Global Arithmetic Constraints on Local Supersymmetric Systems: L-Function Decomposition and the Spectral Realization of the Restricted Goldbach Conjecture

We analyze the Restricted Goldbach Conjecture (GC𝑅) through the construction of a sequence of finite-dimensional supersymmetric quantum mechanical systems associated with even integers 2𝑛 > 4, defined on the Goldbach Complexes 𝐾2𝑛. We establish that GC𝑅(2𝑛) is equivalent to the nontriviality of the Goldbach-Sonin space 𝒮𝐺(2𝑛), derived from the non-commuting logarithmic energy operator 𝐿 and the connectivity Laplacian 𝐻0. Recognizing the limitations of continuous deformation arguments between these discrete systems, we embed them into a global structure defined on the Hilbert space ℓ2(ℙ). We demonstrate that the sequence of local Hamiltonians exhibits non-trivial statistical correlations arising from the shared underlying arithmetic structure (the ”spine”), realized by the global operator 𝐿𝒰. We introduce the Global Dirichlet Series 𝐷𝐺(𝑠), constructed from the partition numbers of the local systems, serving as the global unifying object. We demonstrate that the analysis of 𝐷𝐺(𝑠) is governed by the properties of Dirichlet L-functions 𝐿(𝑠, 𝜒), utilizing the methodology of the Hardy-Littlewood circle method. This realizes the sequence of Goldbach spaces within the hierarchy of arithmetic L-functions, analogous to the decomposition of a Dedekind Zeta function. We establish that global constraints on the L-functions dictate the asymptotic behavior of the partition numbers, thereby imposing constraints on the stability of the local supersymmetric systems.