Unreinjected Residual Source Theorem

TA39 isolates the portion of the circulated residual phase that fails admissible reinjection into the coherent carrier sector. Following the forced extraction, mediated circulation, and admissible reinjection blocks derived in TA36-TA38, the theorem defines the complementary residual component that cannot return through the \(Y\)-admissible reinjection channel. The non-reinjection projector\[QB = IH_R - Pᵣₑᵢₙ\]selects the portion of the residual sector lying outside the admissibly returnable subspace, giving the unreinjected residual state\[ψᵤₙᵣₑᵢₙ(t)=QB\,e^{tEB}\,DB\,ψ_M\]and the residual source term\[Gᵣₑₛ(t)=\|QB\,e^{tEB}\,DB\,ψ_M\|^2.\]This quantity is strictly positive whenever the extraction and circulation blocks generate residual load that admissibility cannot return to the carrier sector. The theorem further shows that the unreinjected residual remains dynamically active rather than simply discarded: the conserved mode of \(EB\) preserves a non-decaying component of \(ψᵤₙᵣₑᵢₙ(t)\) for all \(t ≥ 0\). No identification with physical gravity is made. Instead, \(Gᵣₑₛ(t)\) is defined as the candidate residual source object for the surface-propagation analysis developed in TA40-TA42. Status: solid as a residual-sector decomposition and source-term definition; solid for strict positivity under the stated hypotheses; conditional for the explicit form of \(QB\) pending computation of the leakage generator \(A_L\); speculative for any identification with physical gravitational source terms.