Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v5)

The Cascade Framework (v5) derives all Standard Model coupling constants, fermion mass ratios, quark mixing parameters, Newton’s gravitational constant, and the cosmological constant from the single cubic polynomial x³ = x² + 1 with zero free parameters. 28 observables are reproduced at sub-0. 1% accuracy. 39 exact algebraic theorems are proven from the polynomial (T34 and T35 remain conjectured pending formal proof). The Langlands-dual automorphic object is the unique weight-1 cuspidal newform 31. 1. b. a on Γ₀ (31). The central result: the negative prime discriminant −31 forces the maximal Galois group S₃, mandates complex roots, and makes the entire structure of the Standard Model and cosmological constant consequences of a single sign. Theorem 37 proves from two lines of Vieta’s formulas that cosθ₃ = −Ω_Λ^3/2/2 exactly — unifying the 68. 23% dark energy fraction and the strength of time’s arrow as one algebraic fact. Three fermion generations are a topological invariant of the A₂ singularity via the McKay correspondence. The quasi-closure orbit is a (44, 13) torus knot whose Alexander polynomial gap set encodes the Fibonacci-Lucas structure of the framework. New in v5 — Paper 3 (Dark Energy) and Computation Log T38: Version 4 predicted the DESI DR1 hint of evolving dark energy would not persist. Version 5 supersedes this with a more complete picture. The cascade quasi-closure orbit is a Sturmian quasi-time crystal: its 13-station beat word ababaabaababa encodes a quasi-periodic H (z) oscillation around H_ΛCDM. Theorem T38 establishes n (z) = N₀ − α·ln (1+z) with α = 3/ln (ψₛ) = 7. 8484 (exact, zero free parameters). Corollary T38a proves matter-dark energy equality occurs exactly Δn = 2 cascade steps before today, confirmed to 15 significant figures. All six DESI DR2 tracer bins show the predicted sign of H (z) relative to ΛCDM with N₀ = 55 derived independently of the data (6/6 match, p = 1. 6%, 2. 4σ). The six signs map directly onto the Sturmian sub-word aabaab. Six H₀ recovery redshifts are predicted where H (z) = H_ΛCDM (z) exactly: z = 0. 241, 0. 540, 0. 910, 1. 370, 1. 940, 2. 647. The apparent DESI evidence for evolving dark energy is identified as a model-selection artifact of fitting a smooth w₀wₐ template to a quasi-crystalline signal. The Hubble tension is identified as an orbital phase artifact of the same structure. The T38 Computation Log (03. 1) provides full numerical verification of T38, T38a, Theorem 22 (Sturmian defect symmetry to 1. 26×10⁻⁹ rad), the Fibonacci-Comma identity, the 6/6 DESI sign table, and all six H₀ recovery redshifts. New in v5 — Cross-paper integration: The Sturmian orbit theorems (T22–T25), proven in Paper 7 as pure algebraic geometry, receive their first direct physical interpretation in v5: as the dark energy balance law, the cosmic beat pattern, and the structural origin of Bₒsc = C·π = 1. 146%. Footnotes connecting the cosmological dynamic picture to the electroweak, fermion, and arithmetic geometry papers are added throughout.