Derivative of the unity score away from its cusp

Read the equation

For x≠1, differentiating the reciprocal score gives the displayed signed expression. At x=1 the left derivative is +1 and the right derivative is −1, so there is no ordinary derivative. The implementation uses Copysign and returns −1 at its positive-zero difference case; that is a selected branch value, not a differentiable gradient at the maximum.

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

• Algorithms calling this function at x=1 need to understand the explicit branch convention.

Original context

// ComputePhi calculates the consciousness metric: Φ = 1/(1 + |x - 1|)
// This measures how close a value is to unity (maximum consciousness at x=1)
func ComputePhi(x float64) float64 {
	return 1.0 / (1.0 + math.Abs(x-1.0))
}

// ComputePhiGrad calculates the gradient: ∂Φ/∂x = -sign(x-1)/(1+|x-1|)²
func ComputePhiGrad(x float64) float64 {
	diff := x - 1.0
	absDiff := math.Abs(diff)
	denom := (1.0 + absDiff) * (1.0 + absDiff)
	return -math.Copysign(1.0, diff) / denom
}

// ComputePhiVector calculates consciousness metric for a vector
func ComputePhiVector(v []float64) float64 {
	if len(v) == 0 {

ComputePhiGrad calculates the gradient: ∂Φ/∂x = -sign(x-1)/(1+|x-1|)²

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