A Conditional Full Conservative Derivation of the Two-Body 4PN Sector from the Unified 4D Toy Model

We present the full conservative fourth post-Newtonian (4PN) assembly for the 4D toy-model program within the same declared closure hierarchy used at lower orders. The strict Schwarzschild one-body gate, exact quartic Legendre compiler, Hamiltonian-first generic-frame lift, canonical comparable-mass local residual, aligned-seed correction, and referee reconstruction together show that the entire local instantaneous 4PN sector already has an exact generic-frame ordinary representative. We then isolate the hereditary sector and show that its coefficient is not a new independent datum, but is fixed by the same canonically normalized orbital/worldtube STF quadrupole branch that controls the 2.5PN Burke--Thorne channel, through Ctail=(GM/2c3)γquadeff. Hence 4PN opens no new normalization constant: if γquadeff=2G/(5c5), then Ctail=G2M/(5c8) and the full conservative 4PN local+tail structure matches the standard GR target. The remaining gap is therefore the same passive/outgoing quadrupole-normalization theorem on the moving-throat branch already isolated by the 2.5PN program.