Compressing the Riemann Hypothesis into a Single Quadratic Form

We prove that the Riemann Hypothesis is equivalent to the estimate Q(A) ≪ A^ε for every ε > 0, where Q(A) is a manifestly positive quadratic form in short-interval Möbius running sums. Starting from a truncated von Mangoldt signal on dyadic blocks, whose Fejér energy is classically equivalent to RH, we apply the Vaughan identity and a geometric centering that isolates low-frequency content into a separate mean sector. We pass to the spatial L² formulation via the Fejér/Cesàro identity, connect the Type II variance to the nontrivial zeros of ζ(s) through an explicit formula, and reduce the problem to a continuous weighted second moment of the Vaughan–Möbius Dirichlet polynomial on the critical line. A three-band spectral decomposition controls the high frequencies unconditionally. For the intermediate band, we derive a running-sum identity that perfectly annihilates the O(log³ A) diagonal of the bilinear form, leaving the manifestly positive quadratic form Q(A). We formulate a stronger Shell Conjecture: Q(A) ≪ (log A)^C. This polylogarithmic bound is strictly stronger than RH; it implies M(x) ≪ √x (log x)^{C'} on the Mertens function, in the direction of Gonek's conjecture M(x) ≪ √x (log log x)^{5/4}. GPU experiments (X = 10⁸ to 10¹³) verify the Shell Conjecture with a stable 4× margin across six decades.