# CSM_RH Paper 70 v0.1

Main results:

- Tests Paper 69's F-RH-021 on the canonical smooth Mertens boundary mode.
- Synthetic mode:
  M_rho(x)=x^rho, m_rho(n)=n^rho-(n-1)^rho.
- Prime sampling gives:
  C_rho,X(0) ~ X^rho/log X,
  C_rho,X(h)-C_rho,X(0) ~ -h^rho/log h for h->infinity, h=o(X).
- Thus the boundary signal is mainly a shift-constant component.
- For H=X^(1-tau):
  signed mean retains critical relative amplitude X^-d.
- Centered variance is only
  H^(2 beta+1)/log^2 H
  = H pi(X)^2 X^-2[d+tau(1-d)+o(1)].
- Therefore Paper 69's centered lower target X^-2d is too strong for the smooth boundary mode.
- Original F-RH-021 is closed and corrected.
- Revised high-cost bridge candidate F-RH-021R:
  boundary zero => signed shifted-prime Möbius mean of critical size.
- No formal Mellin pole-transport identity exists for this additive prime-sampling transform; bridge is unproved.
- F-RH-020 remains auxiliary and is de-prioritized as an RH root route.
- Canonical campaign returns to F-RH-017-v3.

No RH proof is claimed.