Further Consequences of the Minimal Quantum State Space: Higgs Mechanism, Fermion Mass Ratios, Conformal Gravity, and the Division Algebra Bridge

In a companion paper 1, we showed that the eigenmode spectra of 𝑆2 and 𝑆3 — the state space and symmetry group of the minimal quantum system — produce the Standard Model gauge group SU(3) × SU(2) × U(1), 12 gauge bosons, the GUT-scale Weinberg angle sin2 𝜃𝑊 = 3/8, three generations of chiral fermions in the 5 ⊕ 10 ⊕ 1 of SU(5), Yang–Mills dynamics, 4-dimensional Minkowski spacetime, and the PMNS solar mixing angle sin2 𝜃12 = 4/13, all from two inputs with zero adjustable parameters. Here we derive four further consequences of the same two inputs. (I) The Higgs scalar doublet emerges as the space of holomorphic sections 𝐻0(ℂℙ1, 𝒪(1)), with all Standard Model quantum numbers derived and the quartic potential 𝑉 = 𝜆|𝜑|4 (𝜇2 = 0) selected by the no-mass-scale argument; the symmetry breaking pattern SU(2) × U(1) → U(1)em is the Hopf fibration read as a symmetry-breaking pattern. (II) A charge–mass complementarity formula √ 𝑚/𝑀 = 𝑑0 + 𝑑1(1 − |𝑄|), using only eigenmode degeneracies and derived electric charges, produces the first-generation mass hierarchy √𝑚𝑒 ∶ √𝑚𝑢 ∶ √𝑚𝑑 = 1 ∶ 2 ∶ 3, matching experiment within 3%. Combined with the Koide formula (𝑄 = 2/3, identified in 1 as a consequence of ‖𝐴‖GNS = ‖𝐼‖GNS on 𝑀2(ℂ)) and the exceptional Jordan algebra 𝐽3(𝕆ℂ) reached via Adams’ Hopf classification, all eight independent zero-parameter predictions for charged fermion mass ratios match data, seven within 2𝜎 and one (the muon-electron ratio) at the 2% level, with deviations increasing systematically with QCD coupling strength. (III) The Galois conjugation 𝜎 ∶ 𝑖 ↦ −𝑖, which produces the Born rule 1, §1.1, forces 𝜃QCD = 0 via a Nelson–Barr mechanism and predicts sin2 𝜃23 = 1/2 (atmospheric mixing) and sin2 𝜃13 ≈ 0 (reactor mixing) at leading order. (IV) The same action-selection logic that uniquely produces Yang–Mills 1, §5.7.4 applied to the spin connection uniquely selects conformal (Weyl) gravity ∫ 𝐶2√𝑔 𝑑4𝑥, with Einstein–Hilbert rigorously excluded at the fundamental level. The gravitational coupling 𝑓2 is universally asymptotically free, with an exactly derived beta coefficient 𝑏2 = 109/12 from the framework’s derived particle content, adding zero free parameters. Ghost states are algebraically forbidden by the same conjugation symmetry that produces the Born rule (𝑧𝑧 ̄ ≥ 0 → GNS positivity). Einstein gravity emerges below the Planck scale through a strong-coupling confinement transition analogous to QCD. All negative results are reported. The framework reduces the Standard Model’s ∼26 free parameters to ∼5 firmly derived constraints, ∼8 approximately derived quantities, and ∼9 undetermined parameters plus one mass scale.