Two-Energy Theory (TDE) v17.0 — Triangular Lepton–Neutrino Geometry, Beta Lattice Gap, and a Sub-eV Neutrino Window (Addendum to v16.9)

This deposit is a focused v17.0 addendum to the published TDE v16.9 mass program. The hadron/PDG engine remains unchanged: all v16.9 hadron-sector tables, plots, and strict forward-only predictions remain intact. v17.0 extends the same frozen geometric inputs to the lepton–neutrino sector using a self-contained construction built from two dimensionless constants: α⁻¹ = 137.035999084 k = 4π + 1 = 13.566370614 From these, the energy units are fixed as E₁ = α⁻¹/4 = 34.259 MeV, E₀ = kE₁, and E₂ = E₁/k = 2.525 MeV. Key Findings (v17.0): Beta Decay Lattice Gap (30 keV Precision): The v16.9 nucleon vectors imply a half-integer lattice step in the mass map, producing a discrete beta-decay lattice gap M_gap = (1/2)E₂ = 1.2626 MeV. This agrees with the measured neutron–proton mass difference (m_n − m_p = 1.2933 MeV) with an absolute error of ≈ 0.0307 MeV. Triangular Lepton Ladder: Charged leptons are tested against a discrete barrier-suppressed closure ansatz: m_ℓ = (aE₁ + bE₂) / k^g. The best-fit barrier depths align exactly with triangular numbers g = T_n = n(n+1)/2, yielding a compact ladder: Tau (n=0, g=0): Error 3.5×10⁻² MeV Muon (n=1, g=1): Error 5.6×10⁻³ MeV Electron (n=2, g=3): Error 1.4×10⁻⁵ MeV Sub-eV Neutrino Window: Extending the triangular ladder to the next rung (n = 3 ⇒ g = 6) implies a falsifiable sub-eV neutrino mass window. The characteristic scale is E₂/k⁷ ≈ 0.03 eV, with a benchmark sum prediction Σm_ν ≈ 3E₂/k⁷ = 0.0896 eV, consistent with Planck18 bounds. Cosmology Cross-Check: An independent cosmology check follows from the same geometry: (Ω_c/Ω_b)_geo = exp(k/8) = 5.451, closely matching the Planck18 ratio (Ω_c h²)/(Ω_b h²) ≈ 5.357. Conclusion: This result establishes a compact, checkable bridge from the frozen constants of v16.9 to a discrete beta gap, a triangular charged-lepton barrier ladder, and a neutrino scale — all without modifying the core v16.9 hadron engine.