Weighted Farey Laplacians: Bounded Operators, Finite Approximation, a Universal Projection Identity for the Mertens Function, and Near-Sharp Gap Bounds

This paper develops a rigorous denominator-weighted Laplacian on the Farey graph. It proves the exact decay threshold for operator existence, boundedness, controlled finite approximation, gauge structure, a uniform commutator estimate, and near-sharp spectral-gap bounds. The finite constant mode recovers the classical Mertens sum. A central correction shows that the commutator formula is a universal projection identity, so the Riemann Hypothesis equivalence restates the classical Mertens criterion and does not prove the hypothesis. Version 2 preserves the defensible earlier content, removes unsupported numerical claims, reproduces the lower-gap proof, and supplies numbered equations and deterministic validation.