A Base-Independent Primality Characterization by Concatenated Consecutive-Divisor Gaps

This note introduces an integer-valued function obtained from the gaps between consecutive positive divisors of an integer. Let n≥2n ≥ 2, and write its positive divisors in increasing order as 1=d1 nD_b(n)>n when nn is composite. The proof separates into two elementary facts: the final consecutive-divisor gap of a composite integer nn is at least n/2n/2, and concatenating any nonempty positive block before a positive integer produces a value greater than twice the final block. Their combination yields the result for every positional base b≥2b≥2. The result is intended as a structural characterization of prime numbers rather than as a computationally efficient primality test.