First-Principles Proofs of Four IHC Theorems: T3, T4, T5, and T8

This paper provides complete first-principles proofs of four theorems in Inverted Hypersphere Cosmology (IHC), expanding on derivations established in the Prequel and companion papers.

T8: The N=33 shell count follows from a vacuum self-consistency condition, not from dynamical stability. On RP4 with a φ-scaled vacuum, a harmonic degree l is self-consistent if and only if its multiplicity d(S4,l) is a Fibonacci number. Proved from the Binet formula and the Hurwitz theorem alone. No KAM theory is required or relevant.

T4: The cohesion field Ψ must satisfy Ψ(−x) = −Ψ(x) on RP4. This follows because RP4 is non-orientable and the vacuum has L_net = −1/2, which forces the field into the twisted scalar sector of the orientation bundle. Conformal coupling ξ = 1/6 follows from the conformal flatness of S4.

T3: The UV–IR seesaw ρ_Λ = √(ρ_UV|ρ_IR|) is independent of regularisation scheme because Z^reg(−1) = −631/30 is an exact rational number fixed by the analytic continuation of a spectral zeta function. Analytic continuation is unique; there is no scheme to choose.

T5: θ_QCD = 0 exactly on RP4. The antipodal map reverses the orientation of S4, which forces ∫Tr(F∧F) = 0 for every equivariant gauge bundle. There is only one topological sector. The strong-CP problem is not resolved — it is dissolved. No Atiyah–Patodi–Singer theory is needed.