Derivation from Underlying Dual‑Primitive‑Element‑Degree‑of‑Freedom Scaling to Emergent Planck Action and Planck's Constant

This work presents a first‑principle derivation of the emergent Planck constant h based on a vacuum primitive‑element elastic cluster model built upon four‑dimensional honeycomb topology. Starting from a set of physical axioms including space‑filling rigidity, particle constraint topology, cluster damping and elastic deformation thresholds, the paper constructs two independent derivation paths for the magnification coefficient: a purely theoretical path without inputting experimental values of the Planck constant or measured speed of light, and an observational cross‑check path. Two distinct statistical accounting schemes for effective primitive‑element numbers are distinguished: physical‑particle counting for bulk spatial filling, and dynamical‑mode‑degree‑of‑freedom counting for local vibrational coupling. Global correction coefficients including the mass calibration factor A_mass and the seven‑cavity topology factor C_7 are introduced to maintain internal self‑consistency of the framework. Through layered superposition of core compression gain, vacuum‑polarization gain and many‑body synergistic interference gain, the fine‑structure‑related topological constant C_topo=137.036 is analytically obtained from double‑layer spherical close‑pack topology. The theoretically derived Planck constant yields a residual relative deviation of approximately 0.92 % compared with experimental measurement. This offset is interpreted as systematic reading bias caused by outer‑layer polarization cloud contamination in conventional mass‑radius measurements, rather than an intrinsic defect of the underlying vacuum topology. Within this model, the Planck constant originates from the intrinsic action quantity of dual primitive‑element pairing units; coherent cluster‑interference scaling cancels the dependence on cluster size, which explains the universality of h across micro‑atomic and macroscopic coherent systems. This paper adopts the vacuum‑scale‑dependent field model as an established working framework. It focuses on complete analytical deduction and internal self‑consistency verification, while in‑depth justification of underlying postulates and systematic comparison with alternative theories are deferred to future work. As an application of the same theoretical system, the vacuum‑compression mechanism accounting for the proton‑radius puzzle is recapitulated, where the optimal constrained parameter η_0'=0.0337 is uniquely fixed by four groups of independent experimental constraints.