The Tensor Form

Same method as section 4, applied at the covariant level.

8.1 — The constraints

Constraint 1 (General covariance). The identity must hold in all coordinate systems. Under EH, the manifold has no preferred coordinates. A tensor equation.

Constraint 2 (Symmetric rank-2 tensor). The source is the stress-energy tensor T_μν — the covariant generalisation of ρ. The geometry side must match: symmetric rank-2 tensor built from the metric.

Constraint 3 (Divergence-free source; bridge compatibility). To write a consistent local covariant field identity with matter coupling on the embedded manifold, the source tensor is required to satisfy ∇^μ T_μν = 0. Motivated by Axiom R: because records cannot be created or destroyed locally, the total number of records in any region changes only due to flow across the boundary, and in the continuum limit this conservation law translates to the vanishing of the covariant divergence of the stress-energy tensor. The exact map from record monotonicity to covariant local conservation is a bridge step, not a completed derivation in this paper (KS-I.8).

Constraint 4 (Second-order in the metric). The curvature involves second derivatives of g (from QRA). The geometric tensor must be built from g_μν and its first and second derivatives.

Constraint 5 (Four-dimensional). The manifold has dimension 3+1. N = 3 is derived from the four independent axioms (AP10, The Dimension). KS-I.6 is CLOSED — contingent on AP10’s KS-D.3 (one axiom, one face / dimension-minimal embedding) remaining unfired. With N = 3 derived under that contingency, the dimensional precondition for Lovelock’s 3+1 uniqueness theorem is met.

8.2 — Lovelock’s theorem

Theorem (Lovelock, 1971; uniqueness in 3+1 specifically, Lovelock 1972). In four spacetime dimensions, the only symmetric, divergence-free, rank-2 tensor constructed from the metric tensor and its first and second derivatives is:

a · G_μν + b · g_μν

where G_μν = R_μν − ½Rg_μν is the Einstein tensor, g_μν is the metric tensor, and a, b are constants.

The tensorial analogue of the uniqueness of the Laplacian.

The proof is constructive and independent of physical content — pure differential geometry. Lovelock 1971 derives the result in general N; Lovelock 1972 establishes the special uniqueness of 3+1 from which the form above follows. AP08 invokes both as external mathematical input, tracked by KS-I.2.

8.3 — The identity in tensor form

Constraints 1–5 from the axioms, plus Lovelock’s theorem, force the full identity:

G_μν + Λg_μν = (8πG/c⁴) T_μν (8.1)

G_μν is the Einstein tensor — the unique divergence-free, symmetric, rank-2 curvature tensor in four dimensions.

T_μν is the stress-energy tensor — the covariant record density.

Λ is the cosmological constant. Predicted to exist by Lovelock’s theorem (allowed by the constraints). Its value is not determined by the derivation (KS-I.7).

8πG/c⁴ is the coupling. The G inside is the same G identified in section 5 through The Lock.

Einstein’s field equations with cosmological constant.

Derived from four axioms, symmetry constraints, EH + QRA, and Lovelock’s theorem.

With AP10’s closure of N = 3 (contingent on KS-D.3 remaining unfired), the derivation is unconditional in the dimensional sense — unconditional relative to the dimensional precondition, with the standing contingency on AP10’s KS-D.3 named explicitly throughout.

The cosmological constant is not added by hand. Its existence is predicted by the same uniqueness theorem that gives the Einstein tensor.