The function takes a string as input and applies a Caesar cipher-like transformation on each character. Each lowercase letter in the input string is shifted a fixed number of positions in the alphabet. Characters outside the lowercase letter range are unchanged.

[CODE]:

```python
def f(input_string):
    shift = X  # The fixed number of positions each character is shifted

    output_string = ''
    for char in input_string:
        if 'a' <= char <= 'z':
            shifted_char = chr((ord(char) - ord('a') + shift) % 26 + ord('a'))
        else:
            shifted_char = char
        output_string += shifted_char

    return output_string
```

Please note that the value of `shift` (denoted as X in the code above) needs to be determined based on further experimentation. By testing the function on additional input strings, we can gather more data to determine the exact value of the shift.