# CSM_RH Paper 59 v0.1

Campaign 46 improved direct exceptional-set amplifier.

Main results:

- General seeded residue-chain Lp inequality.
- Exceptional-set theorem at H=N^alpha gives global PNT-error Lp exponent
  d'_p < min(nu, kappa/2+alpha-1+c/p, 1-alpha(1-kappa/2)).
- Mellin analyticity converts this directly to a zero-free strip and PESC exponent.
- p=1 is optimal among all p>=1.
- L1 PESC output:
  kappa' < min(2nu, kappa+2c+2alpha-2, 2-2alpha+alpha*kappa).
- Strict gate:
  nu>kappa/2 and c>1-alpha.
- Near macroscopic H=N^(1-tau):
  eta < min(2nu-kappa, 2(c-tau), (2-kappa)tau).
- For fixed c and non-bottlenecking threshold:
  tau*=2c/(4-kappa),
  eta*=2c(2-kappa)/(4-kappa).
- This halves the exceptional-set rarity requirement of Paper 56.
- Gafni-Tao published theorem fixes delta and J before X->infinity; direct substitution delta=X^-nu is not licensed.
- Fixed-H removes one covering cost, but finite moment + Markov remains structurally blocked at nu=kappa/2.
- Preferred frontier updated to F-RH-017-v2.

No RH proof is claimed.