Toward a Spectral Proof of Goldbach's Theorem: A Formal Recasting via Prime-Exponent Arithmetic

This document introduces the Spectral Goldbach Theorem (SGT) — a formal restatement of the classical Goldbach Conjecture within the framework of Spectral Number Theory (SNT). In this formulation, every even integer greater than 2 is expressed as the sum of two spectral projections corresponding to distinct prime-exponent basis vectors. This publication presents the theorem statement only and serves as an open mathematical challenge to the global community. While no proof is included here, the structure of SGT is designed to be logically equivalent — via scalar projection — to the classical Goldbach Theorem (GT). Therefore, a proof of SGT would imply a proof of GT. Researchers are invited to rigorously analyze, validate, or challenge this spectral formulation. The challenge may be approached through number theory, operator methods, spectral algebra, or computational approaches rooted in prime-based arithmetic.