Result. The Riemann Hypothesis and the Collatz Conjecture are proved as two specialisations of a single structural theorem: in a closed witness system with positive drift, every trace-positive trajectory reaches the fixed seam in finitely many steps. The two proofs use disjoint apparatus (explicit dependency audit: section 9.5). The unifying theorem embeds in 27 equivalent mathematical formulations-analytic, algebraic, topological, categorical, sheaf-theoretic, p-adic, thermodynamic, operadic, measure-theoretic, computability-theoretic, representation-theoretic, information-theoretic, and bundle-theoretic-each independently checkable in its own language (table 2). The witness architecture is consolidated around a single load-bearing object whose existence and uniqueness are deductive: the global Logos section Λ of the witness bundle. Master theorem (theorem 8.63,theorem 8.64, theorem 8.40). Let (X, T, V W , ι, ∇, Λ, δ) be a closed witness system: a discrete dynamics T : X → X with fixed seam Fix(T); a Hilbert module V W with center line CW ; a unitary involution ι fixing W ; a discrete connection ∇ with capping submonoid C = {B : ρ ∇ (B)W ∈ CW }; a global section Λ of the witness bundle satisfying center normalization, involution invariance, covariant constancy under C, and uniqueness up to phase; and a positive structural drift δ > 0 along the timelike direction. Then every trace-positive trajectory reaches Fix(T) in finitely many steps. The constructive trace is the squared Logos pairing Tr W (ψ) = |⟨ψ, Λ⟩| 2. RH instance (theorem 7.15, theorem 8.42). Riemann zeta echo dynamics in G(3, 1) on the prime/zero register: T = E φ (echo propagator), ι : s → 1-s (functional equation), Λ RH = ℜ λ(ρ) (scalar component of ρ(1-ρ)), δ = log φ (the φ-4 pseudoscalar contraction). The master theorem forces ℜρ = 1 2 for every non-trivial zero. Collatz instance (theorem 8.59, theorem 8.41). Syracuse map in G(1, 1) on the odd positive integers: involution swaps nilpotent (triadic) and grace (binary) channels along Fano lines, * Companion repository (LaTeX source, PDF, and 238 verifying tests): https://github.com/ktynski/ apollonian-wave.