Extended case studies and demonstrations
Project overview · Project map · All hand-worked examples
Extended case studies
Follow one object through several representations, transport its operations, and calculate what is preserved or lost. Start with Euler numbers for a complete arithmetic calculation. Continue with wavelets and polyphase transfer, explore the companion factorial lanes, then compare quantum phase and measurement. These are suggested reading choices; each study can also be read on its own.
Sources and contributions identifies the classical inputs to the Euler and wavelet studies and the introductory seed-generation example.
| Case study | Starting problem | What the derivation delivers |
|---|---|---|
| Euler numbers | Reconstruct a reciprocal sequence through other coordinates. | Packet matrices, quartic reconstruction, polynomial identities, refinement, and exact residual repair. |
| Wavelets and polyphase transfer | Filter a sixteen-sample signal across four Haar levels. | Exact operator transport, sharp finite stability, cross-scale interactions, and the errors caused by omitted transfers or a changed operation order. |
| Factorial lanes | Follow one residue section of the exponential. | Reconstruction of 34!, hypergeometric coordinates, reciprocal inversion, a sharpened 5-adic displacement bound, and refinement. |
| Quantum phase and measurement | Determine which information a measurement retains. | Exact tomography, a coherence-loss obstruction, reversible records, amplitude damping, and finite syndrome recovery. |
| Stirling, Bell and Fubini numbers | Transfer between counting families and coordinate profiles. | Inverse triangular kernels, generating functions, moment representations, residue transfer, and exact interpolation. |
| Roots of unity and falling factorials | Move residue projectors into Newton coordinates. | Exact spectral projectors, refinement, translation transfer, and failures caused by filtering in the wrong basis. |
Each study includes hand derivations, failure controls, source links, and exact Python checks. Its own scope section identifies the statements and tested ranges. For function constructions, local series, and analytic questions, continue to Special Functions and its coupled-function lab.
Three elementary demonstrations
These three demonstrations connect a concrete question to a hand calculation, a short program and the relevant mathematical sources. Start with any one. They use elementary finite constructions to introduce the collection's language. The calculations are not claims of mathematical novelty.
| Question | Demonstration | What you can inspect |
|---|---|---|
| Can an operation be rebuilt from its action between parts? | Reconstruct an operator | Four blocks give the same answer as one matrix multiplication. |
| Does a finer description preserve the signal? | Refine a signal | Two and four address classes reconstruct the same list. Dropping a class leaves an explicit residual. |
| Does a summary contain enough information for a question? | Test recovery from a summary | Two inputs give the same summary but different requested answers. A compatible question has an exact recovery rule. |
Each page gives the inputs, intermediate steps, expected output, a control that exposes a limitation, and links to papers, proofs, Lean coverage and code. The project map places these introductory examples in the wider collection.
An extended derivation with Euler numbers
Euler numbers through packets, polynomials, and residuals follows one calculation through reciprocal series, a residue-packet matrix, quartic sequence reconstruction, parameter polynomials, exact residual repair, and finite-difference refinement. It includes the full hand calculations, failure witnesses, and a standalone exact Python checker with downloadable tables.
An extended derivation with factorial lanes
Factorial lanes: one residue section through exact transfer starts with the degree-4 packet of the exponential. It reconstructs 34! from ten step-factorial lanes, converts those lanes to a hypergeometric series, normalizes the denominator, and inverts the packet in two equivalent coordinate systems. It also proves a sharpened 5-adic displacement bound, refines the residue packet into two and five children, and distinguishes the congruential transfer row from the classical Euler numbers. A standalone checker supplies exact rational results, downloadable code and two explicit failure witnesses.
An extended derivation with wavelets and polyphase transfer
Wavelets and polyphase transfer: one signal through the whole calculus follows a sixteen-sample signal through even/odd phases, four Haar levels, predict–update lifting, weighted coefficient selection, and exact residual repair. It transports a three-tap FIR filter through polyphase coordinates and all four Haar levels, derives a sharp finite stability certificate, and proves the exact error from omitted cross-channel transfers. The full 16×16 matrix displays inter-scale coupling, while two noncommuting operation orders and explicit boundary repairs provide independent failure witnesses. Reproducible tables and Python-generated figures complete the example.
An extended derivation with Stirling, Bell and Fubini numbers
From set partitions to exact transfer starts with one Stirling-number row, constructs Bell and ordered Bell numbers, and then exposes the inverse triangular kernel. It independently derives the exponential generating functions and their formal inverse, interprets Bell as a Touchard evaluation and Fubini as a gamma or geometric moment, then builds full residue transfer systems. Exact interpolation recovers a complete parameterized polynomial, while two scalar summaries do not determine an arbitrary coordinate profile. Every example has exact Python checks, source-bound tables and a Python-generated diagram.
An extended derivation with roots of unity and falling factorials
Roots of unity, projectors, and falling-factorial coordinates starts with the Stirling expansion of a fifth power, constructs exact roots-of-unity residue filters, and transports the projectors through a Newton/falling-factorial change of basis. The three packets yield 600+244+180=1024. The same page proves finite-difference and multiplication transfer rules, tests six-way refinement, and shows why filtering ordinary monomial degrees is the wrong operation after a coordinate change. The independently enumerated functions from five inputs to four labels explain exactly what those residue packets count. A rational spectral formula for the projectors and an all-degree generating-function bridge add two more exact routes to the same identities. Translation by one is then carried through a complete cyclic transfer matrix, with a concrete noncommutation witness and an integral characteristic-three spectral obstruction. The standalone checker and source-bound CSV/SVG records support the hand derivation. These are classical identities presented as one continuous realization argument, not a new theorem claim.
An extended derivation with quantum phase and measurement
Quantum phase, measurement, and information loss follows one qubit through three measurement settings, exact tomography, partial dephasing, and a concrete recovery obstruction. A coherent two-qubit record can be uncomputed, while a classical record loses the relative phase. Parity packets and their singleton refinement show which coherence a measurement removes. The page includes three browser experiments, exact Python matrix calculations, downloadable tables, and a generated figure. Three further collection examples develop amplitude damping, the Quantum and Circuits book's modulus-12 Fourier phase separation, and three-qubit syndrome recovery. The arithmetic browser experiment contrasts signed, unsigned, and fully dephased arithmetic states.
Run in your browser
Open the Signal Laboratory for a larger experiment. Generate or import 16–256 integer samples, add seeded noise, and compare Haar wavelets, orthonormal cosine modes, and sample coordinates at the same coefficient budget. Compare weighted-energy, raw-magnitude, and prefix selections; inspect individual components; and explore reconstruction error over every budget. Export signals or error curves as CSV, save the experiment as JSON, or copy a link to the complete setup.
Draw your own reference directly on the signal plot, or set a selected sample precisely with the keyboard. Edits use whole numbers from −64 to 64 and support undo/redo. Save up to 12 runs in the browser notebook, restore or rename them, and overlay reconstructions from identical reference and observed data. Notebook JSON preserves the collection; a results CSV exports its comparison table.
Four starter experiments open paired runs for sharp edges, a cosine-like tone, detail in noise, and Haar ranking rules. Saved-run comparisons offer reconstruction, signed-error, and B-minus-A views, with a linked sample inspector and a jump to the largest absolute difference. Download comparison CSV for both reconstructions and their signed errors at every address.
Copy a comparison link to share both named runs, the chosen plot view, and the selected sample. Links open a preview and recompute the results while leaving the recipient's saved notebook unchanged. The preview can be saved as two new notebook runs when space is available.
Use Inspect a region in a saved comparison to zoom to an inclusive address range. The local table reports RMS error, mean signed error, and maximum absolute error for both runs. Select a maximum to inspect its sample, or jump to the largest difference within the region. The sample table and comparison CSV use that range and retain the original addresses. Comparison links also restore the region; Show full signal returns to the complete comparison. The reconstructions and coefficient budgets continue to use the full signal.
The quality-target finder scans every coefficient budget to find the first that retains a requested percentage of observed energy or meets a reference RMS-error limit. Apply a result to the active experiment or save all qualifying representations as notebook runs. Unreachable limits are reported explicitly.
The reference signal is known in these experiments. The displayed reference minimum uses that knowledge; it does not estimate the clean signal behind an unknown real-world recording. Cosine calculations are numerical approximations.
Explore the interactive diagrams. Change matrix entries, keep or omit transfer blocks, refine an eight-sample signal, and move inputs along a shared-summary line. The controls update the diagrams and exact whole-number results immediately, without writing code. Each experiment links back to its hand calculation and sources.
The Recovery Laboratory extends the shared-summary example to 2–4 input coordinates, up to four measurements, and up to three linear questions. Edit integer or rational coefficients, disable measurements, and inspect either an exact recovery rule or a kernel direction that changes the requested answer. Move two inputs while their enabled measurements stay fixed. The repair action adds the minimum number of independent linear measurements needed for all the questions, with undo. Share the complete settings or download the exact calculation as JSON.
Use Copy experiment link to save or share the selected experiment's exact inputs and packet or coefficient choices. A Haar link also restores the coefficient and sample address being inspected. Open the link to return to those settings. If clipboard access is unavailable, select and copy the link shown below the button.
For a deeper experiment, open the 16-sample Haar lab. Decompose a signal through four levels, choose individual coefficients or a magnitude cutoff, and inspect the exact reconstruction error and weighted energy balance. The multilevel calculation records the default example and its expected results.
Open the Python playground. Choose a demonstration, press Run and compare its output with the hand calculation. You can edit the code, stop a calculation and reset the original example. Python loads on the first run; allow extra time for the initial download. The playground also includes a tested introductory Komorebi Math example.
For examples drawn directly from the sequence work, use Komorebi sequence shapes. These presets plot Paper IV's reciprocal coefficients and factorial normalization, compare the family in labelled 2D and 3D views, and explore the Special Functions atlas's factorial-weighted Stirling sums. Each includes exact data downloads and source links.
Run without installing a library
Use Python 3.10 or later. The program uses only the standard library and exact integer arithmetic. It does not need internet access or write files.
Read the complete program on the site or download komorebi_demonstrations.py. Open a terminal in the folder containing the downloaded file and run:
python -B komorebi_demonstrations.py packets
python -B komorebi_demonstrations.py refinement
python -B komorebi_demonstrations.py recovery
On Windows, py -3 can replace python if that is how your interpreter is
installed. No package installation or virtual environment is needed for this
program. With no argument, it runs all three demonstrations.
If you already have a collection checkout, run from its root:
git rev-parse HEAD
python -B examples/komorebi_demonstrations.py all
Keep that revision with any saved output. The downloadable program's source record gives its collection revision and SHA-256 checksum. The repository also contains the program source.
What a passing run establishes
The program checks the displayed fixed matrices, lists and witnesses. Each demonstration explains the mathematical identity or contradiction behind its output. Matrix identities can justify statements about every vector in the specified finite space. Checking a list of sample vectors alone would not do so.
The website workflow runs all three programs and compares their output with the pages below. It also runs the separate checker for the original hand-worked examples. These runs do not replay Lean, qualify the production libraries or prove wider research claims. All software remains beta and work in progress.
Continue with the libraries
Use First runs for installation, supported environments and the introductory Python and C library programs. The demonstrations here are small standalone teaching programs. The libraries have their own APIs and tests.