This paper establishes the arithmetic foundation of the CASCADE n-stroke polynomial family Pₙ(x) = xⁿ − xⁿ⁻¹ − 1 through a hierarchy of exact identities emanating from a single three-line proof. The opening theorem is the universal k-stroke lifting identity: (xᵏ+1)·Pₙ = Pₙ₊ₖ + xᵏ·Pₙ₋ₖ, valid for all n > k ≥ 1. Its k = 1 case recovers the CASCADE polynomial recurrence Pₙ = (x+1)Pₙ₋₁ − xPₙ₋₂; its evaluation at the bridge root ψₛ yields the mirror identity P₃₊ₖ(ψₛ) = −ψₛᵏ·P₃₋ₖ(ψₛ), identifying ψₛ as the universal power scaling base of the family — the CASCADE analog of φ in the Fibonacci sequence, but with the fundamental distinction that the scaling law is exact rather than asymptotic. The difference formula Pₙ − Pₙ₋₁ = xⁿ⁻²(x−1)² establishes quadratic convergence of Perron roots to 1. Physical and arithmetic applications follow as corollaries. The Standard Model gauge group SU(3)×SU(2)×U(1) is identified as the prime factorization of the GUT group SU(5), with gauge group stability corresponding to the primality of N in SU(N). The n = 3 level is the unique ±1 arithmetic fixed point of the n-stroke family, explaining why QCD never breaks while the electroweak and Pati-Salam groups do. The baryon winding m₃ = 2 is the k-stroke number of the i-rotation; the dark matter winding m₁ = 11 = K/4 is the orbit step to the i-phase. The closing theorem identifies the quasi-closure comma C as the evaluation of the CASCADE polynomial recurrence at the orbit’s near-fixed-point. The (x+1) lifting factor is proposed as the polynomial incarnation of the Riemann functional equation ξ(s) = ξ(1−s), with P₃(ψₛ) = 0 as the exact algebraic zero that forces all zeros of L(s, 31.1.b.a) onto Re(s) = 1/2. This is Paper 19 of the CASCADE Framework.
Joshua Breault (Wed,) studied this question.