We introduce the record filter, the sequence of denominators producing the longest decimal period seen so far and prove that every record denominator is prime. We prove that the record sequence does not terminate, establishing that multiplicative orders grow without bound. The main result is a separation theorem: unbounded growth of ordₐ (p) is logically independent from the condition ordₐ (p) = p − 1 for infinitely many primes. The separation is structural, arising from the divisor lattice of p − 1, not from quantitative bounds. A corollary constrains the space of viable strategies toward Artin's conjecture: any proof must exploit the interaction between a and the multiplicative structure of p − 1, not merely the growth of orders. The analysis is elementary and self-contained (no GRH required).
davide lugli (Tue,) studied this question.