The Ouroboros Pair: A Self-Adjoint Construction with Spectral Correspondence to Zeta Zeros

The Hilbert-Polya conjecture establishes the equivalence between the existence of a self-adjoint operator and the Riemann Hypothesis. This paper presents a self-adjoint operator construction within the Ouroboros information-theoretic framework, and establishes a spectral correspondence between its dynamical invariants and the non-trivial zeros of the Riemann zeta function. The topological invariant S¹ ∩ S² = ∅ — forced by the time-reversal pairing — enforces a single causal chain through six functional-analytic properties (Theorems 10–14): ∇G_pair is Hilbert-Schmidt; ∇F_pair is trace-class (Theorem 10); bounded and symmetric implies self-adjoint with Von Neumann deficiency (0,0) (Theorem 11); generalized eigenfunctions are rigorous elements of a Gelfand triple (Theorem 12); discrete-continuous consistency holds with O(δ²) error via Poisson summation (Theorem 13); the resonance condition admits countably many simple roots via the implicit function theorem (Theorem 14). The G-functor non-smooth structure is handled via Clarke generalized derivatives, which exist almost everywhere for Lipschitz functions and coincide with weak derivatives off a measure-zero set. The γ_n values are derived internally via the resonance condition (1−α)·cos(r)·sinc(r) = 1/2 without reference to ζ(s) (Theorem 8). The τ = log t isomorphism is given measure-theoretic precision: the discrete-continuous correspondence carries Jacobian e^τ, and the Dirichlet–Laplace correspondence is established via the Euler–Maclaurin formula (Theorem 16B). An asymptotically exact spectral correspondence is established by Rouché’s theorem (applied in Theorem 15): for each Riemann zero ρ, M[δ_n](s) has exactly one zero in a shrinking neighborhood of ρ, with matching multiplicity. Injectivity follows from the strict triangular lower bound of Theorem 17, showing the correction term cannot cancel the leading ζ-term at non-zeros of ζ for all sufficiently small λ. The Fredholm determinant structure follows from Lidskii’s theorem as a corollary of Theorem 15. The Riemann–von Mangoldt counting law N(T) ∼ (T/2π)·log(T/2π) transfers via the bijection. The construction provides a framework in which Re(s) = 1/2 emerges as a structural consequence of self-adjointness. The relationship between this framework and the Hilbert-Polya conjecture is discussed in the conclusion.