Interactive lab · where the S-matrix numbers come from
The S-matrix looks like a pile of irrational decimals. It isn't. Every number in it is built from one equation, and two identities make all of it collapse to whole numbers.
A worksheet, not a result. The mathematics on this page is standard topological quantum computation, roughly 1984–2009. None of it is new. “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.” These sheets exist to move a reader from the first state to the second, and they make no claim of their own.
The method is predict–then–check. Every input box, drill and gate here is built for one move: write down what you think the number is before you reveal it. Numbers are sourced to a paper where a paper says them, and marked unchecked where none does — a sheet whose whole method is check it yourself cannot ask you to take its own arithmetic on trust.
If you are coming from an operator formalism, you may want to ask of every quantity here what the declared observable is, and what mapping puts it into the domain before the algebra acts. That is a fair question, and these sheets sit upstream of it — the arithmetic of d² = d + 1 is the same arithmetic whichever way it is answered. They take no position on it. The full note is on the index.
Prerequisites: none. This is where the series starts — about ten minutes with a pencil.
On screen you see 0.525731 and 0.850651 and nothing suggests where they came from. Written exactly they are 1/𝒟 and φ/𝒟 — and once you can multiply things like φ³ in your head, the messy sums turn into exactly 1 and exactly 0 without a calculator.
That skill is this page. It takes about ten minutes.
Everything starts here. The Fibonacci fusion rule forces the quantum dimension to satisfy:
Solve it and you get φ = (1+√5)/2 = 1.618034… and the other root −1/φ = −0.618034…. But you will almost never need the decimal, because the equation itself is the useful form. Rearranged, it gives two identities that do all the work:
The second follows from the first: divide φ² = φ + 1 by φ to get φ = 1 + 1/φ, then rearrange. Check it numerically: 1.618034 − 1 = 0.618034, and 1 ÷ 1.618034 = 0.618034. Same number.
Because a square collapses to something linear, every power of φ collapses too. You never need φ⁵ as a decimal — it is just some whole number plus some whole number times φ. Try it below.
Type a power. The page reduces it using nothing but φ² = φ + 1, applied over and over.
Notice the coefficients: they are Fibonacci numbers. φⁿ = Fn−1 + Fnφ. That is the same sequence you watched appear in the power-iteration table — the arithmetic and the matrix are the same fact.
For a rank-2 theory the unnormalised S-matrix is built directly from the quantum dimensions:
First row: the dimensions of the two objects, 1 and d. That is all. The −1 in the corner is what makes the two rows independent.
S has to be orthogonal — its rows must have length 1. Row 1 has length √(1² + d²), so divide everything by that. Call it 𝒟:
That is where 𝒟 comes from. It is not a mysterious constant — it is exactly the number that makes the rows unit length, and for Fibonacci it happens to equal √(2+φ) because 1 + φ² = 1 + (φ+1) = 2+φ.
Three checks. Each one is a sum you can do on paper using only the two identities.
| check | what you compute | exact | decimal |
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Row 1 · Row 1 = (1/𝒟)² + (φ/𝒟)² = (1 + φ²)/𝒟². And 𝒟² = 1 + φ² by definition. So the fraction is something over itself = 1. No decimals were harmed.
Row 1 · Row 2 = (1·φ + φ·(−1))/𝒟² = 0/𝒟² = 0. The φ's cancel before you divide by anything.
The quantum dimension is defined as a ratio of first-row entries:
𝒟 cancels. That is why the normalisation never affects the physics — every quantity that means something is a ratio, and the scale divides out. It is also why the Yang–Lee button in the other lab changes the matrix but not the fusion integers.
Now the sum that produced 1.170820 and −0.170820 in the other lab. Watch it in exact form:
The numerator came out φ + 2. The denominator is 𝒟² = 1 + φ² = 2 + φ. They are the same number, so the answer is exactly 1 — not 0.99999. The two ugly decimals were φ³/𝒟² and −1/(φ𝒟²), and they were only ugly because they were being shown separately.
Reduce each to the form a + bφ. Type the two whole numbers.