Format the panel and admin date and clock in the stored application timezone

The panel snapshot now carries the admin-saved `timezone` ("" when unset);
the coordinator threads it onto the household due-today sensor, and the
Party Board and Quest Log cards format their kicker, header and dock
date and clock in it, falling back to Home Assistant's zone only when
none is stored. The admin app loads the stored zone once
(AppTimezoneProvider) and uses it for completion clocks, History day
grouping, clocks and lateness, and "today" in Schedule, History and new
task defaults; Settings publishes a saved zone app-wide at once.

Resolves review finding TZ-001 (task 7e173881).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

admin/src/today/TodayTab.test.tsx

@@ -5,6 +5,7 @@ import type { AdminSnapshot, SnapshotInstance } from "../api/snapshot";
 import { TransitionsProvider } from "../api/useTransitions";
 import { frame, testStream, type TestStream } from "../api/testStream";
 import { storeTokens } from "../auth/oidc";
+import { AppTimezoneProvider } from "../time/AppTimezone";
 
 function instance(overrides: Partial<SnapshotInstance>): SnapshotInstance {
   return {
@@ -406,3 +407,56 @@ describe("TodayTab Undo and live updates", () => {
     await waitFor(() => expect(streams[0].cancelled()).toBe(true));
   });
 });
+
+describe("TodayTab in the stored application timezone", () => {
+  function renderWithZone(timezone: string) {
+    fetchMock.mockImplementation((url: string) =>
+      Promise.resolve(
+        String(url).endsWith("/api/v1/admin/settings")
+          ? jsonResponse({ timezone })
+          : jsonResponse(SNAPSHOT),
+      ),
+    );
+    render(
+      <AppTimezoneProvider>
+        <TodayTab />
+      </AppTimezoneProvider>,
+    );
+  }
+
+  beforeAll(() => {
+    // The browser runs in UTC; the household lives in Auckland (UTC+12 on Sep 23).
+    vi.stubEnv("TZ", "UTC");
+  });
+
+  afterAll(() => {
+    vi.unstubAllEnvs();
+  });
+
+  beforeEach(() => {
+    sessionStorage.clear();
+    storeTokens({ access_token: "at-123", expires_in: 600 });
+    fetchMock = vi.fn();
+    vi.stubGlobal("fetch", fetchMock);
+  });
+
+  afterEach(() => {
+    cleanup();
+    vi.unstubAllGlobals();
+  });
+
+  it("shows the completion clock in the stored zone, not the browser's", async () => {
+    renderWithZone("Pacific/Auckland");
+    const completed = await screen.findByTestId("attention-12");
+    // 11:04 UTC is 11:04 PM in Auckland.
+    expect(within(completed).getByText("Completed 11:04 PM")).toBeTruthy();
+    // The header's day is the API's household-local today_iso, as-is.
+    expect(screen.getByText("Wednesday, September 23 · cycle day 5 of 14")).toBeTruthy();
+  });
+
+  it("an empty stored zone keeps the browser's local clock", async () => {
+    renderWithZone("");
+    const completed = await screen.findByTestId("attention-12");
+    expect(within(completed).getByText("Completed 11:04 AM")).toBeTruthy();
+  });
+});