# CSM_RH Paper 51 v0.1

Campaign 44 / PK5 progress package.

Main results:

- Z1 closed: truncated explicit formula with `T=N^kappa` has middle-band energy remainder `O(N^{-kappa} log^4 N)`.
- Z2 closed: exact discrete zero response kernel with exact phase `m^{i(gamma-t)}`.
- Kusmin–Landau localization:
  `|K_rho,N(t)| << N^{beta-1}[(1+|t|)/(|rho|(1+|gamma-t|)) + 1/(|rho|(1+|gamma|))]`.
- Centering gives `|K_rho,N(t)| << |t| N^{beta-1}/|rho|` for `|t|<=1`.
- Conditional theorem: RH implies the full PK5 middle-band target for every fixed `0<kappa<=1`.
- Unconditional PK5 remains open at Z3/Z4.

No RH proof is claimed.