModel(),
+		);
+
+		expect(output.stopReason).toBe("stop");
+		expect(output.stopDetails).toEqual({ type: "pause_turn" });
+	});
+
 	test("finishes a hosted web search when the response includes visible output", async () => {
 		const output = makeOutput();
 		const stream = { push: () => {}, end: () => {} } as never;