Golden integers

The coordinates of the 120 icosian rotations contain the golden ratio, an irrational number, so a computer would normally round them. There is a way to avoid rounding altogether: keep every coordinate as a pair of whole numbers \((a, b)\) that stands for \(a + b\varphi\). Adding and multiplying such pairs gives another pair, exactly. This chapter explains that arithmetic, called \(\mathbb{Z}[\varphi]\), and says where the project uses it. Terms are also defined in the glossary.

Exact arithmetic in \(\mathbb{Z}[\varphi]\), with the coordinates of unit icosians scaled by one half.

The arithmetic

The golden ratio \(\varphi = (1+\sqrt5)/2\) satisfies

\[\varphi^2 = \varphi + 1 .\]

The set \(\mathbb{Z}[\varphi] = \{a + b\varphi : a, b \text{ integers}\}\) is closed under addition and multiplication, so it supports exact pair arithmetic. Addition adds the pairs. Multiplication uses the rule above to remove \(\varphi^2\):

\[(a+b\varphi)(c+d\varphi) = (ac+bd) + (ad+bc+bd)\,\varphi .\]

This is a proof by expanding and substituting \(\varphi^2=\varphi+1\). Two worked examples:

• \(\varphi\cdot\varphi\) is \((0,1)\cdot(0,1) = (1,1)\), that is \(1+\varphi\).
• \(\varphi^{-1} = \varphi - 1\) is the pair \((-1,1)\), and \(\varphi\cdot(\varphi-1)=\varphi^2-\varphi=1\).

The other root of \(x^2=x+1\) is \(1-\varphi=-\varphi^{-1}\). Swapping \(\varphi\) for it is the Galois conjugate, \(a+b\varphi \mapsto (a+b)-b\varphi\). Multiplying a number by its conjugate gives the integer norm \(N(a+b\varphi)=a^2+ab-b^2\).

Icosian coordinates need the factor one half

The unit icosian coordinates are \((a+b\varphi)/2\), which lie in \(\tfrac12\mathbb{Z}[\varphi]\). The factor one half must be kept. Scaled integer pairs represent them without rounding. For example, the icosian \(\tfrac12(\varphi,\,1,\,\varphi^{-1},\,0)\) has doubled coordinates \((0,1)\), \((1,0)\), \((-1,1)\) and \((0,0)\), and its length checks out exactly: \(\varphi^2+1+\varphi^{-2}=4\).

anchors.rs documents this scale explicitly. Its norm numerators have denominator 4. The same file stores one H4 root as \([a_0,\dots,a_3,b_0,\dots,b_3]\) plus a Galois companion, and warns that this invertible coefficient representation does not by itself establish an orthogonal Euclidean E8 embedding or independent extra state.

Fixed ZPhi arithmetic and exact anchor orientation tables support the research. Chirality and polarity are exact signs on \(\mathbb{Z}[\varphi]\), held as fixed integer tables in anchors.rs. They are used by the retained native geometric learner, not by the current M1 path.

A caution about addresses: equality addresses for complete tuples are not semantic distances between class numbers. Two entries being exactly equal tells you they are the same entry. It does not tell you how alike two different entries are.

Where the model uses this

The two ends of the table matter. The exact golden arithmetic is real, tested code, but the current model's own carrier is built from ordinary integers. Whether exact golden coordinates help a language model has not been measured.

Where it stands

Go deeper

• Code: anchors.rs and prime_route_attention.rs.
• The forced link from the icosians to \(\mathbb{Z}[\varphi]\) is checked in verify_connections.py.
• Next: The Hopf map and the kept fibre. Back to the overview.