Exact $P_{70}$ Boundary-Escape Certificate

The $N=4$ continuous toy model's first rigorous crossing of 70% — continuing from 08's numerical frontier, it takes the exact dual $c_0=1.12269224$, $y_1=-0.38437941$, $y_2=-0.25114540$, $y_3=-0.11796917$, $\mu=-0.03068556$ and runs exact Bernstein subdivision on each of the three multiplicity patterns in turn: the $(2,2)$ pattern passes on all 8 terminal intervals; the $(2,1,1)$ pattern passes on all 62 terminal boxes; the $(1,1,1,1)$ pattern, after using Newton's identities to reduce to three real variables, passes a 3D Bernstein subdivision of 180 terminal boxes on a superset box larger than the true root configuration, with minimum terminal coefficient $5\times10^{-5}>0$. With all three branches proved positive, this yields $\boxed{B_{cert}=11254781/3068556=3.667777612662112\ldots}$, rigorously giving $\mathbb E[p]\ge70\%$ for all $B\le B_{cert}$. The document repeatedly emphasizes that this is an $N=4$ toy marked-configuration theorem defined by this research, not a new theorem about the zeros of the Riemann zeta function — and it also spells out the relationship to Claude's actual paper very clearly: the real problem needs the Fourier support to expand from 1 to about 1.04 to reach 70%, and this toy theorem cannot derive 1.04, but it does rigorously prove the corresponding mechanism claim — once an observable that constrains the boundary spike is added, the bandwidth-one ceiling can be broken through.