Every earlier recovery lesson fixed equilibrium and scanned start or recovery fraction; this scan holds those fixed and scans equilibrium itself. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Every earlier recovery row held equilibrium fixed
Every earlier recovery lesson in this book fixed the equilibrium index at 16 or 12 and only ever varied the start or the recovery fraction. Here start stays 0 and the recovery fraction stays 1/2, but the equilibrium itself drops to 8, leaving gap 8 and final index 4.
8−0=8,21⋅8=4
Equilibrium itself is a source factor for the gap
Start and recovery fraction never change across these rows. Equilibrium alone rescales the gap and, through it, the checked final index.
12−0=12
The book's own equilibrium closes the scan
At equilibrium 16, the same start and recovery fraction give gap 16 and final index 8. This row reproduces the book's own first recovery lesson exactly. Three rows separate equilibrium from the start and fraction factors already scanned elsewhere in this book.