Obstruction Theorems for Farey-Spectral Approaches to the Riemann Hypothesis

A spectral program can fail in two different ways. Its proposed operator may not exist on the intended domain, or the operator may exist while the statistic being read from it loses the arithmetic variation one hoped to recover. This paper proves both kinds of obstruction for a natural family built on the full Farey graph. First, every Farey-graph vertex has infinite degree. With unit edge weights and arbitrary magnetic phases, the quadratic form is infinite on every nonzero finitely supported vector, so the proposed Laplacian does not exist over the standard core. Second, denominator damping by w e=(q u q v)-2σ creates a sharp real-exponent existence boundary: the positive operator exists for σ 1/2, fails over the core for σ≤1/2, and is trace class in the admissible half-line with tr L(σ)=2ζ(2σ)^2/ζ(4σ). The exact trace identity produces a third result: spectral mass collapses toward zero as the Farey truncation grows. For every fixed Lipschitz test function, the global trace has the rigid form g(0)m N+O(1); for every fixed continuous test function its normalized trace tends to g(0). A self-contained Farey-counting argument proves the needed unconditional discrepancy estimate D(N)=O(N log N)=o(m N). It follows unconditionally that any proposed affine trace identity must cancel its extensive term. Complete nonexistence in the remaining bounded or sub-power branch is stated only as a conditional corollary under separately named growth hypotheses; those hypotheses are not claimed as results of this paper. The paper therefore supplies an operator-existence obstruction, an exact trace-collapse theorem, and a lossless map of what remains open. It makes no claim to prove the Riemann Hypothesis or full physical Super Information Theory.