gh-eq-6e39a9d6de9c64f1documented-equation
Read the equation
The update is 0.9x+0.1d, with d=dx_norm. It retains 90% of the previous scalar state and moves 10% toward the supplied target. For a fixed d, the error after n updates is 0.9ⁿ(x₀−d), so the state converges geometrically to d.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| x | Current scalar family state |
| dx_norm | Supplied normalized input d |
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
An ODE interpretation additionally needs a time step and a specified derivative.
Original context
Read the surrounding source passage
## 3.1 Family 0: Continuous Dynamics
```
f₀(x) = x + 0.1 · (dx_norm - x)
```
**Equation Type**: First-order ODE approximation: x' = F(x,t)
**Explanation**: Models smooth, continuous change toward the input signal. The 0.1 coefficient creates a gradual approach, preventing sudden jumps. This family handles flowing, analog-like processes.
## 3.2 Family 1: Discrete Dynamics
```
f₁(x) = 0.9 · x + 0.1 · utility
```
**Equation Type**: Discrete state transition: x_{t+1} = F(x_t, u_t)
**Explanation**: Implements exponential smoothing between current state and utility signal. The 90/10 split creates memory of past states while responding to new inputs. Used for step-by-step reasoning.
## 3.3 Family 2: Reaction-Diffusion
```
f₂(x) = x + 0.05 · (softsign(dx_norm) - x)Exact original notation
f₀(x) = x + 0.1 · (dx_norm - x)
Related builds
Sources & provenance
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