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In multidimensional systems with matrix-valued noise amplitudes $\\sigma(x)$, this dependence includes a local Jacobian contribution that is absent from the scalar examples most often used to build intuition. We formulate a finite-step path-likelihood framework for $\\theta$-discretized diffusions and show that its short-time expansion isolates the scalar $J_\\sigma=\\partial_j\\sigma_{ik}\\partial_i\\sigma_{jk}-(\\partial_i\\sigma_{ik})(\\partial_l\\sigma_{lk})$. 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In multidimensional systems with matrix-valued noise amplitudes $\\sigma(x)$, this dependence includes a local Jacobian contribution that is absent from the scalar examples most often used to build intuition. We formulate a finite-step path-likelihood framework for $\\theta$-discretized diffusions and show that its short-time expansion isolates the scalar $J_\\sigma=\\partial_j\\sigma_{ik}\\partial_i\\sigma_{jk}-(\\partial_i\\sigma_{ik})(\\partial_l\\sigma_{lk})$. 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