  Relativistic plasma jets are observed in many accreting black holes.
According to theory, coiled magnetic fields close to the black hole accelerate
and collimate the plasma, leading to a jet being launched. Isolating emission
from this acceleration and collimation zone is key to measuring its size and
understanding jet formation physics. But this is challenging because emission
from the jet base cannot be easily disentangled from other accreting
components. Here, we show that rapid optical flux variations from a Galactic
black-hole binary are delayed with respect to X-rays radiated from close to the
black hole by ~0.1 seconds, and that this delayed signal appears together with
a brightening radio jet. The origin of these sub-second optical variations has
hitherto been controversial. Not only does our work strongly support a jet
origin for the optical variations, it also sets a characteristic elevation of
<~10$^3$ Schwarzschild radii for the main inner optical emission zone above the
black hole, constraining both internal shock and magnetohydrodynamic models.
Similarities with blazars suggest that jet structure and launching physics
could potentially be unified under mass-invariant models. Two of the
best-studied jetted black hole binaries show very similar optical lags, so this
size scale may be a defining feature of such systems.
