Abstract
The binary Goldbach conjecture states that every even integer greater than 2 is a sum of two primes. This paper studies the smallest prime component of such a representation. For an even integer E, Pmin(E) is defined as the smallest prime p for which E - p is also prime. BOUSSOLE-Pmin combines an ordered FIRST search with an empirical prediction of the likely boundary distance. A longitudinal reference series of 100 selected integers of the form 10^N + 2 was studied up to N = 4000. A separate frozen prediction was then tested on 2000 new integers of the form 10^N + c, using 20 values of N from 20 to 400 and 100 even offsets at each level. The predictive scale was D(E) = log(E) log log(E), with frozen levels 0.258D, 0.790D and 1.088D. Observed coverages were 51.5%, 79.4% and 85.4%. The independent median Pmin/D was 0.24893, close to the frozen median coefficient 0.258, while the upper tail was substantially heavier than predicted. The method therefore uses prediction as a search compass, not as a stopping rule. Finally, Pmin gives a direct reformulation of Goldbach: a future proof can target uniform positivity of the number of prime pairs with the smaller prime restricted to a controlled boundary window.