What the First Overtone Must Earn: An Amplitude and Signal-to-Noise Budget for the n=1 Mode in GW150914

Claims that the first overtone of the fundamental angular mode has been seen in GW150914 have been made and disputed in the literature, and the dispute turns less on principle than on arithmetic. This paper computes that arithmetic directly: how fast the overtone decays relative to the fundamental, what initial amplitude ratio it needs in order to contribute a stated signal-to-noise ratio of its own, and how that requirement moves with the choice of analysis start time. The fundamental mode has frequency 249.1 Hz and damping time 4.05 ms; the first overtone is assigned 245.4 Hz and 1.33 ms, a damping-time ratio of 0.328. At equal starting amplitudes the overtone carries a fraction 0.573 of the fundamental's signal-to-noise ratio, purely because it decays faster. For the overtone to contribute a signal-to-noise ratio of 3 alongside a fundamental contributing 8, its starting amplitude must be at least 0.65 of the fundamental's starting amplitude. The overtone decays 0.33 times as fast in damping time, needs a starting amplitude of at least 0.65 of the fundamental to contribute a signal-to-noise ratio of 3 beside a fundamental of 8, and loses a factor 0.13 in relative amplitude within the first 4 ms. This study is computed from stated assumptions and published catalogue values; it is simulated, not measured.