The Coherence Chain: One Mechanism, Every Row, the Compiler Proved

Paper 136 introduced the coherence thread (a rational point visible at every prime, creating drainage in the Euler product) as the mechanism of the BSD conjecture. Paper 137 introduced the coherence cycle (an algebraic divisor visible at every prime, creating inflation) as the mechanism of the Tate conjecture. This paper unifies both into a single definition: the coherence chain. A coherence chain is an algebraic cycle defined over Q that reduces to a well-defined cycle at every prime of good reduction. The thread (degree 0, acting on beta_1) and the cycle (degree 1, acting on beta_2) are special cases. A parity principle governs the direction: odd Betti slots produce drainage and zeros (BSD, Bloch-Kato); even…