# CSM_RH Paper 84 v0.1

Prime-error / Mertens mixed-state screening.

Main results:

- Exponential smoothing:
  P(X)=sum Lambda(n)e^{-n/X}-X,
  M_mu(X)=sum mu(n)e^{-n/X}.
- At a simple zero rho:
  prime coefficient A_rho=-Gamma(rho);
  Mertens coefficient B_rho=Gamma(rho)/zeta'(rho).
- Same-zero cross atom contains 1/zeta'(rho), so it is genuinely new spectral data beyond prime-error C2.
- No uniform lower bound on |zeta'(rho)| is needed for fixed-power exclusion: each hypothetical zero supplies a fixed nonzero constant.
- Multiple zeros strengthen Mertens forcing by (log X)^(m-1), without changing the horizontal X^beta power.
- Degree-two same-zero cross amplitude has saving exponent 2(1-beta); hence maximal horizontal cross exponent is
  2(1-beta_*)=kappa_*.
- Squaring to a positive energy doubles both degree and threshold, producing no per-copy gain.
- Exact convolution collapse:
  (Lambda * mu)(n) = -mu(n) log n.
- Exact Mellin gradient:
  (-zeta'/zeta)|1/zeta|^2 = F' conjugate(F), F=1/zeta.
  Its real/imaginary parts are derivatives of |1/zeta|^2 in sigma/t.
- Mertens fixed-power bounds are themselves zero-strip/root-level.
- Weak Mertens implies RH, simple zeros, and convergence of sum 1/|rho zeta'(rho)|^2.
- Recent negative-moment literature confirms derivative-sensitive Mertens information remains difficult.
- Verdict: prime/Mertens cross spectrum has vertical derivative novelty but no new horizontal exponent; closed as standalone PESC amplifier.
- Next: formalize the common-power multi-field barrier for finite zeta-derived linear fields.

No RH proof is claimed.