Eight interactive worksheets · pencil, then keyboard
The algebra of anyon theories is usually read, not computed. These sheets are for computing it — three labs, a paper course, and four tutorials, each built around numbers you work out before the page shows you the answer.
These are worksheets, not research. The mathematics on them is standard topological quantum computation, roughly 1984 to 2009 — Arovas–Schrieffer–Wilczek on fractional statistics, Verlinde and Moore–Seiberg on fusion from braiding, Rowell–Stong–Wang on classification, and the Yang–Lee edge singularity, which is older still. None of it is new.
None of the above is new mathematics — it is standard topological quantum computation, roughly 1984–2009. It was simply new to my hands. I had read it for years and never computed with it, and those turn out to be different states of knowing.
That is the whole reason these exist. Nothing here is a result, a finding, or a contribution, and no sheet argues for anything. Where a page calls something surprising, it means surprising to work through — not new to the literature. If you find a claim on one of these pages that reads as a discovery, it is a failure of my wording rather than a claim I am making.
The method is predict, then check. Every input box, drill and gate on these pages exists for one move: write down what you think the number is before you reveal it. Reading the worked answer first feels faster and teaches almost nothing — the two operations spoil each other. Each sheet keeps your ticks and answers in your own browser, so you can put one down mid-derivation and come back to it.
Numbers are sourced, or they are marked. Each sheet carries a short sources block saying which attributions were checked against the paper itself and which are stated on reputation only. That distinction matters more here than usual: language models are reliable on this mathematics and unreliable on its attributions. Ask one for the Yang–Lee central charge or conformal weight and it will give you numbers and cite Cardy — but Cardy's 1985 paper does not contain them, and neither does his 2023 review. That was checked against the actual PDFs. The values are probably right; the citations are folklore. A worksheet whose whole method is check it yourself cannot ask you to take its own arithmetic on trust, so where a number has no source it is labelled unsourced on the page that shows it.
On declarations and observables. A reader coming from an operator formalism may reasonably want to ask, of every quantity here: what is the declared observable, and what measurement mapping puts it into the domain before the algebra acts? That is a fair question, and a good one. These sheets take no position on it. They sit upstream of it — the arithmetic of d² = d + 1 is the same arithmetic whichever way the question is answered — and nothing here is an argument for or against any particular framework's answer. If you want to run that reading over these pages as you go, nothing in them will fight you.
The labs came after the tutorials and turned out to be the ones that worked. They start from a number rather than an axiom. If you only do one thing on this page, do these three in order.
Lab 1 Golden Arithmetic The S-matrix looks like a pile of irrational decimals. It isn't. Every number in it is built from d² = d + 1, and two identities collapse the messy sums to exactly 1 and exactly 0 with no calculator. Exact arithmetic in ℤ[φ], with drills. Prerequisites: none. This is where the series starts. Lab 2 Row Times Column Matrix multiplication is one rule applied four times. Click any answer cell and the page lights up the row and column that made it. Runs on to repeated application, the Fibonacci numbers appearing on their own, and eigenvectors as the directions that survive. Prerequisites: none. Pairs with Lab 1; either order works. Lab 3 Verlinde by Hand Two objects, a 2×2 matrix, and a sum with two terms. Predict a fusion coefficient, then show the working. Nudge the quantum dimension by one percent and watch every integer fall off its value — the rigidity is the point. Prerequisites: Labs 1 and 2.These four came first and were written at research-paper level — axioms before numbers. That was a misjudgement, and it is why the labs exist. Each now opens with a number and says plainly what it assumes.
Tutorial 1 One Modulus, Two Arguments Two different physical theories share the Fibonacci fusion rule — one a universal quantum computer, the other a critical point — and the fusion algebra cannot tell them apart. Fusion rings, quantum dimensions, twists, why Reidemeister I fails for anyons, and a Gauss sum that produces a central charge from two arrows. Prerequisites: Lab 1. Derivations 1 and 3 open Sections 3 and 6. No Berry phase. Tutorial 2 The Phase You Pick Up Arovas, Schrieffer and Wilczek, 1984: four pages that computed what the statistics actually are in a real material. Drag a quasihole around a loop, put a second one inside, and subtract — what is left is the exchange statistics. Predicted 1984, measured 2020. Prerequisites: Sections 1, 4 and 5 stand alone. Section 3 needs Derivation 5 (Berry phase), which is weeks rather than hours. Tutorial 3 What the Modulus Forgets Braiding determines fusion; fusion does not determine braiding. The map runs one way, with a four-element fibre sitting over the Fibonacci ring, and a theorem closing off the obvious physical route to the non-unitary half. Ends where complex zeros pinch the real axis and phase data becomes a measurable magnitude. Prerequisites: Tutorial 1, Lab 3, Derivation 1. Section 3 assumes Derivation 4. Tutorial 4 Where the Integers Come From Counting how anyons fuse is combinatorics. How a theory behaves on a torus is analysis. Verlinde's formula turns one into the other, and Section 3 is twenty minutes of pencil work in which a matrix of braiding phases produces the integer 1 — exactly, twice. Also: how to cite the result honestly, which is not by saying "Verlinde proved". Prerequisites: Lab 1 and Derivation 2. Lab 3 turns the same crank with the numbers filled in live. No Berry phase.