Start with a question

Visual introduction

Useful background: vectors, probabilities, and the idea of a change of basis. The pictures provide an entry point; follow their linked text when a definition is unfamiliar.

1. One selector, three realizations

   Follow the same finite selector through arithmetic, Fourier coordinates, and quantum language.

2. Amplitudes and probabilities

   Compare the amplitude picture with the measurement histogram.

3. Change the measurement basis

   Follow the visual example of a distinction that becomes visible after a basis change.

4. A symmetry example

   Inspect the weight-two benchmark and the costs retained in its model.

5. A restored scratch wire

   See the boundary-restoration requirement in a circuit example.

Checkpoint: explain what a representation changes, what a measurement records, and what must be restored. Then choose a treatment in the full-study route.

Explore both complete visual sequences: all 50 pages.

Try the existing phase and measurement walkthrough, then change the phase in the browser.

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Full study

Useful background: complex linear algebra and finite Fourier transforms for arithmetic and symmetry; gate costs for catalysis; Boolean polynomials over the binary field for circuits.

1. Chapters 1–2 · Arithmetic realizations

   Follow the selector, Fourier projector, and normalized state. Work through the modulus-twelve calculation before the finite bridge theorem.

2. Chapters 3–7 · Symmetry collapse

   Track the packet coordinates, updates, and measurement costs alongside the dimension of the compressed representation.

3. Chapters 8–12 · Catalytic gate sets

   Keep the restricted gate alphabet, allowed excursion, cost convention, and restored boundary state explicit.

4. Chapters 13–21 · Circuit obstructions

   Read the circuit model before the cut comparison, safe covers, projection, and visual examples.

Checkpoint: for a result you want to use, write down its model, hypotheses, resource convention, and conclusion. Use the formal companion to inspect its recorded coverage.

Preface and volume guide · Shared glossary

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Formal companion

Useful background: the corresponding written result and enough Lean notation to read a declaration type. Begin with coverage before selecting a source file.

1. Read the verification boundaries

   Start with Appendix F to distinguish the retained formal records, written arguments, and finite computations.

2. Inspect the arithmetic companion

   Read the trust boundary in Appendix A, then follow its proof architecture and individual formal statements.

3. Inspect the circuit model

   Compare the definitions in Appendix D with the written boundary and restoration conditions.

4. Find a circuit declaration

   Follow the register to its full type, source location, block, and axiom set.

5. Return to the written result

   Use Appendix E to open a numbered statement in HTML or at its printed PDF page.

6. Read the reproduction instructions

   Identify the authoritative files and recorded execution requirements before attempting a replay.

Checkpoint: match a declaration’s full type to the written statement and its hypotheses. The inventories include supporting declarations and do not certify the entire volume.

For the other treatments, consult the symmetry evidence ledger and catalytic obligation ledger.

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