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

We present a controlled derivation of the full conservative two-body first post-Newtonian (1PN) sector from the four-spatial-dimensional brane–bulk toy model. Starting from the exact bulk action, projection map, open-system balance laws, and quasi-static near-zone Poisson hook on the brane, we use a coherent-defect / small-body worldtube reduction to obtain the Newtonian point-particle limit and then assemble the conservative 1PN ledger sector by sector. Newtonian matching fixes the scalar mass-dressing coefficient κρ = 1 (historically written q = 1 in earlier gravity-side bookkeeping), weak-field optical consistency fixes the polytropic exponent n = 5, and wave-supported rest energy yields the universal free-particle v4 coefficient. Local mass scaling plus pair counting generate the static G2/r2 term, exterior potential-flow topology fixes the added-mass coefficient κadd = 1/2 for a w-uniform throat, and a constructive isotropic Fourier-space wake basis reproduces the Einstein–Infeld–Hoffmann coefficients C∥ = −7/2 and CL = −1/2 with α2 = 3/4, aH = 0, and Kvec = 2/π2. Under an explicit adiabatic one-degree-of-freedom breathing closure we further obtain κPV = 3/2, so that β1PN = κρ + κadd + κPV = 3. The assembled reduced two-body Lagrangian is then algebraically identical to the standard conservative EIH form, and the test-mass limit reproduces the standard perihelion shift. The result is a full conservative 1PN derivation within a declared closure hierarchy that makes explicit which ingredients are fixed, protocol-fixed, and still symbolic, with remaining freedom confined to generalized internal response, finite-size structure, and beyond-1PN open-system effects.