Completed-Zeta Scattering Quotient

Abstract

A real spectral parameter gives a unit-modulus quotient of the classical completed-zeta reading.

Theorem 1.1 (Real spectral scattering quotient has unit norm).

$$\forall t\in \mathbb{R},\mathrm{completedZetaReading}\left(\frac{1}{2}+t\cdot i\right)\ne 0\Rightarrow \Vert \frac{\mathrm{completedZetaReading}\left(\frac{1}{2}-t\cdot i\right)}{\mathrm{completedZetaReading}\left(\frac{1}{2}+t\cdot i\right)}\Vert =1$$

Proof. Machine-checked in Lean as D5/S3/Weil/Scattering/CompletedZetaScatteringQuotient.real_spectral_scattering_quotient_norm (✓ std3). ∎

Source. Repository-derived.

Commentary.

This is the leading unitary clause of the source scattering form. For a real spectral parameter, the reflected completed-zeta values coincide by the functional equation, so their quotient has norm one when the denominator is nonzero.

The declaration is an honest partial closure. It does not define a scattering matrix, phase branch, zero-counting function, phase jumps, Wigner delay, numerical certificate, or physical interpretation; all of those source clauses remain unresolved.

References

• Truth anchor: D5/S3/Weil/Scattering/CompletedZetaScatteringQuotient.real_spectral_scattering_quotient_norm
• Dependency: D5/S3/Zeros/CompletedZeta