Circuit cutting allows quantum circuits to be evaluated on smaller devices at the cost of additional sampling and classical reconstruction. For Hamiltonian-simulation circuits, the target unitary $\exp[it\mathbf H]$ connects this decomposition problem to the structure of the generator. We develop a Hamiltonian-level certificate that assesses exact separability and the accuracy of independent module evolution before a gate decomposition is chosen. For a time-independent Hamiltonian and a fixed module partition, the Hilbert--Schmidt projection onto sums of module-local terms isolates a cross-module residual $H_{ns}$. Its vanishing is necessary and sufficient for product evolution at all times; for Pauli-list inputs, this condition is checked in one pass over the combined coefficients. When $H_{ns}\ne0$, the projected evolution has operator-norm error at most $|t| ||H_{ns}||_{op}$. Across a bipartition, an explicit first-order algebraic correction has the same operator-Schmidt rank as $H_{ns}$ and leaves a second-order residual. The same Hamiltonian residual connects this approximation to short-time entanglement, determining the leading state-space and operator-space entangling powers and, through its Schmidt spectrum, the leading Choi tripartite-information terms. Numerical experiments test these relations and demonstrate block-factorised simulation up to $20$ qubits.

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