 Solution

From order $k+1$, set $z=a_{k+1}^{\prime }$, remove that final coefficient, retain $a_i=a_i^{\prime }$ for $1\le i\le k$, and restore the intercept byKeep $v$ fixed. This deterministically recovers the lower-order state and the auxiliary variable that would reproduce the higher-order state under a birth. The inverse map also has absolute Jacobian determinant one. In particular, simply deleting the highest coefficient without restoring the intercept is not the reverse of the supplied move.