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

Abstract 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): ∇Gₚair is Hilbert-Schmidt; ∇Fₚair 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 γₙ 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, Mδₙ (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. Keywords: self-referential operator, Hilbert-Polya operator, trace-class operator, Gelfand triple, Rouché theorem, complete spectral equivalence, bidirectional closure, Riemann zeta function, Fredholm determinant, resonance condition, Clarke generalized derivative, measure-theoretic isomorphism