| Normalization of a cumulative margin bound | statement | lemma B.4 | arxiv |
| Uniform concentration of frozen exponential averages | statement | lemma E.3 | arxiv |
| Deterministic row-norm rigidity | statement | proposition C.6 | arxiv |
| Uniform radius-to-margin ratios | statement | proposition D.4 | arxiv |
| Residual-exogenous optimizer limits | statement | lemma F.2 | arxiv |
| Stochastic optimizer accuracy ordering | statement | proposition 3.4 | arxiv |
| All-aspect spectral input | statement | proposition C.12 | arxiv |
| Full-batch max--sum dominated margin convergence | statement | theorem B.6 | arxiv |
| Max-norm boundary separation | statement | proposition C.10 | arxiv |
| Adam-step and signal stability | statement | lemma E.2 | arxiv |
| All-aspect deterministic margin geometry | statement | theorem C.2 | arxiv |
| Uniform clipped-Gaussian angular barrier | statement | lemma E.7 | arxiv |
| Row-normalized full-batch margin convergence | statement | proposition B.14 | arxiv |
| Freezing one epoch | statement | lemma E.4 | arxiv |
| Geometry-adapted random-reshuffling momentum control | statement | proposition B.11 | arxiv |
| Geometry-adapted softmax and gradient stability | statement | lemma B.9 | arxiv |
| Cumulative remainders for general learning rates | statement | lemma B.3 | arxiv |
| AdamProxy margin characterization | statement | proposition E.10 | arxiv |
| Population-error continuity along a direction set | statement | lemma C.21 | arxiv |
| One-epoch \(\beta_1,\beta_2\to1\) RR-Adam approximation | statement | proposition 3.5 | arxiv |
| Finite-dimensional geometry-to-error transfer | statement | proposition C.19 | arxiv |
| Deterministic optimizer accuracy ordering | statement | theorem 3.1 | arxiv |
| Opposite Row--Max orderings before and after a common Haar rotation | statement | proposition G.7 | arxiv |
| RR-RowSGDM versus AdamProxy accuracy ordering | statement | proposition 3.6 | arxiv |
| Selected mean geometry | statement | lemma F.3 | arxiv |
| Uniform asymptotics of the clipped problem | statement | lemma C.9 | arxiv |
| Full-batch Adam under the present learning-rate assumptions | statement | theorem B.7 | arxiv |
| C2M1SV5VVJ | statement | theorem | arxiv |
| Uniform stochastic near-margin geometry | statement | proposition D.5 | arxiv |
| Exact and near spectral minimizers | statement | proposition C.15 | arxiv |
| Pairwise Gaussian error bounds | statement | lemma C.17 | arxiv |
| Test-noise energies for selected directions | statement | lemma F.4 | arxiv |
| Random-reshuffling row-wise normalized SGDM margin convergence | statement | proposition B.15 | arxiv |
| Population-error exponents of deterministic optimizers | statement | proposition C.27 | arxiv |
| Weighted trace gap and hard-edge truncation | statement | lemma C.13 | arxiv |
| Effective margin implies direction-set convergence | statement | lemma B.1 | arxiv |
| Consequences of max--sum domination | statement | lemma B.5 | arxiv |
| Exact margin implies direction-set convergence | statement | lemma B.2 | arxiv |
| Spectral-norm boundary separation | statement | proposition C.16 | arxiv |
| Deterministic population-error exponents | statement | proposition C.26 | arxiv |
| Near-minimizer stability | statement | proposition D.3 | arxiv |
| Accuracy ordering under anisotropy with independent orientations | statement | theorem 4.2 | arxiv |
| Uniform row-norm alignment | statement | proposition C.7 | arxiv |
| Directed-pair exponent controls multiclass error | statement | lemma C.24 | arxiv |
| Stretched-exponential common input | statement | proposition C.4 | arxiv |
| Finite-population batch moments and relative scale | statement | lemma E.1 | arxiv |
| Geometry-adapted cross-entropy bounds | statement | lemma B.8 | arxiv |
| Selected-direction ordering | statement | proposition F.5 | arxiv |
| Positive margin growth and a trajectory-level coordinate bound | statement | proposition E.6 | arxiv |
| Quantitative sign statistics under a Haar rotation | statement | lemma G.5 | arxiv |
| Capped AdamProxy trajectories and boundary separation | statement | proposition E.8 | arxiv |
| Empirical-to-population polar transfer | statement | lemma C.23 | arxiv |
| Geometry-to-accuracy transfer | statement | theorem C.20 | arxiv |
| Boundary metric under direction-set convergence | statement | lemma C.22 | arxiv |
| Gaussian error transfer | statement | lemma F.1 | arxiv |
| Near-margin reduction | statement | lemma D.1 | arxiv |
| Deterministic whole-face stability | statement | lemma G.2 | arxiv |
| Scale-invariant reduction | statement | lemma C.3 | arxiv |
| Continuity of the Adam ratio | statement | lemma E.5 | arxiv |
| Quantitative whole-face geometry in aligned coordinates | statement | lemma G.3 | arxiv |
| Quantitative Gaussian hard-edge count at the square aspect ratio | statement | lemma C.11 | arxiv |
| Sharp directed signals on the Max and Spectral faces | statement | proposition C.25 | arxiv |
| Exact clipped problem | statement | proposition C.8 | arxiv |
| Average normalization transfer | statement | lemma C.14 | arxiv |
| Normalization stability | statement | lemma C.5 | arxiv |
| Full-batch Muon versus Adam | statement | corollary C.28 | arxiv |
| TA2S520HF9 | statement | theorem | arxiv |
| Combined Gaussian-cloud event | statement | lemma D.2 | arxiv |
| Generic common rotation restores the Row coefficient advantage | statement | lemma G.6 | arxiv |
| Geometry-adapted random-reshuffling momentum margin | statement | corollary B.12 | arxiv |
| Coordinate dependence of the RowGDM--Adam ordering | statement | proposition 4.5 | arxiv |
| Quantitative Gaussian radial event | statement | lemma C.18 | arxiv |
| Gaussian coefficient comparison | statement | lemma G.1 | arxiv |
| An exact Row--Max transition under aligned test anisotropy | statement | proposition G.4 | arxiv |
| Spectral dispersion implies a transition | statement | corollary G.8 | arxiv |
| High-momentum Row near-set control | statement | lemma E.9 | arxiv |
| Y48EP91BD4 | definition | definition | arxiv |
| Population-error exponents | statement | corollary 3.3 | arxiv |
| Random-reshuffling noise and cancellation | statement | lemma B.10 | arxiv |
| Row normalization is row-norm steepest descent | statement | proposition B.13 | arxiv |
| Exact invariance under common feature rescaling | statement | lemma C.1 | arxiv |
