# CSM_RH Paper 86 v0.1

Return to F-RH-017-v3 and polynomial-threshold exceptional-set audit.

Main results:

- Gafni-Tao 2026 gives modern fixed-relative exceptional-set exponents from zero density A(sigma) and zero additive energy A*(sigma).
- F-RH-017 instead needs threshold |Delta_H| > H X^-nu with nu>d=kappa/2.
- General pth-moment Markov law:
  shrinking threshold X^-nu costs exactly p*nu in the exceptional exponent.
- Best-case zero-band baseline, even with all zero-density costs deleted:
  mu_p^(nu)(beta) >= 1 + p(nu-(1-beta)).
- Therefore fixed moments have a phase transition at nu=1-beta.
- At a saturated PESC boundary beta=1-d:
  mu_{2k}^(nu) >= 1+2k(nu-d).
  If nu>d, no fixed even-moment Markov method can even certify density-zero exceptions.
- Higher moments steepen the penalty but do not move the transition.
- For hard intervals, precision H X^-nu requires explicit-formula height
  T=X^(tau+nu+o(1)).
- Threshold-aware Gafni-Tao exponents:
  mu2=(tau+nu)(1-sigma)A(sigma)+2sigma-1+2nu;
  mu4=(tau+nu)(1-sigma)A*(sigma)+4sigma-3+4nu.
  At sigma=1-d both remain >1 when nu>d.
- Smooth kernels may reduce truncation cost, but cannot remove the universal baseline wall.
- Fixed-relative almost-all theorems and power exceptional sets for almost primes remain useful calibration but do not reach polynomially shrinking prime-count accuracy.
- F-RH-017-v3 is therefore threshold-native / non-Markov.
- Next: split prime-excess and prime-deficiency tails and audit one-sided arithmetic mechanisms.

No RH proof is claimed.