Z₂ Gauge Theory as a Predictive Agent, with a Structural Relation for the Fine-Structure Constant

Imagine Z₂ lattice gauge theory as a crowd of agents each guessing a binary outcome. Get it right: call that S. Get it wrong: T. The gauge freedom — flipping link variables locally without touching the physics — is just the agent deciding which outcomes to label as errors. At the critical point, the error probability locks at P(T) = (2 − ln 2)/2 ≈ 0.653, and something unexpected falls out: the fine-structure constant emerges as α = P(T|v⁎)·g/(4π), where g is the ring-exchange constant of the U(1) spin liquid on pyrochlore. With g ≈ 0.14, the formula yields α ≈ 1/138.5 — half a percent from the measured value. I am not claiming a derivation. The functional form is posited, the proportionality is an Ansatz, and the value of g is borrowed from a different model. The structural relation is specific enough to falsify, and the coincidence is close enough to notice. One prediction, Γ_c(J_s > 0) > J_t, can be checked on a quantum simulator.