# CSM_RH Paper 73 v0.1

PAIR5 central zero-Gram closure.

Main results:

- Defines a smoothed q=1 central prime-error transform S_W(N,y) and central energy C_W.
- Exact scalar Mellin identity:
  int_1^infty F_{W,phi}(N) N^(-z-1)dN
  = -(zeta'/zeta)(z) Khat(z) - Khat(1)/(z-1).
- If central energy has saving exponent s, then every zeta zero satisfies Re rho <= 1-s/2.
- Conversely the zero-free strip Re rho<=1-s/2 gives the same central-energy exponent.
- Therefore maximal q=1 central-energy saving exponent equals
  2(1-beta_*) = maximal PESC exponent kappa_*.
- Smooth explicit formula gives Hilbert-valued boundary-zero series with rapidly decaying coefficients.
- If the rightmost abscissa Theta is attained, logarithmic-scale Cesaro mean diagonalizes distinct zero ordinates:
  mean ||sum e^{i gamma t}G_rho||^2 = sum multiplicity^2 ||G_rho||^2 >0.
- Hence off-diagonal zero pairs cannot persistently cancel a rightmost-zero diagonal across log scales.
- Vertical pair decorrelation alone cannot yield a central fixed-power exponent beyond the zero-strip exponent.
- Recent pair-correlation-without-RH results are density-level; fixed-power central energy is sensitive to even sparse rightmost off-line zeros.
- F-RH-022 remains open arithmetically, but generic PAIR5 spectral cancellation is exhausted.

No RH proof is claimed.