# CSM_RH Paper 81 v0.1

Nonlinear / multi-copy screening.

Main results:

- For even q, if M_q(X,H)<<X H^q X^-s then any zeta zero satisfies beta<=1-s/q.
  Hence moment exponent s gives PESC exponent <2s/q.
- A saturated PESC(kappa) boundary has critical qth-moment exponent q*kappa/2.
- Gaussian short-interval qth moment has saving q(1-tau)/2.
  Therefore all raw even moments have the identical scale gate 1-tau>kappa.
  Higher moment degree gives no free exponent gain.
- Montgomery-Soundararajan local refined singular-series moments have Gaussian even main terms with power-small error; for k=4 the error is H^(2-1/28+eps).
- Bloom-Kuperberg 2025 gives near-optimal odd refined-singular-series moment bounds.
- Scalar fourth cumulant detects a single conjugate boundary pair: Cum4=-6|A|^4, but may be contaminated by additive relations among multiple zero ordinates.
- Defines stationary connected fourth lag tensor K4.
- Universal self-frequency identity:
  Fourier coefficient K4hat(lambda,lambda,-lambda)=-|a_lambda|^4.
  This is robust to additive relations among other frequencies.
- Multi-scale determinant/wedge candidates are rejected: a single boundary mode is rank-one, so such determinants delete the leading boundary signal.
- Opens F-RH-026: frequency-resolved connected four-prime cumulant power.
- Next: construct exact arithmetic connected-tensor identity and a positive frequency-resolved boundary forcing theorem.

No RH proof is claimed.