A Controlled Derivation of the Full Conservative Second Post-Newtonian Sector from the Unified 4D Toy Model

We present a controlled derivation of the full conservative two-body second post-Newtonian (2PN) sector from the unified 4D toy model. Carrying forward the exact projection / Poisson-hook / worldtube reduction logic and the already-fixed Newtonian+1PN ledger, we close the missing one-body 2PN response slot by a minimal DtN-invariant denominator that preserves the frozen 1PN coefficient and reproduces the isotropic Schwarzschild test-mass target through 2PN. Minimal Bernoulli exactification and quadratic local mass scaling then supply the self/static 2PN seed, with the pure-static isotropic gate fixing λρ = 1/2. The remaining comparable-mass mismatch against the generic-frame ADM target collapses to a finite algebraic residual with a unique solution for the added cross-sector coefficients qi, ti, s1. After assembly, the full reduced two-body Lagrangian Legendre-transforms exactly to the standard generic-frame conservative ADM Hamiltonian through 2PN. The paper therefore gives a full conservative 2PN derivation within a declared closure hierarchy, while the appendices preserve the broader throat-response constructions and the remaining dynamic DtN/PDE observables.