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∇²ₒ℃ 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: Ωₘ = 1/π, Ω_Λ = (π−1) /π — zero free parameters. Quantum sector (θ → 0°, 180°). Linearisation around a topological soliton xₛol produces a Klein–Gordon equation with Pöschl–Teller potential. The non-relativistic limit yields the Schrödinger equation iℏ ∂ₜ ψ = ĤPT ψ, with Hermiticity proved via Kato–Rellich, unitarity via Stone's theorem, and ℏ = Sᵤniverse / Nₘodes. The Born rule follows from Liouville measure equivariance. The Dirac equation follows from the unique Clifford representation Cl (1, 3) ≅ M₄ (ℂ). Exact bridge. Rₛ/λC = 2 (m/mP) 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 cᵤ and cd, 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. Keywords: S³ topology, Poincaré conjecture, bilayer field equation, Pöschl–Teller, Dirac equation, Clifford algebra, Born rule, Koide formula, three fermion generations, Hubble tension, emergent gravity, quantum-gravity unification, Angular World Architecture, AWA, falsifiable predictions.