Topological Slicing and the Kinematic Emergence of the de Broglie Relation: A Helical Representation Study

This work revisits the kinematic foundations of wave–particle duality by re-examining the logic originally proposed by Louis de Broglie within the framework of Special Relativity developed by Albert Einstein. Starting from the assumption that a massive particle possesses an intrinsic periodic degree of freedom associated with its rest energy, the study explores how Lorentz covariance acts on such periodicity. By representing the internal phase on a transverse cylindrical manifold, it is shown that the relativity of simultaneity naturally converts temporal periodicity into a spatial helical phase structure when observed from a moving frame. Within this geometric representation, the longitudinal phase gradient is fixed by relativistic kinematics, leading directly to the relation kz=p/ℏkz=p/ℏ which reproduces the well-known de Broglie relation. In this framework, particle momentum appears as the torsional density of the helical phase field, while the transverse radius forms a Lorentz-invariant geometric seat for the intrinsic structure. The analysis further shows that this transverse scale naturally aligns with the characteristic length Rc=ℏ/mcRc=ℏ/mc previously identified as a structural scale in the author's earlier work on helical phase-field models. The resulting picture suggests a geometric convergence in which the de Broglie wavelength and the characteristic particle radius emerge as the longitudinal and transverse signatures of a single relativistic phase structure. This work is purely kinematic and does not attempt to resolve the detailed dynamical origin of the internal energy partition. The dynamical mechanism responsible for the transverse stabilization and the associated energy coefficients will be addressed in subsequent studies.