Read the equation
The magnitude is tanh(αC) and the phase advances as ωt. Under the intended nonnegative C, the magnitude is nonnegative and bounded below one. Keeping the complex value retains phase information for the next stage; taking its magnitude alone would discard that phase.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| S | Complex signal |
| C | Input score |
| alpha | Magnitude gain |
| omega | Angular frequency |
| t | Time |
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
Read the surrounding source passage
inline double L2_to_L3(double x_prime, double x_c, double sigma = 0.5) {
double diff = std::abs(x_prime - x_c);
double phi_term = math::phi_constraint(diff);
double gauss_term = math::gaussian_kernel(x_prime, x_c, sigma);
return phi_term * gauss_term;
}
// T₃₄: S = tanh(αC)·exp(iωt)
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);Exact original notation
T₃₄: S = tanh(αC)·exp(iωt)