We develop a unified framework of detectors for singular fibers in arithmetic families π : X → S := Spec Z, specialized to curves after shrinking to principal opens. Four classical detectors-algebraic/geometric (discriminant, resultant, Jacobian), étale (the bump), and motivic (the defect motive and Euler jump)-and a derived detector via the cotangent complex are shown to be equivalent on fibers. On the good-prime open U = D(∆) 1 the algebraic/Hensel gates align with the Jacobian and the five detectors agree; they vanish exactly for smooth fibers. For the benchmark model f (x, y) = x pn + y A we give a two-term derived model, align the Hensel/discriminant gates with Jacobian full rank on D(x) ∪ D(y), and provide explicit local-length formulas at the origin that drive the detector values. The results furnish a one-page minimal certificate and an operational checklist for computations on principal opens, realizing the Master Equivalence as a practical tool for arithmetic surfaces.