Spectral Structural Numbers from the β–Yukawa Effective Lagrangian

Description OverviewThis technical manuscript provides a formal effective-Lagrangian interpretation of the structural numbers used in the beta-organized mass program. It addresses the open question of whether the observed fermion spectrum can be organized by a structural branch variable rather than treated as a set of unrelated constants. Core FrameworkThe framework keeps the visible Higgs relation intact, meaning the Higgs vacuum expectation value still converts Yukawa couplings into masses. The additional structure is placed in the beta-dependence of the effective Yukawa operators. The structural number is defined as a logarithmic charge, representing the local branch sensitivity of the Yukawa operator. Key Scientific Contributions- The structural number of a fermion state is defined as the logarithmic beta-sensitivity of the corresponding Yukawa operator.- A spectral refinement identifies the leading integer part of the structural number with the nodal label of an internal branch eigenfunction.- This interpretation provides a route to discreteness in the fermion spectrum without requiring the scalar Hessian alone to determine all masses. - Level connections, such as flavor mixing and transitions, are represented as mixing or exchange terms in the effective Lagrangian. - The manuscript demonstrates how the structural field can modulate Higgs-fermion couplings through Yukawa form factors. Benchmark DataThe work includes a representative table of first-pass structural numbers for quarks, charged leptons, and neutrinos. These values record the observed target pattern for the effective branch charges based on benchmark mass data. StatusThis construction establishes an effective chain linking the visible action to discrete nodal labels. [cite: 198] It serves as an intermediate closure that is more precise than a numerical parametrization but leaves the exact microscopic spectral operator as part of an open completion.