gh-eq-0cf3dd70006d127adocumented-equation
Read the equation
For each index k, this function maps the scalar x to sin(a)+cos(a), where a=2πφᵏx. The two terms combine into √2 sin(a+π/4), so each component has amplitude √2 and range [−√2,√2]. Different k values provide different frequencies; when x comes from a string hash, these are numerical features of that hash rather than a semantic understanding of the string.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| ψ_k | Feature at index k |
| k | Index from 0 through 15 |
| x | Normalized scalar input |
| 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 amplitude is √2, not 2.
Original context
Read the surrounding source passage
## 2.1 Golden Basis Function ``` ψ_k(x) = sin(2π · φ^k · x) + cos(2π · φ^k · x) ``` **Where**: - `k` = basis index [0, 15] for the 16 cognitive families - `x` = input value normalized to [0, 1] - `φ` = golden ratio **Explanation**: This is the core encoding function for holographic memory. Each basis function creates a unique frequency pattern based on powers of the golden ratio. When all 16 bases are combined, they create a holographic representation where information is distributed across all dimensions. The sum of sine and cosine creates a phase-shifted wave that captures both magnitude and phase information. ## 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.
Exact original notation
ψ_k(x) = sin(2π · φ^k · x) + cos(2π · φ^k · x)
Related builds
Sources & provenance
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