# CSM_RH Paper 30 v0.1

Campaign 29 audits generalized von Mangoldt / higher-order Selberg amplification.

Main results:
- fixed-k generalized Selberg summatory formula is x R_k(log x)+O_k(x);
- a surviving zeta-zero contribution to (-1)^k zeta^(k)/zeta retains exponent x^rho for every fixed k;
- fixed-order filters change residues, not zero exponents;
- finite combinations can cancel known polynomial main terms but current theorems do not certify cancellation among the O_k(x) remainders;
- powers of log become a fixed X-power only at k ~ log X/log log X;
- at exactly that order, Lambda_k prime spikes and factorial-type proof constants become polynomial in X;
- existing fixed-k formulae provide no uniformity at this critical order;
- higher-order Selberg is a real log/subpower amplifier, consistent with Bombieri-Wirsing and Diamond-Steinig history, but no certified fixed-power route is obtained;
- Campaign 30 returns to parity-sensitive signed PESC rather than another almost-prime smoothing layer.

No proof or disproof of RH is claimed.