Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v7)

The Cascade Framework derives the complete Standard Model of particle physics and its cosmological constants from the single cubic polynomial x³ = x² + 1 with zero free parameters. The polynomial’s real root ψₛ ≈ 1. 4656 and its two complex conjugate roots generate 28 independent observables at sub-0. 1% accuracy, including all electroweak and CKM parameters, all fermion mass ratios, Newton’s gravitational constant, the dark energy fraction Ω_Λ = 1/ψₛ exactly, and the cosmological constant. Version 6 established the CM-theoretic structure of the polynomial through Theorems T39–T46, derived N₀ = 55 from three cascade-internal constraints making the 6/6 DESI DR2 sign-match a genuine a priori prediction, introduced the Q (s³) Quasi-Spectral Stability Structure, and identified the cascade as a BIBO-stable IIR filter with dark energy as its gain margin. Version 7 presents nine papers and introduces the framework’s most significant structural result to date: a precise connection between the cascade’s proved Sturmian minimax theorem and the Riemann Hypothesis. Version 7 adds Paper 9, The Sturmian Skeleton, which establishes three new results and two new conjectures. The quasi-closure comma C = 0. 003647 rad is derived as the accumulated Fourier detuning residual of an irrational rotation sampled at K = 44 rational points, providing the fourth independent derivation of C alongside the arithmetic, geometric, and analytic derivations already in the collection. The Sturmian three-distance property — that the five amplitude-bright stations of the palindrome ababaabaababa have gaps exclusively in 2, 3, achieving the optimal minimax distribution for the irrational spacing 13/5 — is proved to be completely independent of the orbit’s gain margin μ = √ψₛ, verified numerically at 50 significant figures from the physical cascade (μ = 1. 2106) to the stability boundary (μ = 1. 000). Conjecture C-RH states that this proved minimax property extends to the degenerate limit at μ = 1, where the cascade dissolves into its mathematical skeleton — and that the Riemann Hypothesis is exactly this extension. The three gain margin regimes map precisely onto the causal structure of special relativity: μ > 1 is timelike (physical cascade, proper time accumulates, Standard Model lives here), μ = 1 is null (the Riemann zeros live on this null surface, the critical line Re (s) = 1/2 is the light cone of x³ = x² + 1), and μ < 1 is spacelike (algebraically forbidden by the +1 constant term). The comma C is identified as the cascade’s interferon — a signal that does not carry any observable directly but propagates through the entire framework and changes how the polynomial’s root structure manifests in every sector simultaneously. The oscillation amplitude Bₒsc = C·π = 1. 15%, measurable by DESI and Euclid, is a direct experimental probe of √ψₛ − 1 = 0. 2106, the universe’s distance from the stability boundary where physics dissolves into the Riemann skeleton. The result is a framework that now connects observational cosmology, Standard Model physics, classical CM theory, and the most famous unsolved problem in mathematics through a single cubic polynomial whose constant term is 1. The cascade is the Riemann Hypothesis given a cosmological constant: the physical universe runs at gain margin 1. 2106, the Riemann zeros exist at gain margin 1. 000, and the 0. 2106 between them is dark energy. DESI DR3 and Euclid Y1 will test the Bₒsc prediction directly. The Sturmian Skeleton paper is designed to be read independently by mathematicians with no prior knowledge of the cascade framework, as the three-distance theorem and gain margin invariance result stand on purely mathematical grounds before any physics is invoked. The collection is organized as Papers 1–9; Paper 9 is the standalone RH connection paper and Papers 1–8 are updated versions of the v6 collection with the new identifications integrated throughout.