Surface Propagation of Unreinjected Residual

TA40 introduces the first boundary-field formulation of the residual transport architecture. Building on TA39, the theorem studies the portion of circulated residual phase that fails admissible reinjection into the coherent carrier sector and remains dynamically active in the complement/bulk-access sector. A boundary propagation operator\[ P∂(τ): H_R→ L^2(∂ B)\]is defined to map unreinjected residual states to fields on the barrier/brane-facing boundary surface. This gives the propagated residual field\[Φᵣₑₛ(τ,t)= P∂(τ)(QB\,e^{tEB}\,DB\,ψ_M),\]together with the residual source density\[ρᵣₑₛ(t)=\|QB\,e^{tEB}\,DB\,ψ_M\|^2.\]Under the positivity conditions established in TA39 and a non-degeneracy assumption on the boundary propagation operator, the theorem proves that the propagated boundary field is nonzero:\[ρᵣₑₛ(t)>0Φᵣₑₛ(τ,t)≠0.\]The result establishes that unreinjected residual phase load does not disappear from the transport ecology, but instead propagates as a dynamically active boundary-surface field. No identification with physical gravity is made here. Rather, \(Φᵣₑₛ(τ,t)\) is defined as the field object to be analyzed in the weak-field propagation framework of TA41. Status: solid as a boundary-field construction and propagation theorem under the stated assumptions; conditional on non-degeneracy of \( P∂(τ)\) and explicit computation of the leakage generator \(A_L\); speculative for any interpretation as a physical gravitational field.