Companion note : The mass Spectrum of particles within the q-dRIS Framework

This companion note refines the particle-sector formulation of the q-dRIS framework byreplacing the initial analytic interpretation—based on poles, zeros, and branch exponentsof a transfer function—with a fully geometric and spectral description. While the transferfunction remains a valid analytic projection of the dynamics, it is no longer the fundamentalobject from which particle properties are inferred. Instead, particles arise as geometricmodes of a quaternionic spectral operator ‘Q‘, whose squared eigenvalues ‘Q2‘ determine thephysical mass spectrum.The two quaternionic layers ‘H1‘ and ‘H2‘, together with the geometric angles ‘θ‘ and‘α‘, define a unique geometric invariant ‘K‘ that governs all fermionic and bosonic struc-tures: mass hierarchies, mixing matrices, family structure, and gauge symmetries. Fermionscorrespond to real modes of ‘Q‘, neutrinos to oscillatory modes evolving in an internal timeinduced by the fundamental cycle ‘h0‘, and gauge bosons to null or imaginary modes. TheCKM and PMNS matrices arise from relative rotations of internal trihedra in ‘Im(H)‘, whileSU(3), SU(2), and U(1) emerge from geometric degeneracies and topological residues of the‘H1 ↔ H2‘ projection.This note provides a coherent, predictive, and parameter-minimal formulation of theparticle spectrum within q-dRIS, fully consistent with the geometric–spectral ontology ofthe framework and independent of the analytic transfer-function approach used in earlierversions.