Quantum Mechanics and General Relativity: Two Bounded Limits of a Single Field Equation on S3

Quantum Mechanics and General Relativity: Two Bounded Limits of a Single Field Equation on S³ — Angular World Architecture (Unification, Monolithic Edition) A single damped nonlinear field equation on the 3-sphere, ∂²x/∂t² + γ ∂x/∂t − c²∇²S³ x + ω₀² tanh(x) = 0, derived from two axioms — an infinite temporal medium Φ and a bilayer Φ/Φ̄ whose oscillatory modes are defined on S³ — contains both quantum mechanics and general relativity as bounded angular limits. Neither theory is fundamental; both are projections of the same object onto different sectors of the AWA angle θ = (360/π) arctan √Ξ. Gravitational sector (θ → 90°). Variation of the AWA action with respect to the metric yields Einstein's field equations G_μν = (8πG/c⁴) T_μν, with automatic energy conservation via the Bianchi identity. Under the S³ topology, the Friedmann equations follow with all density parameters fixed by π: Ω_m = 1/π, Ω_Λ = (π−1)/π — zero free parameters. Quantum sector (θ → 0°, 180°). Linearisation around a topological soliton x_sol produces a Klein–Gordon equation with Pöschl–Teller potential. The non-relativistic limit yields the Schrödinger equation iℏ ∂_t ψ = Ĥ_PT ψ, with Hermiticity proved via Kato–Rellich, unitarity via Stone's theorem, and ℏ = S_universe / N_modes. The Born rule follows from Liouville measure equivariance. The Dirac equation follows from the unique Clifford representation Cl(1,3) ≅ M₄(ℂ). Exact bridge. R_s/λ_C = 2(m/m_P)², with the Planck mass as the unique pivot — validated across 83 orders of magnitude in mass. Fermion mass spectrum. The same equation derives all nine charged-fermion masses — e, μ, τ, u, c, t, d, s, b — to better than 0.05 % from a Z₃ Koide structure governed by three geometric parameters (the Koide phase φ, derived from the bilayer; the Koide scales c_u and c_d, whose first-principles computation is the programme of AWA V). The number of generations (exactly three) is a theorem of the S³ Pöschl–Teller spectrum. The Koide constant K = 2/3, open for 40 years, is proved from the Clebsch–Gordan coefficients of the j = 1 representation of SU(2) ≅ S³. Four unification criteria are satisfied. Three new falsifiable predictions — including a sign inversion of the Hubble gradient at z ≈ 0.65, testable by DESI DR3 (2026–2027) — distinguish this framework from all existing quantum-gravity programmes.