Mathematical Foundations of Structured Space Theory (SST) — Supplement 2: General-Relativistic Recovery via Regge Calculus

Closing the missing gravitational field equation for the SST lattice

This supplement shows that the SST gravitational lattice admits a Regge-calculus realization, supplying the missing discrete gravitational field equation and extending the framework across weak-field, strong-field, rotating, and homogeneous cosmological regimes. It positions the gravitational sector of SST as formally complete, at the framework and continuum-limit level, within its Regge realization.

We show that the SST gravitational lattice class admits a Regge-calculus realization. For the working FCC/tetrahedral–octahedral model, the construction is explicit, and SST lattice equilibrium is identified with Regge stationarity. We construct the simplicial decomposition and four-dimensional spacetime extension, derive the discrete Laplacian from the z = 12 cuboctahedral coordination shell, and prove that the linearized Regge equations reduce to the Poisson equation ∇²Φ = 4πGρ. In the weak-field regime, the SST response laws (E7–E8) are shown to parameterize the corresponding class of edge-length variations that extremize the linearized Regge action, establishing variational equivalence between SST equilibrium and the linearized Einstein field equations.

This closes the missing discrete gravitational field equation for the SST lattice (Open Problem 6 of the Unified Pattern Framework), with the matter source term — including angular-momentum current for rotating sources — derived in the continuum limit from the UPF mass and centroid-circulation identifications. The vector Laplacian on the FCC lattice correctly discretizes all metric components, including the off-diagonal terms that encode frame dragging, thereby establishing the weak-field rotating sector. For homogeneous expansion, the Friedmann equations follow by applying the Regge cosmology result to the SST simplicial lattice, with flat spatial curvature (k = 0) fixed by the FCC topology rather than imposed as a boundary condition. In the strong-field regime, the full nonlinear Regge equations converge to the Schwarzschild solution, identifying the exponential form of E7–E8 as the weak-field approximation. Taken together, these results make the gravitational sector of SST formally complete, at the framework and continuum-limit level, within its Regge realization.