Inverted Hypersphere Cosmology VI: The Neutron–Proton Mass Difference from the Chain Casimir Spectrum

We derive the neutron–proton mass difference m_n − m_p from the Casimir spectrum of the 22-site co-rotating tridiagonal chain intrinsic to the RP⁴ nested-tori architecture of Inverted Hypersphere Cosmology (IHC). The chain eigenvalues λ_j = 2cos(πj/23), j=1,…,22, partition into three classes (G, B, R) whose zero-point energies satisfy E_B = E_R exactly, proven by the bijection j↦(23−j). The d quark (shell k=5, Class B, chain site p=4) couples to the chain with Casimir energy E_cas(p=4) = 0.6467 J, while the u quark (shell k=3, Class R, counter-rotating) sits at the chain boundary where the coupling vanishes exactly. These boundary conditions determine the hadronic matrix element B = cos(π/23) − cos(π/3) = β_coh/6 − 1/2 = 0.490686, giving the main result m_n − m_p = (m_d − m_u) × B = 2.6268 × 0.490686 = 1.2890 MeV, against the observed 1.29333 MeV (error −0.34%). The residual 0.00438 MeV is the O(α) electromagnetic contribution. The quantity cos(π/23) = β_coh/6 connects this result to the cosmological constant (Paper I) and the fourth-generation lepton shell k_4th = 23 (Papers IV–V), revealing a common spectral origin in the Fibonacci self-termination condition d(S⁴,4) = 55 = F₁₀.