# CSM_RH Paper 58 v0.1

Campaign 46 / SG4 standard-method closure package.

Main results:

- Standard positive-coefficient Euler-product / nonnegative-trigonometric-polynomial zero-dropping methods have an intrinsic `O(1/log t)` zero-distance scale and cannot amplify a fixed PESC seed.
- Higher logarithmic derivatives keep positive Dirichlet coefficients but their zero kernels change sign for every derivative order m>=1, destroying the one-sign zero-dropping mechanism.
- Raw positivity of prime counts is blind to a boundary mode of relative size `N^(-kappa/2)`.
- Existing Heath-Brown componentwise seed-scale estimates cannot produce a fixed-power joint improvement after exact cross-j recoupling.
- A strict Heath-Brown amplifier would require a genuinely new joint cross-j covariance deficit.
- Current 2026 positivity / zero-free-region literature is consistent with these scope limits.
- No new equivalent criterion is introduced.
- Preferred direct frontier remains F-RH-017; F-RH-018 remains secondary.
- SG4D nonlinear prime coherence remains open but unspecified.

No RH proof is claimed.