# CSM_RH Paper 88 v0.1

Extremal-family audit for the prime-excess tail.

Main results:

- Elementary polynomial-threshold inertia:
  |Delta_H(y)-Delta_H(x)| <= 2(|y-x|+1)log(4X).
  Hence an excess H X^-nu persists at half-threshold on
  L=H X^-nu / log X = X^(1-tau-nu+o(1)).
- Bazzanella-Perelli's published inertia theorem has a threshold-gap condition >=exp(-sqrt(log X)); the elementary argument extends the power-scale content to polynomial thresholds.
- If |E_nu^+| >= X^(1-c), a maximal L-separated family has
  J >= X^(tau+nu-c-o(1)).
- A greedy H-separated subfamily has
  K >= X^(tau-c-o(1)).
- Thus many disjoint H-intervals are forced only when c<tau.
  The family method is naturally strip-dominated d<tau.
- Combining d<tau with d<nu<(1-tau)/2 is possible iff d<1/3, i.e. kappa<2/3.
- For the multiset union of J intervals:
  A_q=JH/q+O(J).
  Aggregate divisor remainder to D is JD polylog; relative level is D/H.
  J cancels completely.
- Summed Brun-Titchmarsh retains the constant 2/(1-tau), far above 1+X^-nu.
- Inertia grid has X^(tau+nu+o(1)) candidate clusters, but target exceptional fraction remains X^-c.
  Representative-wise concentration still requires order X^nu log X.
- At target relative precision X^-nu, deterministic Type-I information only reaches
  D << H X^-nu,
  exponent gamma_th=1-tau-nu.
- Therefore the only remaining family-specific leverage is balanced Type-II cancellation.
- Opens F-RH-028+:
  threshold-accurate Type-II estimates for the data-dependent extremal interval sequence.
- Next: derive unconditional Type-II baseline for arbitrary separated interval families and test whether any gain grows with J.

No RH proof is claimed.