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This work aims to derive various observables from the cubic polynomial equation with no free parameters.

April 17, 2026Open Access

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

Key Points

This work aims to derive various observables from the cubic polynomial equation with no free parameters.
Developed a framework overview and complete scorecard for Standard Model predictions.
Predicted dark energy and provided forecasts for validation through DESI/Euclid.
Used cascade arithmetic to determine fermion mass ratios.
Analyzed electroweak sector, CKM matrix, and neutrino masses.
Derived 27 total observables with 19 achieving sub-0.1% accuracy.
Predicted dark energy density Ω_Λ = 0.6823.
Validated 8/8 predictions in electroweak sector and 5/5 in CKM matrix.
Proven fourteen exact algebraic theorems supporting the framework.

Abstract

A theoretical framework deriving 27 Standard Model and cosmological observables from the single cubic polynomial x³ = x² + 1 with zero free parameters. The four papers cover: (1) framework overview and complete scorecard (19/27 sub-0. 1%), (2) dark energy prediction Ω_Λ = 0. 6823 with falsifiable DESI/Euclid forecasts, (3) fermion mass ratios via cascade arithmetic, and (4) electroweak sector (8/8 sub-0. 1%), CKM matrix (5/5 sub-0. 1%), and neutrino masses. Fourteen exact algebraic theorems are proven. The sub-leading correction structure classifies into geometric, algebraic, and hybrid branches.

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Joshua Breault

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

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