# CSM_RH Paper 61 v0.1

Correction and strengthening of Papers 59–60.

Key correction:
- Paper 60's statement `c>tau is globally sharp` is too strong when `tau>kappa/2`.
- Actual local von Mangoldt increments satisfy
  `|U_H| << min(N^(1-kappa/2+o(1)), H N^o(1))`.

Corrected seeded Lp exponent:
`d'_p < min(nu, (d-tau)_+ + c/p, d+tau(1-d))`,
where `d=kappa/2`.

Consequences:
- p=1 remains optimal.
- Correct strict gate:
  `nu>d` and `c>min(d,tau)`.
- PESC gain:
  `eta < 2 min(nu-d, c-min(d,tau), tau(1-d))`.

Sharpness:
- polynomial-scale Pintz boundary forcing plus the local envelope gives critical exceptional mass
  `N^(1-min(d,tau)-o(1))`;
- a local-increment-constrained residue-chain model gives the same piecewise critical count.

Scope correction:
- Pintz short-interval adaptation is only claimed/certified for fixed polynomial scales `h=Y^-tau`, `0<tau<1`, not arbitrary tiny h.

Preferred frontier becomes F-RH-017-v3.

No RH proof is claimed.