Appendix A — Notation Covenant
Notation is a public interface between this book and its sibling. Core symbols retain their meanings; this volume adds symbols when the new diagnostic objects require them.
A.1 Inherited core
| Object | Notation | Contract |
|---|---|---|
| Scalar | \(x\) | Italic lowercase unless a field convention requires otherwise |
| Vector | \(\vect{x}\) | Bold; a column on paper |
| Matrix or batched array | \(\matr{A}\) | Bold; shape completes the type |
| Aggregate objective | \(\loss\) | A local loss is \(\ell_i\) |
| Expectation | \(\E\) | Population and conditioning are stated |
| Variance | \(\var\) | Denominator convention is stated |
| Covariance | \(\cov\) | Sample and population objects are distinguished |
| Norm | \(\norm{\cdot}\) | The subscript names the norm when ambiguity matters |
| Definition | \(:=\) | Equality remains \(=\) |
| Approximation | \(\approx\) | Never used as silent equality |
| Algorithmic update | \(\leftarrow\) | Distinct from a mathematical identity |
Code stores a batch by rows unless a chapter explicitly declares otherwise. Every non-obvious matrix product states dimensions first, and every reduction names its axes.
A.2 New in this volume
| Object | Notation | First use |
|---|---|---|
| Centered sum of squares | \(M_{2,n}\) | Streaming state |
| Operator norm | \(\norm{\matr{A}}_{\mathrm{op}}\) | Random operators |
| Largest singular value | \(\sigma_{\max}(\matr{A})\) | Random operators |
| Smallest singular value | \(\sigma_{\min}(\matr{A})\) | Random operators |
| Frobenius norm | \(\norm{\matr{A}}_{\mathrm{F}}\) | Random operators |
| Epsilon-net | \(\mathcal{N}_{\epsilon}\) | Continuous extrema |
| Covering number | \(N(K,d,\epsilon)\) | Continuous extrema |
| Sub-Gaussian Orlicz norm | \(\norm{X}_{\psi_2}\) | Safe-zone concentration |
| Sub-exponential Orlicz norm | \(\norm{X}_{\psi_1}\) | Products and squares |
| Unit roundoff | \(u\) | Floating-point contract |
| Unit in the last place | \(\operatorname{ulp}(x)\) | Floating-point contract |
| Spectral condition number | \(\kappa\) | Quadratic modes |
| Spectral radius | \(\rho(\matr{A})\) | Quadratic modes |
| Sub-Gaussian MGF proxy scale | \(\sigma\) | Safe-zone concentration |
| Sample covariance | \(\widehat{\matr{\Sigma}}\) | Spectral nulls |
| Empirical spectral distribution | \(\mu_{\matr{M}}\) | Spectral nulls |
| Aspect ratio | \(\gamma=n/m\) | Spectral nulls |
| Marchenko–Pastur edges | \(\lambda_-,\lambda_+\) | Spectral nulls |
| Population spike | \(\beta\) | Spectral nulls |
| Effective rank | \(r_{\mathrm{eff}}(\matr{\Sigma})\) | Effective dimension |
| Stable rank | \(\operatorname{sr}(\matr{A})\) | Effective dimension |
| Jacobian | \(\matr{J}_F(\vect{x})\) | Reverse accumulation |
| Jacobian–vector product | \(\matr{J}_F(\vect{x})\vect{v}\) | Reverse accumulation |
| Vector–Jacobian product | \(\vect{u}^{\mathsf T}\matr{J}_F(\vect{x})\) | Reverse accumulation |
| Node cotangent | \(\bar z_v\) | Reverse accumulation |
| Realized Hessian | \(\matr{H}\) | Curvature objects |
| Generalized Gauss–Newton matrix | \(\matr{G}\) | Curvature objects |
| Model Fisher information | \(\matr{F}\) | Curvature objects |
| Empirical Fisher | \(\widetilde{\matr{F}}\) | Curvature objects |
| Model-curvature residual | \(\matr{R}\) | Curvature objects |
| Arithmetic intensity | \(I=F/Q\) | Runtime model |
| Machine balance | \(I_*=P_{\max}/B\) | Runtime model |
| Conditional noise-to-signal ratio | \(\chi_k\) | Stochastic regimes |
| Centering projector | \(\matr{P}\) | Coordinate-wise normalization |
| Nuclear norm | \(\norm{\matr{A}}_*\) | Matrix update geometry |
| Polar factor | \(\operatorname{polar}(\matr{A})\) | Matrix update geometry |
The machine-readable source is contracts/notation-covenant.yml. A notation change is incomplete until that covenant, the HTML macros, the PDF macros, and this appendix agree.