Pushing one row inward toward a vertex induces a graph contraction

Let $K=\mathrm{Hpoly}(A,B)$ be bounded with $B_i>0$, let $p,q$ be vertices, row $r$ tight at $p$ and slack at $q$, and $h$ a direction with $\langle h,p\rangle=0$ such that on the face $\{x\in K:\langle A_r,x\rangle=B_r\}$ one has $\langle h,x\rangle\le 0$ with equality only at $x=p$ (an 'inward push toward $p$'). Replace $A_r$ by $A_r+s\,h$ for $0<s<\eta$. Then for some $\eta>0$: the perturbed polytope is bounded, $p$ and $q$ remain vertices, and there is a map $\varphi$ from vertices of the perturbed polytope to vertices of $K$ fixing $p$ and $q$ that sends every perturbed edge to an edge of $K$ or collapses it to a point.

This is the primal form of Santos' Lemma 2.2 (pushing a vertex does not decrease dual distances). The proof classifies vertices of the perturbed polytope by their tight bases: bases invertible at $s=0$ give branches converging to old vertices, and bases that become invertible only for $s>0$ are constant branches lying on the old $r$-face with $\langle h,x\rangle=0$, hence equal to $p$. Edges are handled by the rank of their common tight rows, with an explicit finite-minor bound on $\eta$.