```python
"""A block-allocation bitmap: 32-bit words, LSB-first, a set bit is occupied."""

WORD_BITS = 32
WORD_MAX = 0xFFFFFFFF


def _whole(value):
    return not isinstance(value, bool) and isinstance(value, int)


def _check_words(words):
    for word in words:
        if not _whole(word) or word < 0 or word > WORD_MAX:
            raise ValueError("bitmap words must be integers from 0 to 4294967295")


def _bit_of(words, index):
    return (words[index // WORD_BITS] >> (index % WORD_BITS)) & 1


def bit_at(words, index):
    _check_words(words)
    if not _whole(index):
        raise ValueError("bit index must be an integer")
    if index < 0 or index >= len(words) * WORD_BITS:
        raise ValueError("bit index is outside the words")
    return _bit_of(words, index)


def occupy_run(words, start, length):
    _check_words(words)
    if not _whole(start) or start < 0:
        raise ValueError("run start must be a non-negative integer")
    if not _whole(length) or length <= 0:
        raise ValueError("run length must be a positive integer")
    if start + length > len(words) * WORD_BITS:
        raise ValueError("run leaves the words")
    for i in range(start, start + length):
        if _bit_of(words, i) == 1:
            raise ValueError("run touches an occupied block")
    marked = list(words)
    for i in range(start, start + length):
        marked[i // WORD_BITS] |= 1 << (i % WORD_BITS)
    return marked


def find_free_run(words: list, capacity: int, length: int) -> int:
    if not isinstance(capacity, int) or capacity <= 0:
        raise ValueError("capacity must be a positive integer")
    if len(words) * WORD_BITS < capacity:
        raise ValueError("word count does not match the capacity")
    if not isinstance(length, int) or length <= 0:
        raise ValueError("run length must be a positive integer")

    # Check if there are enough words to hold the capacity
    if len(words) != (capacity + WORD_BITS - 1) // WORD_BITS:
        raise ValueError("word count does not match the capacity")

    # Calculate the number of full words and the remaining bits
    full_words = capacity // WORD_BITS
    remaining_bits = capacity % WORD_BITS

    # Check for stray bits at or beyond the capacity
    if len(words) * WORD_BITS > capacity:
        for i in range(full_words, len(words)):
            if _bit_of(words, i * WORD_BITS):
                raise ValueError("stray bit at or beyond the capacity")

    # Find a free run of length below the capacity
    for start in range(len(words) * WORD_BITS - length + 1):
        if all(not _bit_of(words, i) for i in range(start, start + length)):
            return start

    return -1
```