We present a framework for embedding arbitrary real linear generators into continuous-variable quantum systems through deterministic first-moment dynamics of a quadratic Hamiltonian. We show that any matrix $A\in {\mathbb{R}}^{N\times N}$ can be realized exactly on an invariant locking manifold, where the embedded dynamics evolves as $\langle \dot{\mathbf{x}}\rangle =A\langle \mathbf{x}\rangle$. Reversibility of the full symplectic evolution enforces a conjugate sector evolving under $-A^{\mathsf{T}}$, establishing a fundamental trade-off: Any contraction generated by $A$ must be compensated by expansion in the adjoint sector. For dissipative generators, this renders the locking manifold transversely unstable, requiring periodic resets. We derive resource bounds, including squeezing requirements that scale as $O\left(N^{2}t\right)$ for diffusive dynamics, and classify generators by their stability properties: Dissipative generators require active stabilization, conservative generators are marginally stable, and amplifying generators self-stabilize. Numerical validation of three discretized partial differential equations, including heat, advection, and backward heat, confirms the theoretical predictions with relative errors below ${10}^{-8}\%$.

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• Quantum information processing with continuous variables
• Quantum simulation
• Quantum-to-classical transition