Topological String-Gradient Theory: Metric Replication of Space-Time and Cosmological Evolution

We present the Topological String-Gradient Theory (TSGT), a framework in which space-time is discrete at the Planck scale and gravity is the gradient of the spacing betweendiscrete elements (“pixels”) rather than a fundamental force. Two kinematic postulates—a spatial pixel strain set by the Newtonian potential and an equal and opposite tempo-ral strain—reproduce the weak-field Schwarzschild metric, and hence Newtonian accelera-tion, gravitational redshift, the Shapiro delay and the full general-relativistic light deflec-tion ∆θ = 4GM/c2b. We stress that this reproduction is by construction rather than aprediction. The discrete substrate is taken to be a random Poisson sprinkling of densityL−4p rather than a regular lattice. This choice preserves Lorentz invariance statistically,and thereby removes the preferred-frame and anisotropy problems that afflict lattice for-mulations; it also withdraws the energy-dependent photon dispersion, gravitational-wavedispersion and propagation-anisotropy signatures claimed in earlier versions of this frame-work, which we retract explicitly. In exchange, TSGT acquires a quantitative cosmologicalconsequence. If cosmic expansion proceeds by the creation of new pixels rather than bythe stretching of existing ones—the only option compatible with an invariant Lp—then thenumber–volume correspondence carries Poisson fluctuations δN ∼ √N , and the cosmologicalterm conjugate to the four-volume inherits a residual magnitude Λ ∼ V −1/2. Numericallythis gives ρΛ ≃ 2.8 × 10−27 kg m−3 against an observed 5.8 × 10−27 kg m−3: agreement withina factor of two, compared with the 10123 discrepancy of the naive Planck-density estimate.This argument is not original to TSGT—it is Sorkin’s heuristic, developed within causal settheory—but it follows from TSGT’s own postulates without additional assumptions, andit carries a signature that current surveys can test: the dark energy term is not constantbut fluctuates, so w̸ = −1. We then show that the two kinematic postulates need not bepostulated at all. Written locally, the number–volume correspondence reads √−g = L4pn(x),which is the unimodular constraint; enforcing it in an Einstein–Hilbert action yields thetrace-free Einstein equations, with Λ appearing as an integration constant rather than as aparameter, and with the weak-field metric recovered rather than assumed. The price is statedplainly: TSGT thereby becomes provably equivalent to General Relativity at the classicallevel, so whatever independent content it possesses lies entirely in the integer character ofthe pixel count. Unresolved problems, including the absence of a dynamical equation for thepixel density and the difficulty of deriving excitations on an irregular substrate, are listedexplicitly.