The Toy $P_{70}$ Minimal Boundary-Escape Frontier

The open-band toy knows only $S(1),S(2),S(3)$; this round adds one minimal piece of extra information, $\mathbb E[S(4)]\le B$, and re-solves for $p_{\min}(B)$, with the goal of finding $B^*_{70}=\sup\{B:p_{\min}(B)\ge0.70\}$. The continuous column-generation numerics give $B=3.67\Rightarrow69.9982\%$ and $B=3.65\Rightarrow70.0596\%$, bracketing $B^*_{70}$ between $3.65$ and $3.67$; linear interpolation yields an exploratory candidate $B^*_{70}\approx3.66941$ — the document explicitly flags that “this 3.66941 is not a theorem, only the target the next exact certificate should aim at.” The finding with real conceptual value is a comparison of scale: the open-band optimum's own boundary row is about $S(4)\approx3.73$, but pushing the floor from 69.82% past 70% does not require forcing the boundary row all the way down to the CUE value (i.e., 1) — it only requires ruling out the most extreme sliver of spike freedom. The information needed to break 70% is far less than fully knowing the boundary row. The document also supplies a concrete numerical dual candidate for the next certification target (after retaining a $5\times10^{-5}$ safety margin, the dual objective still comes to 70.0546%), paving the way for the exact certificate in the next round (09).