RH Toolkit 2 — The Möbius Arithmetic Operator

This paper develops a self-contained operator-theoretic framework for the Mobius function and the canonical linear operators it induces on discrete Hilbert spaces. It analyzes three operator classes: the Mobius multiplication operator M, the Mobius Dirichlet convolution operator C = mu *, and Mobius-weighted finite-range kernel operators. These operators separate three arithmetic mechanisms: sign modulation, squarefree projection, and full prime-power recovery. The multiplication operator M acts diagonally by mu(n). It is bounded and self-adjoint on l2(N), has spectrum {-1, 0, 1}, and satisfies M^2 = P, where P is the squarefree projection. The convolution operator C = mu * implements Mobius inversion and recovers the von Mangoldt function from the logarithmic divisor profile: Lambda = mu * ell, where ell(n) = log n. This distinction matters because P removes nonsquarefree support, while Lambda includes all prime powers. The paper also studies Mobius-weighted kernels J(m,n) = A(m,n) mu(|m-n|). When A is symmetric and Hilbert-Schmidt, the associated operators are bounded, self-adjoint, Hilbert-Schmidt, and compact. They therefore admit the standard compact self-adjoint spectral resolution: nonzero spectral values are real eigenvalues of finite multiplicity, with possible accumulation only at zero. The appendices provide a finite-window Mobius propagator model, weighted arithmetic-kernel realization criteria, bounded-plus-Hilbert-Schmidt decompositions for extracted arithmetic kernels, the exact von Mangoldt recovery interface, and a zero-aware squarefree positivity-compatible sector. The final appendix addition relates this sector to the upstream theta-convex absorption interface, while emphasizing that positivity, residual absorption, and zero-exhaustion remain conditional downstream tasks rather than conclusions of this paper.