The β_PPN = 1 Condition in Directional Anisotropy Gravity: A Derivation in the Solar-System Regime

In Directional Anisotropy Gravity (DAG) v9, the post-Newtonian parameter γ_PPN = 1 follows algebraically from the quadratic spatial metric g_kk = −Ā_k², but β_PPN = 1 had to be imposed as a separate consistency condition on the temporal coupling B(Ā). This note derives β_PPN = 1 from the DAG field equations within the static, isotropic, weak-field sector, under the effective-massless-potential approximation V_A ≈ 0 and the reduced-sector closure (both justified at solar-system scales r ≲ 100 AU). The derivation has three steps: (1) the trace-reversed stress tensor of any static massless scalar vanishes identically in four spacetime dimensions (an algebraic lemma); (2) this forces the (0,0) Ricci equation to read R₀₀ = 0, and 2PN expansion then gives β_PPN = 1 with corrections O(U_⊙) ≲ 10⁻⁶; (3) the same constraint solved in closed form yields the temporal coupling B(Ā_k) = (1 − ln Ā_k)², whose Taylor expansion reproduces the v9 condition to second order. The residual correction to β from the non-zero potential V_A is bounded by |δβ| ≲ 10⁻¹³ at solar-system scales, well below the Lunar Laser Ranging bound |β_PPN − 1| < 8 × 10⁻⁵. This document is a companion technical note to the DAG gravity-sector paper (Zenodo concept DOI: 10.5281/zenodo.19389365).