Scalar Waves and Sound under the Logic of Existence (LoE): Frequency as Emergent from Relational Topology and a Clean Bridge to α

This work explores scalar waves and sound through the framework of the Logic of Existence (LoE), presenting frequency as an emergent property of relational topology rather than a predefined parameter. Using the Laplacian spectrum of a graph to model relational coupling, modal frequencies arise as ωk2=Ω02+κλk(L)\omega_k^2=\Omega_0^2+\kappa\lambda_k(L)ωk2=Ω02+κλk(L), revealing a direct link between topology, connectivity, and phase coherence. The study introduces a dimensionless relational coupling constant αR(ω)=ZR(ω)/(2RK)\alpha_R(\omega)=Z_R(\omega)/(2R_K)αR(ω)=ZR(ω)/(2RK), analogous to the fine-structure constant α=Z0/(2RK)\alpha=Z_0/(2R_K)α=Z0/(2RK), where ZR(ω)Z_R(\omega)ZR(ω) denotes an effective impedance of the relational network. Changes in network structure modify the impedance spectrum, producing a measurable “running” of αR\alpha_RαR with frequency. Practical verification pathways are outlined for acoustic metamaterials, mass–spring lattices, and numerical graph simulations. The results provide a testable, physically grounded bridge between topology, impedance, and emergent coupling in LoE systems.