function pt = block_phase_transition(N, M)
    tau_min = 0;
    tau_max = 10;
    tau_interval = 0.05;
    tau_range = tau_min:tau_interval:tau_max;
    
    K_min = 1;
    K_max = 25;
    K_range = K_min:K_max;
    
    pt = zeros(25, 1);
    
    cache_filename = "I.mat";
    
    if exist(cache_filename, "file")
        load(cache_filename);
    else
        I = zeros(length(tau_range), 0);
    
        for tau_idx = 1:length(tau_range)
            tau = tau_range(tau_idx);
            I(tau_idx) = calc_block_integral(tau, 4);
        end
        save I.mat
    end
    
    
    for K_idx = 1:length(K_range)
        K = K_range(K_idx);
        f_set = zeros(length(tau_range), 1);
    
        for tau_idx = 1:length(tau_range)
            tau = tau_range(tau_idx);
            f_set(tau_idx) = 1/2 * (K * (2 * M + tau ^ 2) + (N - K) * I(tau_idx));
        end
    
        pt(K_idx) = min(f_set);
    end
end