Mathematical Foundations of Structured Space Theory (SST) — Supplement 3: Closure of the UPF Problem Set and Geometric Extensions from FCC Lattice Geometry

This supplement derives six zero-parameter closures for the UPF Open Problems plus geometric extensions to the charged-lepton sector, quark sector, CKM mixing, gauge-coupling unification, Higgs mass, and proton stability, all within the present FCC/SpS worked realization.

First, the proton–neutron mass splitting is shown to equal Δm = m_n(R−1)/[z·ln(d²_NNN/d²_NN)], predicting 1.2934 MeV versus 1.2933 MeV observed [Navas et al. 2024] (0.004%).

Second, the neutrino mass-squared splitting ratio is derived from a shared-node constraint mechanism, giving Δm²₃₂/Δm²₂₁ = z(z−1)/|F| = 33, matching the observed 32.6 ± 0.9 [Navas et al. 2024] (0.5σ). Three neutrino flavors emerge from z/|F| = 3 octahedral axes. The absolute mass m₃ = (z+1)/(|F|z·12⁹) = 49.25 meV (0.5%), the PMNS mixing angles sin²θ₁₂ = 4/13 (0.2%), sin²θ₂₃ = 7/13 (1.4%), sin²θ₁₃ = 1/44 (3.3%), and the CP phase δ = −π/2 (maximal, from the C₄ axis chirality) are all derived.

Third, the absolute proton mass is derived as m_p = 24·m_Pl/12¹⁹ × R², where the correction factor R² = (K_h/K_n)² arises from gravitational self-coupling. The prediction is 938.281 MeV versus 938.272 MeV observed [Navas et al. 2024] (0.001%). With this result, the isotope bridge equation becomes fully zero-parameter.

Fourth, the emission probability for a p6 pattern factorizes as P = /24, with  = 0 for the ground state (proton stable) and = 1/4 for an excited state (τ ≈ 10⁻²¹ s, nuclear γ timescale).

Fifth, the neutron drip line is , where S is the FCC surface void count, s = 2 for stable cores and s = 3 for unstable rank-0 cores. This reproduces the experimentally confirmed neutron drip lines tested here (Z ≤ 10; data from [Thoennessen 2016, Ahn et al. 2022]) within ±2 neutrons.

Sixth, the Weinberg angle is derived as sin²  = (z/|F|)/(z+1) = 3/13 = 0.2308, matching the observed 0.2312 [Navas et al. 2024] to 0.19%. The electromagnetic coupling normalization α⁻¹(0) = z² − (|F| + z/|F|) = 137 (0.026%) and its running to  (0.08%) are derived from the two-step bond-response space and its excluded face-state and screening sectors. All three gauge couplings unify at the Z mass with universal denominator z²−|F|² = 128:  = 1/128 (0.08%),  = (|F|²−1)/128 = 15/128 (0.1%), and the effective weak coupling  =  /sin²θ_W = 13/384 (0.2%), where the bare weak numerator  = |F| = 4 acquires the Weinberg mixing factor (z+1)/(z/|F|) = 13/3. The W-boson mass  GeV (0.7%), with √3 from the octahedral Laplacian eigenvalue λ = 4 multiplicity 3, and the Z-boson mass  = 91.0 GeV (0.2%) follow directly.

Beyond the original UPF §20 problem set, the charged-lepton sector is also derived: the electron, muon, and tau masses follow from three stable delocalization topologies on the FCC lattice, with the universal factor |F|² + 1 = 17 (the vertex response algebra dimension) connecting all three generations. The proton-to-electron mass ratio  matches observation to 0.008%, and the Koide ratio is satisfied to 0.01% with the geometric value 2/3 = 1 − |F|/z as the natural lattice interpretation. Fractional quark charges are derived from octahedral winding ( ), all six quark masses follow from a charge-dependent coupling rule (0.6–1.3%), and the full CKM matrix is determined by four lattice parameters (λ = 1/√20, A = 5/6, δ = arctan(5/2), r = √(3/20)), unifying with the PMNS matrix as external vs internal views of the same octahedral geometry. Proton stability is proved (δ_eff = 0 by construction), and the Higgs mass  GeV (0.24%) completes the electroweak sector.