# CSM_RH Paper 66 v0.1

F-RH-019 averaged-character progress.

Main results:

- For coefficients supported on interval length H, dyadic rational arcs q~R of width ~1/(RK) satisfy:
  int_{M_R} |A(alpha)|^2 d alpha
  << (H+R^2)/(R K) * sum |a_n|^2.
- With K~H and R^2<=H:
  major-arc L2 mass << R^-1 * coefficient energy.
- Therefore every polynomial conductor block R=X^u has fixed-power X^-u suppression, with no individual Dirichlet-L zero-free region.
- This can be viewed as Gauss normalization + character orthogonality + arc-width dilution / large sieve.
- Large-sieve averaging has a low-conductor floor: R=X^o(1) gives only subpower; fixed q gives no fixed power.
- PESC seed controls principal low-conductor characters, but not nonprincipal ones.
- Jutila zero-density calibration agrees: weighted bad-character counts become power-small only once a positive conductor exponent is present.
- F-RH-019 partially closes:
  HIGH conductor part certified;
  LOW-conductor nonprincipal core open.
- PT6F opens: low-conductor nonprincipal character annihilation.

No RH proof is claimed.