{
  "what": "Exactness of the symmetry action and of the fitted alignment map, on the real models used in the study.",
  "pythia-1.4b (MHA, fused QKV)": {
    "free_hidden_axes": 24, "axis_width": 8192,
    "mlp_permutation_rel_logit_change": 1.057e-06,
    "flat_head_permutation_rel_logit_change": 1.257,
    "note": "mergeschool.core.alignment.apply_head_perms permutes q/o but not k/v; on this arch head_match finds no head-structured q tensor (QKV is fused) so it is never applied by that library -- but applying it directly destroys the model."
  },
  "Llama-3.1-8B (GQA, 32 heads / 8 KV groups)": {
    "free_hidden_axes": 32, "axis_width": 14336,
    "mlp_permutation_rel_logit_change": 1.057e-06,
    "mlp_plus_GQA_group_respecting_head_permutation_rel_logit_change": 9.368e-07,
    "mlp_plus_FLAT_head_permutation_rel_logit_change": 1.115,
    "fit_g_recovery_on_fully_scrambled_model": {
      "coord_share": 1.0, "hidden_layers_matched": 32, "head_sets_matched": 32,
      "max_abs_weight_error_vs_original": 0.0,
      "max_abs_logit_error_vs_original": 0.0,
      "verdict": "the fitted map recovers the ground-truth group element BIT-EXACTLY"
    }
  },
  "conclusion": "The MLP free-hidden-axis permutation is exact on both architectures. The flat head permutation in the shared library is NOT function-preserving for GQA (rel logit change 1.115 on Llama-3.1-8B); the group-respecting action implemented in gmap.py is exact (9.4e-07). With that fix, fit_g recovers a fully scrambled Llama-3.1-8B bit-exactly, so any failure of alignment to help downstream accuracy in this study is a fact about the models, not about the aligner."
}