Cayley Unitarity Defect
Abstract
The zero-indexed Cayley operator is unitary exactly when every source zero is on the midline.
Theorem 1.1 (Cayley unitarity defect formula).
Proof. Machine-checked in Lean as D5/S3/Midline/Cayley/CayleyUnitarityDefect.cayley_unitarity_defect_formula (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let Z be the repository’s exhaustive duplicate-free enumeration of classical zeta zeros in the open strip. For each indexed zero rho_n, the coefficient c_n is constructed as (rho_n - 1)/rho_n. The operator C is the diagonal operator with these coefficients, its star conjugates them coordinatewise, and e_n is the coordinate basis vector.
On every coordinate, the star-unitarity defect sends e_n to delta_n e_n. The public statement identifies delta_n both as |c_n|^2 - 1 and as (1 - 2 Re(rho_n))/|rho_n|^2. Positivity of the real part in the source carrier makes every denominator nonzero.
Consequently, all enumerated zeros lie on the real-part-one-half midline if and only if every Cayley coefficient has norm one, if and only if C* C is the identity, if and only if C has its coordinatewise star as a two-sided inverse. The statement covers the full countable carrier and does not replace it with a finite matrix or a selected zero.
References
- Truth anchor:
D5/S3/Midline/Cayley/CayleyUnitarityDefect.cayley_unitarity_defect_formula