We derive the neutron–proton mass difference mₙ − mₚ 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 λⱼ = 2cos (πj/23), j=1, …, 22, partition into three classes (G, B, R) whose zero-point energies satisfy EB = ER 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 Ecas (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) = βcoh/6 − 1/2 = 0. 490686, giving the main result mₙ − mₚ = (md − mᵤ) × 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) = βcoh/6 connects this result to the cosmological constant (Paper I) and the fourth-generation lepton shell k₄th = 23 (Papers IV–V), revealing a common spectral origin in the Fibonacci self-termination condition d (S⁴, 4) = 55 = F₁₀.
Peacock et al. (Wed,) studied this question.