The Exact-Rational Bernstein Certificate for the N=4 Continuous Toy PairCeiling

Elevates a numerical candidate to a rigorous, machine-recheckable lower bound: within the toy configuration class defined by $N=4$, $m_i\in\{1,2\}$, positions anywhere on the unit circle, and constraints only on the open band $j=1,2,3$, an exact-rational dual $(c_0,y_1,y_2,y_3)$ is chosen, and positivity is proved for all three multiplicity patterns — the $(1,1,1,1)$ pattern directly by symbolic checking, the $(2,2)$ pattern by a 3-step exact Bernstein subdivision (minimum terminal coefficient $\approx0.00481$), and the $(2,1,1)$ pattern by an exact bivariate Bernstein subdivision over 77 internal boxes (minimum terminal coefficient $\approx5.7\times10^{-6}$) — yielding the rigorous lower bound $p_{\min}\ge0.6982110925=69.82110925\%$. The document stresses that this is not a lower bound from a numerical optimiser but the result of weak duality plus an exact positivity proof; the candidate floor from the previous round's numerical column generation was about 69.82311%, leaving only a very narrow gap to this rigorous lower bound — the upper side of that gap is still a numerical candidate, not a rigorous upper bound. The document also lays out the full pipeline — “numerical discovery → rationalize with margin → derive exact polynomial → Bernstein subdivision → exact positivity certificate” — clearly, to serve as a fixed template for later certificates of the same kind.