Hyperbolic Rapidity Parametrization, the Leptonic Mass Law, and the Algebraic Correspondence with the Higgs Polynomial via Vacuum Inversion
Abstract
We present an analytical parametrization of the charged lepton mass spectrum and examine its structural relation to the Higgs potential within the framework of hyperbolic vacuum rapidity. Starting from the phase accumulation of spinor states across Riemann sheets and its relation to the empirical Koide ratio (Q = 2/3), we show that extending the bounded circular projection into a complex angle ϑ = a + ib yields the exact modulus identity |sin(a + ib)|² = sinh² b + sin² a. The circular phase is anchored at a = π/4, the angle of the hyperbolic asymptote, which coincides with the Minkowski null light-cone direction. This anchoring produces the Leptonic Mass Law m(b) = κ(sinh² b + ½) = (κ/2) cosh(2b), which is structurally analogous to the exterior vacuum zero-point energy E₀ = ½ m cosh β. For the electron rapidity bₑ = ½ ln 2, the hyperbolic functions take the exact rational values cosh(2bₑ) = 5/4 and sinh(2bₑ) = 3/4, and the product of the boosted energy E(φ) ∝ cosh φ with the Lorentz-contracted Gaussian width σ(φ) = σ₀ sech φ yields the rapidity-invariant action product E(φ)σ(φ) = (5/4) ℏ c. Finally, we show that differentiating the reciprocal amplitude ψ(β) = sech β and the conic order parameter u(β) = tanh β with respect to rapidity generates equations algebraically identical to the Ginzburg–Landau/Higgs field equation ψ″ − ψ + 2ψ³ = 0 and to the symmetric double-well polynomial (u′)² = (1 − u²)² = 1 − 2u² + u⁴. In the asymptotic interior limit β → ∞, the zero-point energy scale vanishes (Eₛ → 0), a limit that, under the present interpretation, corresponds to an asymptotic restoration of the Higgs symmetry without point singularities.
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BibTeX
@misc{barros2026hyperbolic,
title = {{Hyperbolic Rapidity Parametrization, the Leptonic Mass Law, and the Algebraic Correspondence with the Higgs Polynomial via Vacuum Inversion}},
author = {Rodrigo Silva Barros},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23595839},
url = {https://zenodo.org/records/23595839},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Rapidity Parametrization, the Leptonic Mass Law, and the Algebraic Correspondence with the Higgs Polynomial via Vacuum Inversion
AU - Rodrigo Silva Barros
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23595839
UR - https://zenodo.org/records/23595839
AB - We present an analytical parametrization of the charged lepton mass spectrum and examine its structural relation to the Higgs potential within the framework of hyperbolic vacuum rapidity. Starting from the phase accumulation of spinor states across Riemann sheets and its relation to the empirical Koide ratio (Q = 2/3), we show that extending the bounded circular projection into a complex angle ϑ = a + ib yields the exact modulus identity |sin(a + ib)|² = sinh² b + sin² a. The circular phase is anchored at a = π/4, the angle of the hyperbolic asymptote, which coincides with the Minkowski null light-cone direction. This anchoring produces the Leptonic Mass Law m(b) = κ(sinh² b + ½) = (κ/2) cosh(2b), which is structurally analogous to the exterior vacuum zero-point energy E₀ = ½ m cosh β. For the electron rapidity bₑ = ½ ln 2, the hyperbolic functions take the exact rational values cosh(2bₑ) = 5/4 and sinh(2bₑ) = 3/4, and the product of the boosted energy E(φ) ∝ cosh φ with the Lorentz-contracted Gaussian width σ(φ) = σ₀ sech φ yields the rapidity-invariant action product E(φ)σ(φ) = (5/4) ℏ c. Finally, we show that differentiating the reciprocal amplitude ψ(β) = sech β and the conic order parameter u(β) = tanh β with respect to rapidity generates equations algebraically identical to the Ginzburg–Landau/Higgs field equation ψ″ − ψ + 2ψ³ = 0 and to the symmetric double-well polynomial (u′)² = (1 − u²)² = 1 − 2u² + u⁴. In the asymptotic interior limit β → ∞, the zero-point energy scale vanishes (Eₛ → 0), a limit that, under the present interpretation, corresponds to an asymptotic restoration of the Higgs symmetry without point singularities.
ER -