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The increment is 0.05 softsign(dx_norm−x), so its magnitude is less than 0.05 and its direction points toward the input. For a fixed target, this reduces the distance to that target without overshoot. The displayed scalar rule does not by itself implement separation, alignment, cohesion, or other multi-agent swarm interactions.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| x | Current state |
| dx_norm | Attraction target |
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Original context
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## 3.10 Family 9: Swarm Interaction
```
f₉(x) = x + 0.05 · softsign(dx_norm - x)
```
**Equation Type**: Swarm force: x_i = Σ f(x_j - x_i)
**Explanation**: Models attraction/repulsion dynamics between agents. The softsign creates bounded forces that prevent collapse or explosion.
## 3.11 Family 10: Field Accumulator
```
ρ = 1 / (1 + φ)
f₁₀(x) = (1 - ρ) · x + ρ · dx_norm
```
**Equation Type**: Leaky integrator: φ_t = (1-ρ)φ_{t-1} + Δφ
**Explanation**: Accumulates input over time with golden-ratio-based decay. The decay rate ρ ≈ 0.382 creates balanced memory that favors neither too much nor too little history.
## 3.12 Family 11: Variational Free Energy
```Exact original notation
f₉(x) = x + 0.05 · softsign(dx_norm - x)