# CSM_RH Paper 72 v0.1

Main conclusions:

- Audits Matomaki-Radziwill-Tao's averaged Hardy-Littlewood proof under a PESC seed.
- Their final theorem has arbitrary log savings, but hard Type d3/d4 substeps contain genuine fixed-power margins; noncentral machinery is not the root boundary.
- For the Fejer-weighted pair aggregate, polylog q>=2 major-arc approximation errors have only polylog Fejer weight and are fixed-power negligible relative to N H^2.
- PESC seed d=kappa/2 gives q=1 central exponential-sum error |S_0(beta)| << N^(1-d+o(1)).
- On |beta|<=c/N, D_H(beta)~H, so central energy is only bounded by
  N H^2 N^(-kappa+o(1)).
- A smooth boundary mode saturates this scale.
- Parseval shows frequencies |alpha|>=N^-v have saving 2(1-tau-v); all seed-critical difficulty is confined to ultra-low additive frequencies.
- Therefore perfect improvement of MRT Type d3/d4 and other noncentral minor arcs cannot amplify while the q=1 central arc remains at seed scale.
- Classical fixed-strip Selberg estimate, even with Density Hypothesis, gives only saving kappa(1-tau)<kappa.
- F-RH-022 remains open, but naive seed insertion into existing averaged-pair proof is closed.
- New active track PAIR5: principal-arc / low-frequency pair-cancellation excess.

No RH proof is claimed.