# CSM_RH Paper 35 v0.1

Campaign 34 generalizes the rank-one low-frequency decomposition to finite-rank scale-local projections.

Main results:
- exact PESC orthogonal decomposition into low-rank and residual components;
- exact finite correction matrix m_c^* G^{-1} m_B;
- for a smooth Mellin drift B(x)=x^beta, every fixed-rank component carries the same N^(2 beta+1) exponent;
- fixed rank therefore changes only coefficients, not the RH-strength exponent;
- polynomial multi-moment recentering exports the hard drift into a finite PESC-scale correction matrix;
- Bernstein approximation theory gives best polynomial approximation error for u^beta of order r^(-2 beta);
- any uniform polynomial residual-suppression strategy that needs N^(-delta) therefore requires polynomial rank r >= N^(delta/(2 beta)+o(1));
- this is more expensive than the formal growing-order Selberg threshold;
- finite-rank low-frequency projection is closed as a lower-strength route;
- Campaign 35 audits rational and Müntz filters, whose approximation complexity is genuinely different.

No proof or disproof of RH is claimed.