```python
def fold_machine(machine: dict) -> dict:
    alphabet = machine.get("alphabet")
    states = machine.get("states")
    start = machine.get("start")
    accepting = machine.get("accepting")
    moves = machine.get("moves")

    if not alphabet or not states or not start or not moves:
        raise ValueError("Machine must have non-empty alphabet, states, start, and moves.")

    if len(alphabet) != len(set(alphabet)):
        raise ValueError("Alphabet must contain distinct symbols.")
    if len(states) != len(set(states)):
        raise ValueError("States must contain distinct names.")
    if not all(isinstance(s, str) and s for s in alphabet + states):
        raise ValueError("All symbols and state names must be non-empty strings.")

    if start not in states:
        raise ValueError(f"Start state '{start}' is not a listed state.")

    if any(state not in states or accepting.count(state) > 1 for state in accepting):
        raise ValueError("Accepting states must be distinct and listed states.")

    valid_moves = set((s, sym, t) for s, sym, t in moves if s in states and sym in alphabet and t in states)
    if len(valid_moves) != len(alphabet) * len(states):
        raise ValueError("Each state-symbol pair must have exactly one move.")

    reachable_states = {start}
    queue = [start]
    while queue:
        current = queue.pop()
        for s, sym, t in moves:
            if s == current and t not in reachable_states:
                reachable_states.add(t)
                queue.append(t)

    unreachable_states = set(states) - reachable_states
    machine["moves"] = [(s, sym, t) for s, sym, t in moves if s not in unreachable_states and t not in unreachable_states]

    indistinguishable_groups = {state: None for state in states}
    group_number = 0

    def dfs(state):
        nonlocal group_number
        if indistinguishable_groups[state] is None:
            indistinguishable_groups[state] = group_number
            group_number += 1
            for sym, t in sorted((sym, t) for s, sym, t in moves if s == state):
                dfs(t)

    dfs(start)
    machine["moves"] = [(indistinguishable_groups[s], sym, indistinguishable_groups[t]) for s, sym, t in machine["moves"]]
    machine["states"] = list(indistinguishable_groups.values())
    machine["accepting"] = sorted(indistinguishable_groups[state] for state in accepting if indistinguishable_groups[state] is not None)
    machine["size"] = len(machine["states"])
    machine["start"] = 0

    return machine
```