The Deterministic Structure of the Prime Numbers: Guard Watches, Boundary Phases, and Zero Randomness

We prove that the prime numbers are generated by a closed deterministic recursion with zero randomness at any step. The construction partitions the positive integers into guard watches [q², nextprime(q)²), where every integer coprime to all primes ≤ q is prime. Each prime is determined exactly by its position within an eligible gap cycle and a computable boundary phase β(q) = q² mod primorial(q). The apparent randomness attributed to primes for 90 years (Cramér model, Hardy-Littlewood heuristics, random matrix theory) is pseudorandomness arising from β(q), whose outputs look random because primorial(q) grows as e^q. Verified at 200 boundary transitions with 100% agreement. The framework reframes the Riemann Hypothesis as an equidistribution problem for a specific deterministic function.