{
  "corollary_3_4": {
    "dual": "sum_k ||M_k||_nuclear",
    "muon_norm": "max_k ||W_k||_spectral"
  },
  "corollary_3_5": {
    "hybrid_norm": "max{||W||_msp,||u||_infinity}"
  },
  "definition_5_1": {
    "approximate_sd": "N(t)=integral nu diverges; norm(theta)/N bounded; liminf alignment r(t)>=1"
  },
  "theorem_3_1": {
    "learning_rate": "integral_0^infinity eta(t) dt=infinity",
    "soft_margin": "varphi^{-1}(log(1/L(theta)))/||theta||^L"
  },
  "theorem_3_2": {
    "conclusion": "normalized-direction limit points are directions of KKT points for the same norm"
  },
  "theorem_3_6": {
    "adam": "epsilon=0, c1>=c2 (beta1<=beta2), decaying nonsummable eta",
    "margin_norm": "l_infinity"
  }
}