The Riemann Hypothesis via a Mean-Square Collision of a Fixed Observable

It is shown that the Riemann Hypothesis can be reduced to a rigid and unavoidable mean-square incompatibility for a single, fixed exponential observable. Fix a normalized window ψL(x)=η(x/L)ex/2 and consider the associated logarithmically localized sum A_ψ(U,L;β) = ∑u∈Λ w(u)\,ψL(u-U)\,e-iβ u, together with its intrinsic energy scale M_ψ(U,L). A central feature of the framework is that this observable, its normalization, and its energy scale are fixed once and for all; all constants appearing in the analysis are independent of the window length L. It is proved that if there exists a nontrivial zero ρ₀=σ₀+iγ₀ of ζ(s) with σ₀>1/2, then the Weil–Guinand explicit formula forces exponential amplification of the same observable A_ψ(U,L;β) at the resonant frequency β=γ₀. A quantitative localization argument converts this pointwise amplification into a localized mean-square lower bound in the frequency variable, with constants uniform in the scale parameter. On the other hand, when the summation set Λ satisfies a structural nonresonance condition—in particular, the square-free filter Λ=log n:μ²(n)=1\—unconditional theory enforces uniform mean-square upper bounds for A_ψ(U,L;β) with at most polynomial growth in L, via classical large sieve inequalities. Crucially, both bounds apply to the same observable, with the same normalization and in the same intrinsic energy units M_ψ(U,L). As L→∞, the exponential lower bound forced by an off-critical zero is therefore incompatible with the unconditional polynomial upper bound. This incompatibility excludes the existence of nontrivial zeros with (ρ)>1/2. By the functional equation of ζ(s), all nontrivial zeros of ζ(s) lie on the critical line (s)=1/2. No new hypotheses are introduced beyond classical results, and no new analytic estimates are required. The argument rests entirely on fixing a single observable and exposing an irreconcilable mean-square collision between explicit-formula amplification and unconditional structural control.