The same letter can refer to a mathematical operator or to a particular stored representation. Keep those levels separate when reading an algorithm or a benchmark row.
| Symbol | Object | Question to ask |
|---|---|---|
| \(P_\alpha\) | Coordinate projector | Which input or output coordinates does it select? |
| \(A_{\beta\alpha}\) | A block map | What are its source and target dimensions? |
| \(E\) | Active block addresses | Is this exact support, or only a structural upper bound? |
| \(D+UV^*\) | Factored operator | Is the desired output a vector, factors, or a dense array? |
| \(Q_L,G,Q_R\) | Range factors and core | Are the outer columns orthonormal, with exact reconstruction? |
| \(E_j\) | An error operator | Is it a factor perturbation or an additive error after a product? |
| \(s_e,\epsilon_e\) | Sensitivity and tolerance | Is the budget input-dependent or uniform over unit inputs? |
| \((\kappa_v,\epsilon_v,\sigma_v)\) | Candidate-tree data | Do costs, errors, and split overheads really add? |
| \(F_v(b)\) | A budget table | Does \(b\) mean exact weight, or an upper bound? |
| \(R\) | Reuse count | Which one-time costs have been included? |
Conventions used in the examples
Matrices act on column vectors. A block \(A_{\beta\alpha}\) goes from source \(\alpha\) to target \(\beta\). Products act from right to left. Real examples use transpose; complex examples use adjoint unless a vectorization identity explicitly requires transpose. Column-major vectorization stacks the columns of a matrix.
An arithmetic count follows the paper's modeled multiply/add convention. It is not a count of executed machine instructions. The tiny scheduling and amortization examples use declared synthetic units, not measured GPU times. The original benchmark corpus remains historical data in the preserved source tree.
Evidence levels
The original theorem ledger records its inherited finite formal scope. The new exact checks validate these particular examples. A plot samples a formula. A historical timing describes its recorded case and machine. None of these evidence types can silently substitute for another.
Read this as
Before interpreting a number, identify its object, norm, output contract, and timing scope. Most apparent contradictions disappear once those four fields are explicit.