A Full Conservative Derivation of the Two-Body 3PN Sector from the Unified 4D Toy Model

We present the full conservative two-body 3PN assembly for the 4D toy-model program within the same declared closure hierarchy used at 1PN and 2PN. Starting from the exact 4D reduction backbone, the frozen Newtonian/1PN/2PN ledger, and the 2.5PN narrowing to the grouped real P20 ⊕ P21 ⊕ P22 quadrupole bundle, we close the complete conservative near-zone ledger through order c−6 in the fixed ADM chart. The strict Schwarzschild one-body gate fixes a minimal three-parameter repair ledger (μρ,3, d3, s24) = (1/4, −45/4, −1/16). The comparable-mass sector then splits exactly into three lanes: the ordinary-Lagrangian counterimage of the universal free two-body relativistic Hamiltonian, an exact grouped real P2 middle-block closure, and a unique geometry-side static completion. Equivalently, ΔL̂3GR = Δl1v8 + LP2mid + Δl15(g)U4, with ΔH3 = −ΔL3 in the fixed ADM chart. The apparent pure-kinetic remainder collapses to the compiler image of the free 3PN Hamiltonian, while the remaining static slot is uniquely geometry-side after sigma-collapse. Thus the conservative 3PN bottleneck is no longer algebraic: within the declared hierarchy, the two-body 3PN ledger is fully assigned, and the live unresolved problem is the deeper moving-throat derivation of the grouped-P2 and geometry constitutive lanes together with the already-narrow outgoing quadrupole-normalization gap inherited from the 2.5PN program.