Hand-worked examples
Work through these calculations with pencil and paper. Every example gives its inputs, intermediate steps, result and a short exercise with an answer. You do not need to install software.
For a guided path from calculation to runnable code and source evidence, start with the three complete demonstrations.
Start with the first example if matrices are new to you. A matrix acts on a column vector by taking the dot product of each row with that vector. The glossary explains the notation and the distinctions used here.
Use the topic map to find the related papers, written proofs, Lean coverage and code. Each example has these routes at the end, and the related work pages link back to the examples.
Already reading a paper? Find its examples in the paper index. For a book or another work, use the monograph and work index. The coverage list shows which works still need an introductory tutorial in this guide.
| Example | What you will work out | Background |
|---|---|---|
| 1. Split and reconstruct a vector | Projectors, complete packet families and an oblique decomposition | Addition and multiplication |
| 2. Generate a space from one seed | Operator orbits, span and dependence on the allowed action | Vectors and span |
| 3. Change coordinates and transport an operator | An inverse coordinate change and the same calculation in two systems | Small matrix products |
| 4. Assemble an operator from transfer blocks | Source packets, target packets and reconstruction of an operation | Example 1 |
| 5. Refine a signal into smaller packets | Even and odd samples, residue classes and a discarded component | Indexing a list |
| 6. Reconstruct from redundant coordinates | A left inverse and components that are not projectors | Examples 1 and 3 |
| 7. Find an obstruction to recovering information | Two inputs with the same coarse value and different requested answers | Two-variable arithmetic |
| 8. Average with unequal weights | Conditional expectation and orthogonality in a weighted inner product | Weighted means |
| 9. Round a binary number by hand | Precision, ties to even, error direction and local spacing | Fractions and powers of two |
| 10. Find a kernel and its fibers | Rank, nullity and the information a map erases | Linear equations |
| 11. Build spectral projectors | Eigenvectors, polynomial identities and oblique packets | Examples 1 and 2 |
| 12. Eliminate one block | Schur complements, back substitution and singular cases | Two linear equations |
| 13. Reconstruct from residues | Modular inverses, CRT and residue selectors | Integer remainders |
| 14. Separate averages and differences | Two Haar levels, reconstruction and energy accounting | Averages and squares |
| 15. Compose typed operations | Legal paths, one-sided inverses and a commutator | Matrix dimensions and products |
| 16. Split reciprocal coefficients | Paper IV: coefficient recursion, cyclic projection and truncation | Polynomial multiplication |
| 17. Calculate quantum measurement | Paper VII: phases, density matrices, probabilities and dephasing | Complex numbers and small matrices |
| 18. Follow modular lift branches | Paper X: reduction compatibility and branches that stop | Integer remainders |
| 19. Compute finite-field Frobenius | Paper XI: a quadratic field, trace, norm and an intertwiner | Modular arithmetic and matrices |
| 20. Separate support from multiplicity | Paper XII: module presentations, fibers and nilpotent structure | Quotients and linear maps |
| 21. Choose a proof certificate | Proof Path Theory: shared premises, minimum cost and incomparable certificates | Sets and small tables |
| 22. Recover hidden memory | Sufficient Resolution and Machine Societies: visible summaries and delayed feedback | Two-coordinate updates |
| 23. Find a reusable calibration | Grounded Abstraction: known inputs, compatible models and an invariant core | Basis vectors and matrix products |
| 24. Calculate feedback coefficients | Special Functions: contracted arguments and a nonlinear coefficient recurrence | Polynomial products and differentiation |
| 25. Choose repair directions | Constructive Repair: free repair, weighted menus and an obstruction witness | Linear equations and span |
| 26. Update a Gaussian model | Certified Inference: one observation, posterior covariance and a candidate gain | Small matrices and Gaussian densities |
Extended research examples
The Euler-number derivation, factorial-lanes derivation, and wavelet and polyphase transfer follow longer chains of exact representations, transfer laws, and explicit failure witnesses. The wavelet case includes a three-tap discrete filter, four levels of Haar coefficients, exact predict–update lifting, a weighted best-detail selection, sharp finite stability, a full wavelet transfer matrix, a filtering-versus-coarsening counterexample, boundary repair and downloadable results. Each extended example retains its own finite evidence scope.
The Stirling, Bell and Fubini example adds a combinatorial flagship: one triangular counting kernel with two distinct input weightings, a signed inverse, a general formal-series substitution, Touchard polynomial and moment operations, complete residue packet transfers, and an exact interpolation-versus-aggregation boundary. It includes a standalone exact Python checker and downloadable source-bound records.
The roots-of-unity and falling-factorial example connects the earlier arithmetic, Euler and Stirling examples. It carries a character projector through a Stirling/Newton change of basis and independently checks the resulting operation against actual function-image enumeration, rational spectral projectors, filtered generating functions, finite differences, general translation and cyclic carry blocks, and transported multiplication. Wrong-basis and characteristic-three spectral counterexamples keep the assumptions explicit. Eleven exact CSV tables and three Python-generated SVG diagrams accompany the source.
Conventions and scope
Vectors are columns unless a displayed signal is explicitly a list. Matrix products act from right to left. All fractions in the calculations are exact. Example 9 uses exact fractions to describe a specified rounding operation.
These are introductory finite examples, not new results about the research programs. The pages state the identities they use and link to the relevant source guides. A successful example does not replace the hypotheses of a general theorem or establish a new Lean correspondence.
Check the arithmetic or run a program
The exact arithmetic checker checks the 26 examples with Python's standard library and exact rational, modular and finite-set calculations. From the collection checkout, run:
python -B tools/check_hand_examples.py
Expected result:
PASS: 26 hand-worked examples
The checker covers the displayed finite calculations. It does not validate every sentence in the tutorials, review a research proof or run Lean.
For executable library examples, continue to First runs or the Komorebi Math user guide. The hand calculations remain useful without either library.