What question did this study set out to answer?

This paper aims to provide a first-principles explanation for the nonzero Higgs vacuum expectation value based on a cubic polynomial's algebraic properties.

May 30, 2026Open Access

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

Key Points

This paper aims to provide a first-principles explanation for the nonzero Higgs vacuum expectation value based on a cubic polynomial's algebraic properties.
Utilized the cubic polynomial x³ = x² + 1 to derive relationships between physical constants and algebraic constraints.
Proposed a canonical duality between nonzero Higgs vacuum and quasi-closure comma derived from polynomial roots.
Established that the Higgs quartic coupling runs to exactly zero at the Planck scale, which implies vacuum stability.
Derived critical top quark mass, aligning with experimental values from the Standard Model, with a minimal deviation of 0.020%.
Predicted the stability of the Higgs vacuum and the absence of supersymmetric partners due to algebraic constraints.

Abstract

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 ψₛ = 1. 4656… is the super golden ratio. Its complex conjugate roots ω, ω̄ satisfy |ω|² = 1/ψₛ 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 Cₑff → 0 and therefore vₑff → 0. This means the Higgs quartic coupling runs to exactly zero at the Planck scale — λ (MPl) = 0 — not approximately, not coincidentally, but algebraically. The Higgs vacuum stability boundary is derived as: mₜ, 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 mₜ ≈ 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

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