# CSM_RH Paper 82 v0.1

Nonlinear branch correction and closure of F-RH-026 as a standalone root amplifier.

Main results:

- Defines positive degree-4 diagonal functional:
  D4 = -sum_lambda K4hat(lambda,lambda,-lambda)
     = sum_lambda |a_lambda|^4 >=0.
- Exact collapse:
  D4 = mean_u |C2(u)|^2.
  Thus the robust connected-fourth self detector is completely determined by the two-point spectrum.
- For a single conjugate boundary pair
  F(t)=A e^{i gamma t}+conj(A)e^{-i gamma t},
  C2(u)=2|A|^2 cos(gamma u).
  C2 determines both amplitude and frequency.
- Every stationary finite-degree polynomial moment/cumulant of the single-pair process is determined by C2.
  Scalar even moments:
  mean F^(2m)=binom(2m,m)|A|^(2m).
- RH failure can logically occur with a single conjugate rightmost zero pair.
  Therefore finite-degree polynomial multi-copy statistics of one linear prime-error field do not give new universal spectral coercivity beyond second order.
- Genuine off-diagonal fourth information may be new, but is not universally forced by a single boundary pair.
- Degree-4 diagonal exponent remains neutral:
  boundary D4~X^(-2 kappa);
  a D4 saving s implies PESC exponent <s/2.
- Corrects Paper81: Montgomery-Soundararajan local R4 is an additive-shift local theorem, while exact zeta-ordinate resolution lives in log-scale/Mellin coordinates; it does not directly estimate the frequency-resolved self atom.
- Leung 2026 multiscale Gaussian law assumes RH + LI, so it cannot be used as root input.
- F-RH-026 closed as standalone universal root amplifier.
- Future candidates must mix in arithmetic state not determined by C2: factorization-conditioned, Boolean/threshold, or mixed additive-multiplicative observables.

No RH proof is claimed.