The Mixing Matrix Architecture: A Cascade Geometric Derivation of the CKM and PMNS Parameters — With the First Geometric Origin of the Quark-Lepton Complementarity Relation

The CKM and PMNS mixing matrices encode the flavour mixing of quarks and neutrinos and have been measured to high precision, but no derivation from first principles exists. This paper derives five of the six Standard Model mixing angles from the two Feigenbaum constants α = 2. 502907875 and δ = 4. 669201609 using the Universal Cascade Theorem. The PMNS (neutrino) sector operates in the T-regime of the cascade (T = δ/α): tan²θ₂₃ = α²/δ → sin²θ₂₃ = 0. 57296 NuFit 5. 3: 0. 572, error 0. 167%; tan²θ₁₂ = T·sin²θW = √δ/ (2α) → sin²θ₁₂ = 0. 30151 0. 491%; tan²θ₁₃ = 1/ (2δ²) 1. 81%. The CKM (quark) sector operates in the δ-regime: sinθ₁₂ = 1/δ = 0. 21417 4. 61% at low energy, GUT-scale candidate. The Quark-Lepton Complementarity relation — that the Cabibbo angle and the solar neutrino angle sum to approximately 45° — has been observed for thirty years without derivation. This paper presents the first geometric derivation: arcsin (1/δ) + arctan (√ (T·sin²θW) ) = 45. 67° 1. 49%. The profound asymmetry between CKM (hierarchical, suppressed) and PMNS (near-maximal, large angles) is a structural prediction of the cascade: no scalar survives a cascade level unchanged. Paper 48 of the Resonance Theory series.