We study the Diophantine equation 1/p_1 = 1/p_2 + 1/p_3 + 1/p_4 for distinct primes p_1, p_2, p_3, p_4, a special case of the Sierpiński—Schinzel unit-fraction equations. We introduce a spectral coherence threshold R, derived from an exact infimum formula for a normalized exponential sum over the primes involved via the Kronecker—Weyl equidistribution theorem, and show that R = 1/2 is its sharp transition value. For three primes, we establish an exact identity linking R to prime gaps, giving a new proof that no three consecutive primes satisfy the analogous three-term equation. For four primes, we combine this classification with an exact divisor-based algorithm to search over 6 times 10^8 configurations, finding no solution for p_1 le 5431 but reporting the closest known near-solution. We prove that no solution can occur with the three larger denominators in arithmetic progression, and record a short, fully general divisibility argument showing that no such equation admits a solution in distinct primes for any number of terms — clarifying explicitly that this fact concerns single-prime denominators only, not the composite-denominator setting central to the Sierpiński—Schinzel conjectures.