$J v$ is exact, not a finite difference: it is computed by lucid.autograd.jvp in two reverse passes (the double-VJP trick), or — through an op whose backward has no graph mode, such as LSTM or a padded avg_pool — in one reverse pass per output element.

The results join the caller's autograd graph as the reference's do. Under no_grad neither does. Otherwise the primal output is attached when it depends on a tensor that requires grad — a primal, or a parameter func captures — and the tangent is attached when a primal, a tangent or such a parameter requires grad. Through an op with no graph-mode backward the tangent cannot be attached: when only captured parameters would attach it, it comes back detached with a warning.

func runs once when grad mode is off or a primal requires grad. With grad mode on and no primal requiring grad it runs twice: once at the primals, which is the only way to learn whether the output reaches the caller's graph through tensors func captures, and once for the tangent. The second run replays the first one's random draws, so a dropout mask is the same in both, but any other side effect happens twice — a BatchNorm in training mode updates its running statistics twice, and an in-place buffer update is applied twice.