fix(chat): preserve tool duration across session switches (#1712)

* fix(chat): preserve tool duration across session switches

Fix #1636: ToolPart.tsx reset pinnedTime to empty on unmount/remount,
causing LiveDuration to not render on first paint. Now initializes
pinnedTime from server-provided time?.start/time?.end in the useState
initializer, eliminating the one-frame gap.

* fix(sync): preserve tool state.time in materialization merge

---------

Co-authored-by: Leonid Skorobogatyy <bash@opencode.itc.local>

packages/ui/src/sync/__tests__/materialization.test.ts

@@ -122,6 +122,37 @@ describe("materializeSessionSnapshots", () => {
 
     expect(result.part.msg_1).toEqual([serverPart])
   })
+
+  test("preserves state.time from existing part when snapshot drops it", () => {
+    const livePart = {
+      id: "prt_1",
+      messageID: "msg_1",
+      sessionID: "ses_1",
+      type: "tool",
+      state: { status: "completed", time: { start: 1000, end: 2000 } },
+    } as unknown as Part
+    const snapshotPart = {
+      id: "prt_1",
+      messageID: "msg_1",
+      sessionID: "ses_1",
+      type: "tool",
+      state: { status: "completed" },
+    } as unknown as Part
+    const state = {
+      message: { ses_1: [message("msg_1")] },
+      part: { msg_1: [livePart] },
+    }
+
+    const result = materializeSessionSnapshots(
+      state,
+      "ses_1",
+      [{ info: message("msg_1"), parts: [snapshotPart] }],
+    )
+
+    const mergedPart = result.part.msg_1[0] as { state?: { time?: { start?: number; end?: number } } }
+    expect(mergedPart.state?.time?.start).toBe(1000)
+    expect(mergedPart.state?.time?.end).toBe(2000)
+  })
 })
 
 describe("getSessionMaterializationStatus", () => {