Self-similarity of Fibonacci period structures at prime powers

For the Fibonacci recurrence and its parity transform modulo prime powers, Self-Similarity at Prime Powers. At each transition $p^k\to p^{k+1}$, all existing periods are preserved, and each period of length $\ell>1$ at $p^k$ generates new periods of length $p\ell$ at $p^{k+1}$, with multiplicities multiplied by $p$. For class B2 primes, the multiplicity of the middle period remains exactly $2\alpha$ for every power $p^k$. The supplied text gives examples but no resolution status.