Newtonian-Type Limit of Residual Admissibility Fields

This theorem closes two obligations left open by the TA24–TA27 arc. First, it derives the prepared-projector perturbation bound \( ‖δΠ_Y(M_k)‖ = O(εn) \), where \( ε = 1/320 \) is the primitive transport scale (TA13), by showing that the prepared gate projector is obtained from the static projector by a phase rotation \( U_k = e^{εnA_k} \) in the low-mode regime. Expanding to first order gives \( δΠ_Y(M_k) = εn[A_k, Π_Y] + O(ε²n²) \), with uniform boundedness over \( k ∈ ℤ_32 \) following from the finiteness of the phase-slot lattice. This resolves the open derivation obligation of T137 (seam lemma), upgrading the \( S_res ~ n² r\) eduction from a physically motivated assumption to a controlled perturbative result. Second, under three assumptions (weak deformation, linear (additive) response, and effective three-dimensional conserved propagation) the accumulated residual source density \( C = Σ η_i n_i² \) sources an admissibility-deformation field Φ satisfying a Newtonian-type field equation\( ∇²Φ = αC \). For a localized source, the divergence theorem and radial symmetry give the inverse-square gradient falloff \( |∇Φ(r)| = αC_total / (4πr²) \), reproducing the formal structure of a Newtonian-type field law. Corollaries establish: effective source-strength identification (M_eff ∝ C, as a structural analogue only); linear superposition of sources; and a strong-field deviation criterion identifying the regime where Newtonian behaviour fails. The coupling constant α is not identified with Newton's constant G. Relativistic corrections and strong-field behaviour are not addressed. The structural parallel to Newtonian gravity is noted; physical identification remains open.