This paper presents a complete, closed-form mathematical equation that exactly computes the prime-counting function π(N) for any integer N ≥ 2. Unlike existing methods which are either asymptotic approximations or recursive algorithms, our formulation is a single evaluable expression. The equation operates in two distinct modes: (1) using a sequence of known primes, or (2) using the simple sequence Ju2081 = 2, Ju2099 = (n-1)-th odd integer ≥ 3 for n ≥ 2, with an intrinsic primality test μu2099 = ⌈∏u2093u208cu2081u207fu207b¹ (Ju2099/Ju2093 - ⌊Ju2099/Ju2093⌋)⌉ where μu2099 = 1 if and only if Ju2099 is prime. The formula directly yields π(N) through elementary arithmetic operations without recursion, iteration, or algorithmic procedures. The implications of this formula are explored in comparison to existing prime counting functions and its potential impact on the study of prime distribution, it is an explicit sieve-theoretic expression and a self-contained rewriting. This complements classical exact prime-counting methods (Meissel—Lehmer and descendants), which are vastly more efficient for computation.