Quadratic input penalty

Read the equation

Squaring dx_norm makes equal positive and negative inputs incur the same cost. For λ≥0, the penalty is nonnegative and grows quadratically, so doubling input magnitude quadruples the cost. λ determines how strongly this contribution competes with the other objective terms.

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

• A negative λ rewards large magnitude instead of penalizing it.

Original context

## 5.5 Penalty Component

```
Penalty = λ · dx_norm²
```

**Where**:
- `λ` = penalty weight (default 0.1)

**Explanation**: Penalizes high-change environments, encouraging the system to seek stable conditions.

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# 6. DNA Holographic Memory

## 6.1 Learning Equation

```
θ_k^{new} = (1-α) · θ_k^{old} + α · w · ψ_k(h(item)) · Φ(importance)
```

**Where**:
- `θ_k` = k-th component of the 16-dimensional knowledge vector
- `α` = learning rate (default 0.1)

Penalty = λ · dx_norm²

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