The Topological Chain Hypothesis: Topological Hidden Variables from Dimensional Emergence

We propose a framework in which physical reality is fundamentally a two-dimensional topological surface M², three-dimensional space emerges as a stable projection, and quantum correlations—including Bell-inequality violations—arise from topological hidden variables whose structure is explicitly specified. The hidden variable is the topological triple λ = (n, g, φ_T): winding number, genus, and topological phase. This yields a specific correction to quantum predictions: C_TCH(a,b) = −cos θ + ε·f(n,g,θ), where f is a harmonic function numerically determined from Fibonacci anyon simulations and ε ~ 10⁻³⁵. The dimensional hierarchy 0D → 1D → 2D → 3D_projected is derived from three premises—existence, finitude, and distinguishability—through a universal transition principle. Numerical evidence from companion papers demonstrates that topological sectors carry statistically significant mutual information with CHSH values, that Bell violation is generic in Fibonacci anyon braiding (64.6%–91.8% of sequences at L=9), and that a single topological generator controls entanglement as an exact on/off switch. We present eight falsifiable predictions, including autocorrelation in Bell-test detection order (ACF(1) ∈ [0.24, 0.50]) and spectral fingerprints that distinguish topological orders. TCH specifies the hidden-variable structure explicitly, yielding falsifiable predictions through the harmonic pattern of f(n,g,θ). v1.3: Separability/free-choice clarification with Vieira et al. 2025 comparison; ACF qualifier and range updated to [0.24, 0.59]; SIM-01/02 numerical verification footnotes added.