Incidence Obstructions for Power Products in Elliptic Divisibility Sequences

Let $E/\mathbb Q$ be an elliptic curve, let $P\in E(\mathbb Q)$ be non-torsion, and let $(D_n)$ be the associated elliptic divisibility sequence. We study when a product $$\prod_{i=1}^k D_{n_i}$$ can be a $\rho$-th power, where $\rho$ is a fixed prime. Our obstructions concern the incidence of prime divisors among the indices $n_i$, rather than the prime factorisation of individual EDS terms. Assuming that $D_1$ is divisible by $2$ or $3$, we show that sufficiently large primes $\ell$ occurring as simple largest prime divisors of the indices, with $n_i/\ell$ $B$-smooth whenever $\ell\mid n_i$, must occur in $\rho$-balanced incidence blocks. In particular, for a finite family of such primes, the associated incidence rows are disjoint, linearly independent over $\mathbb F_\rho$, and satisfy $$|\Lambda^\ast|\le \lfloor k/\rho\rfloor .$$