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The old stored quantity is multiplied by 1−ρ and a new increment is added. When 0<ρ≤1, earlier contributions decay geometrically. This reference uses an arbitrary increment Δφ, whereas the neighboring implemented family specifically uses ρ times its input.
Terms & definitions
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Original context
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## 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
```
f₁₁(x) = x - 0.05 · |dx_norm| + 0.05 · utility
```
**Equation Type**: Free energy: F = E[ln q(z) - ln p(x,z)]Exact original notation
Leaky integrator: φ_t = (1-ρ)φ_{t-1} + Δφ