Twin Primes Are Not Random A Complete Characterization via Four Sedimentation Equations, Zero Division Operations, and a Structural Bridge to Quantum Correlations

We present a complete, deterministic, and division-free characterization of twin prime pairs (p, p+2) with p = 6k−1 > 3. The method rests on a single structural observation: the digit sequence 1–9 admits a twisted array 2 3 1 5 6 4 8 9 7 that partitions all integers coprime to 6 into two orthogonal digital-root chambers — a left rail {2, 5, 8} carrying numbers of the form 6k−1, and a right rail {1, 4, 7} carrying numbers of the form 6k+1. We prove that every composite on the 6k±1 manifold is generated by exactly one of four sedimentation equations, each expressible through additions and multiplications alone. The predicate J(k), defined as the simultaneous non-satisfaction of all four equations, provides a necessary and sufficient condition: the pair (6k−1, 6k+1) consists of twin primes if and only if J(k) holds. Two algorithms are derived from this characterization: (1) ARC-ALGO, a division-free enumeration sieve that achieves 1.3–2.4× speedup over the classical Sieve of Eratosthenes by operating on only one-third of the number line, and (2) an individual primality test requiring zero division, zero modular exponentiation, and zero probabilistic rounds. Both are experimentally validated to 100% accuracy up to N = 1,000,000. We further observe that the structural constants governing this framework — the inversion symmetry φ(x) = 9−x, the barycenter 4.5, and the projection constant k_arc = 1/2 — are identical to those from which the Tsirelson bound 2√2 derives in quantum mechanics, suggesting that prime distribution and quantum correlations are two projections of a single discrete topology. Keywords: Twin primes, 6k±1 manifold, sedimentation equations, division-free algorithm, digital root, inversion topology, projection constant, Tsirelson bound.