Within the Harvey space framework—a five-dimensional complex-time manifold with coordinates (t, τ, x, y, z) whose imaginary-time coordinate τ is topologically compactified onto a circle S¹ of radius r_τ = ℏ / E—we give an emergent derivation path for General Relativity (GR). The central identification is to equate the metric lapse with a physical τ-evolution-rate field V_τ (x): ω (x) / ω₀ = V_τ / c = √gₜt On this basis, three central results follow: 1. Geometrisation of Gravity and Acceleration The gravitational acceleration is rewritten exactly as the logarithmic gradient of the evolution field: a = c² ∇ ln (V_τ / c) (which reduces to the weak-field form a = −c ∇V_τ). This unifies time dilation, gravitational redshift, and free fall as intrinsic properties of a single scalar field; a symbolic check confirms it equals the proper acceleration of a static observer in the Schwarzschild field term by term. 2. Reconstruction of the Einstein Tensor and Parameter Rigidity Field Equations: In the continuum limit of the node sea, a graph-Laplacian update law reconstructs the time component of the Einstein tensor under linearisation, while the non-linear closure is uniquely fixed by the Lovelock theorem. Coupling Constant Deconstruction: The dimensionless factor in the Einstein coupling 8πG / c⁴ is decomposed geometrically as 8π = 2 × Ω₂, where Ω₂ = 4π is the area of the unit 2-sphere—locked to the three-dimensional space ℝ³ of Harvey space—and the factor 2 arises from the trace-reversal structure of a rank-2 (spin-2) source. Rigidity: While the absolute value of G is not derivable on dimensional grounds, the gravitational sector carries no free dimensionless parameters beyond this geometric 8π (parameter rigidity). 3. Entanglement Locking and Falsifiable Cosmic Evolution Taking the speed of light c as strictly invariant and the node-hopping time as dictated by the vacuum entanglement density S, the Planck-length identity lP² = ℏG / c³ forces the product ℏG to be entanglement-locked: ℏG ∝ 1 / S². This yields a distinct, falsifiable prediction: if fundamental "constants" evolve with cosmic time, the fine-structure constant α and Newton's gravitational constant G must be anti-correlated. Confronted with empirical data from quasar absorption lines, the Oklo natural nuclear reactor, atomic clocks, Big Bang Nucleosynthesis (BBN), and lunar laser ranging, the allowed present-day variation is extremely constrained (a monotonic power law gives |n| ≲ 10⁻⁷). The framework predicts the constants to be nearly stable at current precision; the only realistic discriminating window lies in next-generation optical-clock searches for an α drift. The framework reproduces—rather than modifies—GR; its major increments are conceptual unification, a purely geometric origin for the coupling constant, and a rigid constraint on the co-evolution of fundamental constants.
Harvey Sang (Mon,) studied this question.