fix(sdk): carry strict through the custom-tool definition bridge

customToolToDefinition rebuilt ToolDefinition without strict, so the
Task proxies' (and startup MCP tools') explicit strict:false was dropped
before RegisteredToolAdapter and the OpenAI-family serializers saw it.
Declare strict on ToolDefinition, copy it in the bridge, and cover the
proxy -> definition -> registered adapter path in the parity test.

packages/coding-agent/src/extensibility/extensions/types.ts

@@ -543,6 +543,9 @@ export interface ToolDefinition<TParams extends TSchema = TSchema, TDetails = un
 	/** Tool approval tier. Defaults to `"exec"` when omitted.
 	 *  `"read"`: read-only operations. `"write"`: mutations. `"exec"`: code execution. */
 	approval?: ToolApproval;
+	/** Structured-output strict grammar opt-in/out. `false` is meaningful: OpenAI-family
+	 *  serializers preserve an explicit `strict: false` on the wire (#4336/#4340). */
+	strict?: boolean;
 	/** MCP server name for discovery/search metadata when this tool fronts an MCP server. */
 	mcpServerName?: string;
 	/** Original MCP tool name for discovery/search metadata. */

packages/coding-agent/src/sdk.ts

@@ -907,7 +907,7 @@ function registerEvalCleanup(): void {
 	postmortem.register("julia-cleanup", disposeAllJuliaKernelSessions);
 }
 
-function customToolToDefinition(tool: CustomTool): ToolDefinition {
+export function customToolToDefinition(tool: CustomTool): ToolDefinition {
 	const definition: ToolDefinition & { [TOOL_DEFINITION_MARKER]: true } = {
 		name: tool.name,
 		label: tool.label,
@@ -917,6 +917,9 @@ function customToolToDefinition(tool: CustomTool): ToolDefinition {
 		loadMode: defaultLoadModeForToolName(tool.name, tool.loadMode),
 		deferrable: tool.deferrable,
 		approval: typeof tool.approval === "function" ? tool.approval.bind(tool) : tool.approval,
+		// Preserved through RegisteredToolAdapter so MCP-backed tools' explicit
+		// `strict: false` (#4336/#4340) survives the custom-tool → definition bridge.
+		strict: tool.strict,
 		mcpServerName: tool.mcpServerName,
 		mcpToolName: tool.mcpToolName,
 		execute: (toolCallId, params, signal, onUpdate, ctx) =>

packages/coding-agent/test/task-executor-mcp-parity.test.ts

@@ -1,9 +1,13 @@
 import { describe, expect, it, vi } from "bun:test";
 import { INTENT_FIELD } from "@oh-my-pi/pi-wire";
 import type { CustomToolContext } from "../src/extensibility/custom-tools/types";
+import type { ExtensionRunner } from "../src/extensibility/extensions/runner";
+import type { RegisteredTool } from "../src/extensibility/extensions/types";
+import { wrapRegisteredTool } from "../src/extensibility/extensions/wrapper";
 import { MCPManager } from "../src/mcp/manager";
 import { DeferredMCPTool, MCPTool } from "../src/mcp/tool-bridge";
 import type { MCPServerConnection, MCPToolDefinition } from "../src/mcp/types";
+import { customToolToDefinition } from "../src/sdk";
 import { createMCPProxyTools } from "../src/task/executor";
 import { createMockConnection, createMockTransport } from "./mcp-test-utils";
 
@@ -52,6 +56,23 @@ describe("MCP tool strict declaration", () => {
 		const [proxy] = createMCPProxyTools(manager);
 		expect(proxy?.strict).toBe(false);
 	});
+
+	it("survives the custom-tool → definition bridge into the registered session tool", () => {
+		const manager = new MCPManager(process.cwd());
+		vi.spyOn(manager, "getTools").mockReturnValue([new MCPTool(createCapturedConnection([]), STRICT_TOOL)]);
+		const [proxy] = createMCPProxyTools(manager);
+		if (!proxy) {
+			expect.unreachable("no proxy tool created");
+			return;
+		}
+		const definition = customToolToDefinition(proxy);
+		expect(definition.strict).toBe(false);
+		const adapter = wrapRegisteredTool(
+			{ definition, extensionPath: "<sdk>" } as RegisteredTool,
+			{ createContext: () => ({}) } as unknown as ExtensionRunner,
+		);
+		expect(adapter.strict).toBe(false);
+	});
 });
 
 describe("Task MCP proxy parity", () => {