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

In Directional Anisotropy Gravity (DAG) v9, the post-Newtonian parameter γPPN = 1 follows algebraically from the quadratic spatial metric gₖk = −Āₖ², but βPPN = 1 had to be imposed as a separate consistency condition on the temporal coupling B (Ā). This note derives βPPN = 1 from the DAG field equations within the static, isotropic, weak-field sector, under the effective-massless-potential approximation VA ≈ 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 βPPN = 1 with corrections O (U_⊙) ≲ 10⁻⁶; (3) the same constraint solved in closed form yields the temporal coupling B (Āₖ) = (1 − ln Āₖ) ², whose Taylor expansion reproduces the v9 condition to second order. The residual correction to β from the non-zero potential VA is bounded by |δβ| ≲ 10⁻¹³ at solar-system scales, well below the Lunar Laser Ranging bound |βPPN − 1| < 8 × 10⁻⁵. This document is a companion technical note to the DAG gravity-sector paper (Zenodo concept DOI: 10. 5281/zenodo. 19389365).