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We revisit the topic of invisible neutrino decay in the precision cosmological context, via a first-principles approach to understanding the cosmic microwave background and large-scale structure phenomenology of such a non-standard physics scenario. Assuming an effective Lagrangian in which a heavier standard-model neutrino $ν_H$ couples to a lighter one $ν_l$ and a massless scalar particle $ϕ$ via a Yukawa interaction, we derive from first principles the complete set of Boltzmann equations, at both the spatially homogeneous and the first-order inhomogeneous levels, for the phase space densities of $ν_H$, $ν_l$, and $ϕ$ in the presence of the relevant decay and inverse decay processes. With this set of equations in hand, we perform a critical survey of recent works on cosmological invisible neutrino decay in both limits of decay while $ν_H$ is ultra-relativistic and non-relativistic. Our two main findings are: (i) in the non-relativistic limit, the effective equations of motion used to describe perturbations in the neutrino--scalar system in the existing literature formally violate momentum conservation and gauge invariance, and (ii) in the ultra-relativistic limit, exponential damping of the anisotropic stress does not occur at the commonly-used rate $Γ_{\rm T} =(1/τ_0) (m_{νH}/E_{νH})^3$, but at a rate $\sim (1/τ_0) (m_{νH}/E_{νH})^5$. Both results are model-independent. The impact of the former finding on the cosmology of invisible neutrino decay is likely small. The latter, however, implies a significant revision of the cosmological limit on the neutrino lifetime $τ_0$ from $τ_0^{\rm old} \gtrsim 1.2 \times 10^9\, {\rm s}\, (m_{νH}/50\, {\rm meV})^3$ to $τ_0 \gtrsim (4 \times 10^5 \to 4 \times 10^6)\, {\rm s}\, (m_{νH}/50 \, {\rm meV})^5$.