First-impact step for a one-dimensional unilateral contact

Input a gap $q\ge 0$, velocity $v\in \mathbb{R}$, constant acceleration $a\in \mathbb{R}$, step horizon $h>0$, mass $m>0$, restitution $e\in [0,1]$, and a candidate-root budget $B\in \mathbb{N}$. Use exact real arithmetic and comparisons. Define $g(s)=q+sv+\frac{1}{2}a{s}^2$ and $w(s)=v+as$ on $[0,h]$. Return impact, free-flight, or unresolved by these instructions.

• If $q=0$ and $v<0$, return impact at $s=0$ with contact gap $0$, $v^{-}=v$, $J=-m(1+e)v^{-}$, and $v^{+}=-e{v}^{-}$.
• Otherwise form the complete candidate set $R$ of real roots of $g$. If $q=v=a=0$, set $R=\mathrm{\varnothing }$. If $a=0$ and $v\mathrm{\ne }0$, set $R=\{-q\mathrm{/}v\}$. If $a=0$, $v=0$, and $q>0$, set $R=\mathrm{\varnothing }$. If $a\mathrm{\ne }0$, put $D=v^2-2aq$: for $D<0$ set $R=\mathrm{\varnothing }$; for $D=0$ set $R=\{-v\mathrm{/}a\}$; and for $D>0$ set $R=\{(-v-\sqrt{D})\mathrm{/}a,(-v+\sqrt{D})\mathrm{/}a\}$, with duplicate values removed. If $\mathrm{\mid }R\mathrm{\mid }>B$, return unresolved. From $R$, retain exactly the values $s$ satisfying $0<s\le h$ and $w(s)<0$, and order them increasingly.
• If the retained list is nonempty, take its least value $s$, set $v^{-}=w(s)$, $J=-m(1+e)v^{-}$, and $v^{+}=-e{v}^{-}$, and return impact with time $s$, contact gap $0$, impulse $J$, and post-impact velocity $v^{+}$.
• If the retained list is empty, compute the minimum of $g$ on $[0,h]$ by evaluating its two endpoints and, when $a>0$ and $-v\mathrm{/}a\in (0,h)$, its vertex $-v\mathrm{/}a$. If this minimum is negative, return unresolved; otherwise return free-flight with endpoint $(g(h),w(h))$ and impulse $0$.

The candidate set contains every root of the nonzero polynomial $g$, so the retained list is complete. The vertex check certifies nonpenetration over the whole step, including a grazing zero. The procedure stops after this first event and returns at the event time; it does not integrate the post-impact mode, sustained contact, or repeated impacts. A numerical implementation that cannot certify an exact comparison returns unresolved.