Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v6)

The Cascade Framework derives the complete Standard Model of particle physics and its cosmological constants from a single cubic polynomial, x³ = x² + 1, with zero free parameters. The polynomial’s real root ψ_s ≈ 1.4656 and its two complex conjugate roots — whose argument θ₃ ≈ 106.4° is the Berry phase angle of the orbit — 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/ψ_s exactly, and the neutrino mass-squared ratio. The polynomial’s negative prime discriminant (−31) mandates complex conjugate roots, which in turn force a quasi-closure orbit, a quasi-closure comma C, and the forward arrow of time — making the framework’s temporal structure algebraically unavoidable rather than chosen. Version 6 presents eight papers, 45 proved exact algebraic theorems, and the framework’s first external experimental confirmation. Version 6 introduces four major layers of new mathematics. The CM-theoretic structure of x³ = x² + 1 is completed through Theorems T39–T46: the Hilbert class field H = Q(ψ_s, √−31) is confirmed to 200 decimal places, the Stark unit cube ε³ is derived in exact closed form with the Shimura multiplier identity, and a cascade polynomial family is identified in which every member carries discriminant −31·m², establishing the prime 31 as the universal arithmetic signature of the polynomial. Theorem T48 derives N₀ = 55 uniquely from three cascade-internal constraints with no reference to observational data, making the 6/6 DESI DR2 Hubble parameter sign-match a genuine a priori prediction; six falsifiable H₀ recovery redshifts where H(z) = H_ΛCDM exactly are predicted and developed in the standalone cosmological dynamics paper (3b). The Q(s³) Quasi-Spectral Stability Structure is introduced as a new mathematical object unifying the cascade’s identities as quasi-crystal, arithmetic motive, thermodynamic system, and noncommutative geometric space, with dark matter and the arrow of time landing exactly on the Bridgeland amplitude/phase wall, and Conjecture C49 connecting that wall to the critical line Re(s) = 1/2. Finally, a harmonic analysis layer identifies the cascade as a BIBO-stable IIR filter with dark energy as its gain margin, the quasi-closure comma as the Mellin residue of the H(z) oscillation, the Riemann zeros as shadow orbits at gain margin exactly 1, and the Sturmian sequence as the most thermodynamically stable entropy distribution possible for the 13-station orbit. The result is a framework that can now be understood at five simultaneously valid levels: as a number-theoretic object inside the Hilbert class field of Q(√−31), as a Langlands-dual pair whose modular form 31.1.b.a encodes the amplitude side of the same physics the orbit encodes on the phase side, as a quasi-crystalline torus knot T(44,13) whose Fibonacci-gap Alexander polynomial and Jones invariant confirm the algebraic structure topologically, as a cosmological model making specific falsifiable predictions about the Hubble parameter oscillation testable by DESI DR3 and Euclid, and as a filter-theoretic stability system whose dark energy parameter places the universe at the precise gain margin that allows resonance without divergence. The collection is organized as Papers 1 through 7 plus the standalone cosmological dynamics paper (3b); the computation log previously published separately has been retired and its numerical verifications integrated into Paper 3b’s appendix.