Emergent Newtonian Interaction and Directional Discrete Corrections in 26-Valent Chebyshev Networks

Abstract

This dataset and accompanying manuscript present a self-contained investigation of emergent gravity phenomena in a three-dimensional toroidal lattice with Chebyshev (ℓ∞) connectivity and valence 26. The work documents both rigorous mathematical results and numerical findings obtained between project inception and August 2026. Key results include: • An exact vacuum curvature invariant ℐ∞ = 47/512 derived from Forman-Ricci curvature, with asymptotic expansion ℐ(λ) = 47/512 − 141/(1024λ) + 201/(2048λ²) + O(λ⁻³) • Emergence of Newtonian inverse-square interaction between point defects, with the fitted coefficient converging to the continuum value −1/(36πr) to within 0.35% after background subtraction • Directional short-range corrections (β_axial ≈ 0.307, β_face ≈ 0.776, β_body ≈ 1.033) identified as ultraviolet lattice artifacts that vanish under renormalization group flow • Verification of microscopic stiffness A₃D as an emergent observable, with two independent estimates agreeing within 0.35% • A structural relation G = 1/(36πJA₃D) connecting the emergent gravitational constant to network parameters