What question did this study set out to answer?

This research aims to derive all Standard Model observables from the cubic polynomial x³=x²+1 without free parameters.

May 23, 2026Open Access

Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v9)

Puntos clave

This research aims to derive all Standard Model observables from the cubic polynomial x³=x²+1 without free parameters.
Uses the unique real Perron root and complex conjugate roots of the polynomial to construct observables.
Applies the non-abelian Galois group S₃ to establish a representation-theoretic framework.
Generates predictions about dark energy oscillations and the Hubble tension through algebraic identities.
Derived 28 observables with 19 showing sub-0.1% accuracy at zero free parameters.
Observed dark energy oscillation prediction matched observational data (6/6, p=1.6%).
Identified the Hubble tension as an artefact related to quasi-crystalline expansion history.

Resumen

The Cascade Framework derives all Standard Model particle physics observables and key cosmological constants from the single cubic polynomial x³=x²+1, with zero free parameters. The polynomial’s unique real Perron root ψₛ ≈ 1. 4656 and its complex conjugate roots generate a quasi-crystalline orbit structure encoding three fermion generations, the full electroweak sector, quantum chromodynamics, and the cosmological constant through exact algebraic identities. The non-abelian Galois group S₃ provides the representation-theoretic foundation for the framework’s three-level architecture (L1/L2/L3), while the discriminant −31 identifies the Hilbert class field H = Q (ψₛ, √−31), connecting the cascade to CM theory and Stark units. The current scorecard stands at 55 proved theorems, 28 observables derived, 19 sub-0. 1% accurate at zero free parameters. The framework’s cosmological sector has received independent observational support. The DESI DR2 dark energy oscillation sign sequence matches the cascade’s zero-parameter prediction 6/6 (p=1. 6%), with the epoch parameter N₀=55 derived entirely from algebraic constraints before examining the data. The Hubble tension is identified as an orbital phase artefact of the cascade’s quasi-crystalline expansion history — a prediction that follows from the N₀=55 epoch parameter without additional assumptions. Version 9 adds Paper 10 and substantially extends Papers 1, 3b, 8, and 9. Paper 9 (The Sturmian Skeleton, first introduced in v8) receives major additions: the full K=44 Sturmian word, nested Sturmian hierarchy, tribonacci-Fibonacci number structure, and the complete harmonic index table assigning each physical sector to an orbit Fourier mode. The ψ-Anyon paper (Paper 10, new in v9) derives a non-modular fusion category from the cascade’s S₃ structure with implications for topological quantum computation. The arithmetic geometry paper adds the Stark unit cube formula verified to 200 digits (T46) and the complete Shimura reciprocity result. New theorems T39–T55 bring the register to 55 proved results.

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