Logarithmic Spectral Calibration and Closure Composition A second companion to Paper 3

A scalar spectrum can supply a ruler that does not depend on the gauge response being tested. In this model, multiplying the preparation source multiplies a scalar mass and every heavy fermion mass by the same factor. The logarithm of the scalar mass ratio then records successive multiplication as addition. The familiar logarithmic threshold correction becomes linear in that measured coordinate. The result establishes agreement with an exponential to the order actually calculated. An exact exponential would also fix the higher-order terms. Those terms have not been calculated here. Moreover, relabelling an existing preparation operation and changing the operation to add measured readings are physically different procedures. The next-order test must say which procedure it uses. We construct a local effective field model in which an independently measured logarithmic scalar mass ratio yields a nonzero homogeneous gauge response through one-loop order. Two neutral scalars have a homogeneous quartic interaction and an external radial preparation source. Their positive equilibrium branch has one fixed scalar mass and one mass proportional to the prepared radius. Vectorlike fermion masses generated by a Yukawa interaction scale with the same radius. Standard threshold matching therefore gives a logarithmic gauge response linear in the spectral coordinate. This construction changes the leading calibration assumed in the earlier affine-matrix obstruction. It supplies a conditional one-loop example, without an exact symmetry protecting the full response. At the next order, both gauge matching and scalar pole calibration contribute. We derive the corresponding composition conditions, distinguish a fixed source protocol from addition imposed through the measured coordinate, and obtain a necessary compatibility condition shared by several gauge sectors.