# CSM_RH Paper 78 v0.1

F-RH-024 zero-spectrum audit.

Main results:

- Balanced coefficient:
  c_{U,V}(n)=sum_{ek=n,e>V,k>U} Lambda(e)b_U(k).
- Exact Dirichlet series:
  C_{U,V}(s)=(-zeta'/zeta-L_V(s))(zeta(s)M_U(s)-1).
- Paper77 counterterm is coefficientwise:
  q_{U,V}(n)=M_U log n-M_U L_V-J_U-1.
- Its Dirichlet series Q exactly removes the double and simple poles of C at s=1.
- Renormalized H=C-Q is holomorphic at s=1.
- At every nontrivial zeta zero rho of multiplicity m:
  Res H(s)|_{rho}=+m, independent of U,V.
- Exact spectral split:
  H_{U,V}=zeta'/zeta+E_{U,V},
  where E has no nontrivial zeta-zero poles.
  Thus h_{U,V}=-Lambda+e_{U,V}, with e a zero-pole-free Type-I approximant.
- Any two Vaughan parameter choices differ by a coefficient with no nontrivial zero poles; the root-hard zero spectrum is parameter invariant.
- If the rightmost abscissa is attained, the smoothed central transforms of h and the prime error have opposite rightmost-zero responses and a strictly negative log-scale cross-Gram mean.
- Critical cross exponent is 2(1-Theta); at a saturated PESC(kappa) boundary it is exactly kappa.
- Generic Titchmarsh/divisor-shift power-saving theorems do not automatically apply because h retains the universal zeta-zero poles.
- Ford's shifted-prime Kubilius transference controls small prime factors in the vanishing-TV regime y=x^{o(1)}; the polynomial balanced factors needed by F-RH-024 remain outside that regime.
- Next: dyadically localize which polynomial factor ranges carry the universal zero-pole mass.

No RH proof is claimed.