Local Holographic Reduction of Vacuum Action: The Topological Postulate n=D-1 via the Generalized Stokes Theorem

We demonstrate that the integration of the vacuum action in confined quantum systems undergoes a strict, deterministic dimensional reduction. By applying Morse Theory to the bounded phase space of an isotropic confined state, we establish that the relevant spatial phase manifold MD is diffeomorphic to the closed D-dimensional ball BD. Furthermore, because gauge invariance enforces transversality via the Ward–Takahashi identity (qμΠμν(q) = 0), the effective vacuum action is an exact differential D-form. Invoking the Generalized Stokes Theorem, we prove that the macroscopic bulk integral collapses identically to its topological boundary, the (D−1)-dimensional hypersphere SD−1. This establishes the dimensional selection rule: not as an axiomatic phenomenological parameter, but as a rigorous geometric necessity. The derivation thereby formalizes the Holographic Principle at the level of the local quantum phase space.