Add logout functionality and Garmin sync structured logging

- Add POST /api/auth/logout endpoint with tests (5 tests)
- Add logout button to settings page (5 tests)
- Add structured logging to garmin-sync cron (sync start/complete/failure)
- Update IMPLEMENTATION_PLAN.md with spec gap analysis findings
- Total: 835 tests passing across 44 test files

Closes spec gaps from authentication.md (logout) and observability.md (logging)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

src/app/settings/page.test.tsx

@@ -296,7 +296,7 @@ describe("SettingsPage", () => {
         expect(cycleLengthInput).toBeDisabled();
         expect(screen.getByLabelText(/notification time/i)).toBeDisabled();
         expect(screen.getByLabelText(/timezone/i)).toBeDisabled();
-        expect(screen.getByRole("button")).toBeDisabled();
+        expect(screen.getByRole("button", { name: /saving/i })).toBeDisabled();
       });
 
       resolveSave(mockUser);
@@ -525,4 +525,150 @@ describe("SettingsPage", () => {
       });
     });
   });
+
+  describe("logout", () => {
+    it("renders a logout button", async () => {
+      render(<SettingsPage />);
+
+      await waitFor(() => {
+        expect(
+          screen.getByRole("button", { name: /log out/i }),
+        ).toBeInTheDocument();
+      });
+    });
+
+    it("calls POST /api/auth/logout when logout button clicked", async () => {
+      mockFetch
+        .mockResolvedValueOnce({
+          ok: true,
+          json: () => Promise.resolve(mockUser),
+        })
+        .mockResolvedValueOnce({
+          ok: true,
+          json: () =>
+            Promise.resolve({
+              success: true,
+              message: "Logged out successfully",
+              redirectTo: "/login",
+            }),
+        });
+
+      render(<SettingsPage />);
+
+      await waitFor(() => {
+        expect(
+          screen.getByRole("button", { name: /log out/i }),
+        ).toBeInTheDocument();
+      });
+
+      const logoutButton = screen.getByRole("button", { name: /log out/i });
+      fireEvent.click(logoutButton);
+
+      await waitFor(() => {
+        expect(mockFetch).toHaveBeenCalledWith("/api/auth/logout", {
+          method: "POST",
+        });
+      });
+    });
+
+    it("redirects to login page after logout", async () => {
+      mockFetch
+        .mockResolvedValueOnce({
+          ok: true,
+          json: () => Promise.resolve(mockUser),
+        })
+        .mockResolvedValueOnce({
+          ok: true,
+          json: () =>
+            Promise.resolve({
+              success: true,
+              message: "Logged out successfully",
+              redirectTo: "/login",
+            }),
+        });
+
+      render(<SettingsPage />);
+
+      await waitFor(() => {
+        expect(
+          screen.getByRole("button", { name: /log out/i }),
+        ).toBeInTheDocument();
+      });
+
+      const logoutButton = screen.getByRole("button", { name: /log out/i });
+      fireEvent.click(logoutButton);
+
+      await waitFor(() => {
+        expect(mockPush).toHaveBeenCalledWith("/login");
+      });
+    });
+
+    it("shows loading state while logging out", async () => {
+      let resolveLogout: (value: unknown) => void = () => {};
+      const logoutPromise = new Promise((resolve) => {
+        resolveLogout = resolve;
+      });
+
+      mockFetch
+        .mockResolvedValueOnce({
+          ok: true,
+          json: () => Promise.resolve(mockUser),
+        })
+        .mockReturnValueOnce({
+          ok: true,
+          json: () => logoutPromise,
+        });
+
+      render(<SettingsPage />);
+
+      await waitFor(() => {
+        expect(
+          screen.getByRole("button", { name: /log out/i }),
+        ).toBeInTheDocument();
+      });
+
+      const logoutButton = screen.getByRole("button", { name: /log out/i });
+      fireEvent.click(logoutButton);
+
+      await waitFor(() => {
+        expect(
+          screen.getByRole("button", { name: /logging out/i }),
+        ).toBeInTheDocument();
+      });
+
+      resolveLogout({
+        success: true,
+        message: "Logged out successfully",
+        redirectTo: "/login",
+      });
+    });
+
+    it("shows error if logout fails", async () => {
+      mockFetch
+        .mockResolvedValueOnce({
+          ok: true,
+          json: () => Promise.resolve(mockUser),
+        })
+        .mockResolvedValueOnce({
+          ok: false,
+          json: () => Promise.resolve({ error: "Logout failed" }),
+        });
+
+      render(<SettingsPage />);
+
+      await waitFor(() => {
+        expect(
+          screen.getByRole("button", { name: /log out/i }),
+        ).toBeInTheDocument();
+      });
+
+      const logoutButton = screen.getByRole("button", { name: /log out/i });
+      fireEvent.click(logoutButton);
+
+      await waitFor(() => {
+        expect(screen.getByRole("alert")).toBeInTheDocument();
+        expect(screen.getByText(/logout failed/i)).toBeInTheDocument();
+      });
+    });
+  });
 });