  Searches for spacetime variations of fundamental constants have entered an
era of unprecedented precision. New, high quality quasar spectra require
increasingly refined analytic methods. In this article, a continuation in a
series to establish robust and unbiased methodologies, we explore how
convergence criteria in non-linear least squares optimisation impact on quasar
absorption system measurements of the fine structure constant alpha. Given
previous claims for high-precision constraints, we critically examine the
veracity of a so-called ``blinding'' approach, in which alpha is fixed at the
terrestrial value during the model building process, releasing it as a free
parameter only after the ``final'' absorption system kinematic structure has
been obtained. We show that this approach results in an extended flat canyon in
chi squared-alpha space, such that convergence is unlikely to be reached, even
after as many as 1000 iterations. The fix is straightforward: alpha must be
treated as a free parameter from the earliest possible stages of absorption
system model building. The implication of the results presented here is that
all previous measurements that have used initially-fixed alpha should be
reworked.
