Two-Sided Fortunate Numbers and Goldbach Amplification for Primorial Multiples

Let p_k be the k-th prime, let p_k# = ∏_{p ≤ p_k} p, and consider

     N = R · p_k#,

where every prime factor of R is at most p_k. The least offsets m > 1 for which N ± m is prime are generalized two-sided Fortunate numbers. Because N is divisible by every prime at most p_k, such offsets are themselves prime whenever m < p_{k+1}².

For each sign, let J± be the number of successive prime offsets m ≥ p_{k+1} inspected before N ± m is prime. We conjecture that the normalized first-arrival index

     T± = J± · (N/φ(N)) / log N

converges to the unit exponential distribution. A uniform Hardy–Littlewood hypothesis implies this law, while the corresponding singular series has the exact form

     S(R · p_k#) = (p_k# / φ(p_k#)) · ∏_{p > p_k} (1 - 1/(p-1)²).

Thus the familiar Mertens factor

     p_k# / φ(p_k#) ~ e^γ · log p_k

appears directly as a prime-arrival amplification.

Using fourteen choices of R over ranges extending to k = 2000, we obtain 8,946 first-arrival observations, counting the two signs separately. The exponential model fits well below k ≈ 1200 and again beyond k = 1732. Between these ranges the sampled primorial multiples exhibit a localized increase in the first-arrival rate of about 6%. The effect is bounded on both sides by prospectively specified controls and is not explained by the finite-k corrections appearing in the derivation.