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The mixing coefficient ρ=1/(1+φ)=1/φ²≈0.381966. Each update retains about 61.8% of the old state and takes about 38.2% of dx_norm. This is an exponential moving average: with a constant target, the remaining error is multiplied by 1−ρ=1/φ each time.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| ρ | Input mixing weight |
| x | Current stored field value |
| dx_norm | New target input |
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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)]
**Explanation**: Minimizes surprise (absolute change) while maximizing utility. This implements the free energy principle from theoretical neuroscience.
## 3.13 Family 12: Compression / Rate Control
```
f₁₂(x) = x - 0.05 · |dx_norm|Exact original notation
ρ = 1 / (1 + φ) f₁₀(x) = (1 - ρ) · x + ρ · dx_norm