Weak-Field Residual Gradient Theorem

TA41 derives the weak-field gradient structure generated by the propagated residual field introduced in TA40. Under three operative assumptions (weak linear surface-to-bulk propagation, localized residual source, and radial symmetry) the residual field satisfies the weak-field propagation equation\[∇^2Φᵣₑₛ=α\,ρᵣₑₛ,\]where \(α>0\) is the residual-coupling constant and\[Mᵣₑₛ=∫ ρᵣₑₛ\,dV\]is the total integrated residual source load. Applying the divergence theorem and radial symmetry yields the inverse-square residual gradient\[|∇Φᵣₑₛ(r)|={α\,Mᵣₑₛ}{4π r^2}.\]The theorem therefore shows that if the unreinjected residual phase propagates as a weak linear field in an effective three-dimensional boundary geometry, then its gradient necessarily has inverse-square form. This establishes the first gravity-like scaling structure in the residual architecture while maintaining explicit separation between mathematical structure and physical interpretation. The result is derived entirely from standard weak-field potential theory applied to the residual field object constructed in TA39-TA40. TA41 does not prove physical gravity, derive Newton’s constant, or establish equivalence with general relativity. Rather, it isolates the precise assumptions under which the residual propagation framework reproduces inverse-square weak-field behaviour. Status: solid as a conditional weak-field propagation theorem under the stated assumptions; conditional on weak linear propagation, localization, and radial symmetry assumptions not yet derived from Q5 geometry; speculative for any identification of \(Φᵣₑₛ\) or \(α Mᵣₑₛ\) with physical gravitational fields or source terms.