Emergent Electron Mass from Two-Space Boundary

We formulate a candidate rest--frame closure for the electron branch as the stationary equilibrium of a locked Gaussian ring satisfying \(Rω=c\) in a two--space kinematics, with generator space \(C\) separated from propagation space \(R³\). Two independent closures are evaluated on the same locked shape data. Path~A is the scalar vacuum--in--volume model, RLVM, while Path~B is the tangential vector--tension model, RLTM. The only microscopic length inserted into the closure prefactors is \(P=√{ G/c³}\), and all remaining factors are dimensionless functions of the locked geometry, \(α\), and the assigned scalar or vector completion. For each path the stationary radius is obtained in closed form, and the branch mass is read from \(m•=D/(cR•)=mP C•(α)exp[-π/(8α)]\), with \(mP=√ c/G\) and \(•∈\{ A, B\}\). In the current numerical run the two rest--energy equivalents are \(mAc²=0.5109977017083316\,MeV\) and \(mBc²=0.5109977017184913\,MeV\). Relative to the CODATA electron rest--energy reference, these undershoot by \(-2.44419615096268\,ppm\) and \(-2.444176268804592\,ppm\), respectively, below the \(11\,ppm\) one--sigma uncertainty propagated from the present uncertainty of \(G\). Equating the two closed masses gives the scale--free diagnostic \(α_{}=0.00729735256488327037\), with \((α_{}-α)/α=0.0799290435808\,ppb\). Since the Planck prefactor gives \(m•∝( c/G)1/2\) at fixed dimensionless brackets, the same formulas can be algebraically inverted to define an internal \(G\)-diagnostic from \((m_e,α)\), \({G}geo⁽ᵉ⁾=6.6742673735758264×10⁻¹¹\,{m³\,kg⁻¹\,s⁻²}\). This inversion is a consistency check, not a measurement of \(G\). The construction therefore provides a reproducible rest--frame audit of electron inertia within the stated Relator closure assumptions.