= Solution
The strongest evidence for large-scale <cosmological isotropy> is the nearly direction-independent blackbody temperature of the <cosmic microwave background>. After removing the dipole associated mainly with our <peculiar velocity>, its residual temperature fluctuations are of order $10^{-5}$. Nearly isotropic counts of distant radio sources and <galaxies>, after correcting for sky coverage and selection effects, provide independent evidence. Local <galaxies>, clusters and voids clearly are not isotropic structures; the claim concerns a statistical description on sufficiently large scales.
Large redshift surveys show that the mean <galaxy> density stabilizes as increasingly large volumes are sampled, and that the excess pair correlations weaken at large separations. This supports <statistical homogeneity> beyond the characteristic clustering scales. The consistency of large-scale observations in different directions and the success of a common <FRW metric> description are additional checks.
<Isotropy> around one observer does not alone prove <spatial homogeneity>: a radially varying distribution centered on that observer could look isotropic. The <Copernican principle> supplies the assumption that we occupy no preferred position; comparable isotropy about typical locations then supports the <cosmological principle>. Surveys observe a finite <past light cone>, with source evolution and selection, so they cannot establish homogeneity everywhere on a simultaneous spatial slice. \b[The evidence supports large-scale statistical isotropy and homogeneity, with substantial small-scale structure and an explicit typical-observer assumption.]
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