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Replacing |x−1| with √((x−1)²+T) removes its cusp when T>0. The resulting score has a zero derivative at x=1, but its maximum is 1/(1+√T), which is below one; with T=0.01 the maximum is about 0.90909. The smooth and original score therefore have different peak calibration.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| T | Positive smoothing parameter |
| x | Activation input |
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Original context
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inline float compute_phi_f(float activation) {
return 1.0f / (1.0f + std::abs(activation - 1.0f));
}
/**
* compute_phi_smooth - Differentiable approximation of phi
*
* Uses soft absolute value: |x| ≈ sqrt(x^2 + T)
* This allows gradient flow through the consciousness metric.
*
* @param activation The activation value
* @param temperature Smoothing temperature (smaller = sharper)
* @return Smoothed consciousness level
*/
inline double compute_phi_smooth(double activation, double temperature = TEMPERATURE_DEFAULT) {
double diff = activation - 1.0;
double smooth_abs = std::sqrt(diff * diff + temperature);
return 1.0 / (1.0 + smooth_abs);Exact original notation
Uses soft absolute value: |x| ≈ sqrt(x^2 + T)