What question did this study set out to answer?

The aim is to derive Standard Model and cosmological observables from a cubic polynomial with zero free parameters.

May 2, 2026Open Access

Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v3)

Key Points

The aim is to derive Standard Model and cosmological observables from a cubic polynomial with zero free parameters.
Derivation of 28 observables from x³ = x² + 1 using algebraic theorems, including Vieta's formulas.
Introduction of major theorems such as Theorem 37 and Theorem 32, focusing on dark energy and gravitational constants.
Development of a v3 package comprising six papers covering various aspects of the framework.
Theorem 37 proves a relationship between dark energy density and the arrow of time, showing cos θ₃ = −Ω_Λ^(3/2)/2.
Newton’s gravitational constant is accurately derived at 0.002% accuracy and added to the scorecard.
Significant precision improvement on the CP-violating phase δ_CP from 0.15% to 0.003%.

Abstract

Version 3 of the Cascade Framework derives 28 Standard Model and cosmological observables from the single cubic polynomial x³ = x² + 1 with zero free parameters and 32 exact algebraic theorems. Major advances over v2 include: Theorem 37 (Euler Form of the Temporal Sector), which proves that the 68. 23% dark energy density and the arrow of time are the same algebraic fact — cos θ₃ = −Ω_Λ^ (3/2) /2, exact from Vieta’s formulas; Theorem 32, which formally derives Newton’s gravitational constant at 0. 002% accuracy (new to the scorecard) ; a 50× precision improvement on the CP-violating phase δCP (0. 15% → 0. 003%) ; and a complete physical identification of the polynomial’s three roots as Being, Gravity, and Becoming. This v3 package comprises six papers: Framework Overview, Temporal Sector (new), Dark Energy, Electroweak/CKM, Fermion Masses, and Arithmetic Geometry.

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