The Adelic Simplicial Architecture (ASA) organises computation across the four normed division algebras $\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O}$. The $\mathbb{C}$-rung hosts associative quantum memory (surface codes); the $\mathbb{O}$-rung hosts non-associative transversal execution (Fano-line gates). Bridging these two extremes poses a precise physical challenge: the octonion associator penalty $|\mathcal{A}| = 2$ for non-Fano triples can backscatter into the associative memory grid, producing correlated error bursts that no surface-code decoder anticipates. This paper introduces the Quaternionic Virtual Machine (Q-VM): the formal $\mathbb{H}$-rung routing layer. The Q-VM exploits two structural properties of the quaternions unavailable at either adjacent rung. First, the non-commutativity $AB \neq BA$ of $\mathbb{H}$ enforces a Topological Diode — a directed acyclic graph (DAG) in which logical flow is strictly causal and cannot reverse. Second, the left-right chiral symmetry $\mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R \oplus \mathfrak{su}(2)\mathrm{spin}$ of $\mathrm{Sp}(2) \cong \mathrm{Spin}(5)$ provides a spinorial decomposition connecting directly to Penrose's twistor programme and to the $\mathrm{SU}(2)$ phase geometry of TRS vortons. The Q-VM's primary hardware role is as a backscatter valve in Fibrational Lattice Surgery (LS2.0): a buffer zone at the $\mathbb{C}/\mathbb{O}$ boundary that maintains $|\mathcal{A}| = 0$ on all boundary Fano faces, converting $|\mathcal{A}| = 2$ noise into heraldable erasures before it can corrupt associative memory. The paper also establishes the DVC $\tau_3 = 0$ boundary as the formal routing condition: W-class states are handled by the $\mathbb{H}$-rung; GHZ-class states ($\tau_3 > 0$) are forwarded to the $\mathbb{O}$-rung. The BOIL→SNAP phase transition of TRS vorton dynamics is identified as spontaneous chiral symmetry breaking in the $\mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$ decomposition, grounding the empirical $Q_L/Q_R$ asymmetric schedule of TRS v68.0.1 in the quaternionic algebra.