The algebra inside

The same operation, with the usual tools

The operation of a book about processes turns out to be a nine-bit XOR. That connects it with three centuries of mathematics.

1. A landa is nine bits

Each dimension is a number from 0 to 3, that is, two bits (Ξ = 00, Θ = 01, Φ = 10, Ω = 11). The V₄ table is exactly the XOR of those two bits: 1 ⊕ 2 = 3 because 01 XOR 10 = 11 [8]. The hemisphere of the wrapper is one more bit (yang = 0, yin = 1).
So a landa is 1 + 4 × 2 = 9 bits, and there are 2⁹ = 512. With that encoding, the book's operation is a ∗ b = a XOR b XOR rest: the Kosmacrux library computes it literally like that, in one line.

Try it with two landas

rest

a ∗ b

hemisphere · Ξ/Θ/Φ/Ω of level 1 · level 2 · level 3 · level 4

Column by column, a XOR b XOR rest. The library gives the same landa:

they match

a and b differ in {n} bits (Hamming distance).

The book defines the coordinate of a landa relative to the rest, c(a) = (t; x₁, x₂, x₃, x₄), with t ∈ {0, 1} the hemisphere and xᵢ ∈ V₄. The two rules of the operation say exactly that c(a ∗ b) = c(a) ⊕ c(b), component by component.
That is: the 512 landas with ∗ are isomorphic to ℤ₂ × V₄⁴ = (ℤ₂)⁹, with the rest as neutral element. That is the elementary abelian group of order 2⁹ [4]: abelian, with every non-neutral element of order 2. Its elements are the vectors of a vector space of dimension 9 over the field with two elements [5].
All the properties of the theorem are inherited from there: commutativity, associativity, the neutral element and the involution are those of XOR. The book's proof is exactly that. The isomorphism is checked in «Check it», over the 262,144 pairs.

The table of a finite group is a Latin square: every row and every column is a permutation of its elements [6]. It is the «lossless» property of the theorem: from a ∗ b and a, b is recovered. And since the group is abelian, the table is symmetric about the diagonal.
Not every Latin square comes from a group [7], but the one of the 512 landas does. The full table would have 262,144 cells; that of V₄, its building block, has 16, and is a 4 × 4 Latin square.

The difference between two landas is canonical: whether the hemisphere changes and which swap of V₄ there is at each level. The movements form the group T = ℤ₂ × V₄⁴, with 512 elements, all of order 2. But the landas themselves are not that group: they are a space on which T acts freely and transitively. In mathematics that is called a torsor, or principal homogeneous space [10].
It is the same difference as between the points and the vectors of an affine space: a torsor is the group with its neutral element forgotten. To turn a landa (a state) into a movement (an action) you have to choose an origin. That origin is the rest, and that is what «the void as reference» means.

The XOR of heaps of counters is the nim sum: binary addition without carrying [9]. Charles Bouton proved in 1901 that the winning strategy in the game of Nim is always to leave a nim sum of zero [17].
What a ∗ b and the nim sum share is this: two objects combine without carrying anything, and combining an object with itself gives zero. It is an analogy: Nim has nothing to do with landas.

Chapter 6 of the book puts on V₄ a canonical bilinear form ω(u, v) = u₁v₂ ⊕ u₂v₁ (a determinant) and defines, for each landa w, the character χ_w(x) = (−1)^⟨w, x⟩, which is +1 or −1. It satisfies χ_w(a ∗ b) = χ_w(a) · χ_w(b), and the 512 characters are distinct and orthogonal.
In a finite abelian group the irreducible representations have degree 1 and are precisely the characters; over the Boolean group (ℤ/2ℤ)ⁿ, the Fourier transform is the Hadamard transform [12]. This is a generalized Fourier transform, equivalent to a multidimensional discrete transform of size 2 × 2 × … × 2, and its matrix has entries (−1)^(k·n) with the bitwise dot product [11].
That is why «every landa is also a frequency», and piloting becomes multiplication by signs. Here the book follows Terras [16]. Multiplicativity, orthogonality and the exact inverse are checked in «Check it».

Another way of seeing the 512 landas: as the vertices of a hypercube of dimension 9, the graph whose vertices are the 9-bit strings, adjacent when they differ in a single bit [13][14]. Composing with a fixed landa m —x ↦ x ∗ m— flips the same bits of every vertex, so it preserves Hamming distances: it is a translation of the hypercube.
Careful: this 9-bit hypercube is not the book's characteristic graph, the one of the eight poles on the tesseract. They only share the idea that the geometry of the bits organizes the system.

The theorems, with proofs, are in chapters 4 and 6 of the book. Here they are only shown and checked.