# CSM_RH Paper 75 v0.1

Campaign 47 opens under the rule: explicit q=1 fixed-power inequality first, framework second.

Main results:

- Define triangular shift-averaged prime sequence
  a_{N,H}(n)=W(n/N) sum_r omega_H(r) Lambda(n+r).
- Total mass A(N,H)~N H int W.
- Deterministic bounded-variation lattice sampling gives, uniformly in d,
  A_d=A/d+O_W(N).
- Hence high-level sieve remainder:
  sum_{d<=D} mu^2(d) tau_5(d)|r_d|
  << N D log^O(1) N,
  with relative ratio D/H.
- Choosing D=N^(2/3+eps), H=N^(1-tau) gives fixed local remainder exponent
  eta_R=1/3-tau-eps.
- This removes the local-distribution hurdle after shift averaging, provided tau<1/3.
- First Campaign-47 inequality F-RH-023:
  a fixed-power version of the Friedlander-Iwaniec parity-breaking bilinear axiom
  for the special sequence a_{N,H}.
- The axiom uses gamma(n,C), mu(mn), and the ordinary additive translate mn+r.
- Selberg's parity sequence calibrates that local remainder information alone does not imply this bilinear cancellation.
- Paper64 pseudo-prime properties do not imply it; generic Beurling systems do not possess the same ordinary mn / mu(mn) / mn+r structure.
- Important obstruction: the published generic asymptotic sieve outputs only relative O(log delta/log Delta), hence only logarithmic prime-detection accuracy even if inputs are power-small.
- Campaign47 splits:
  C47-A prove the fixed-power bilinear axiom;
  C47-B first derive a power-output exact prime-detection identity tailored to the dense shift-averaged sequence.
- Exponent budget limits this first bootstrap mechanism to roughly kappa<1/3.

No RH proof is claimed.