Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v8)

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 observables through an algebraic hierarchy: a Sturmian quasi-crystalline orbit, a CM-theoretic connection to the Hilbert class field H = Q (ψₛ, √−31), and a causal spacetime structure that maps the three gain margin regimes onto the timelike, null, and spacelike sectors of special relativity. Version 8 presents nine papers and 55 proved theorems across particle physics, cosmology, algebraic number theory, dynamical systems, signal processing, and the Riemann Hypothesis connection established in v7. Version 8’s central result is the derivation of the complete electroweak mass hierarchy from first principles. Theorem T57 (MZ Phase-Sector Correction) derives the Z boson mass as MZ = MZᵗree × (1 − C/μ), where C is the quasi-closure comma and μ = √ψₛ is the gain margin, achieving 0. 0009% accuracy — within the PDG experimental uncertainty of ±0. 0021 GeV. Theorem T58 (Higgs Mass Formula) then derives the Higgs boson mass as MH = √2 · MZ · (1 − 8C) = 125. 195 GeV, achieving 0. 004% accuracy and within PDG uncertainty of ±0. 110 GeV. All three electroweak mass scales — MW, MZ, and MH — are now cascade-derived within current experimental measurement precision with zero free parameters. The structural key: MW at bright Sturmian station 4 (SU (2) alone) receives an upward Branch 3 silver correction (1+√2·C) ; MZ at shadow station 6 (SU (2) +U (1) mix) receives a downward phase-sector correction (1−C/μ) ; MH inherits √2 from the silver sector with a collective shadow correction (1−F₆·C) where F₆ = 8 is the number of shadow stations. Version 8 also formalizes eight additional theorems established after v7’s launch. The wall-crossing sequence (T49) proves the Bridgeland wall is crossed at N = 11 (2k+1) with spacing 22 = K/2, and the wall deviation formula (T50) proves the kth crossing deviates from exact 90° by (2k+1) ·C/4 radians — confirming N₀ = 55 places the current cosmological epoch at 0. 26° from the Bridgeland wall, within the amplitude/phase boundary. The DESI-Null Bridge (T51) proves Bₒsc = 2C·ccascade, connecting DESI’s dark energy oscillation amplitude to the cascade speed of light ccascade = π/2 (T52). These combine into the v8 collection’s most striking cross-domain result: MH = √2 · MZ · (1 − 4 · Bₒsc/ccascade), meaning DESI DR3 measuring Bₒsc to 0. 1% precision will constrain the Higgs mass to ±0. 004 GeV — approximately 30 times tighter than the current PDG uncertainty. The cascade speed of light is the structural bridge between cosmological dark energy measurements and particle physics.