A Descartes Companion Identity at Q = 2/3 and a Geometric Light-Fermion Mass Cascade

Three coherent wave sources at 120^◦ separation produce hexagonal interference patterns whose lattice constant reaches a minimum of a = 2λ/3, the rational fraction that also defines the charged-lepton Koide ratio Q = 2/3. Integer-wavelength resonances occur at sinθ = 2/(3N); at N = 3, exactly three wave lengths fit per cell, suggesting a geometric interpretation of the same rational value 2/9 that appears as Shulga’s compact-cycle offset in the charged-lepton spectrum. The geometric-mean interpolation between adjacent resonance scales is modeled on the Dykhne effective-medium theorem for two-phase Z₂-symmetric boundaries. These condensed-matter structures (C₃ phase cancellation, standing-wave resonance, and effective-medium interpolation) motivate a companion identity (the Fidentity) that reduces to the classical Descartes circle theorem at Q = 2/3. Applied to charged-lepton masses with a single input µ_⋆ = m_ℓ = 1883.1 MeV, the construction yields closed-form candidate expressions for three light-quark masses (m_s, m_d, m_u), with internal self-consistency at 0.06% and an up-quark value m_u = 2.22 MeV (+0.23σ from PDG, within current uncertainties). Whether these correspondences between wave interference geometry and the fermion mass spectrum reflect an underlying physical connection or are structural parallels between C₃ × Z₂ systems is an open question that we propose merits investigation.