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This work investigates how a single cubic polynomial can derive multiple fundamental observables in physics.

May 8, 2026Open Access

Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v4)

Key Points

This work investigates how a single cubic polynomial can derive multiple fundamental observables in physics.
Derivation of 28 standard model and cosmological observables from the polynomial x³ = x² + 1.
Application of 37 algebraic theorems, with some conjectured but not yet proven.
Analysis of quantum optical negative atomic excitation based on recent experimental measurements.
Identified the quasi-closure orbit as a (44,13) torus knot with precise polynomial properties.
Demonstrated geometric relationships through Theorem DH-1 and DH-3, contributing to topological understanding in physics.
Predicted atomic excitation saturation values and optimal probe detuning closely matching reported measurements.

Abstract

Version 4 of the Cascade Framework derives 28 Standard Model and cosmological observables from the single cubic polynomial x³ = x² + 1 with zero free parameters and 37 exact algebraic theorems (theorems 34 and 35 conjectured pending formal proof). Major advances over v3 include: the quasi-closure orbit identified as a (44, 13) torus knot on T², whose Alexander polynomial gap set is F₃, …, F₇ and whose Jones polynomial satisfies V (e²πi/⁵) = 1 exactly; the Penrose–Zeldovich dual holography identification, establishing the cascade simultaneously as a Penrose cut-and-project scheme (n=2→n=3) and a Zeldovich catastrophe-unfolding system (n=5→n=4) ; Theorem DH-1, giving the quasi-closure comma C its first geometric derivation as a commutator residual; Theorem DH-3 (ADE completeness via McKay), proving three fermion generations a topological invariant; and Theorem 36 (V4 theta-anyon correspondence for Fib⊗Silver). A seventh paper applies the framework to the quantum optical negative atomic excitation time results of Angulo et al. (arXiv 2024, Phys. Rev. Lett. 2026), predicting both saturation values (+0. 5414 and −0. 8232 in units of τ₀) and the optimal probe detuning (20. 8 MHz) from first principles, within 0. 01σ of reported measurements. This v4 package comprises seven papers: Framework Overview, Negative Atomic Excitation Time (new), Dark Energy, Temporal Sector, Electroweak/CKM, Fermion Masses, and Arithmetic Geometry.

Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v4)

Key Points

Abstract

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