Apollo signature: golden-base hyperbolic cosine

Read the equation

The sum φˣ+φ⁻ˣ equals 2cosh(x logφ). It is even in x, has minimum two at zero, and grows with |x|. A downstream clamp to [0,1] can therefore remove much of this signature’s variation unless combined terms first change its scale.

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

func FAthena(x float64) float64 {
	// Simplified: find max of utility function U(x,y) = -y² + 2xy
	// argmax_y: dU/dy = -2y + 2x = 0 => y* = x
	// U(x, x) = -x² + 2x² = x²
	return x * x
}

// FApollo: F_Apollo(x) = φ^x + φ^{-x} (Mathematics - golden ratio)
func FApollo(x float64) float64 {
	return math.Pow(PHI, x) + math.Pow(PHI, -x)
}

// FHermes: F_Hermes(x) = dx/dt approximation (Communication - rate of change)
func FHermes(x float64) float64 {
	// Velocity function: assume x represents phase, return angular velocity
	return AngularFrequency * x
}

FApollo: F_Apollo(x) = φ^x + φ^{-x} (Mathematics - golden ratio)

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