Drive and Damping Are Finite Ledgers

Off-Frequency Kicks Do Not Claim Growth

Visible kicks are not enough when the drive frequency mismatches. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Off-frequency drive closes the growth claim

The natural frequency is 4 hertz, but the drive frequency is 3 hertz.

fnat=4 Hz,fdrive=3 Hzf_\mathrm{nat}=4\ \mathrm{Hz},\quad f_\mathrm{drive}=3\ \mathrm{Hz}
Off-frequency driveThe match flag is no.f0=4 Hzfdrive=3 HzkickTotal=8 Jaccepted=0 Jmatch=nokick1=2 Jkick2=2 Jkick3=2 Jkick4=2 J

Visible kicks do not imply coherent growth

Even with four visible kicks, the accepted growth is exactly 0 joules.

accepted=0 J\mathrm{accepted}=0\ \mathrm{J}
Rejected growth ledgerThe toy ledger rejects off-frequency growth.f0=4 Hzfdrive=3 HzkickTotal=8 Jaccepted=0 Jmatch=nokick1=2 Jkick2=2 Jkick3=2 Jkick4=2 J