# CSM_RH Paper 71 v0.1

Main results:

- Expands the root short-interval second moment using Lambda_0=Lambda-1.
- Montgomery-Soundararajan's exact modified-singular-series average gives:
  R_2(H)=-H log H + A H + O(H^(1/2+eps)).
- Diagonal contributes NH log N.
- Therefore:
  M_2(N,H)=NH log(N/H)+B NH + R_HL^(2)(N,H)+lower terms,
  where R_HL^(2) is the complete aggregate error of actual prime-pair correlations around the modified singular series.
- If H=N^(1-tau) and
  |R_HL^(2)| << N H^2 N^(-xi+o(1)),
  then lag exponent delta=min(1-tau,xi).
- Paper 54 gives:
  kappa' < min(1-tau, xi, kappa+tau(2-kappa)).
- Strict amplifier iff:
  tau<1-kappa and xi>kappa.
- Gain:
  eta<min(1-tau-kappa, xi-kappa, tau(2-kappa)).
- Opens F-RH-022:
  averaged Hardy-Littlewood pair residual excess.
- This requires only aggregate error cancellation, not individual twin-prime asymptotics.
- Current averaged prime-pair theorems give arbitrary log-power savings over shifts, but no fixed X^-eta power.
- Local singular-series cancellation is already solved; only global pair-error cancellation remains.
- F-RH-017-v3 remains canonical general root frontier; F-RH-022 is preferred ordinary-prime arithmetic subfrontier.

No RH proof is claimed.