tokens: each ii / jj term and ij / ji term. We start with ii / jj . For a fixed baseline that involves antenna i , there will be at most N ant additional baselines that are redundant with and do not involve gain i . From equations ( 39 ) and ( 40 ), the weighting of the covariance between two different baselines in which only one involves antenna i goes as . Thus, the presence of redundancy adds no more than N ant terms involving antenna i but not antenna j and vice versa, for each ii summand in equation ( 35 ). Multiplying this overall factor of by N ant to account for the N ant different ii sums leads to a contribution to the noise amplitude of the order of , similar to the level of the noise without redundancy. As a result, redundancy changes the modelling error amplitude by a factor of order unity in the diagonal terms. Next, we consider the ij terms in equation ( 35 ). For a given and i j , there will be at most N ant redundant baselines that do not involve the i th or j th gains, causing the weighting of each unique variance term to go as rather than in the non-redundant case. Since there are unique baselines that do not involve i or j , the overall sum of these terms goes as . As a result, the ij terms in equation ( 35 ) will have a similar magnitude as the ii / jj terms but the overall impact on the amplitude of the modelling noise described in equation ( 35 ) still changes the amplitude by a factor of order unity. We confirm these arguments with a numerical comparison between the amplitude of the modelling noise with and without redundancy taken into account for two redundant arrays of 91 and 331 hexagonally packed 14m apertures and S min equal to the naturally weighted confusion limit. We compute the off-diagonal elements of by numerically computing the beam integral in equation ( 5 ) for all with Airy beams and perform the full matrix inversions prescribed in equations ( 16 ) and ( 17 ). We compare our results to the same calculation where all off-diagonal elements of are set to zero (Fig. A1 ) and find that the difference in amplitude is essentially a factor of order unity, leading to a negligible increase in the effective . This calculation confirms our argument for HERA-scale arrays. Figure A1. We compare the amplitude of modelling noise on a short baseline when is assumed to be diagonal (orange line) and the off-diagonal terms of are explicitly included for a 91-element (dashed lines) and 331-element (solid lines) hexagonally packed array of 14m apertures. We find that even in a maximally redundant array, the effect of redundancy is to change the overall amplitude of the modelling noise by a factor of a few. This only has a small impact (10percent) on , the smallest k where the 21cm signal (black line) dominates over the modelling noise, as computed from equation ( 51 ) that ignores the effect of redundancy. APPENDIX B: The Point Source Approximation For Modelled Foregrounds For analytic tractability, we assumed that the modelled
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