Layer four to five: magnitude response with phase modulation

Read the equation

The body takes sigmoid(w|S|) and multiplies it by 1+0.2sin(arg S). The phase multiplier ranges from 0.8 to 1.2, so the output can exceed one. The source’s phase_alignment placeholder is therefore this specific sinusoidal modulation in the implementation.

Terms & definitions

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

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

    inline std::complex<double> L3_to_L4(double coherence, double time,
                                          double alpha = 2.0, double omega = 0.01) {
        double magnitude = math::tanh_stable(alpha * coherence);
        double phase = omega * time;
        return std::polar(magnitude, phase);
    }

    // T₄₅: P = σ(W·|S|)·(1 + phase_alignment)
    inline double L4_to_L5(const std::complex<double>& sync, double w = 1.5) {
        double mag = std::abs(sync);
        double phase = std::arg(sync) / PI;
        double base = math::sigmoid(w * mag);
        return base * (1.0 + 0.2 * std::sin(phase * PI));
    }

    // T₅₆: D = softmax(W·P + b)
    inline double L5_to_L6(double predictive, double w = 2.0, double b = 0.0) {
        double z = w * predictive + b;

T₄₅: P = σ(W·|S|)·(1 + phase_alignment)

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