# Claim 6

**Result: verified.** Direct, high-accuracy execution of Eq. (23) at
the paper Figure-6 parameters p=64, H=256 yields
ρ* = **1.7497342724**. The first-integral margin is positive at
0.95ρ* (no critical root) and negative at 1.05ρ*, where a finite critical
argument is found. Reversing those classifications is rejected.

The result is deliberately described as Eq. (23) threshold execution, not as a
raw finite-p ν=4 gradient-ascent rerun.

## Registered wording

For the nu=4 symmetric case, Section 10 derives a gradient-ascent solution identifying convergent low-noise and divergent high-noise regimes separated by an explicit threshold given in Equation 23.