The Dual Structure of Vacuum Stability

A structural framework is proposed in which quantum and classical physics emerge from the incompatibility between continuous geometric scaling and discrete arithmetic spectral constraints. Quantum behavior corresponds to a non-vanishing obstruction to their joint representation, while classical spacetime arises when this obstruction becomes topologically quantized via an Atiyah-Patodi-Singer-type index theorem. The framework introduces a consistency principle rather than new dynamics and treats the construction of the arithmetic operator as an open problem related to the Hilbert-Pólya program, drawing conceptually on ideas from Connes’ noncommutative geometry without assuming a specific spectral realization.