The difference between two landas, read from the void

02 · Kosmacrux

Kosmacrux is the formal expression of the Daturoma. Its fundamental operation, a ∗ b, combines any two landas into a third: always, in a single way, and by rules that fit on one line.

Compute an operation

Understand it step by step

The idea, in one line

• a ∗ b is a moved by the movement that takes the void to b. Put another way: it is the difference between a and b, read from the void.
• 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 book's is Λ(·≻=≻·≻=): inhibition and normalization alternating, nothing circulates, nothing fluctuates.

Calculator of the operation

Choose two landas —a hemisphere and four dimensions each— and see how they combine. A landa is determined by those five data, and every combination is a landa. The derivation follows the book's format and is computed by the book's own library.

First landa (a)

Second landa (b)

The result, a ∗ b

Derivation

Level by level: d = dᵃ ⊕ dᵇ ⊕ (2, 3, 2, 3), with Ξ = 0, Θ = 1, Φ = 2 and Ω = 3. Then the hemispheres are derived again: they change exactly where the dimension repeats.

Seven properties

The theorem of chapter 4 of the book says that ∗ is:

Total and single-valued

It is defined for all 262,144 pairs of landas, with a single result.

Commutative and associative

a ∗ b = b ∗ a, and (a ∗ b) ∗ c = a ∗ (b ∗ c): neither the order of the factors nor the parentheses matter.

With a neutral element: the void

a ∗ rest = a. The rest changes nothing.

An involution

a ∗ a is the rest: between a thing and itself there is no otherness, hence no movement.

Lossless

From a ∗ b and a, b is recovered, because b = a ∗ (a ∗ b). For each a, the map b ↦ a ∗ b is a bijection of the 512 landas: a Latin square.

No inert data

Changing any datum of either operand (the hemisphere or one of the four dimensions) changes the result.

Invariant

If a and b are moved by the same movement, a ∗ b does not change: it only depends on the movement between a and b.

Check them yourself

What is built on top

The operation is the foundation of the book's later chapters:

Piloting and systences

A directive of landas pilots an entity: Lᵢ₊₁ = Lᵢ ∗ D(i mod T). The balance of the directive is D₀ ∗ … ∗ D(T−1): if it is the void, the entity closes in T steps; if not, in 2T.

Fourier

Every landa is also a frequency: seen from those frequencies, piloting becomes multiplication by signs. The transform is computed in nine stages, like any Walsh–Hadamard transform.

Becoming

The surplus per dimension of one turn (yang +1, yin −1) tells which pole dominates; the dominant pole wraps the next start. A tie produces no becoming.

This page explains chapter 4 of the Kosmacrux book (v1). The full book proves the theorems and includes the exhaustive check over the 512 landas.