A Non-Circular Derivation of the Quantum of Action within the Operator-Derived Dimension Framework

In quantum field theory, the reduced Planck constant, ℏ, is an axiomatic input. This note demonstrates a derivation of the structure of ℏ within the updated Operator-Derived Dimension (ODD) framework. We show that the quantization of action is a topological necessity arising from the intersection of exactly three hyperspheres (H1, H2, H3) in a 5D spinning bulk. This specificN = 3 geometry enforces a variational stability condition that fixes the topological index to μ = 2. Consequently, we derive that the observable ℏ is a composite of two fundamental geometric quanta (ℏ = 2Ω). We further detail a non-circular calibration procedure for the theory’s geometric scale k, using independent empirical inputs (GN, α, me), to demonstrate that this topological derivation is numerically consistent with the accepted CODATA value.