The Coherence Cycle: How Algebraic Divisors Pin Frobenius Eigenvalues on K3 Surfaces

Paper 136 identified the coherence thread as the unit of the BSD conjecture: a rational point on an elliptic curve reduces modulo every prime, creating a systematic bias that drains the Euler product to zero. This paper applies the same mechanism one Betti slot higher. On a K3 surface, the coherence thread becomes a coherence cycle: an algebraic divisor defined over Q that reduces to a divisor at every prime, pinning one normalised Frobenius eigenvalue to 1 and creating a pole rather than a zero. Each independent cycle pins one eigenvalue; the Picard number counts the cycles; the order of the pole counts the pinned eigenvalues. The BSD mechanism drained via (1-epsilon)^infinity = 0; the Tate mechanism inflates via (1+epsilon)^infinity = infinity — mirror images driven by the same divergence of the sum of 1/p. Same compiler, same engine, reading beta_2 instead of beta_1. The Tate conjecture is the beta_2 row of the compilation table introduced in Paper 135. Numerical verification is provided for the Fermat quartic (Picard number at least 2, confirmed by exact Frobenius trace Tr = 2p at all tested primes congruent to 3 mod 4) and a generic quartic (Picard number 1, confirmed by fluctuating transcendental trace). Part of the Bootstrap Universe Programme.