  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 component 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 and describe how it accounts
for previously unexplained scatter seen in the majority of published
measurements. Extensive supercomputer calculations are reported, revealing new
systematic effects that guide future analyses: (i) systematic errors
significantly increase if line broadening models are turbulent but are
minimised if gas temperature is included as a free parameter; modelling quasar
absorption systems using turbulent broadening should be avoided and compound
broadening is preferable. (ii) The general overfitting tendency of AICc
dramatically increases non-uniqueness and hence the overall error budget on
estimates of \alpha variations. The newly introduced Spectral Information
Criterion (SpIC) statistic is more suitable and substantially decreases
non-uniqueness compared to AICc-based models.
