Spin as an Emergent Degree of Freedom from Sub–Planckian Coarse-Graining

This work explores the possibility that quantum spin is not a fundamental property, but an emergent phenomenon arising from sub–Planckian structures filtered through the Planck-scale limit of measurability. Starting from the relativistic length contraction and the hypothesis of a minimal observable length, we argue that internal degrees of freedom existing below the Planck scale cannot be directly accessed by an external observer. Instead, coarse-graining at the Planck length acts as a filter that erases most fine details while preserving global symmetries such as parity. By constructing a toy model with Gaussian substructures, we show that symmetric and antisymmetric combinations survive coarse-graining as two distinguishable states, naturally forming a binary degree of freedom. This mechanism provides a physical interpretation of spin-$1/2$ as the residual imprint of hidden sub–Planckian dynamics. The proposal links with frameworks such as Doubly Special Relativity, the Generalized Uncertainty Principle, and holography, all of which imply a minimal length scale. Implications include a new perspective on the origin of spin, possible extensions to higher spins, and connections with emergent internal symmetries.