Controlled pagination pill with lock & fit-scroll

Make the floating pagination pill a controlled component and add lock/fit-aware scroll behavior. UnifiedPagination now accepts activeIndex and onDominantSectionChange, reports observer-determined dominant section (parent may ignore when locked) and only shows/hides based on footer visibility. HomePage implements controlled state (activeIndex, lockedTo) with Prev/Next/jump locking, release on wheel/touch/key or 30s safety timeout, and dot clicks that always navigate and release locks. Extracted scroll math to src/lib/utils/paginationScroll.ts (decideScrollForPageChange) so paging avoids scrolling when a section fits below the sticky header and clamps targets; added unit tests and updated component tests and docs to reflect the new behavior. Removed now-unused onPageChange prop from HomeSection.

tests/app/home.page.test.tsx

@@ -66,6 +66,8 @@ vi.mock('@/components/ui/UnifiedPagination', () => ({
       label: string;
       onPageChange: (page: number) => void;
     }>;
+    activeIndex: number;
+    onDominantSectionChange: (idx: number) => void;
   }) => (
     <div>
       {sections.map((s) => (