Gravity as a Cross-Term Effective Gradient: Source-Position Dependence and Spherical-Symmetry Breaking in an A5-Seeded Finite Medium

This paper studies a restricted precursor to gravitational modelling: where a two-body direction sign can first be carried in a finite, static, symmetric, linear-response, quadratic medium with no primitive direction. The main result is a distance-kernel localization theorem. For a vertex- and distance-transitive medium with equivariant single-source response uₐ and equivariant symmetric Q, BQ (a, b) = depends only on the separation d (a, b). Hence the self-energy is source-position independent, while the nontrivial two-source dependence is confined to a cross-distance kernel hQ (d). Within this restricted grammar and without external witnesses or higher-order/dissipative terms, the cross-distance kernel is the natural carrier of the two-body source-pair dependence; this scope statement is not advanced as an independent uniqueness theorem. The physical reading is made through the source-coupled Green variational form, for which the on-shell interaction has the attractive sign. Here, readout means an admissible finite extraction of effective observables, not an arbitrary external projection. The force channel is the coarse readout of an energy-lowering update tendency on the active substrate, not a passive measurement of a primitive force; the cross-distance kernel supplies the source-pair carrier whose separation gradient feeds it. As a minimal finite demonstration, the icosahedral A5 seed yields a zero-mode-removed Green profile that verifies the attractive ordering. The same pairwise-kernel grammar also gives a finite N-source frustration hierarchy on the icosahedron. The paper does not derive Newton's constant, the inverse-square law, the equivalence principle, the continuum limit, general relativity, or quantum gravity; these are registered as open bridges.