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 state, we establish its topology as a -dimensional ball (). Furthermore, because gauge invariance enforces transversality via the Ward identity, the effective vacuum action operates strictly as an exact differential form. Utilizing the Generalized Stokes Theorem, we prove that the macroscopic bulk physics collapses identically to its topological boundary, the -dimensional hypersphere (). This completely dictates the accessible vacuum measure, establishing the dimensional selection rule not as an axiomatic phenomenological parameter, but as a rigorous geometric necessity. Ultimately, this derivation formalizes the Holographic Principle at the local quantum phase space level.
Luis Rodrigues (Sun,) studied this question.