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 Betti slots produce inflation and poles (Tate). Both directions are driven by the divergence of the sum of 1/p. The entry route also alternates: even cohomology receives algebraic cycles directly via the cycle class map (exact pinning); odd cohomology receives them indirectly via Abel-Jacobi or Kummer maps (statistical drainage). The beta_3 row is computed for the Fermat quintic threefold — the most studied Calabi-Yau in string theory — revealing that the Frobenius trace on H^3 vanishes exactly at primes p not congruent to 1 mod 5: the quintic wall as an arithmetic gate. The degree sequence of the compilation table is 3 (elliptic curve), 4 (K3 surface), 5 (Calabi-Yau threefold) — the Calabi-Yau condition d=n+1 forces the deepest row to be literally the quintic. Proving the coherence chain mechanism for general degree subsumes BSD, Tate, and Bloch-Kato as corollaries. Part of the Bootstrap Universe Programme.