Universal Flavour Geometry (ERB Framework) Complete Algebraic Closure: Quarks, Bosons, and CKM Mixing

The Emergent Resonant Brane (ERB) framework presents a purely geometric derivation of the fundamental parameters of the Standard Model. This work achieves a complete algebraic closure, reducing the 22 free parameters of the flavour sector—including gauge boson and Higgs masses—to a single geometric seed: the hexagonal vacuum attractor (ξ = 3/2). By modeling the vacuum as a collision of two branes (NB=2), we derive the three fermion generations as stationary SO(3) modes within a spherical cavity. Key findings include: Mass Derivations: The Top-quark mass is identified as a natural Euler-decay (mₜ = Λ/e) of the energy scale Λ. Gauge Sector: $W, Z,$ and $Higgs$ masses are calculated as torsional responses to vacuum strain, matching experimental data with $>99%$ accuracy. Mixing Dynamics: CKM and PMNS matrices are derived as geometric projections, identifying the CP-violating phase as a hexagonal lock-in (60^∘) shifted by Berry curvature. The framework is validated via the ERB Diamond Master Engine (v66), a Python-based implementation that reproduces all 13 primary constants without the use of free fit-parameters.