The ∗ operation, step by step

Chapter 4 of the book

Thirty spokes converge on the hub of the wheel; but it is its emptiness that makes it useful.

Lao Tzu, Tao Te Ching, XI

1. A landa is a hemisphere and four dimensions

A landa Λ(p₁≻p₂≻p₃≻p₄) is a chain of four poles in which each pole is a neighbor of the next. Reading a landa pole by pole hides a regularity: every pole is a dimension (Ξ, Θ, Φ, Ω) in a hemisphere (yang or yin).
The book's structure theorem says that a landa is determined by the hemisphere of the wrapper and the sequence of its four dimensions, and that every combination gives a landa. The rule that rebuilds the other hemispheres is a single one: the hemisphere changes exactly when the dimension repeats.
That is why there are 2 · 4⁴ = 512 landas, and 2 · 4! = 48 hamiltonians, the ones that repeat no dimension.

Three landas, taken apart

hemisphere of the wrapper

dimensions

repeats

repeats no dimension: it is hamiltonian

Let us number the dimensions Ξ = 0, Θ = 1, Φ = 2, Ω = 3, and let ⊕ be the operation of the following table, that of the Klein group V₄ [1][2]. Subtracting two dimensions makes sense: d ⊕ d′ is the swap that takes d to d′, and d ⊕ v is the result of applying the swap v to d.
It is the table of a group in which every element is its own inverse [3]. The three non-trivial swaps are (ΞΘ)(ΦΩ), (ΞΦ)(ΘΩ) and (ΞΩ)(ΘΦ).
The book explains why four: to subtract dimensions as an object you need a regular, normal subgroup of the permutations of the dimensions. It exists for 2, 3 and 4 dimensions (ℤ₂, A₃ and V₄) and not for 5, so four is the largest number of dimensions with this canonical algebra. With four, the normal subgroups of S₄ are 1, V₄, A₄ and S₄, and the only one of order 4 is V₄, which is regular [15].

The ⊕ table: d ⊕ d′ = the swap that takes d to d′

A landa is a state; a movement is a change. For one landa to act on another it has to be read as a movement, and that requires deciding which landa counts as «not moving». That reference is a convention.
The one the book adopts is what Kosmagest describes as the absence of all dynamism: the rest, or the void, Λ(·≻=≻·≻=). Inhibition and normalization alternating —nothing circulates, nothing fluctuates— and the dimensions Ξ and Θ absent: it says neither what there is nor how it connects. Its dimensions are (2, 3, 2, 3) and its hemisphere is yin.

In coordinates the rule is short. For two landas a and b, the landa a ∗ b is obtained with two rules:

Level by level, d(a ∗ b) = d(a) ⊕ d(b) ⊕ (2, 3, 2, 3).

yang ⊕ yang = yin, yang ⊕ yin = yang, yin ⊕ yin = yin. The other three hemispheres are derived again with the change rule.

Six products, with their derivation

Each line is computed by the book's library; the results match those of the printed book.

The void changes nothing

The mirror inverts the hemispheres

A landa with itself gives the void

Any two landas, with a dimension that repeats

The bud of inflammation with its resolution gives pure excitation

Two pure oscillations give a pure oscillation

The book's definition is symmetric: a ∗ b is a moved by the movement that takes the void to b; equivalently, b moved by the movement that takes the void to a.
A movement is written with five data: whether the hemisphere changes and, for each level, a swap of V₄.

An example, step by step

The movement that takes the void to b

a moved by that movement

a ∗ b, with the rule above

They are the same landa.

yes

The landa Λ(☆≻◇≻☆≻◇) —the void with its hemisphere inverted— inverts every hemisphere of any landa: x ∗ mirror is x with each pole swapped for its dimensional opposite. Its dimensions are (2, 3, 2, 3), those of the void, so the dimensions of x do not change; only the wrapper changes hemisphere, and since the other hemispheres are derived again from the same repetitions, they all change.
The void and pure excitation are the two extremes of the algebra: the first moves nothing; the second flips everything without changing any dimension.

If another landa e′ is taken as rest, the new operation is a ∗′ b = (a ∗ b) ∗ e′: every result is shifted by a fixed landa. What is structural does not change: ∗′ has the same properties, with e′ as its neutral element. The only thing that changes is which landa is called neutral.
The book uses the void because the neutral element must be what changes nothing. In «Check it» you can verify this with 50 alternative rests.

The void changes nothing: Λ(○≻+≻☆≻◇) ∗ Λ(·≻=≻·≻=): (0, 1, 2, 3) ⊕ (2, 3, 2, 3) ⊕ (2, 3, 2, 3) = (0, 1, 2, 3) = Ξ, Θ, Φ, Ω ; yang ⊕ yin = yang → Λ(○≻+≻☆≻◇)
The mirror inverts the hemispheres: Λ(○≻+≻☆≻◇) ∗ Λ(☆≻◇≻☆≻◇): (0, 1, 2, 3) ⊕ (2, 3, 2, 3) ⊕ (2, 3, 2, 3) = (0, 1, 2, 3) = Ξ, Θ, Φ, Ω ; yang ⊕ yang = yin → Λ(△≻□≻·≻=)
A landa with itself gives the void: Λ(○≻+≻☆≻◇) ∗ Λ(○≻+≻☆≻◇): (0, 1, 2, 3) ⊕ (0, 1, 2, 3) ⊕ (2, 3, 2, 3) = (2, 3, 2, 3) = Φ, Ω, Φ, Ω ; yang ⊕ yang = yin → Λ(·≻=≻·≻=)
Any two landas, with a dimension that repeats: Λ(△≻·≻☆≻◇) ∗ Λ(◇≻+≻○≻◇): (0, 2, 2, 3) ⊕ (3, 1, 0, 3) ⊕ (2, 3, 2, 3) = (1, 0, 0, 3) = Θ, Ξ, Ξ, Ω ; yin ⊕ yang = yang ; Ξ repeats → Λ(+≻○≻△≻=)
The bud of inflammation with its resolution gives pure excitation: Λ(◇≻☆≻+≻○) ∗ Λ(=≻·≻□≻△): (3, 2, 1, 0) ⊕ (3, 2, 1, 0) ⊕ (2, 3, 2, 3) = (2, 3, 2, 3) = Φ, Ω, Φ, Ω ; yang ⊕ yin = yang → Λ(☆≻◇≻☆≻◇)
Two pure oscillations give a pure oscillation: Λ(○≻△≻○≻△) ∗ Λ(+≻□≻+≻□): (0, 0, 0, 0) ⊕ (1, 1, 1, 1) ⊕ (2, 3, 2, 3) = (3, 2, 3, 2) = Ω, Φ, Ω, Φ ; yang ⊕ yang = yin → Λ(=≻·≻=≻·)