across the 280-sample pull, beta_c_estimate stayed sharply interior, usually about 1.89 to 2.56 with 4.23e-12 brackets at N=1024 and about 1.64e-14 brackets at N=2048. But the terminal detection fraction was 0.0 in every sampled payload, spike overlap stayed tiny at only about 0.014 to 0.036, and the top gap ratio was often enormous at about 200 to 2500, so the family is repeatedly producing dramatic spectral gaps without actual spike recovery.