Pole Rank-One Decomposition

Abstract

The completed-zeta pole pair of a convolution square is one positive boundary observation energy.

Theorem 1.1 (The pole pair is one boundary observation energy).

$$\forall f\in \mathcal{W},\mathrm{poleTerm}(\mathrm{convolutionSquare}(f))=2\Vert {\int }_{\mathbb{R}}\exp (\frac{x}{2})f(x)dx{\Vert }^2$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaBridge/PoleRankOneDecomposition.pole_rank_one_decomposition (✓ std3). ∎

Source. Repository-derived.

Commentary.

Here W is the frozen carrier of even smooth compactly supported complex functions on the real line. The convolution square and pole term are the existing canonical objects. Evenness identifies the two half-frequency boundary readings, while the frozen complex-frequency convolution factorization turns each pole evaluation into the squared modulus of the displayed integral.

References

• Truth anchor: D5/S3/Weil/ZetaBridge/PoleRankOneDecomposition.pole_rank_one_decomposition
• Dependency: D5/S3/Weil/ZetaBridge/ConvolutionSquareOrbitBounds