# CSM_RH Paper 87 v0.1

One-sided polynomial exceptional-set audit.

Main results:

- Split E_nu into prime-excess E_nu^+ and prime-deficiency E_nu^-.
- For nu<min(1/2,1-tau), weighted psi deviations translate to prime-count relative deviation delta=X^-nu around lambda=H/log X.
- Near-Poisson falling factorial transfer:
  if E(N)_r <= lambda^r exp(epsilon_r r) and r<=delta lambda/2,
  then
  P(N>=(1+delta)lambda)
  <= exp(-(delta/3-epsilon_r)r).
- To obtain exceptional probability X^-c requires
  r ~ delta^-1 log X = X^nu log X
  and per-copy log loss epsilon_r=O(delta).
- Thus polynomial relative resolution requires polynomially growing moment complexity and near-exact prime-tuple precision.
- Compatibility r<<delta lambda is equivalent to 2nu<1-tau.
  A window d<nu<(1-tau)/2 exists iff kappa<1-tau, exactly the root amplifier scale gate.
- Ideal Poisson tail has exponent delta^2 lambda=X^(1-tau-2nu)/log X, leaving enormous conjectural margin.
- Kuperberg's unconditional Selberg-sieve moments are available only for r=o((log X)^(1/4)) and carry large r-dependent overhead, far below the needed X^nu complexity and precision.
- Gallagher/Kuperberg upper-tail sieve results are effective for coarse/extreme tails, not 1+X^-nu deviations.
- Jha 2026 conditional Poisson-tail results concern slowly growing lambda under strong Hardy-Littlewood, far below polynomial lambda=X^(1-tau)/log X.
- Upper tail: resolution-hard but not intrinsically parity-hard.
- Lower tail: upper factorial moments do not apply; prime lower bounds additionally encounter the classical sieve parity barrier.
- Matomaki's P2 almost-all theorem is a control experiment showing strong lower-tail sieve results for almost primes but not primes.
- Opens auxiliary F-RH-027+ and F-RH-027-.
- F-RH-017-v3 follows if both one-sided exponents beat min(d,tau).

No RH proof is claimed.