Diagonal-plus-cyclic kernel matrix

Read the equation

The matrix places cos(θₖ) on its diagonal and 0.1sin(θₖ) on a cyclic neighbor diagonal, with zeros elsewhere. Multiplication combines each component with its cyclic neighbor. Although called a rotation in the source, this is generally not an orthogonal rotation: when cos(θₖ)=0, it scales a cyclic permutation by 0.1 and changes vector norms.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

• Do not substitute the different angle sequence from section 2.3.

Original context

## 10.1 Kernel Rotation

```
Registry[k][i][j] = cos(θ_k) if i=j, sin(θ_k)·0.1 if j=(i+1) mod 16, else 0
```

**Where**:
- `θ_k = 2π · k / φ`

**Explanation**: Each of the 161 kernels is a rotation matrix with golden-ratio-based angles, creating rich transformation dynamics.

## 10.2 Strand State Update

```
new_state[i] = Σ_j Registry[k][i][j] · state[j]
state[i] ← state[i] · (1 - dt) + (new_state[i] + 0.1 · core_state[i]) · dt
```

**Explanation**: Each strand's state evolves through kernel transformation blended with core state influence.

## 10.3 Strand Energy

```
energy = √(Σ_i state[i]²)

Registry[k][i][j] = cos(θ_k) if i=j, sin(θ_k)·0.1 if j=(i+1) mod 16, else 0

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