Cayley Ratio and the Critical Line

Abstract

The scalar Cayley ratio has unit modulus exactly on the critical line.

Theorem 1.1 (Unit circle corresponds to the critical line).

$$\forall s\in \mathbb{C},|\frac{s-1}{s}|=1\iff \Re (s)=\frac{1}{2}.$$

Proof. Machine-checked in Lean as D5/S3/Midline/CayleyCriticalLine.cayley_ratio_norm_one_iff_critical_line (✓ std3). ∎

Source. Repository-derived.

Commentary.

The proof treats the totalized division value at zero separately. For a nonzero parameter, the squared norm defect is (1 - 2 Re(s)) divided by the norm square of s.

Thus unit modulus is equivalent to vanishing horizontal displacement from real part one half.

References

• Truth anchor: D5/S3/Midline/CayleyCriticalLine.cayley_ratio_norm_one_iff_critical_line