def _online_network_benchmarking_truly_strict(network, path_list, bounces, C_budget):
    candidate_set = list(path_list)
    s = 0
    C_const = 0.01
    delta = 0.1
    cost = 0
    estimated_fidelities = {}
    

    cost_per_sample_unit = sum(bounces)
    if cost_per_sample_unit == 0:
        cost_per_sample_unit = 1

    while cost < C_budget and len(candidate_set) > 1:
        s += 1
        Ns = math.ceil(C_const * 2**(2 * s) * math.log2(2**s * len(candidate_set) / delta))
        if Ns < 4:
            Ns = 4
        
        cost_needed_for_one_path = Ns * cost_per_sample_unit
        if cost + cost_needed_for_one_path > C_budget and s > 1:
             break

        p_s = {}
        sample_times = {i: Ns for i in bounces}
        for path in candidate_set:
            if cost + cost_needed_for_one_path > C_budget:
                continue
            p, bounces_num = network.benchmark_path(path, bounces, sample_times)
            cost += bounces_num
            estimated_fidelity = p + (1 - p) / 2
            estimated_fidelities[path] = estimated_fidelity
            p_s[path] = p
        if not p_s:
            break
        p_max = max(p_s.values())
        new_candidate_set = [path for path in candidate_set if path in p_s and p_s[path] + 2**(-s) > p_max - 2**(-s)]
        candidate_set = new_candidate_set
        
    if not estimated_fidelities:

    best_path = max(estimated_fidelities, key=estimated_fidelities.get)
    best_path_fidelity = estimated_fidelities[best_path]
    correctness = (best_path == network.best_path)