Conditional Reduction to General Relativity from Zero-Sum Constraints: Lovelock Uniqueness and the Group-Theoretic Origin of Dimension

This work establishes a conditional reduction chain from a discrete ontological substrate to general relativity. The method does not assemble the Einstein field equations piece by piece; instead, it compresses the field equations into three structural properties—locality $(L)$, second-order $(O)$, and conserved source $(C')$—and then invokes the Lovelock uniqueness theorem to force them into Einstein form. The algebraic skeleton: zero-sum constraints fix codimension 1; the incidence matrix of a connected graph spans the zero-sum hyperplane and gives integer reachability; discrete Dirichlet energy is positive semidefinite with kernel the constants; continuum approximation combined with metric-volume consistency recovers the spatial metric; reversible/irreversible splitting directly gives signature $(1,m-1)$. The metric construction: closed-walk counting gives a parameter-free edge weight $w_{ij}\propto\phi_i\phi_jA_{ij}$; finite truncation $k=n$ restores locality. The dimension account is the weakest link: the core postulates do not constrain dimension (proposition: for every $m\ge2$ there exists a model satisfying the postulates); $D=4$ follows from two independent routes—physically, "propagating gravitons exist" requires the unique positive solution of $\frac{D(D-3)}2=2$, and geometrically, the closure–motion correspondence principle $\dim\ker B=\dim\operatorname{im}B$ combined with edge-transitivity uniquely selects $K_4$. The premises of both routes are not consequences of the core postulates; dimension is therefore a conditional input. **Keywords**: Lovelock theorem; general relativity; zero-sum constraint; discrete geometry; Hodge duality; dimension problem ---