**Hexadic cycles** are defined within the **DASHI Geometry Architecture** as a specific algebraic and temporal regime known as the **Hexadic Universe**, a six-beat wrap-around dimension where space-time "loops" every sixth turn. This regime is formally managed through the `HexTruth` data type, which consists of six discrete states (`hex-0` through `hex-5`) and a rotation function (`rotateHex`) proven to close in exactly six steps ($rotateHex^6 = id$).
### 1. Structure and Generation
A hexadic cycle is not merely a collection of states but is generated through the interaction of lower-order symmetries:
* **Compositional Origin:** The cycle is generated by combining a **triadic rotation** (order 3, `rotateTri`) and an **orientation involution** (order 2, `inv`).
* **The Rotation-Reversing Involution:** Formally, if $r$ is the triadic rotation and $s$ is the involution, the pair $\langle r, s \rangle$ generates a 6-cycle provided that $s \circ r \circ s = r^{-1}$. In this context, the involution is not an independent bit but a "sign flip" that reverses the direction of the rotation.
* **CLOCK vs. DASHI:** In the system's hierarchy, **DASHI** represents the triadic phase, while the **CLOCK** represents the hexadic cycle (phase + orientation). The triadic phase is extracted as a projection of the hexadic cycle every two ticks ($\mathbb{Z}/6 \to \mathbb{Z}/3$).
### 2. Physical and Linear Realization
The hexadic cycle has a rigorous mapping to linear and complex representations:
* **Eigenvalue Plane:** A 6-cycle corresponds to a 2D rotation plane by an angle $\theta = \pi/3$ (60°). This is represented by the eigenvalues $e^{\pm i\pi/3}$ over $\mathbb{C}$.
* **Roots of Unity:** While DASHI is modeled by cube roots of unity, the hexadic CLOCK is modeled by **sixth roots of unity** ($e^{ik\pi/3}$). Squaring the hexadic lift recovers the triadic DASHI phase step.
* **Symmetry Loops:** Hexadic beats are used to synchronize parallel compute threads across **6-fold symmetry loops**, ensuring consistency as abstract logic is reduced to granular operations.
### 3. Monster Group and FRACTRAN Context
Hexadic structures appear at the highest levels of the system's mathematical ontology:
* **Conjugacy Classes:** The **Monster Group ($\mathbb{M}$)** contains six conjugacy classes of order 6, labeled **6A through 6F**. For example, the **6A-orbifold** partition function involves eta-quotients that reflect the bipartite prime factoring of $6 = 2 \times 3$.
* **The Moonshine Machine:** In the **FRACTRAN** implementation of monstrous moonshine, a hexadic state (such as Class 6A) is represented by a specific integer register value (e.g., $2^2 \cdot 3 \cdot 5^2 \cdot 13^3$). Reaching the identity state from a 6-class input requires a compound trajectory that accounts for these prime factors.
* **TradeWars Gamestate:** In multiversal gameplay, the hexadic cycle serves as the "beat" for the 6-beat parallel universe, determining the temporal logic for a ship's state transitions.