  High resolution spectra of quasar absorption systems provide the best
constraints on temporal or spatial changes of fundamental constants in the
early universe. An important systematic that has never before been quantified
concerns model non-uniqueness. The absorption structure is generally
complicated, comprising many blended lines. This characteristic means any given
system can be fitted equally well by many slightly different models, each
having a different value of \alpha, the fine structure constant. We use AI
Monte Carlo modelling to quantify non-uniqueness. Extensive supercomputer
calculations are reported, revealing new systematic effects that guide future
analyses: (i) Whilst higher signal to noise and improved spectral resolution
produces a smaller statistical uncertainty for \alpha, model non-uniqueness
adds a significant additional uncertainty. (ii) Non-uniqueness depends on the
line broadening mechanism used. We show that modelling the spectral data using
turbulent line broadening results in far greater non-uniqueness, hence this
should no longer be done. Instead, for varying \alpha studies, it is important
to use the more physically appropriate compound broadening. (iii) We have
studied two absorption systems in detail. Generalising thus requires caution.
Nevertheless, if non-uniqueness is present in all or most quasar absorption
systems, it seems unavoidable that attempts to determine the existence (or
non-existence) of spacetime variations of fundamental constants is best
approached using a statistical sample.
