Why the Higgs Field Must Exist: Electroweak Symmetry Breaking as an Algebraic Necessity from x³ = x² + 1

The Standard Model accepts electroweak symmetry breaking as an empirical input — the Higgs vacuum expectation value v = 246.22 GeV is nonzero, but no first-principles explanation exists for why. This paper proposes that the answer lies in the algebraic structure of the cubic polynomial x³ = x² + 1. The polynomial’s unique real root ψ_s = 1.4656… is the super golden ratio. Its complex conjugate roots ω, ω̄ satisfy |ω|² = 1/ψ_s exactly (a Vieta product constraint), and the accumulated phase of ω⁴⁴ defines a quasi-closure comma C = 0.003647 rad that is nonzero by algebraic necessity: x³ = x² + 1 is a shadow polynomial, and shadow polynomials cannot have C = 0. The central result is a canonical duality: v ≠ 0 ⟺ C ≠ 0 These are not analogous — they are the same statement in two languages. The physical consequence is immediate: at Planck-scale energy density, the orbit of ω can no longer sustain its K = 44 quasi-crystalline structure, forcing C_eff → 0 and therefore v_eff → 0. This means the Higgs quartic coupling runs to exactly zero at the Planck scale — λ(M_Pl) = 0 — not approximately, not coincidentally, but algebraically. The Higgs vacuum stability boundary is derived as: m_t,crit = v/√2 × (1 − m₃²·C) = 171.56 GeV where m₃ = 2 is the baryon winding number of the same polynomial. This matches the Buttazzo et al. three-loop Standard Model calculation of 171.53 ± 0.42 GeV to 0.020% with zero free parameters. Additional predictions: the Higgs vacuum is exactly stable (not metastable); supersymmetric partners will never be found because SUSY corresponds to the C = 0 skeleton state, algebraically inaccessible from our C ≠ 0 universe; and FCC-ee threshold measurement of the top quark mass will find m_t ≈ 171.5 GeV in a scheme-independent mass. This paper is self-contained. The broader 15 page suite — Cascade Framework, which derives more than 35 Standard Model, cosmological, and observational predictions from x³ = x² + 1 with zero free parameters, is available at https://doi.org/10.5281/zenodo.19592471