# CSM_RH Paper 53 v0.1

Campaign 44 / PK5-Z4 structural closure package.

Main results:

- Re-centers the campaign on the exact Paper 48 identity:
  `J_N^theta = sum_{N<=n<2N} (theta(n)-n)^2`.
- Proves prime-power mean-square equivalence between theta and psi at every PESC scale `0<kappa<=1`.
- Proves discrete / continuous dyadic mean-square equivalence.
- Uses the exact Mellin identity
  `s int_1^infty (psi(x)-x)x^{-s-1}dx = -zeta'/zeta(s)-s/(s-1)`.
- Proves:
  `PESC(kappa) => zeta(s) != 0 for Re(s)>1-kappa/2`.
- By functional-equation symmetry:
  `kappa/2 <= beta <= 1-kappa/2`.
- At `kappa=1`, proves exponent-level equivalence:
  `PESC(1) <=> RH`.
- Z4 closes structurally, but PK5 unconditional endpoint admission remains open and is now explicitly RH-equivalent.
- Recommends next campaign: exponent amplification or MLEPG bootstrap.

No RH proof is claimed.