# CSM_RH Paper 69 v0.1

Main results:

- Exact centered decomposition:
  E2 = H|Cbar|^2 + V2.
- PESC seed d=kappa/2 and H=X^(1-tau), tau<d, gives
  H|Cbar|^2 << H pi(X)^2 X^(-2(d-tau)+o(1)).
  Thus F-RH-020 is a centered variance problem for eta<d-tau.
- Defines prime-shift Hankel matrix M_{h,p}=mu(p+h):
  V2=||P0 M 1_P||_2^2.
- Exact centered covariance kernel and exact Fourier/Hankel double-integral kernel are derived.
- Lichtman's theorem already implies
  E2 << H pi(X)^2 log^(-1/3+delta) X.
- Its typical-factorization component has arbitrary fixed log-power energy saving.
- Seed gives fixed-power short-interval Mobius Fourier control at alpha=0.
- Fixed-power centered variance remains open because nonzero additive frequencies retain low-conductor character barriers.
- No certified root bridge exists: small prime-Mobius cross-spectrum does not imply small prime spectrum without Mobius coercivity.
- Opens F-RH-021 as an auxiliary bridge candidate:
  boundary zeta zero => critical lower bound for centered prime-Mobius Hankel variance.

No RH proof is claimed.