The Causal-Set Substrate: Regge Action, Tetrahedral Defect, Scalar Impossibility Theorem

This article is self-contained and serves as the causal-substrate layer of the G■ programme. Starting from the minimal discrete geometric space G■=(Z²,d■) with the cost-asymmetric metric, we equip the integer lattice with a causal (partial) order induced by the asymmetry, turning it into a locally finite causal set in the sense of Bombelli–Lee–Meyer–Sorkin. We prove three results: (T1) the angular defect concentrated at the temporal vertex equals 4·arctan(½)=2·arctan(4/3), an exact closed form; (T2) on the flat bulk the zeroth-order Regge action vanishes, so the substrate is dynamically flat; (T3) the Scalar Impossibility Theorem — no scalar finite-difference Laplacian can reproduce the target eigenvalue list, so a scalar operator is structurally excluded, motivating the matrix operator of the companion article. All claims are proven in full and verified numerically.