| analyze_top_combinations | Analyze Top Operation Combinations |
| apply_operations | Apply Sequence of Operations |
| apply_operations_batch_gpu | Apply operations to batch of states on GPU |
| bidirectional_bfs | Bidirectional BFS Shortest Path |
| breakpoint_distance | Breakpoint Distance Between Two States |
| calculate_angular_distance_z | Angular Distance Between Two Celestial Points |
| calculate_differences | Calculate Manhattan Distances for All States |
| calculate_midpoint_z | Midpoint Between Two Celestial Coordinates |
| cayley_gpu_available | Check if GPU acceleration is available |
| cayley_gpu_free | Free GPU backend resources |
| cayley_gpu_init | Initialize GPU backend |
| cayley_gpu_status | Get GPU status information |
| check_duplicates | Find Duplicate States Between Two Tables |
| convert_digits | Convert String to Integer Vector of Digits |
| convert_LRX_to_celestial | Convert LRX Counts to Celestial Coordinates |
| find_best_random_combinations | Find Best Random Operation Sequences |
| find_closest_to_coords | Find Closest State to Target Coordinates |
| find_combination_in_states | Find a State in Reachable States Table |
| find_path_bfs | Find Path via BFS Highways |
| find_path_iterative | Iterative Path Finder Between Permutation States |
| generate_state | Generate Reachable Random State |
| generate_unique_states_df | Generate Data Frame of Unique Random States |
| get_reachable_states | Find Cycle in Permutation Group |
| get_reachable_states_light | Find Cycle Length (Lightweight Version) |
| invert_path | Invert a Path of Operations |
| manhattan_distance | Manhattan Distance Between Two States |
| manhattan_distance_matrix_gpu | Compute Pairwise Manhattan Distance Matrix on GPU |
| reconstruct_bfs_path | Reconstruct path from sparse BFS result |
| reverse_prefix | Reverse First k Elements (with Coordinates) |
| reverse_prefix_simple | Reverse First k Elements (Simple) |
| save_bridge_states | Save Bridge States to CSV |
| select_unique | Select Unique States by V-columns |
| shift_left | Shift State Left (with Coordinates) |
| shift_left_simple | Shift State Left (Simple) |
| shift_right | Shift State Right (with Coordinates) |
| shift_right_simple | Shift State Right (Simple) |
| short_path_bfs | Shorten Path via Greedy BFS Hopping |
| short_position | Simplify Operation Path |
| sparse_bfs | Sparse BFS with Look-ahead and Hybrid Selection |
| validate_and_simplify_path | Validate and Simplify a Path |