Small-$N$ Primal Toy LP and Boundary-Spike Escape

The first independent reproduction of the bandwidth-one adversarial-law mechanism — the document states up front that this is not a reproduction of Anthropic's $N=256$ exact-rational LP, and positions are restricted to a discrete circular grid. For an explicit mixture with $N=4,M=24$, the mixed result gives $\bar S(1)=1/4,\bar S(2)=1/2,\bar S(3)=3/4$ (an exact match to the open-band CUE target), yet $\bar p=70.18\%$ — already reproducing, in a toy class far narrower than the official one, the phenomenon that “open-band pair rows matching CUE exactly ⇏ simple fraction near 1.” The same solution also makes the constrained $j<N$ rows fit CUE almost perfectly, while producing a huge spike at the unobserved boundary row ($j=256$) ($S(256)\approx211.43$, consistent with the official kernel-checked bound $|D(1)|\le0.82395317$) — this is defined as the Boundary-Spike Obstruction (BSO). The document then runs two sets of experiments to test how actionable this mechanism is: adding just one boundary constraint $\mathbb E[S(N)]\le B$ pushes the adversarial floor all the way from 70.18% (unconstrained) to 81.91% ($B=1.0$); a support-extension experiment points in the same direction, but the document explicitly warns that these percentages should not be matched one-to-one against Claude's real support thresholds $1.04,1.26,1.70$ — the toy grid is too narrow and the row sampling too coarse.