Variational Backbone and Regime Closures X: General Relativity as a Sector in R3 Cone invariance, metric regime, and the geodesic instance

We formulate general relativity as a sectorial closure of the metric channel under the R3reading, rather than as an additional structural axiom. The characteristic cone (and limitingspeed c∗) is treated as a pre-instance structural invariant under admissible eliminations, andthe primary time instance is fixed by signal-cone synchronization. On a space–time domainU ⊂ M × R (or U ⊂ Rd × R in a chart), we define a metric regime sector GεGR (U) by (i)Lorentzian nondegeneracy, (ii) cone–metric compatibility (the inherited characteristic cone isthe null cone of g, up to declared conformal/rail choices), and (iii) smallness of an Einsteintype residual measured by ΞGR := |RE|g/ΛGR with REµν := Gµν(g) − κGRTµν. Within aregime-valid region (g, T) ∈ GεGR (U), Einstein-type closure holds with controlled remainder,and energy–momentum balance is controlled in residual form by the contracted Bianchiidentity. Geodesic motion is formulated as an explicit representation protocol, licensed onlywhile regime validity persists (a stopping rule); under a weak-field/slow-variation gate, theNewtonian narrative is recovered as a licensed sub-instance with an explicit remainder bound.