= Solution
From order $k+1$, set $z=a'_{k+1}$, remove that final coefficient, retain $a_i=a'_i$ for $1\leq i\leq k$, and restore the intercept by
$$
\boxed{a_0=a'_0+z\left(\frac1n\sum_{i=1}^n x_i^{k+1}\right).}
$$
Keep $v$ 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.
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