Abstract
Stirling transforms inherit modular periodicity from their fixed columns only when the weights suppress the additional prime-power factors in the column periods. We establish a local p-adic valuation criterion for an integer-weight Stirling transform and prove that, for every modulus greater than two, Euler’s totient is an eventual period. The proof involves truncating the transform modulo each prime power, applying Kwong’s fixed-column period bounds and handling the exceptional third column at the prime two on its own. We then apply the criterion to the signed binomial-recurrence sequence; the formal exponential generating function of this sequence yields integral Stirling weights such that the odd and even prime valuations satisfy the criterion, thereby proving Bala’s modular-periodicity conjecture for this sequence. For every integral substitution G(e^x-1), we obtain the stronger universal eventual period given by the Carmichael function.