Cross-Stage Macrostate Compatibility Under Network Pruning: Exchange-Residual Evidence and Group-Action Diagnostics in Z₇ and S₃ Tasks

This is the rivised version of https://zenodo.org/records/23763578, and it was finished by Yancheng Guo(email: yanchengguo423@gmail.com), and Yuanfei Hu(email: yuanfeihu_duke@outlook.com). Neural networks can preserve a task solution while their hidden representations change, but task accuracy alone does not reveal whether a compressed macro-description remains compatible across stages. This study tests a narrow version of that question in finite-group operation networks. A width-32, two-hidden-layer multilayer perceptron was trained on either addition in the cyclic group Z₇ or composition in S₃, using one-hot operands plus eight Gaussian nuisance features. For each of 30 paired seeds per group, we compared a matched continued-training control with 60% magnitude pruning followed by additional training. Stage-specific, label-free PCA encoders with dimensions 4, 6, and 8 defined early and late macrostates. Separate predictors estimated an early-micro-to-late-micro map F and an early-macro-to-late-macro map G; a held-out empirical exchange residual compared φ₁(F̂(h₀)) with Ĝ(φ₀(h₀)). Five of six planned group-by-dimension contrasts showed an increase in residual after pruning, with Benjamini–Hochberg-adjusted p values below .001; the S₃, dimension-4 contrast did not. Task accuracy and output-equivariance accuracy were 100% in all six matched cells, while macrostate task-readout accuracy remained high for S₃ dimensions 6 and 8 and Z₇ dimension 8. However, none of the six cells passed the full prespecified joint screen for macro-action quality. In particular, inverse-composition errors of the fitted linear group-action operators remained high, despite successful synthetic calibration on the designed control scenarios. The results therefore support a limited empirical claim: task behavior and fitted cross-stage macrostate compatibility can dissociate under pruning. They do not establish that a valid macro-level group representation is preserved, that every possible macro-encoding fails to commute, or that the observed pattern generalizes to biological development.