Blend family output with the two-state interpolant

gh-eq-c31c8f63f7e968a5documented-equation

Ψ''_i(t+1) = 0.5 · Ψ'_i(t+1) + 0.5 · (Ψ_i(t-1) + (φ-1) · (Ψ_i(t) - Ψ_i(t-1)))

Read the equation

Half the result comes from the weighted family update. The other half is an interpolant between the previous and current states, with weights 2−φ and φ−1; these weights are positive and sum to one. The displayed term is therefore interpolation within the two prior values, although the source calls it inertial correction.

Terms & definitions

Term	Meaning in this source
Ψ_prime	Weighted family output
Ψ_t	Current state
Ψ_previous	Previous state

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

The full transition can still amplify values through its family functions or weights.

Original context

Read the surrounding source passage

## 4.3 φ-Correction (Inertial Blending)

```
Ψ''_i(t+1) = 0.5 · Ψ'_i(t+1) + 0.5 · (Ψ_i(t-1) + (φ-1) · (Ψ_i(t) - Ψ_i(t-1)))
```

**Explanation**: This equation blends the new state with a momentum term based on the golden ratio. The (φ-1) ≈ 0.618 factor creates natural oscillation damping while preserving meaningful trends.

## 4.4 Stability Gating

```
γ = γ_min + (1 - γ_min) · clamp₀₁(stability)
Ψ'''_i(t+1) = Ψ_i(t) + γ · (Ψ''_i(t+1) - Ψ_i(t))
```

**Where**:
- `γ_min` = minimum gate value (default 0.1)
- `stability` = external stability signal

**Explanation**: When stability is low, updates are damped (γ approaches γ_min). This prevents erratic behavior during uncertain conditions.

## 4.5 Final Normalization

```

Exact original notation

Ψ''_i(t+1) = 0.5 · Ψ'_i(t+1) + 0.5 · (Ψ_i(t-1) + (φ-1) · (Ψ_i(t) - Ψ_i(t-1)))

Related builds

Sources & provenance

That AI Guy Core — Mathematical Reference v1.0.0

Lines 288–288