Paths in an Implicit Fibonacci Tree

Question

Define a Fibonacci tree of order `k` as a root whose left subtree has order `k-1` and whose right subtree has order `k-2`. Nodes are labeled by their one-based preorder position. Compute paths without materializing the tree.

```python
def tree_size(order: int) -> int: ...
def fib_path(order: int, start: int, end: int) -> list: ...
def fib_directions(order: int, start: int, end: int) -> list: ...
def fib_lca(order: int, start: int, end: int) -> int: ...
```

## Requirements

- Use `tree_size(k) = F(k+2) - 1`, with `F(0)=0` and `F(1)=1`.
- `fib_path` returns node labels from `start` to `end`, inclusive.
- `fib_directions` returns `"Up"` steps to the lowest common ancestor, then `"Left"` or `"Right"` steps downward.
- Validate that labels lie within the requested tree.
- Order zero is an empty tree, order one is a single root, and subtrees of order zero or below are empty. Public orders must be non-negative non-boolean integers; labels must be non-boolean integers in `1..tree_size(order)`. Raise `ValueError` otherwise, including path queries on an empty tree. Equal endpoints return a single-label path and no directions.
- The supplied `run_case` is fixture dispatch glue. Underscored helpers are optional; implement the four listed interfaces.
- Time and memory should depend on tree height, not total node count.

For example, `fib_path(4, 4, 7)` returns `[4, 3, 2, 1, 6, 7]`.

## Design discussion (not graded)

1. Support very large orders where sizes exceed fixed-width integers.
2. Return only path length without constructing the path.
3. Answer many path or LCA queries for the same order.