Scattering-Ratio Completion

Abstract

Scattering-ratio readings together with right-shift normalization determine the function.

Theorem 1.1 (Scattering ratios determine a normalized function).

$$\forall F:\mathbb{C}\to \mathbb{C},\forall G:\mathbb{C}\to \mathbb{C},\left(\forall z\in \mathbb{C},F(z)\ne 0\right)\wedge \left(\left(\forall z\in \mathbb{C},G(z)\ne 0\right)\wedge \left(\left(\forall s\in \mathbb{C},\frac{F(2\cdot s-1)}{F(2\cdot s)}=\frac{G(2\cdot s-1)}{G(2\cdot s)}\right)\wedge \left(\forall z\in \mathbb{C},\underset{n\to \infty }{\lim }\frac{F(z+n)}{G(z+n)}=1\right)\right)\right)\Rightarrow F=G$$

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

Source. Repository-derived.

Commentary.

The displayed hypotheses keep the source ratio observation explicit: the values of F and G are nonzero, their shifted ratios agree, and the quotient F(z+n)/G(z+n) tends to one along every right shift.

Evaluating the ratio identity at (z+1)/2 gives one-step periodicity of F/G. Iteration and the right-shift limit force that quotient to equal one at every z, hence F=G.

References

• Truth anchor: D5/S3/Weil/Scattering/ScatteringRatioCompletion.scattering_ratio_completion