The CASCADE Rosetta Theorem: Pisot Trace Lattice, Exactness Spectrum, and the Four Shadows of One Identity

This paper establishes the CASCADE Rosetta Theorem: the universal k-stroke lifting identity (xᵏ+1) ·Pₙ = Pₙ₊ₖ + xᵏ·Pₙ₋ₖ, when evaluated in four distinct ways — at a single root, as a Galois orbit sum, in the reciprocal variable, and modulo primes — generates a complete translation dictionary between four mathematical languages: the irrational mirror identity, the integer trace lattice, the Ruelle zeta functional relation, and the Frobenius congruences. The paper proves that the CASCADE framework’s fundamental constants h, m₁, m₃, W, q, K are not empirical inputs but the discrete exactness spectrum of this structure — the specific indices where the Galois-summed scaling law achieves integer exactness. Key results include: (1) an exact closed-form formula for the quasi-closure comma, C = 4·arcsin (ψₛ^ (−33/2) /2), derived from the integer trace identity Tr (M^ (−11) ) = 0, establishing zero free parameters across the entire framework; (2) the five-constant identity 284K = q²W + h connecting the Baker–Stark discriminant bound to all four primary CASCADE constants; (3) a structural proof that the coefficient 7 in δ₄ = φ⁻² + C/2 − 7C² equals 2h+1 via the modular identity 9W ≡ K/2 + (2h+1) (mod K) ; and (4) a complete defect audit confirming that no core near-identity in the CASCADE suite is a coincidence. The paper further establishes the Index-21 Theorem: for both CASCADE cubic SL₃ (ℤ) Pisot families — the supergolden (x³−x²−1, disc=−31) and the plastic (x³−x−1, disc=−23) — the arithmetic conductor appears as an inverse trace at index 21 = h (2h+1), proved by direct integer recurrence. Cross-family structure is developed: the boundary factorization P₅ = Φ₆· (x³−x−1) delivers both classical conductor-3 dihedral L-functions within the single CASCADE family, and the beacon product Tr (M₃¹1) ×Tr (M₄¹2) = 67×47 = P₃ (15) demonstrates exact interlock between lattices at different polynomial levels. The recipe (a+b) /D² is identified as the Verlinde fusion formula with D²=4 (Ising total quantum dimension squared), from which m₃ = d²_σ (baryon winding equals σ-anyon quantum dimension squared) and m₁ = K/D² (topological period of the CASCADE σ-anyon braiding phase). The paper closes with a precise formulation of the dynamical-to-arithmetic bridge separating the proved Ruelle mirror identity from a proof of GRH for L (s, 31. 1. b. a), and identifies the Deninger determinant framework as the required machinery. This is Paper 20 of the CASCADE Framework.