# CSM_RH Paper 36 v0.1

Campaign 35 audits rational and Müntz low-frequency filters.

Main results:
- fractional powers admit an exact positive Stieltjes/resolvent representation;
- best rational approximation to u^beta has root-exponential error, so N^(-delta) shape approximation formally needs only O((log N)^2) rank;
- fixed resolvent coordinates retain the full N^(beta+2) / N^(beta+1) drift scaling in the PESC moment vectors;
- fixed finite rational projection is therefore exponent-neutral;
- uniform function approximation does not control the derivative/correlation error needed by PESC;
- exponentially clustered poles that enable root-exponential approximation create polynomial derivative conditioning at the fixed-power rank scale;
- rational correction matrices require arithmetic control of the same hard prime-error moments; approximation theory does not provide it;
- Müntz completeness is representational, while its basis functions are themselves Mellin coordinates with hard arithmetic moments;
- rational/Müntz filtering is closed as a lower-strength route;
- Campaign 36 reopens only genuinely new fixed-power arithmetic estimates.

No proof or disproof of RH is claimed.