Read the equation
The multiplier b decreases as the squared Euclidean norm of Ψ grows: b=1 at the zero vector and approaches zero as the norm becomes large. If b is computed from the same vector it rescales, the output norm is r/(1+r²/φ), whose maximum is √φ/2 at r=√φ. This is a radial contraction rule with a precise bound, rather than a division that makes every nonzero vector have unit norm.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| b | Scale factor |
| Ψ | State vector |
| r | Euclidean norm of Ψ |
| φ | Golden ratio |
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 bound assumes the multiplier and the rescaled vector use the same state snapshot.
Original context
Read the surrounding source passage
## 2.2 Normalization Function ``` b(t) = 1 / (1 + ||Ψ||² / φ) ``` **Where**: - `||Ψ||²` = sum of squares of all 16 family activations - `φ` = golden ratio **Explanation**: This normalizer prevents unbounded growth in the meta-state while preserving relative magnitudes. The golden ratio in the denominator creates a soft ceiling that allows the system to approach but never exceed natural limits. The resulting value `b` is always between 0 and 1. ## 2.3 Golden Spiral Rotation ``` θ_k = 2π / φ^k ``` **Explanation**: Used in the Lattice Engine to rotate transformation kernels. Each strand operates at a different angular frequency, creating rich interference patterns that enable emergent computation. --- # 3. G^16 Family Functions
Exact original notation
b(t) = 1 / (1 + ||Ψ||² / φ)