fix(ui): the page's end scrolls out from under Back to top

The button floats over the bottom-right corner, and at the end of a long table it covered the closing
pager's '100' and 'All'. While it is up, PageScroller ends the page a button's height later (only
then, so a full-height layout gets no scrollbar). Its shown state is a shared store read through a
use… wrapper: a store's .use() called straight from a component is memoized by the React Compiler
and never re-renders (contract: store.contract.test.ts).

frontend/src/components/ui/PageScroller.dom.test.tsx

@@ -7,7 +7,7 @@
 // through window.history.state (what ensureNavId/peekNavId read) — firing the nav bus (notifyNav)
 // stands in for a Back/Forward that doesn't re-render the scroller.
 import { describe, it, expect, vi, beforeAll, beforeEach, afterEach } from "vitest";
-import { render, fireEvent, cleanup } from "@testing-library/react";
+import { render, fireEvent, cleanup, act } from "@testing-library/react";
 
 vi.mock("react-i18next", async (importOriginal) => {
   const actual = await importOriginal<typeof import("react-i18next")>();
@@ -15,6 +15,7 @@ vi.mock("react-i18next", async (importOriginal) => {
 });
 
 import PageScroller from "./PageScroller";
+import BackToTop from "./BackToTop";
 import { notifyNav } from "../../lib/history";
 
 beforeAll(() => {
@@ -97,3 +98,27 @@ describe("PageScroller scroll memory", () => {
     expect(main.scrollTop).toBe(300);
   });
 });
+
+// E28 UAT: Back to top floats over the bottom-right corner, and at the end of a long table it covered
+// the closing pager's "100" and "All". While it is up, the page ends a button's height later.
+describe("PageScroller · room for Back to top", () => {
+  it("ends the page later while the button is up, and not otherwise", () => {
+    const { container } = render(
+      <>
+        <PageScroller scrollKey="room">
+          <p>rows</p>
+        </PageScroller>
+        <BackToTop />
+      </>,
+    );
+    const main = container.querySelector("main")!;
+    const spacer = () => main.querySelector(':scope > [aria-hidden="true"]');
+    expect(spacer()).toBeNull();
+    main.scrollTop = 900;
+    act(() => void main.dispatchEvent(new Event("scroll")));
+    expect(spacer()?.className).toBe("h-18");
+    main.scrollTop = 0;
+    act(() => void main.dispatchEvent(new Event("scroll")));
+    expect(spacer()).toBeNull();
+  });
+});