Systematic next-to-leading-order expansion of the conformal scalar response equation and numerical control benchmarks

We present a systematic next-to-leading-order (NLO) expansion of the conformal scalar field equation within the Vacuum Field Theory (VFT) framework. Motivated by empirical tensions in the leading-order (LO) monopolar model, we derive the exact non-linear perturbation equation from the covariant action and isolate the quadratic NLO source functional. We demonstrate analytically that this functional cleanly separates into a renormalisable monopolar sector and an irreducible non-monopolar (ℓ≥2) channel. This non-monopolar remainder provides the rigorous theoretical origin for the previously identified morphology-dependent response (Q-channel), replacing ad-hoc operational proxies. We conduct numerical control benchmarks in the canonical-kinetic limit across a range of baryonic source geometries. The benchmarks establish that the NLO correction is perturbatively controlled and exhibits a strict morphology-coupling factorisation: Spherical Control: For perfectly spherical sources, the non-monopolar NLO correction vanishes identically. Aspherical Activation: For axisymmetric toy models and regularised bulge+disk geometries, the irreducible quadrupolar fraction is monotonically activated by baryonic disk flattening and disk dominance. Coupling Modulation: The absolute amplitude of the activated channel is scaled by the coupling-gradient coefficient (α1). Solar-System Viability: A local benchmark confirms that the small solar oblateness (J2) generates a negligible NLO acceleration (∼10−6gN at 1 AU), avoiding immediate conflict with precision-gravity constraints like Cassini. These results establish the existence, local viability, and astrophysical robustness of the morphology-activated NLO scalar response. They provide a fully covariant, structured foundation for upcoming quantitative confrontations with galactic rotation curves.