The Cascade Exponent — Turbulence as Quasicrystal in Scale Space

Kolmogorov's −5/3 energy spectrum exponent for fully developed turbulence has been understood since 1941 as a consequence of dimensional analysis. We derive −5/3 from the geometry of S³/2I without dimensional arguments. The Steinbach polynomial (the unique cyclic cubic of smallest discriminant, t³ − 7t/3 + 7/27 = 0) contains only the primes {3, 7} in its coefficients. Its Newton-Girard power sums generate the prime 5 at exactly k = 5 (p₅ = −5 × 7²/3⁴) — the first prime not present in the polynomial's own coefficients. The cascade exponent is −5/3: the quintic emergence order divided by the eigenvalue count. The decomposition 5 = 3 + 2 identifies turbulence as a quasicrystal in scale space, with 3D physical strain space plus 2D perpendicular intermittency space. The Newton-Girard route explains why Kolmogorov's dimensional analysis works: its inputs (D = 3, quadratic energy) are outputs of the Steinbach algebra. It also predicts what dimensional analysis cannot: Fibonacci structure, the quintic wall blocking the next convergent, and two independent intermittency modes from the cut-and-project decomposition. The closing prime pattern (p₇ = −7⁴/3⁵, pure heptagonal) is verified to be unique among non-trivial cyclic cubics. Paper 166 of the Bootstrap Universe Programme.