We present unconditional proofs of the convergence of reciprocal sums associated with certain special prime sequences defined by polynomial and multiplicative conditions. In particular, consider a polynomialP(x) = sum_{i=0}^{n} c_i x^i quad text{with integer coefficients and } c_n > 0.Define S_P = { p : p text{ prime and } P(p) text{ is prime} }. We prove thatsum_{p in S_P} frac{1}{p} < infty. Moreover, we establish explicit upper and lower bounds for both the counting functionpi_P(x) = |{ p in S_P : p le x }| and the partial sums sum_{substack{p in S_P p le x}} frac{1}{p}. Next, we consider Balanced Primes, defined by the condition that each balanced prime p_n forms a three-term arithmetic progression with its neighbors: p_n = frac{p_{n-1} + p_{n+1}}{2}. Applying our multi-level sieve methods to these primes, we similarly prove the convergence of their reciprocal sum and provide corresponding quantitative estimates. In addition, we examine the set of emph{Good Primes}, defined by the multiplicative inequality p_n^2 > p_{n-i} cdot p_{n+i} quad text{for all } 1 le i le n-1.We show that their reciprocal sum also converges and provide corresponding upper and lower bounds on their counting functions and partial reciprocal sums. Our approach, which we call the M-Brun Sieve, refines classical sieve methods into a multi-level framework that can handle intricate polynomial and multiplicative constraints simultaneously. Notably, our results do not rely on any unproven conjectures. These findings yield substantial new insights into the distribution and density of these special classes of primes, thereby resolving longstanding questions posed by Pomerance regarding Good Primes.