Replication Descent and $J$-Finality of Replicable Functions

Abstract

We show that replication carries congruence symmetry down an explicit level tower. If a normalized replicable function $f$, holomorphic on the upper half-plane, is invariant under ${\Gamma }_0(N)$, then its $n$th replicate is invariant under ${\Gamma }_0(N/(N,n))$. Thus every replicate whose index is divisible by $N$ is the normalized modular invariant $J=j-744$. This gives a direct and classification-free proof of $J$-finality for congruence-invariant replicable functions. The argument uses only the replication identities and an elementary generation theorem for congruence subgroups; it requires neither complete replicability nor arithmetic hypotheses on the Fourier coefficients. We then apply the descent law to completely replicable functions of finite replication order. Their replication towers have a canonical terminal replicate, and the only possible terminal functions are $J$, $q^{-1}$, and $q^{-1}+q$. Moreover, a finite-order completely replicable function is $J$-final precisely when its terminal replicate is $J$, and any symmetry beyond translations forces this alternative.