Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings

We study determinant-compatible lifts of invertible 2×2 matrices over odd prime-power rings through an affine first-variation formula for matrix order defects and cyclic adjoint transfer on the traceless tangent space. We classify the full-order transfer for scalar, semisimple, and Jordan reductions and give an explicit three-stratum perturbative-depth classification for the specified Jordan family. In characteristic three, for nonscalar reductions, the resulting order-lifting dynamics obey PRE → 9 PRE + 18 ACT and ACT → 27 ACT. Two independent, offline, standard-library Python scripts accompany the manuscript to reproduce finite checks of transfer ranks, Jordan polynomial coefficients, and PRE/ACT branching. The manuscript is a pre-referee research preprint, not peer reviewed. Its historical priority and novelty have not been independently certified. The manuscript and its LaTeX source are licensed under CC BY 4.0. The accompanying Python verification scripts are licensed under the MIT License. See FILE_LICENSE_MAP.md for details.