Generative Quantum Gravity: The Metric as an Emergent Signature of Phase Coupling

Abstract Classical general relativity treats the metric as an a priori geometric stage, and quantum gravity theories attempt to "quantize" this stage, yet fall into the conceptual dilemma of "turning a classical tensor field into an operator." Based on the generative mathematics system, this paper proposes a fundamentally different ontology: the metric is not a presupposed background, but the macroscopic signature of the phase coupling of a discrete pixel network under emergent continuity. The core proposition of this paper is: the metric is generated, not to act upon. Starting from the discrete local topological charge density ρ_I(i) of the Axioms Paper, via the intrinsic mechanisms of isoperimetric optimality and topological convergence, an emergent bridge to the continuous information density field ρ_I(x) is rigorously established. The information density field directly determines the potential term V[κ] of the operator flow equation, which in turn emerges as the spacetime metric g_μν. Mass m = (ℏ/c²) · I · ν_c, as the joint signature of the topological charge and the coupling frequency, is essentially the spatial integral of information density at the particle scale. This paper establishes the generative construction of the covariant derivative: the discrete phase difference Δ_κ φ(i,j) = κ_ij · Δφ_ij signs as the emergent derivative ∂_μ under emergent continuity; the antisymmetry of the coupling kernel K(-Δ) = -K(Δ) guarantees the closure of parallel transport, signing as the covariant derivative ∇_μ under emergent continuity; the Riemann curvature R^ρ_σμν is the accumulated phase difference when information propagates along different paths in the pixel network. The metric signature (-+++) is naturally derived from the separation of slow and fast variables in spiral spacetime. In the multi-dimensional generalization, the symmetry and convexity of one-dimensional isoperimetric optimality are naturally inherited, guaranteeing the uniqueness of the steady state. This paper establishes an explicit comparison with linearized general relativity: in the weak-field limit, the generative perturbation equation has exactly the same form as the standard linear GR wave equation, and the two independent polarization modes of gravitational waves correspond to the two independent spiral components of the information density field. The nonlinear corrections of generativism arise from the incompleteness of emergent continuity at finite pixel numbers. This paper derives seven testable predictions from first principles, and provides a complete comparison table of order-of-magnitude estimates, observational windows, and experimental upper limits. Keywords: generative quantum gravity; operator flow; information density; emergent metric; phase coupling; covariant derivative; testable predictions