Prime Jump Decomposition

Abstract

The finite prime-power term is coherent mass minus a nonnegative translation energy, and that energy is the quadratic form of the arithmetic jump Laplacian.

Theorem 1.1 (Prime jump decomposition).

$$\forall f\in \mathcal{W},L\in \mathbb{R},hSupport:\mathrm{tsupport}\left(f\right)\subseteq [-L,L]\Rightarrow \mathrm{primeTerm}\left(\mathrm{convolutionSquare}\left(f\right)\right)=2\cdot \mathrm{totalPrimeWeight}\left(L\right)\cdot l2Mass\left(f\right)-\mathrm{arithmeticJumpEnergy}\left(L,f\right)\wedge \left(0\le \mathrm{arithmeticJumpEnergy}\left(L,f\right)\wedge \mathrm{arithmeticJumpEnergy}\left(L,f\right)=\Re ({\int }_{\mathbb{R}}\mathrm{conj}\left(f\left(y\right)\right)\cdot \mathrm{arithmeticJumpLaplacian}\left(L,f,y\right)dy)\right)$$

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

Source. Repository-derived.

Commentary.

The test function lies in the canonical carrier of even smooth compactly supported complex-valued functions. The displayed support witness places its support inside the interval from minus L to L.

The active channels are the prime powers below exp(2L), filtered by nonzero von Mangoldt weight. Their critical-line weights are summed to form the coherent mass and weight the squared translation displacements in the arithmetic energy.

All three source clauses are public: the exact complex prime-term decomposition, nonnegativity of the independently constructed energy, and its equality with the real part of the explicit Laplacian quadratic form.

References

• Truth anchor: D5/S3/Weil/ZetaBridge/PrimeJumpDecomposition.prime_jump_decomposition
• Dependency: D5/S3/Weil/ZetaGamma/ArchimedeanJumpDecomposition