Deriving Key Post-Newtonian Coefficients from the Unified 4D Toy Model

This paper is a derivation supplement to the unified 4D throat model, written to make explicit which coefficients used earlier in the series are fixed by the 4D structure (and which must remain symbolic pending additional closure). We show that a defect uniform in the extra coordinate behaves as a throat with hypercylindrical topology, for which the exterior potential–flow problem yields the classical added-mass coefficient κadd = 1/2 (whereas a 4D bubble would instead give 1/3), providing a sharp topology discriminator. We then show that matching the weak-field refractive coefficient uniquely selects the stiff polytropic exponent n = 5. In the 1PN interaction sector, EIH matching produces a one-parameter family of couplings, which collapses uniquely to α2 = 3/4 and Kvec = 2/π2 once the n = 5 thermodynamic relation is imposed. Treating rest mass as wave-supported energy reproduces the special-relativistic expansion E = γE0, including the universal v4 coefficient 3/8. We also summarize a controlled reduction to brane-effective equations, making leakage/source terms explicit, and we outline the additional response modeling required to determine the remaining undetermined coefficients.