Mixed Fermion Thresholds and Closure Composition A companion to Paper 3

We test whether mixing between heavy fermions evades the one-loop obstruction to homogeneous gauge response found in Paper 3. A finite family of vectorlike fermions with masses affine in the controller is described by matrix-valued Yukawa couplings. The parity-even gauge threshold depends on the logarithm of a normalized mass determinant. General mixing can reverse the local curvature, so the diagonal positivity argument does not extend to every mass matrix. Nevertheless, a polynomial identity proves a broader interval result: within the stated matching and calibration assumptions, an affine one-loop logarithmic response must be constant. Positive threshold weights then require every charged block to have a constant determinant magnitude. Explicit examples show that masses can vary while this response vanishes, and that any finite order of local agreement with a nonzero affine response can be arranged without the interval identity. Smooth perturbative recalibration preserves the leading result. The claims concern a specified class of quantum field models at one-loop order and fixed probe scale.