This technical note records the derivation and numerical audit of a prime-endpoint envelope for the reciprocal-prime harmonic sum. For prime endpoints x = p_k, the work studies the representation sum_{j <= k} 1/p_j = log log p_k + B + C_osc(p_k)/(sqrt(p_k) log p_k), where B is the Meissel-Mertens constant for primes. The accompanying scripts numerically test upper envelopes of the form log log p_k + B + C/(sqrt(p_k) log p_k) and report threshold constants observed up to p_k <= 10^9. The note is connected with the Robin/MVDC research framework and should be read as an experimental and computationally supported mathematical note. It does not claim a proof of the Riemann Hypothesis or an infinite-range proof of the proposed envelope.