The ER → MOND Derivation Chain: Quadratic Fixed Point, Local Virial Closure, and Radial Relaxation

This companion document provides a step-by-step validation of the derivation chain from Entangled Relativity, based on the \(L_m^2/R\) action, to the MOND simple interpolation function. The result is organized in two protected tiers. Tier 1 is an exact theorem within the quadratic-relay class: under Newtonian synchronisation, absence of autonomous offset, local stability and BTFR scaling, the unique closure is \(P(\nu)=\nu(\nu-1)\), whose stable fixed point is the MOND simple function. Tier 2 shows how the weak-field ER relay equation reduces to this quadratic class through the scalar force identification, Minazzoli decoupling, weak-field analyticity and controlled truncation. The document also gives the geometric origin of the Milgrom acceleration scale, \[ a_0 = m_0 c/3, \] with no adjustable parameter, and includes a local ER/KK virial closure yielding \(A_1=-1/2\). This coefficient fixes the sign and normalization of the relaxation channel, while leaving the MOND fixed point independent of \(A_1\). A final appendix derives the local elliptic-to-radial relaxation reduction and clarifies that the residual source is not inserted phenomenologically: it is the dynamical form of the Tier-1 quadratic closure. The relaxation length \(\tau_r\) is therefore a local spatial relaxation scale, not a new MOND parameter. The companion is intended as a referee-facing technical guide: it separates exact results, controlled weak-field reductions, local virial closures and remaining open issues such as full galaxy-by-galaxy elliptic solutions, non-spherical geometries and non-equilibrium transients.