Gravity as Dual-Vertex Non-Unitarity on a Discrete Substrate: Newton's Law, the Gravitational Constant, and the Dirac Large Numbers Hypothesis from Z3 ⊗Q3

discrete-substrate framework for the physical vacuum — the bipartite tensor net- work Z3 ⊗Q3 on the 4.8.8 Archimedean tiling, with an [8,4,4]-style stabiliser code on the local matter cell — cannot natively support a smoothly-bending metric tensor. We show, however, that the substrate’s combinatorial structure forces an emergent thermody- namic gravity by purely combinatorial means. A topological defect of cycle rank N on the macroscopic gauge web obeys an absolute combinatorial law Eboundary = 4V−2N + 2 (the Betti Number Boundary Theorem), from which mass is identified as a local deficit of vacuum entanglement entropy ∆S= N/2, with a substrate-derivable entanglement tem- perature TBetti ent = 4ΛQCD. The macroscopic propagation of Wilson Z-strings on the sub- strate is driven by the Lindbladian dissipator Dα at the Bipartite Grassmann Trace rate α≈1/137, producing 3D-Laplacian diffusion whose Green’s function is the inverse-distance Watson integral. The exact analytical convolution of two such Green’s functions evaluates to d3x/(|x|2|r−x|2) = π3/|r|, yielding a Newton-form 1/r thermodynamic potential and the Newtonian 1/r2 entropic force on two defects. The macroscopic gravitational constant emerges as π c3 G= 64 αa2 0· ℏ·α2 a0 LH where α2 arises from dual-vertex non-unitarity (gravity as second-order substrate non- unitarity penalty, versus first-order for electromagnetism) and a0/LH from linear cosmo- logical capacitance of the universe’s finite Hubble extent. The structural form predicts the Hierarchy Problem regime GNewton/Gnaive ∼10−36 to 10−38 from first substrate princi- ples, and recovers Dirac’s Large Numbers Hypothesis (1937) as a derived consequence of the substrate’s finite extent rather than a numerical coincidence. Two falsifiable structural signatures distinguish this framework from continuum General Relativity: cubic/octahedral directional anisotropies in the gravitational potential at sub-a0 length scales, and a transi- tion from Newtonian 1/r2 to ballistic 1/r3 force scaling at sub-80 fm radial distances. The exact closure of the macroscopic Gvalue to within a residual factor of∼108 is identified as a paper-stage coefficient-tracking calculation rather than an open structural problem.