# CSM_RH Paper 80 v0.1

Campaign-47 Vaughan/Buchstab closure.

Main results:

- B_U(s)=zeta(s)M_U(s)-1.
- Cutoff flow:
  B_{U2}-B_{U1}=zeta(s) sum_{U1<d<=U2} mu(d)d^-s.
  Therefore B_U(rho)=-1 is conserved at every zeta zero.
- Empty-divisor decomposition:
  B_U=(zeta-1)+zeta(M_U-1).
  The universal zero value -1 comes entirely from d=1.
  Every nonempty Möbius-divisor correction vanishes at zeta zeros.
- In the balanced coefficient C=A_V B_U, all nontrivial zero poles already occur in the d=1 branch; Buchstab shell corrections are zero-pole-free.
- Squarefree complementary identity:
  b_U(k)=-mu(k) sum_{q|k,q<k/U}mu(q).
  Cutoff motion transports parity rather than destroying it.
- Paper78 split H=zeta'/zeta+E can be sharpened:
  R=E-zeta has entire analytic continuation.
- Coefficientwise:
  h_{U,V}=-(Lambda-1)+r_{U,V},
  where r has entire Dirichlet-series continuation.
- Exact correlation:
  sum f r = E_I, and E_I is already Type-I power-small.
- Hence:
  Vren_{U,V}=-R_W+E_I.
- Therefore F-RH-024 is root-equivalent at fixed-power resolution once Type I is supercritical.
- F-RH-025 Buchstab/cross-scale telescoping cannot create a new root gain; its nonempty-divisor sector is precisely the regular correction sector.
- Campaign47 Vaughan/Buchstab branch is structurally closed at the empty-divisor hard core.
- Future linear transforms should first be rejected if their Dirichlet series equals c zeta'/zeta plus a zero-pole-free correction.
- Next genuinely new route should be nonlinear / multi-copy ordinary-prime boundary breaking.

No RH proof is claimed.