Limiting Betti-number distributions in preferential-attachment clique complexes

We study the finite clique complexes of sequential affine preferential-attachment graphs. For every $q\ge2$, $m\ge2q$ and $\delta>-m$ with $x=(m+\delta)/(2m+\delta)<1/(2q)$, the $q$th rational Betti number, divided by its expectation, converges to a nondegenerate random variable. We identify its law as a normalized sum of products of attachment weights over induced octahedral subgraphs in an explicit independent-Beta construction. The limit is positive almost surely, has zero essential infimum and has every positive moment. In particular, its upper tail is not regularly varying with any finite positive exponent. The proof compares all octahedral link contributions with finite-time homology; larger link cores have lower order, and an earlier cone makes each fixed base contribute to subsequent births.