Relational Geometric Mechanics XII: Electroweak Vacuum Stability — The Higgs Mass and Top Quark Mass as a Geometric Fixed Point of the RGM Foam

This paper derives the Higgs boson mass (125. 270 GeV) and top quark mass (172. 740 GeV) as the unique electroweak fixed point of the Relational Geometric Mechanics quantum foam. The derivation requires no fitted parameters. The central mechanism is a four-step chain. (1) The SU (3) Casimir invariant CF = 4/3 projected onto the off-axis angular factor sin² (4π/3) = 3/4 at the top quark Wannier site yields exactly unity — the strong force geometrically anchors the top quark and contributes zero net shift to the Yukawa coupling. (2) The residual correction is purely electromagnetic: yₜ = 1 − αEM (mH), where the EM coupling is evaluated at the Higgs mass scale because the Higgs is the foam's restoring pressure against which the top quark is an excitation. (3) The Higgs mass formula mH = vH/2 + mₜop/mW (from λH = 1/8 Wannier ring projection and the W-mediated cross-term) feeds back into the EM coupling evaluation, creating a self-referential loop. (4) This loop has a unique stable fixed point, convergent from any seed in the physical range in three iterations: mH* = 125. 270 GeV (0. 016% from PDG) and mₜop* = 172. 740 GeV (0. 012% from PDG). The paper resolves the Standard Model hierarchy problem within RGM: the Higgs mass is not an independent parameter subject to fine-tuning but the inevitable fixed point of the foam geometry. The complete derivation traces from the single algebraic fact t = √2 (SU (2) j=1 raising operator) through the Koide formula, Higgs VEV, Wannier ring normalization, Weinberg angle, and SU (3) Casimir exhaustion to both electroweak masses. Builds on Papers I–XI and Documents 12–13 of the RGM series.