It seems that you're trying to implement a variation of the binary search algorithm in Python to find the index of a given number `a` in a sorted list `x`. The description you've provided suggests modifications to a typical binary search, particularly around how the list is sliced during the search process. Below is an implementation of the algorithm according to your specification:

```python
def modified_binary_search(x, a):
    left, right = 0, len(x) - 1
    
    while right - left > 1:  # Continue while the length of the sublist is greater than 2
        mid = (left + right) // 2
        if x[mid] == a:
            return mid

        if x[mid] < a:
            left = mid  # Consider the sublist starting from mid to right
        else:
            right = mid  # Consider the sublist starting from left to mid

    # If the loop exits and the length of the list is 2 or less:
    if x[left] == a:
        return left
    if right != left and x[right] == a:
        return right
    return -1  # If a is not found

# Example usage:
sorted_list = [1, 3, 4, 6, 8, 10, 15]
target_number = 6

result = modified_binary_search(sorted_list, target_number)
print(f'The index of {target_number} in the list is {result}.')
```

Explanation of the Code:
- The function `modified_binary_search` takes a sorted list `x` and a target value `a`.
- It uses two pointers, `left` and `right`, which initially point to the start and end of the list.
- A while loop continues to execute until the difference between `right` and `left` is more than 1, i.e., the sublist is reduced to a length of 2 or less.
- Inside the loop, it calculates the middle index (`mid`), checks if the value at `mid` equals `a`, and returns `mid` if a match is found.
- If no match is found, it updates the `left` or `right` pointer based on the comparison between `x[mid]` and `a`, thus slicing the list either from `mid` to `right` or from `left` to `mid`.
- If the loop exits because the sublist is of length 2 or less, it performs an additional check on the values at `left` and `right` positions.
- If `a` is not found, it returns -1.

This implementation optimizes the search for the element by progressively narrowing down the sublist based on the value comparison, similar to binary search but adapting the sublist boundaries non-traditionally.