Pre-Cauchy Barriers for Signed Quadratic Root Packets

We study a signed bilinear model built from quadratic congruence root packets at the critical scales $D = K^2$. The long variable carries a von Mangoldt–Dirichlet coefficient and the short variable a Möbius–Dirichlet coefficient. A first Cauchy–Schwarz or Hilbert-space completion generally destroys the signed long-variable information; in a natural shift–character implementation, separate square-root estimates for the resulting character family lead to a $3/4$-scale output. We isolate one exceptional mechanism that does better. Projection onto the moving quadratic character produces, before Cauchy–Schwarz, an exact reciprocal coefficient of size $d^{-1/2}$. A tensor energy estimate then gives the endpoint bound $Q \ll D^{1/2}K^{1/2}(DK)^\epsilon$. We show at the same time that this quadratic mode removes only $O(d^{-1})$ of the centered local energy: the generic high-order packet remains asymptotically intact. We then prove several exact structural barriers. Injective local transforms and affine root reparameterizations preserve the conductor-sized orbit dimension; root-blind factor-ratio Mellin transforms preserve the real part of the distinguished Dirichlet singularity; and Hensel-window sparsity has no polynomial interpolation capable of producing a power gain at the critical scale. We also give an exact Salié-convolution formula for the Kloosterman root-pushforward and show that the convolution restores the full generic spectrum. The surviving primitive block is therefore a genuinely signed moving-modulus correlation problem. Its simplest fixed-parameter analogue lies on the horizontal prime-modulus Kloosterman frontier. We formulate a quadratic-root-orbit variant that sits strictly between the classical fixed-parameter Kloosterman problem and the explicitly solvable quadratic/Salié family. No claim is made of a new exponent below $3/4$, of the Riemann hypothesis, or of a universal impossibility theorem for all pre-Cauchy transforms.