The Dual Structure of Vacuum Stability: From Logical Commutators to Topological Indices

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.