# CSM_RH Paper 34 v0.1

Campaign 33 audits the low-frequency rank-one scalar exposed by mean-zero PESC recentering.

Main results:
- exact weighted covariance + rank-one decomposition of PESC;
- endpoint Mellin kernel I(s)=(2^(s+2)-1)/((s+1)(s+2));
- first- and second-primitive Mellin symbols are nonzero throughout the critical strip;
- one isolated conjugate off-line zero gives rank-one response of order N^(2 beta+1) on arbitrarily large scales;
- for the real smooth drift B(x)=x^beta, both the rank-one term and the centered covariance retain exponent N^(2 beta+1);
- the covariance is strictly negative for every fixed 0<beta<1, while the rank-one term is positive;
- as beta approaches 1 the covariance coefficient tends to zero, so the rank-one term captures most near-linear drift, but not at a better exponent;
- rank-one control alone is therefore not a complete PESC theorem and is not promoted to a new frontier;
- Campaign 34 tests finite-rank low-frequency projections.

No proof or disproof of RH is claimed.