# CSM_RH Paper 83 v0.1

Mixed arithmetic-state screening.

Main results:

- For any periodic shifted factor state g(n+a) modulo Q:
  P_g(s)=sum_chi ghat_a(chi)(-L'/L(s,chi))+finite correction.
- Principal coefficient:
  pi_g=(1/phi(Q)) sum_{r in units mod Q} g(r+a).
- Exact principal projector:
  P_g(s)=pi_g(-zeta'/zeta)+nonprincipal-L / finite-Euler remainder.
- Therefore:
  uncentered periodic factor state retains a scalar copy of the original zeta hard core;
  principal-centered state deletes that hard core.
- Every finite divisor-sieve state is periodic, hence covered.
- Finite roughness gate density:
  pi_{z,a}=product_{p<=z,p not dividing a}(p-2)/(p-1).
- For any finite factor-state partition, the principal zeta spectrum is rank one in direction pi=(pi_j).
  Orthogonal anatomy fluctuations have no principal zeta projector.
- Physical-space decomposition:
  E_j=N_j-pi_j X = pi_j(psi-X)+Delta_j,
  with sum Delta_j=0.
  Conditional anatomy can be estimated sharply without improving the total PNT error.
- Ford 2025 and Bharadwaj-Rodgers 2026 provide strong conditional shifted-prime factor-anatomy calibration, fully compatible with the rank-one theorem.
- Periodic / small-factor mixed-state route closed as principal-rank-one or root-blind.
- First surviving class for next screen:
  global prime-error / global Mertens-state cross spectrum, whose simple-zero coefficient contains 1/zeta'(rho) and is not determined by prime-error C2.

No RH proof is claimed.