# CSM_RH Paper 85 v0.1

General common-power multi-field barrier.

Main results:

- Defines common-defect normalized fields with boundary response
  c_j(rho) X^{-(1-beta)} log^m X e^{i gamma log X}.
- At deterministic scale X^theta_j, the defect becomes X^{-theta_j(1-beta)}.
- A weighted monomial of total scale/copy weight
  W=sum n_j theta_j
  has boundary exponent W(1-beta).
- A fixed-power upper X^-s therefore forces
  beta_*<=1-s/W
  and PESC exponent <2s/W.
- General finite polynomial: the smallest nonvanishing weighted degree W_* controls the full horizontal forcing.
- Fixed analytic observable: Taylor expansion reduces to the same first nonzero weighted degree.
- Log-scale derivatives, fixed scale differences, compact Mellin smoothings, and antiderivatives change residues/log factors but not normalized horizontal slope.
- Ratios may expose residue data such as 1/zeta'(rho), but cancel the horizontal X power and therefore do not locate beta by themselves.
- Fixed thresholds are eventually blind because normalized boundary fields tend to zero.
- Scale-adaptive threshold |Y|>X^-d escapes analytic homogeneity: if 1-beta<d, the boundary mode exceeds the threshold on positive log-scale mass.
- This is exactly the exceptional-set geometry of F-RH-017-v3.
- Structural conclusion:
  finite zeta-derived analytic multi-field route is closed as common-power degree-neutral.
- The first robust escape returns to F-RH-017-v3.
- Future alternatives require dynamic/growing-complexity observables or a field with genuinely different horizontal slope.

No RH proof is claimed.